Acoustics Module
User´s Guide
VERSION 4.3b
Acoustics Module User’s Guide
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Version:
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COMSOL 4.3b
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Part No. CM020201
C o n t e n t s
Chapter 1: Introduction
Acoustics Module Physics Guide
2
Where Do I Access the Documentation and Model Library?
8
Overview of the User’s Guide
11
Chapter 2: Modeling with the Acoustics Module
Acoustics Module Capabilities
16
What Can the Acoustics Module Do? . . . . . . . . . . . . . . . 16
What are the Application Areas? . . . . . . . . . . . . . . . . . 17
Which Problems Can You Solve? . . . . . . . . . . . . . . . . . 19
Fundamentals of Acoustics Modeling
20
Acoustics Explained . . . . . . . . . . . . . . . . . . . . . . 20
Examples of Standard Acoustics Problems . . . . . . . . . . . . . 21
Mathematical Models for Acoustic Analysis . . . . . . . . . . . . . 23
Resolving the Waves . . . . . . . . . . . . . . . . . . . . . 25
Damping . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Artificial Boundaries
. . . . . . . . . . . . . . . . . . . . . 29
A Note About Perfectly Matched Layers (PMLs) . . . . . . . . . . . 30
Evaluating the Acoustic Field in the Far-Field Region . . . . . . . . . 31
The Far Field Plots . . . . . . . . . . . . . . . . . . . . . . 32
Solving Large Acoustics Problems Using Iterative Solvers. . . . . . . . 33
About the Material Databases for the Acoustics Module . . . . . . . . 35
The Acoustics Module Study Types
36
Stationary Study . . . . . . . . . . . . . . . . . . . . . . . 36
Frequency Domain Study . . . . . . . . . . . . . . . . . . . . 37
Eigenfrequency Study . . . . . . . . . . . . . . . . . . . . . 38
CONTENTS
|i
Mode Analysis Study . . . . . . . . . . . . . . . . . . . . . 39
Time Dependent Study . . . . . . . . . . . . . . . . . . . . 40
Frequency Domain Modal and Time-Dependent Modal Studies . . . . . 41
Modal Reduced Order Model . . . . . . . . . . . . . . . . . . 41
Additional Analysis Capabilities . . . . . . . . . . . . . . . . . 41
Special Variables in the Acoustics Module
43
Intensity Variables . . . . . . . . . . . . . . . . . . . . . . 43
Power Dissipation Variables . . . . . . . . . . . . . . . . . . . 45
Boundary Mode Acoustics Variables . . . . . . . . . . . . . . . 47
Reference for the Acoustics Module Special Variables . . . . . . . . . 49
Chapter 3: The Pressure Acoustics Branch
The Pressure Acoustics, Frequency Domain User Interface
52
Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure
Acoustics, Frequency Domain User Interface . . . . . . . . . . . 57
Monopole Source
. . . . . . . . . . . . . . . . . . . . . . 58
Dipole Source . . . . . . . . . . . . . . . . . . . . . . . . 59
Initial Values. . . . . . . . . . . . . . . . . . . . . . . . . 59
Sound Hard Boundary (Wall) . . . . . . . . . . . . . . . . . . 60
Normal Acceleration . . . . . . . . . . . . . . . . . . . . . 60
Sound Soft Boundary . . . . . . . . . . . . . . . . . . . . . 61
Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Impedance . . . . . . . . . . . . . . . . . . . . . . . . . 62
Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 63
Plane Wave Radiation . . . . . . . . . . . . . . . . . . . . . 63
Spherical Wave Radiation. . . . . . . . . . . . . . . . . . . . 64
Cylindrical Wave Radiation . . . . . . . . . . . . . . . . . . . 65
Incident Pressure Field. . . . . . . . . . . . . . . . . . . . . 66
Periodic Condition . . . . . . . . . . . . . . . . . . . . . . 67
Interior Sound Hard Boundary (Wall) . . . . . . . . . . . . . . . 69
Axial Symmetry . . . . . . . . . . . . . . . . . . . . . . . 70
Continuity . . . . . . . . . . . . . . . . . . . . . . . . . 70
Pressure Acoustics Model . . . . . . . . . . . . . . . . . . . 71
Background Pressure Field . . . . . . . . . . . . . . . . . . . 78
ii | C O N T E N T S
Matched Boundary . . . . . . . . . . . . . . . . . . . . . . 79
Far-Field Calculation
. . . . . . . . . . . . . . . . . . . . . 80
Interior Normal Acceleration . . . . . . . . . . . . . . . . . . 82
Interior Impedance/Pair Impedance . . . . . . . . . . . . . . . . 83
Interior Perforated Plate . . . . . . . . . . . . . . . . . . . . 84
Line Source . . . . . . . . . . . . . . . . . . . . . . . . . 85
Line Source on Axis. . . . . . . . . . . . . . . . . . . . . . 88
Monopole Point Source . . . . . . . . . . . . . . . . . . . . 89
Point Source . . . . . . . . . . . . . . . . . . . . . . . . 91
Circular Source . . . . . . . . . . . . . . . . . . . . . . . 93
The Pressure Acoustics, Transient User Interface
95
Domain, Boundary, Edge, and Point Nodes for the Pressure
Acoustics, Transient User Interface . . . . . . . . . . . . . . . 96
Transient Pressure Acoustics Model. . . . . . . . . . . . . . . . 96
The Gaussian Pulse Source Type . . . . . . . . . . . . . . . . . 98
The Boundary Mode Acoustics User Interface
99
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
100
Boundary, Edge, Point, and Pair Nodes for the Boundary Mode
Acoustics User Interface . . . . . . . . . . . . . . . . . .
101
Theory Background for the Pressure Acoustics Branch
102
The Governing Equations. . . . . . . . . . . . . . . . . . .
102
Pressure Acoustics, Frequency Domain Equations . . . . . . . . .
105
Pressure Acoustics, Transient Equations . . . . . . . . . . . . .
109
Boundary Mode Acoustics Equations . . . . . . . . . . . . . .
109
Theory for the Plane, Spherical, and Cylindrical Radiation Boundary
Conditions . . . . . . . . . . . . . . . . . . . . . . .
111
Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral .
113
Theory for the Pressure Acoustics Fluid Models
118
Introduction to the Pressure Acoustics Fluid Models . . . . . . . .
118
About the Linear Elastic with Attenuation Fluid Model
. . . . . . .
119
. . . . . . . . .
121
About the Macroscopic Empirical Porous Models
About the Viscous Fluid Model . . . . . . . . . . . . . . . .
121
About the Thermally Conducting Fluid Model . . . . . . . . . . .
122
About the Thermally Conducting and Viscous Fluid Model . . . . . .
123
CONTENTS
| iii
About the Biot Equivalent Fluid Models . . . . . . . . . . . . .
123
About the Boundary-Layer Absorption Fluid Model . . . . . . . . .
125
References for the Pressure Acoustics Branch
128
Chapter 4: Acoustic-Structure Interaction
The Acoustic-Solid Interaction, Frequency Domain User
Interface
130
Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Solid Interaction, Frequency Domain User Interface . . . .
132
Acoustic-Structure Boundary . . . . . . . . . . . . . . . . .
134
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
135
Flow Line Source on Axis
135
. . . . . . . . . . . . . . . . . .
Intensity Line Source on Axis . . . . . . . . . . . . . . . . .
136
Power Line Source on Axis . . . . . . . . . . . . . . . . . .
137
Intensity Edge Source . . . . . . . . . . . . . . . . . . . .
138
Power Edge Source . . . . . . . . . . . . . . . . . . . . .
138
Intensity Point Source . . . . . . . . . . . . . . . . . . . .
139
Power Point Source . . . . . . . . . . . . . . . . . . . . .
140
The Acoustic-Solid Interaction, Transient User Interface
142
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
143
The Acoustic-Piezoelectric Interaction, Frequency Domain
User Interface
144
Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Piezoelectric Interaction, Frequency Domain User Interface .
146
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
148
The Acoustic-Piezoelectric Interaction, Transient User
Interface
iv | C O N T E N T S
149
The Elastic Waves and Poroelastic Waves User Interfaces
152
The Elastic Waves User Interface. . . . . . . . . . . . . . . .
152
The Poroelastic Waves User Interface . . . . . . . . . . . . . .
154
Domain, Boundary, and Shared Nodes for the Elastic Waves and
the Poroelastic Waves User Interface . . . . . . . . . . . . .
155
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
157
Poroelastic Material . . . . . . . . . . . . . . . . . . . . .
157
Porous, Fixed Constraint . . . . . . . . . . . . . . . . . . .
161
Porous, Free . . . . . . . . . . . . . . . . . . . . . . .
161
Porous, Pressure . . . . . . . . . . . . . . . . . . . . . .
162
Porous, Prescribed Displacement. . . . . . . . . . . . . . . .
162
Porous, Prescribed Velocity . . . . . . . . . . . . . . . . . .
164
Porous, Prescribed Acceleration . . . . . . . . . . . . . . . .
165
Porous, Roller . . . . . . . . . . . . . . . . . . . . . . .
166
Porous, Septum Boundary Load . . . . . . . . . . . . . . . .
166
Continuity . . . . . . . . . . . . . . . . . . . . . . . .
167
Theory for the Elastic Waves and Poroelastic Waves User
Interfaces
169
About Elastic Waves . . . . . . . . . . . . . . . . . . . .
169
About Poroelastic Waves. . . . . . . . . . . . . . . . . . .
170
About the Boundary Conditions for Poroelastic Waves . . . . . . .
174
References for the Elastic Waves and Poroelastic Waves User
Interfaces . . . . . . . . . . . . . . . . . . . . . . . .
176
The Acoustic-Shell Interaction, Frequency Domain User
Interface
178
Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Shell Interaction, Frequency Domain User Interface . . . .
180
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
182
Initial Values (Boundary) . . . . . . . . . . . . . . . . . . .
183
Exterior Shell . . . . . . . . . . . . . . . . . . . . . . .
183
Interior Shell . . . . . . . . . . . . . . . . . . . . . . .
184
Uncoupled Shell . . . . . . . . . . . . . . . . . . . . . .
185
The Acoustic-Shell Interaction, Transient User Interface
186
The Pipe Acoustics User Interfaces
188
The Pipe Acoustics, Frequency Domain User Interface . . . . . . .
188
The Pipe Acoustics, Transient User Interface . . . . . . . . . . .
189
Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics User
CONTENTS
|v
Interfaces . . . . . . . . . . . . . . . . . . . . . . . .
190
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
191
Fluid Properties . . . . . . . . . . . . . . . . . . . . . .
192
Pipe Properties . . . . . . . . . . . . . . . . . . . . . .
193
Closed. . . . . . . . . . . . . . . . . . . . . . . . . .
194
Pressure . . . . . . . . . . . . . . . . . . . . . . . . .
194
Velocity . . . . . . . . . . . . . . . . . . . . . . . . .
195
End Impedance
196
. . . . . . . . . . . . . . . . . . . . . .
Theory for the Pipe Acoustics User Interfaces
199
Governing Equations . . . . . . . . . . . . . . . . . . . .
199
Theory for the Pipe Acoustics Boundary Conditions . . . . . . . .
203
Solving Transient Problems . . . . . . . . . . . . . . . . . .
206
Cut-off Frequency . . . . . . . . . . . . . . . . . . . . .
207
References for the Pipe Acoustics User Interfaces . . . . . . . . .
208
Swirl Correction Factor   
Chapter 5: The Aeroacoustics Branch
The Aeroacoustics, Frequency Domain User Interface
210
Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics, Frequency Domain User Interface . . . . . . . .
vi | C O N T E N T S
212
Aeroacoustics Model . . . . . . . . . . . . . . . . . . . .
213
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
214
Sound Hard Boundary (Wall) . . . . . . . . . . . . . . . . .
214
Velocity Potential . . . . . . . . . . . . . . . . . . . . . .
215
Normal Mass Flow . . . . . . . . . . . . . . . . . . . . .
216
Plane Wave Radiation . . . . . . . . . . . . . . . . . . . .
216
Incident Velocity Potential . . . . . . . . . . . . . . . . . .
217
Sound Soft Boundary . . . . . . . . . . . . . . . . . . . .
218
Normal Velocity . . . . . . . . . . . . . . . . . . . . . .
218
Impedance and Pair Impedance . . . . . . . . . . . . . . . .
219
Vortex Sheet . . . . . . . . . . . . . . . . . . . . . . .
219
Interior Sound Hard Boundary (Wall) . . . . . . . . . . . . . .
220
Continuity . . . . . . . . . . . . . . . . . . . . . . . .
220
Axial Symmetry . . . . . . . . . . . . . . . . . . . . . .
221
Mass Flow Line Source on Axis . . . . . . . . . . . . . . . .
222
Mass Flow Edge Source . . . . . . . . . . . . . . . . . . .
222
Mass Flow Point Source . . . . . . . . . . . . . . . . . . .
222
Mass Flow Circular Source . . . . . . . . . . . . . . . . . .
223
The Aeroacoustics, Transient User Interface
224
Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics, Transient User Interface . . . . . . . . . . . .
The Boundary Mode Aeroacoustics User Interface
225
226
Boundary, Edge, Point, and Pair Nodes for the Boundary Mode
Aeroacoustics User Interface . . . . . . . . . . . . . . . .
The Compressible Potential Flow User Interface
227
229
Domain, Boundary, and Pair Nodes for the Compressible Potential
Flow User Interface . . . . . . . . . . . . . . . . . . . .
230
Compressible Potential Flow Model. . . . . . . . . . . . . . .
231
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
232
Slip Velocity . . . . . . . . . . . . . . . . . . . . . . . .
232
Normal Flow . . . . . . . . . . . . . . . . . . . . . . .
232
Mass Flow . . . . . . . . . . . . . . . . . . . . . . . .
233
The Aeroacoustics with Flow, Frequency Domain User
Interface
234
Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics with Flow, Frequency Domain User Interface . . . .
236
Aeroacoustics with Flow Model . . . . . . . . . . . . . . . .
237
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
237
The Aeroacoustics with Flow, Transient User Interface
239
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
240
Theory Background for the Aeroacoustics Branch
241
Linearized Potential Flow Aeroacoustics . . . . . . . . . . . . .
242
Compressible Potential Flow . . . . . . . . . . . . . . . . .
243
Frequency Domain Equations . . . . . . . . . . . . . . . . .
245
Time Dependent Equation . . . . . . . . . . . . . . . . . .
246
Mode Analysis Study in Boundary Mode Aeroacoustics . . . . . . .
247
CONTENTS
| vii
Chapter 6: The Thermoacoustics Branch
The Thermoacoustics, Frequency Domain User Interface
250
Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustics, Frequency Domain User Interface . . . . . . .
254
Thermoacoustics Model . . . . . . . . . . . . . . . . . . .
255
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
259
Heat Source. . . . . . . . . . . . . . . . . . . . . . . .
259
Sound Hard Wall . . . . . . . . . . . . . . . . . . . . . .
259
Isothermal . . . . . . . . . . . . . . . . . . . . . . . .
260
Acoustic-Thermoacoustic Boundary . . . . . . . . . . . . . .
260
Pressure (Adiabatic). . . . . . . . . . . . . . . . . . . . .
261
Wall. . . . . . . . . . . . . . . . . . . . . . . . . . .
261
Symmetry . . . . . . . . . . . . . . . . . . . . . . . .
261
Velocity . . . . . . . . . . . . . . . . . . . . . . . . .
262
Slip . . . . . . . . . . . . . . . . . . . . . . . . . . .
262
Stress . . . . . . . . . . . . . . . . . . . . . . . . . .
263
No Stress
263
. . . . . . . . . . . . . . . . . . . . . . . .
Normal Stress . . . . . . . . . . . . . . . . . . . . . . .
263
Normal Impedance . . . . . . . . . . . . . . . . . . . . .
264
Adiabatic . . . . . . . . . . . . . . . . . . . . . . . . .
264
Temperature Variation . . . . . . . . . . . . . . . . . . . .
265
The Thermoacoustic-Solid Interaction, Frequency Domain
User Interface
266
Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustic-Solid Interaction, Frequency Domain User Interface .
268
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
270
Continuity . . . . . . . . . . . . . . . . . . . . . . . .
270
Symmetry . . . . . . . . . . . . . . . . . . . . . . . .
271
The Thermoacoustic-Shell Interaction, Frequency Domain
User Interface
273
Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustic-Shell Interaction, Frequency Domain User Interface .
viii | C O N T E N T S
276
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
279
Initial Values (Boundary) . . . . . . . . . . . . . . . . . . .
279
Exterior Shell . . . . . . . . . . . . . . . . . . . . . . .
280
Interior Shell . . . . . . . . . . . . . . . . . . . . . . .
280
Uncoupled Shell . . . . . . . . . . . . . . . . . . . . . .
281
Theory Background for the Thermoacoustics Branch
282
The Viscous and Thermal Boundary Layers . . . . . . . . . . . .
282
General Linearized Compressible Flow Equations . . . . . . . . .
283
Formulation for Eigenfrequency Studies . . . . . . . . . . . . .
289
Formulation for Mode Analysis
290
. . . . . . . . . . . . . . . .
References for the Thermoacoustics, Frequency Domain User
Interface . . . . . . . . . . . . . . . . . . . . . . . .
291
Chapter 7: The Structural Mechanics Branch
The Solid Mechanics User Interface
294
Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics . .
297
Linear Elastic Material . . . . . . . . . . . . . . . . . . . .
298
Change Thickness . . . . . . . . . . . . . . . . . . . . .
301
Damping . . . . . . . . . . . . . . . . . . . . . . . . .
302
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
305
About the Body, Boundary, Edge, and Point Loads . . . . . . . . .
305
Body Load . . . . . . . . . . . . . . . . . . . . . . . .
305
Boundary Load
. . . . . . . . . . . . . . . . . . . . . .
306
Edge Load . . . . . . . . . . . . . . . . . . . . . . . .
307
Point Load . . . . . . . . . . . . . . . . . . . . . . . .
308
Fixed Constraint . . . . . . . . . . . . . . . . . . . . . .
308
Prescribed Displacement . . . . . . . . . . . . . . . . . . .
309
Free. . . . . . . . . . . . . . . . . . . . . . . . . . .
311
Symmetry . . . . . . . . . . . . . . . . . . . . . . . .
311
Antisymmetry . . . . . . . . . . . . . . . . . . . . . . .
312
Roller . . . . . . . . . . . . . . . . . . . . . . . . . .
312
Periodic Condition . . . . . . . . . . . . . . . . . . . . .
312
Initial Stress and Strain. . . . . . . . . . . . . . . . . . . .
314
Phase . . . . . . . . . . . . . . . . . . . . . . . . . .
314
Prescribed Velocity . . . . . . . . . . . . . . . . . . . . .
315
Prescribed Acceleration . . . . . . . . . . . . . . . . . . .
316
CONTENTS
| ix
Spring Foundation . . . . . . . . . . . . . . . . . . . . .
317
Pre-Deformation . . . . . . . . . . . . . . . . . . . . . .
319
Thin Elastic Layer. . . . . . . . . . . . . . . . . . . . . .
319
Added Mass . . . . . . . . . . . . . . . . . . . . . . . .
321
Low-Reflecting Boundary . . . . . . . . . . . . . . . . . . .
322
Theory for the Solid Mechanics User Interface
323
Material and Spatial Coordinates . . . . . . . . . . . . . . . .
323
Coordinate Systems. . . . . . . . . . . . . . . . . . . . .
324
Lagrangian Formulation . . . . . . . . . . . . . . . . . . .
325
About Linear Elastic Materials . . . . . . . . . . . . . . . . .
326
Strain-Displacement Relationship . . . . . . . . . . . . . . . .
333
Stress-Strain Relationship. . . . . . . . . . . . . . . . . . .
336
Plane Strain and Plane Stress Cases . . . . . . . . . . . . . . .
336
Axial Symmetry . . . . . . . . . . . . . . . . . . . . . .
337
Loads . . . . . . . . . . . . . . . . . . . . . . . . . .
339
Pressure Loads
341
. . . . . . . . . . . . . . . . . . . . . .
Equation Implementation . . . . . . . . . . . . . . . . . . .
341
Setting up Equations for Different Studies . . . . . . . . . . . .
342
Damping Models . . . . . . . . . . . . . . . . . . . . . .
344
Initial Stresses and Strains . . . . . . . . . . . . . . . . . .
347
About Spring Foundations and Thin Elastic Layers . . . . . . . . .
348
About Added Mass . . . . . . . . . . . . . . . . . . . . .
350
Geometric Nonlinearity Theory for the Solid Mechanics User
x | CONTENTS
Interface . . . . . . . . . . . . . . . . . . . . . . . .
351
About the Low-Reflecting Boundary Condition . . . . . . . . . .
355
Cyclic Symmetry and Floquet Periodic Conditions . . . . . . . . .
356
Calculating Reaction Forces
358
Using Predefined Variables to Evaluate Reaction Forces . . . . . . .
358
Using Weak Constraints to Evaluate Reaction Forces . . . . . . . .
359
Using Surface Traction to Evaluate Reaction Forces . . . . . . . . .
360
Evaluating Surface Traction Forces on Internal Boundaries . . . . . .
361
Geometric Nonlinearity, Frames, and the ALE Method
362
Reference for Geometric Nonlinearity. . . . . . . . . . . . . .
364
Springs and Dampers
365
Damping and Loss
367
Overview of Damping and Loss . . . . . . . . . . . . . . . .
367
Linear Viscoelastic Materials . . . . . . . . . . . . . . . . .
371
Rayleigh Damping. . . . . . . . . . . . . . . . . . . . . .
371
Equivalent Viscous Damping. . . . . . . . . . . . . . . . . .
372
Loss Factor Damping . . . . . . . . . . . . . . . . . . . .
373
Explicit Damping . . . . . . . . . . . . . . . . . . . . . .
373
Chapter 8: The Piezoelectric Devices User Interface
The Piezoelectric Devices User Interface
376
Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric
Devices User Interface . . . . . . . . . . . . . . . . . . .
378
Piezoelectric Material . . . . . . . . . . . . . . . . . . . .
380
Electrical Material Model . . . . . . . . . . . . . . . . . . .
381
Electrical Conductivity (Time-Harmonic) . . . . . . . . . . . . .
383
Damping and Loss . . . . . . . . . . . . . . . . . . . . .
384
Remanent Electric Displacement . . . . . . . . . . . . . . . .
385
Dielectric Loss. . . . . . . . . . . . . . . . . . . . . . .
385
Initial Values. . . . . . . . . . . . . . . . . . . . . . . .
386
Periodic Condition . . . . . . . . . . . . . . . . . . . . .
386
Theory for the Piezoelectric Devices User Interface
388
The Piezoelectric Effect . . . . . . . . . . . . . . . . . . .
388
Piezoelectric Constitutive Relations . . . . . . . . . . . . . . .
389
Piezoelectric Material . . . . . . . . . . . . . . . . . . . .
391
Piezoelectric Dissipation . . . . . . . . . . . . . . . . . . .
391
Initial Stress, Strain, and Electric Displacement. . . . . . . . . . .
391
Geometric Nonlinearity for the Piezoelectric Devices User Interface . .
392
Damping and Losses Theory . . . . . . . . . . . . . . . . .
394
References for the Piezoelectric Devices User Interface . . . . . . .
397
Piezoelectric Damping
399
About Piezoelectric Materials . . . . . . . . . . . . . . . . .
399
CONTENTS
| xi
Piezoelectric Material Orientation . . . . . . . . . . . . . . .
400
Piezoelectric Losses. . . . . . . . . . . . . . . . . . . . .
406
References for Piezoelectric Damping . . . . . . . . . . . . . .
409
Chapter 9: Glossary
Glossary of Terms
xii | C O N T E N T S
412
2
Introduction
The Acoustics Module is an optional package that extends the COMSOL
Multiphysics® environment with customized interfaces and functionality
optimized for the analysis of acoustics and vibration problems.
This module solves problems in the general areas of acoustics, acoustic-structure
interaction, aeroacoustics, thermoacoustics, pressure and elastic waves in porous
materials, and vibrations. The physics interfaces included are fully multiphysics
enabled, making it possible to couple them to any other physics interface in
COMSOL Multiphysics. Explicit demonstrations of these capabilities are supplied
with the product in a library (the Model Library) of ready-to-run models that make
it quicker and easier to get introduced to discipline-specific problems. One example
being a model of a loudspeaker involving both electromechanical and
acoustic-structural couplings.
This chapter is an introduction to the capabilities of this module. A summary of the
physics interfaces and where you can find documentation and model examples is
also included. The last section is a brief overview with links to each chapter in this
guide.
1
Acoustics Module Physics Guide
The Acoustics Module extends the functionality of the physics user interfaces of the
COMSOL Multiphysics® base package. The details of the physics user interfaces and
study types for the Acoustics Module are listed in the table below.
In the COMSOL Multiphysics Reference Manual:
• Studies and the Study Nodes
• The Physics User Interfaces
• See Physics Guide for a list of all the interfaces included with the
COMSOL Multiphysics basic license.
PHYSICS USER INTERFACE
ICON
TAG
SPACE
DIMENSION
AVAILABLE PRESET STUDY TYPE
Pressure Acoustics,
Frequency Domain*
acpr
all dimensions
eigenfrequency; frequency
domain; frequency-domain
modal; mode analysis (2D and
1D axisymmetric models only)
Pressure Acoustics,
Transient
actd
all dimensions
eigenfrequency; frequency
domain; frequency-domain
modal; time dependent;
time-dependent modal; modal
reduced order model; mode
analysis (2D and 1D
axisymmetric models only)
Boundary Mode
Acoustics
acbm
3D, 2D
axisymmetric
mode analysis
acsl
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency-domain
modal
Acoustics
Pressure Acoustics
Acoustic-Structure Interaction
Acoustic-Solid
Interaction,
Frequency Domain
2 |
CHAPTER 2: INTRODUCTION
PHYSICS USER INTERFACE
ICON
TAG
SPACE
DIMENSION
AVAILABLE PRESET STUDY TYPE
Acoustic-Solid
Interaction, Transient
astd
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency-domain
modal; time dependent;
time-dependent modal; modal
reduced order model
Acoustic-Shell
Interaction,
Frequency Domain1
acsh
3D
eigenfrequency; frequency
domain; frequency-domain
modal
Acoustic-Shell
Interaction,
Transient1
acshtd
3D
eigenfrequency; frequency
domain; frequency-domain
modal; time dependent;
time-dependent modal; modal
reduced order model
Acoustic-Piezoelectri
c Interaction,
Frequency Domain
acpz
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency-domain
modal
Acoustic-Piezoelectri
c Interaction,
Transient
acpztd
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency-domain
modal; time dependent;
time-dependent modal; modal
reduced order model
Elastic Waves
elw
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency-domain
modal
Poroelastic Waves
elw
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency-domain
modal
Pipe Acoustics,
Frequency Domain2
pafd
3D, 2D
eigenfrequency; frequency
domain
Pipe Acoustics,
Transient2
patd
3D, 2D
time dependent
ae
all dimensions
frequency domain; mode
analysis (2D and 1D
axisymmetric models only)
Aeroacoustics
Aeroacoustics,
Frequency Domain
ACOUSTICS MODULE PHYSICS GUIDE
|
3
PHYSICS USER INTERFACE
ICON
TAG
SPACE
DIMENSION
AVAILABLE PRESET STUDY TYPE
Aeroacoustics,
Transient
aetd
all dimensions
frequency domain; time
dependent mode analysis (2D
and 1D axisymmetric models
only)
Boundary Mode
Aeroacoustics
aebm
3D, 2D
axisymmetric
mode analysis
Aeroacoustics with
Flow, Frequency
Domain
aepf
all dimensions
frequency domain
Aeroacoustics with
Flow, Transient
atpf
all dimensions
frequency domain; time
dependent
Compressible
Potential Flow
cpf
all dimensions
stationary; time dependent
Thermoacoustics,
Frequency Domain
ta
all dimensions
eigenfrequency; frequency
domain; frequency domain
modal; mode analysis (2D and
1D axisymmetric models only)
Thermoacoustic-Soli
d Interaction,
Frequency Domain
tas
3D, 2D, 2D
axisymmetric
eigenfrequency; frequency
domain; frequency domain
modal
Thermoacoustic-Shell
Interaction1
tash
3D
eigenfrequency; frequency
domain; frequency domain
modal
solid
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency;
prestressed analysis,
eigenfrequency; time
dependent; time dependent
modal; frequency domain;
frequency-domain modal;
prestressed analysis,
frequency domain; modal
reduced order model
Thermoacoustics
Structural Mechanics
Solid Mechanics*
4 |
CHAPTER 2: INTRODUCTION
PHYSICS USER INTERFACE
Piezoelectric Devices
ICON
TAG
SPACE
DIMENSION
AVAILABLE PRESET STUDY TYPE
pzd
3D, 2D, 2D
axisymmetric
stationary; eigenfrequency;
time dependent;
time-dependent modal;
frequency domain; frequency
domain modal; modal reduced
order model
* This is an enhanced interface, which is included with the base COMSOL package but has
added functionality for this module.
1
Requires both the Structural Mechanics Module and the Acoustics Module.
2 Requires both the Pipe Flow Module and the Acoustics Module.
SHOW MORE PHYSICS OPTIONS
There are several general options available for the physics user interfaces and for
individual nodes. This section is a short overview of these options, and includes links
to additional information when available.
The links to the features described in the COMSOL Multiphysics
Reference Manual (or any external guide) do not work in the PDF, only
from within the online help.
To locate and search all the documentation for this information, in
COMSOL Multiphysics, select Help>Documentation from the main menu
and either enter a search term or look under a specific module in the
documentation tree.
To display additional options for the physics interfaces and other parts of the model
) on the Model Builder and then select the applicable
tree, click the Show button (
option.
), additional sections get displayed on the settings
After clicking the Show button (
window when a node is clicked and additional nodes are available from the context
menu when a node is right-clicked. For each, the additional sections that can be
displayed include Equation, Advanced Settings, Discretization, Consistent Stabilization,
and Inconsistent Stabilization.
ACOUSTICS MODULE PHYSICS GUIDE
|
5
You can also click the Expand Sections button (
) in the Model Builder to always show
some sections or click the Show button (
) and select Reset to Default to reset to
display only the Equation and Override and Contribution sections.
For most nodes, both the Equation and Override and Contribution sections are always
available. Click the Show button (
) and then select Equation View to display the
Equation View node under all nodes in the Model Builder.
Availability of each node, and whether it is described for a particular node, is based on
the individual selected. For example, the Discretization, Advanced Settings, Consistent
Stabilization, and Inconsistent Stabilization sections are often described individually
throughout the documentation as there are unique settings.
SECTION
CROSS REFERENCE
Show More Options and
Expand Sections
Advanced Physics Sections
Discretization
Show Discretization
The Model Wizard and Model Builder
Discretization (Node)
Discretization—Splitting of
complex variables
Compile Equations
Consistent and
Inconsistent Stabilization
Show Stabilization
Constraint Settings
Weak Constraints and Constraint Settings
Override and Contribution
Physics Exclusive and Contributing Node Types
Numerical Stabilization
OTHER COMMON SETTINGS
At the main level, some of the common settings found (in addition to the Show
options) are the Interface Identifier, Domain, Boundary, or Edge Selection, and
Dependent Variables.
At the nodes’ level, some of the common settings found (in addition to the Show
options) are Domain, Boundary, Edge, or Point Selection, Material Type, Coordinate
6 |
CHAPTER 2: INTRODUCTION
System Selection, and Model Inputs. Other sections are common based on application
area and are not included here.
SECTION
CROSS REFERENCE
Coordinate System
Selection
Coordinate Systems
Domain, Boundary, Edge,
and Point Selection
About Geometric Entities
Interface Identifier
Predefined Physics Variables
About Selecting Geometric Entities
Variable Naming Convention and Scope
Viewing Node Names, Identifiers, Types, and Tags
Material Type
Materials
Model Inputs
About Materials and Material Properties
Selecting Physics
Adding Multiphysics Couplings
Pair Selection
Identity and Contact Pairs
Continuity on Interior Boundaries
ACOUSTICS MODULE PHYSICS GUIDE
|
7
Where Do I Access the
Documentation and Model Library?
A number of Internet resources provide more information about COMSOL, including
licensing and technical information. The electronic documentation, context help, and
the Model Library are all accessed through the COMSOL Desktop.
If you are reading the documentation as a PDF file on your computer, the
blue links do not work to open a model or content referenced in a
different guide. However, if you are using the online help in COMSOL
Multiphysics, these links work to other modules, model examples, and
documentation sets.
THE DOCUMENTATION
The COMSOL Multiphysics Reference Manual describes all user interfaces and
functionality included with the basic COMSOL Multiphysics license. This book also
has instructions about how to use COMSOL and how to access the documentation
electronically through the COMSOL Help Desk.
To locate and search all the documentation, in COMSOL Multiphysics:
• Press F1 or select Help>Help (
8 |
CHAPTER 2: INTRODUCTION
) from the main menu for context help.
• Press Ctrl+F1 or select Help>Documentation (
) from the main menu for opening
the main documentation window with access to all COMSOL documentation.
• Click the corresponding buttons (
or
) on the main toolbar.
and then either enter a search term or look under a specific module in the
documentation tree.
If you have added a node to a model you are working on, click the Help
button (
) in the node’s settings window or press F1 to learn more
about it. Under More results in the Help window there is a link with a
search string for the node’s name. Click the link to find all occurrences of
the node’s name in the documentation, including model documentation
and the external COMSOL website. This can help you find more
information about the use of the node’s functionality as well as model
examples where the node is used.
THE MODEL LIBRARY
Each model comes with documentation that includes a theoretical background and
step-by-step instructions to create the model. The models are available in COMSOL
as MPH-files that you can open for further investigation. You can use the step-by-step
instructions and the actual models as a template for your own modeling and
applications.
In most models, SI units are used to describe the relevant properties, parameters, and
dimensions in most examples, but other unit systems are available.
) from the main menu, and
To open the Model Library, select View>Model Library (
then search by model name or browse under a module folder name. Click to highlight
any model of interest, and select Open Model and PDF to open both the model and the
documentation explaining how to build the model. Alternatively, click the Help
button (
) or select Help>Documentation in COMSOL to search by name or browse
by module.
The model libraries are updated on a regular basis by COMSOL in order to add new
models and to improve existing models. Choose View>Model Library Update (
) to
update your model library to include the latest versions of the model examples.
If you have any feedback or suggestions for additional models for the library (including
those developed by you), feel free to contact us at info@comsol.com.
WHERE DO I ACCESS THE DOCUMENTATION AND MODEL LIBRARY?
|
9
CONT ACT ING COMSOL BY EMAIL
For general product information, contact COMSOL at info@comsol.com.
To receive technical support from COMSOL for the COMSOL products, please
contact your local COMSOL representative or send your questions to
support@comsol.com. An automatic notification and case number is sent to you by
email.
COMSOL WEBSITES
10 |
COMSOL website
www.comsol.com
Contact COMSOL
www.comsol.com/contact
Support Center
www.comsol.com/support
Download COMSOL
www.comsol.com/support/download
Support Knowledge Base
www.comsol.com/support/knowledgebase
Product Updates
www.comsol.com/support/updates
COMSOL Community
www.comsol.com/community
CHAPTER 2: INTRODUCTION
Overview of the User’s Guide
The Acoustics Module User’s Guide gets you started with modeling using COMSOL
Multiphysics®. The information in this guide is specific to this module. Instructions
on how to use COMSOL in general are included with the COMSOL Multiphysics
Reference Manual.
As detailed in the section Where Do I Access the Documentation and
Model Library? this information can also be searched from the COMSOL
Multiphysics software Help menu.
TA B L E O F C O N T E N T S , G L O S S A R Y, A N D I N D E X
To help you navigate through this guide, see the Contents, Glossary, and Index.
MODELING WITH THE ACOUSTICS MODULE
The Modeling with the Acoustics Module chapter introduces you to Acoustics Module
Capabilities and the Fundamentals of Acoustics Modeling including illustrative models
and information that serves as a reference source for more advanced modeling. The
Acoustics Module Study Types briefly describes the available study types and the
Fundamentals of Acoustics Modeling has some background theory and equations.
THE PRESSURE ACOUSTICS BRANCH
The Pressure Acoustics Branch chapter describes the interfaces found under the
Pressure Acoustics (
) branch on the Add Physics page of the Model Wizard.
The Pressure Acoustics, Frequency Domain User Interface is the core interface which
models the sound waves in the frequency domain and The Pressure Acoustics,
Transient User Interface is the core interface which models the sound waves in the time
domain. The Boundary Mode Acoustics User Interface solves for modes that
propagate through a cross section of your geometry.
THE ACOUSTIC-STRUCTURE INTERACTION BRANCH
Acoustic-Structure Interaction chapter describes the interfaces found under the
Acoustic-Structure Interaction (
) branch on the Add Physics page of the Model
Wizard.
The Acoustic-Solid Interaction, Frequency Domain User Interface is a combination of
pressure acoustics and solid mechanics with predefined couplings and The
O V E R V I E W O F T H E U S E R ’S G U I D E
|
11
Acoustic-Solid Interaction, Transient User Interface is a combination of transient
pressure acoustics and solid mechanics with predefined couplings.
The Acoustic-Piezoelectric Interaction, Frequency Domain User Interface is a
combination of pressure acoustics and piezoelectric effects with a predefined coupling
for the boundary between the acoustic domain and the piezoelectric device. The
Acoustic-Piezoelectric Interaction, Transient User Interface combines Pressure
Acoustics, Transient, Solid Mechanics, Electrostatics, and the Piezoelectric Devices
interface features.
The Elastic Waves User Interface combines pressure acoustics and solid mechanics to
connect the fluid pressure with the structural deformation in solids. It also features The
Poroelastic Waves User Interface, which can be seen as linear elastic waves coupled to
pressure waves in porous elastic materials damped by a pore fluid.
The Acoustic-Shell Interaction, Frequency Domain User Interface requires a
Structural Mechanics Module license and is found under the Acoustic-Structure
Interaction (
) branch on the Add Physics page of the Model Wizard. It uses the
features from the Pressure Acoustics, Frequency Domain and the Shell interfaces to
connect the acoustics pressure waves in a fluid domain with the structural deformation
in a shell. The interface is available for 3D geometry only.
The Acoustic-Shell Interaction, Transient User Interface, which also requires a
Structural Mechanics Module license, uses the features from the Pressure Acoustics,
Transient and the Shell interfaces to connect the transient pressure acoustics in a fluid
domain with the structural deformation of shell boundary. The interface is available for
3D geometry only.
The Pipe Acoustics User Interfaces (
), which require both the Pipe Flow Module
and the Acoustics Module, have the equations and boundary conditions for modeling
the propagation of sound waves in flexible pipe systems. The equations are formulated
in a general way to include the possibility of a stationary background flow. There are
two interfaces, one for transient analysis and one for frequency domain studies.
THE AEROACOUSTICS BRANCH
The Aeroacoustics Branch chapter describes the interfaces found under the
Aeroacoustics (
) branch on the Add Physics page of the Model Wizard.
The Aeroacoustics, Frequency Domain User Interface models acoustic waves in
potential flow in the frequency domain and The Aeroacoustics, Transient User
Interface models acoustic waves in potential flow in the time domain. The Boundary
Mode Aeroacoustics User Interface solves for modes that propagate through a cross
12 |
CHAPTER 2: INTRODUCTION
section of your geometry.
The Aeroacoustics with Flow, Frequency Domain User Interface is a combination of
aeroacoustics and compressible potential flow with predefined couplings and The
Aeroacoustics with Flow, Transient User Interface is a combination of transient
aeroacoustics and compressible potential flow with predefined couplings. Finally, The
Compressible Potential Flow User Interface models irrotational flow.
THE THERMOACOUSTICS BRANCH
The Thermoacoustics Branch chapter describes The Thermoacoustics, Frequency
Domain User Interface, which is necessary when modeling acoustics accurately in
geometries with small dimensions. Near walls viscosity and thermal conduction
become important because they create a viscous and a thermal boundary layer where
losses are significant.
The Thermoacoustic-Solid Interaction, Frequency Domain User Interface is also
described here. This physics interface combines features from pressure acoustics,
thermoacoustics and solid mechanics with predefined couplings between all three
physics.
The Thermoacoustic-Shell Interaction, Frequency Domain User Interface requires a
Structural Mechanics Module license. The interface uses the features from the
Thermoacoustics, Frequency Domain and the Shell interfaces to connect wave
propagation in pressure acoustic domains and thermoacoustic domains with the
structural deformation of shell boundaries.
THE STRUCTURAL MECHANICS BRANCH
The Structural Mechanics Branch chapter provides information about the Solid
Mechanics interface for modeling, for example, the structural part of
acoustic-structure interaction. This is an extension of the Solid Mechanics interface in
) branch
COMSOL Multiphysics, and you find it under the Structural Mechanics (
on the Add Physics page of the Model Wizard. The theory for this interface is also
included.
PIEZOELECTRIC DEVICES
The Piezoelectric Devices User Interface chapter provides information about
modeling piezoelectric effects using the Piezoelectric Devices interface. You find it
) branch on the Add Physics page of the Model
under the Structural Mechanics (
Wizard.
O V E R V I E W O F T H E U S E R ’S G U I D E
|
13
14 |
CHAPTER 2: INTRODUCTION
3
Modeling with the Acoustics Module
This chapter introduces you to the Acoustic Module modeling stages, including a
short description of the module capabilities and application areas. Information that
serves as a reference source for more advanced modeling is presented, along with
the basic governing equations and the different study types.
In this chapter:
• Acoustics Module Capabilities
• Fundamentals of Acoustics Modeling
• The Acoustics Module Study Types
• Special Variables in the Acoustics Module
|
15
Acoustics Module Capabilities
In this section:
• What Can the Acoustics Module Do?
• What are the Application Areas?
• Which Problems Can You Solve?
What Can the Acoustics Module Do?
The Acoustics Module is a collection of physics user interfaces for COMSOL
Multiphysics adapted to a broad category of acoustics simulations in fluids and solids.
This module is useful even if you are not familiar with computational techniques. It
can serve equally well as an excellent tool for educational purposes.
The module supports time-harmonic (frequency domain), modal, and transient
studies for fluid pressure as well as static, transient, eigenfrequency, and
frequency-response analyses for structures. The available physics interfaces include the
following functionality:
• Pressure acoustics: model the propagation of sound waves (pressure waves) in the
frequency domain and in the time domain.
• Acoustic-structure interactions: combine pressure waves in the fluid with elastic
waves in the solid. The interface provides predefined multiphysics couplings at the
fluid-solid interface.
• Boundary mode acoustics: find propagating and evanescent modes in ducts and
waveguides.
• Thermoacoustics: model the detailed propagation of sound in geometries with small
length scales. This is acoustics including thermal and viscous losses explicitly. Also
known as visco-thermal acoustics, thermo-viscous acoustics, or linearized
compressible Navier-Stokes.
• Aeroacoustics: model the influence a compressible potential background flow has
on the propagation of sound waves.
• Compressible potential flow: determine the flow of a compressible, irrotational, and
inviscid fluid.
• Solid mechanics: solve structural mechanics problems.
16 |
CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
• Piezoelectricity: model the behavior of piezoelectric materials in a multiphysics
environment solving for the electric field and the coupling to the solid structure.
• Poroelastic and elastic waves: model the propagation of elastic waves in solids and
in porous materials. In the latter case the coupled behavior of the porous matrix and
the saturating fluid is modeled in detail.
• Pipe acoustics: use this interface to model the propagation of sound waves in pipe
systems including the elastic properties of the pipe. The equations are formulated in
1D for fast computation and may include a stationary background flow. This
functionality requires the addition of the Pipe Flow Module.
All the physics interfaces include a large number of boundary conditions. For the
pressure acoustics applications you can choose to analyze the scattered wave in
addition to the total wave. Perfectly matched layers (PMLs) provide accurate
simulations of open pipes and other models with unbounded domains. The modeling
domain can include dipole sources as well as monopole sources and it is easy to specify
a monopole point source. The module also includes modeling support for several types
of damping and losses that occur in porous materials or that are due to viscous and
thermal losses. For results evaluation of pressure acoustics models, you can compute
the far field (phase and magnitude) and plot it in predefined far-field plots.
What are the Application Areas?
The Acoustics Module can be used in all areas of engineering and physics to model the
propagation of sound waves in fluids. The module also includes several multiphysics
interfaces because it is common for many application areas involving sound to also have
interaction between fluid and solid structures, have electric fields in piezo materials,
have heat generation, or require modeling of electro-acoustic transducers. Typical
application areas for the Acoustics Module include:
• Automotive applications such as mufflers, particulate filters, and car interiors.
• Sound scattering and sound emission problems.
• Civil engineering applications such as characterization of sound insulation and
sound scatterers. Vibration control and sound transmission problems. Pipe acoustics
for HVAC type of systems.
• Modeling of loudspeakers, microphones, and other transducers.
• Mobile applications such as feedback analysis, optimized transducer placement, and
directivity assessment.
• Aeroacoustics for jet engine noise.
ACOUSTICS MODULE CAPABILITIES
|
17
• Ultrasound piezo transducers for sonar applications.
• Musical instruments.
• Bioacoustic applications with ultrasound and more.
• Underwater acoustics, for example, ultrasound.
• Pressure waves in geophysics.
• Advanced multiphysics applications such as photoacoustics, optoacoustics,
thermoacoustic cooling, acoustofluidics, acoustic streaming and radiation, and
combustion instabilities.
Figure 3-1: An application example is the modeling of mufflers. Here a pressure isosurface
plot from the Absorptive Muffler model from the COMSOL Multiphysics Model Library.
Using the full multiphysics couplings within the COMSOL Multiphysics environment,
you can couple the acoustic waves to, for example, an electromagnetic analysis or a
structural analysis for acoustic-structure interaction. The module smoothly integrates
with all of the COMSOL Multiphysics functionality.
18 |
CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
Which Problems Can You Solve?
The Acoustics Module physics user interfaces handle acoustics in fluids and solids. The
physics interfaces for acoustics in fluids support transient, eigenfrequency, frequency
domain, mode analysis, and boundary mode analysis in pressure acoustics and
aeroacoustics. Thermoacoustic problems, that involve thermal and viscous losses, have
support for eigenfrequency and frequency domain analysis. The study of elastic and
poroelastic waves in solids also has support for eigenfrequency and frequency domain
analysis. The physics interfaces for solids support static, transient, eigenfrequency, and
frequency response analysis. Further, by using the predefined couplings between fluid
and solid physics interfaces, you can solve problems involving acoustic-structure
interaction including the coupling to piezoelectric materials.
All categories are available as 2D, 2D axisymmetric, and 3D models, with the following
differences.
• The Acoustic-Shell Interaction interfaces are only supported in 3D and also require
the addition of the Structural Mechanics Module.
• The Pipe Acoustics interfaces, which require the Pipe Flow Module, exist in edges
in 2D and 3D.
• In 2D the module offers in-plane physics interfaces for problems with a planar
symmetry as well as axisymmetric physics interfaces for problems with a cylindrical
symmetry.
• Use the fluid acoustics physics interfaces with 1D and 1D axisymmetric geometries.
When using the axisymmetric models, the horizontal axis represents the r
direction and the vertical axis the z direction. The geometry in the right
half plane; that is, the geometry must be created and is valid only for
positive r.
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19
Fundamentals of Acoustics Modeling
There are certain difficulties that often arise when modeling acoustics, such as the
rather severe requirements on the mesh resolution, the modeling of artificial
boundaries, and the modeling of real-world damping materials. This section also
includes a brief introduction to acoustics, gives some examples of standard acoustics
problems, and provides a short introduction to the mathematical formulation of the
governing equations.
In this section:
• Acoustics Explained
• Examples of Standard Acoustics Problems
• Mathematical Models for Acoustic Analysis
• Resolving the Waves
• Damping
• Artificial Boundaries
• A Note About Perfectly Matched Layers (PMLs)
• Evaluating the Acoustic Field in the Far-Field Region
• The Far Field Plots
• Solving Large Acoustics Problems Using Iterative Solvers
• About the Material Databases for the Acoustics Module
Overview of the Physics and Building a COMSOL Model in the
COMSOL Multiphysics Reference Manual
Acoustics Explained
Acoustics is the physics of sound. Sound is the sensation, as detected by the ear, of very
small rapid changes in the air pressure above and below a static value. This static value
is the atmospheric pressure (about 100,000 pascals), which varies slowly. Associated
with a sound pressure wave is a flow of energy—the intensity. Physically, sound in air
is a longitudinal wave where the wave motion is in the direction of the movement of
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
energy. The wave crests are the pressure maxima, while the troughs represent the
pressure minima.
Sound results when the air is disturbed by some source. An example is a vibrating
object, such as a speaker cone in a sound system. It is possible to see the movement of
a bass speaker cone when it generates sound at a very low frequency. As the cone moves
forward it compresses the air in front of it, causing an increase in air pressure. Then it
moves back past its resting position and causes a reduction in air pressure. This process
continues, radiating a wave of alternating high and low pressure propagating at the
speed of sound.
The propagation of sound in solids happens through small-amplitude elastic
oscillations of its shape. These elastic waves are transmitted to surrounding fluids as
ordinary sound waves. The elastic sound waves in the solid are the counter part to the
pressure waves or compressible waves propagating in the fluid.
Examples of Standard Acoustics Problems
Depending on the basic dependent variable used to model the acoustic field, the
acoustical interfaces can be divided into the following main categories.
• Pressure acoustics—The dependent variable is the acoustic pressure p.
• Acoustic-solid interaction—The dependent variables are the pressure p and the
displacement field u in the solid.
• Poroelastic waves—The dependent variables are the pressure p inside the saturating
fluid and the total displacement u of the porous matrix.
• Aeroacoustics—The dependent variable is the potential  for the acoustic
particle-velocity field v=   . In the typical situation, the fluid is in motion with a
total velocity vtot  V  v, split into a stationary background-flow velocity V and
the particle velocity v associated with the acoustic waves.
• Thermoacoustics—The dependent variables are the acoustic pressure p, the
particle-velocity field v, and the acoustic temperature variation T. This is a detailed
acoustic model solving the full set of linearized equations for a compressible flow:
Navier-Stokes (momentum conservation), continuity (mass conservation), and
energy conservation equations.
These standard problems or scenarios occur frequently when analyzing acoustics:
FUNDAMENTALS OF ACOUSTICS MODELING
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21
THE RADIATION PROBLEM
A vibrating structure (a speaker, for example) radiates sound into the surrounding
space. A radiation boundary condition or a PML (perfectly matched layer) is necessary
to model the unbounded open domain.
THE SCATTERING PROBLEM
An incident wave impinges on a body and creates a scattered wave. A radiation
boundary condition or a PML is necessary. This could be a sonar application in
underwater acoustics or an analysis of the scattered sound field around a human head.
THE SOUND FIELD IN AN INTERIOR SPACE
The acoustic waves stay in a finite volume so no radiation condition is necessary. This
could be the sound inside a room or a car interior. A more advanced example is the
sound inside a transducer like a microphone; in this case, the acoustic field should be
solved with the Thermoacoustics interface.
COUPLED FLUID-ELASTIC STRUCTURE INTERACTION (STRUCTURAL
ACOUSTICS)
If the radiating or scattering structure consists of an elastic material, the interaction
must be considered between the body and the surrounding fluid. In the multiphysics
coupling, the acoustic analysis provides a load (the sound pressure) to the structural
analysis, and the structural analysis provides accelerations to the acoustic analysis.
T H E TR A N S M I S S I O N P RO B L E M
An incident sound wave propagates into a body, which can have different acoustic
properties. Pressure and acceleration are continuous on the boundary. A typical
transmission problem is that of modeling the behavior of mufflers.
AEROACOUSTICS PROBLEMS
The sound (noise) field is influenced by a background flow. This could be the
propagating sound from a jet engine.
POROELASTIC WAVES PROBLEM
If the acoustic waves are propagating inside the saturating fluid of porous material the
detailed coupling between the fluid pressure and the solid displacement need to be
taken into account. In cases where only the fluid pressure is of interest the porous
material may be modeled using an equivalent fluid model.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
TR A N S D U C E R P R O B L E M S
Transducers transformation of one form of energy to another type (electrical,
mechanical, or acoustical). This type of problem is common in acoustics and is a true
multiphysics problem involving electric, structural, and acoustic interfaces. Typical
problems of this type involve modeling loudspeakers, microphones, and piezo
transducers.
Mathematical Models for Acoustic Analysis
Standard acoustic problems involve solving for the small acoustic pressure variations p
on top of the stationary background pressure p0. Mathematically this represents a
linearization (small parameter expansion) around the stationary quiescent values.
The governing equations for a compressible lossless (no thermal conduction and no
viscosity) fluid flow problem are the momentum equation (Euler's equation) and the
continuity equation. These are given by:
u1
-----+  u   u = – --- p

t

------ +    u  = 0
t
where  is the total density, p is the total pressure, and u is the velocity field. In classical
pressure acoustics all thermodynamic processes are assumed reversible and adiabatic,
known as an isentropic process. The small parameter expansion is performed on a
stationary fluid of density 0 (SI unit: kg/m3) and at pressure p0 (SI unit: Pa) such
that:
p = p 0 + p'
 =  0 + '
p' « p 0
with
' «  0
u = 0 + u'
where the primed variables represent the small acoustic variations. Inserting these into
the governing equations and only retaining terms linear in the primed variables yields:
u'
1-------- = – ----p'
0
t
'
------- +  0    u'  = 0
t
FUNDAMENTALS OF ACOUSTICS MODELING
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23
One of the dependent variables, the density, is removed by expressing it in terms of the
pressure using a Taylor expansion (linearization):
 0
1
' = --------- p' = ----2- p'
p s
cs
where cs is recognized as the (isentropic) speed of sound (SI unit: m/s) at constant
entropy s. The subscripts s and 0 are dropped in the following. Finally, rearranging the
equations (divergence of momentum equation inserted into the continuity equation)
and dropping the primes yields the wave equation for sound waves in a lossless
medium:
2
1-  p
1
------+    – ---  p – q d  = Q m
2
2

c  t
(3-1)
The speed of sound is related to the compressibility of the fluid where the waves are
propagating. The combination  c2 is called the bulk modulus, commonly denoted K
(SI unit: N /m2). The equation is further extended with two optional source terms:
the dipole source qd (SI unit: N/m3) and the monopole source Qm (SI unit: 1/s2).
A special case is a time-harmonic wave, for which the pressure varies with time as
p  x t  = p  x  e
it
where 2f (SI unit: rad/s) is the angular frequency and f (SI unit: Hz) is denoting
the frequency. Assuming the same harmonic time-dependence for the source terms,
the wave equation for acoustic waves reduces to an inhomogeneous Helmholtz
equation:
2
1
 p
   – ---  p – q d  – ---------2- = Q m

c
(3-2)
With the two source terms removed, this equation can also be treated as an eigenvalue
PDE to solve for eigenmodes and eigenfrequencies.
Typical boundary conditions for the wave equation and the Helmholtz equation are:
• Sound-hard boundaries (walls)
• Sound-soft boundaries
• Impedance boundary conditions
• Radiation boundary conditions
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
These are described in more detail in the next sections.
In lossy media, an additional term of first order in the time derivative needs to be
introduced to model attenuation of the sound waves:
2
1-  p
p
1
------– d a +    – ---  p – q d  = Q m
2
2
 


t
 c t
where da is the damping coefficient. Note also that even when the sound waves
propagate in a lossless medium, attenuation frequently occurs by interaction with the
surroundings at the boundaries of the system.
A detailed derivation of the governing equations for the propagation of
compressional (acoustic) waves in a viscous and thermally conducting
fluid is given in Theory Background for the Thermoacoustics Branch.
Resolving the Waves
Solutions to acoustic problems are wavelike. The waves are characterized by a
wavelength  in space, whose value depends on the frequency and speed of sound c in
the medium according to cf. This wavelength has to be resolved by the mesh.
For the solution on the discrete grid to have any meaning at all there has to be at least
two degrees of freedom (DOFs) per wavelength in the direction of propagation, but
such coarse a solution is useless in practice. In reality, the lower limit for a fully reliable
solution lies at about ten to twelve degrees of freedom per wavelength.
Because the direction of propagation is generally not known beforehand, it is good
practice to aim for an isotropic mesh with about twelve DOFs per wavelength on
average, independently of the direction. Therefore the number of DOFs in a
sufficiently resolved mesh is about
• 1728 times the model volume measured in wavelengths cubed in 3D
• 144 times the model area measured in wavelengths squared in 2D
• 12 times the model length measured in wavelengths in 1D
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Before starting a new model, try to estimate the required number of DOFs using these
guidelines. The maximum number of DOFs that can be solved for differs between
computer systems, but a 32-bit system can usually deal with somewhere from a few
hundred thousand up to a million DOFs. Even on a 64-bit system models with more
than a few million DOFs are cumbersome to handle.
USING LAGRANGE ELEMENTS
When creating an unstructured mesh for use with the default 2nd-order Lagrange
elements, set the maximum element size hmax to about  or smaller. Because all
elements in the constructed mesh are smaller than hmax, the limit is set larger than the
actual required element size. After meshing the model, check the total number of
DOFs against the model volume and the above guidelines. If the mesh turns out, on
average, to be too coarse or too fine, try to change hmax accordingly.
Unstructured meshes are generally better than structured meshes for
wave problems where the direction of wave propagation is not known
everywhere in advance. The reason is that in a structured mesh, the
average resolution typically differs significantly between directions parallel
to the grid lines and directions rotated 45 degrees about one of the axes.
Meshing in the COMSOL Multiphysics Reference Manual
Damping
Fluids with a dynamic viscosity in the same range as air or water—by far the most
common media in acoustics simulations—exhibit practically no internal damping over
the number of wavelengths that can be resolved on current computers. Instead,
damping takes place through interaction with solids, either because of friction between
the fluid and a porous material filling the domain, or because acoustic energy is
transferred to a surrounding solid where it is absorbed. In systems with small length
scales, significant losses may occur in the viscous and thermal acoustic boundary layer
at walls.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
POROUS ABSORBING MATERIALS
For frequency-domain modeling, the most convenient and compact description of a
damping material (material refers to the homogenization of a fluid and a porous solid)
is given by its complex wave number k and complex impedance Z, both functions of
frequency. Knowing these properties, define a complex speed of sound as cck and
a complex density as ckZ. Defining c and cc results in a so-called equivalent-fluid
model or fluid model.
It is possible to directly measure the complex wave number and impedance in an
impedance tube in order to produce curves of the real and imaginary parts (the
resistance and reactance, respectively) as functions of frequency. These data can be used
directly as input to COMSOL Multiphysics interpolation functions to define k and Z.
Sometimes acoustic properties cannot be obtained directly for a material you want to
try in a model. In that case you must resort to knowledge about basic material
properties independent of frequency. Several empirical or semi-empirical models exist
in COMSOL and can estimate the complex wave number and impedance as function
of material parameters. They are the Biot equivalent model (also known as the
Johnson-Champoux-Allard model) and the empirical porous models (including the
well known Delany-Bazley and the Miki fluid models), the latter uses frequency and
flow resistivity as input.
COMSOL includes a series of fluid models that are described in Pressure
Acoustics Model and Theory for the Pressure Acoustics Fluid Models. In
addition, The Poroelastic Waves User Interface can be used for detailed
modeling of the propagation of coupled pressure and elastic waves in
porous materials.
BOUNDARY LAYER ABSORPTION (THERMOACOUSTICS)
In systems of small dimensions (or at low frequencies) the size of the acoustic
boundary layer (the viscous and thermal acoustic penetration depth) that exists at all
walls may become comparable to the physical dimensions of the modeled system. In
air the boundary layer thickness is 0.22 mm at 100 Hz. This is typically the case inside
miniature transducers, condenser microphones, in MEMS systems, in tubing for
hearing aids, or in narrow gaps of vibrating structures.
For such systems it is often necessary to use a more detailed model for the propagation
of the acoustics waves. This model is implemented in the Thermoacoustics interface in
COMSOL. In simple cases for sound propagating in long ducts of constant cross
FUNDAMENTALS OF ACOUSTICS MODELING
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27
sections, the losses occurring at the boundaries may be smeared out on the fluid using
one of the boundary-layer absorption fluid models.
More details on the detailed acoustic model for viscous and thermal losses
are described in The Thermoacoustics Branch. The boundary-layer
absorption fluid model is described in About the Boundary-Layer
Absorption Fluid Model.
DAMPING AT BOUNDARIES
Acoustics in closed ducts and cavities appears to be easier to deal with than exterior
problems because no artificial boundary condition is necessary. On the other hand,
real-world cavity walls are usually either treated in some way (lined) or elastic in
themselves.
The problem is that a liner typically reflects part of the wave and does so not at the
interface with the domain but somewhere inside the liner or at its back wall against
whatever structure is outside. This means that a liner boundary condition must contain
more information about the outside world than an absorbing boundary. It also means
that a real-world liner cannot be adequately described by a local boundary condition
because waves at oblique incidence cause waves to propagate in the tangential direction
inside the liner layer.
In fact, there seems to be no final answer as to how the process inside a porous liner is
most accurately modeled. Various assumptions can be made about the interaction
between the fluid pressure waves and the liner material and about boundary conditions
between liner and free fluid and at the back of the liner. The most accurate ways to deal
with the situation include modeling the actual liner layer. It is only possible to use a
general impedance boundary condition for thin liners and when the angle of incidence
is known for a liner that cannot be assumed locally reacting, an assumption that rarely
holds with any justification.
Porous Absorber: Model Library path Acoustics_Module/Industrial_Models/
porous_absorber.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
Artificial Boundaries
In most cases, the acoustic wave pattern that is to be simulated is not contained in a
closed cavity. That is, there are boundaries in the model that do not represent a physical
wall or limit of any kind. Instead, the boundary condition has to represent the
interaction between the wave pattern inside the model and everything outside.
Conditions of this kind are generically referred to as artificial boundary conditions
(ABCs).
Such conditions should ideally contain complete information about the outside world,
but this is not practical. After all, the artificial boundary was introduced to avoid
spending degrees of freedom (DOFs) on modeling whatever is outside. The solution
lies in trying to approximate the behavior of waves outside the domain using only
information from the boundary itself. This is difficult in general for obvious reasons.
One particular case that occurs frequently in acoustics concerns boundaries that can be
assumed to let wave energy propagate out from the domain without reflections. This
leads to the introduction of a particular group of artificial boundary conditions known
as non-reflecting boundary conditions (NRBCs), of which two kinds are available in
this module: matched boundary conditions and radiation boundary conditions. The
radiation boundary conditions apply primarily to wave guide ports connected to a
cavity, while the matched boundary conditions approximate the boundary at infinity
in an exterior problem. A drawback to these boundary conditions is that they are not
perfectly nonreflecting when subjected to a general incoming wave. This is described
in more detail in Theory for the Plane, Spherical, and Cylindrical Radiation Boundary
Conditions and related sections.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu in COMSOL
Multiphysics and either enter a search term or look under a specific
module in the documentation tree.
Another way to model an open nonreflecting boundary is to add a so-called perfectly
matched layer (PML) domain. This domain dampens all outgoing waves with no or
FUNDAMENTALS OF ACOUSTICS MODELING
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29
minimal reflections. See A Note About Perfectly Matched Layers (PMLs) for more
information.
The Absorptive Muffler and Muffler with Perforates models both use a
nonreflecting boundary condition of the radiation type.
A Note About Perfectly Matched Layers (PMLs)
The perfectly matched layer (PML) is a domain or layer that is added to an acoustic
model to mimic an open and non-reflecting infinite domain. It sets up a perfectly
absorbing domain as an alternative to non-reflecting boundary conditions. The PML
works with all types of waves, not only plane waves. It is also efficient at very oblique
angles of incidence. The PML imposes a complex-valued coordinate transformation to
the selected domain that effectively makes it absorbing at a maintained wave
impedance, and thus eliminating reflections at the interface.
Infinite Element Domains and Perfectly Matched Layers in the COMSOL
Multiphysics Reference Manual
A Perfectly Matched Layers node is added to the model from the Model>Definitions
node. The PMLs can be used for the Pressure Acoustics, Acoustic-Structure
Interaction, Aeroacoustics, and Thermoacoustics interfaces.
The PMLs damp a certain wavelength existing in the system. The
wavelength is deducted from the frequency and a reference wave speed
cref. The wave speed is defined in the Typical Wave Speed section. Set cref
equal to the speed of sound of the material in the PML.
If the PML spans both a solid and a fluid domain try to set an average
value of the reference wave speed cref based on the speed in the fluid and
the speed for compressional waves in the solid.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
Evaluating the Acoustic Field in the Far-Field Region
The Acoustics Module has functionality to evaluate the acoustic pressure field in the
far-field region outside of the computational domain. This is the Far-Field Calculation
feature available for pressure acoustics problems. This section gives some general
advice for analyzing the far field.
THE NEAR-FIELD AND FAR-FIELD REGIONS
The solution domain for a scattering or radiation problem can be divided into two
zones, reflecting the behavior of the solution at various distances from objects and
sources. In the far-field region, scattered or emitted waves are locally planar, velocity
and pressure are in phase with each other, and the ratio between pressure and velocity
approaches the free-space impedance of a plane wave.
Moving closer to the sources into the near-field region, pressure and velocity
gradually slide out of phase. This means that the acoustic field contains energy that
does not travel outward or radiate. These evanescent wave components are effectively
trapped close to the source. Looking at the sound pressure level, local maxima and
minima are apparent in the near-field region.
Naturally, the boundary between the near-field and far-field regions is not sharp. A
general guideline is that the far-field region is that beyond the last local energy
maximum, that is, the region where the pressure amplitude drops monotonously at a
rate inversely proportional to the distance from any source or object, R.
A similar definition of the far-field region is the region where the radiation pattern—
the locations of local minima and maxima in space—is independent of the distance to
the wave source. This is equivalent to the criterion for Fraunhofer diffraction in optics,
which occurs for Fresnel numbers, Fa2/R, much smaller than 1. For engineering
purposes, this definition of the far-field region can be applied:
8a 2
8
R  ---------- = ------ ka 2

2
(3-3)
In Equation 3-3, a is the radius of a sphere enclosing all objects and sources,  is the
wavelength, and k is the wave number. Another way to write the expression leads to
the useful observation that the size of the near-field region expressed in source-radius
units is proportional to the dimensionless number k a, with a prefactor slightly larger
than one.
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31
Knowing the extent of the near-field region is useful when applying radiation
boundary conditions because these are accurate only in the far-field region. PMLs, on
the other hand, can be used to truncate a domain already inside the near-field region.
THE HELMHOLTZ-KIRCHHOFF INTEGRAL REPRESENTATION
In many cases, solving the acoustic Helmholtz equation everywhere in the domain
where results are requested is neither practical nor necessary. For homogeneous media,
the solution anywhere outside a closed surface containing all sources and scatterers can
be written as a boundary integral in terms of quantities evaluated on the surface. To
evaluate this Helmholtz-Kirchhoff integral, it is necessary to know both Dirichlet and
Neumann values on the surface. Applied to acoustics, this means that if the pressure
and its normal derivative is known on a closed surface, the acoustic field can be
calculated at any point outside, including amplitude and phase. This functionality is
included in the Far-Field Calculation feature. The feature has two options for the
evaluation, one full integral and one that only looks in the extreme far field. See the
section Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral for
further details.
FULL INTEGRAL
To evaluate the full Helmholtz-Kirchhoff integral use the Full integral option in the
settings for the far-field variables. The full Helmholtz-Kirchhoff integral gives the
pressure at any point at a finite distance from the source surface, but the numerical
integration tends to lose accuracy at large distances. See Far-Field Calculation.
THE FAR-FIELD LIMIT
In many applications, the quantity of interest is the far-field radiation pattern, which
can be defined as the limit of r | p | when r goes to infinity in a given direction. To
evaluate the pressure in the far-field limit use the Integral approximation at r  
option in the settings for the far-field variables See Far-Field Calculation.
The Far Field Plots
Evaluating the acoustic pressure in the far field is essential for the development of
several acoustic devices. This is especially true as it is essential to reduce the
computational domain while still being able to determine the pressure in the near-field
to far-field. Application areas range from underwater acoustic transducers and
loudspeakers, to determining the spatial sensitivity of microphone systems (for
example, using reciprocity).
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The Far Field plots are specially designed for easy evaluation of the far-field variables,
that is, the acoustics far-field pressure and the far-field sound pressure level.
The variables are plotted for a selected number of angles on a unit circle (in 2D) or a
unit sphere (in 3D). The angle interval and the number of angles can be manually
specified. Also the circle origin and radius of the circle (2D) or sphere (3D) can be
specified. For 3D Far Field plots you also specify an expression for the surface color.
The main advantage with the Far Field plot, as compared to making a Line Graph, is
that the unit circle/sphere that you use for defining the plot directions, is not part of
your geometry for the solution. Thus, the number of plotting directions is decoupled
from the discretization of the solution domain.
Default Far Field plots are automatically added to any model that uses
far-field calculations.
• Evaluating the Acoustic Field in the Far-Field Region
• Far Field and Results Analysis and Plots in the COMSOL Multiphysics
Reference Manual
• For a 3D example, see Bessel Panel: Model Library path
Acoustics_Module/Tutorial_Models/bessel_panel.
• For a 2D axisymmetric example, see Cylindrical Subwoofer: Model
Library path Acoustics_Module/Tutorial_Models/cylindrical_subwoofer.
Solving Large Acoustics Problems Using Iterative Solvers
This section has some guidance for solving large acoustics problems. For smaller
problems using a direct solver like MUMPS is often the best choice. For larger
problems, especially in 3D, the only option is often to use an iterative method such as
multigrid.
The underlying equation for many of the problems within acoustics is the Helmholtz
equation. For high frequencies (or wave numbers) the matrix resulting from a
finite-element discretization becomes highly indefinite. In such situations it can be
problematic to use geometric multigrid (GMG) with simple smoothers such as Jacobi
or SOR (the default smoother). Fortunately, there exist robust and memory-efficient
FUNDAMENTALS OF ACOUSTICS MODELING
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33
approaches that circumvent many of the difficulties associated with solving the
Helmholtz equation using geometric multigrid.
When using a geometric multigrid as a linear system solver together with simple
smoothers, the Nyquist criterion must be fulfilled on the coarsest mesh. If the Nyquist
criterion is not satisfied, the geometric multigrid solver might not converge. One way
to get around this problem is to use GMRES or FGMRES as a linear system solver with
geometric multigrid as a preconditioner. The default preconditioner is the incomplete
LU, right-click the Iterative solver node and select Multigrid. Even if the Nyquist
criterion is not fulfilled for the coarse meshes of the multigrid preconditioner, such a
scheme is more likely to converge. For problems with high frequencies this approach
might, however, lead to a large number of iterations. Then it might be advantageous
to use either:
• Geometric multigrid as a linear system solver (set the Solver selection to Use
preconditioner) with GMRES as a smoother. Under the Multigrid node right-click
the Presmoother and Postsmoother nodes and select the Krylov Preconditioner with
the Solver selection to GMRES.
• FGMRES as a linear system solver (set the Solver selection to FGMRES) with
geometric multigrid as a preconditioner (where GMRES is used as a smoother, as
above).
Using GMRES/FGMRES as an outer iteration and smoother removes the
requirements on the coarsest mesh. When GMRES is used as a smoother for the
multigrid preconditioner, FGMRES must be used for the outer iterations because such
a preconditioner is not constant.
See Y. Saad, “A Flexible Inner-outer Preconditioned GMRES
Algorithm,” SIAM J. Sci. Statist. Comput., vol. 14, pp. 461–469, 1993.
Use GMRES as a smoother only if necessary because GMRES smoothing is very timeand memory-consuming on fine meshes, especially for many smoothing steps.
When solving large acoustics problems, the following options, in increasing order of
robustness and memory requirements, may be of use:
• If the Nyquist criterion is fulfilled on the coarsest mesh, try to use geometric
multigrid as a linear system solver (set Multigrid as preconditioner and set the linear
system solver to Use preconditioner) with default smoothers. The default smoothers
are fast and have small memory requirements.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
• An option more robust than the first point is to use GMRES as a linear system solver
with geometric multigrid as a preconditioner (where default SOR smoothers are
used). GMRES requires memory for storing search vectors. This option can
sometimes be used successfully even when the Nyquist criterion is not fulfilled on
coarser meshes. Because GMRES is not used as a smoother, this option might find
a solution faster than the next two options even if a large number of outer iterations
are needed for convergence.
• If the above suggestion does not work, try to use geometric multigrid as a linear
system solver with GMRES as a smoother.
• If the solver still has problems converging, try to use FGMRES as a linear system
solver with geometric multigrid as a preconditioner (where GMRES is used as a
smoother).
• Try to use as many multigrid levels as needed to produce a coarse mesh for which a
direct method can solve the problem without using a substantial amount of
memory.
• If the coarse mesh is still too fine for a direct solver, try using an iterative solver with
5–10 iterations as coarse solver.
In the COMSOL Multiphysics Reference Manual:
• Studies and Solvers
• Multigrid
About the Material Databases for the Acoustics Module
The Acoustics Module includes two material databases: Liquids and Gases, with
temperature-dependent fluid dynamic and thermal properties, and a Piezoelectric
Materials database with over 20 common piezoelectric materials.
For detailed information about Materials, the Liquids and Gases Material
Database, and the Piezoelectric Materials Database see the COMSOL
Multiphysics Reference Manual.
FUNDAMENTALS OF ACOUSTICS MODELING
|
35
The Acoustics Module Study Types
The Acoustics Module is primarily designed for frequency-domain simulations,
including related eigenvalue and mode analysis problems. Transient analysis is possible
but less efficient from the computational point of view. The Thermoacoustics
interfaces only support the frequency-domain type analysis. The Compressible
Potential Flow interface is tailored to model a stationary background flow to be used
in a subsequent time-harmonic aeroacoustics simulation. In the Solid Mechanics
interface, the static analysis type is also included and can be use to model the stationary
state of pre-stressed systems subject to time harmonic vibrations.
The analysis types require different solvers and equations. The following study types,
briefly discussed in this section, can help you find good candidates for the application:
• Stationary Study
• Frequency Domain Study
• Eigenfrequency Study
• Mode Analysis Study
• Time Dependent Study
• Frequency Domain Modal and Time-Dependent Modal Studies
• Modal Reduced Order Model
• Additional Analysis Capabilities
Studies and Solvers and Harmonic Perturbation—Exclusive and
Contributing Nodes in the COMSOL Multiphysics Reference Manual
Stationary Study
A stationary analysis solves for stationary displacements or a steady-state condition. All
loads and constraints are constant. For a stationary analysis, use a Stationary study
type (
). For all pure acoustic and vibration problems this type of analysis yields the
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
zero solution as, by definition, these represent and describe propagating varying
fields—either time dependent or time harmonic in the frequency domain.
Stationary in the COMSOL Multiphysics Reference Manual
Frequency Domain Study
Wave propagation is modeled by equations from linearized fluid dynamics (pressure
waves) and structural dynamics (elastic waves). The full equations are time dependent,
but noting that a harmonic excitation of the field u has a time dependence of the form
u = ue it
gives rise to an equally harmonic response with the same frequency; the time can be
eliminated completely from the equations. Instead the angular frequency f,
enters as a parameter where f is the frequency.
This procedure is often referred to as working in the frequency domain or Fourier
domain as opposed to the time domain. From the mathematical point of view, the
time-harmonic equation is a Fourier transform of the original time-dependent
equations and its solution as function of  is the Fourier transform of a full transient
solution. It is therefore possible to synthesize a time-dependent solution from a
frequency-domain simulation by applying an inverse Fourier transform.
COMSOL Multiphysics and the Acoustics Module are based on the finite element
method; a frequency domain simulation suits this method very well. Therefore, choose
) over a time dependent study whenever possible.
the Frequency Domain study type (
Certain important software features, notably PMLs and damping due to porous media
or boundary layer absorption, are only present when using the frequency domain
physics interfaces.
The result of a frequency domain analysis is a complex time dependent field u, which
can be interpreted as an amplitude uamp = abs(u) and a phase angle uphase = arg(u).
The actual displacement at any point in time is the real part of the solution:
u = u amp cos  2f  t + u phase 
Visualize the amplitudes and phases as well as the solution at a specific angle (time).
When using the Solution data sets, the solution at angle (phase) parameter makes this
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37
task easy. When plotting the solution, COMSOL Multiphysics multiplies it by ei,
where  is the angle in radians that corresponds to the angle (specified in degrees) in
the Solution at angle field. The plot shows the real part of the evaluated expression:
u = u amp cos   + u phase 
The angle  is available as the variable phase (in radians) and is allowed in plot
expressions. Both the frequency freq and angular frequency omega are available
variables.
In a frequency domain study almost everything is treated as harmonic—
prescribed pressures and displacements, velocities, and accelerations—not
only the forces and dependent fields. Notable exceptions are certain
quantities, such as the sound pressure level, which by definition are time
averages.
Frequency Domain and Solution (data sets) in the COMSOL
Multiphysics Reference Manual
Eigenfrequency Study
If all sources are removed from a frequency-domain equation, its solution becomes
zero for all but a discrete set of angular frequencies , where the solution has a
well-defined shape but undefined magnitude. These solutions are known as
eigenmodes and the corresponding frequencies as eigenfrequencies.
The eigenmodes and eigenfrequencies have many interesting mathematical properties,
but also direct physical significance because they identify the resonance frequency (or
frequencies) of the structure. When approaching a resonance frequency in a
harmonically-driven problem, a weaker and weaker source is needed to maintain a
given response level. At the actual eigenfrequency, the time-harmonic problem breaks
down and lacks solution for a nonzero excitation.
) when you are interested in the resonance
Select the Eigenfrequency study type (
frequencies of the acoustic domain or the structure, whether you want to exploit them,
as in a musical instrument, or avoid them, as in a reactive muffler or inside a hifi speaker
system. To an engineer, the distribution of eigenfrequencies and the shape of
eigenmodes can also give a good first impression about the behavior of a system.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
An eigenfrequency analysis solves for the eigenfrequencies and the shape of the
eigenmodes. When performing an eigenfrequency analysis, specify whether to look at
the mathematically more fundamental eigenvalue  (available as the variable lambda)
or the eigenfrequency f which is more commonly used in an acoustics context:
–
f = --------2i
Eigenfrequency in the COMSOL Multiphysics Reference Manual
Mode Analysis Study
The Mode Analysis study (
) is available with the Pressure Acoustics, Frequency
Domain, Aeroacoustics, Frequency Domain, and Thermoacoustics, Frequency
Domain interfaces in plane 2D and axially symmetric 1D acoustics interfaces. The
Boundary Mode Acoustics and Boundary Mode Aeroacoustics are special interfaces for
more advanced Mode Analysis studies on boundaries in 3D and 2D axisymmetry.
Acoustic waves can propagate over large distances in ducts and pipes, with a generic
name referred to as waveguides. After some distance of propagation in a waveguide of
uniform cross section, such guided waves can be described as a sum of just a few
discrete propagating modes, each with its own shape and phase speed. The equation
governing these modes can be obtained as a spatial Fourier transform of the
time-harmonic equation in the waveguide axial z direction, or more easily by inserting
the assumption that the mode is harmonic in space,
u = ue –ikz z
and eliminating all out-of-plane z dependence.
The axial wave number kz is a parameter in the 2D acoustics physics
interfaces.
Similar to the full time-harmonic equation, the transformed equation can be solved at
a given frequency with a nonzero excitation for most axial wave numbers kz. But at
certain discrete values the equation breaks down. These values are the propagation
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constants or wave numbers of the propagating or evanescent waveguide modes. The
eigenvalue solver can solve for these propagation constants together with the
corresponding mode shapes.
The propagating wave number is a function of the frequency. The relation
between the two is commonly referred to as a dispersion curve.
The most common use for the Mode Analysis is to define sources for a subsequent
time-harmonic simulation. If there is a component with one or more waveguide
connections, its behavior can be described by simulating its response to the discrete set
of propagating modes on the waveguide port cross sections. In thermoacoustics a
Mode Analysis study also provides information about the absorption coefficient for the
propagating modes, which is the imaginary part of the wave number.
Mode Analysis in the COMSOL Multiphysics Reference Manual
Jet Pipe: Model Library path Acoustics_Module/Tutorial_Models/jet_pipe
Time Dependent Study
The complete equations behind the theory of acoustic wave propagation are time
dependent, as discussed in the Frequency Domain Study section. Solving time-domain
equations is more complicated from a numerical point of view and should therefore be
avoided when possible. Short-term transient processes like step and impulse responses
can benefit from modeling in the time domain, if not for efficiency so for convenience.
Some central modeling techniques, such as the use of PMLs, are not
available for the Time Dependent study type. Be careful when defining your
sources to avoid, as far as possible, exciting waves at frequencies that the
mesh cannot resolve.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
Time Dependent in the COMSOL Multiphysics Reference Manual
Frequency Domain Modal and Time-Dependent Modal Studies
The Frequency Domain Modal study type (
) is used to do modal analysis in the
) is used to do
frequency domain and the Time-Dependent Modal study (
time-dependent modal analysis.
Frequency Domain Modal and Time-Dependent Modal in the COMSOL
Multiphysics Reference Manual
Modal Reduced Order Model
The Modal Reduced Order Model study type (
) is used to obtain the data necessary
to construct reduced-order models from a COMSOL Multiphysics simulation. This
study step is added after an existing Eigenvalue study step by right-clicking the Study
1 node and selecting Study Steps>Modal Reduced Order Model. After solving the model,
right-click the Derived Values node (under Results) and select System Matrix. In the
output section select the Matrix to display and the Format. Using the Matrix settings it
is possible to access the stiffness, damping, and mass matrices of the system, for
example.
In the COMSOL Multiphysics Reference Manual:
• Modal Reduced Order Model and Working with Studies and Solvers
• System Matrix and Results Analysis and Plots
Additional Analysis Capabilities
In a multiphysics interface you might want to use different analysis types for the
different dependent variables. This can be done by adding an Empty Study (
), and
then adding different study steps to this study. Also perform parametric analyses by
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41
using the Parametric Sweep study node (
). Typical parameters to vary include
geometric properties, the frequency, and the out-of-plane or axial wave number.
Parametric Sweep in the COMSOL Multiphysics Reference Manual
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
Special Variables in the Acoustics
Module
Several specialized variables specific to acoustics are predefined in the Acoustics
Module and can be used when analyzing the results of an acoustic simulation. The
variables are available from the expressions selection menus when plotting. In this
section:
• Intensity Variables
• Power Dissipation Variables
• Boundary Mode Acoustics Variables
• Reference for the Acoustics Module Special Variables
In the COMSOL Multiphysics Reference Manual:
• Results Analysis and Plots
• Operators, Functions, and Constants
Intensity Variables
The propagation of an acoustic wave is associated with a flow of energy in the direction
of the wave motion, the intensity vector I. The sound intensity in a specific direction
(through a specific boundary) is defined as the time average of energy flow per unit
area in the direction of the normal to that area.
Knowledge of the intensity is important when characterizing the strength of a sound
source—that is, the power emitted by the source. The power is given by the integral
of n·I on a surface surrounding the source, where n is the surface normal. The intensity
is also important when characterizing transmission phenomena, for example, when
determining transmission loss or insertion loss curves.
The acoustic intensity vector I (SI unit: W/m2) is defined as the time average, or root
mean square (RMS), of the instantaneous energy flow per unit area pu, such that
T
1
I = ---- pu dt
T

0
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43
where p is the pressure and u the particle velocity. In the frequency domain (harmonic
time dependence) the integral reduces to
1
1
I = --- Re  pu  = ---  pu + p u 
4
2
The instantaneous value of the intensity is defined as
I inst = pu = Re  p Re  u 
Both the intensity (RMS) and the instantaneous intensity are available as results and
analysis variables and can be selected from the expressions menus when plotting. The
variables are defined for The Pressure Acoustics, Frequency Domain User Interface,
The Aeroacoustics, Frequency Domain User Interface, and The Thermoacoustics,
Frequency Domain User Interface. The variables are defined in Table 3-1, Table 3-2,
and Table 3-3. In the variable names, phys_id represents the physics interface
identifier (for example, acpr for a Pressure Acoustics interface or ta for
Thermoacoustics).
TABLE 3-1: INTENSITY VARIABLES IN 3D
VARIABLE
DESCRIPTION
phys_id.I_rms
Magnitude of the intensity vector
phys_id.Ix
x-component of the intensity vector
phys_id.Iy
y-component of the intensity vector
phys_id.Iz
z-component of the intensity vector
phys_id.I_inst
Magnitude of the instantaneous intensity vector
phys_id.Iix
x-component of the instantaneous intensity vector
phys_id.Iiy
y-component of the instantaneous intensity vector
phys_id.Iiz
z-component of the instantaneous intensity vector
TABLE 3-2: INTENSITY VARIABLES IN 2D AXISYMMETRIC
44 |
VARIABLE
DESCRIPTION
phys_id.I_rms
Magnitude of the intensity vector
phys_id.Ir
r-component of the intensity vector
phys_id.Iz
z-component of the intensity vector
phys_id.I_inst
Magnitude of the instantaneous intensity vector
phys_id.Iir
r-component of the instantaneous intensity vector
phys_id.Iiz
z-component of the instantaneous intensity vector
CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
TABLE 3-3: INTENSITY VARIABLES IN 2D
VARIABLE
DESCRIPTION
phys_id.I_rms
Magnitude of the intensity vector
phys_id.Ix
x-component of the intensity vector
phys_id.Iy
y-component of the intensity vector
phys_id.I_inst
Magnitude of the instantaneous intensity vector
phys_id.Iix
x-component of the instantaneous intensity vector
phys_id.Iiy
y-component of the instantaneous intensity vector
In the COMSOL Multiphysics Reference Manual:
• Results Analysis and Plots
• Expressions and Predefined Quantities
Power Dissipation Variables
Common to all the pressure acoustics viscous and thermally conducting fluid models
(and The Thermoacoustics, Frequency Domain User Interface) is that all the interfaces
model some energy dissipation process, which stem from viscous and thermal
dissipation processes. The amount of dissipated energy can be of interest as a results
analysis variable or as a source term for a multiphysics problem. An example could be
to determine the amount of heating in the human tissue when using ultrasound.
The energy conservation-dissipation corollary describes the transport and dissipation
of energy in a system (see Ref. 1 pp. 516). In linear acoustics, this equation is derived
by taking the dot product (scalar product) of the momentum and the velocity v,
adding it to the continuity equation, and then adding the entropy. After some
manipulation and integration, the use of Gauss’ theorem yields Equation 3-4
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 w dV
+
t


  I  n  dA

= –  dV

or

w
------+   I = –
t
2
2 1 p
1 0 T0 2
1
w = ---  0 v + --- -----------2 + --- ------------- s
2 c
2 Cp
2
0 0
I = pv – 
(3-4)
k
T T
 ej vi ij – -----T0
ij
k
2
 = :v + ------  T  =  v +  t
T0
where:
• w is the disturbance energy of the control volume
• v = |v| is the velocity
• T is the temperature variations
• p is the acoustic pressure
• p0 is the static pressure
• T0 the static temperature
• 0 the static density
• c0 the isentropic speed of sound
• Cp the heat capacity at constant pressure
• k the coefficient of thermal conduction
• I is the flux of energy out of a control volume
•  is the dissipated energy per unit volume and time 
(SI unit: Pa/s = J/(m3s) = W/m3)
• s is the entropy
• :v is the viscous dissipation function, and
• v and t are the viscous and thermal contributions to the dissipation function.
In the Acoustics Module special variables exist for the dissipation term .
For the case of a plane wave propagating in the bulk of a fluid (the general thermal and
viscous fluid models described in About the Thermally Conducting and Viscous Fluid
Model) the dissipation is
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
1
4 b
k   – 1  p 2
 = ------------------2-   --- + ------ + --------------------  ------
 t 


3 
Cp
2
 0 c0 
and in the frequency domain after averaging over one period
2
1
4 b
k  – 1  
 = ------------------2-   --- + ------ + -------------------- ------ pp


2
3 
Cp
2
 0 c0 
(3-5)
where * in Equation 3-5 is the complex conjugate operator.
For the case of the Thermoacoustics, Frequency Domain interface, the dissipation
term is directly given by the RMS value of the tensor expression
1
 v =  :v = ---   :v + :  v  
4
(3-6)
where: in Equation 3-5 is the double dot operator. In the above expressions, the time
averaged expressions for a product in the frequency domain is defined as:
 AB =  Re  Ae
it
Re  Be
it
1
 = ---  A B + AB 
4
The power dissipation variables are defined in Table 3-4. In the variable names,
phys_id represents the physics interface identifier (acpr, for example, for a Pressure
Acoustics interface).
TABLE 3-4: POWER DISSIPATION VARIABLES
VARIABLE
DESCRIPTION
phys_id.diss_therm
Thermal power dissipation density
phys_id.diss_visc
Viscous power dissipation density
phys_id.diss_tot
Total thermal and viscous power dissipation density
Boundary Mode Acoustics Variables
A series of special variables exist for postprocessing after solving a boundary mode
acoustics problem. They include in-plane and out-of-plane components of the velocity
v and acceleration a.
The in-plane (ip) and out-of-plane (op) components to the acceleration and velocity
are defined as
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a ip = a –  a  n n
v ip = v –  v  n n
a op =  a  n n
v op =  v  n n
where n is the normal to the surface being modeled. The velocity and acceleration are
defined in terms of the gradient of the pressure p as follows
p =  t p – ik n pn
in 3D
m
p =  t p – ik n p  n r 0 n z  + p  0 – i ----- 0

r 
in 2D axisymmetry
and
i
v = ------- p

and
a = iv
where kn is the out-of-plane wave number solved for, m is a possible radial wave mode
number, and  t is the tangential derivative along the boundary.
The boundary mode acoustics variables are defined in Table 3-5. In the variable names,
phys_id represents the physics interface identifier (acbm, for example, for a Boundary
Mode Acoustics interface).
TABLE 3-5: BOUNDARY MODE ACOUSTICS VARIABLES IN 3D
48 |
VARIABLE
DESCRIPTION
phys_id.vipx
In-plane velocity, x-component
phys_id.vipy
In-plane velocity, y-component
phys_id.vipz
In-plane velocity, z-component
phys_id.vip_rms
In-plane velocity RMS value
phys_id.aipx
In-plane acceleration, x-component
phys_id.aipy
In-plane acceleration, y-component
phys_id.aipz
In-plane acceleration, z-component
phys_id.aip_rms
In-plane acceleration RMS value
phys_id.vopx
Out-of-plane velocity, x-component
phys_id.vopy
Out-of-plane velocity, y-component
phys_id.vopz
Out-of-plane velocity, z-component
phys_id.vop_rms
Out-of-plane velocity RMS value
CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
TABLE 3-5: BOUNDARY MODE ACOUSTICS VARIABLES IN 3D
VARIABLE
DESCRIPTION
phys_id.aopx
Out-of-plane acceleration, x-component
phys_id.aopy
Out-of-plane acceleration, y-component
phys_id.aopz
Out-of-plane acceleration, z-component
phys_id.aop_rms
Out-of-plane acceleration RMS value
Reference for the Acoustics Module Special Variables
1. A. D. Pierce, Acoustics: An Introduction to its Physical Principles and
Applications, Acoustic Society of America, Melville, New York, 1991.
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CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE
4
The Pressure Acoustics Branch
This chapter describes the Acoustics Module background theory and physics
interfaces found under the Pressure Acoustics branch (
) in the Model Wizard.
• The Pressure Acoustics, Frequency Domain User Interface
• The Pressure Acoustics, Transient User Interface
• The Boundary Mode Acoustics User Interface
• Theory Background for the Pressure Acoustics Branch
• Theory for the Pressure Acoustics Fluid Models
• References for the Pressure Acoustics Branch
51
The Pressure Acoustics, Frequency
Domain User Interface
The Pressure Acoustics, Frequency Domain (acpr) user interface (
) has the equations,
boundary conditions, and sources for modeling acoustics, solving for the sound
) in the Model
pressure. Select the interface from the Pressure Acoustics branch (
Wizard. The interface is designed for the analysis of various types of pressure acoustics
problems in the frequency domain, all concerning pressure waves in a fluid. An
acoustics model can be part of a larger multiphysics model that describes, for example,
the interactions between structures and acoustic waves. This interface is suitable for
modeling acoustics phenomena that do not involve fluid flow.
The sound pressure p, which is solved for in pressure acoustics, represents the acoustic
variations (or excess pressure) to the ambient pressure. The ambient pressure is in the
absence of flow simply the static absolute pressure. In the presence of a background
acoustic pressure wave pb the total acoustic pressure pt is the sum of the pressure
solved for p and the background pressure wave. The governing equations are
formulated using the total pressure in a a so-called scattered field formulation. The
equations hence contain the information about the background pressure, which for
example could be a user defined incident wave or a plane wave.
When the geometrical dimensions of the acoustic problems are reduced from 3D to
2D (planar symmetry or axisymmetric) or to 1D axisymmetric, it is possible to specify
an out-of-plane wave number kz and a circumferential wave number m, when
applicable. The wave number used in the equations keq contains both the ordinary
wave number k as well as the out-of-plane wave number and circumferential wave
number, when applicable.
The Pressure Acoustics interface solves the full acoustic problem including a priori
knowledge about the acoustic problem, in the form of background pressure fields and
symmetries.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
The following table lists the names and SI units for the most important physical
quantities in the Pressure Acoustics, Frequency Domain interface:
TABLE 4-1: PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE PHYSICAL QUANTITIES
QUANTITY
SYMBOL
SI UNIT
Pressure
p
pascal
Pa
Density

kilogram/meter3
kg/m3
Frequency
f
hertz
Hz
Wave number
k
1/meter
Dipole source
qd
ABBREVIATION
1/m
3
newton/meter
2
N/m3
Monopole source
Qm
1/second
1/s2
Speed of sound
c
meter/second
m/s
Acoustic impedance
Z
pascal-second/meter
Pa·s/m
2
Normal acceleration
an
meter/second
m/s2
Source location
r0
meter
m
Wave direction
nk
(dimensionless)
1
In the following descriptions of the functionality in this interface, the subscript c in c
and cc (the density and speed of sound, respectively) denotes that these can be
complex-valued quantities in models with damping.
When this interface is added, these default nodes are also added to the Model Builder—
Pressure Acoustics Model, Sound Hard Boundary (Wall), and Initial Values.
Right-click the Pressure Acoustics node to add other features that implement, for
example, boundary conditions.
Physics Nodes—Equation Section in the COMSOL Multiphysics
Reference Manual
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
|
53
The default identifier (for the first interface in the model) is acpr.
DOMAIN SELECTION
The default setting is to include All domains in the model to define a sound pressure
field and the associated acoustics equation. To choose specific domains, select Manual
from the Selection list.
EQUATION
Expand the Equation section to see the equations solved for with the Equation form
specified. The default selection is Equation form is set to Study controlled. The available
studies are selected under Show equations assuming.
When the Equation form is set to Study controlled, the scaling and
non-reflecting boundary settings are optimized for the numerical
performance of the different solvers.
To display the Pressure Acoustics Equation Settings section for 2D and 1D
models, click to expand the Equation section, then select Frequency domain
as the Equation form and enter the settings as described in Scaling Factor
and Non-reflecting Boundary Condition Approximation.
PRESSURE ACOUSTICS EQUATION SETTINGS
For 1D axisymmetric models, the Circumferential wave number m
(dimensionless) default is 0 and the Out-of-plane wave number kz (SI unit:
rad/m) default is 0 rad/m. Enter different values or expressions as
required.
For 2D axisymmetric models, the Circumferential wave number m
(dimensionless) default is 0. Enter a different value or expression as
required.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
For 2D models, the Out-of-plane wave number kz (SI unit: rad/m) default
is 0 rad/m. Enter a different value or expression as required.
Scaling Factor and Non-reflecting Boundary Condition Approximation
For all model dimensions, and if required, click to expand the Equation section, then
select Frequency domain as the Equation form and enter the settings as described below.
The default Scaling factor is 1/2 and Non-reflecting boundary condition
approximation is Second order. These values correspond to the equations for a
Frequency Domain study when the equations are study controlled.
To get the equations corresponding to an Eigenfrequency study, change the Scaling
factor  to 1 and the Non-reflecting boundary conditions approximation to First order.
SOUND PRESSURE LEVEL SETTINGS
The zero level on the dB scale varies with the type of fluid. That value is a reference
pressure that corresponds to 0 dB. This variable occurs in calculations of the sound
pressure level Lp based on the root mean square (rms) pressure prms, such that
p rms
L p = 20 log  ----------
 p ref 
with
p rms =
1
--- p p
2
where pref is the reference pressure and the star (*) represents the complex conjugate.
This is an expression valid for the case of harmonically time-varying acoustic pressure
p.
Based on the fluid type, select a Reference pressure for the sound pressure level. Select:
• Use reference pressure for air to use a reference pressure of 20 Pa (20·106 Pa).
• Use reference pressure for water to use a reference pressure of 1 Pa (1·106 Pa).
• User-defined reference pressure to enter a reference pressure pref, SPL (SI unit: Pa).
The default value is the same as for air, 20 Pa.
TY P I C A L W AV E S P E E D
Enter a value or expression for the Typical wave speed for perfectly matched layers cref
(SI unit m/s). The default is 343 m/s.
DEPENDENT VA RIA BLES
This interface defines one dependent variable (field), the Pressure p. If required, edit
the name, which changes both the field name and the dependent variable name. If the
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new field name coincides with the name of another pressure field in the model, the
interfaces will share degrees of freedom and dependent variable name. The new field
name must not coincide with the name of a field of another type, or with a component
name belonging to some other field.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Quadratic (the default), Linear, Cubic, Quartic, or Quintic for the Pressure. Specify the
Value type when using splitting of complex variables—Real or Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure
Acoustics, Frequency Domain User Interface
• Theory Background for the Pressure Acoustics Branch
Eigenmodes of a Room: Model Library path COMSOL_Multiphysics/
Acoustics/eigenmodes_of_room
This model also requires the Particle Tracing Module—Acoustic
Levitator: Model Library path Acoustics_Module/Tutorial_Models/
acoustic_levitator
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure
Acoustics, Frequency Domain User Interface
The Pressure Acoustics, Frequency Domain User Interface has these domain,
boundary, edge, point, and pair nodes available and listed in alphabetical order.
• Background Pressure Field
• Line Source
• Circular Source
• Line Source on Axis
• Continuity
• Matched Boundary
• Cylindrical Wave Radiation
• Monopole Point Source
• Dipole Source
• Monopole Source
• Destination Selection in the COMSOL
Multiphysics Reference Manual
• Normal Acceleration
• Far-Field Calculation
• Plane Wave Radiation
• Impedance
• Point Source
• Incident Pressure Field
• Pressure Acoustics Model
• Interior Normal Acceleration
• Pressure
• Interior Impedance/Pair Impedance
• Sound Hard Boundary (Wall)
• Interior Perforated Plate
• Sound Soft Boundary
• Interior Sound Hard Boundary (Wall)
• Spherical Wave Radiation
• Initial Values
• Symmetry
• Periodic Condition
Continuity in the total pressure is the default condition on interior
boundaries.
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The Pressure Acoustics, Transient User Interface also shares these nodes,
with some additional features described in Domain, Boundary, Edge, and
Point Nodes for the Pressure Acoustics, Transient User Interface.
The Boundary Mode Acoustics User Interface also shares these nodes,
with one additional feature described in Boundary, Edge, Point, and Pair
Nodes for the Boundary Mode Acoustics User Interface. For the
Boundary Mode Acoustics interface, apply the feature to boundaries
instead of domains for 3D models.
For axisymmetric models, COMSOL Multiphysics takes the axial
symmetry boundaries (at r = 0) into account and automatically adds an
Axial Symmetry node to the model that is valid on the axial symmetry
boundaries only.
To locate and search all the documentation, in COMSOL, select
Help>Documentation from the main menu and either enter a search term
or look under a specific module in the documentation tree.
Monopole Source
Use the Monopole Source node to add a the domain source term Qm to the governing
equation. A monopole source added to a domain has a uniform strength in all
directions. In advanced models this source term can, for example, be used to represent
a domain heat source causing pressure variations.
DOMAIN SELECTION
From the Selection list, choose the domains to define.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
MONOPOLE SOURCE
Enter a Monopole source Qm (SI unit: 1/s2). The default is 0 1/s2.
In a transient model the Monopole Source may be used to add
nonlinearities to the governing equation. See the model Nonlinear
Acoustics: Modeling of the 1D Westervelt Equation for such an example:
Model Library path Acoustics_Module/Tutorial_Models/
nonlinear_acoustics_westervelt_1d.
Dipole Source
Use the Dipole Source node to add the domain source term qd to the governing
equation. This source will is typically stronger in two opposite directions. In advanced
models this term may, for example, be used to represent a uniform constant
background flow convecting the sound field.
DOMAIN SELECTION
From the Selection list, choose the domains to define.
DIPOLE SOURCE
Enter coordinates for the Dipole source qd (SI unit: N/m3). These are the individual
components of the dipole source vector. The defaults are 0 N/m3.
Initial Values
The Initial Values node adds initial values for the sound pressure and the pressure time
derivative that can serve as an initial guess for a nonlinear solver. If more than one
initial value is needed, right-click the interface node to add more Initial Values nodes.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
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INITIAL VALUES
Enter a value or expression for the initial values for the Pressure p (SI unit: Pa) and the
Pressure, first time derivative, p/t (SI unit: Pa/s). The defaults are 0 Pa and 0 Pa/s,
respectively.
Sound Hard Boundary (Wall)
The Sound Hard Boundary (Wall) adds a boundary condition for a sound hard boundary
or wall, which is a boundary at which the normal component of the acceleration is
zero:
1
– n   – ------  p – q d  = 0
0
For zero dipole source and constant fluid density, this means that the normal derivative
of the pressure is zero at the boundary:
p
= 0
n
Sound-hard boundaries are available for all study types. Note that this condition is
identical to the Symmetry condition.
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
Normal Acceleration
The Normal Acceleration adds an inward normal acceleration an:
1
– n   – ------  p – q d  = a n
 0

Alternatively, specify the acceleration a0 of the boundary. The part in the normal
direction is used to define the boundary condition:
1
n   – ------  p – q d  = n  a 0
0
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This feature represents an external source term. It can also be used to manually couple
acoustics with a structural analysis for modeling acoustic-structure interaction.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
NORMAL ACCELERATION
Select a Type—Inward Acceleration (the default) or Acceleration.
• If Inward Acceleration is chosen, enter the value of the Inward acceleration an
(SI unit: m/s2). The default is 0 m/s2. Use a positive value for inward acceleration
or a negative value for outward acceleration.
• If Acceleration is chosen, enter values for the components of the Acceleration a0
(SI unit: m/s2). The defaults are 0 m/s2.
Sound Soft Boundary
The Sound Soft Boundary adds a boundary condition for a sound soft boundary, where
the acoustic pressure vanishes: p = 0. It is an appropriate approximation for a liquid-gas
interface and in some cases for external waveguide ports.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. If the node is selected from
the Pairs submenu, this list cannot be edited and it shows the boundaries in the selected
pairs.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
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Pressure
The Pressure node creates a boundary condition that acts as a pressure source at the
boundary, which means a constant acoustic pressure p0 is specified and maintained at
the boundary: p = p0. In the frequency domain, p0 is the amplitude of a harmonic
pressure source.
The node is also available from the Pairs submenu as an option at interfaces between
parts in an assembly.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. If the node is selected from
the Pairs submenu, this list cannot be edited and it shows the boundaries in the selected
pairs.
PRESSURE
Enter the value of the Pressure p0 (SI unit: Pa). The default is 0 Pa.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
These are the same settings as for Sound Soft Boundary.
Impedance
The Impedance node adds an impedance boundary condition, which is a generalization
of the sound-hard and sound-soft boundary conditions:
ip t
1
– n   – -----  p t – q d  = – -----------c
Zi
In the Pressure Acoustics, Transient interface using a time-dependent study, the
impedance boundary condition is the following:
1
1 p t
– n   – -----  p – q d  = ---- c

Zi  t
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Here Zi is the acoustic input impedance of the external domain and it has the unit of
a specific acoustic impedance. From a physical point of view, the acoustic input
impedance is the ratio between the local pressure and local normal particle velocity.
The Impedance boundary condition is a good approximation for a locally reacting
surface—a surface for which the normal velocity at any point depends only on the
pressure at that exact point.
In the two opposite limits Zi and Zi0, this boundary condition is
identical to the Sound Hard Boundary (Wall) condition and the Sound
Soft Boundary condition, respectively.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
IMPEDANCE
Enter the value of the Impedance Zi (SI unit: Pa·s/m). The default value is set to the
specific impedance of air 1.2 kg/m3·343 m/s.
Symmetry
The Symmetry node adds a boundary condition where there is symmetry in the
pressure. Use this condition to reduce the size of a model by cutting it in half where
there are symmetries. In pressure acoustics this boundary condition is mathematically
identical to the Sound Hard Boundary (Wall) condition.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. If the node is selected from
the Pairs submenu, this list cannot be edited and it shows the boundaries in the selected
pairs.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
Plane Wave Radiation
The Plane Wave Radiation node adds a radiation boundary condition for a plane wave.
If required, right-click the main node to add an Incident Pressure Field to model an
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incoming wave. This radiation condition allows an outgoing plane wave to leave the
modeling domain with minimal reflections, when the angle of incidence is near to
normal.
The plane wave type is suitable for both far-field boundaries and ports. Because many
waveguide structures are only interesting in the plane-wave region, it is particularly
relevant for ports. When using the radiation condition on an open far-field boundary
it is recommended to construct the boundary such that the incidence angle is near to
normal, this of course requires a priory knowledge of the problem and the solution.
See the theory section Theory for the Plane, Spherical, and Cylindrical Radiation
Boundary Conditions for details about the equations and the formulation of this
non-reflecting boundary condition.
An estimate of the reflection coefficient Rs, for the spurious waves
reflecting off the plane wave radiation boundary, is, for incident plane
waves at angle , given by the expression:
cos  – 1R s = --------------------cos  + 1
N
where N is the order of the boundary condition (here 1 or 2). So at
normal incidence (= 0) there are no spurious reflections, while, for
example, at an incidence angle of 30o for N = 2 (plane wave radiation in
the frequency domain) the amplitude of the spurious reflected wave is
0.5 % of the incident.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
• Acoustics of a Muffler: Model Library path COMSOL_Multiphysics/
Acoustics/automotive_muffler
• Absorptive Muffler: Model Library path Acoustics_Module/
Industrial_Models/absorptive_muffler
Spherical Wave Radiation
The Spherical Wave Radiation node adds a radiation boundary condition for a spherical
wave, for which you define the source location. If required, right-click the main node
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
to add an Incident Pressure Field to model an incoming wave. This radiation condition
allows an outgoing spherical wave to leave the modeling domain with minimal
reflections. The geometry of the modeling domain should be adapted to have the
outgoing spherical waves coincide with the boundary, this is in order to minimize
spurious reflections.
See the theory section Theory for the Plane, Spherical, and Cylindrical Radiation
Boundary Conditions for details about the equations and the formulation of this
non-reflecting boundary condition.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
SPHERICAL WAVE RADIATION
Enter coordinates for the Source location r0 (SI unit: m). The defaults are 0 m.
Bessel Panel: Model Library path Acoustics_Module/Tutorial_Models/
bessel_panel
Cylindrical Wave Radiation
The Cylindrical Wave Radiation node adds a radiation boundary condition for a
cylindrical wave, for which you define the source location and the source axis direction
If required, right-click the main node to add an Incident Pressure Field to model an
incoming wave. This radiation condition allow an outgoing cylindrical wave to leave
the modeling domain with minimal reflections. The geometry of the modeling domain
should be adapted to have the outgoing cylindrical waves coincide with the boundary,
this is in order to minimize spurious reflections.
See the theory section Theory for the Plane, Spherical, and Cylindrical Radiation
Boundary Conditions for details about the equations and the formulation of this
non-reflecting boundary condition.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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CYLINDRICAL WAVE R ADIATION
Enter coordinates for the Source location r0 (SI unit: m) (the defaults are 0 m) and the
Source axis direction raxis (dimensionless) (the defaults are 0).
Acoustic Cloaking: Model Library path Acoustics_Module/Tutorial_Models/
acoustic_cloaking
Incident Pressure Field
The Incident Pressure Field node is a subnode to all non-reflecting boundary conditions
(plane, cylindrical, spherical wave radiation, and matched boundary). Right-click the
Matched Boundary, Plane Wave Radiation, Spherical Wave Radiation, or Cylindrical
Wave Radiation nodes to add this subnode. If the incident pressure field pi is a
predefined plane wave, it is of the type:
pi = p0 e
–i  k  r 
= p0 e
re
– i k eq  ---------------k
 e 
k
where p0 is the wave amplitude, k is the wave vector (with amplitude keq=|k| and
wave direction vector ek), and r is the location on the boundary. The incident pressure
field can also be a user-defined value or expression.
In transient analysis the incident pressure field is only of the user defined
type. In this case the incident pressure field needs to be defined as a
traveling wave of the form
f  t – k  x 
where  is the angular frequency and k is the wave vector. The function
f is any function, for example, a sine function. This is a requirement for
the radiation boundary condition to work properly.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to include an incident pressure field pi
in the boundary condition. By default, this feature node inherits the selection from its
parent node, and only a selection that is a subset of the parent node’s selection can be
used.
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INCIDENT PRESSURE FIELD
From the Incident pressure field type list, select Plane wave to define an incident
pressure field of plane wave type. Then enter a Pressure amplitude p0 (SI unit: Pa) (the
default is 0 Pa) and Wave direction ek (SI unit: m).
Select User defined to enter the expression for the Incident pressure field pi (SI unit: Pa)
as a function of space. The default is 0 Pa.
Periodic Condition
The Periodic Condition node adds a periodic boundary condition that can be used to
reduce the model size by using symmetries and periodicities in the geometry and
physics being modeled.
The Porous Absorber model uses Floquet periodic boundary conditions
to model an infinite porous absorber used for sound proofing. The Model
Library path is Acoustics_Module/Industrial_Models/porous_absorber
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. The software automatically
identifies the boundaries as either source boundaries or destination boundaries.
This feature works well for cases like opposing parallel boundaries. In
other cases use a Destination Selection subnode to control the destination.
By default it contains the selection that COMSOL Multiphysics identifies.
PERIODICITY SETTINGS
Select a Type of periodicity—Continuity (the default), Floquet periodicity, Cyclic
symmetry, User defined, or Antiperiodicity. For the Aeroacoustics interfaces, only
Continuity and Antiperiodicity are available as the Type of periodicity.
• If Floquet periodicity is selected, enter a k-vector for Floquet periodicity kF (SI unit:
rad/m) for the x, y, and z coordinates (3D models), or the r and z coordinates (2D
axisymmetric models), or x and y coordinates (2D models). This condition is used
for modeling infinite periodic structures with non-normal incident pressure fields or
excitations. Use this condition to model, for example, a large perforated plate with
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67
an oblique incident wave with wave vector k (and set kF = k) by only analyzing one
hole or one subset of holes that is periodic.
• If Cyclic symmetry is selected, select a Sector angle—Automatic (the default), or User
defined. If User defined is selected, enter a value for S (SI unit: rad). For any
selection, also enter an Azimuthal mode number m (dimensionless). This condition is
used to model any geometry that has a cyclic periodic structure like, for example, a
microphone or a loudspeaker driver. Setting the azimuthal mode number m
determines which mode is analyzed. The response of the full system to an external
excitation will in general be a linear combination of many different modes.
When an acoustics interface uses a Solid Mechanics or Piezoelectric
Devices interface, User defined is often available as a Type of periodicity.
For the case of the Acoustic-Solid Interaction, Frequency Domain, the
Acoustic-Piezoelectric Interaction, Frequency Domain, and the Thermoacoustic-Solid
Interaction, Frequency Domain interfaces:
• If User defined is selected, select the Periodic in u, Periodic in v (for 3D and 2D
models), and Periodic in w (for 3D and 2D axisymmetric models) check boxes as
required. Then for each selection, choose the Type of periodicity—Continuity (the
default) or Antiperiodicity.
The available check boxes for the User defined option are based on the interface
• Acoustic-Solid Interaction, Frequency Domain: Periodic in p, and Periodic in u
(component wise).
• Acoustic-Piezoelectric Interaction, Frequency Domain and Acoustic-Piezoelectric
Interaction, Transient: Periodic in p, Periodic in V, and Periodic in u (component
wise).
• Thermoacoustic-Solid Interaction, Frequency Domain: Periodic in p, Periodic in
u_fluid (component wise), Periodic in T, and Periodic in u_solid (component wise).
In the time domain both the Cyclic symmetry and the Floquet periodicity
boundary conditions reduce to the continuity condition.
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To optimize the performance of the Floquet periodicity and the Cyclic
symmetry conditions it is recommended that the source and destination
meshes are identical. This can be achieved by first meshing the source
boundary or edge and then copy the mesh to the destination boundary
or edge.
CONSTRAINT SETTINGS
These are the same settings as for Sound Soft Boundary.
In the COMSOL Multiphysics Reference Manual:
• Periodic Condition and Destination Selection
• Periodic Boundary Conditions
Interior Sound Hard Boundary (Wall)
The Interior Sound Hard Boundary (Wall) node adds a boundary condition for a sound
hard boundary or wall on interior boundaries. A sound-hard boundary is a boundary
at which the normal component of the acceleration is zero:
1
– n  –  -----  p t – q d  = 0
 c

1
1
– n    – -----  p t – q d  = 0
   c

2
where the subscripts 1 and 2 represent the two sides of the boundary. For zero dipole
charge and constant fluid density, this means that the normal derivative of the pressure
is zero at the boundary. On an interior sound hard boundary the pressure is not
continuous but is treated as a so-called slit feature.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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Axial Symmetry
The Axial Symmetry node is a default node added for all 2D and 1D
axisymmetric models. The boundary condition is active on all boundaries
on the symmetry axis.
BOUNDARY SELECTION
The boundaries section shows on which boundaries the node is active. All
boundaries on the symmetry axis are automatically selected.
Continuity
Continuity is available as an option at interfaces between parts in a pair.
In the COMSOL Multiphysics Reference Manual:
• Continuity on Interior Boundaries
• Identity and Contact Pairs
This condition gives continuity in total pressure and in the normal acceleration over
the pair (subscripts 1 and 2 in the equation refer to the two sides in the pair):
1
1
– n  –  -----  p t – q d  –  – -----  p t – q d 
= 0

c
c
1
2
BOUNDARY SELECTION
This list cannot be edited. It shows the boundaries in the selected pairs.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
PAIR SELECTION
When this node is selected from the Pairs menu, choose the pair to define. An identity
pair has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
These are the same settings as for Sound Soft Boundary.
Pressure Acoustics Model
The Pressure Acoustics Model node adds the equations for time-harmonic and
eigenfrequency acoustics modeling in the frequency domain. In the settings window,
define the properties for the acoustics model and model inputs including the
background pressure and temperature.
For more details about each of the available fluid models, see Theory for
the Pressure Acoustics Fluid Models.
For more information about using variables during the results analysis, see
Special Variables in the Acoustics Module.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains to compute the acoustic pressure field and
the equation that defines it, or select All domains as required.
MODEL INPUTS
For all fluid models enter a Temperature T and an Absolute pressure p:
• Select User defined to enter a value or an expression for the absolute pressure
(SI unit: Pa) and the temperature (SI unit: K) in the field. This input is always
available.
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• In addition, select a temperature field defined by, for example, a heat transfer
interface or a non-isothermal fluid flow interface (if any).
Non-isothermal flow requires the addition of the Heat Transfer Module
or CFD Module.
• If applicable, select a pressure defined by a fluid-flow interface present in the model.
For example, select Pressure (spf/fp1) to use the pressure defined by the Fluid
Properties node fp1 in a Single-Phase Flow interface spf. Selecting a pressure variable
also activates a check box for defining the reference pressure, where 1[atm] is the
default value. This makes it possible to use a system-based (gauge) pressure, while
automatically including the reference pressure in the absolute pressure.
The input to these fields influences the value of the material parameters in the model.
Typically, the density and the speed of sound c in the model are dependent on the
absolute pressure and/or the temperature. Picking up any of those from another
interface typically results in xand c = cx to be specially varying.
PRESSURE ACOUSTICS MODEL
To define the properties of the bulk fluid, select a Fluid model from the list. The fluid
models represent different loss mechanisms applied in a homogenized way to the bulk
of the fluid. This type of fluid model is sometimes referred to as an equivalent fluid
model. The model may be a theoretical or a phenomenological model that accounts
for the losses due to viscosity and thermal conduction, for example, when acoustic
waves propagate in porous materials.
Use the Linear elastic selection, which is already the default, to specify a linearly elastic
fluid using either the density and speed of sound or the impedance and wave number.
When the material parameters are real values this corresponds to a lossless compressible
fluid. Go to Defining a Linear Elastic Fluid Model.
The theory for the fluid models is in the section Theory for the Pressure
Acoustics Fluid Models.
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The fluid models may be roughly divided into these categories:
• General fluids:
- Linear elastic. Go to Defining a Linear Elastic Fluid Model.
- Linear elastic with attenuation. Go to Defining a Linear Elastic with Attenuation
Fluid Model.
- Ideal gas. Go to Defining an Ideal Gas Fluid Model.
• Porous material fluid models:
- Macroscopic empirical porous models (Delany-Bazely or Miki models). Go to
Defining Macroscopic Empirical Porous Fluid Models.
- Biot equivalents (limp porous matrix or rigid porous matrix, the latter is also
known as the Johnson-Champoux-Allard model). Go to Defining a Biot
Equivalent Fluid Model.
• Fluids with bulk viscous and thermal losses:
- Viscous. Go to Defining a Viscous Fluid Model.
- Thermally conducting. Go to Defining a Thermally Conducting Fluid Model.
- Thermally conducting and viscous. Go to Defining a Thermally Conducting and
Viscous Fluid Model.
• Fluids models for viscous and thermal boundary-layer induced losses in channels
and ducts:
- Boundary-layer absorption (narrow or wide duct). Go to Defining a
Boundary-Layer Absorption Fluid Model.
Defining a Linear Elastic Fluid Model
To specify the properties as complex-valued data, from the Specify list, select Density
and speed of sound (the default) or Impedance and wave number.
• If Density and speed of sound is selected, the default Speed of sound c (SI unit: m/s)
and Density  (SI unit: kg/m3) values are taken From material. Select User defined to
enter other values or expressions.
• If Impedance and wave number is selected, enter a Wave number k (SI unit: rad/m).
By default the Characteristic acoustic impedance Z (SI unit: Pa·s/m) is the value
taken From material for the fluid. Select User defined to enter other values or
expressions.
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Defining a Linear Elastic with Attenuation Fluid Model
The default Speed of sound c (SI unit: m/s) and Density  (SI unit: kg/m3) values are
taken From material. Select User defined to enter other values or expressions for one or
both options.
Select an Attenuation type—Attenuation coefficient Np per unit length to define an
attenuation coefficient  in Np/m (nepers per meter), Attenuation coefficient dB per
unit length to define an attenuation coefficientin dB/m (decibel per meter), or
Attenuation coefficient dB per wavelength to define an attenuation coefficient in dB/
 (decibel per wavelength). For any selection, enter a value or expression in the
Attenuation coefficient  field.
About the Linear Elastic with Attenuation Fluid Model
Defining Macroscopic Empirical Porous Fluid Models
The default Speed of sound c (SI unit: m/s) and Density  (SI unit: kg/m3) values are
taken From material. Select User defined to enter other values or expressions for one or
both options.
Enter a Flow resistivity Rf (SI unit: Pa·s/m2). Flow resistivity is easy to measure and is
independent of frequency.
Select an option from the Constants list—Delany-Bazely (the default), Miki, or User
defined. If User defined is selected, enter values in the C1 to C8 fields.
About the Macroscopic Empirical Porous Models
Defining a Viscous Fluid Model
For each of the following, the default values are taken From material. Select User defined
to enter other values or expressions for any or all options.
• Speed of sound c (SI unit: m/s)
• Density  (SI unit: kg/m3)
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• Dynamic viscosity  (SI unit: Pa·s)
• Bulk viscosity B (SI unit: Pa·s)
About the Viscous Fluid Model
Defining a Thermally Conducting Fluid Model
For each of the following, the default values are taken From material. Select User defined
to enter other values or expressions for any or all options.
• Speed of sound c (SI unit: m/s)
• Density  (SI unit: kg/m3)
• Heat capacity at constant pressure Cp (SI unit: J/(kg·K))
• Ratio of specific heats  (SI unit: 1)
• Thermal conductivity k (SI unit: W/(m·K))
About the Thermally Conducting Fluid Model
Defining a Thermally Conducting and Viscous Fluid Model
For each of the following, the default values are taken From material. Select User defined
to enter other values or expressions for any or all options.
• Speed of sound c (SI unit: m/s)
• Density  (SI unit: kg/m3)
• Heat capacity at constant pressure Cp (SI unit: J/(kg·K))
• Ratio of specific heats  (SI unit: 1)
• Thermal conductivity k (SI unit: W/(m·K))
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• Dynamic viscosity  (SI unit: Pa·s)
• Bulk viscosity B (SI unit: Pa·s)
It is possible to assess the magnitude of the losses due to thermal
conduction and viscosity, that is, the power dissipation density (SI unit:
W/m3). This is done during the analysis process by plotting the variables
for:
• the viscous power dissipation density (diss_visc),
• the thermal power dissipation density (diss_therm), or
• the combined total power dissipation density (diss_tot).
About the Thermally Conducting and Viscous Fluid Model
Defining an Ideal Gas Fluid Model
Specify the fluid properties by selecting a gas constant type and selecting between
entering the heat capacity at constant pressure or the ratio of specific heats.
• Select a Gas constant type—Specific gas constant Rs (SI unit: J/(kg·K) or Mean molar
mass Mn (SI unit: kg/mol). For both options, the default values are taken From
material. Select User defined to enter other values or expressions for one or both
options. If Mean molar mass is selected, the molar gas constant (universal gas
constant) R  8.314 J/(mol·K), is used as the built-in physical constant.
• From the Specify Cp or  list, select Heat capacity at constant pressure Cp (SI unit: J /
(kg·K)) or Ratio of specific heats . For both options, the default values are taken
From material. Select User defined to enter other values or expressions for one or
both options. For common diatomic gases such as air, 1.4 is the standard value.
Defining a Biot Equivalent Fluid Model
By default, the Fluid material uses the Domain material.
For each of the following, the default values are taken From material. Select User defined
to enter other values or expressions for any or all options.
• Speed of sound c (SI unit: m/s)
• Density f (SI unit: kg/m3)
• Heat capacity at constant pressure Cp (SI unit: J/(kg·K))
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• Ratio of specific heats  (SI unit: 1)
• Thermal conductivity k (SI unit: W/(m·K))
• Dynamic viscosity  (SI unit: Pa·s)
The section Porous Model is also displays when Biot Equivalents is selected as the fluid
model. The default Porous elastic material uses the Domain material (the material
defined for the domain). Select another material as required.
Select a Porous matrix approximation—Limp (the default) or Rigid.
• If Limp is selected, the default value for Drained density of porous material d (SI unit:
kg/m3) is taken From material. Select User defined to enter another value or
expression.
• For both Limp and Rigid porous matrix approximations, the Porosity p
(dimensionless) value is taken From material. Select User defined to enter other values
or expressions for one or both options. Then enter values or expressions for each of
the following:
- Flow resistivity Rf (SI unit: Pa·s/m2). The default is 0 Pa·s/m2.
- From the Specify list, select Viscous characteristic length parameter to enter a value
for s (dimensionless) (the default is 1), or select Viscous characteristic length to
directly enter an expression for Lv (SI unit: m). The default expression for the
Viscous characteristic length is sqrt(acpr.mu*acpr.tau*8/
(acpr.Rf*acpr.epsilon_p)), which corresponds to s = 1.
- Thermal characteristic length Lth (SI unit: m). The default expression is
2*acpr.Lv.
- Tortuosity factor  (dimensionless). The default is 1.
The Biot equivalents model with a rigid porous matrix is also often
referred to as the Johnson-Champoux-Allard (JCA) model with a
motionless skeleton.
About the Biot Equivalent Fluid Models
Defining a Boundary-Layer Absorption Fluid Model
By default, the Fluid material uses the Domain material.
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Select a Duct type—Wide duct (the default) or Narrow duct. For either selection, enter
a Hydraulic diameter Hd (SI unit: m). The default is 0 m. Then for each of the
following, the default values are taken From material. Select User defined to enter other
values or expressions for any or all options.
• Speed of sound c (SI unit: m/s)
• Density  (SI unit: kg/m3)
• Ratio of specific heats  (SI unit: 1)
• Dynamic viscosity  (SI unit: Pa·s)
The following are available for Wide duct only:
• Heat capacity at constant pressure Cp (SI unit: J/(kg·K))
• Thermal conductivity k (SI unit: W/(m·K))
The boundary-layer absorption model adds the viscous and thermal losses
effect of the acoustic boundary layer to the bulk of the fluid. This
equivalent-fluid model may be used in long tubes of constant cross
section instead of a full detailed thermoacoustic model.
About the Boundary-Layer Absorption Fluid Model
Background Pressure Field
Add a Background Pressure Field node to model an incident pressure wave or to study
the scattered pressure field ps, which is defined as the difference between the total
acoustic pressure pt and the background pressure field pb:
pt = pb + ps
This feature sets up the equations in a so-called scattered field formulation where the
dependent variable is the scattered field p = ps. In a model where the background
pressure field is not defined on all acoustic domains (or it is different) continuity is
automatically applied in the total field pt on internal boundaries between domains.
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The background pressure field can be a function of space and, for The Pressure
Acoustics, Transient User Interface using a Time Dependent study, a function of time.
For a Frequency Domain study type, the frequency of the background
pressure field is the same as for the dependent variable p.
DOMAIN SELECTION
From the Selection list, choose the domains to define.
BACKGROUND PRESSURE FIELD
Select a Background pressure field type—Plane wave (the default) or User defined. The
Plane wave option results in a background pressure field of the type:
pp = p0 e
  r  e k -
– i ---- ----------------c ek
where r is the spatial coordinate, ek specifies the wave direction, p0 is the wave pressure
amplitude,  is the angular frequency, c the speed of sound, and   c is equal to the
wave number k of the background plane wave.
• If Plane wave is selected, enter values for the Pressure amplitude p0 (SI unit: Pa) and
Wave direction, ek (dimensionless). Select to define the Speed of sound c (SI unit: m/
s) either From material or User defined (default is 0 m/s).
• Select User defined to enter the expression for the Background pressure field pb
(SI unit: Pa). The default is 0 Pa.
• Acoustic Cloaking: Model Library path Acoustics_Module/
Tutorial_Models/acoustic_cloaking
• Acoustic Scattering off an Ellipsoid: Model Library path
Acoustics_Module/Tutorial_Models/acoustic_scattering
Matched Boundary
The Matched Boundary node adds a matched boundary condition. Like the radiation
boundary conditions, it belongs to the class of non-reflecting boundary conditions
(NRBCs). If required, right-click the main node to add an Incident Pressure Field.
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Properly set up, the matched boundary condition allows one mode with wave number
k1 (set k2k1), or two modes with wave numbers k1 and k2, to leave the modeling
domain with minimal reflections. The equation is given by
 2
i   ---- + k 1 k 2 p + i T p



c c
1
– n   – -----  p – q d  + --------------------------------------------------------------c
c  k1 + k2 
 2
i   ---- + k 1 k 2 p i + i T p i
  c c

1
= ------------------------------------------------------------------- + n  ----- p i
c
c  k1 + k2 
Here T, for a given point on the boundary, refers to the Laplace operator in the
tangential plane at that point, while pi is the amplitude of an optional incoming plane
wave with wave vector k. In addition to pi, specify the propagation direction, nk,
whereas the wave number is defined by keq /cc in 3D and in 2D.
 2
2
2
k eq =  ---- – k z
 c c
The matched boundary condition is particularly useful for modeling acoustic waves in
ducts and waveguides at frequencies below the cutoff frequency for the second excited
transverse mode. In such situations set k1  /cc and k2  1/cc, where 12f1,
and f1 is the cutoff frequency for the first excited mode. The cutoff frequency or wave
number may be found using a Boundary Mode Acoustics model. When k1  k2  /
cc, the matched boundary condition reduces to the time-harmonic plane-wave
radiation boundary condition.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
MATCHED BOUNDARY
From the Match list, select One mode (the default) or Two modes. Then enter a Wave
number (SI unit rad/m) based on the selection: k1 if One mode is selected and k1 and
k2 if Two modes is selected.
Far-Field Calculation
Use the Far-Field Calculation node to apply the source boundaries for the
near-to-far-field transformation and to specify a name for the acoustic far-field variable.
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This feature allows the calculation of the pressure field outside the computational
domain. The far-field boundary need to enclose all sources and scatterers.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to specify the source aperture for the far
field.
FAR-FIELD CALCULATION
Enter a Far-field variable name for the far-field acoustic pressure field (the default is
pfar)
Select a Type of integral—Integral approximation for r (the default) to compute the
value in The Far-Field Limit or Full integral to compute The Helmholtz-Kirchhoff
Integral Representation.
If required, use symmetry planes in your model when calculating the far-field variable.
The symmetry planes have to coincide with one of the Cartesian coordinate planes. For
each of these planes, select the type of symmetry check boxes—Symmetry in the x=0
plane, Symmetry in the y=0 plane, or Symmetry in the z=0 plane. This selection should
match the boundary condition used for the symmetry boundary. With these settings,
the parts of the geometry that are not in the model for symmetry reasons can be
included in the far-field analysis.
ADVANCED SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
The option Use polynomial-preserving recovery for the normal gradient is selected per
default. This means that the far-field feature automatically uses the
polynomial-preserving recovery operator ppr() to get an enhanced evaluation of the
normal derivative of the pressure, on internal boundaries. This increases the precision
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81
of the far-field calculation. If you click to clear this check box this removes all instances
of the operator from the equations.
The ppr() operator is not added when the far-field calculation is
performed on an external boundary or a boundary adjacent to a perfectly
matched layer (PML) domain. In the latter case, the down() or up()
operator is added in order to retrieve values of variables from the physical
domain only.
In these cases, use a single boundary layer mesh on the inside of the outer
boundary or on the inside of the PML layer to enhance the precision of
the far-field calculation.
• The Far Field Plots
• Evaluating the Acoustic Field in the Far-Field Region
• ppr and pprint and up and down (operators) in the COMSOL
Multiphysics Reference Manual
• Acoustic Scattering off an Ellipsoid: Model Library path
Acoustics_Module/Tutorial_Models/acoustic_scattering
• Bessel Panel: Model Library path Acoustics_Module/Tutorial_Models/
bessel_panel
• Cylindrical Subwoofer: Model Library path Acoustics_Module/
Tutorial_Models/cylindrical_subwoofer
Interior Normal Acceleration
The Interior Normal Acceleration node adds a normal acceleration on an interior
boundary and ensures that the pressure is non-continuous here. The pressure has a
so-called slit condition on this boundary. This boundary condition can be used to
model sources as, for example, the movement of a speaker cone modeled as a
boundary. The condition adds the normal part of an acceleration a0:
1
– n   – -----  p t – q d 
 c

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= n  a0
CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
1
– n   – -----  p t – q d 
 c

2
= n  a0
Alternatively, specify the inward acceleration an. The normal of the boundary is
interpreted as pointing outward.
1
– n   – -----  p t – q d 
 c

1
= an
1
– n   – -----  p t – q d 
 c

2
= an
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
INTERIOR NORMAL ACCELERATION
Select a Type—Acceleration (the default) or Inward acceleration.
• If Acceleration is chosen, enter values for the components of the Acceleration a0
(SI unit: m/s2). The defaults are 0 m/s2.
• If Inward Acceleration is chosen, enter the value of the Inward acceleration an
(SI unit: m/s2). The default is 0 m/s2. Use a positive value for inward acceleration
or a negative value for outward acceleration. The normal of the boundary points
outward.
• Cylindrical Subwoofer: Model Library path Acoustics_Module/
Tutorial_Models/cylindrical_subwoofer
• Lumped Loudspeaker Driver: Model Library path Acoustics_Module/
Industrial_Models/lumped_loudspeaker_driver
Interior Impedance/Pair Impedance
The Interior Impedance and Pair Impedance nodes add an impedance boundary
condition on interior boundaries or boundaries between the parts of pairs. This
condition is a generalization of the sound-hard and sound-soft boundary conditions.
The condition corresponds to a transfer impedance condition, relating the pressure
drop across the boundary p t1 – p t2 to the velocity at the boundary. In the frequency
domain, it imposes the following equations:
–i 
1
– n   – -----  p t – q d  =  p t1 – p t2  --------c
Z
1
–i 
1
– n   – -----  p t – q d  =  p t1 – p t2  -------- c

Z
2
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For a time-dependent study (time domain), the boundary condition uses the following
equations:
1
1
– n   – ---  p t – q d  = ---- -----  p t1 – p t2 
 
1
Z t
1
1
– n   – ---  p t – q d  = ---- -----  p t1 – p t2 
 
2
Z t
Z is the impedance, which from a physical point is the ratio between pressure and
normal particle velocity.
In the two opposite limits Z and Z0, this boundary condition is
identical to the Sound Hard boundary condition and the Sound Soft
boundary condition, respectively. Additional information is found in
Identity and Contact Pairs in the COMSOL Multiphysics Reference
Manual.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. For the Pair Impedance node,
this list is not editable and shows the boundaries in the selected pairs.
PAIR SELECTION
If Pair Impedance is selected from the Pairs menu, choose the pair to define. An identity
pair has to be created first. Ctrl-click to deselect.
INTERIOR IMPEDANCE/PAIR IMPEDANCE
Enter the value of the Impedance Zi (SI unit: Pa·s/m). The default is 0 Pa·s/m.
Interior Perforated Plate
The Interior Perforated Plate node provide the possibility of specifying the characteristic
properties for a perforated plate. COMSOL Multiphysics then calculates the transfer
impedance using the following model expression (Ref. 3):
tp
k eq
 1 8k eq 

Z---------=  --- --------------- 1 + ------ +  f + i --------  t p +  h 




d
c cc

c


h
c c
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. For the Pair Perforated Plate
node, this list is not editable and shows the boundaries in the selected pairs.
PAIR SELECTION
If Pair Perforated Plate is selected from the Pairs menu, choose the pair to define. An
identity pair has to be created first. Ctrl-click to deselect.
INTERIOR PERFORATED PLATE
The equation above includes the properties listed for the perforated plate. Specify the
properties of the perforated plate and enter the following:
• Dynamic viscosity  (SI unit: Pa·s). The default is 1.8·105 Pa·s.
• Area porosity , that is, the holes’ fraction of the boundary surface area—a
dimensionless number between 0 and 1. The default is 0.1; that is, 10% of the plate’s
area consists of holes.
• Plate thickness tp (SI unit: m). The default is 0 m.
• Hole diameter dh (SI unit: m). The default is 1 mm (103 m).
• End correction h to the reactance (SI unit: m). The default is 0.25dh (a quarter of
the hole diameter).
• Flow resistance f a contribution to the resistive part of the impedance that can be
used, for example, to include the effects of a mean flow or non-linear effect at large
sound pressure levels. The default is 0.
The transfer impedance model implemented here is only one of many
engineering relations that exist for perforated plates (perforates in
general). Use the Interior Impedance/Pair Impedance condition to enter
a user defined model.
Line Source
Use the Line Source node to add a source on a line/edge in 3D models. This type of
source corresponds a radially vibrating cylinder in the limit where its radius tends to
zero. The line source adds a source term to the right hand side of the governing
Helmholtz equation such that:
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85
2
k eq p t
1
1
3
   – -----  p t – q d  – ------------- = ----- 4S  r – r 0 dl
 c

c
c
3
where   r – r 0  is the delta function in 3D that adds the source on the edge where
rr0 and dl is the line element along the edge (SI unit: m). The monopole amplitude
S (SI unit: N/m2) depends on the source type selected, as discussed below.
• Frequency Domain Study
• Solution (data sets) in the COMSOL Multiphysics Reference Manual
For the Pressure Acoustics, Transient interface, only the Flow (no phase
specification), User defined, and the Gaussian pulse source types are
available. The Gaussian pulse source type has no effect in the frequency
domain.
EDGE SELECTION
From the Selection list, choose the edges to define.
LINE SOURCE
Select a Type—Flow (the default), Intensity, Power, or User defined. If User defined is
selected, enter a Monopole amplitude, S = Suser (SI unit: N/m2). The default is 0 N/
m2. Otherwise, follow these instructions.
Flow
Select Flow to add an edge source located at rr0 defined in terms of the volume flow
rate per unit length out from source QS and the phase  of the source. The flow edge
source defines the following monopole amplitude:
S = e
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
i
i
-----------c- Q S
4
A flow edge source with the strength QS represents an area flow out from the source
(the source is a very thin cylinder with a surface that pulsates).
• Enter a Volume flow rate per unit length out from source, QS (SI unit: m2/s) for the
source-strength amplitude in the field. The default is 0 m2/s.
• Enter a Phase (SI unit: rad). The default is 0 rad.
When defining a Solution data set and plotting the results, specify a
nonzero phase to produce a nonzero result when visualizing the
resulting pressure field using the default value (0) in the Solution at angle
(phase).
Intensity
Select Intensity to add an edge source located at rr0 defined in terms of the source
intensity radiated Irms and the phase of the source. Set a desired free space reference
intensity (RMS) Irms at a specified distance dsrc from the source. In a homogeneous
medium, the specified intensity is obtained when the edge is a straight line (this is the
reference). With other objects and boundaries present, or if the edge is curved, the
actual radiated intensity is different. This source type defines the following monopole
amplitude:
i d src
S = e --------------- 2I ref  c c c
L edge
where Ledge is the length of the source line (automatically determined), dsrc is the
distance from the source where free space reference intensity (RMS) Irms is specified.
Enter values or expressions for:
• Free space reference intensity (RMS), Irms (SI unit: W/m2). The default is 0 W/m2.
• Distance from source center dsrc (SI unit: m). The default is 0 m.
• Phase (SI unit: rad). The default is 0 rad.
Power
Select Power to add an edge source located at rr0 specified in terms of the source’s
reference RMS strength by stating the total power Prms a straight line source would
radiate into a homogeneous medium. This source type defines the following monopole
amplitude:
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87
i
 c c c P rms
e
S = -------------- --------------------L edge
2
where Ledge is the length of the source line (automatically determined) and Prms
denotes the free space reference RMS power (in the reference homogeneous case) per
unit length measured in W/m. Enter values or expressions for:
• Free space reference power (RMS), Prms (SI unit: W). The default is 0 W.
• Phase (SI unit: rad). The default is 0 rad.
Line Source on Axis
Use the Line Source on Axis node to add a source on the axis of symmetry in 2D
axisymmetric models. This type of source corresponds a radially vibrating cylinder in
the limit where its radius tends to zero. The line source adds a source term to the
right-hand side of the governing Helmholtz equation such that:
2
k eq p t
1
1
3
   – -----  p t – q d  – ------------- = ----- 4S  z – z 0 dz
 c

c
c
3
where   z – z 0  is the delta function in 3D that adds the source on the axis of
symmetry where zz0 and r = 0, and dz is the line element along the z-axis (SI unit:
m). The monopole amplitude S (SI unit: N/m2) depends on the source type selected
and is the same as discussed in the 3D case for a Line Source.
• Frequency Domain Study
• Solution (data sets) in the COMSOL Multiphysics Reference Manual
For the Pressure Acoustics, Transient interface, only the Flow (no phase
specification), User defined, and the Gaussian pulse source types are
available. The Gaussian pulse source type has no effect in the frequency
domain.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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LINE SOURCE ON AXIS
Select a Type—Flow (the default), Intensity, Power, or User defined. See the options and
expression for Line Source. The sources are the same but in 2D axisymmetric models
they are only applicable on the symmetry axis at r = 0.
Monopole Point Source
Use the Monopole Point Source node to as a monopole point source in 3D models on
any point and in 2D axisymmetric models on points on the axis of symmetry. This is a
source that is uniform and equally strong in all directions. A monopole represents a
radially pulsating sphere in the limit where the radius tends to zero. The monopole
point source adds a point source term to the right hand side of the governing
Helmholtz equation such that:
2
k eq p t
1
1
3
   – -----  p t – q d  – ------------- = ----- 4S  r – r 0 
 c

c
c
3
where   r – r 0  is the delta function in three dimensions and adds the source at the
point where rr0. The monopole amplitude S (SI unit: N/m2) depends on the
source type selected, as discussed below.
For the Pressure Acoustics, Transient interface, only the Flow (no phase
specification), User defined, and the Gaussian pulse source types are
available. The Gaussian pulse source type has no effect in the frequency
domain.
POINT S EL EC TION
From the Selection list, choose the points to define.
MONOPOLE POINT SOURCE
Select a Type—Flow (the default), Intensity, Power, or User defined. If User defined is
chosen, enter a Monopole amplitude, S = Suser (SI unit: N/m). The default is 0 N/m.
Flow
Select Flow to add an monopole point source located at rr0 defined in terms of the
volume flow rate out from source QS and the phase  of the source. The source defines
the following monopole amplitude:
THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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89
S = e
i
i
-----------c- Q S
4
Enter values or expressions for:
• Volume flow rate out from source, QS (SI unit: m3/s). The default is 0 m3/s.
• Phase (SI unit: rad). The default is 0 rad.
Intensity
Select Intensity to define the source in terms of the free space reference RMS intensity
Irms it radiates. In a homogeneous medium the specified intensity is obtained (the
reference), but with other objects and boundaries present the actual intensity is
different. The source defines the following monopole amplitude:
i
S = e d src 2 c c c I rms
where dsrc is the distance from the source where the intensity Irms is specified and 
is the phase of the source. Enter values or expressions for:
• Free space reference intensity (RMS), Irms (SI unit: W/m2). The default is 0 W/m2.
• Distance from source center dsrc (SI unit: m). The default is 0 m.
• Phase (SI unit: rad). The default is 0 rad.
Power
When Power is selected, specify the source’s reference RMS strength by stating the
power it radiates. In a homogeneous medium the specified power is obtained (the
reference), but with other objects and boundaries present the actual power is different.
The source defines the following monopole amplitude:
S = e
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
i
 c c c P rms
--------------------2
where Pref denotes the radiated RMS power per unit length measured in W/m. Enter
values or expressions for:
• Free space reference power (RMS), Prms (SI unit: W). The default is 0 W.
• Phase (SI unit: rad). The default is 0 rad.
• Bessel Panel: Model Library path Acoustics_Module/Tutorial_Models/
bessel_panel
• Hollow Cylinder: Model Library path Acoustics_Module/
Tutorial_Models/hollow_cylinder
Point Source
Use the Point Source node to add a point source to a 2D model. This source
corresponds to an infinite line source in the out-of-plane direction. The source is
uniform and equally strong in all the in-plane directions. The point source adds a point
source term to the right hand side of the governing Helmholtz equation such that:
2
k eq p t
1
1
2
   – -----  p t – q d  – ------------- = ----- 4S  r – r 0 
 c

c
c
2
where   r – r 0  is the delta function in 2D and adds the source at the point where
(x,y) = rr0. The monopole amplitude S (SI unit: N/m2) depends on the source
type selected, as discussed below.
For the Pressure Acoustics, Transient interface, only the Flow (no phase
specification), User defined, and the Gaussian pulse source types are
available. The Gaussian pulse source type has no effect in the frequency
domain.
POINT S EL EC TION
From the Selection list, choose the points to define.
MONOPOLE POINT SOURCE
Select a Type—Flow (the default), Intensity, Power, or User defined. If User defined is
chosen, enter a Monopole amplitude, Suser (SI unit: N/m). The default is 0 N/m.
THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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91
Flow
Select Flow to add an monopole point source located at rr0 defined in terms of the
volume flow rate per unit length out from source QS and the phase  of the source.
The source defines the following monopole amplitude:
S = e
i
i
-----------c- Q S
4
Enter values or expressions for:
• Volume flow rate out from source, QS (SI unit: m3/s). The default is 0 m3/s.
• Phase (SI unit: rad). The default is 0 rad.
Intensity
Select Intensity to define the source in terms of the free space reference RMS intensity
Irms it radiates. In a homogeneous medium the specified intensity is obtained (the
reference), but with other objects and boundaries present the actual intensity is
different. The source defines the following monopole amplitude:
S = e
i
2 c I rms d src
--------------------------------2
where dsrc is the distance from the source where the intensity Irms is specified and 
is the phase of the source. Enter values or expressions for:
• Free space reference intensity (RMS), Irms (SI unit: W/m2). The default is 0 W/m2.
• Distance from source center dsrc (SI unit: m). The default is 0 m.
• Phase (SI unit: rad). The default is 0 rad.
Power
When Power is selected, specify the source’s reference RMS strength by stating the
power per unit length it radiates. In a homogeneous medium the specified power is
obtained (the reference), but with other objects and boundaries present the actual
source power is different. The source defines the following monopole amplitude:
S = e
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
i
2 c P rms
------------------------2
 2 
where Prms denotes the free space RMS reference power per unit length measured in
W/m and  is the phase of the source. Enter values or expressions for:
• Free space reference power (RMS) per unit length, Prms (SI unit: W/m). The default
is 0 W.
• Phase (SI unit: rad). The default is 0 rad.
Circular Source
Use the Circular Source node to add a source in a 2D axisymmetric models on points
off the axis of symmetry. Such points correspond to circular sources or ring sources.
This type of source is, for example, used to mimic source terms from rotors. The
circular source adds a point source term to the right hand side of the governing
Helmholtz equation such that:
2
k eq p t
1
1
3
   – -----  p t – q d  – ------------- = ----- 4S  r – r 0 rd
 c

c
c
3
where   r – r 0  is the delta function that adds the source at the point where rr0
and rd is the line element around the circular source (SI unit: m). The monopole
amplitude S (SI unit: N/m2) depends on the source type selected, as discussed below
For the Pressure Acoustics, Transient interface the Flow (no phase
specification), User defined, and the Gaussian pulse source types are
available. The Gaussian pulse source type has no effect in the frequency
domain.
POINT S EL EC TION
From the Selection list, choose the points to define.
Select a Type—Flow (the default) or User defined. If User defined is selected, enter a
Monopole amplitude, S = Suser (SI unit: N/m2). The default is 0 N/m2.
Flow
When Flow is selected the source is defined in terms of the volume flow rate QS per
unit length out form the source. The flow circular source defined the following
monopole amplitude:
S = e
i
i
-----------c- Q S
4
THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
|
93
Enter values or expressions for:
• Volume flow rate per unit length out from source, QS (SI unit: m2/s). The default is
0 m2/s.
• Phase (SI unit: rad). The default is 0 rad.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
The Pressure Acoustics, Transient
U s e r Inte r f a c e
The Pressure Acoustics, Transient (actd) user interface (
), found under the
Acoustics>Pressure Acoustics branch (
) in the Model Wizard, has the equations,
boundary conditions, and sources for modeling transient acoustic phenomena. The
interface solves the wave equation in terms of the sound pressure.
When this interface is added, these default nodes are also added to the Model Builder—
Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), and Initial Values.
Right-click the Pressure Acoustics, Transient node to add other features that implement
boundary conditions and sources.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is actd.
The remainder of the settings window is shared with The Pressure
Acoustics, Frequency Domain User Interface.
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure
Acoustics, Frequency Domain User Interface
• Domain, Boundary, Edge, and Point Nodes for the Pressure Acoustics,
Transient User Interface
• Theory Background for the Pressure Acoustics Branch
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95
Transient Gaussian Explosion: Model Library path Acoustics_Module/
Tutorial_Models/gaussian_explosion
Domain, Boundary, Edge, and Point Nodes for the Pressure Acoustics,
Transient User Interface
The Pressure Acoustics, Transient User Interface shares most of its nodes with the
Pressure Acoustics, Frequency Domain interface, except the following:
• Transient Pressure Acoustics Model
• The Gaussian Pulse Source Type
Domain, Boundary, Edge, Point, and Pair Nodes for the Pressure
Acoustics, Frequency Domain User Interface
To locate and search all the documentation, in COMSOL Multiphysics
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
Transient Pressure Acoustics Model
The Transient Pressure Acoustics Model node adds the equations for primarily
time-dependent (transient) acoustics modeling. This is the wave equation
2
 pt
1 - ----------------- +   –1
---  p t – q d  = Q m
2
2

c t
where pt is the total acoustic pressure,  is the fluid density, c is the speed of sound, qd
is the Dipole Source, and Qm is the Monopole Source.
In the settings window, define the properties for the acoustics model and model inputs
including temperature.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains to compute the acoustic pressure field and
the equation that defines it, or select All domains as required.
TR A N S I E N T P RE S S U R E A C O U S T I C S M O D E L
See Pressure Acoustics Model for details of the fluid model equations.
Select a Fluid model—Linear elastic (the default), Viscous, Thermally conducting,
Thermally conducting or viscous, or Ideal Gas. Then see the descriptions for The Pressure
Acoustics, Frequency Domain User Interface:
• Defining a Linear Elastic Fluid Model
• Defining a Viscous Fluid Model
• Defining a Thermally Conducting Fluid Model
• Defining a Thermally Conducting and Viscous Fluid Model
• Defining an Ideal Gas Fluid Model
MODEL INPUTS (IDEAL GAS ONLY)
If Ideal gas is selected as the Fluid model, enter a Temperature T (which can be a constant
temperature or a temperature field from a heat transfer interface) and an Absolute
pressure pA:
• Select User defined to enter a value or an expression for the absolute pressure
(SI unit: Pa) in the field that appears. This input is always available.
• In addition, select a pressure defined by a fluid-flow interface present in the model
(if any). For example, then select Pressure (spf/fp1) to use the pressure defined by the
Fluid Properties node fp1 in a Single-Phase Flow interface spf. Selecting a pressure
variable also activates a check box for defining the reference pressure, where 1
[atm] has been automatically included. This allows the use of a system-based
(gauge) pressure, while automatically including the reference pressure in the
absolute pressure.
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97
The Gaussian Pulse Source Type
In transient models the Gaussian pulse exists as a source type in a Line Source, Line
Source on Axis, Monopole Point Source, Point Source, and Circular Source. This type
adds a source with a Gaussian time profile defined in terms of its amplitude A, its
frequency bandwidth f0, and the pulse peak time tp. Using this source type results in
solving a wave equation of the type:
2
pt
4
1 - ----------------- +   –1
---  p t – q d  = ------ S  n   r – r 0 
2
2


c t
2 2
2
2 2
2
– f0  t – tp 
– f0  t – tp 
 
 2
S = ------ -----  Ae
 = – A  --- f 0  t – t p e
4 t
2
where S is the source strength and the superscript n in the delta function depends on
the dimension, n = 2 in 2D and n = 3 in 3D models.
In 3D models (for Line Source), in 2D axisymmetric models (for Line Source on Axis
and Circular Source), and 2D models (for Point Source), all of which are effectively
line sources, enter the following values or expressions:
• The value of the pulse Amplitude A (SI unit: m2/s).
• Frequency bandwidth f0 (SI unit: Hz).
• Pulse peak time tp (SI unit: s) for the duration of the pulse.
In 3D and 2D axisymmetric models for the Monopole Point Source, enter the
following values or expressions:
• The value of the pulse Amplitude A (SI unit: m3/s).
• Frequency bandwidth f0 (SI unit: Hz).
• Pulse peak time tp (SI unit: s) for the duration of the pulse.
Transient Gaussian Explosion: Model Library path Acoustics_Module/
Tutorial_Models/gaussian_explosion
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
The Boundary Mode Acoustics User
Interface
The Boundary Mode Acoustics (acbm) user interface (
), found under the
Acoustics>Pressure Acoustics branch (
) in the Model Wizard, has the equations,
boundary conditions, and sources for modeling boundary mode acoustics, solving for
the sound pressure and the propagation constant. The interface is useful for analysis of
transverse acoustic modes in, for example, waveguides and ducts.
When this interface is added, these default nodes are also added to the Model Builder—
Pressure Acoustics Model, Sound Hard Boundary (Wall), and Initial Values. Right-click the
Boundary Mode Acoustics node to add other features that implement boundary
conditions and sources.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is acbm.
BOUNDARY SELECTION
The default setting is to include All boundaries in the model to define a sound pressure
field and the associated acoustic boundary mode equation. To choose specific
boundaries, select Manual from the Selection list.
THE BOUNDARY MODE ACOUSTICS USER INTERFACE
|
99
EQUATION
For 2D axisymmetric models, the Circumferential wave number m is by
default 0. It is an integer entering the axisymmetric expression for the
pressure:
p  r z   = p  r  e
– i  k z z + m 
Change the value as required. Also see Physics Nodes—Equation Section
in the COMSOL Multiphysics Reference Manual.
SOUND PRESSURE LEVEL SETTINGS
See the settings for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
DEPENDENT VARIABLES
This interface defines one dependent variable (field), the Pressure p. The name can be
changed but the names of fields and dependent variables must be unique within a
model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Quadratic (the default), Linear, Cubic, or Quartic for the Pressure. Specify the Value type
when using splitting of complex variables—Real or Complex (the default).
• Show More Physics Options
• Initial Values
• Boundary, Edge, Point, and Pair Nodes for the Boundary Mode
Acoustics User Interface
• Boundary Mode Acoustics Equations
Initial Values
The Initial Values node adds initial values for the sound pressure. Right-click to add
more Initial Values nodes.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter a value or expression for the Pressure p (SI unit: Pa) initial value. The default is
0 Pa.
Special post-processing variables exist for the Boundary Mode Acoustics
interface. They are described in Boundary Mode Acoustics Variables.
Boundary, Edge, Point, and Pair Nodes for the Boundary Mode
Acoustics User Interface
Except for Initial Values, The Boundary Mode Acoustics User Interface
shares all of its feature nodes with the Pressure Acoustics, Frequency
Domain interface. See Domain, Boundary, Edge, Point, and Pair Nodes
for the Pressure Acoustics, Frequency Domain User Interface.
Also, for the Boundary Mode Acoustics interface, apply the features to
boundaries instead of domains for 3D models.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
THE BOUNDARY MODE ACOUSTICS USER INTERFACE
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101
Theory Background for the Pressure
Acoustics Branch
This section describes the governing equations and the mathematical formulation of
the governing equations as used in the Pressure Acoustics branch of the Acoustics
Module. Details are also given regarding some of the boundary conditions, among
which the radiation boundary conditions as well as the far-field calculation feature. The
section starts with a general introduction to the governing equations used in pressure
acoustics.
In this sections:
• The Governing Equations
• Pressure Acoustics, Frequency Domain Equations
• Pressure Acoustics, Transient Equations
• Boundary Mode Acoustics Equations
• Theory for the Plane, Spherical, and Cylindrical Radiation Boundary Conditions
• Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff Integral
• References for the Pressure Acoustics Branch
The Governing Equations
Pressure acoustic problems involve solving for the small acoustic pressure variations p
on top of the stationary background pressure p0. Mathematically, this represents a
linearization (small parameter expansion) of the dependent variables around the
stationary quiescent values.
The governing equations for a compressible lossless (no thermal conduction and no
viscosity) fluid flow problem are the momentum conservation equation (Euler's
equation) and the mass conservation equation (continuity equation). These are given
by:
u1
-----+  u   u = – --- p
t

----+    u  = 0
t
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
where  is the total density, p is the total pressure, and u is the velocity field. In classical
pressure acoustics, all thermodynamic processes are assumed to be reversible and
adiabatic, that is, isentropic processes. The small parameter expansion is performed on
a stationary fluid (u0 = 0) of density 0 (SI unit: kg/m3) and at pressure p0 (SI unit:
Pa) such that:
p = p 0 + p'
 =  0 + '
with
u = 0 + u'
p' « p 0
' «  0
where the primed variables represent the small acoustic variations. Inserting these into
the governing equations and only retaining terms linear in the primed variables yields:
u'
1
-------- = – ----- p'
t
0
'
------- +  0    u'  = 0
t
(4-1)
One of the dependent variables, the density, is removed by expressing it in terms of the
pressure using a Taylor expansion (linearization) in the small parameters:
 0
1
' = --------- p' = ----2- p'
p s
cs
(4-2)
where cs is recognized as the (isentropic) speed of sound (SI unit: m/s) at constant
entropy s. This expression gives a useful condition that needs to be fulfilled for the
linear acoustic equations to hold:
2
p' «  0 c s
The subscript s is dropped in the following a long with the subscript 0 on the
background density 0. Finally, rearranging Equation 4-1 and Equation 4-2
(divergence of momentum equation inserted into the continuity equation) and
dropping the primes yields the wave equation for sound waves in a lossless medium:
2
1-  p
1
------+    – ---  p – q d  = Q m
2
2

c  t
(4-3)
Here  (SI unit: kg/m3) refers to the density, and c (SI unit: m/s) denotes the speed
of sound. The equation is further extended with two optional source terms: The dipole
source qd (SI unit: N/m3) and the monopole source Qm (SI unit: 1/s2).
THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH
|
103
The combination c2 is called the adiabatic bulk modulus, commonly denoted K (SI
unit: Pa). The bulk modulus is equal to the one over the adiabatic compressibility
coefficient 1/K (SI unit: 1/Pa).
In Equation 4-3 both the speed of sound c = c(x) and the density (x)may be
dependent on the spatial coordinates x while they are independent of time, or only
slowly varying.
Some classical references on acoustics for further reading are found in
Ref. 4, Ref. 5, Ref. 6, Ref. 7, and Ref. 8.
An important special case is a time-harmonic wave, for which the pressure varies with
time as
p  x t  = p  x  e
i t
where = 2f (rad/s) is the angular frequency and f (SI unit: Hz) is the frequency.
Assuming the same harmonic time dependence for the source terms, the wave
equation for acoustic waves reduces to an inhomogeneous Helmholtz equation:
2
1
 p
   – -----  p – q d  – ------------2- = Q m
 c

c cc
(4-4)
In this equation the subscript c on the density and the speed of sound refers to that
they may be complex valued. Lossy media, like porous materials or highly viscous
fluids, can be modeled by using the complex valued speed of sound and density. A
selection of such fluid models is available in The Pressure Acoustics, Frequency
Domain User Interface. The attenuation in these fluid models is frequency dependent
in different ways, depending on the physical origin of the damping. A description of
the different fluid models is given in Theory for the Pressure Acoustics Fluid Models.
In the time domain, only certain frequency dependencies can be modeled, which limits
the number of fluid models that can be used in The Pressure Acoustics, Transient User
Interface. One way to model damping in the time domain is to introduce an additional
term of first order in the time derivative to account for attenuation of the sound waves:
2
1-  p
p
1
------– d a +    – ---  p – q d  = Q m
2
2
 

t
 c t
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
(4-5)
The damping term in Equation 4-5 is absent from the standard PDE formulations in
the Pressure Acoustics, Transient interface, but it corresponds to a monopole source
proportional to the time derivative of the pressure. This approach is, however, not used
in the viscous and thermally conducting fluid models for transient acoustics (see About
the Viscous Fluid Model, About the Thermally Conducting Fluid Model, and About
the Thermally Conducting and Viscous Fluid Model). The damping is here introduced
via a dipole like source.
Even when sound waves propagate in a lossless medium, attenuation can
occur by interaction with the surroundings at the system boundaries. In
particular, this applies to the impedance boundary conditions.
Alternatively. treat the Helmholtz Equation 4-4 as an eigenvalue PDE to solve for
eigenmodes and eigenfrequencies, see the Eigenfrequency and Mode Analysis in 2D
and 1D axisymmetric sections below.
In order to solve the governing equations, boundary conditions are necessary. Typical
boundary conditions used in acoustics are:
• Sound Hard Boundary (Wall)
• Sound Soft Boundary (zero acoustic pressure)
• Specified acoustic Pressure
• Specified Normal Acceleration
• Impedance boundary conditions
• Plane Wave Radiation
• Spherical Wave Radiation
• Cylindrical Wave Radiation
Pressure Acoustics, Frequency Domain Equations
The Pressure Acoustics, Frequency Domain User Interface exists for several types of
studies. Here the equations are presented for the frequency domain, eigenfrequency,
and modal studies. All the interfaces solve for the acoustic pressure p. It is available in
all space dimensions—for 3D, 2D, and 1D Cartesian geometries as well as for 2D and
1D axisymmetric geometries.
THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH
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105
FREQUENCY DOMAIN
The frequency domain, or time-harmonic, formulation uses the inhomogeneous
Helmholtz equation:
2
k eq p t
1
   – -----  p t – q d  – ------------- = Q m
 c

c
(4-6)
This is Equation 4-4 repeated with the introduction of the wave number keq used in
the equations. It contains both the ordinary wave number k as well as out-of-plane and
circumferential contributions, when applicable. Note also that the pressure is here the
total pressure pt which is the sum of a possible Background Pressure Field pb and the
scattered field ps. This enables for a so-called scattered field formulation of the
equations. If no background field is present pt = ps = p.
In this equation, p  p (x,) = p(x)eit (the dependence on  is henceforth not
explicitly indicated). Compute the frequency response by doing a parametric sweep
over a frequency range using harmonic loads and sources.
When there is damping, c and cc are complex-valued quantities. The available
damping models and how to apply them is described in the sections Pressure Acoustics
Model and Theory for the Pressure Acoustics Fluid Models.
Equation 4-6 is the equation that the software solves for 3D geometries. In
lower-dimensional and axisymmetric cases, restrictions on the coordinate dependence
mean that the equations differ from case to case. Here is a brief summary of the
situation.
2D
In 2D, the pressure is of the form
p  r  = p  x y e
– i kz z
which inserted in Equation 4-6 gives
2
k eq
1
   – -----  p t – q d  – -------- p t = Q m
 c
 c
2
k eq
(4-7)
 22
= -----– kz
c c2
The out-of-plane wave number kz can be set on the Pressure Acoustics page. By
default its value is 0. In the mode analysis type ikz is used as the eigenvalue .
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2D Axisymmetry
For 2D axisymmetric geometries the independent variables are the radial coordinate r
and the axial coordinate z. The only dependence allowed on the azimuthal coordinate
 is through a phase factor,
p  r  z  = p  r z e
– im
(4-8)
where m denotes the circumferential wave number. Because the azimuthal
coordinate is periodic m must be an integer. Just like kz in the 2D case, m can be set
on the Pressure Acoustics settings window.
As a result of Equation 4-8, the equation to solve for the acoustic pressure in 2D
axisymmetric geometries becomes
2
k eq

1 p
 ---r p
– -  – q r + r
– -----  – q z – -------- rp = rQm




 z c  z
 r c  r
c
 2 m 2
2
k eq =  ------ –  -----
 c c
 r
1D Axisymmetry
In 1D axisymmetric geometries,
p  r  z  = p  r e
– i  kz z + m 
leading to the radial equation
2
k eq

r p
– -----  – q r – -------- rp = rQ m
 r c   r
c
 2 m
2
2
k eq =  ------ –  ----- – k z
cc
r
2
where both the circumferential wave number m, and the axial wave number kz,
appear as parameters.
1D
The equation for the 1D case is obtained by letting the pressure depend on a single
Cartesian coordinate x:
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107
2
k eq
d  1  dp
– -----  – q d  – -------- p = Q m

d x c d x
c
 2
2
k eq =  ------
cc
EIGENFREQUENCY
In the eigenfrequency formulation, the source terms are absent; the eigenmodes and
eigenfrequencies are solved for:
2
1
 p
   – ----- p + ----------2- = 0
 c 
c cc
(4-9)
The eigenvalue  introduced in this equation is related to the eigenfrequency f, and
the angular frequency , through i2fi. Because they are independent of the
pressure, the solver ignores any dipole and monopole sources unless a coupled
eigenvalue problem is being solved.
Equation 4-9 applies to the 3D case. The equations solved in eigenfrequency studies
in lower dimensions and for axisymmetric geometries are obtained from their
time-harmonic counterparts, given in the previous subsection, by the substitution
22.
Switch between specifying the eigenvalues, the eigenfrequencies, or the angular
frequencies by selecting from the Eigenvalue transformation list in the solver
sequence’s Eigenvalue feature node’s settings window.
Vibrations of a Disk Backed by an Air-Filled Cylinder: Model Library path
Acoustics_Module/Verification_Models/coupled_vibrations_acsh
MODE ANALYSIS IN 2D AND 1D AXISYMMETRIC
See Mode Analysis Study in the Boundary Mode Acoustics Equations section. The
mode analysis study type is only available for the Pressure Acoustics, Frequency
Domain interface in 2D and 1D axisymmetric models. Where the solver solves for the
eigenvalues =ikz for a given frequency. Here kz is the out-of-plane wave number
of a given mode and the resulting pressure field p represents the mode on the cross
section of an infinite wave guide or duct.
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Pressure Acoustics, Transient Equations
Use the Time Dependent study type to model transient acoustic phenomena in a
stationary fluid and to solve the wave equation
2
1 - --------p1
-------+    – ---  p – q d  = Q m
 

c 2 t 2
for the acoustic pressure, p  px, t. Here c is the speed of sound and  denotes the
equilibrium density, while qd and Qm are dipole and monopole sources, respectively.
The density and speed of sound can both be non constant in space. In contrast, they
are assumed to vary with time on scales much larger than the period for the acoustic
waves and are therefore considered time independent in the previous equation. This
interface is available for 3D, 2D, and 1D Cartesian geometries as well as for 2D and
1D axisymmetric geometries.
Boundary Mode Acoustics Equations
When an acoustic wave of a given angular frequency  is fed into a waveguide or a duct,
only a finite set of shapes, or modes, for the transverse pressure field can propagate over
long distances inside the structure. The higher the frequency, the higher the number
of sustainable modes.
Take, as an example, a uniform straight duct whose axis is in the z-direction. The
acoustic field in such a duct can be written as a sum of the form
N
p r =
 pj  x y e
– i k zj z
j=0
The constant kzj is the axial wave number of the jth propagating transverse mode,
pj(x, y). These transverse modes and their associated axial wave numbers are solutions
to an eigenvalue problem defined on the duct’s cross section. The mode analysis
capabilities in The Boundary Mode Acoustics User Interface makes it possible to solve
such eigenvalue problems. The interface is available for 3D Cartesian and 2D
axisymmetric geometries and solves for the transverse eigenmodes for the acoustic
pressure p and the associated propagation constants kz. The Mode Analysis Study is
briefly discussed.
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109
MODE ANALYSIS STUDY
The eigenvalue solver computes a specified number of solutions pj, j to the
equation
2
  2 k n
1
   – -----  p – q d  –  ----------2- – ------ p = Q m
c
 c cc c
(4-10)
defined on a 2D boundary of the modeling domain (in 3D) or on the 2D domain
itself, with =ikn as the eigenvalue. In this equation, p is the in-plane pressure, c is
the density, cc is the speed of sound,  is the angular frequency, and kn is the
propagation constant in the direction normal to the surface, in this context also
referred to as the out-of-plane wave number.
The out-of-plane wave number is denoted kn, and is in the normal
direction to the two-dimensional surface on which Equation 4-10. As for
a mode analysis study in the frequency domain the propagation direction
does not necessarily have to be normal to the z-axis for 3D geometries.
Special post-processing variables exist for the Boundary Mode Acoustics
interface. They are described in Boundary Mode Acoustics Variables.
Notice that the above equation is identical to the time-harmonic equation for pressure
acoustics, except that kn is interpreted as an eigenvalue and not as a parameter.
For axisymmetric geometries, the relevant eigenvalue equation to solve for the radial
pressure modes and the eigenvalues  is
d  ---r- dp
 2
2 m 2 rp
+  ------ +  –  ----- ------ = 0




 r  c
d r c d r
cc
Here m, the circumferential wave number, is an integer-valued parameter. The
equation is defined on the interval r1rr2. The eigenvalue  is defined in terms of
the axial wave number kz through the equation ikz
Absorptive Muffler: Model Library path Acoustics_Module/
Industrial_Models/absorptive_muffler
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Theory for the Plane, Spherical, and Cylindrical Radiation Boundary
Conditions
Specify a Plane Wave Radiation, Spherical Wave Radiation, or Cylindrical Wave
Radiation boundary condition to allow an outgoing wave to leave the modeling
domain with minimal reflections. The condition can be adapted to the geometry of the
modeling domain. The plane wave type is suitable for both far-field boundaries and
ports. Because many waveguide structures are only interesting in the plane-wave
region, it is particularly relevant for ports.
Radiation boundary conditions are available for all types of studies. For the Frequency
domain study, Givoli and Neta’s reformulation of the Higdon conditions (Ref. 1) for
plane waves has been implemented to the second order. For cylindrical and spherical
waves, COMSOL Multiphysics uses the corresponding 2nd-order expressions from
Bayliss, Gunzburger, and Turkel (Ref. 2). The Transient, Mode analysis, and
Eigenfrequency studies implement the same expansions to the first order.
The first-order radiation boundary conditions in the frequency domain read
pi
p i
1
p
– n   – -----  p t – q d  +  ik +   r   ----- =  ik +   r   ----- + n   ---------

 c

c
c
c 
where k is the wave number and  ( r ) is a function whose form depends on the wave
type:
• Plane wave: ( r )0
• Cylindrical wave: ( r )1(2 r)
• Spherical wave: ( r )1r
In the cylindrical and spherical wave cases, r is the shortest distance from the point
r(x, y, z) on the boundary to the source. The right-hand side of the equation
represents an optional incoming pressure field pi (see Incident Pressure Field).
The second-order radiation boundary conditions in the frequency domain are defined
below. In these equations, T at a given point on the boundary denotes the Laplace
operator in the tangent plane at that particular point.
PLANE WAVE
i
k
i
k
1
1
– n   – -----  p t – q d  + i ----- p + ------------  T p = ------------  T p i + i ----- p i + n  ----- p i
c
2k c
c
2k c
c
c
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111
In the notation of Givoli and Neta (Ref. 1), the above expressions correspond to the
parameter choices C0  C1  C2  /k. For normally incident waves, this gives a
vanishing reflection coefficient.
C Y L I N D R I C A L WAVE
1
– n   – ----- p t – q d =
 c

 pi – p 
 r T p i – r T p 
1- ---------------------------------1
1 ik + ----- ------------------ + n  ---–
p i + ---------------------------------------- eq 2r 8r  1 + ik eq r   c
2  1 + ik eq r 
c
c
The cylindrical wave boundary condition is based on a series expansion of the
outgoing wave in cylindrical coordinates (Ref. 2), and it assumes that the field is
independent of the axial coordinate. Specify the axis of this coordinate system by giving
an orientation (nx, ny, nz) and a point (x0, y0, z0) on the axis. In axisymmetric
geometries, the symmetry axis is the natural and only choice.
SPHERICAL WAVE
r T p
1
1 p
– n   – -----  p t – q d  +  ik eq + --- ----- – -------------------------------------
c
r   c 2 c  ik eq r + 1 
r T p i
1
1 pi
= – ----------------------------------------- +  ik eq + --- ----- + n  ----- p i

2 0c  ik eq r + 1 
r  c
c
Use a spherical wave to allow a radiated or scattered wave—emanating from an object
centered at the point (x0, y0, z0) that is specified—to leave the modeling domain
without reflections. The boundary condition is based on an expansion in spherical
coordinates from Bayliss, Gunzburger, and Turkel (Ref. 2), implemented to the
second order.
TR A N S I E N T A N A L Y S I S
The transient radiation boundary condition is the first-order expression
1 1 p
1 1 p i
1
– n    – ---  p t – q d  + ---  --- +   r  p = ---  --+   r  p i + n  p i

 c t
 c t
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
where  ( r ) is the same wave-type dependent function as for the eigenfrequency case
and pi the optional Incident Pressure Field.
An estimate of the refection coefficient Rs for spurious waves off the plane
wave radiation boundary, for incident plane waves at angle  is given by
the expression:
cos  – 1- N
R s = --------------------cos  + 1
where N is the order of the boundary condition (here 1 or 2). So at
normal incidence (= 0) there are no spurious refections, while, for
example, at an incidence angle of 30o for N = 2 (plane wave radiation in
the frequency domain) the amplitude of the spurious reflected wave is
0.5 % of the incident.
Theory for the Far-Field Calculation: The Helmholtz-Kirchhoff
Integral
The Acoustics Module has functionality to evaluate the acoustic pressure field in the
far-field region. This section gives the relevant definitions and mathematical
background as well as some general advice for analyzing the far field. Details about
how to use the far-field functionality is described in Far-Field Calculation.
THE NEAR-FIELD AND FAR-FIELD REGIONS
The solution domain for a scattering or radiation problem can be divided into two
zones, reflecting the behavior of the solution at various distances from objects and
sources. In the far-field region, scattered or emitted waves are locally planar, velocity
and pressure are in phase with each other, and the ratio between pressure and velocity
approaches the free-space impedance of a plane wave.
Moving closer to the sources into the near-field region, pressure and velocity
gradually slide out of phase. This means that the acoustic field contains energy that
does not travel outward or radiate. These evanescent wave components are effectively
trapped close to the source. Looking at the sound pressure level, local maxima and
minima are apparent in the near-field region.
Naturally, the boundary between the near-field and far-field regions is not sharp. A
general guideline is that the far-field region is that beyond the last local energy
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113
maximum, that is, the region where the pressure amplitude drops monotonously at a
rate inversely proportional to the distance from any source or object, R.
A similar definition of the far-field region is the region where the radiation pattern—
the locations of local minima and maxima in space—is independent of the distance to
the wave source. This is equivalent to the criterion for Fraunhofer diffraction in optics,
which occurs for Fresnel numbers, Fa2/R, much smaller than 1. For engineering
purposes, this definition of the far-field region can be applied:
8a 2
8
R  ---------- = ------ ka 2

2
(4-11)
In Equation 4-11, a is the radius of a sphere enclosing all objects and sources,  is the
wavelength, and k is the wave number. Another way to write the expression leads to
the useful observation that the size of the near-field region, expressed in source-radius
units, is proportional to the dimensionless number k a, with a prefactor slightly larger
than one.
Knowing the extent of the near-field region is useful when applying radiation
boundary conditions because these are accurate only in the far-field region. Perfectly
matched layers (PMLs), on the other hand, can be used to truncate a domain already
inside the near-field region.
THE HELMHOLTZ-KIRCHHOFF INTEGRAL REPRESENTATION
In many cases, solving the acoustic Helmholtz equation everywhere in the domain
where results are requested is neither practical nor necessary. For homogeneous media,
the solution anywhere outside a closed surface containing all sources can be written as
a boundary integral in terms of quantities evaluated on the surface. To evaluate this
Helmholtz-Kirchhoff integral, it is necessary to know both Dirichlet and Neumann
values on the surface. Applied to acoustics, this means that if the pressure and its
normal derivative (which is related to the normal velocity) is known on a closed
surface, the acoustic field can be calculated at any point outside.
In general, the solution p to Helmholtz’ equation
–   p – k 2 p = 0
in the homogeneous domain exterior to a closed surface, S, can be explicitly expressed
in terms of the values of p and its normal derivative on S:
pR =
  G  R r  p  r  – G  R r  p  r    n dS
S
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
Here the coordinate vector r parameterizes S. The unit vector n is the outward normal
to the exterior infinite domain; thus, n points into the domain that S encloses. The
function G (R, r) is a Green’s function satisfying
–   G  R r  – k 2 G  R r  = 
3
R – r
This essentially means that the Green’s function, seen as a function of r, is an outgoing
traveling wave excited by a simple source at R. In 3D, the Green’s function is
therefore:
e – ik r – R
G  R r  = -----------------------4 r – R
In 2D, the Green’s function contains a Hankel function instead of the exponential:
i 2
G  R r  = --- H 0  k r – R 
4
Inserting the 3D Green’s function in the general representation formula gives:
 1 + ik r – R 
1 e – ik r – R
-  r – R   n dS
p  R  = ------ -----------------------  p  r  + p  r  -----------------------------------
r–R
4
r–R2

(4-12)
S
while in 2D, the Hankel function leads to a slightly different expression:
2
H1  k r – R 

i  2
p  R  = – ---  H 0  k r – R  p  r  + kp  r  -------------------------------------  r – R   n dS (4-13)
4 
r–R


S
For axially symmetric geometries, the full 3D integral must be evaluated. The
Acoustics Module uses an adaptive numerical quadrature in the azimuthal direction on
a fictitious revolved geometry in addition to the standard mesh-based quadrature in
the rz-plane.
To evaluate the full Helmholtz-Kirchhoff integral in Equation 4-12 and
Equation 4-13, use the Full integral option in the settings for the far-field variables.
See Far-Field Calculation.
THE FAR-FIELD LIMIT
The full Helmholtz-Kirchhoff integral gives the pressure at any point at a finite
distance from the source surface, but the numerical integration tends to lose accuracy
at large distances. At the same time, in many applications, the quantity of interest is the
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115
far-field radiation pattern, which can be defined as the limit of r | p | when r goes to
infinity in a given direction.
Taking the limit of Equation 4-12 when | R | goes to infinity and ignoring the rapidly
oscillating phase factor, the far field, pfar is defined as
rR
1 ik -----------R
p far  R  = – ------ e R  p  r  – ikp  r  -------  n dS

4
R

S
The relevant quantity is | pfar| rather than pfar because the phase of the
latter is undefined. For the same reason, only the direction of R is
important, not its magnitude.
Because Hankel functions asymptotically approach exponential, the limiting 2D
integral is remarkably similar to that in the 3D case:
rR
1 – i ik -----------R
p far  R  = -------------- e R  p  r  – ikp  r  -------  n dS
R
4 k

S
For axially symmetric geometries, the azimuthal integral of the limiting 3D case can be
handled analytically, which leads to a rather complicated expression but avoids the
numerical quadrature required in the general case. For the circumferential wave
number m0, the expression is:
zZ
ik ------1
krR
p far  R   – --- re R  J 0  ----------- p  r   n –
 R 
2

S
(4-14)
ikp
r
krR
krR
----------------in RJ 1  ----------- + n z ZJ 0  -----------  dS
 R 
 R 
R  r
In this integral, r and z are the radial and axial components of r, while R and Z are the
radial and axial components of R.
To evaluate the pressure in the far-field limit according to the equations in this section,
use the Integral approximation at r  option in the settings window for the far-field
variables See Far-Field Calculation.
THE ELKERNEL ELEMENT
These integrals can be implemented as integration coupling variables in COMSOL
Multiphysics. However, such an approach is very inefficient because then the simple
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
structure of the integration kernels cannot be exploited. In the Acoustics Module,
convolution integrals of this type are therefore evaluated in optimized codes that hides
all details from the user.
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Theory for the Pressure Acoustics
Fluid Models
In this section:
• Introduction to the Pressure Acoustics Fluid Models
• About the Linear Elastic with Attenuation Fluid Model
• About the Macroscopic Empirical Porous Models
• About the Viscous Fluid Model
• About the Thermally Conducting Fluid Model
• About the Thermally Conducting and Viscous Fluid Model
• About the Biot Equivalent Fluid Models
• About the Boundary-Layer Absorption Fluid Model
Introduction to the Pressure Acoustics Fluid Models
The Pressure Acoustics Model node is used to define the attenuation properties of the
bulk fluid by specifying the fluid model. The fluid model is also known as an equivalent
fluid model as it models losses in a homogenized way. Losses and damping occur when
acoustic waves propagate in a porous material (material refers to the homogenization
of a fluid and a porous solid), because of bulk viscous and thermal properties, or
because of thermal and viscous losses in the acoustic boundary layer at walls in narrow
ducts. The purpose of the fluid model is to mimic a special loss behavior by defining a
complex valued density c and speed of sound cc. These are often frequency
dependent.
The default Linear elastic fluid model (see Defining a Linear Elastic Fluid Model)
enables you to specify a linearly elastic fluid using either the density  and speed of
sound c or the impedance Z and wave number k. When any of these material
parameters are complex valued damping is introduced.
It is possible to directly measure the complex wave number and impedance in an
impedance tube in order to produce curves of the real and imaginary parts (the
resistance and reactance, respectively) as functions of frequency. These data can be used
directly as input to COMSOL Multiphysics interpolation functions to define k and Z.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
The linear elastic fluid model is thus the most general fluid model as all user defined
expressions may be entered here; analytical expressions or measurement data.
The following more specific fluid models are described in this section (settings options
detailed for the Pressure Acoustics Model node). They can be divided into these
categories:
• General fluid:
- Linear elastic
- Linear elastic with attenuation
- Ideal gas (not described here)
• Propagation in porous materials:
- Macroscopic empirical porous model (Delany-Bazely or Miki)
- Biot equivalents (Limp porous matrix or Rigid porous matrix)
• Viscous and thermally conducting fluids:
- Viscous
- Thermally conducting
- Thermally conducting and viscous
• Propagation in narrow tubes or waveguides:
- Boundary-layer absorption
Ideal gas is also available as an option, but is not described here. This fluid
model is used to specify the fluid properties by selecting a gas constant
type and selecting between entering the heat capacity at constant pressure
or the ratio of specific heats. See Defining an Ideal Gas Fluid Model for
details.
About the Linear Elastic with Attenuation Fluid Model
Use the linear elastic attenuation fluid model to specify a linearly elastic fluid with
attenuation using the density and speed of sound and to account for damping of
acoustic waves using an attenuation coefficient . There are different attenuation types
THEORY FOR THE PRESSURE ACOUSTICS FLUID MODELS
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119
to choose from as the fluid model—Attenuation coefficient Np/m, Attenuation
coefficient dB/m, or Attenuation coefficient dB/.
Defining a Linear Elastic with Attenuation Fluid Model
Select Attenuation coefficient Np/m to define an attenuation coefficient in Np/m
(nepers per meter):

k = ---- – i
c

c c = ---k
2
c  c = -------2
cc
Select Attenuation coefficient dB/m to define an attenuation coefficient in dB/m
(decibel per meter):


k = ---- – i ln  10  -----20
c

c c = ---k
2
c  c = -------2
cc
Select Attenuation coefficient dB/ to define an attenuation coefficient in dB/ (decibel
per wavelength):


k = ----  1 – i ln  10  ------------------
c
2  20

c c = ---k
c = 
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
About the Macroscopic Empirical Porous Models
The macroscopic empirical porous model is an equivalent fluid model that mimics the
bulk losses in certain porous/fibrous materials. The model represents a porous
medium with the following complex propagation constants:
 0 f –C2
 0 f –C 4

k c = ---- 1 + C 1  --------
– iC 3  --------
 Rf 
 Rf 
c
 0 f –C6
 0 f –C 8
Z c =  0 c 1 + C 5  --------
– iC 7  --------
 Rf 
 Rf 
Two predefined sets of the coefficients Ci exist, one representing the Delany-Bazley
model, and one set representing the Miki model (see Ref. 11, section 2.5).
There are restrictions on the applicability of the Delany-Bazley model:
• The porosity of the material should be close to 1.
• The value of   0 f   R f should lie between 0.01 and 1.0.
• The flow resistivity Rf should lie between 1000 and 50,000 Pa·s/m2.
Defining Macroscopic Empirical Porous Fluid Models
Absorptive Muffler: Model Library path Acoustics_Module/
Industrial_Models/absorptive_muffler
About the Viscous Fluid Model
The viscous model is an equivalent-fluid model that mimics the propagation of sound
in a fluid including viscous losses occurring in the bulk of the fluid. The elastic fluid
model with viscous losses is defined by:
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ib –1
 c =   1 + ---------2-


c
1
---
ib 2
c c = c  1 + ---------2-


c
4
b =  ---  +  B
3

where  is the dynamic viscosity and B is the bulk viscosity (see Ref. 4 or Ref. 10
chapter 9). This choice is only appropriate for situations where the damping takes place
in free space and is not related to interaction between the fluid and a solid skeleton or
a wall. These losses, in most fluids, occur over long distances or at very high
frequencies.
Defining a Viscous Fluid Model
About the Thermally Conducting Fluid Model
The thermally conducting model is an equivalent-fluid model that mimics the
propagation of sound in a fluid including losses due to thermal conduction in the bulk.
The elastic fluid model with thermal losses is defined by:
ib –1
 c =   1 + ---------2-


c
1
---
ib 2
c c = c  1 + ---------2-


c
  – 1 k
b =  --------------------
 Cp 
where  is the ratio of specific heats, Cp is the specific heat at constant pressure, and k
is the thermal conductivity (see Ref. 10 chapter 9). This choice is only appropriate for
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
situations where the damping takes place in free space and is not related to interaction
between the fluid and a solid skeleton or a wall.
Defining a Thermally Conducting Fluid Model
About the Thermally Conducting and Viscous Fluid Model
The thermally conducting and viscous model is an equivalent-fluid model that mimics
the propagation of sound in a fluid including losses due to thermal conduction and
viscosity in the bulk of the fluid. The elastic fluid model with thermal and viscous losses
is defined by:
ib – 1
 c =   1 + ---------2-


c
1
---
ib 2
c c = c  1 + ---------2-


c
4
  – 1 k
b =   ---  +  B + --------------------

 3
Cp 
where  is the dynamic viscosity and B is the bulk viscosity,  is the ratio of specific
heats, Cp is the specific heat at constant pressure, and k is the thermal conductivity (see
Ref. 10 chapter 9). This choice is only appropriate for situations where the damping
takes place in free space and is not related to interaction between the fluid and a solid
skeleton or a wall.
Defining a Thermally Conducting and Viscous Fluid Model
About the Biot Equivalent Fluid Models
The Biot equivalent fluid models are models that mimic two limiting behaviors of the
full Poroelastic Material model. The first is the limp porous matrix model and the
second is the rigid porous matrix model. These are both equivalent fluid models and
are thus computationally less demanding than the full poroelastic model. However,
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123
they are only physically correct for certain choices of material parameters. Both models
are based on describing the, frequency dependent, effective density () and the
effective bulk modulus K() of the saturating fluid inside the porous matrix.
LIMP AND RIGID PO RO US MATRIX MODELS
The limp porous matrix model models materials whose solid phases (the porous
matrix) are so weak that they cannot support free, structure-borne wave propagation
(neither longitudinal nor transverse). That is, their drained bulk stiffness (in vacuo bulk
stiffness) is very small compared to air such that the solid phase motion becomes
acoustically significant. If it is light enough, the solid phase still moves because it is
“dragged along” by the fluid motion. So, a limp porous material model is also an
equivalent fluid model because it only features a single longitudinal wave type.
Typically, the limp model can be used to model very light weight fibrous materials (say
less than 10 kg/m3) if they are not specifically stiffened by the injection of binder
material.
The rigid porous matrix model is at the opposite end of the limp model, in that the
matrix is assumed to be so stiff that it does not move (sometimes referred to as a
motionless skeleton model). The present rigid porous model is also often referred to
as the Johnson-Champoux-Allard (JCA) model (see Ref. 11 section 5.5.1).
The limp (subscript “limp”) and rigid (subscript “rig”) porous matrix models are
defined by the following equivalent densities, ()and equivalent bulk moduli, K():
2
 rig
 f
Rf p
4i  f
= ------- 1 + -------------- 1 + -----------------------2 2 2
p
i f 
R L 
f
K eq
v p
2

P 0
iL th Pr f
8
--------------------------= ---------  –   – 1   1 + --------------------------+
1
2

p
16 
iL th Pr f


–1 –1
2
 rig  av –  f
 limp = ---------------------------------- av +  rig – 2 f
 av =  d +  p  f
1 8
L v = --- ----------s p Rf
2V p
L th = ----------  2L v
Sp
Here  is the tortuosity factor, f is the fluid density, p is the porosity, Rf is the flow
resistivity,  is the dynamic viscosity, P0 is the quiescent pressure,  is the ratio of
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
specific heats, Lv is the viscous characteristic length, Lth is the thermal characteristic
length, Pr is the Prandtl number, d is the drained porous matrix density, Sp and Vp
are the surface area and volume of the pores, s is a pore geometry dependent factor
between 0.3 and 3.0 (for example 1 for circular pores, 0.78 for slits), av is the average
effective density, and limp is the effective limp density. See Ref. 11 and Chapter 5 in
Ref. 12 for further details.
This results in the following propagation parameters
k c, limp =   limp  K eq
k c, rig =   rig  K eq

c c = ----kc
c0
 c =  f  -----
cc
2
 2
2
2
k eq =  ---- – k z
cc
The expression given for the geometry dependent pore factor s is only
valid for values of s close to 1. If this is not the case it is possible to enter
the viscous characteristic length Lv directly into the model.
Defining a Biot Equivalent Fluid Model
Porous Absorber: Model Library path Acoustics_Module/Industrial_Models/
porous_absorber
About the Boundary-Layer Absorption Fluid Model
The boundary-layer absorption fluid models are used to mimic the thermal and viscous
losses that exist in narrow tubes where the tube cross-section length-scale is
THEORY FOR THE PRESSURE ACOUSTICS FLUID MODELS
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125
comparable to the thermal and viscous boundary layer thickness. Including these losses
is essential in order to get correct results.
The boundary-layer absorption models are commonly used in situations where solving
a full detailed thermoacoustic model becomes computationally costly. This is, for
example, the case in long narrow ducts/tubes of constant cross section where it is
possible to add or smear the losses associated with the boundary layer onto the bulk of
the fluid—this is an equivalent fluid model.
In more complex geometries where thermal and viscous losses are
important, see The Thermoacoustics, Frequency Domain User Interface,
which is more fundamental and detailed.
Two fluid models exist: one for wide ducts, where the duct width is larger than the
acoustic boundary layer thickness, and one for (very) narrow ducts, where the duct
width is much smaller than the acoustic boundary layer thickness.
WIDE DUCTS
For a relatively wide duct, the losses introduced in the acoustic boundary layer may be
studied by adding them as an effective wall shear force. This approach is used in
Blackstock (Ref. 10) and results in equivalent fluid complex wave number kc defined
by

1

B 
k c = ----- -----------------------------  -----  1 + ---- ----
c0
c
2 i
0

1 – B -----i
C p
–1
S
4

Pr = ---------H d = 4 ---B = ------- ---------  1 + -----------
k
C
H d  0
Pr
(4-15)
where Hd is the hydraulic diameter of the duct, S is the duct cross-section area, C is
the duct circumference,  is the dynamic viscosity,  is the ratio of specific heats, Cp is
the specific heat at constant pressure, k is the thermal conductivity, and Pr is the
Prandtl number. For a cylindrical duct example, Hd = 2a, where a is the radius. The
approximation in Equation 4-15 is only valid for systems where the effective radius
Hd/2 is larger than the boundary layer but not so small that mainstream thermal and
viscous losses are important. Thus requiring
2
Hd c0 1
d visc  -------  ------2 ---------2  d visc
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
d visc =
-----
where dvisc is the characteristic thickness of the viscous boundary layer (the viscous
penetration depth), c0 is the speed of sound, and  is the angular frequency.
NARROW DUCTS
In the other limit where the duct diameter is sufficiently small or the frequency
sufficiently low, the boundary layer thickness becomes much larger than the duct cross
section a. This is the case when
2
 0 a
a « d visc  ----------------- « 1

2
 0 a C p
a « d therm  ------------------------ « 1
k
where dtherm is the characteristic thickness of the thermal boundary layer (thermal
penetration depth). In this case, see Pierce (Ref. 5), the system may be seen as
isothermal and the acoustic temperature variation is zero everywhere in the duct T = 0.
In this case the fluid complex wave number kc is defined by
1
---
1
---
  4  2  4 - 2
k c = ------  ----------------2- – i  -----------------
c T   a 
  c 2 a 2
0
Hd
a = ------2
0 T
(4-16)
c0
c T = -----
where cT is the isothermal speed of sound and a is the tube radius. Note that setting
a = Hd/2 is a further approximation. The theory is derived for ducts of circular cross
section and, thus, the model is only applicable for systems with small variations away
from a circular cross section.
For both fluid models, the relation between the complex wave number and the
complex density and speed of sound is given by the usual

c c = ----kc
c0
 c =   -----
cc
2
 2
2
2
k eq =  ---- – k z
cc
Defining a Boundary-Layer Absorption Fluid Model
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127
References for the Pressure Acoustics
Branch
1. D. Givoli and B. Neta, High-order Non-reflecting Boundary Scheme for
Time-dependent Waves, J. Comput. Phys., vol. 186, pp. 24–46, 2004.
2. A. Bayliss, M. Gunzburger, and E. Turkel, Boundary Conditions for the Numerical
Solution of Elliptic Equations in Exterior Regions, SIAM J. Appl. Math., vol. 42,
no. 2, pp. 430–451, 1982.
3. A. B. Bauer, Impedance Theory and Measurements on Porous Acoustic Liners,
J. Aircr., vol. 14, pp. 720–728, 1977.
4. S. Temkin, Elements of Acoustics, Acoustical Society of America, 2001.
5. A. D. Pierce, Acoustics: An Introduction to its Physical Principles and
Applications, Acoustical Society of America (second print), 1991.
6. D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley and Sons, Inc.,
2000.
7. P. M. Morse and K. U. Ignard, Theoretical Acoustics, Princeton University Press,
1986.
8. L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Course of Theoretical Physics,
Volume 6, Butterworth-Heinemann, 2003.
9. S. Temkin, Elements of Acoustics, Acoustical Society of America, 2001.
10. D.T. Blackstock, Fundamentals of Physical Acoustics, John Wiley and Sons, Inc.,
2000.
11. J.F. Allard and N. Atalla, Propagation of Sound in Porous Media, John Wiley and
Sons, Ltd., 2009.
12. R. Panneton, “Comment on the Limp Frame Equivalent Fluid Model for Porous
Media,” J. Acoust. Soc. Am. vol. 122, no. 6, pp. EL217–EL222, 2007.
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CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH
5
Acoustic-Structure Interaction
This chapter describes the multiphysics interfaces that are used for modeling
acoustic-structure interaction. These interfaces are selected from the
Acoustic-Structure Interaction branch (
) in the Model Wizard.
• The Acoustic-Solid Interaction, Frequency Domain User Interface
• The Acoustic-Solid Interaction, Transient User Interface
• The Acoustic-Piezoelectric Interaction, Frequency Domain User Interface
• The Acoustic-Piezoelectric Interaction, Transient User Interface
• The Elastic Waves and Poroelastic Waves User Interfaces
• Theory for the Elastic Waves and Poroelastic Waves User Interfaces
• The Acoustic-Shell Interaction, Frequency Domain User Interface
• The Acoustic-Shell Interaction, Transient User Interface
• The Pipe Acoustics User Interfaces
• Theory for the Pipe Acoustics User Interfaces
129
The Acoustic-Solid Interaction,
Frequency Domain User Interface
The Acoustic-Solid Interaction, Frequency Domain (acsl) user interface (
), found
under the Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard,
combines Pressure Acoustics, Frequency Domain and Solid Mechanics, connecting the
acoustics pressure in a fluid domain with the structural deformation in a solid domain.
Special interface conditions define the fluid loads on the solid domain and the
structural acceleration’s effect on the fluid. Acoustic-structure interaction refers to a
multiphysics phenomenon where the fluid’s pressure causes a fluid load on the solid
domain, and the structural acceleration affects the fluid domain as a normal
acceleration across the fluid-solid boundary.
The interface is available for 3D, 2D, and 2D axisymmetric geometries and has the
capability to model pressure acoustics and solid mechanics in the frequency domain,
including a special acoustic-solid boundary condition for the fluid-solid interaction.
When this interface is added, these default nodes are also added to the Model Builder—
Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Acoustic-Structure Boundary,
Linear Elastic Material, and Initial Values. For 2D axisymmetric models an Axial
Symmetry node is also added.
Right-click the Acoustic-Solid Interaction, Frequency Domain node to add other features
that implement boundary conditions and sources.The following sections provide
information about all nodes specific to this multiphysics interface, but all other
acoustics- and solid mechanics-specific features are found under the Pressure Acoustics
and Solid Mechanics interfaces, respectively.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is acsl.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
DOMAIN SELECTION
The default setting is to include All domains in the model to define the dependent
variables and the equations. To choose specific domains, select Manual from the
Selection list.
SOUND PRESSURE LEVEL SETTINGS
The settings are the same for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
TY P I C A L W AV E S P E E D
Enter a value or expression for the Typical wave speed for perfectly matched layers cref
(SI unit m/s). The default value is the speed of sound in air, 343 m/s.
THICKNESS
For 2D models, define the thickness d by entering a value or expression
(SI unit: m). The default value of 1 m is suitable for plane strain models,
where it represents a a unit-depth slice, for example. In rare cases, when
the thickness needs to be changed in parts of the geometry; then use the
Change Thickness node.
REFERENCE POINT FOR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
The resulting moments (applied or as reactions) are then computed relative to this
reference point. During the results and analysis stage, the coordinates can be changed
in the Parameters section in the result nodes.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Linear, Quadratic (the default), Cubic, or Quartic as the element order for the Pressure
and Displacement field. Specify the Value type when using splitting of complex variables—
Real or Complex (the default).
THE ACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE
|
131
DEPENDENT VARIABLES
This interface defines these dependent variables (fields), the Pressure p and the
Displacement field u and its components. The name can be changed but the names of
fields and dependent variables must be unique within a model.
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Solid Interaction, Frequency Domain User Interface
• Theory for the Solid Mechanics User Interface
• Theory Background for the Pressure Acoustics Branch
• Acoustic-Structure Interaction: Model Library path Acoustics_Module/
Tutorial_Models/acoustic_structure
• Loudspeaker Driver: Model Library path Acoustics_Module/
Industrial_Models/loudspeaker_driver
Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Solid Interaction, Frequency Domain User Interface
Because The Acoustic-Solid Interaction, Frequency Domain User Interface is a
multiphysics interface, almost every node is shared with, and described for, other
interfaces. Below are links to the domain, boundary, edge, point, and pair nodes as
indicated.
Some functionality differs slightly between the pure Pressure Acoustics
interfaces and the functionality in the multiphysics interfaces. This mainly
concerns the acoustic point and edge sources.
The Acoustic-Solid Interaction, Transient User Interface also shares the
same nodes as listed below, with its Initial Values node being the
exception.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
These nodes are described in this section:
• Acoustic-Structure Boundary
• Intensity Edge Source
• Initial Values
• Power Edge Source
• Flow Line Source on Axis
• Intensity Point Source
• Intensity Line Source on Axis
• Power Point Source
• Power Line Source on Axis
These nodes are described for the Pressure Acoustics, Frequency Domain or Pressure
Acoustics, Transient interface (listed in alphabetical order):
• Background Pressure Field
• Matched Boundary
• Continuity
• Monopole Source
• Cylindrical Wave Radiation
• Normal Acceleration
• Dipole Source
• Periodic Condition
• Far-Field Calculation
• Plane Wave Radiation
• Impedance
• Pressure
• Interior Impedance/Pair Impedance
• Pressure Acoustics Model
• Interior Normal Acceleration
• Sound Hard Boundary (Wall)
• Interior Perforated Plate
• Sound Soft Boundary
• Interior Sound Hard Boundary (Wall)
• Spherical Wave Radiation
THE ACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE
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133
These nodes are described for the Solid Mechanics interface (listed in alphabetical
order):
• Added Mass
• Pre-Deformation
• Antisymmetry
• Prescribed Acceleration
• Body Load
• Prescribed Displacement
• Boundary Load
• Prescribed Velocity
• Edge Load
• Roller
• Fixed Constraint
• Spring Foundation
• Free
• Symmetry
• Linear Elastic Material
• Thin Elastic Layer
• Point Load
Acoustic-Structure Boundary
The Acoustic-Structure Boundary node is available for The Acoustic-Solid Interaction,
Frequency Domain User Interface and The Acoustic-Solid Interaction, Transient User
Interface. This boundary condition includes the fluid load and structural acceleration
for use on the fluid-solid boundaries, where it is the default boundary condition.
For the acoustic-solid interaction, this boundary condition is the default on the
boundaries between the fluid and the solid. It is not applicable on other boundaries.
This boundary condition includes the following interaction from fluid to solid and vice
versa:
• A pressure load (force per unit area) Fpn p on the boundaries where the fluid
interacts with the solid. In this expression, n is the outward-pointing unit normal
vector seen from inside the solid domain.
• A structural acceleration acting on the boundaries between the solid and the fluid.
This makes the normal acceleration for the acoustic pressure on the boundary equal
to the acceleration based on the second derivatives of the structural displacements
u with respect to time: ann · utt.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
For the node that contains the acoustic-structure boundary condition as
a default condition, the boundary selection automatically becomes the
boundaries between the fluid domain and the solid domain, and manual
selection is not available.
If additional Acoustic-Structure Boundary nodes are added, which is
normally not necessary, select the boundaries where this condition then
becomes active.
Initial Values
The Initial Values node adds initial values for the sound pressure and the displacement
field. Right-click the interface node to add more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter a value or expression for the initial values for Pressure p (SI unit: Pa) and the
Displacement field u (SI unit: m). The default values are 0.
Flow Line Source on Axis
Use the Flow Line Source on Axis node to add a line source along the
symmetry axis in 2D axisymmetry.
For the Flow Line Source on Axis node, both the source amplitude and its complex phase
can be specified. This is done by defining the source amplitude as a complex number.
This can be useful if there are two or more sources mutually out of phase. The
following term is added to the right-hand side of the equation for the acoustic
pressure:
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|
135
i Q S 
2
 r – r0 
where QS is the source amplitude defined as the flow rate out from the source per unit
length.
BOUNDARY SELECTION
From the Selection list, choose the boundaries on the symmetry axis to define the flow
line source.
FLOW LINE SOURCE ON AXIS
Enter a Flow rate out from source per unit length QS (SI unit: m2/s) for the value for
the source-strength amplitude.
When defining a Solution data set and plotting the results, specify an
imaginary source-strength to produce a nonzero result when visualizing
the resulting pressure field using the default value (0) in the Solution at
angle (phase).
• Frequency Domain Study
• Solution (data sets) in the COMSOL Multiphysics Reference Manual
Intensity Line Source on Axis
Use the Intensity Line Source on Axis node to add a line source located
along the symmetry axis in 2D axisymmetry.
Using the Intensity Line Source on Axis node, set a desired reference root mean square
(RMS) intensity Iref at a specified distance dsrc from the source. In a homogeneous
medium, the specified RMS intensity is obtained (this is the reference), but when other
objects and boundaries are present, the actual intensity is different. The term that this
feature node adds to the right-hand side of the equation is given by
I ref c c  2 
d src
4 ------------- 2 -------------   r – r 0 
L edge
c
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
where Ledge is the length of the source line (automatically determined). The reference
RMS intensity Iref (SI unit: W/m2). The delta function

2
 r – r0 
has the dimension of a length element dl (Si unit m).
BOUNDARY SELECTION
From the Selection list, choose the boundaries on the symmetry axis to define the
intensity line source.
INTENSITY LINE SOURCE ON AXIS
Enter the Reference intensity (RMS) Iref (SI unit: W/m2) of the source and the Distance
from source center dsrc (SI unit: m).
Power Line Source on Axis
Use the Power Line Source on Axis node to add a line source along the
symmetry axis in 2D axisymmetry.
In this case, specify the source’s strength by stating the total power it would radiate
into a homogeneous medium. The Power Line Source on Axis node adds the following
term to the right-hand side of the equation:
L edge P ref c c  2 
2 -----------2 ----------------------------   r – r 0 
c
L edge
where Ledge is the length of the source line (automatically determined) and Pref now
denotes the radiated RMS power (in the reference homogeneous case) per unit length
measured in W/m.
BOUNDARY SELECTION
From the Selection list, choose the boundaries on the symmetry axis to define the
power line source.
POWER LINE SOURCE ON AXIS
Enter the Reference power per unit length (RMS) Pref (SI unit: W/m) radiating from the
line source.
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Intensity Edge Source
Use an Intensity Edge Source node to add an edge source located at rr0 for 3D
models. Set a desired reference RMS intensity Iref at a specified distance dsrc from the
source. In a homogeneous medium the specified intensity is obtained when the edge
is a straight line (this is the reference). With other objects and boundaries present, or
if the edge is curved, the actual intensity is different. This node adds the following term
to the right-hand side of the equation for the acoustic pressure:
d src
I ref c c  2 
4 ------------- 2 -------------   r – r 0 
L edge
c
where Ledge is the length of the source line (automatically determined). The reference
RMS intensity Iref (SI unit: W/m2). The delta function

2
 r – r0 
has the dimension of a length element dl (Si unit m).
EDGE SELECTION
From the Selection list, choose the edges to define.
INTENSITY EDGE SOURCE
Enter the Reference intensity (RMS) Iref (SI unit: W/m2) of the source and the Distance
from source center dsrc (SI unit: m).
Power Edge Source
Use a Power Edge Source node to add an edge source located at rr0 for 3D models.
Specify the source’s reference RMS strength by stating the total power a straight line
source would radiate into a homogeneous medium. The node adds the following term
to the right-hand side of the equation:
L edge P ref c c  2 
2 -----------2 ----------------------------   r – r 0 
c
L edge
where Ledge is the length of the source line (automatically determined) and Pref
denotes the radiated RMS power (in the reference homogeneous case) per unit length
measured in W/m.
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EDGE SELECTION
From the Selection list, choose the edges to define.
POWER EDGE SOURCE
Specify the power radiating from the edge source. Enter the value for the Reference
power (RMS) per unit length Pref (SI unit: W/m).
Intensity Point Source
For 3D and 2D models, use an Intensity Point Source node to add a point
source located at rr0.
For 2D axisymmetric models, it is added on the symmetry axis at r(z,r)
 (z0,
Set a desired reference RMS intensity Iref at a specified distance dsrc from the source.
In a homogeneous medium (the reference) the specified intensity is obtained, but with
other objects and boundaries present the actual intensity is different. The term that this
node adds to the right-hand side of the equation for the acoustic pressure differs
depending on the space dimension.
For 3D and 2D axisymmetric points, the following term defines the source:
I ref c c  3 
4d src 2 -------------   r – r 0 
c
For a point in 2D the corresponding terms is
f I ref d src  2 
4 2 --------------------   r – r 0 
c
POINT S EL EC TION
From the Selection list, choose the points to define.
INTENS IT Y PO INT S OURCE
Enter the Reference intensity (RMS) Iref (SI unit: W/m2) and the Distance from source
center dsrc (SI unit: m).
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Power Point Source
For 3D and 2D models, use a Power Point Source node to add a point
source located at rr0.
For 2D axisymmetric models, it is added on the symmetry axis at r(z,r)
 (z0,
Specify the source’s strength by stating the total RMS power it would radiate into a
homogeneous medium (the reference). The expression for the source differs slightly
between 2D points and 3D points.
For a point in 3D, the feature adds the following term:
2P ref c c  3 
2 ---------------------   r – r0 
c
Here Pref is the reference radiated RMS power (SI unit: W).
For a point in 2D, the feature uses the following term:
2P ref   2 
2 ----------------   r – r0 
c
where Pref now denotes the reference radiated RMS power per unit length measured
in W/m.
POINT SELECTION
From the Selection list, choose the points to define.
POWER PO IN T S OURCE
Specify the power radiating from the point source. In 2D, enter the value
for the Reference power (RMS) per unit length Pref (SI unit: W/m).
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For 3D and 2D axisymmetric models, specify the power radiating from
the point source. Enter the value for the Reference power (RMS), Pref (SI
unit: W).
Hollow Cylinder: Model Library path Acoustics_Module/Tutorial_Models/
hollow_cylinder
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The Acoustic-Solid Interaction,
Transient User Interface
The Acoustic-Solid Interaction, Transient (astd) user interface (
), found under the
Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard, combines
the Pressure Acoustics, Transient and Solid Mechanics interfaces to connect the
acoustics pressure in a fluid domain with the structural deformation in a solid domain.
Special interface conditions define the fluid loads on the solid domain and the
structural acceleration’s effect on the fluid. The interface is available for 3D, 2D, and
2D axisymmetric geometries. Acoustic-structure interaction refers to a multiphysics
phenomenon where the fluid’s pressure causes a fluid load on the solid domain, and
the structural acceleration affects the fluid domain as a normal acceleration across the
fluid-solid boundary.
When this interface is added, these default nodes are also added to the Model Builder—
Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Acoustic-Structure
Boundary, Linear Elastic Material, and Initial Values. For 2D axisymmetric models an
Axial Symmetry node is also added. Right-click the Acoustic-Solid Interaction, Transient
node to add other features that implement boundary conditions and sources.
For modeling of acoustic-structure interaction in the frequency domain,
The Acoustic-Solid Interaction, Frequency Domain User Interface
contains additional functionality that is not applicable for modeling in the
time domain.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
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The default identifier (for the first interface in the model) is astd.
See The Acoustic-Solid Interaction, Frequency Domain User Interface
for the rest of the settings for this interface. Links to all the feature nodes
(except Initial Values in this section) are found in Domain, Boundary,
Edge, Point, and Pair Nodes for the Acoustic-Solid Interaction,
Frequency Domain User Interface.
• Show More Physics Options
• Theory for the Solid Mechanics User Interface
• Theory Background for the Pressure Acoustics Branch
Initial Values
The Initial Values node adds initial values for the pressure, the displacement field, the
structural velocity field and the pressure, first time derivative. Right-click the interface
node to add more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter a value or expression for the initial values.
• Pressure p (SI unit: Pa). The default is 0 Pa.
• Pressure, first time derivative p/t (SI unit: Pa/s). The default is 0 Pa/s.
• Displacement field u (SI unit: m). The default is 0 m.
• Structural velocity field u/t (SI unit: m/s). The default is 0 m/s.
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The Acoustic-Piezoelectric
Inte r a c t i o n, Freq u en c y D om ai n U ser
Interface
The Acoustic-Piezoelectric Interaction, Frequency Domain (acpz) user interface (
),
found in the Model Wizard under the Acoustics>Acoustic-Structure Interaction
branch (
), combines the Pressure Acoustics, Frequency Domain, Solid Mechanics,
Electrostatics, and Piezoelectric Devices interfaces.
When this interface is added, these default nodes are also added to the Model Builder—
Pressure Acoustics Model, Piezoelectric Material, Sound Hard Boundary (Wall), Free,
Acoustic-Structure Boundary, Zero Charge, and Initial Values. For 2D axisymmetric
models an Axial Symmetry node is also added.
Right-click the Acoustic-Piezoelectric Interaction, Frequency Domain node to add other
features that implement boundary conditions and sources.The following sections
provide information about all nodes specific to this multiphysics interface, but all other
acoustics-, solid mechanics-, and electrostatics-specific nodes are found under the
Acoustical, Structural, and Electrical menus, respectively.
The equations solved in the solid and fluid domains can be found in
Theory for the Piezoelectric Devices User Interface and Theory
Background for the Pressure Acoustics Branch, respectively.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is acpz.
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DOMAIN SELECTION
The default setting is to include All domains in the model. To choose specific domains,
select Manual from the Selection list.
SOUND PRESSURE LEVEL SETTINGS
The settings are the same for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
THICKNESS
For 2D models, define the thickness d by entering a value or expression
(SI unit: m). The default value of 1 m is suitable for plane strain models,
where it represents a a unit-depth slice, for example.
In rare cases, if the thickness needs to be changed in parts of the geometry;
then use the Change Thickness node.
REFERENCE POINT FOR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
The resulting moments (applied or as reactions) are then computed relative to this
reference point. During the results and analysis stage, the coordinates can be changed
in the Parameters section in the result nodes.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Linear, Quadratic (the default), Cubic, or Quartic as the element order for the Pressure,
Displacement field, and Electric potential. Specify the Value type when using splitting of
complex variables—Real or Complex (the default).
DEPENDENT VA RIA BLES
This interface defines these dependent variables (fields), the Pressure p, the
Displacement field u and its components, and the Electric potential V. The name can be
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changed but the names of fields and dependent variables must be unique within a
model.
• Show More Physics Options
• Initial Values
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Piezoelectric Interaction, Frequency Domain User Interface
• Theory for the Piezoelectric Devices User Interface
• Theory for the Solid Mechanics User Interface
• Theory Background for the Pressure Acoustics Branch
Piezoacoustic Transducer: Model Library path Acoustics_Module/
Tutorial_Models/piezoacoustic_transducer
Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Piezoelectric Interaction, Frequency Domain User Interface
Because The Acoustic-Piezoelectric Interaction, Frequency Domain User Interface is
a multiphysics interface, almost every node, except Initial Values, is shared with, and
described for, other interfaces. Below are links to the domain, boundary, edge, point,
and pair nodes as indicated.
The Acoustic-Piezoelectric Interaction, Transient User Interface also
shares the same nodes.
The point end edge sources are described in the The Acoustic-Solid
Interaction, Frequency Domain User Interface section.
The continuity condition between the fluid domain and the solid domain
is described in the Acoustic-Structure Boundary section under the
Acoustic-Solid Interaction User Interface.
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To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
ACOUSTICAL MENU
These nodes are described for the Pressure Acoustics, Frequency Domain interface
(listed in alphabetical order):
• Background Pressure Field
• Matched Boundary
• Continuity
• Monopole Source
• Cylindrical Wave Radiation
• Normal Acceleration
• Dipole Source
• Periodic Condition
• Far-Field Calculation
• Pressure
• Impedance
• Pressure Acoustics Model
• Interior Impedance/Pair Impedance
• Plane Wave Radiation
• Interior Normal Acceleration
• Sound Hard Boundary (Wall)
• Interior Perforated Plate
• Sound Soft Boundary
• Interior Sound Hard Boundary (Wall)
• Spherical Wave Radiation
STRUCTURAL MENU
These nodes are described for the Solid Mechanics interface (listed in alphabetical
order):
• Added Mass
• Point Load
• Antisymmetry
• Prescribed Acceleration
• Body Load
• Prescribed Displacement
• Boundary Load
• Prescribed Velocity
• Edge Load
• Roller
• Fixed Constraint
• Spring Foundation
• Free
• Symmetry
• Linear Elastic Material
• Thin Elastic Layer
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PIEZOELECTRIC DEVICES MENU
These nodes are described for the Piezoelectric Devices interface:
• Electrical Material Model
• Piezoelectric Material
ELECTRICAL MENU
These nodes are described for the Electrostatics interface in the COMSOL
Multiphysics Reference Manual (listed in alphabetical order):
• Electric Potential
• Point Charge
• Electric Displacement Field
• Space Charge Density
• Ground
• Thin Low Permittivity Gap
• Line Charge
• Zero Charge
The links to the COMSOL Multiphysics Reference Manual do not work
in the PDF, only from the on line help in COMSOL Multiphysics.
Initial Values
The Initial Values node adds an initial value for the pressure, displacement field, and the
electric potential.Right-click the interface node to add additional Initial Values nodes.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter the initial values as values or expressions for the Pressure p (SI unit: Pa),
Displacement field u (SI unit: m) and the Electric potential V (SI unit: V).
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The Acoustic-Piezoelectric
Interaction, Transient User Interface
The Acoustic-Piezoelectric Interaction, Transient (acpztd) user interface (
), found in
the Model Wizard under the Acoustics>Acoustic-Structure Interaction branch (
),
combines Pressure Acoustics, Transient, Solid Mechanics, Electrostatics, and the
Piezoelectric Devices interface features.
When this interface is added, these default nodes are also added to the Model Builder—
Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Zero Charge,
Acoustic-Structure Boundary, Initial Values, and Piezoelectric Material. For 2D
axisymmetric models an Axial Symmetry node is also added.
Right-click the Acoustic-Piezoelectric Interaction, Transient node to add other nodes
that implement boundary conditions and sources.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is acpztd.
DOMAIN SELECTION
The default setting is to include All domains in the model. To choose specific domains,
select Manual from the Selection list.
SOUND PRESSURE LEVEL SETTINGS
The settings are the same for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
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THICKNESS
For 2D models, define the thickness d by entering a value or expression
(SI unit: m). The default value of 1 m is suitable for plane strain models,
where it represents a a unit-depth slice, for example.
In rare cases, where the thickness needs to be changed in parts of the
geometry; then use the Change Thickness node.
REFERENCE POINT F OR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
The resulting moments (applied or as reactions) are then computed relative to this
reference point. During the results and analysis stage, the coordinates can be changed
in the Parameters section in the result nodes.
TO T A L E N E R G Y A N D L O S S
This section contains the variables Wtot, the total energy (SI unit: J), and Qh,tot, the
total loss (SI unit: W). The default variables are pzd.W_int and pzd.Qh_int,
respectively. A model might contain other components of loss and stored energy,
which do not appear in the integral over the interface. Add those contributions in the
fields for Wtott and Qh,tot to get the quality factor WtotQh,tot correct.
DEPENDENT VARIABLES
This interface defines these dependent variables (fields), the Pressure p, the
Displacement field u and its components, and the Electric potential V. The name can be
changed but the names of fields and dependent variables must be unique within a
model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Linear, Quadratic (the default), Cubic, or Quartic as the element order for the Pressure,
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Displacement field, and Electric potential. Specify the Value type when using splitting of
complex variables—Real or Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Piezoelectric Interaction, Frequency Domain User Interface
• Theory for the Piezoelectric Devices User Interface
• Theory for the Solid Mechanics User Interface
• Theory Background for the Pressure Acoustics Branch
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T he E la s t i c Wav es an d Poroel ast i c
Waves User Interfaces
In this section:
• The Elastic Waves User Interface
• The Poroelastic Waves User Interface
• Domain, Boundary, and Shared Nodes for the Elastic Waves and the Poroelastic
Waves User Interface
The Elastic Waves User Interface
The Elastic Waves (elw) user interface (
), found under the
Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard, combines
Pressure Acoustics and Solid Mechanics to connect the fluid pressure with the
)
structural deformation in solids. It also features the Poroelastic Waves interface (
which can be seen as linear elastic waves in porous elastic materials damped by a pore
fluid. This interface is based on Biot’s theory for porous materials. It has the
capabilities for modeling waves in the frequency domain. The interface is available for
3D, 2D, and 2D axisymmetric geometries.
This interface allows adding fluid or porous domains adjacent to the default solid
domain. Dedicated boundary conditions define the loads between fluid, solid and
porous domains, including special sets of boundary conditions for the fluid-solid,
fluid-porous and porous-solid interactions.
When the Elastic Waves interface is added, these default nodes are also added to the
Model Builder— Linear Elastic Material, Sound Hard Boundary (Wall), Free, Continuity, and
Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added.
When a Poroelastic Waves interface is added, a Poroelastic Waves Material node is added
to the Model Builder, instead of a Linear Elastic Material.
Right-click the Elastic Waves node or the Poroelastic Waves node to add other features
that implement boundary conditions and sources. The following sections provide
information about all nodes specific to these multiphysics interfaces, see details about
other features under the Pressure Acoustics, Frequency Domain or Solid Mechanics
submenus, respectively.
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INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier for the first interface in the model is elw.
DOMAIN SELECTION
The default setting is to include All domains in the model to define the dependent
variables and the equations. To choose specific domains, select Manual from the
Selection list.
SOUND PRESSURE LEVEL SETTINGS
The settings are the same for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
REFERENCE POINT FOR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
The resulting moments (applied or as reactions) are then computed relative to this
reference point. During the results and analysis stage, the coordinates can be changed
in the Parameters section in the result nodes.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Linear, Quadratic (the default), Cubic, or Quartic for the Pressure and Displacement field.
Specify the Value type when using splitting of complex variables—Real or Complex (the
default).
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DEPENDENT VARIABLES
This interface defines these dependent variables (fields), the Pressure p and the
Displacement field u and its components. The name can be changed but the names of
fields and dependent variables must be unique within a model.
• Theory for the Solid Mechanics User Interface
• Theory Background for the Pressure Acoustics Branch
• The Poroelastic Waves User Interface
• Domain, Boundary, and Shared Nodes for the Elastic Waves and the
Poroelastic Waves User Interface
The Poroelastic Waves User Interface
The Poroelastic Waves (elw) user interface (
), found under the
Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard, has most of
the same settings as The Elastic Waves User Interface.
When the Poroelastic Waves interface is added, these default nodes are also added to
the Model Builder— Poroelastic Material, Sound Hard Boundary (Wall), Free, Continuity,
and Initial Values. For 2D axisymmetric models an Axial Symmetry node is also added.
Right-click the Poroelastic Waves node to add other features that implement boundary
conditions and sources. The following sections provide information about all nodes
specific to these multiphysics interfaces, see details about other nodes under the
Pressure Acoustics, Frequency Domain or Solid Mechanics submenus, respectively.
See Domain, Boundary, and Shared Nodes for the Elastic Waves and the
Poroelastic Waves User Interface for all settings and to follow the links for
domain, boundary, pair, edge, and point conditions.
• Theory for the Solid Mechanics User Interface
• Theory Background for the Pressure Acoustics Branch
• The Elastic Waves User Interface
• Domain, Boundary, and Shared Nodes for the Elastic Waves and the
Poroelastic Waves User Interface
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Acoustics of a Particulate-Filter-Like System: Model Library path
Acoustics_Module/Tutorial_Models/acoustics_particulate_filter
Domain, Boundary, and Shared Nodes for the Elastic Waves and the
Poroelastic Waves User Interface
Because The Elastic Waves and Poroelastic Waves User Interfaces are multiphysics
interfaces, almost every node is shared with, and described for, other interfaces. Below
are links to the domain, boundary, edge, pair, and point nodes as indicated.
Some functionality differs slightly between the pure Pressure Acoustics
interfaces and the functionality in the multiphysics interfaces. This mainly
concerns the acoustic point and edge sources, which are described in The
Acoustic-Solid Interaction, Frequency Domain User Interface.
These porous material nodes are described in this section (listed in alphabetical order):
• Continuity
• Porous, Prescribed Acceleration
• Initial Values
• Porous, Prescribed Displacement
• Porous, Fixed Constraint
• Porous, Prescribed Velocity
• Porous, Free
• Porous, Roller
• Poroelastic Material
• Porous, Septum Boundary Load
• Porous, Pressure
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These nodes are described for the Pressure Acoustics, Frequency Domain or Pressure
Acoustics, Transient interface (listed in alphabetical order):
• Background Pressure Field
• Matched Boundary
• Cylindrical Wave Radiation
• Monopole Source
• Dipole Source
• Normal Acceleration
• Far-Field Calculation
• Periodic Condition
• Impedance
• Plane Wave Radiation
• Interior Impedance/Pair
Impedance
• Pressure
• Interior Normal Acceleration
• Spherical Wave Radiation
• Interior Perforated Plate
• Sound Hard Boundary (Wall) (the
default)
• Interior Sound Hard Boundary
(Wall)
• Pressure Acoustics Model
• Sound Soft Boundary
These nodes are described for the Solid Mechanics interface (listed in alphabetical
order):
• Antisymmetry
• Linear Elastic Material
• Body Load
• Point Load
• Boundary Load
• Prescribed Acceleration
• Edge Load
• Prescribed Displacement
• Fixed Constraint
• Prescribed Velocity
• Free (the default boundary condition
for Linear elastic materials)
• Roller
• Symmetry
Destination Selection in the COMSOL Multiphysics Reference Manual
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Initial Values
The Initial Values node adds initial values for the pressure and displacement field.
Right-click the interface node to add more Initial Values nodes.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter a value or expression for the Pressure p (SI unit: Pa) and Displacement field u
(SI unit: m) initial values. The default is 0 Pa for the pressure and 0 m for the
displacement field.
Poroelastic Material
Use the Poroelastic Material node to define the poroelastic material and fluid
properties, that is the properties of the porous matrix and the saturating fluid. Also
right-click the node to add an Initial Stress and Strain node.
DOMAIN SELECTION
From the Selection list, choose the domains to define.
MODEL INPUTS
This section contains field variables that appear as model inputs, if the current settings
include such model inputs. By default, this section is empty. If a linear temperature
relation for the conductivity is added, then define the source for the temperature T.
From the Temperature list, select an existing temperature variable (from another
physics interface) if available, or select User defined to define a value or expression for
the temperature (SI unit: K) in the field that appears underneath the list.
To define the Absolute Pressure, see the settings for the Heat Transfer in
Fluids node as described in the COMSOL Multiphysics Reference
Manual.
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COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
POROELASTIC MATERIAL
The default Porous elastic material uses the Domain material (the material defined for
the domain). Select another material as required.
Select a Porous model—Drained matrix, isotropic, Drained matrix, orthotropic, or Drained
matrix, anisotropic. Then enter or select the settings as described.
Porous Model for Drained Matrix, Isotropic
If Drained matrix, isotropic is selected from the Porous model list, select a pair of elastic
properties to describe an isotropic drained porous material. From the Specify list, select:
• Bulk modulus and shear modulus (the default) to specify the drained bulk modulus K
(SI unit: Pa) and the drained shear modulus G (SI unit: Pa). The bulk drained
modulus is a measure of the solid porous matrix’s resistance to volume changes. The
shear modulus is a measure of the solid porous matrix’s resistance to shear
deformations.
• Young’s modulus and Poisson’s ratio to specify drained Young’s modulus (elastic
modulus) E (SI unit: Pa) and Poisson’s ratio  (dimensionless). For an isotropic
material Young’s modulus is the spring stiffness in Hooke’s law, which in 1D form
is Ewhere  is the stress and  is the strain. Poisson’s ratio defines the normal
strain in the perpendicular direction, generated from a normal strain in the other
direction and follows the equation  = 
• Lamé parameters to specify the drained Lamé parameters  (SI unit: Pa) and 
(SI unit: Pa).
• Pressure-wave and shear-wave speeds to specify the drained pressure-wave speed cp
(SI unit: m/s) and the shear-wave speed cs (SI unit: m/s).
For each pair of properties, select from the applicable list to use the value From material
or enter a User defined value or expression. Each of these pairs define the drained elastic
properties and it is possible to convert from one set of properties to another.
Porous Model for Drained Matrix, Orthotropic
When Drained matrix, orthotropic is selected from the Porous model list, the material
properties of the solid porous matrix vary in orthogonal directions only.
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The default properties take values From material. Select User defined to enter values or
expressions for the drained Young’s modulus E (SI unit: Pa), the drained Poisson’s ratio
 (dimensionless), and the drained Shear modulus G (SI unit: Pa).
Porous Model for Drained Matrix, Anisotropic
When Drained matrix, anisotropic is selected from the Porous model list, the material
properties of the solid porous matrix vary in all directions, and the stiffness comes from
the symmetric Elasticity matrix, D (SI unit: Pa). The default uses values From material.
Select User defined to enter values in the 6-by-6 symmetric matrix that displays.
Parameters for All Porous Models
Enter the following remaining parameters necessary to defined the properties of a
porous material for all the Porous models selected above. The defaults use values From
material. Select User defined to enter other values or expressions as required.
• Drained density of porous material to specify the drained density of the porous
material in vacuum d (SI unit: kg/m3). The drained density d is equal to
(1  p) s where s is the density of the solid material from which the matrix is
made and p is the porosity.
• Permeability to specify the permeability of the porous material p (SI unit: m2). The
permeability is a measure of the ability of the porous material to let fluid pass
through it. It hence gives some measure of the pore size and thus correlates to the
viscous damping experienced by pressure waves propagating in the saturating fluid.
• Porosity to specify the porosity of the material p (dimensionless). It defines the
amount of void volume inside the porous matrix and takes values between 0 (no
porous material only fluid) and 1 (fully solid material no fluid).
• Biot-Willis coefficient to specify the Biot-Willis coefficient B (dimensionless). This
coefficient relates the bulk modulus (compressibility) of the drained porous matrix
to a block of solid material. It is defined as
Kd
 B = 1 – ------Ks
where Kd is the drained bulk modulus (named K here) and Ks is the bulk module
of a block of solid material. The drained bulk modulus is related to the stiffness of
the porous matrix, while the solid bulk modulus is related to the compressibility of
the material or grains from which the porous matrix is made. The Biot-Willis
coefficient is bound by  p   B  1 . A rigid porous matrix (Voigt upper bound) has
 B =  p and a soft or limp porous matrix (Reuss lower bound) has  B = 1 .
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• Tortuosity factor or the structural form factor  (dimensionless). This is a purely
geometrical factor that depends on the microscopic geometry and distribution of
the pores inside the porous material. It is independent of the fluid and solid
properties and is normally >1. The default is 2. The more complex the propagation
path through the material, the higher is the absorption. The tortuosity partly
represents this complexity.
FLUID PROPERTIES
Define the properties of the saturating fluid in terms of its density, viscosity and
compressibility but also the viscosity model. The defaults use values for the material
parameters are From material. Select User defined to enter other values or expressions
as required.
• Density defines the density of the saturating fluid f (SI unit: kg/m3).
• Dynamic viscosity to define the dynamic viscosity of the saturating fluid f (SI unit:
Pa·s). The parameter is important for the amount of viscous damping experienced
by the acoustic waves.
• Compressibility of the saturating fluid f (SI unit: 1/Pa). The compressibility of the
fluid enters the expression for Biot’s module M, give by
Ks
M = ---------------------------------------------------Kd
1 –  p – ------- +  p K s  f
Ks
Kd
K s = ---------------1 – B
It should be noted that Biot-Willis coefficient only depends on the properties of the
porous matrix while Biot’s module depends on both fluid and porous matrix
properties.
Select a Viscosity Model, either Biot’s low frequency range or Biot’s high frequency range.
• Biot’s low frequency range models damping at low frequencies where the acoustic
boundary layer (the viscous penetration depth) is assumed to span the full width of
the pores.
• If Biot’s high frequency range is selected, then also enter a Reference frequency fc (SI
unit: Hz). This model implements a correction factor to the viscosity that accounts
for the relative scale difference between a typical pore diameter and the acoustic
boundary layer thickness. The modified viscosity is of the form
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̃ f =  f F  ---f-
fc
f
f c = -----------------2
2a  f
where fc is the reference frequency and a is a characteristic size of the pores. The
expression for fc is one typically used in literature but it is often measured or
empirically determined. The expression for fc corresponds to finding the frequency
at which the viscous boundary layer thickness is of the scale a.
See High Frequency Correction for more details.
Porous, Fixed Constraint
The Porous, Fixed Constraint node adds a condition that makes the porous matrix fixed
(fully constrained); that is, the displacements are zero in all directions. This boundary
condition also sets an impervious (sound-hard) boundary for the fluid pressure.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
Porous, Fixed Constraint (Sound-Hard Boundary) Equations
Porous, Free
The Porous, Free node is the default boundary condition. It means that there are no
constraints and no loads acting on the porous matrix, and a sound-soft boundary for
the fluid pressure.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
Porous, Free (Sound-Soft Boundary) Theory
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Porous, Pressure
The Porous, Pressure node creates a boundary condition that acts as a pressure source
at the boundary, which means a constant acoustic pressure p = p0 is specified. In the
frequency domain, p0 is the amplitude of a harmonic pressure source.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PRESSURE
Enter the value of the Pressure p0 (SI unit: Pa) at the boundary.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Porous, Pressure Equations
Porous, Prescribed Displacement
The Porous, Prescribed Displacement node adds a condition where the displacements
are prescribed in one or more directions to the porous matrix boundary.
If a displacement is prescribed in one direction, this leaves the porous matrix free to
deform in the other directions. Also define more general displacements as a linear
combination of the displacements in each direction.
• If a prescribed displacement is not activated in any direction, this is the same as a
Free constraint.
• If a zero displacement is applied in all directions, this is the same as a Fixed
Constraint.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
PRESCRIBED DISPLACEMENT
Define the prescribed displacements using a Standard notation or a General notation.
Standard Notation
To define the displacements individually, click the Standard notation button (the
default).
To define a prescribed displacement for each space direction (x, y, and z
for 3D), select one or more of the Prescribed in x direction, Prescribed in y
direction, and Prescribed in z direction check boxes. Then enter a value or
expression for the prescribed displacements u0, v0, or w0 (SI unit: m).
For 2D axisymmetric models and to define a prescribed displacement for
each space direction (r and z), select one or both of the Prescribed in r
direction and Prescribed in z direction check boxes. Then enter a value or
expression for the prescribed displacements u0, or w0 (SI unit: m).
General Notation
To specify the displacements using a General notation that includes any linear
combination of displacement components, click the General notation button.
Enter values in the H matrix and R vector fields. For the H matrix, also select an
Isotropic, Diagonal, Symmetric, or Anisotropic matrix and enter values as required.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
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variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Porous, Prescribed Displacement Equations
Porous, Prescribed Velocity
The Porous, Prescribed Velocity node adds a boundary condition where the velocity of
the porous matrix is prescribed in one or more directions. With this boundary
condition it is possible to prescribe a velocity in one direction, leaving the solid free in
the other directions.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. Coordinate systems with
directions which change with time should not be used.
PO RO U S , P RE S C R I B E D VE L O C I T Y
To define a porous, prescribed velocity for each space direction (x, y, and
z for 3D), select one or all of the Prescribed in x direction, Prescribed in y
direction, and Prescribed in z direction check boxes. Then enter a value or
expression for the components vx, vy, and vz (SI unit: m/s).
For 2D axisymmetric models and to define a prescribed velocity for each
space direction (r and z), select one or both of the Prescribed in r direction
and Prescribed in z direction check boxes. Then enter a value or expression
for vr and vz (SI unit: m/s).
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Porous, Prescribed Velocity Equations
Porous, Prescribed Acceleration
The Porous, Prescribed Acceleration node adds a boundary condition, where the
acceleration of the porous matrix is prescribed in one or more directions. With this
boundary condition, it is possible to prescribe a acceleration in one direction, leaving
the solid free in the other directions.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. Coordinate systems with
directions which change with time should not be used.
POROUS, PRESCRIBED ACCELERATION
To define a porous, prescribed acceleration for each space direction (x, y,
and z for 3D), select one or all of the Prescribed in x direction, Prescribed
in y direction, and Prescribed in z direction check boxes. Enter a value or
expression for the prescribed acceleration ax, ay, and az (SI unit: m/s2).
For 2D axisymmetric models and to define a porous, prescribed
acceleration for each space direction (r and z), select one or both of the
Prescribed in r direction and Prescribed in z direction check boxes. Then
enter a value or expression for the prescribed acceleration ar and az
(SI unit: m/s2).
Porous, Prescribed Acceleration Equations
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Porous, Roller
The Porous, Roller node adds a roller (sliding wall) constraint as the boundary
condition; that is, the porous matrix displacement is zero in the direction
perpendicular (normal) to the boundary, but the porous matrix is free to move in the
tangential direction. This boundary condition also sets an impervious (sound-hard)
boundary for the fluid pressure.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Porous, Roller Equations
Porous, Septum Boundary Load
Add a Poroelastic Septum Boundary Load to boundaries for a pressure acting on the
porous matrix through a septum layer. Right-click the node to add a Phase.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
POROUS, SEPTUM BOUNDAR Y LOAD
Enter a Surface density sep (SI unit: kg/m3). The default is 0 kg/m3. Enter
coordinates for the Load FA (SI unit: N/m2).
Porous, Septum Boundary Load Equations
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Continuity
The Continuity boundary condition includes the fluid load and structural acceleration
for use on the fluid-solid, fluid-porous, and solid-porous boundaries, where it is the
default boundary condition. It is not applicable on other boundaries.
For fluid-solid boundaries, this boundary condition includes the following interaction
between fluid and solid domains:
• A pressure load (force per unit area) Fpn p on the boundaries where the fluid
interacts with the solid. In this expression, n is the outward-pointing unit normal
vector seen from inside the solid domain.
• A structural acceleration acting on the boundaries between the solid and the fluid.
This makes the normal acceleration for the acoustic pressure on the boundary equal
to the acceleration based on the second derivatives of the structural displacements
u with respect to time: ann · -2u.
For fluid-porous boundaries, this boundary condition includes the following
interaction between fluid and porous domains:
• Continuity of the fluid pressure on the boundaries where the fluid interacts with the
porous domain. The pore pressure in the porous domain is set equal to the total
pressure in the fluid domain: ppore = pt.
• A pressure load for the elastic waves in the porous material
n    d –  B pI  = – np t
where pt is the total acoustic pressure in the fluid domain and the left hand side
represents the total stress for the saturated porous domain.
• The pressure acoustic domain experiences a normal acceleration that depends both
on the acceleration of the porous matrix skeleton but also on the pore pressure.
Because of the pressure boundary condition, which is a bidirectional constraint, this
condition reduces to the fluid experiencing a normal acceleration: an = (i)2u.
For solid-porous boundaries, this boundary condition includes the following
interaction between solid and porous domains:
• Continuity of the displacement field on the boundaries where the solid interacts
with the porous domain.
• A normal acceleration acting on the boundaries between the porous material and
the fluid. This makes the normal acceleration for the acoustic pressure on the
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boundary equal to the acceleration based on the second derivatives of the structural
displacements u with respect to time: ann · -2u.
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the boundary selection automatically becomes the default selection
between the fluid, solid or porous domains. Manual selection is not available.
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Theory for the Elastic Waves and
Poroelastic Waves User Interfaces
The Elastic Waves and Poroelastic Waves User Interfaces theory is described in this
section:
• About Elastic Waves
• About Poroelastic Waves
• About the Boundary Conditions for Poroelastic Waves
• References for the Elastic Waves and Poroelastic Waves User Interfaces
About Elastic Waves
The most general linear relation (see About Linear Elastic Materials) between the stress
and strain tensors in solid materials can be written as
 ij = c ijkl  kl
here,  is the Cauchy’s stress tensor,  is the strain tensor, and cijkl is a fourth-order
elasticity tensor. For small deformations, the strain tensor is defined as
1
  u  = ---  u + u T 
2
where u represents the displacement vector.
The elastic wave equation is then obtained from Newton’s second law
2
 -------2- u –      u  – s 0  = F
t
here,  is the medium density, and s0 and F represent source terms.
An important case is the time-harmonic wave, for which the displacement varies with
time as
u  x t  = u  x  e
it
with f (SI unit: Hz) denoting the frequency and 2f (SI unit: rad/s) the angular
frequency. Assuming the same time-harmonic dependency for the source terms s0 and
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F, the wave equation for linear elastic waves reduces to an inhomogeneous Helmholtz
equation:
–  2 u –      u  –s0  = F
(5-1)
Alternatively, treat this equation as an eigenvalue PDE to solve for eigenmodes and
eigenfrequencies as described in Setting up Equations for Different Studies. Also add
damping as described in Damping Models.
About Poroelastic Waves
In his seminal work, Biot extended the classical theory of linear elasticity to porous
media saturated with fluids (Ref. 1, Ref. 2, Ref. 3).
In Biot’s theory, the bulk moduli and compressibilities are independent of the wave
frequency, and can be treated as constant parameters. The porous matrix is described
by linear elasticity and damping is introduced by considering the viscosity of the fluid
in the pores, which can be frequency dependent.
Consider Biot’s expressions for poroelastic waves (Ref. 3, Ref. 4, Ref. 6)
2
2
 av -------2- u +  f -------2- w –    = 0
t
t
2
f 
 2

 f -------2- u + ----- ----- w + -----  f -------2- w + p f = 0
 t
 p t
t
(5-2)
here, u is the displacement of the porous material,  is the total stress tensor (fluid and
porous material), w is the fluid displacement with respect to the porous matrix, f and
f are the fluid’s density and viscosity,  is the tortuosity, p is the porosity, pf is the
fluid pore pressure,  is the permeability and av the average density. The average
density is the total density (porous material plus pore fluid) av = dr + pf.
Assuming a time-harmonic dependency for the variables, u(x,t) = u(x)eit,
w(x,t) = w(x)eit, the time derivatives can be removed, so the system in Equation 5-2
becomes
–  av  2 u +  f  2 w –    = 0
–  f  2 u –  2  c   w + p f = 0
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(5-3)
here, the complex density c() (Ref. 5) accounts for the tortuosity, porosity and fluid
density, and the viscous drag on the porous matrix
f

 c    = -----  f + ---------p
i
(5-4)
HIGH FREQUENCY CORRECTION
When Biot’s high frequency range is selected from the Viscosity model list, Equation 5-4
is implemented with a frequency-dependent viscosity c(f) (Ref. 2, Ref. 3, Ref. 5)
 c  f  =  f  F c  ---f-
 f
c
here, fc is a reference frequency (SI unit: Hz) which determines the low-frequency
range f << fc and the high-frequency range f >> fc.
The reference frequency fc can be interpreted as the limit when viscous forces equal
inertial forces in the fluid motion. In the low-frequency limit, viscous effects dominate,
while in the high-frequency limit, inertial effects dominate fluid motion in the pores.
In Biot’s low frequency range, 0 and Fc = 1.
In order to account for a frequency dependence on the viscous drag, Biot defined the
operator Fc() as
1
T   
F c    = ---  --------------------------------------
4  1 + 2iT     
here, T() is related to the Kelvin functions Ber() and Bei()
– –i J1  –i  
Ber'    + iBei'   
T    = ------------------------------------------------- = -------------------------------------Ber    + iBei   
J0  –i  
and J0 and J1 are Bessel functions of the first kind.
U-P FORMULATION
The formulation in terms of the displacements u and w is not optimal from the
numerical viewpoint, since it requires to solve for two displacement fields (Ref. 7, Ref.
8, Ref. 9). The Poroelastic Waves interface solves for the fluid pore pressure variable pf
instead of the fluid displacement field w.
The second row in Equation 5-3 is simplified to
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1
-  p f –  f  2 u 
w = --------------------2 c   
so the first row in Equation 5-3 becomes
f
–  av  2 u – --------------  p f –  f  2 u  –    = 0
c   
(5-5)
The total stress tensor  is then divided into the contributions from the elastic porous
(drained) matrix and from the pore fluid
  u p f  =  dr  u  –  B p f I
here, the identity tensor I means that the pore pressure pf only contributes to the
diagonal of the total stress tensor . The parameter B is the so-called Biot-Willis
coefficient. The drained, elastic stress tensor is written as drc: when  is the strain
tensor of the porous matrix, and the elasticity tensor c contains the drained porous
matrix’s elastic properties (see About Linear Elastic Materials).
Finally, arrange Equation 5-5 in terms of the variables u and p:
 f2 
f

–   av – --------------  2 u –     dr  u  –  B p f I  = -------------- p f
c   
 c   

(5-6)
The next Biot’s equation comes from taking the divergence of the second row in
Equation 5-3, previously divided by c()
f
1
 2    -------------- u +  2   w +    – -------------- p f = 0
  c   
 c    
(5-7)
Using the expressions for the volumetric strain vol·u and fluid displacement (Ref.
3, Ref. 4),
pf
–   w = ----- +  B  vol
M
Biot’s modulus M is calculated from the porosity p, fluid compressibility f,
Biot-Willis coefficient B and the drained bulk modulus of the porous matrix Kdr
B – p
1
-  1 – B 
----- =  p  f + ----------------K dr
M
so Equation 5-7 simplifies to
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f
1
1
 2    -------------- u –  2  ----- p f +  B  vol +    – -------------- p f = 0
M

  c   
 c    
(5-8)
and Biot’s wave equations (Equation 5-6 and Equation 5-8) can be written in terms
of the variable u and pf as
 f2 
f

–  2   av – -------------- u –     dr  u  –  B p f I  = -------------- p f







c
c
(5-9)
1
2
– ------- p f +   – --------------  p f –  2  f u  =  2  B  vol
M
c   
The saturated (also called Gassmann) modulus can be obtained from the
drained bulk modulus Kdr, Biot modulus M, and Biot-Willis coefficient
B as KsatKdr B2M (Ref. 5).
Further arranging the first row in Equation 5-9 to fit the formulation in the Elastic
Waves interface (Equation 5-1) gives
 f2 

–  2   av – -------------- u –     dr  u  – s 0  = F
 c   

(5-10)
The body load F depends on the angular frequency and the gradient of fluid pressure
and the fluid pressure acts as a spherical contribution to the diagonal of Cauchy stress
tensor
f
F = -------------- p f
c   
s0 = B pf I
Arranging the second row in Equation 5-9 to fit the implementation of the Pressure
Acoustics, Frequency Domain interface gives (see Theory Background for the Pressure
Acoustics Branch)
1
2
– ------- p f +   –  --------------  p f – q d  = Q m
 c   

M
(5-11)
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The monopole source Qm (SI unit: 1/s2) and the dipole source qd (SI unit: N/m3)
depend on the angular frequency , the displacement of the porous matrix u, the fluid
density and Biot-Willis coefficient B
Q m =  2  B  vol
qd =  2 f u
About the Boundary Conditions for Poroelastic Waves
Although boundary conditions can be set up for the porous matrix and fluid
independently of each other, there exist a few common boundary conditions which
deserve special attention. The following sections refer to the boundary conditions for
the system written in Equation 5-10 and Equation 5-11. See derivation in Ref. 7, Ref.
8, and Ref. 9.
POROUS, FREE (SOUND-SOFT BOUNDAR Y) THEOR Y
The free boundary condition is the default for the Poroelastic Waves interface. It
means that the displacement of the porous matrix in Equation 5-10 is unconstrained,
so it can move freely without experiencing any loads.
The sound soft boundary condition for acoustics creates a boundary condition for
Equation 5-11 where the acoustic pressure vanishes, so it sets pf0.
POROUS, FIXED CONSTRAINT (SOUND-HARD BOUNDAR Y) EQUATIONS
For simulating a poroelastic medium bounded by a rigid impervious wall, impose a
Fixed Constraint node for the porous matrix displacement in Equation 5-10, u 
and a sound-hard boundary condition for the pore pressure in Equation 5-11:
1
n   --------------  p f – q d  = 0
c   
POROUS, PRESSURE EQUATIONS
For a given fluid pressure p0 on the boundary, set the pressure in Equation 5-11 to
p f = p 0 . Since the fluid pressure is set to p0, the normal stress on the porous matrix
in Equation 5-10 reduces to
n   dr  u  = n   B – 1 p 0
For a rigid porous matrix Bp, the load is equivalent to
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n   dr  u  = n   p – 1 p 0
and for a soft porous matrix B1, there is no load since
n   dr  u  = 0
POROUS, PRESCRIBED DISPLA CEMENT EQUATIONS
For a prescribed displacement u0 at the boundary, set the displacement of the porous
matrix in Equation 5-10 as u u0 and assume a sound-hard (impervious) boundary
for the fluid pressure in Equation 5-11:
1
n   --------------  p f – q d  = 0
c   
PO R O U S , P R E S C R I B E D VE L O C I T Y E Q U A T I O N S
For a prescribed velocity v0 at the boundary, set the displacement of the porous matrix
in Equation 5-10 as
1
u = ------ v 0
i
and assume a sound-hard (impervious) boundary condition for the fluid pressure in
Equation 5-11
1
n   --------------  p f – q d  = 0
c   
POROUS, PRESCRIBED ACCELERATION EQUATIONS
For a prescribed acceleration a0 at the boundary, set the displacement of the porous
matrix in Equation 5-10 as
1
u = ---------2- a 0
–
and assume a sound-hard (impervious) boundary condition for the fluid pressure in
Equation 5-11
1
n   --------------  p f – q d  = 0
c   
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POROUS, ROLLER EQUATIONS
The roller, or sliding wall boundary, means that the boundary is impervious
(sound-hard) to fluid displacements, but it allows tangential displacements of the
porous matrix.
The normal displacement of the porous matrix in Equation 5-10 is constrained, but
the porous matrix is free to move in the tangential direction
nu = 0
The impervious (sound hard) boundary condition for the fluid pressure in
Equation 5-11 is obtained from
1
n   --------------  p f – q d  = 0
 c   

POROUS, SEPTUM BOUNDAR Y LOAD EQUATIONS
For a prescribed load FA at the boundary, suppose that one side of the septum is fixed
to the porous matrix and the other side bears the load.
A septum is a very limp and thin impervious layer with surface density sep. Since the
septum can be seen as a boundary mass density, this boundary condition is achieved by
setting an effective load FSFAsep2u on the porous matrix, so the normal stress
in Equation 5-10 reduces to
n    dr  u  –  B p f I  = F S
and a sound-hard (impervious) boundary condition is applied for the fluid pressure in
Equation 5-11
1
n   --------------  p f – q d  = 0
 c   

References for the Elastic Waves and Poroelastic Waves User Interfaces
1. M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-saturated Porous
Solid. I. Low-frequency Range,” J. Acoust. Soc. Am., vol. 28, pp 168–178, 1956.
2. M. A. Biot, “Theory of Propagation of Elastic Waves in a Fluid-saturated Porous
Solid. II. Higher Frequency Range,” J. Acoust. Soc. Am., vol. 28, pp 179–191, 1956.
3. M. A. Biot, “Generalized Theory of Acoustic Propagation in Porous Dissipative
Media,” J. Acoust. Soc. Am., vol. 34, pp 1254–1264, 1962.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
4. M. A. Biot, “Mechanics of Deformation and Acoustic Propagation in Porous
Media.” J. Appl. Phys., vol. 33, pp 1482–1498, 1962.
5. G. Mavko and others, The Rock Physics Handbook, 2nd ed., Cambridge University
Press, 2009.
6. J.M. Carcione, Wave Fields in Real Media: Wave Propagation in Anisotropic,
Anelastic, Porous and Electromagnetic Media, 2nd ed. Elsevier (Handbook of
Geophysical Exploration, vol. 38, Seismic Exploration), 2007.
7. P. Debergue, R. Panneton, and N. Atalla, “Boundary Conditions for the Weak
Formulation of the Mixed (u,p) Poroelasticity Problem,” J. Acoust. Soc. Am., 106,
pp 2383–2390, 1999.
8. N. Atalla, M.A. Hamdi, and R. Panneton, “Enhanced Weak Integral Formulation
for the Mixed (u,p) Poroelastic Equations,” J. Acoust. Soc. Am., vol. 109, pp 3065–
3068, 2001.
9. J.F. Allard and N. Atalla, Propagation of Sound in Porous Media, 2nd ed., Wiley,
2009.
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T he Ac o us t i c - S h el l In t eract i on ,
Frequency Domain User Interface
This interface requires a Structural Mechanics Module license. For theory
and interface feature descriptions relating to the Shell interface, see the
Structural Mechanics Module User’s Guide.
The interface is only available for 3D geometries, and it is capable of
modeling the coupled pressure acoustics and shell vibrations in the
frequency domain.
The Acoustic-Shell Interaction, Frequency Domain (acsh) user interface (
), found
under the Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard,
uses the features from the Pressure Acoustics, Frequency Domain and the Shell
interfaces to connect the acoustics pressure in a fluid domain with the structural
deformation in a shell boundary.
Special conditions define the fluid loads on the shell boundary, and the structural
acceleration’s effect on the fluid.
When this interface is added, these default nodes are also added to the Model Builder—
Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Exterior Shell, and Initial
Values.
Right-click the Acoustic-Shell Interaction, Frequency Domain node to add other features
that implement boundary conditions and sources. The following sections provide
information about all nodes specific to this multiphysics interface, but all other
acoustics-specific nodes are found under the Pressure Acoustics, Frequency Domain
interface.
For transient simulations, use The Acoustic-Solid Interaction, Transient
User Interface instead.
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INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is acsh.
DOMAIN SELECTION
The default setting is to include All domains in the model to define the dependent
variables and the equations. To choose specific domains, select Manual from the
Selection list.
SOUND PRESSURE LEVEL SETTINGS
The settings are the same for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
THICKNESS
Define the Thickness d by entering a value or expression (SI unit: m) in the field. The
default is 0.01 m. Use the Change Thickness node to define a different thickness in
parts of the shell or plate. The thickness can be variable if an expression is used.
REFERENCE POINT FOR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
The resulting moments (applied or as reactions) are then computed relative to this
reference point. During the results and analysis stage, the coordinates can be changed
in the Parameters section in the result nodes.
FOLD-LINE LIMIT ANGLE
Enter a value for (SI unit: radians). The default is 0.05 radians.
HEIGHT OF EVALUATION IN SHELL, [-1,1]
Enter a value for z(dimensionless). The default is 1.
DEPENDENT VA RIA BLES
This interface defines these dependent variables (fields), the Pressure p and
Displacement field, u, and its components. The name can be changed but the names of
fields and dependent variables must be unique within a model.
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ADVANCED SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
Normally these settings do not need to be changed. The Use MITC interpolation check
box is selected by default, and this interpolation, which is part of the MITC shell
formulation, should normally always be active.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Linear, Quadratic (the default), Cubic, or Quartic for the Pressure. Select Linear or
Quadratic (the default) for the Displacement field. Specify the Value type when using
splitting of complex variables—Real or Complex (the default).
The links to the nodes described in the Structural Mechanics Module
User’s Guide do not work in the PDF, only from the online help.
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Shell Interaction, Frequency Domain User Interface
• Theory Background for the Pressure Acoustics Branch
• Theory for the Shell and Plate User Interfaces in the Structural
Mechanics Module User’s Guide
• Loudspeaker Driver in a Vented Enclosure: Model Library path
Acoustics_Module/Industrial_Models/vented_loudspeaker_enclosure
• Baffled Membrane: Model Library path Acoustics_Module/
Tutorial_Models/baffled_membrane
Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Shell Interaction, Frequency Domain User Interface
Because The Acoustic-Shell Interaction, Frequency Domain User Interface is a
multiphysics interface, almost every node is shared with, and described for, other
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interfaces. Below are links to the domain, boundary, edge, point, and pair nodes as
indicated.
The Acoustic-Shell Interaction, Transient User Interface also shares the
same features as listed below, with an additional domain node: Transient
Pressure Acoustics Model as described for the Pressure Acoustics,
Transient interface.
These nodes are described specifically for this interface:
• Exterior Shell
• Initial Values
• Initial Values (Boundary)
• Interior Shell
• Uncoupled Shell
PRESSURE ACOUSTICS, FREQUENCY DOMAIN MENU
These nodes are described for the Pressure Acoustics, Frequency Domain interface
(listed in alphabetical order):
• Background Pressure Field
• Matched Boundary
• Continuity
• Monopole Source
• Cylindrical Wave Radiation
• Normal Acceleration
• Dipole Source
• Periodic Condition
• Far-Field Calculation
• Plane Wave Radiation
• Impedance
• Pressure
• Interior Impedance/Pair Impedance
• Pressure Acoustics Model
• Interior Normal Acceleration
• Sound Hard Boundary (Wall)
• Interior Perforated Plate
• Sound Soft Boundary
• Interior Sound Hard Boundary (Wall)
• Spherical Wave Radiation
SOLID MECHANICS MENU
These nodes are described for the Solid Mechanics interface and described in this
guide:
• Added Mass
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181
• Fixed Constraint
• Free
• Pre-Deformation
• Spring Foundation
SHELL MENU
These nodes are described for the Shell interface.
Both the Shell interface and its nodes are described in the Structural
Mechanics Module User’s Guide as this interface requires the Structural
Mechanics Module. For that reason, these links do not work in the PDF.
Applied Force, Applied Moment, and Mass and Moment of Inertia are
also described in the Structural Mechanics Module User’s Guide, but for
the Solid Mechanics interface.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
• Antisymmetry
• Pinned
• Body Load
• Prescribed Acceleration
• Change Thickness
• Prescribed Displacement/Rotation
• Edge Load
• Prescribed Velocity
• Face Load
• Point Load
• No Rotation
• Symmetry
Initial Values
The Initial Values node adds initial values for the sound pressure. Right-click the
interface node to add more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
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interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter a value or expression for the Pressure p (SI unit: Pa) initial value. The default is
0 Pa.
Initial Values (Boundary)
The Initial Values node adds initial values for the displacement field and the
displacement of shell normals.
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
INIT IA L VA LUES
Based on space dimension, enter coordinate values for the Displacement field u
(SI unit: m) and the Displacement of shell normals ar (dimensionless).
Exterior Shell
Use the Exterior Shell boundary condition to model any deformable shell boundary,
only one side of which is adjacent to the acoustic domain.
The normal acceleration for the acoustic pressure on the boundary equals the
acceleration based on the second time derivative of the shell displacement
1
– n   – ------  p – qd  = n  u tt
0
In addition, the pressure load (force per unit area) on the shell is: Fpn p.
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BOUNDARY SELECTION
To choose specific boundaries, select Manual from the Selection list, or select All
boundaries.
A Linear Elastic Material node is automatically added to this boundary
condition. For the default node, the setting inherits the selection from the
parent node, and cannot be edited; that is, the selection is automatically
selected and is the same as for the interface. When nodes are added from
the context menu, you can select Manual from the Selection list to choose
specific boundaries or select All boundaries as required.
Interior Shell
Use the Interior Shell boundary condition to model any deformable shell with both
sides adjacent to the acoustic domains.
The normal accelerations for the acoustic pressure on both sides equal to the
acceleration based on the second time derivative of the shell displacement
1
– n   – ------  p – qd  = n  u tt
0
1
1
– n   – ------  p – qd  = n  u tt
0
2
where 1 and 2 subscripts stand for two adjacent domains on different sides of the shell.
In addition, the pressure load (force per unit area) on the shell is: Fp (np)1(np)2.
An Elastic Material node is automatically added to this boundary
condition. Right-click to add more if required.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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Uncoupled Shell
Use the Uncoupled Shell boundary condition to model deformable shells that are not
adjacent to the acoustic domains.
An Elastic Material node is automatically added to this boundary
condition. Right-click to add more if required.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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T he Ac o us t i c - S h el l In t eract i on ,
Transient User Interface
This interface requires a Structural Mechanics license. For theory and
interface feature descriptions relating to the Shell interface, see the
Structural Mechanics Module User’s Guide.
The interface is available for 3D geometry only.
The Acoustic-Shell Interaction, Transient (acshtd) user interface (
), found under the
Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard, uses the
features from the Pressure Acoustics, Frequency Domain and the Shell interfaces to
connect the transient pressure acoustics in a fluid domain with the structural
deformation of shell boundary.
When this interface is added, these default nodes are also added to the Model Builder—
Transient Pressure Acoustics Model, Sound Hard Boundary (Wall), Free, Acoustic-Structure
Boundary, and Initial Values.
Right-click the Acoustic-Shell Interaction, Transient node to add other features that
implement boundary conditions and sources. The following sections provide
information about all nodes specific to this multiphysics interface, but all other nodes
are found under the Pressure Acoustics, Transient and Shell interfaces, respectively.
The Acoustic-Solid Interaction, Frequency Domain User Interface
contains additional functionality that is not applicable for modeling in the
time domain.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
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Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is acshtd.
See The Acoustic-Solid Interaction, Frequency Domain User Interface
for the rest of the interface settings.
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Acoustic-Shell Interaction, Frequency Domain User Interface
• Theory Background for the Pressure Acoustics Branch
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The Pipe Acoustics User Interfaces
In this section:
• The Pipe Acoustics, Frequency Domain User Interface
• The Pipe Acoustics, Transient User Interface
• For links to all the physics features, go to Edge, Boundary, Point, and Pair Nodes
for the Pipe Acoustics User Interfaces
These physics user interfaces require both the Pipe Flow Module and the
Acoustics Module.
The Pipe Acoustics, Frequency Domain User Interface
The Pipe Acoustics, Frequency Domain (pafd) user interface (
), found under the
Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard, has the
equations and boundary conditions for modeling the propagation sound waves in
flexible pipe systems, with the assumption of harmonic vibrations. Thus the equations
can be solved in the frequency domain. The equations are formulated in a general way
to include the possibility of a stationary background flow. The interface is available in
3D on edges and points, and in 2D on boundaries and points.
When this interface is added, these default nodes are also added to the Model Builder—
Fluid Properties, Pipe Properties, Closed, and Initial Values. Right-click the Pipe Acoustics,
Frequency Domain node to add other pipe acoustics nodes.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is pafd.
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EDGE OR BOUNDARY SELECTION
The default setting includes All edges or All boundaries in the model. To choose specific
edges or boundaries, select Manual from the Selection list.
DEPENDENT VA RIA BLES
This section is used to define the dependent variables (fields) for Pressure p (SI unit:
Pa) and Tangential velocity u (SI unit: m/s). If required, edit the name, but dependent
variables must be unique within a model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. It controls
the element types used in the finite element formulation.
Select discretization options from the Pressure and Tangential velocity lists—Linear,
Quadratic, Cubic, Quartic, or Quintic. The defaults is quadratic for the pressure and linear
for the tangential velocity.
For each Dependent variable in the table under Value types when using splitting of
complex variables, choose either a Complex or Real Value type. Click the cell to select
from a drop-down list.
• Show More Physics Options
• Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics User
Interfaces
• Theory for the Pipe Acoustics User Interfaces
The Pipe Acoustics, Transient User Interface
The Pipe Acoustics, Transient (patd) user interface (
), found under the
Acoustics>Acoustic-Structure Interaction branch (
) in the Model Wizard, has the
equations and boundary conditions for modeling the propagation of transient sound
waves in flexible pipe systems. The equations are formulated so to include the
possibility of a stationary background flow. The interface is available in 3D on edges
and points, and in 2D on boundaries and points.
When this interface is added, these default nodes are also added to the Model Builder—
Fluid Properties, Pipe Properties, Closed, and Initial Values. Right-click the Pipe Acoustics,
Transient node to add other pipe acoustics nodes.
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189
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is patd.
The rest of the settings are the same as for The Pipe Acoustics, Frequency
Domain User Interface.
• Show More Physics Options
• Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics User
Interfaces
• Theory for the Pipe Acoustics User Interfaces
Edge, Boundary, Point, and Pair Nodes for the Pipe Acoustics User
Interfaces
The Pipe Acoustics, Transient User Interface has these edge, boundary, point, and pair
physics nodes available and listed in alphabetical order:
• Closed
• Pipe Properties
• End Impedance
• Pressure
• Fluid Properties
• Velocity
• Initial Values
Volume Force is described for the Pipe Flow interface in the Pipe Flow
User’s Guide.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
In the COMSOL Multiphysics Reference Manual:
• Continuity on Interior Boundaries
• Identity and Contact Pairs
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
Initial Values
The Initial Values node adds initial values for the pressure and tangential velocity that
can serve as an initial condition for a transient simulation or as an initial guess for a
nonlinear solver.
EDGE OR BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific edges or boundaries or select All edges or All
boundaries as required.
INIT IA L VA LUES
Enter values or expressions for the initial value of the Pressure p (SI unit: Pa) and the
Tangential Velocity u (SI unit: m/s).
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191
Fluid Properties
The Fluid Properties node adds the momentum and continuity equations solved by the
interface, except for volume forces which are added by the Volume Force node. The
node also provides an interface for defining the material properties of the fluid.
Volume Force is described for the Pipe Flow interface in the Pipe Flow
User’s Guide.
EDGE OR BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific edges or boundaries or select All edges or All
boundaries as required.
MODEL INPUTS
Edit input variables to the fluid-flow equations if required. For fluid flow, these are
typically introduced when a material requiring inputs has been applied.
BACKGROUND PROPERTIES
Enter a value or expression for the Background velocity u0 (SI unit: m/s) and
Background pressure p0 (SI unit: Pa).
Physically sound background property variables for the pressure p0 and
velocity u0 may be obtained by solving a Pipe Flow model on the same
geometry.
PHYSICAL PROPERTIES
Select a Fluid model—Linear elastic (the default).
The default Density  (SI unit: kg/m3) uses the value From material. Select User defined
to enter a different value or expression.
The default Speed of sound cs (SI unit: m/s) uses the value From material. Select User
defined to enter a different value or expression.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
Pipe Properties
The Pipe Properties node is used to define the pipe shape, pipe model, wall drag force,
and swirl correction.
EDGE OR BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific edges or boundaries or select All edges or All
boundaries as required.
PIPE SHAPE
Select a pipe shape from the list—Not set (the default), Round, Square, Rectangular, or
User defined.
• If Round is selected, enter a value or expression for the Inner diameter di (SI unit: m).
The default is 10 cm (0.01 m).
• If Square is selected, enter a value or expression for the Inner width wi (SI unit: m).
The default is 5 cm (0.005 m).
• If Rectangular is selected, enter a value or expression for the Inner width wi
(SI unit: m; the default is 5 cm) and Inner height hi (SI unit: m; the default is 10 cm).
• If User defined is selected, enter a value or expression for the Cross sectional area A
(SI unit: m2; the default is 0.01 m2) and Wetted perimeter Z (SI unit: m; the default
is 0.4 m).
PIPE MODEL
Select a Pipe model—Incompressible cross section (the default), Zero axial stress,
Anchored at one end, or Anchored at both ends.
When Zero axial stress, Anchored at one end, or Anchored at both ends is chosen, select
an option from the Young’s modulus E (SI unit: Pa) and Wall thickness  w lists—Not
set (the default) or User defined. If User defined is selected in either case, enter different
values or expressions.
For Anchored at one end or Anchored at both ends also select an option from the
Poisson’s ratio v (dimensionless) list—Not set (the default) or User defined. If User
defined is selected, enter a value or expression.
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193
WA LL DRA G FO RCE
Enter a value or expression for w (SI unit: N/m2). The default is 0 N/m2.
SWIRL CORRECTION
Enter a value or expression for  (dimensionless). The default is 1. For most practical
applications this factor will be 1 as the propagating waves are assumed plane and
uniform. This value should typically be changed if a wall drag force is introduces or if
a background flow field is used.
Swirl Correction Factor b
Closed
Use the Closed node to impose zero velocity. This is the default condition added on all
end points.
POINT SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific points or select All points as required.
Theory for the Pipe Acoustics Boundary Conditions
Pressure
Use the Pressure node to define the boundary pressure at the pipe ends.
POINT SELECTION
From the Selection list choose the points where you want to define a boundary
pressure.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
PRESSURE
Enter a value or expression for the Pressure p (SI unit: Pa). The default is 0 Pa.
In the frequency domain p represents the amplitude an phase (as it is
complex valued) of a harmonic pressure source.
In the time domain enter an expression for the pressure p, for example, a
forward moving sinusoidal wave of amplitude 1 Pa may be written as
1[Pa]*sin(omega*t-k*x), where omega and k are parameters defining
the angular frequency and wave number.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Theory for the Pipe Acoustics Boundary Conditions
Velocity
Use the Velocity node to prescribe a velocity at the pipe ends.
POINT S EL EC TION
From the Selection list, choose the points to define.
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VE L O C I T Y
Enter a value or expression for the velocity uin (SI unit: m/s) at the inlet and/or outlet
of a pipe. The default is 0 m/s.
The velocity uin is defined relative to background flow u0 and thus in the
tangential coordinate system. Enable Show physics symbols under
Options>Preferences>Graphics in order to visualize the boundary or edge
tangent direction. Click on the Fluid Properties node to see the tangents
as a red arrows.
Theory for the Pipe Acoustics Boundary Conditions
End Impedance
Use the End Impedance node to model conditions at the end of a pipe. The condition
may either model an infinite pipe and thus represent the characteristic impedance of
the pipe system at that point. This will result in a zero reflection condition.
Alternatively the condition may represent the radiation impedance of an open pipe in
either a flanged (in an infinite baffle) or unflanged (a pipe ending in free open space).
The end impedance may also be user defined and could represent modeled or
experimental values for a specific pipe configuration.]
Note that the wave speed c in the pipe may be different from the speed of
sound cs in an open space. It then depends on the elastic properties of the
pipe structure. It is defined in Equation 5-16 in the Governing Equations
section.
The wave speed may be evaluated as sqrt(1/patd.invc2) or sqrt(1/
pafd.invc2) during the analysis and results stage.
POINT SELECTION
From the Selection list, choose the points to define.
END IMPEDANCE
Select an End impedance.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
For The Pipe Acoustics, Transient User Interface choose Infinite pipe (low Mach number
limit) (the default) or User defined. The Infinite pipe (low Mach number limit) models
and infinite pipe by specifying the characteristic impedance at that point. This
condition creates anon reflecting boundary. The expression is valid for small values of
the Mach number Ma = u0/c. If User defined is selected, enter an End impedance Zend
(SI unit: Pas/m). The default is 0.
For The Pipe Acoustics, Frequency Domain User Interface choose from the
following—Infinite pipe (low Mach number limit) (the default), Infinite pipe, Flanged
pipe, circular, Flanged pipe, rectangular, Unflanged pipe, circular (low ka limit), Unflanged
pipe, circular, or User defined.
• If Infinite pipe is selected, enter a Wave number k (SI unit: rad/m). The default
expression is pafd.omega*(sqrt(pafd.invc2)). This end impedance models the
infinite pipe using the full (non-linear) dispersion relation. It is valid for all Mach
numbers but require the additional input of the wave number k.
• If Flanged pipe, circular is selected, enter an Inner radius a (SI unit: m). The default
expression is pafd.dh/2. This end impedance models the radiation impedance of a
circular pipe terminated in an infinite baffle. It is an exact analytical result valid for
all frequencies and pipe radii. In the low frequency limit it reduces to the classical
results:
2
1
Z end = c  ---  ka  + i  0,8216  ka 
2
• If Flanged pipe, rectangular is selected, enter an Inner width wi (SI unit: m). The
default is 5 cm (0.005 m). Also enter an Inner height hi (SI unit: m). The default is
10cm (0.01 m). This end impedance models the radiation impedance of a pipe of
rectangular cross section terminated in an infinite baffle. The model is only valid in
the low frequency range where kwi « 1 and kh i « 1 .
• If Unflanged pipe, circular (low ka limit) or Unflanged pipe, circular is selected, enter
an Inner radius a (SI unit: m). The default expression is pafd.dh/2. These two end
impedance models prescribe the radiation impedance of an unflanged circular pipe
(a pipe ending in free open space). The first model is the classical low frequency
approximation valid for ka « 1 . While the second model extends the frequency
range to ka  3,83 .
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• If User defined is selected, enter an End impedance Zend (SI unit: Pas/m). The
default expression is pafd.rho*(sqrt(1/pafd.invc2)).
For a detailed review of the end impedance models see: Theory for the
Pipe Acoustics Boundary Conditions.
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Theory for the Pipe Acoustics User
Interfaces
The equations governing the propagation of sound in pipes stem from considering
momentum, mass, and energy balances for a control volume of a piece of pipe. The
resulting equations are expressed in the cross-sectional averaged variables and reduces
the equations to a 1D model with scalar dependent variables. The present theory
assumes no thermal conduction and thus no losses due to thermal conduction
(isentropic sound propagation).
The Pipe Acoustics, Transient and the Pipe Acoustics, Frequency Domain
interfaces require both the Pipe Flow Module and the Acoustics Module.
Governing Equations
The continuity equation derived for a control volume is given by
-------------- A +    Au  = 0
t
(5-12)
and the corresponding momentum balance equation is
------------------ Au -
2
+   Au  = – Ap +  w Z + AF
t
(5-13)
where Z is the inner circumference of the pipe and A = A(x,p,...) is the inner wetted
cross-sectional area. u is the area-averaged mean velocity, which is also defined in the
tangential direction u = uet, p is the mean pressure along the pipe, w is the wall drag
force, and F is a volume force. The gradient is taken in the tangential direction et. The
term  is a swirl-correction factor relating the mean of the squared total velocity to the
square of the mean velocity. Such that
1 ˜
u = ---- u
 dA
A

1
p = ---- p˜ dA
A

2
1
2
 =  ----  u˜ dA   u
A


(5-14)
where
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p˜ = p˜  x  and u˜ = u˜  x 
are the local non-averaged parameters. Again p and u are the are the area-averaged
dependent variables.
LINEARIZATION
The governing equations are now linearized, that is, all variables are expanded to first
order assuming stationary zero (0th) order values (steady-state background
properties). The acoustic variations of the dependent variables are assumed small and
on top of the background values. This is done according to the following scheme:
u  x t  = u 0  x  + u 1  x t 
p  x t  = p 0  x  + p 1  x t 
  x t  =  0  x  +  1  x t 
A  x t  = A 0  x  + A 1  x t 
where A0 is often only function of x; however, A0 may be changed by external factors
such as heating or structural deformation, thus the time dependency. The 1st order
terms represent small perturbations on top of the background values (0th order). They
are valid for
1 « 0
2
p1 « 0 c0
u1 « c0
A1 « A0
Moreover, the perturbations for the fluid density and cross-sectional area are expanded
to first order in p0 in a Taylor series such that

 1 =  –  0 =  p – p 0  -----p
A
A 1 = A – A 0 =  p – p 0  ------p
s
s
0
0
where the subscript s refers to constant entropy; that is, the processes are isentropic.
The relations for the fluid compressibility and the cross-sectional area-compressibility
are
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
1
1
1 
 0 = ------- = ------ = ------ ----- 0 p
K0
Ks
1 A
 A = ------- ------A 0 p
s
0
s
0
1 1
= ------ ----20 c
s
1
= ------KA
Here, 0 is the fluid compressibility at the given reference pressure p0, the isentropic
bulk speed of sound is denoted cs, and 0 is the fluid density at the given reference
temperature and reference pressure. A is the effective compressibility of the pipe’s
cross-sectional A0 due to changes in the inner fluid pressure. The bulk modulus K is
equal to one over the compressibility.
Inserting the above expansions into the governing equations (Equation 5-12 and
Equation 5-13) and retaining only 1st-order terms yield the pipe acoustics equations
including background flow. These are:
u0



1 p 1
A 0 ----2- --------- +    A 0  0  u 1 + -----------2 p 1  = 0
t



0 c
c
2
u0
u 0 p 1
 u 1


p + 2 0 A 0 u 0 u 1
 0 A 0  ---------- + -----------2 --------- +   A 0  -----2 1
t
 t



0 c
c
(5-15)
+ A 0  p 1 + p 1  A p 0  +  w Z + AF = 0
1
1
11 0
---=  0   0 +  A  =  0  ------- + ------- = ----2- + ------2
 K 0 K A
c
cs KA
where c is the effective speed of sound in the pipe (it includes the effect due to the
elastic properties of the pipe defined through KA). The bulk modulus for the
cross-sectional area KA is given by the pipe material properties according to the
so-called Korteweg formula (see Ref. 2). For a system with rigid pipe walls cs = c as KA
tends to infinity.
Using the fact that the velocity is taken along the tangential direction et the governing
equations are rewritten in terms of the scalar values u and p and projected onto the
tangent. The 0 subscript is dropped on the density and area and the 1 subscript is also
dropped on the dependent variables.
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201
u0  
 
1 p
A ----2- ------ +  t  A  u + --------2- p   e t = 0
t
 
c
c  
2
 u u 0 p
 u0

A  ------ + --------2- ------ +  t  A -----2- p + 2Au 0 u  e t
t
t




c
c
(5-16)
+ A  t p + p A t p0   et + w Z + A  F  et  = 0
1
1

11
---=    0 +  A  =   ------- + ------- = ----2- + ------2
 K 0 K A
K
A
c
cs
where  t is the tangential derivative, w is the tangential wall drag force (SI unit: N/
m2) and F is a volume force (SI unit: N/m3).
G OVE R N I N G E Q U A T I O N S F O R T H E P I P E A C O U S T I C S , TR A N S I E N T U S E R
INTERFACE
Finally, the expression for the time derivative of the pressure in the momentum
equation is replaced by spatial derivatives using the continuity equation. This yields the
equations solved in the Pipe Acoustics, Transient interface
u0  
 
1 p
A ----2- ------ +  t  A  u + --------2- p   e t = 0
t
 
c  
c
2
u0  
 
 u0

u
A ------ +  t  A -----2- p + 2Au 0 u  e t + u 0  t  A  u + --------2- p   e t
t
 


c
c  
(5-17)
+ A  t p + p A t p0   et + w Z + A  F  et  = 0
1

1
1
1
----- =    0 +  A  =   ------- + ------- = ----- + ------2
2 K
 K 0 K A
A
c
cs
GOVER NING EQUATIONS FOR THE PIPE ACOUSTICS, FREQUENCY
DOMAIN USER INTERFACE
In the frequency domain all variables are assumed to be time harmonic such that
it
p = p˜  x e
it
u = u˜  x e
(5-18)
inserting this into the governing Equation 5-17 (and dropping the tilde) yields the
equations solved in the Pipe Acoustics, Frequency Domain interface
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
u0  
 
A
i ----2- p +  t  A  u + --------2- p   e t = 0
 
c
c  
2
u0  
 
 u0

iAu +  t  A -----2- p + 2Au 0 u  e t + u 0  t  A  u + --------2- p   e t
 


c
c  
(5-19)
+ A   t p + p A  t p0   et + w Z + A  F  e t  = 0
1
1

11
---=    0 +  A  =   ------- + ------- = ----2- + ------2
 K 0 K A
K
A
c
cs
where  = 2f is the angular frequency and f is the frequency.
Theory for the Pipe Acoustics Boundary Conditions
PRESSURE, OPEN, AND CLOSED CONDITIONS
The simplest boundary conditions to specify are to prescribe the pressure or the
velocity at the pipe ends. These result in the Pressure condition
p = p in
and the Velocity condition
u = u in
and can be set independently of each other leaving the other dependent variable free.
A special subclass of the velocity condition is the Closed condition where
u = 0
this corresponds to the sound-hard wall condition in pressure acoustics. It is also
assumed here that u0 = 0 at a closed boundary.
END IMPEDANCE CONDITION
At the end of pipes the relation between the pressure and the velocity may be defined
in terms of an end impedance Zend. The End Impedance condition is in the pipe
acoustics interface given by
u 0
u0 

 1
A  u + -----2- p = A   ----------- + -----2- p

 Z end c 
c 
(5-20)
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203
where Zend = p/u (Si unit: Pa·s/m). Different models for the end impedance exist in
the Pipe Acoustics interfaces. The variety depend on if the transient or the frequency
domain equations are solved.
Transient End-Impedance Models
In the transient interface the end impedance may be user defined or set to mimic an
infinite long pipe for low Mach number background flow conditions. In this case it is
assumed that the pipe continues with constant cross section A and that there is no
external body force F and drag w. Because the acoustic waves are, by design, always
normal to the pipe ends. In order to define the relation between the pressure and the
velocity (the impedance) the dispersion relation for a plane wave needs to be
determined.
In order to do so insert the assumed plane wave form
i  t – kx 
p = Re  p˜ e

i  t – kx 
˜

u = Re  u e
into the governing Equation 5-17 and solve for the desired relations. After some
manipulation this results in
1u
11 
---------= --- = ----2- ---  ---- – u 0

Z end
p
c  k
with the dispersion relation
u 2
1

---- = u 0  c    – 1   -----0- +  1 – ---  A p 0
 c
 k

k
(5-21)
This dispersion relation is non-linear in k. In the limit where A tends to zero and for
small Mach numbers M (= u0/c) the expression is expanded to
u 2
1

----  u 0  c  1 + ---    – 1   -----0- 


2
c 
k
Hence, the infinite pipe (low Mach number limit) end impedance relation reads
u0 2
1
111
---------= ----2- ---   – 1 u 0  c  1 + ---    – 1   ------ 
 c 

2
Z end
c 
(5-22)
where the sign in front of c depends on the direction of propagation of the wave.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
Frequency Domain End-Impedance Models
In the frequency domain many engineering relations exist for the end impedance or
radiation impedance of a pipe or waveguide. Most of the relations apply only to a
specific geometry or frequency range. The relations available in the Pipe Acoustics,
Frequency Domain interface are:
• Infinite pipe (low Mach number limit): This is the same relation as for the
transient study and the end impedance is given by Equation 5-22. This may be
thought of as the characteristic impedance of the tube.
• Infinite pipe: This relation uses the full dispersion relation given in Equation 5-21
and yields the expression
u 2
11
1
1
----------- = ----- ---   – 1 u 0  c    – 1   -----0- +  1 – ---  A p 0
2




c
k
Z end

c
(5-23)
where the wave number k at the right hand side is a user input. In the frequency
domain a good estimate for this quantity is simply /c.
• Flanged pipe, circular: In the case of a circular pipe terminated in an infinite baffle
(a flanged pipe) an analytical expression exists for the radiation impedance (see Ref.
1),
2H 1  2ka 
2J 1  2ka 
Z end = c  1 – -------------------------- + i ---------------------------

2ka 
2ka
(5-24)
where J1 is the Bessel function of order 1, H1 is the Struve function of order 1, a is
the pipe radius, and k is the wave number. The Struve function is approximated
according to Ref. 3 by
2
16
sin x
36 1 – cos xH 1  x   --- – J 0  x  +  ------ – 5 ------------ +  12 – ------ -------------------2

 x



x
(5-25)
In the low frequency limit (small ka) Equation 5-24 reduces to the classical
expression for the radiation impedance
2
1
Z end = c  ---  ka  + i  0,8216  ka 
2
(5-26)
• Flanged pipe, rectangular: In the case of a pipe of rectangular cross-section (with
sides wi and hi) terminated in an infinite baffle (a flanged pipe) the radiation
impedance may be approximated by
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205
c 2
2
3  2 wi
Z end = ------  k  w i h i  + ik  w i h i  f  ------ 

 hi  
2
f  x  = 2x
12
–1 –1
sinh x
+ 2x
–1  2
kw i « 1 kh i « 1
2 3  2 2 –3  2 2
–1 3  2
sinh x + --- x
+ --- x
– ---  x + x 
3
3
3
(5-27)
–1
see Ref. 4 and Ref. 5.
• Unflanged pipe, circular (low ka limit): In the case of a circular pipe of radius a
ending in free air the classical low ka limit for the radiation impedance is given by
2
1
Z end = c  ---  ka  + i  0,6133  ka 
4
(5-28)
ka « 1
see Ref. 1 and Ref. 5.
• Unflanged pipe, circular: A solution for the unflanged pipe exists for the case when
ka « 3,83 = 1,22 , it is presented in Ref. 6 and is based on solving the
Wiener-Hopf integral, it reads
1+R
Z end = c -------------1–R
–  ka   2 
2
R = e

R =
R = Re
1
4
1
19
1 + ---  ka  ln   --------- + ------ 
  ka 12 
6
kae
– ka 

3 1
1 + ------ --------------2
32  ka  
2ika
=e
0,5772
ka  1
(5-29)
1  ka  3,83
where  is an interpolation function found by numerical integration for ka = 0,
 = 0.6133.
Common for the last four radiation impedance relations is that they do only apply
when there is no background flow present u0 = 0 (or at least when it is very small).
Solving Transient Problems
When solving transient acoustic problems where the wave shape is not necessarily
harmonic it may be necessary to resolve its spatial variations with a fine mesh, say with
a minimal scale dx. Now, in order for the numerical solution of the temporal
development of the acoustic field to be good it is necessary to restrict the maximal time
steps dt taken by the solver. The condition is known as the CFL condition (Courant–
Friedrichs–Lewy condition). For transient acoustic problems it is defined as
C = c
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
dt
dx
where C is the Courant number, and c is the velocity.
For applications where all the shape functions are quadratic the Courant number
should be around 0.2. This condition restricts any acoustic disturbances to propagate
more than 20% of the mesh size dx during one time step dt. In the Pipe Acoustics user
interface where a mixed formulation exists, with linear elements for the pressure and
quadratic elements for the velocity, the condition may have to be tightened such that
C  0.2.
For an example of a model where the CFL condition is used see Water
Hammer: Model Library path Pipe_Flow_Module/Verification_Models/
water_hammer_verification.
Cut-off Frequency
The pipe acoustics interface assumes plane wave propagation. This means that it
cannot model the propagation of the higher order modes that can propagate above
their cut-off frequency fc. In a rectangular pipe of cross section width wi and height hi
the cut-off frequency is
1
m 2
n 2
c
f mn = --- c  ------ +  -----
 h i
2  w i
In a pipe of circular cross section (with radius a) the cut-off frequency is
' mn c
c
f mn = --------------2a
where 'mn is the n’th zero of the differential of the Bessel function J’m(x) or order
m. The first few values are '01 = 0, '02 = 3.83, '11 = 1.84, and '21 = 3.05 (see Ref.
1 and Ref. 5 for further details).
Swirl Correction Factor 
The swirl correction factor  accounts for the ratio of the integrated local square
velocity field to the square of the integrated local velocity field (see Equation 5-14). It
is defined in terms of the total velocity field (background plus acoustic variations).
In the case of no-background flow (u0 = 0)  will, in the absence of a wall drag
coefficient, be 1 as only plane wave modes propagate. If a wall drag force is introduced,
THEORY FOR THE PIPE ACOUSTICS USER INTERFACES
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207
to model some loss mechanism, will start to differ slightly from 1. This may for
example be losses introduced to model viscous and thermal effects in narrow pipes.
In the presence of a background u0 the factor can be set different 1 in order to model
a non-flat velocity profile inside the tube. The value of b (and the actual shape of the
background field) influences the convective momentum transfer balances. The places
where  enter the governing equations are multiplied with either the Mach number or
the Mach number squared, indicating that the effects become important for an
increasing background flow.
References for the Pipe Acoustics User Interfaces
1. D. T. Blackstock, Fundamentals of Physical Acoustics, John Wiley & Sons, 2000.
2. M. S. Ghidaoui, M. Zhao, D. A. McInnis, and D. H. Axworthy, “A Review of Water
Hammer Theory and Practice,” Applied Mechanics Reviews, ASME, 2005.
3. R. M. Aarts and A. J. E. M. Janssen, “Approximation of the Struve Function H1
Occurring in Impedance Calculations,” J. Acoust. Soc. Am., vol. 113, pp. 2635–2637,
2003.
4. O. A. Lindemann, “Radiation Impedance of a Rectangular Piston at Very Low
Frequencies,” J. Acoust. Soc. Am., vol 44, pp. 1738–1739, 1968.
5. A. D. Pierce, Acoustics: An Introduction to Its Physical Principles and
Applications, Acoustics Society of America, 1994.
6. H. Levine and J. Schwinger, “On the Radiation of Sound from an Unflanged
Circular Pipe,” Phys. Rev., vol. 73, pp. 383–406, 1948.
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CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION
6
The Aeroacoustics Branch
This chapter describes the Aeroacoustics interfaces found under the Aeroacoustics
branch
in the Model Wizard.
In this chapter:
• The Aeroacoustics, Frequency Domain User Interface
• The Aeroacoustics, Transient User Interface
• The Boundary Mode Aeroacoustics User Interface
• The Compressible Potential Flow User Interface
• The Aeroacoustics with Flow, Frequency Domain User Interface
• The Aeroacoustics with Flow, Transient User Interface
• Theory Background for the Aeroacoustics Branch
209
The Aeroacoustics, Frequency Domain
U s e r Inte r f a c e
The Aeroacoustics, Frequency Domain (ae) user interface (
), found under the
Acoustics>Aeroacoustics branch (
) in the Model Wizard, has the equations, boundary
conditions, and sources for modeling aeroacoustics in the potential flow limit. That is
the one way coupling between a stationary background potential flow (an irrotational
and inviscid flow) and the acoustic field. This is sometimes also known as flow born
noise. The interface does not include any form of flow induced noise capability. The
interface solves for the velocity potential.
Right-click the Aeroacoustics, Frequency Domain node to add other features that
implement boundary conditions and sources. When this interface is added, these
default nodes are also added to the Model Builder— Aeroacoustics Model, Sound Hard
Boundary (Wall), and Initial Values. For axisymmetric models an Axial Symmetry node is
also added.
This interface is limited to flows with a Mach number M < 1, partly due
to limitations in the potential flow formulation and partly due to the
acoustic boundary settings needed for supersonic flow.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is ae.
DOMAIN SELECTION
The default setting is to include All domains in the model. To choose specific domains,
select Manual from the Selection list.
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CHAPTER 6: THE AEROACOUSTICS BRANCH
AEROACOUSTICS EQUATION SETTINGS
For 1D axisymmetric models, the Circumferential wave number m
(dimensionless) is 0 by default. The Out-of-plane wave number kz (SI unit:
rad/m) is 0 rad/m by default. Enter different values or expressions as
required.
For 2D models, the Out-of-plane wave number kz (SI unit: rad/m) is 0
rad/m by default.
For 2D axisymmetric models the Circumferential wave number m
(dimensionless) is 0 by default.
Enter different values or expressions as required.
SOUND PRESSURE LEVEL SETTINGS
The settings are the same as Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
TY P I C A L WA V E S P E E D
The settings are the same as Typical Wave Speed for the Pressure Acoustics, Frequency
Domain interface.
DEPENDENT VARIABLES
This interface defines one dependent variable (field), the Velocity potential phi. The
name can be changed but the names of fields and dependent variables must be unique
within a model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Set the
element order for the Velocity potential to Linear, Quadratic (the default), Cubic, or
THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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211
Quartic. Specify the Value type when using splitting of complex variables—Real or
Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics, Frequency Domain User Interface
• Theory Background for the Aeroacoustics Branch
• Flow Duct: Model Library path Acoustics_Module/Industrial_Models/
flow_duct
• Doppler Shift: Model Library path Acoustics_Module/Tutorial_Models/
doppler_shift
Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics, Frequency Domain User Interface
The Aeroacoustics, Frequency Domain User Interface has these domain, boundary,
edge, point and pair nodes available and listed in alphabetical order.
• Aeroacoustics Model
• Normal Mass Flow
• Continuity
• Normal Velocity
• Initial Values
• Periodic Condition
• Impedance and Pair Impedance
• Plane Wave Radiation
• Interior Sound Hard Boundary (Wall)
• Sound Hard Boundary (Wall)
• Mass Flow Circular Source
• Sound Soft Boundary
• Mass Flow Edge Source
• Velocity Potential
• Mass Flow Point Source
• Vortex Sheet
For axisymmetric models, COMSOL Multiphysics takes the axial
symmetry (at r = 0) into account and automatically adds an Axial
Symmetry node to the model that is valid on the axial symmetry edges/
points only.
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CHAPTER 6: THE AEROACOUSTICS BRANCH
In the COMSOL Multiphysics Reference Manual:
• Destination Selection
• Continuity on Interior Boundaries
• Identity and Contact Pairs
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
Aeroacoustics Model
The Aeroacoustics Model node adds the equations for frequency domain aeroacoustics
modeling. You here need to enter the material properties as well as the background
mean flow information.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains where the aeroacoustics model is valid and
to compute the velocity potential, or select All domains as required.
AEROACOUSTICS MODEL
The default values for the Density  (SI unit: kg/m3) and the Mean flow speed of sound
cmf (SI unit: m/s) are taken From material. Select User defined to enter other values or
expressions. This could for example bee to select the values taken from a simulation
run using The Compressible Potential Flow User Interface.
Also enter values or expressions for the Mean flow velocity V (SI unit: m/s). The
defaults are 0 m/s. It is here important to know that the velocity field needs to be a
solution to a compressible potential flow simulation. It has to be an irrotational and
THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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213
inviscid flow, for example, a constant flow field V is of this type. Any other type of flow
yields non-physical solutions for this formulation of the governing equations.
Initial Values
The Initial Values node adds initial values for the velocity potential. Right-click the
interface node to add more Initial Values nodes. For The Aeroacoustics, Transient User
Interface it adds initial values for the velocity potential, first time derivative.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter a value or expression for the initial value of the Velocity potential phi
(SI unit: m2/s). The default is 0 m2/s.
For The Aeroacoustics, Transient User Interface enter a Velocity potential, first time
derivative,   t (SI unit: m2/s2). The default is 0 m2/s2.
Sound Hard Boundary (Wall)
Use the Sound Hard Boundary (Wall) condition to model rigid boundary surfaces or
walls. It prescribes a vanishing normal component of the particle velocity at the
boundary. Multiplied by the density, it can equivalently be expressed as a no-flow
condition:


  
– n     – V -------+ V    = 0
2 t



c mf
The sound-hard boundary condition is available for all analysis types. The equation
above applies to the time domain calculations in The Aeroacoustics, Transient User
Interface; to obtain the corresponding condition for frequency domain, simply replace
t by i.
The Boundary Mode Aeroacoustics User Interface the no-flow or wall condition,
known as sound hard, sets the normal acceleration—and thus also the normal
velocity—to zero at the edge.
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CHAPTER 6: THE AEROACOUSTICS BRANCH



– n     – V ----------2-  i +    V   = 0


c mf
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
Velocity Potential
Use the Velocity Potential node when coupling two Aeroacoustics, Frequency Domain
interfaces together because sometimes be necessary to set the velocity potential:
   0.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. When using the pair node, this
list cannot be edited. It shows the boundaries in the selected pairs.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
Use Pairs to couple two Aeroacoustics, Frequency Domain interfaces
together as it can sometimes be necessary to set the velocity
potential    0.
VE L O C I T Y PO T E N T I A L
Enter a Velocity potential 0 (SI unit: m2/s). The default is 0 m2/s.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
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215
Normal Mass Flow
Use the Normal Mass Flow node to set the inward mass flow boundary condition.
For The Aeroacoustics, Frequency Domain User Interface, the natural boundary
condition for the total wave has the meaning of a mass flow through the boundary
surface:


 
– n     – V ----------2-  + V    = m n




c mf  t
For The Boundary Mode Aeroacoustics User Interface, the natural edge condition for
the total wave has the meaning of normal mass flow.



– n     – V ----------2-  i +    V  +  V n  = m n


c mf
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
NORMAL MASS FLOW
Enter an Inward mass flow mn (SI unit: kg/(m2·s)). The default is 0 kg/(m2·s).
Plane Wave Radiation
The Plane Wave Radiation is a class of non-reflecting boundary conditions, which
assume that there is an outgoing plane wave, and optionally also an incoming exciting
wave.
For transient analysis the boundary condition is



  

 
– k n n  V =
– n     – V ----------2-  +    V   – k n – n  V -------2 t



t
t

t


c mf
c mf

 0
 0


   0
 k  n  n k  – n  V -------kk
k
n k  V – k n
+ n  V -------n  V
2
2  nt
t 0 k

t

t
c
c
mf
1
k k = -------------------------------c mf + V  n k
mf
kn
1
= ---------------------------c mf + V  n
while the corresponding time-harmonic equation reads
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CHAPTER 6: THE AEROACOUSTICS BRANCH




– n     – V ----------2-  i +   V  –  ik n  – n  V ------- i – ik n n  V  =
2


c mf
c mf


–i k  r
V
V
 ik k n k  V  – ik n + n  ------- ik n n  V   0 e
  ik k n  n k – n  -------2
2


c mf
c mf

k k = -------------------------------c mf + V  n k

k n = ---------------------------c mf + V  n
k = k nk
ek
n k = -------ek
Specify an Incident Velocity Potential (incoming plane wave)
0 e
– ik  r
by supplying its amplitude,  0, and propagation wave direction vector, ek. The vector
nk is the normalized wave direction vector of unit length.
This boundary condition is most relevant for ports, because many
waveguide structures are only interesting in the plane-wave region.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
Incident Velocity Potential
Right-click the Plane Wave Radiation node to add an Incident Velocity Potential node,
which is then used to add a velocity potential and wave direction.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
I N C I D E N T VE L O C I T Y PO T E N T I A L
Enter a Velocity potential  0 (SI unit: m2/s) and Wave direction ek (SI unit: m). The
default for the velocity potential is 0 m2/s, and for the wave direction, the default is
the inward normal direction of the boundary.
THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
|
217
Sound Soft Boundary
The Sound Soft Boundary creates a boundary condition for a sound soft boundary,
where the acoustic pressure vanishes and p = 0.
This boundary condition is an appropriate approximation for a liquid-gas
interface and in some cases for external waveguide ports.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. If Sound Soft Boundary is
selected from the Pairs menu, this list cannot be edited. It shows the boundaries in the
selected pairs.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
Select the Use weak constraints check box to replace the standard constraints with a
weak implementation.
Normal Velocity
Use the Normal Velocity node in time-harmonic analysis to specify the velocity
component normal to the boundary:



1
n     – V -----------2  i + V    =   v n + ------ V   v n


i


c mf
Here vn denotes the outward normal velocity at the boundary surface, which is
specified in the vn field.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
N O R M A L VE L O C I T Y
Enter a Normal velocity vn (SI unit: m/s). The default is 0 m/s.
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CHAPTER 6: THE AEROACOUSTICS BRANCH
Impedance and Pair Impedance
Use the Impedance or Pair Impedance node in time-harmonic analysis to define the
input impedance of an external domain or at the boundary between parts in an
assembly as the ratio of pressure to normal velocity, Zip(n · v) at the boundary. The
associated impedance boundary condition is


p

p 1
– n     – -----------2  i  + V   V =   ---- + ------  V    ----
i
Z
Z


c mf
For the Pair Impedance node the above equation applies on each side of the boundary.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. For the Pair Impedance node,
this list is not editable and shows the boundaries in the selected pairs.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
IMPEDANCE/PAIR IMPEDANCE
Enter an input Impedance Zi (SI unit: Pa·s/m). The default value is 1Pa·s/m for the
boundary condition and 0 Pa·s/m for the pair condition.
Vortex Sheet
Use the Vortex Sheet boundary condition to model a shear layer that separates a stream
from the free velocity field. Because the velocity potential is discontinuous over this
boundary, use a slit boundary condition or a pair in an assembly. Vortex sheets are only
applicable on interior boundaries.
The equations defining the vortex sheet boundary condition are



n     – V -----------2  i + V   


c mf
p up = p down
=    i + V    w  i
i = up down
i
w up = – w down
where w denotes the outward normal displacement (SI unit: m) of the boundary
surface, which this boundary condition adds as ae.w or aepf.w, using the default
THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
|
219
name for the physics interface. The subscripts “up” and “down” refer to the two sides
of the boundary.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. If selected from the Pairs
submenu, this list is not editable and shows the boundaries in the selected pairs.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
Interior Sound Hard Boundary (Wall)
For The Aeroacoustics, Frequency Domain User Interface and The Aeroacoustics,
Transient User Interface, use the Interior Sound Hard Boundary (Wall) condition to
model interior rigid boundary surfaces, or walls. It prescribes a vanishing normal
component of the particle velocity at the boundary. Multiplied by the density, it can
equivalently be expressed as a no-flow condition:


  
– n     – V -------+ V   
2 t


c mf
= 0
i = up down
i
Here, the subscripts “up” and “down” refer to the two sides of the boundary.
The sound-hard boundary condition is available for all analysis types. The equation
above applies to the time domain calculations in the Aeroacoustics, Transient interface;
to obtain the corresponding condition for the frequency domain, simply replace t
with i.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
Continuity
Continuity is available as an option at interfaces between parts in a pair.
For the Aeroacoustics interfaces, this condition gives continuity in the velocity
potential as well as continuity in the mass flow. It corresponds to a situation where the
boundary has no direct effect on the acoustic velocity potential field (subscripts 1 and
2 in the equation refers to the two sides of the pair):
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CHAPTER 6: THE AEROACOUSTICS BRANCH
 

 

V
V
n      – ----2-  i + V     –     – ----2-  i + V    
 





cs
cs
1
= 0
2
In the COMSOL Multiphysics Reference Manual:
• Continuity on Interior Boundaries
• Identity and Contact Pairs
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Axial Symmetry
The Axial symmetry feature is a default node added for all axisymmetric
models. The boundary condition is active on all boundaries on the
symmetry axis.
BOUNDARY SELECTION
The boundaries section shows on which boundaries the feature is active. All boundaries
on the symmetry axis are automatically selected.
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221
Mass Flow Line Source on Axis
For 2D axisymmetric models, use a Mass Flow Line Source on Axis node to
add a line source along the symmetry axis.



     – V ------2-  i + V    = m'


cs
BOUNDARY SELECTION
From the Selection list, choose the boundaries on the symmetry axis to define.
MASS FLOW LINE SOURCE ON AXIS
Enter a Mass flow rate m' (SI unit: kg/(m·s)). The default is 0 kg/(m·s).
Mass Flow Edge Source
For 3D models, use a Mass Flow Edge Source to specify the mass flow rate on an edge:



     – V -----------2  i + V    = m'


c mf
EDGE SELECTION
From the Selection list, choose the edges to define.
MASS FLOW EDGE SOURCE
Enter a Mass flow rate m' (SI unit: kg/(m2·s)). The default is 0 kg/(m2·s).
Mass Flow Point Source
Add a Mass Flow Point Source node to specify the mass flow rate on a point:



     – V ------2-  i + V    = m'


cs
POINT SELECTION
From the Selection list, choose the points to define.
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MA SS FLOW POINT S OURCE
Enter a Mass flow rate m' (SI unit: kg/s for 3D and 2D axisymmetric models; kg/(m·s)
for 2D models). The default is 0 for all dimensions.
Doppler Shift: Model Library path Acoustics_Module/Tutorial_Models/
doppler_shift
Mass Flow Circular Source
For 2D axisymmetric models, use a Mass Flow Circular Source node to add
a circular source located at rr0.



     – V ------2-  i + V    = m'


cs
POINT S EL EC TION
From the Selection list, choose the points to define.
MASS FLOW CIRCULAR SOURCE
Enter a Mass flow rate m' (SI unit: kg/(m·s). The default is 0 kg/(m·s).
THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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223
The Aeroacoustics, Transient User
Interface
The Aeroacoustics, Transient (aetd) user interface (
), found under the
Acoustics>Aeroacoustics branch (
) in the Model Wizard, has the equations, boundary
conditions, and sources for modeling transient aeroacoustics in the potential flow limit.
That is the one way coupling between a stationary background potential flow (an
irrotational and inviscid flow) and the acoustic field. This is sometimes also known as
flow born noise. The interface does not include any form of flow induced noise
capability. The interface solves for the velocity potential.
When this interface is added, these default nodes are also added to the Model Builder—
Aeroacoustics Model, Sound Hard Boundary (Wall), and Initial Values. For axisymmetric
models an Axial Symmetry node is also added. Right-click the Aeroacoustics, Transient
node to add other features that implement boundary conditions and sources.
This interface is limited to flows with a Mach number M < 1, partly due
to limitations in the potential flow formulation and partly due to the
acoustic boundary settings needed for supersonic flow.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is aetd.
The remainder of the settings are shared with The Aeroacoustics,
Frequency Domain User Interface.
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• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics, Transient User Interface
• Theory Background for the Aeroacoustics Branch
Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics, Transient User Interface
The Aeroacoustics, Transient User Interface shares all its domain, boundary, edge,
point, and pair nodes with the Aeroacoustics, Frequency Domain interface (listed in
alphabetical order):
• Aeroacoustics Model
• Mass Flow Edge Source
• Continuity
• Normal Mass Flow
• Initial Values
• Periodic Condition
• Interior Sound Hard Boundary (Wall)
• Plane Wave Radiation
• Mass Flow Point Source
• Sound Hard Boundary (Wall)
• Mass Flow Circular Source
• Velocity Potential
In the COMSOL Multiphysics Reference Manual:
• Continuity on Interior Boundaries
• Identity and Contact Pairs
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
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The Boundary Mode Aeroacoustics
User Interface
The Boundary Mode Aeroacoustics (aebm) user interface (
), found under the
Acoustics>Aeroacoustics branch (
) in the Model Wizard, has the equations, boundary
conditions, and sources for modeling aeroacoustic mode analysis problems, solving for
the out-of-plane wave number. This interface is often used to determine boundary
conditions for the velocity potential for a full aeroacoustics model.
When this interface is added, these default nodes are also added to the Model Builder—
Aeroacoustics Model, Sound Hard Boundary (Wall), and Initial Values. For 2D
axisymmetric models an Axial Symmetry node is also added.
Right-click the Boundary Mode Aeroacoustics to add other nodes that implement
boundary conditions and sources.
This interface is limited to flows with a Mach number M < 1, partly due
to limitations in the potential flow formulation and partly due to the
acoustic boundary settings needed for supersonic flow.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is aebm.
BOUNDARY SELECTION
The default setting is to include All boundaries in the model. To choose specific
boundaries, select Manual from the Selection list.
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AEROACOUSTICS EQUATION SETTINGS
For 2D axisymmetric models, the Circumferential wave number m
(dimensionless) is 0 by default. It is an integer entering the axisymmetric
expression for the velocity potential:
  r z   =   r  e
– i  k z z + m 
SOUND PRESSURE LEVEL SETTINGS
The settings are the same as Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
DEPENDENT VA RIA BLES
This interface defines one dependent variable (field), the Velocity potential phi. The
name can be changed but the names of fields and dependent variables must be unique
within a model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Quadratic (the default), Linear, Cubic, or Quartic for the Velocity potential. Specify the
Value type when using splitting of complex variables—Real or Complex (the default).
• Show More Physics Options
• Boundary, Edge, Point, and Pair Nodes for the Boundary Mode
Aeroacoustics User Interface
• Theory Background for the Aeroacoustics Branch
Flow Duct: Model Library path Acoustics_Module/Industrial_Models/
flow_duct
Boundary, Edge, Point, and Pair Nodes for the Boundary Mode
Aeroacoustics User Interface
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227
The Boundary Mode Aeroacoustics User Interface shares all its boundary,
edge, point, and pair nodes with the Aeroacoustics, Frequency Domain
interface.
For the Boundary Mode Aeroacoustics interface, apply the features to
boundaries instead of domains for 3D models.
• Aeroacoustics Model
• Sound Hard Boundary (Wall)
• Continuity
• Sound Soft Boundary
• Initial Values
• Velocity Potential
• Normal Mass Flow
In the COMSOL Multiphysics Reference Manual:
• Continuity on Interior Boundaries
• Identity and Contact Pairs
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
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The Compressible Potential Flow User
Interface
The Compressible Potential Flow (cpf) user interface (
), found under the
Acoustics>Aeroacoustics branch (
) in the Model Wizard, has the equations, boundary
conditions, and sources for modeling the mean flow in an ideal barotropic, irrotational
fluid at constant entropy. This interface can be used to model the background flow
used as input for the The Aeroacoustics, Frequency Domain User Interface or the The
Aeroacoustics, Transient User Interface.
When this interface is added, these default nodes are also added to the Model Builder—
Compressible Potential Flow, Slip Velocity, and Initial Values. For axisymmetric models an
Axial Symmetry node is also added.
Right-click the Compressible Potential Flow node to add other features that implement
boundary conditions and sources.
The potential flow formulation for steady compressible flow is in general
not suited for modeling shocks. In the region after a shock the flow is
typically rotational, hence it is only suited for problems with a Mach
number M < 1.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is cpf.
DOMAIN SELECTION
The default setting is to include All domains in the model where the compressible
potential flow model is valid and to compute the density and the mean flow velocity
potential. To choose specific domains, select Manual from the Selection list.
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229
R E F E RE N C E VA L U E S
Edit or enter the values as required:
• Reference pressure pref (SI unit: Pa). The default is 1 atm.
• Reference density ref (SI unit: kg/m3). The default is 1.2 kg/m3.
• Reference velocity ref (SI unit: m/s). The default is 0 m/s.
• Reference force potential ref (SI unit: J/kg). The default is 0 J/kg.
DEPENDENT VARIABLES
This interface defines two dependent variables (field), the Mean flow velocity potential
Phi and the Density, rho. The name can be changed but the names of fields and
dependent variables must be unique within a model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Quadratic (the default), Linear, Cubic, or Quartic for the Mean flow velocity potential and
Density. Specify the Value type when using splitting of complex variables—Real or
Complex (the default).
• Show More Physics Options
• Domain, Boundary, and Pair Nodes for the Compressible Potential
Flow User Interface
• Theory Background for the Aeroacoustics Branch
Flow Duct: Model Library path Acoustics_Module/Industrial_Models/
flow_duct
Domain, Boundary, and Pair Nodes for the Compressible Potential
Flow User Interface
The Compressible Potential Flow User Interface has these domain, boundary, and pair
nodes available (listed in alphabetical order):
• Compressible Potential Flow Model
• Initial Values
• Mass Flow
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• Normal Flow
• Periodic Condition
• Slip Velocity
In the COMSOL Multiphysics Reference Manual:
• Using Symmetries
• Continuity on Interior Boundaries
• Identity and Contact Pairs
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
Compressible Potential Flow Model
The Compressible Potential Flow Model node adds equations for time dependent or
stationary modeling of compressible potential flow.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
COMPRESSIBLE POTENTIAL FLOW MODEL
Enter a Ratio of specific heats  (dimensionless). The default is 1.4. Also enter a Force
potential (SI unit: J/kg). The default is 0 J/kg.
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231
Initial Values
The Initial Values node adds initial values for the mean flow velocity potential and
density variables. Right-click the interface node to add more Initial Values nodes.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter a value or expression for the initial values Mean flow velocity potential Phi
(SI unit: m2/s). The default is 0 m2/s. Enter a Density rho (SI unit: kg/m3). The
default is cpf.rhoref.
Slip Velocity
The Slip Velocity node is the natural condition at a boundary impervious to the flow,
meaning that the velocity normal to the boundary is zero. By multiplying with the
density, this condition can be alternatively be expressed as a vanishing mass flow
through the boundary:
n   = 0
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
Normal Flow
The Normal Flow node implies that the flow is normal to the boundary and thus that
the tangential velocity is zero. This corresponds to a constant velocity potential along
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the boundary. Because the velocity potential is determined only up to a constant,
imposing this condition fixes the arbitrary constant to zero where 
Setting the Normal Flow condition on two or more disjoint boundaries
can result in the wrong features unless symmetry implies that the velocity
potential is equal on the boundaries in question.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
Mass Flow
The Mass Flow node specifies the mass flow through the boundary. The mass flow is
given by the product of two variables: the normal velocity, vn, and the density at the
boundary, bnd:
n   = v n  bnd
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
MASS FLOW
Enter the Normal Velocity vn (SI unit: m/s) and Fluid density at the boundary bnd
(SI unit: kg/m3). The defaults are cpf.vref and cpf.rhoref, respectively.
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233
The Aeroacoustics with Flow,
Frequency Domain User Interface
The Aeroacoustics with Flow, Frequency Domain (aepf) user interface (
), found under
the Acoustics>Aeroacoustics branch (
) in the Model Wizard, combines Aeroacoustics
and Compressible Potential Flow and connects the result from compressible potential
flow simulation with the Aeroacoustic Model. It is where calculations of density and
mean flow velocity potential in an ideal barotropic, irrotational fluid at constant
entropy is combined with aeroacoustic modeling. The interface is available for 3D, 2D,
and 1D as well as 2D and 1D axisymmetric geometries and has the capabilities for
modeling frequency domain aeroacoustics in compressible potential flow.
When this interface is added, these default nodes are also added to the Model Builder—
Aeroacoustics with Flow Model, Sound Hard Boundary (Wall), Slip Velocity, and Initial
Values. For axisymmetric models an Axial Symmetry node is also added.
Right-click the Aeroacoustics with Flow, Frequency Domain node to add other physics
that implement boundary conditions and sources.
The features specific to this multiphysics interface are described below,
but all other nodes are described for The Aeroacoustics, Frequency
Domain User Interface and The Compressible Potential Flow User
Interface, respectively.
This interface is limited to flows with a Mach number M < 1, partly due
to limitations in the potential flow formulation and partly due to the
acoustic boundary settings needed for supersonic flow.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
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The default identifier (for the first interface in the model) is aepf.
DOMAIN SELECTION
The default setting is to include All domains in the model. To choose specific domains,
select Manual from the Selection list.
SOUND PRESSURE LEVEL SETTINGS
See Sound Pressure Level Settings for the Pressure Acoustics, Frequency Domain
interface.
TY P I C A L W AV E S P E E D
See Typical Wave Speed for the Pressure Acoustics, Frequency Domain interface.
REF ERENCE VAL UES
See Reference Values for the Compressible Potential Flow interface.
DEPENDENT VA RIA BLES
This interface defines these dependent variables (fields), the Mean flow velocity
potential Phi, the Density rho and Velocity potential phi. The name can be changed but
the names of fields and dependent variables must be unique within a model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Quadratic (the default), Linear, Cubic, or Quartic for the Velocity potential, Mean flow
velocity potential and Density. Specify the Value type when using splitting of complex
variables—Real or Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the Aeroacoustics
with Flow, Frequency Domain User Interface
• Theory Background for the Aeroacoustics Branch
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
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Domain, Boundary, Edge, Point, and Pair Nodes for the
Aeroacoustics with Flow, Frequency Domain User Interface
Because The Aeroacoustics with Flow, Frequency Domain User Interface is a
multiphysics interface, almost every node is shared with, and described for, other
interfaces. The Aeroacoustics with Flow, Transient User Interface also shares the same
nodes as listed in this section. Below are links to the domain, boundary, edge, point,
and pair nodes as indicated.
These nodes are described in this section:
• Aeroacoustics with Flow Model
• Initial Values
These nodes are described for the Aeroacoustics, Frequency Domain interface:
• Aeroacoustics Model
• Normal Mass Flow
• Continuity
• Normal Velocity
• Impedance and Pair Impedance
• Plane Wave Radiation
• Interior Sound Hard Boundary (Wall)
• Sound Hard Boundary (Wall)
• Mass Flow Point Source
• Sound Soft Boundary
• Mass Flow Circular Source
• Velocity Potential
• Mass Flow Line Source on Axis
• Vortex Sheet
These nodes are described for the Compressible Potential Flow interface:
• Compressible Potential Flow Model
• Mass Flow
• Normal Flow
• Slip Velocity
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• Periodic Condition
In the COMSOL Multiphysics Reference Manual:
• Using Symmetries
• Continuity on Interior Boundaries
• Identity and Contact Pairs
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
Aeroacoustics with Flow Model
The Aeroacoustics with Flow Model node adds equations for modeling aeroacoustics
with compressible potential flow. The background field velocity variables of the
Aeroacoustics physics interface are by default set equal to the flow velocity of the
compressible potential flow.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
AEROACOUSTICS WITH FLOW MODEL
Enter a Ratio of specific heats  (dimensionless). The default is 1.4. Also enter a Force
potential (SI unit: J/kg). The default is 0 J/kg.
Initial Values
The Initial Values node adds initial values for the mean flow velocity potential, the
density, and the velocity potential. Right-click the interface node to add more Initial
Values.
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237
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter a value or expression for the initial values of Velocity potential phi (SI unit: m2/
s) (the default is 0 m2/s), Mean flow velocity potential Phi (SI unit: m2/s) (the default
is 0 m2/s), and Density rho (SI unit: kg/m3) (the default is aepf.rhoref).
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The Aeroacoustics with Flow,
T r a ns i e nt U s e r In t erface
The Aeroacoustics with Flow, Transient (atpf) user interface (
), found under the
Acoustics>Aeroacoustics branch (
) in the Model Wizard, combines Transient
Aeroacoustics and Compressible Potential Flow and connects the result from
compressible potential flow simulation with the Aeroacoustic model. It enables the
calculations of density and mean flow velocity potential in an ideal barotropic,
irrotational fluid at constant entropy is combined with aeroacoustic modeling. The
interface is available for 3D, 2D, 1D, and 2D and 1D axisymmetric geometries.
When this interface is added, these default nodes are also added to the Model Builder:
Aeroacoustics with Flow Model, Sound Hard Boundary (Wall), Slip Velocity, and Initial
Values. For axisymmetric models an Axial Symmetry node is also added. Right-click the
main node to add other features that implement boundary conditions and sources.
This interface is limited to flows with a Mach number M < 1, partly due
to limitations in the potential flow formulation and partly due to the
acoustic boundary settings needed for supersonic flow.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
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The default identifier (for the first interface in the model) is atpf.
The remainder of this settings window, and all the nodes, are shared with
The Aeroacoustics with Flow, Frequency Domain User Interface.
The Initial Values for this multiphysics interface is described in this
section. For links to the other nodes, go to Domain, Boundary, Edge,
Point, and Pair Nodes for the Aeroacoustics with Flow, Frequency
Domain User Interface.
• Show More Physics Options
• Theory Background for the Aeroacoustics Branch
Initial Values
The Initial Values node adds initial values for the mean flow velocity potential, density,
velocity potential, and velocity-potential time derivative. Right-click the interface node
to add more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter a value or expression for the initial values.
• Velocity potential phi (SI unit: m2/s). The default is 0 m2/s.
• Velocity potential, first time derivative phit (SI unit: m2/s2). The default is 0 m2/
s2 .
• Mean flow velocity potential Phi (SI unit: m2/s). The default is 0 m2/s.
• Density rho (SI unit: kg/m3). The default is atpf.rhoref.
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Theory Background for the
Ae r o a c o us t i c s Bran ch
The scientific field of aeroacoustics deals with the interaction between a mean
background flow and an acoustic field propagating in this flow. In general this concerns
both the, very complex, description of the creation of sound by turbulence in the
background flow, but also the influence the mean flow has on the propagation of an
externally created sound field, that is, flow born noise. The capabilities of the
aeroacoustics user interfaces in COMSOL Multiphysics® only cover the flow born
noise situation, and only in the limit where the background flow is a compressible
potential flow. This restricts the applications of the interface to systems where the
background flow is well described by a compressible potential flow, that is a flow that
is inviscid, barotropic, and irrotational. The sound sources also need to be external to
the flow or at least they need to be represented by simple well defined sources.
Application areas typically include modeling of how jet engine noise is influenced by
the mean flow.
In the formulation adopted here to couple the acoustics with the fluid dynamics is
based on the potential field for the particle velocity.
This section presents the basic mathematical framework for the aeroacoustic equations
solved, starting with the equations for the acoustic waves in a given mean-flow velocity
field. Then follows a presentation of the equations describing the dynamics of this
background flow, which is assumed to be compressible, inviscid, barotropic, and
irrotational.
In this section:
• Linearized Potential Flow Aeroacoustics
• Compressible Potential Flow
• Frequency Domain Equations
• Time Dependent Equation
• Mode Analysis Study in Boundary Mode Aeroacoustics
THEORY BACKGROUND FOR THE AEROACOUSTICS BRANCH
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241
Linearized Potential Flow Aeroacoustics
The basic dependent variable is the velocity potential  conventionally defined by the
relationship
v = 
where v  v(r, t) is the particle velocity associated with the acoustic wave motion. The
total particle velocity is given by
v tot  r, t  = V  x  + v  r t 
(6-1)
where V denotes the local mean velocity for the fluid motion. The dynamic equations
for this mean-flow field are described in the next subsection. For now, just assume V
to be a given irrotational background velocity field; hence, also the mean-flow
velocity can be defined in terms of a potential field , by V.
The equation for the velocity potential  , governing acoustic waves in a background
flow with mean velocity, V, mean density, , and mean speed of sound, cmf, is
  
   
– -------+ V   +     – -------+ V   V = 0
2 t t
2 t
c mf
c mf
(6-2)
In deriving this equation, all variables appearing in the full nonlinear fluid-dynamics
equations were first split in time-independent and acoustic parts, in the manner of
Equation 6-1. Then, linearizing the resulting equations in the acoustic perturbation
and eliminating all acoustic variables except the velocity potential gives Equation 6-2.
Thus, the density  in this equation is the time-independent part. The corresponding
acoustic part is ar, tpr, tcmf2 where p is the acoustic pressure, given by

p  r t  = –   + V  
t
Hence, once Equation 6-2 has been solved for the velocity potential the acoustic
pressure can easily be calculated.
When transformed to the frequency domain, the wave Equation 6-2 reads


 i + V   V = 0
– -------i  i + V    +     – -------2
2
c mf
c mf
while the acoustic pressure is
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CHAPTER 6: THE AEROACOUSTICS BRANCH
p  r  = –   i + V   
Typical boundary conditions include:
• Sound-hard boundaries or walls
• Sound-soft boundaries
• Impedance boundary conditions
• Radiation boundary conditions
Compressible Potential Flow
Linearized Potential Flow Aeroacoustics presented the equations for aeroacoustic
waves in a background mean-flow field characterized by its velocity, density, and sound
speed. This section discusses the equations of motion and state for the fluid in some
detail.
Consider a compressible and inviscid fluid in some domain . The motion and state of
the fluid is described by its velocity V, density , pressure p, and total energy per unit
volume e. Its dynamics is governed by the Euler equations, expressing the conservation
of mass, momentum, and energy:

+    V  = 0
t
V

+  V   V + p = f
t

(6-3)
e
----+     e + p V  = 0
t
Here a volume force f has been included on the right-hand side of the momentum
equation, whereas a possible heat-source term on the right-hand side of the energy
equation (the last one) has been set to zero.
To close this system of five equations with six unknowns, an equation of state is
required. Here this is taken to be the equation for an ideal barotropic fluid,
 
p = p 0  ------
  0
where cp cV is the ratio between the specific heats at constant pressure and constant
volume, while p0 and 0 are reference quantities for the pressure and the density,
THEORY BACKGROUND FOR THE AEROACOUSTICS BRANCH
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243
respectively, valid at some point in space. An alternative form of the ideal-fluid state
equation is
p =    – 1 e
The assumption that the fluid is barotropic means that pp. Taking the total time
derivative and using the chain rule, leads to the relation
dp d
2 d
dp
=
c
dt
d dt
dt
where, using the equation of state,
c =
p
 --
defines the speed of sound in the ideal fluid.
Assuming the flow to be irrotational, there exists a velocity potential field, , such that
V. If, in addition, the volume force is assumed to be given by f, where
 is referred to as the force potential, the second of Equation 6-3 can be integrated to
yield the Bernoulli equation
–1
p0 

  1
2

1 2
+  ---  + p 0 ----------------------- +  = --- v 0 + ----------------------- +  0
t  2


–
1  0
2

  – 1 
0
In this equation, two additional reference quantities have entered: the velocity, v0, and
the force potential, 0, both valid at the same reference point as p0 and 0. Note, in
particular, that neither the pressure, p, nor the energy per unit volume, e, appears in
this equation.
TIME DEPENDENT STUDY
Collecting the results, the equations governing the compressible, inviscid, irrotational
flow of an ideal fluid are
2
–1
p0 
v0

2

  1
+  ---  + p 0 ----------------------- +  = -------- + ----------------------- +  0


–
1  0
t  2
2

  – 1 
0

+       = 0
t
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c =
p
 --
  cp  cV
where  is the specific-heat ratio cp/cV and  denotes the force potential, that is, the
potential energy per unit mass measured in J/kg. In this equation, subscript 0 signifies
reference quantities that apply at a specific point or surface. Thus, p0 is a reference
pressure, 0 is a reference density, v0 is a reference velocity, and 0 is a reference force
potential.
STATIONARY STUDY
In a stationary study, the same equation is used, but all time derivatives are set to zero,
such that:
–1
2
p0
p 0
v
  2

 
- +  = -----0 + ---------------------- + 0
 -------------- + ----------- -----------------

–
1
2
2


–
1  0


0
     = 0
Frequency Domain Equations
The Aeroacoustics, Frequency Domain User Interface is designed for the analysis of
aeroacoustics problems in the frequency domain and it solves for the velocity
potential . This section briefly describes the equations solved.
For harmonic waves of the form
  r t  =   r e
it
the governing frequency domain—or time-harmonic—equation is


– -------i  i + V    +     – ------- i + V   V = 0
2
2
c mf
c mf
In 2D, where
  r t  =   x y e
i  t – kz z 
the out-of-plane wave number kz enters the equations when the  operators are
expanded:
THEORY BACKGROUND FOR THE AEROACOUSTICS BRANCH
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245




– i  ----------2-  i + V   – ik z V z   +      – V ----------2-  i + V   – ik zV z  


c mf
c mf

2
+  k z  + ik z V z -----------2  i + V   – ik z V z   = 0
c mf
The default value of the out-of-plane wave number is 0, that is, no wave propagation
perpendicular to the 2D plane. In a mode analysis solve for kz.
For 2D axisymmetric models
  r t  =   r z  e
i  t – m 
the circumferential wave number m similarly appears in the equation as a parameter:




m2
– i  -------2  i + V    +      – V -------2  i + V    +  ----2-  = 0
r


cs
cs
The background velocity field, V, cannot have a circumferential
component because the flow is irrotational.
Time Dependent Equation
The Aeroacoustics, Transient User Interface is designed for the analysis of
aeroacoustics problems in the time domain and it solves for the velocity potential, .
This section briefly describes the equations solved.
For modeling of aeroacoustics in the frequency domain, The
Aeroacoustics, Frequency Domain User Interface contains additional
functionality that is not applicable for modeling in the time domain.
The following equation governs acoustic waves in a mean flow:
   
  
– -------+ V   +     – -------+ V   V = 0
2 
2 t t


c mf
c mf  t
(6-4)
Here  (SI unit: kg/m3) is the density, V (SI unit: m/s) denotes the mean velocity,
and cmf (SI unit: m/s) refers to the speed of sound. The software solves the equation
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CHAPTER 6: THE AEROACOUSTICS BRANCH
for the velocity potential , with SI unit m2/s. The validity of this equation relies on
the assumption that , V, and cmf are approximately constant in time, while they may
be functions of the spatial coordinates.
The background velocity field, V, cannot have a circumferential
component because the flow is irrotational.
Mode Analysis Study in Boundary Mode Aeroacoustics
The Boundary Mode Aeroacoustics User Interface is designed to solve aeroacoustics
mode analysis problems on boundaries in 3D and 2D axisymmetric geometries. This
section briefly describes the Mode Analysis Study in Boundary Mode Aeroacoustics.
The boundary mode analysis type in 3D uses the eigenvalue solver to solve the
equation
(6-5)




– i  ----------2-  i +  V t    – ik zV n   +      – V t ------------  i +  V t    – ik zV n  
2


c mf
c
mf
+
2
 kz 

+ ik zV n -----------2  i +  V    – ik zV n   = 0
c mf
for the eigenmodes, , and eigenvalues, ikz, on a bounded two-dimensional
domain, , given well-posed edge conditions on . In this equation, is the velocity
potential,  is the density, cmf is the speed of sound,  is the angular frequency, and kz
is the out-of-plane wave number or propagation constant. Furthermore, Vt denotes
the mean velocity in the tangential plane while Vn is the mean-velocity component in
the normal direction.
Although the out-of-plane wave number is called kz, the two-dimensional
surface on which Equation 6-5 is defined does not necessarily have to be
normal to the z-axis for 3D geometries.
THEORY BACKGROUND FOR THE AEROACOUSTICS BRANCH
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247
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CHAPTER 6: THE AEROACOUSTICS BRANCH
7
The Thermoacoustics Branch
This chapter describes the multiphysics interfaces that combine pressure acoustics
and thermoacoustics. The interfaces are found under the Thermoacoustics
branch (
) in the Model Wizard.
• The Thermoacoustics, Frequency Domain User Interface
• The Thermoacoustic-Solid Interaction, Frequency Domain User Interface
• The Thermoacoustic-Shell Interaction, Frequency Domain User Interface
• Theory Background for the Thermoacoustics Branch
249
The Thermoacoustics, Frequency
Domain User Interface
The Thermoacoustics, Frequency Domain (ta) user interface (
), found under the
Thermoacoustics branch (
) in the Model Wizard, combines the Pressure Acoustics,
Frequency Domain interface with Thermoacoustics features. The thermoacoustic
feature node options are selected further from Mechanical and Thermal menus. This
interface solves for the acoustic pressure p, the velocity variation u (particle velocity),
and the acoustic temperature variations T. It is available for 3D, 2D, and 1D Cartesian
geometries as well as for 2D and 1D axisymmetric geometries.
When this interface is added, these default nodes are also added to the Model Builder—
Thermoacoustics Model, Sound Hard Wall, Isothermal, Pressure Acoustics Model,
Acoustic-Thermoacoustic Boundary, and Initial Values. Right-click the Thermoacoustics
node to add other features that implement boundary conditions and sources.
The Thermoacoustics, Frequency Domain interface is necessary when modeling
acoustics accurately in geometries with small dimensions. Near walls viscosity and
thermal conduction become important because it creates a viscous and a thermal
boundary layer (also called the viscous and thermal penetration depth) where losses are
significant. For this reason it is necessary to include thermal conduction effects and
viscous losses explicitly in the governing equations. For this reason, the thermoacoustic
interface solves the full linearized Navier-Stokes (momentum), continuity, and energy
equations. It solves for the propagation of compressible linear waves in a general
viscous and thermally conduction fluid. Thermoacoustics is also known as
thermoviscous acoustics or as viscothermal acoustics.
Due to the detailed description necessary when modeling thermoacoustics, the model
simultaneously solves for the acoustic pressure p, the velocity vector u, and the
acoustic temperature variations T.
The length scale at which the thermoacoustic description is necessary is given by the
thickness of the, above-mentioned, viscous boundary layer, which is
v =
----------f 0
and the thickness of the thermal boundary layer
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
t =
k -----------------f 0 C p
The thickness of both boundary layers depends on the frequency f and they decreases
with increasing frequency. The ratio of the two length scales is related to the
non-dimensional Prandtl number Pr, by
v
----- =
t
C
----------p- =
k
Pr
which define the relative importance of the thermal and viscous effects for a given
material. In air at 20oC and 1 atm the viscous boundary layer thickness is 0.22 mm at
100 Hz while it is only 55 m in water under the same conditions. The Prandtl
number is 0.7 in air and 7 in water.
Evaluate the value of the viscous and thermal boundary layer thickness as
well as the Prandtl number in post processing. They are defined by the
variables ta.d_visc, ta.d_therm, and ta.Pr, respectively.
The physical quantities commonly used in the Thermoacoustics interfaces are defined
in Table 7-1 below.
TABLE 7-1: THERMOACOUSTICS, FREQUENCY DOMAIN PHYSICAL QUANTITIES
QUANTITY
SYMBOL
SI UNIT
ABBREVIATION
Pressure
p
pascal
Pa
Temperature
T
kelvin
K
Particle velocity
u = u, v, w
meter/second
m/s
Dynamic viscosity

pascal-second
Pa·s
Bulk viscosity
B
pascal-second
Pa·s
Thermal conductivity
k
watt/meter-kelvin
W/(m·K)
3
Heat capacity at constant
pressure
Cp
joule/meter -kelvin
J/(m3·K)
Isothermal compressibility
T
1/pascal
1/Pa
Coefficient of thermal
expansion
0
1/kelvin
1/K
Ratio of specific heats

(dimensionless)
1
Frequency
f
hertz
Hz
Wave number
k
1/meter
1/m
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251
TABLE 7-1: THERMOACOUSTICS, FREQUENCY DOMAIN PHYSICAL QUANTITIES
QUANTITY
SYMBOL
SI UNIT
ABBREVIATION
Equilibrium pressure
p0
pascal
Pa
Equilibrium density
0
kilogram/meter3
kg/m3
Equilibrium temperature
T0
kelvin
K
Speed of sound
c
meter/second
m/s
Acoustic impedance
Z
pascal-second/meter
Pa·s/m
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is ta.
DOMAIN SELECTION
The default setting is to include All domains in the model. To choose specific domains,
select Manual from the Selection list.
SOUND PRESSURE LEVEL SETTINGS
See the settings for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
TY P I C A L WA V E S P E E D
Enter a value or expression for the Typical wave speed for perfectly matched layers cref
(SI unit: m/s).
DEPENDENT VARIABLES
This interface defines these dependent variables (fields), the Pressure p, the Velocity
field, u and its components, and the Temperature variation, T. The name can be
changed but the names of fields and dependent variables must be unique within a
model.
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. From the
Discretization of fluids list select the element order for the velocity components, the
temperature, and the pressure: P2+P1 (the default) or P3+P2.
• P2+P1 means second-order elements for the velocity components and the
temperature. The pressure field has second-order elements in pressure acoustic
domains and linear elements in thermoacoustic domains.
• P3+P2 means third-order elements for the velocity components and the
temperature. The pressure field has third-order elements in the pressure acoustic
domains and second-order elements in the thermoacoustic domains. This can add
additional accuracy but it also adds additional degrees of freedom compared to
P2+P1 elements.
For both options, the velocity components and the temperature share the
same element order as they vary similarly over the same length scale in the
acoustic boundary layer. Therefore, both require the same spatial
accuracy.
Specify the Value type when using splitting of complex variables—Real or Complex (the
default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustics, Frequency Domain User Interface
• Theory Background for the Thermoacoustics Branch
• Uniform Layer Waveguide: Model Library path Acoustics_Module/
Verification_Models/uniform_layer_waveguide
• Generic 711 Coupler: An Occluded Ear-Canal Simulator: Model
Library path Acoustics_Module/Industrial_Models/generic_711_coupler
THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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253
Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustics, Frequency Domain User Interface
Because The Thermoacoustics, Frequency Domain User Interface is a multiphysics
interface, some nodes are shared with, and described for, other interfaces. Below are
links to the domain, boundary, edge, point, and pair nodes available and listed in
alphabetical order.
Right-click the interface node to add these nodes from the main context menu. Some
nodes are also available from the Mechanical and Thermal submenus (listed in
alphabetical order):
• Acoustic-Thermoacoustic Boundary
• Slip
• Adiabatic
• Sound Hard Wall
• Heat Source
• Stress
• Initial Values
• Symmetry
• Isothermal
• Temperature Variation
• Normal Impedance
• Thermoacoustics Model
• Normal Stress
• Velocity
• No Stress
• Wall
• Pressure (Adiabatic)
The Continuity node with this interface is available as a pair boundary
condition. This gives continuity in pressure, temperature variation,
velocity and in the flux on a pair boundary between thermoacoustic
domains. For a pair boundary between pressure acoustic domains see
Continuity as described for the Pressure Acoustics, Frequency Domain
interface.
For more information about pairs, see Identity and Contact Pairs in the
COMSOL Multiphysics Reference Manual.
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
These nodes are described for the Pressure Acoustics, Frequency Domain interface
(listed in alphabetical order):
• Background Pressure Field
• Interior Sound Hard Boundary (Wall)
• Cylindrical Wave Radiation
• Monopole Source
• Dipole Source
• Normal Acceleration
• Far-Field Calculation
• Periodic Condition
• Impedance
• Plane Wave Radiation
• Incident Pressure Field
• Power Edge Source
• Intensity Edge Source
• Power Point Source
• Intensity Point Source
• Pressure
• Interior Impedance/Pair
Impedance
• Pressure Acoustics Model
• Interior Normal Acceleration
• Spherical Wave Radiation
• Sound Soft Boundary
• Interior Perforated Plate
In the COMSOL Multiphysics Reference Manual:
• Destination Selection
• Continuity on Interior Boundaries
• Identity and Contact Pairs
Thermoacoustics Model
Use the Thermoacoustics Model node to define the model inputs (the background
equilibrium temperature and pressure) and the material properties of the fluid
(dynamic viscosity, bulk viscosity, thermal conductivity, heat capacity at constant
pressure, and equilibrium density) necessary to model the propagation of acoustic
compressible waves in a thermoacoustic context. Extended inputs are available for the
THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE
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255
coefficient of thermal expansion and the compressibility, which enables modeling of
any constitutive relation for the fluid.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
MODEL INPUTS
This section contains field variables that appear as model inputs, if the current settings
include such model inputs. From the Equilibrium temperature T0 (SI unit: K) list, select
an existing temperature variable (from another physics interface) if available, or select
User defined to define a different value or expression. The default is User defined set to
293.15 K (that is 20oC). From the Equilibrium pressure p0 (SI unit: Pa) list, select an
existing pressure variable (from another physics interface) if available, or select User
defined to define a different value or expression. The default is User defined set to 1 atm.
THERMOACOUSTICS MODEL
Define the material parameters of the fluid by selecting an Equilibrium density—Ideal
gas (the default), From material, or User defined.
• If Ideal gas is selected, also select the Gas constant type—select Specific gas constant
Rs (SI unit: J/(kg·K) (the default) or Mean molar mass Mn (SI unit: kg/mol). Both
take the values From material by default, or select User defined to enter different
values. This model
• If From material is selected the equilibrium density, and its dependence on the
equilibrium pressure p0 and temperature T0 is, is taken from the defined material.
• If User defined is selected, enter a value or expression for the Equilibrium density
0(p0, T0) (SI unit: kg/m3). The default is ta.p0/(287[J/kg/K]*ta.T0).
The other thermoacoustic model parameters defaults use values From material. If User
defined is selected, enter another value or expression for:
• Dynamic viscosity (SI unit: Pa·s). The default is 0 Pa·s.
• Bulk viscosity B (SI unit: Pa·s). The default is 0 Pa·s.
• Thermal conductivity k (SI unit: W/(m·K)). The default is 0 W/(m·K).
• Heat capacity at constant pressure Cp (SI unit: J/(kg·K)). The default is 0 J/(kg·K).
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
THERMAL EXPANSION AND COMPRESSIBILITY
One of the main characteristics of an acoustic wave is that it is a compressional wave.
In the detailed thermoacoustic description this property is closely related to the
constitutive relation between the density, the pressure, and also the temperature. This
results in the important (linear) relation for the acoustic density variation:
 = 0  T p – 0 T 
where  is the density variation, p is the acoustic pressure, T is the acoustic temperature
variations, T is the (isothermal) compressibility of the fluid, and 0 the coefficient of
thermal expansion. If this constitutive relation is not correct then no waves propagate
or possibly they propagate at an erroneous speed of sound. When the From equilibrium
density option (the default) is selected for the coefficient of thermal expansion and the
compressibility, both values are derived from the equilibrium density 0(p0,T0) using
their defining relations
1  0
 T = ------ -------- 0 p
T
1  0
 0 = – ------ -------- 0 T
p
If the equilibrium density 0 is a user defined constant value or the material model does
not define both a pressure and temperature dependence for 0 the coefficient of
thermal expansion and the compressibility needs to be set manually, or it evaluates to 0.
For most materials, selected form the material library, it is necessary to set
the coefficient of thermal expansion and the compressibility using one of
the non-default option.
If the material is air the From equilibrium density option works well as the
equilibrium density 0 = 0(p0,T0) is a function of both pressure and
temperature.
For water the coefficient of thermal expansion is well defined as
0 = 0(T0), while the compressibility can easily be defined using the
From speed of sound option.
The Thermal Expansion and Compressibility section displays if: From material or User
defined is selected as the Equilibrium density under Thermoacoustics Model.
Select an option from the Coefficient of thermal expansion 0 list—From equilibrium
density (the default), From material, or User defined. If User defined is chosen, enter a
value for 0 (SI unit: 1/K = K-1). The default is 0 K-1.
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257
Select an option from the Isothermal compressibility T lists—From equilibrium density
(the default), From speed of sound, From isentropic compressibility, or User defined.
• If User defined is chosen, enter a value for T (SI unit: 1/Pa). The default is 0 (1/
Pa).
• If From speed of sound is chosen, the values for the Speed of sound c (SI unit: m/s)
and Ratio of specific heats  (dimensionless) are taken From material. Or select User
defined for one or both of the options and enter a different value or expression.
• If From isentropic compressibility is chosen, the values for the Isentropic
compressibility 0 (SI unit: 1/Pa) and Ratio of specific heats  (dimensionless) are
taken From material. Or select User defined for one or both of the options and enter
a different value or expression.
See the Theory Background for the Thermoacoustics Branch section for
a detailed description of the governing equations and the constitutive
relations.
Visualize the dissipated energy due to viscosity and thermal conduction in
post processing. Three post processing variables exist: 1) the viscous
power dissipation density ta.diss_visc, 2) the thermal power
dissipation density ta.diss_therm, or 3) the total thermo-viscous power
dissipation density ta.diss_tot.
In certain cases it may be interesting to not include thermal conduction
in the model and treat all processes as adiabatic (isentropic). This is, for
example, relevant for fluids where the thermal boundary layer is much
thinner than the viscous. Not solving for the temperature field T also
saves some degrees of freedom (DOF).
This is achieved by setting the Isothermal compressibility to User defined
and here enter the adiabatic value 0 = ·T. Then, in the solver sequence
under Solver Configuration > Solver 1 > Dependent Variables select Define
by study step to User defined and under > Temperature variation (mod1.T)
click to clear the Solver for this field box.
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Initial Values
The Initial Values node adds initial values for the sound pressure, velocity field, and
temperature variation. Right-click the interface node to add more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter values or expressions for the Pressure p (SI unit: Pa) (the default is 0 Pa), Velocity
field u (SI unit: m/s) (the defaults are 0 m/s), and Temperature variation T (SI unit: K)
(the default is 0 K).
Heat Source
Use the Heat Source node to define the heat source for the thermoacoustic model. This
adds a domain heat source Q to the right-hand side of the energy equation.
DOMAIN SELECTION
From the Selection list, choose the domains to define.
HEAT SOURCE
Enter a value for the Heat source Q (SI unit: W/m3). The default is 0 W/m3.
Sound Hard Wall
Use the Sound Hard Wall node to model a physical wall. The boundary condition
corresponds to the no-slip condition in the thermoacoustic domain (Wall condition)
and the Sound Hard Boundary (Wall) condition in the pressure acoustic domain.
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
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Isothermal
Use the Isothermal node to model a wall that is assumed to be a good thermal
conductor and backed by a large heat reservoir kept at constant temperature. This
implies that the harmonic temperature variations vanish: T = 0.
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Acoustic-Thermoacoustic Boundary
Use the Acoustic-Thermoacoustic Boundary node to couple the thermoacoustic domain
to a pressure acoustic domain. As it is only necessary to solve the full thermally
conduction and viscous model near walls in the boundary layer region, it makes sense
to switch to the classical pressure acoustics outside this region. This saves a lot of
memory and solution time due to the reduced number of degrees of freedom. The
coupling is done normally to the interface:
 – pI +   u + u T  –  2
------- –  B    u I n = – p t n

 3


1
– n   – ------  p t – q  = – n  i  u
0
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
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Pressure (Adiabatic)
Use the Pressure (Adiabatic) node to specify a prescribed pressure pbnd, that acts as a
pressure source at the boundary, typically an inlet or outlet. In the frequency domain
pbnd is the amplitude of a harmonic pressure source. The adiabatic condition states
that no heat flows into or out of the boundary:
p = p bnd
 – pI +   u + u T  –  2
------- –  B    u I n = – p bnd n

 3


– n   – k T  = 0
This condition is in general not physically correct on a solid wall because solids are
generally better thermal conductors than air.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PRESSURE
Enter the value of the Pressure pbnd (SI unit: Pa) at the boundary. The default is 0 Pa.
Wall
Use the Wall node to model a sound-hard wall where the no-slip condition applies. All
velocity components are zero and u 0.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
CONSTRAINT SETTINGS
See Isothermal for the settings.
Symmetry
The Symmetry node for The Thermoacoustics, Frequency Domain User Interface adds
a boundary condition that represents symmetry. This corresponds to the Sound Hard
Boundary (Wall) condition in pressure acoustics domains. In the thermoacoustic
domains it corresponds to the Slip condition for the mechanical degrees of freedom
and the Adiabatic condition for the temperature variation.
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261
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
CONSTRAINT SETTINGS
See Isothermal for the settings.
Velocity
Use the Velocity node to define the prescribed velocities u0 on the boundary: u = u0.
This condition is useful, for example, when modeling a vibrating wall.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
VE L O C I T Y
To define a prescribed velocity for each space direction (x and y, plus z for
3D), select one or more of the Prescribed in x direction, Prescribed in y
direction, and Prescribed in z direction check boxes. Then enter a value or
expression for the prescribed velocities u0, v0, or w0 (SI unit: m/s).
To define a prescribed velocity for each space direction (r and z), select
one or both of the Prescribed in r direction and Prescribed in z direction
check boxes. Then enter a value or expression for the prescribed velocities
u0, or w0 (SI unit: m/s).
CONSTRAINT SETTINGS
See Isothermal for the settings.
Slip
Use the Slip node to prescribe a no-penetration condition specifying zero normal
velocity on the boundary
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nu = 0
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
CONSTRAINT SETTINGS
See Isothermal for the settings.
Stress
Use the Stress node to define the coordinates of the stress vector on the boundary:
2
– pI +   u + u T  –  ------- –  B    u I n = 
3
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
STRESS
Enter the Stress  (SI unit: N/m2) coordinates for each space direction (x, y, and z or
r and z for 2D axisymmetric models). The defaults are 0 N/m2.
No Stress
Use the No Stress node to set the total surface stress equal to zero:
2
– pI +   u + u T  –  ------- –  B    u I n = 0
 3

BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
Normal Stress
Use the Normal Stress node to define the inward normal stress, n, on the boundary:
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263
2
– pI +   u + u T  –  ------- –  B    u I n = –  n n
3
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
NORMAL STRESS
Enter a value or expression for the Inward normal stress n (SI unit: N/m2). The
default is 0 N/m2.
Normal Impedance
Use the Normal Impedance node to specify a normal impedance Z0 on a boundary. This
feature is useful outside the viscous boundary layer, as this condition mimics the
behavior of a corresponding Pressure Acoustics Model with a normal impedance
condition. The boundary condition reads:
2
– pI +   u + u T  –  ------- –  B    u I n = – Z 0  u  n n
 3

BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
NORMAL IMPEDANCE
Enter a value or expression for the Normal impedance Z0 (SI unit: Pas/m). The default
is 0 Pas/m.
Adiabatic
Use the Adiabatic node to define a situation with no heat flow into or out of the
boundary:
– n   – k T  = 0
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
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Temperature Variation
Use the Temperature Variation node to define the temperature variation on the
boundary Tbnd. In the frequency domain this is the amplitude of a harmonic
temperature variation:
T = T bnd e
it
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
TE M P E R A T U R E VA R I A T I O N
Enter a value or expression for the Temperature variation Tbnd (SI unit: K). The default
is 0 K.
CONSTRAINT SETTINGS
See Isothermal for the settings.
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The Thermoacoustic-Solid
Inte r a c t i o n, Freq u en c y D om ai n U ser
Interface
The Thermoacoustic-Solid Interaction, Frequency Domain (tas) multiphysics user
) in the Model Wizard,
interface (
), found under the Thermoacoustics branch (
combines the Thermoacoustics, Frequency Domain, and the Solid Mechanics
interfaces. It is available for 3D, 2D, and 2D axisymmetric geometries.
When this interface is added, these default nodes are also added to the Model Builder—
Thermoacoustics Model, Sound Hard Wall, Isothermal, Free, Pressure Acoustics Model,
Linear Elastic Material, Continuity, and Initial Values. Right-click the Thermoacoustic-Solid
Interaction, Frequency Domain node to add other features that implement boundary
conditions and sources.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is tas.
DOMAIN SELECTION
The default setting is to include All domains in the model. To choose specific domains,
select Manual from the Selection list.
SOUND PRESSURE LEVEL SETTINGS
See the settings for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
TY P I C A L WA V E S P E E D
Enter a value or expression for the Typical wave speed for perfectly matched layers cref
(SI unit: m/s).
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REFERENCE POINT FOR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
All moments are then computed relative to this reference point.
DEPENDENT VA RIA BLES
This interface defines these dependent variables (fields), the Pressure p, the Velocity
field ufluid and its components, the Temperature variation, T, and the Displacement field
usolid and its components. The name can be changed but the names of fields and
dependent variables must be unique within a model.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. From the
Discretization of fluids list select the element order for the velocity components, the
temperature, and the pressure: P2+P1 (the default) or P3+P2.
• P2+P1 means second-order elements for the velocity components and the
temperature. The pressure field has second-order elements in pressure acoustic
domains and linear elements in thermoacoustic domains.
• P3+P2 means third-order elements for the velocity components and the
temperature. The pressure field has third-order elements in the pressure acoustic
domains and second-order elements in the thermoacoustic domains. This can add
additional accuracy but it also adds additional degrees of freedom compared to
P2+P1 elements.
For both options, the velocity components and the temperature share the
same element order as they vary similarly over the same length scale in the
acoustic boundary layer. Therefore, both require the same spatial
accuracy.
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From the Displacement field list select the element order for the displacement field in
the solid: Linear, Quadratic (the default), Cubic, Quartic, or (in 2D) Quintic. Specify the
Value type when using splitting of complex variables—Real or Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustic-Solid Interaction, Frequency Domain User Interface
• Theory Background for the Pressure Acoustics Branch
• Theory Background for the Thermoacoustics Branch
• Theory for the Solid Mechanics User Interface
Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustic-Solid Interaction, Frequency Domain User Interface
Because The Thermoacoustic-Solid Interaction, Frequency Domain User Interface is
a multiphysics interface, almost every node (except Initial Values, Continuity, and
Symmetry) is shared with, and described for, other interfaces. Below are links to the
domain, boundary, edge, point, and pair nodes for each interface as indicated.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
Destination Selection in the COMSOL Multiphysics Reference Manual
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
THERMOACOUSTICS, FREQUENCY DOMAIN MENU
These nodes are described for the Thermoacoustics, Frequency Domain interface
(listed in alphabetical order):
• Adiabatic
• Slip
• Heat Source
• Sound Hard Wall
• Isothermal
• Stress
• Normal Impedance
• Temperature Variation
• No Stress
• Thermoacoustics Model
• Normal Stress
• Velocity
• Pressure (Adiabatic)
• Wall
PRESSURE ACOUSTICS, FREQUENCY DOMAIN MENU
These nodes are described for the Pressure Acoustics, Frequency Domain interface
(listed in alphabetical order):
• Background Pressure Field
• Matched Boundary
• Cylindrical Wave Radiation
• Monopole Source
• Dipole Source
• Normal Acceleration
• Far-Field Calculation
• Periodic Condition
• Impedance
• Plane Wave Radiation
• Interior Impedance/Pair Impedance
• Pressure
• Interior Normal Acceleration
• Pressure Acoustics Model
• Interior Perforated Plate
• Sound Soft Boundary
• Interior Sound Hard Boundary (Wall)
• Spherical Wave Radiation
These nodes are described for the Acoustic-Solid Interaction, Frequency Domain
interface:
• Intensity Edge Source
• Power Edge Source
• Intensity Point Source
• Power Point Source
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269
SOLID MECHANICS MENU
These nodes are described for the Solid Mechanics interface (listed in alphabetical
order):
• Added Mass
• Point Load
• Antisymmetry
• Prescribed Acceleration
• Body Load
• Prescribed Displacement
• Boundary Load
• Prescribed Velocity
• Edge Load
• Roller
• Fixed Constraint
• Spring Foundation
• Free
• Symmetry
• Linear Elastic Material
• Thin Elastic Layer
Initial Values
The Initial Values node adds initial values for the sound pressure, velocity field,
temperature variation, and displacement field. Right-click the interface node to add
more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter values or expressions for the Pressure p (SI unit: Pa), Velocity field ufluid
(SI unit: m/s), Temperature variation T (SI unit: K), and Displacement field usolid
(SI unit: m). The defaults are 0.
Continuity
Use the Continuity node condition in The Thermoacoustic-Solid Interaction,
Frequency Domain User Interface to model continuity in a general sense. This
boundary condition is automatically added between pressure acoustics and
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
thermoacoustic domains, between pressure acoustic and solid domains, and also
between thermoacoustic and solid domains.
The finals condition simply states that the fluid velocity should be equal to the
structure velocity at the thermoacoustic-structure interface:
u fluid = iu solid
where usolid is the deformation field (SI unit: m) and ufluid is the fluid velocity (SI
unit: m/s). Multiplication by i corresponds to a time derivative in frequency
domain.
The Continuity pair condition is a multiphysics feature combining the
features from Thermoacoustics, Pressure Acoustics and Solid Mechanics.
See Continuity on Interior Boundaries and Identity and Contact Pairs as
described in the COMSOL Multiphysics Reference Manual.
• Acoustic-Thermoacoustic Boundary
• Acoustic-Structure Boundary
Symmetry
The Symmetry node in The Thermoacoustic-Solid Interaction, Frequency Domain
User Interface adds a boundary condition that represents symmetry. This corresponds
to the Sound Hard Boundary (Wall) condition in pressure acoustics domains. In the
thermoacoustic domains it corresponds to the Slip condition for the mechanical
degrees of freedom and the Adiabatic condition for the temperature variation. In solid
domains is corresponds to the Symmetry condition (described for the Solid Mechanics
interface).
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
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271
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
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The Thermoacoustic-Shell
Interaction, Frequency Domain User
Interface
This interface requires a Structural Mechanics Module license. For theory
and interface feature descriptions relating to the Shell interface, see the
Structural Mechanics Module User’s Guide.
The interface is only available for 3D geometries, and it is capable of
modeling the coupled thermoacoustics and shell vibrations in the
frequency domain.
The Thermoacoustic-Shell Interaction, Frequency Domain (tash) user interface (
),
) in the Model Wizard, uses the
found under the Acoustics>Thermoacoustics branch (
features from the Thermoacoustics, Frequency Domain and the Shell interfaces to connect
wave propagation in pressure acoustic domains and thermoacoustic domains with the
structural deformation of shell boundaries.
Three types of shells are available—exterior shells (on exterior boundaries), interior
shells (on interior boundaries), and uncoupled shells (on boundaries that are not
connected to any acoustic domain).
When this interface is added, these default nodes are also added to the Model Builder—
Thermoacoustics Model, Pressure Acoustics Model, Sound Hard Wall, Isothermal,
Acoustic-Thermoacoustic Boundary, Exterior Shell (including a Linear Elastic Material
default node), Free, and two Initial Values.
Right-click the Thermoacoustic-Shell Interaction, Frequency Domain node to add other
features that implement boundary conditions and sources.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
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273
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is tash.
DOMAIN SELECTION
The default setting is to include All domains in the model to define the dependent
variables and the equations. To choose specific domains, select Manual from the
Selection list.
SOUND PRESSURE LEVEL SETTINGS
See the settings for Sound Pressure Level Settings for the Pressure Acoustics,
Frequency Domain interface.
TY P I C A L WA V E S P E E D
Enter a value or expression for the Typical wave speed for perfectly matched layers cref
(SI unit: m/s).
THICKNESS
Enter a value for the thickness d (SI unit: m). The default is 0.01 m. Select an Offset
definition—No offset (the default), Relative offset, or Physical offset.
• If Relative offset is selected, enter a value for zrel_offset (dimensionless). The default
is 0.
• If Physical offset is selected, enter a value for zoffset (SI unit: m). The default is 0 m.
REFERENCE POINT F OR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
All moments are then computed relative to this reference point.
FOLD-LINE LIMIT ANGLE
Enter a value for (SI unit: radians). The default is 0.05 radians.
HEIGHT OF EVALUATION IN SHELL, [-1,1]
Enter a value for z(dimensionless). The default is 1.
DEPENDENT VARIABLES
This interface defines these dependent variables (fields), the Pressure p, Velocity field
ufluid (and its components), the Temperature variation T, Displacement field, ushell
(and its components), and the Displacement of shell normals ar (and the components).
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The names can be changed but the names of fields and dependent variables must be
unique within a model.
ADVANCED SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
Normally these settings do not need to be changed. The Use MITC interpolation check
box is selected by default, and this interpolation, which is part of the MITC shell
formulation, should normally always be active.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization.
From the Discretization of fluids list select the element order for the velocity
components, the temperature, and the pressure: P2+P1 (the default) or P3+P2.
• P2+P1 means second-order elements for the velocity components and the
temperature. The pressure field has second-order elements in pressure acoustic
domains and linear elements in thermoacoustic domains.
• P3+P2 means third-order elements for the velocity components and the
temperature. The pressure field has third-order elements in the pressure acoustic
domains and second-order elements in the thermoacoustic domains. This can add
additional accuracy but it also adds additional degrees of freedom compared to
P2+P1 elements.
For both options, the velocity components and the temperature share the
same element order as they vary similarly over the same length scale in the
acoustic boundary layer. Therefore, both require the same amount spatial
accuracy.
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275
From the Displacement field list select the element order for the displacement field of
the shell: Linear or Quadratic (the default). Specify the Value type when using splitting
of complex variables—Real or Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustic-Shell Interaction, Frequency Domain User Interface
• Theory Background for the Pressure Acoustics Branch
• Theory Background for the Thermoacoustics Branch
• Theory for the Shell and Plate User Interfaces in the Structural
Mechanics Module User’s Guide
The links to the nodes described in the Structural Mechanics Module
User’s Guide do not work in the PDF, only from the on line help in
COMSOL Multiphysics.
Domain, Boundary, Edge, Point, and Pair Nodes for the
Thermoacoustic-Shell Interaction, Frequency Domain User Interface
Because The Thermoacoustic-Shell Interaction, Frequency Domain User Interface is
a multiphysics interface, almost every node is shared with, and described for, other
interfaces. Below are links to the domain, boundary, edge, point, and pair nodes as
indicated.
The Continuity node with this interface is available as a pair boundary
condition. This gives continuity in pressure, temperature variation,
velocity and in the flux on a pair boundary between thermoacoustic
domains. For a pair boundary between pressure acoustic domains see
Continuity as described for the Pressure Acoustics, Frequency Domain
interface.
For more information, see Identity and Contact Pairs in the COMSOL
Multiphysics Reference Manual.
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
These nodes are described specifically for this interface:
• Exterior Shell
• Interior Shell
• Initial Values
• Uncoupled Shell
• Initial Values (Boundary)
PRESSURE ACOUSTICS, FREQUENCY DOMAIN MENU
These nodes are described for the Pressure Acoustics, Frequency Domain interface
(listed in alphabetical order):
• Background Pressure Field
• Matched Boundary
• Cylindrical Wave Radiation
• Monopole Source
• Dipole Source
• Normal Acceleration
• Far-Field Calculation
• Periodic Condition
• Impedance
• Plane Wave Radiation
• Incident Pressure Field
• Pressure
• Interior Impedance/Pair Impedance
• Pressure Acoustics Model
• Interior Normal Acceleration
• Sound Soft Boundary
• Interior Perforated Plate
• Spherical Wave Radiation
• Interior Sound Hard Boundary (Wall)
These nodes are described for the Acoustic-Solid Interaction, Frequency Domain
interface:
• Intensity Edge Source
• Power Edge Source
• Intensity Point Source
• Power Point Source
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277
THERMOACOUSTICS, FREQUENCY DOMAIN MENU
These nodes are described for the Thermoacoustics, Frequency Domain interface
(listed in alphabetical order):
• Acoustic-Thermoacoustic Boundary
• Slip
• Adiabatic
• Sound Hard Wall
• Heat Source
• Stress
• Isothermal
• Symmetry
• Normal Impedance
• Temperature Variation
• Normal Stress
• Thermoacoustics Model
• No Stress
• Wall
• Pressure (Adiabatic)
SOLID MECHANICS MENU
These nodes are described for the Solid Mechanics interface and described in this
guide:
• Added Mass
• Pre-Deformation
• Fixed Constraint
• Spring Foundation
• Free
SHELL MENU
These nodes are described for the Shell interface.
Both the Shell interface and its nodes are described in the Structural
Mechanics Module User’s Guide as this interface requires the Structural
Mechanics Module. For that reason, these links do not work in the PDF.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
• Antisymmetry
• Phase
• Body Load
• Pinned
• Change Thickness
• Prescribed Acceleration
• Damping
• Prescribed Displacement/Rotation
• Edge Load
• Prescribed Velocity
• Face Load
• Point Load
• Initial Stress and Strain
• Rigid Connector
• Linear Elastic Material
• Symmetry
• No Rotation
• Thermal Expansion
Applied Force, Applied Moment, and Mass and Moment of Inertia are
described for the Solid Mechanics interface in the Structural Mechanics
Module User’s Guide.
Initial Values
The Initial Values node adds initial values for the pressure, velocity field (and its
components), and the temperature variation. Right-click the interface node to add
more Initial Values.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter a value or expression for the Pressure p (SI unit: Pa), Velocity field ufluid (and its
components) (SI unit: m/s), the Temperature variation T (SI unit: K).
Initial Values (Boundary)
The Initial Values node adds initial values for the displacement field (and its
components) and the displacement of shell normals (and the components).
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279
BOUNDARY SELECTION
The default setting is to include All boundaries in the model. To choose specific
boundaries, select Manual from the Selection list.
INITIAL VALUES
Based on space dimension, enter coordinate values for the Displacement field ushell
(SI unit: m) and Displacement of shell normals ar (dimensionless).
Exterior Shell
Use the Exterior Shell boundary condition to model any deformable shell boundary,
only one side of which is adjacent to an acoustic domain. See Exterior Shell for the
Acoustic-Shell Interaction, Frequency Domain interface for a description of a shell
exterior to a pressure acoustic domain.
The condition for a shell exterior to a thermoacoustic domain simply states that the
fluid velocity should be equal to the structure velocity at the thermoacoustic-shell
interface:
u fluid = iu shell
where ushell is the deformation field (SI unit: m) and ufluid is the fluid velocity (SI
unit: m/s). Multiplication by i corresponds to a time derivative in the frequency
domain.
A Linear Elastic Material node is automatically added to this boundary
condition. Right-click to add more if required.
BOUNDARY SELECTION
From the Selection list, choose Manual to select the boundaries to define or select All
boundaries.
Interior Shell
Use the Interior Shell boundary condition to model any deformable shell with both
sides adjacent to the acoustic domains. See Interior Shell for the Acoustic-Shell
Interaction, Frequency Domain interface for a description of a shell between two a
pressure acoustic domains.
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The condition for a shell between two thermoacoustic domains simply states that the
fluid velocity should be equal to the structure velocity at the thermoacoustic-shell
interface:
u fluid = iu shell
where ushell is the deformation field (SI unit: m) and ufluid is the fluid velocity (SI
unit: m/s). Multiplication by i corresponds to a time derivative in the frequency
domain.
When the shell is positioned between a pressure acoustic domain and a thermoacoustic
domain, the same constraint is set on the thermoacoustic side. On the pressure acoustic
side of the shell, the normal acceleration for the acoustic pressure on the boundary
equals the acceleration based on the second time derivative of the shell displacement
1
– n   – ------  p – q = n  u tt
 0

In addition, the pressure load (force per unit area) on the shell is: Fpn p,
where p is the acoustic pressure in the pressure acoustics domain.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
A Linear Elastic Material node is automatically added to this boundary
condition. Right-click to add more if required.
Uncoupled Shell
Use the Uncoupled Shell boundary condition to model deformable shells that are not
adjacent to the acoustic domains.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
A Linear Elastic Material node is automatically added to this boundary
condition. Right-click to add more if required.
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T he o r y B a c k g rou n d for t h e
Thermoacoustics Branch
The Thermoacoustics, Frequency Domain User Interface is designed for the analysis
of acoustics in viscous and thermally conducting, compressible Newtonian fluids. The
interface solves the linearized Navier-Stokes equation, the continuity equation, and the
energy equation. This corresponds to a small parameter expansion of the dependent
variables. The interface solves for the acoustic pressure variations p, the fluid velocity
variations u, and the acoustic temperature variations T.
The Thermoacoustic interfaces are available for 3D, 2D, and 1D Cartesian geometries
as well as for 2D and 1D axisymmetric geometries. The interface solves problems in
the frequency domain, that is, Frequency Domain, Frequency-Domain Modal, and
Eigenfrequency type analysis. In 2D and 1D axisymmetric systems a Mode Analysis
study is also available.
This theory also applies to The Thermoacoustic-Solid Interaction,
Frequency Domain User Interface and The Thermoacoustic-Shell
Interaction, Frequency Domain User Interface, although the Mode
Analysis study is not available for these interfaces.
In this section:
• The Viscous and Thermal Boundary Layers
• General Linearized Compressible Flow Equations
• Formulation for Eigenfrequency Studies
• Formulation for Mode Analysis
• References for the Thermoacoustics, Frequency Domain User Interface
The Viscous and Thermal Boundary Layers
In general, a tangential harmonic oscillation of amplitude u0 and frequency f applied
to a wall at z  0 creates a viscous wave of the form
u  z  = u0 e
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
f
– -----------0  1 + i z

where, f is the frequency, 0 is the static density, and  is the dynamic viscosity. The
viscous shear waves are therefore dispersive with wavelength
 - = 2
L v = 2 ----------v
 0 f
and highly damped since their amplitude decays exponentially with distance from the
boundary (see Ref. 3). In fact, in just one wavelength, the amplitude decreases to
about 1500 of its value at the boundary. Therefore, the viscous boundary layer
thickness can for most purposes be considered to be less than Lv. The length scale v
is the so-called viscous penetration depth or viscous boundary layer thickness.
Similarly, a harmonically oscillating temperature with amplitude T0 and frequency f at
z 0 creates a thermal wave of the form
T  z  = T0 e
f 0 C p
– -----------------  1 + i z
k
where Cp is the heat capacity at constant pressure and k is the thermal conductivity.
The wavelength is here
k - = 2
L t = 2 -----------------t
 0 fC p
and a decay behavior similar to the viscous waves. The length scale t is here the
thermal penetration depth.
The ratio of viscous wavelength to thermal wavelength is a non-dimensional number
related to the Prandtl number Pr, as
Lv
------ =
Lt
C
----------p- =
k
Pr
In air, this ratio is roughly 0.8, while in water, it is closer to 2.7. Thus, at least in these
important cases, the viscous and thermal boundary layers are of the same order of
magnitude. Therefore, if one effect is important for a particular geometry, so is
probably the other.
General Linearized Compressible Flow Equations
In general, the motion of a viscous compressible Newtonian fluid, including the energy
equation, is governed by the set of equations listed in Equation 7-1:
THEORY BACKGROUND FOR THE THERMOACOUSTICS BRANCH
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283

du
= +F
dt
d
+   u = 0
dt
C p
(7-1)
dp
dT
– 0 T
= –    – k T  +     B u  + Q
dt
dt
2
 = – pI +  = – pI +   u +  u  T  –  ------- –  B    u I
 3

where the dependent variables are pressure p, velocity u, temperature T, and density
. The first equation is the momentum equation (the Navier-Stokes equation), the
second is the continuity equation, and the third is the energy equation formulated
using the Fourier heat law. The last equation defines the total stress tensor  and the
viscous stress tensor  - this is a constitutive relation. See, for example, Ref. 1 to 6 for
further details. The material time derivatives d  dt are in the following expanded
according to
dA
 x t A
--------------------= ------- + u  A
dt
t
where A is a dummy variable.
To close the system above, an equation of state must also be added to the ones
displayed. The equation of state relates local values of pressure, density and
temperature and is therefore an algebraic equation or an ODE, rather than a PDE. A
common form of state equations is to know the density as function of pressure and
temperature,   p, TIn the following, it is assumed that the state equation has this
form.
The basic properties of the fluid are the dynamic viscosity and thermal conductivity
k. The coefficient B is the bulk (or second) viscosity and describes losses due to
compressibility (expansion and contraction of the fluid), where  describes losses due
to shear friction. The bulk viscosity can in some cases be used to model an empirically
observed deviation from Stokes’ assumption but is usually negligible compared to 
unless the motion is really irrotational, see Ref. 3 and Ref. 4. These three properties
are taken to be constant or at most weakly temperature-dependent. The specific heat
at constant pressure Cp and the (isobaric) coefficient of volumetric thermal expansion
0
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
1 
 0 = – -- T
p
are both possibly functions of pressure and temperature.
In the energy equation
 =   u :S  u 
is the viscous dissipation function, that is, the scalar contraction of the viscous stress
tensor  with the rate of train tensor S. Both tensors are seen as functions of a velocity
vector. If the mean velocity is zero, this term vanishes in the following linearization
since it is homogenous of second order in the velocity gradients. Otherwise, it acts as
an oscillating source/sink. In the right-hand sides of Equation 7-1, F and Q are a
volume force and a heat source, respectively.
For small harmonic oscillations about a steady state solution, the dependent variables
and sources can be assumed to take on the following form:
u = u 0 + u'e it
p = p 0 + p'e it
T = T 0 + T'e it
 =  0 + 'e it
F = F 0 + F'e it
Q = Q 0 + Q'e it
Assuming zero mean flow u0 = 0 and after inserting into the governing Equation 7-1,
the steady state equations can be subtracted from the system, which is subsequently
linearized to first order by ignoring terms quadratic in the primed variables. Dropping
the primes for readability yields the thermoacoustic equations:
2
i 0 u =    – pI +   u +  u  T  –  ------- –  B    u I + F
3
i +  0    u  = 0
(7-2)
i 0 C p T = –    – k T  + ipT 0  0 + Q
where the unprimed variables are now the acoustic deviation from the steady state. The
density  is expressed in terms of the pressure and the temperature using a first order
Taylor expansion about the steady quiescent values
THEORY BACKGROUND FOR THE THERMOACOUSTICS BRANCH
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285
1  0
 = p ------ -------- 0 p
1  0
+ T ------ -------- 0 T
T
p
The two thermodynamic quantities (the coefficients terms in square brackets) define
the isobaric coefficient of thermal expansion 0 and the isothermal compressibility T,
according to the following relations
1  0
 T = ------ -------- 0 p
1
1 
= ------- = ------ ----2- =  0
KT
0 c
K0
Cp
 = ------- = ------Cv
KT
T
1  0
 0 = – ------ -------- 0 T
(7-3)
p
where K0 is the isentropic bulk modulus, KT the isothermal bulk modulus, Cv is the
heat capacity at constant volume, c is the speed of sound, and  is the ratio of specific
heats (the adiabatic index). The isothermal compressibility T is related to the
isentropic (or adiabatic) compressibility 0 and the coefficient of thermal expansion 0
via the thermodynamic relations
2
0 T0
 0 =  T – ------------0 Cp
 T =  0
(7-4)
it is derived using the Maxwell relations, see, for example, Ref. 5 and Ref. 6.
Using the state equation, both the equilibrium density and the density variation can be
eliminated from the system of equations. Moreover, the volume fore F is set equal to
zero. For clarity, the equilibrium density is retained, though:
2
i 0 u =    – pI +   u +  u  T  –  ------- –  B    u I
3
i +  0    u  = 0
(7-5)
i 0 C p T = –    – k T  + ipT 0  0 + Q
 = 0  T p – 0 T 
This set of equations describes the behavior of a general compressible fluid under small
harmonic oscillations. This is the system of equations implemented in the
Thermoacoustic interface.
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
IDEAL GAS
For an ideal gas, the equation of state p  RT, where R is the specific gas constant,
leads to
1
 T = -----p0
1
 0 = -----T0
and the density
p
T
 =  0  ------ – ------
 p 0 T 0
Inserting these expressions and dividing the continuity equation by the reference
density, the system of equations takes on the following simplified form:
2
i 0 u =    – pI +   u +  u  T  –  ------- –  B    u I
3
p
T
i  ------ – ------ +   u = 0
 p 0 T 0
i 0 C p T = –    – k T  + ip + Q
This is the system of equations implemented in the Thermoacoustic, Frequency
Domain interface when ideal gas law is selected.
ISENTROPIC (ADIABATIC) CASE
If the process is assumed to be adiabatic and reversible, that is isentropic, the thermal
conductivity is effectively zero. Then also the temperature can be eliminated, giving
for an ideal gas:
2
i 0 u =    – pI +   u +  u  T  –  ------- –  B    u I

 3


(7-6)
1
1
i  ------ – -------------------- p +   u = 0
 p0 0 Cp T 
0
Defining the speed of sound c in analogy with the standard assumptions for linear
acoustics (term in front of the pressure in the continuity equation), it is found that
1
R
11
1
1
----------= ------ – -------------------- = ------  1 – ------- = --------Cp
p0 0 Cp T0
p0
p 0
0 c2
or
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|
287
c =
p
--------00
In the case with a general fluid, the corresponding relation is using Equation 7-3 and
Equation 7-4:
1  0
11
----------= ------- = ------ --------2
 0 p
K0
0 c
2
T 0 1  0 2
1 T0 0
– -------------  ------ ---------  = ------- – ------------KT 0 Cp
T  0 C p  0 T p
where K0 is the adiabatic bulk modulus, KTthe isothermal bulk modulus, and 0the
coefficient of thermal expansion.
ISOTHERMAL CASE
If, on the other hand, the thermal conductivity is high, or the thermoacoustic waves
propagate in a narrow space between highly conductive walls, the temperature can be
assumed to be constant (isothermal assumption) and the system of equations for an
ideal gas becomes:
2
i 0 u =    – pI +   u +  u  T  –  ------- –  B    u I
3
1
i  ------ p +   u = 0
p0
which, again comparing to standard assumptions, gives
11
----------= -----p0
0 c2
or equivalently
c =
p0
-----0
Therefore, thermal conductivity and/or conducting walls decrease the apparent speed
of sound in narrow domains.
THE HELMHOLTZ EQUATION
If the thermodynamic processes in the system are assumed to be adiabatic and viscosity
can be neglected Equation 7-6 reduces to
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
i 0 u =    – pI  = p
1
i -----------2- p +   u = 0
0 c
Now, taking the divergence of the momentum equations and inserting the expression
for the divergence of the velocity, taken from the continuity equations, yields the
Helmholtz equation
2
2

 p + -----p = 0
2
c
Formulation for Eigenfrequency Studies
When performing an eigenfrequency study the governing equations (Equation 7-5)
are on the form:
2
–  0 u =    – pI +   u +  u  T  –  ------- –  B    u I

 3


–  +  0    u  = 0
(7-7)
–    0 C p T – pT 0  0  = –    – k T  + Q
 = 0  T p – 0 T 
where the eigenvalue is  = i. It is important to note that there is a difference
between regular pressure acoustics and thermoacoustics in terms of what modes can
exist and which modes are found during an eigenfrequency study. In pressure acoustics
only the pure acoustic modes exist, here the equations and assumptions made ensure
this. In thermoacoustics on the other hand the equations are formulated for all small
signal components that can exist. This means that other non-acoustic modes also exist
they are thermal and vorticity modes.
VO R T I C I T Y A N D T H E R M A L M O D E S
When solving an eigenfrequency problem in thermoacoustics it is important to take a
close look at the obtained eigenfrequencies and assess if they are acoustic or not. The
nature of the solution is of the form
p  x t   p  x e
– t
= p  x e
it
 =  + i
 = –  + i
where  is the eigenvalue. Typically, eigenvalues exist near the positive real axis, where
  0 . These are exponentially decaying non-acoustic (non-oscillating) modes that
THEORY BACKGROUND FOR THE THERMOACOUSTICS BRANCH
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289
stem from the thermal equation or the deviatoric part of the momentum equation (the
non-pressure and non-volume part of the stress tensor) also called the vorticity modes.
The acoustic eigenvalues on the other hand lie close to the imaginary axis and are
oscillating and slightly damped.
OTHER SPURIOUS MODES
Note that other spurious and non-acoustic modes may also exist when for example a
PML layer is used to model an open boundary. These modes stem from non-physical
phenomena and the scaling inside the PML layer. In all cases it is a good idea to have
an a priori knowledge of the location/type of the eigenvalues, maybe from solving an
lossless pressure acoustics model, and also to look at the modes in terms of, for
example, the pressure field.
Formulation for Mode Analysis
The Mode Analysis study type is available for thermoacoustics in 2D and 1D
axisymmetric models. This type of study is used to determine the form of the
propagating acoustic modes in waveguide structures. The analyzed 2D and 1D
axisymmetric geometries may be thought of as the cross sections of a waveguide. The
spatial dependency in the (out-of-plane) axial z-direction along the waveguide is
assumed to be of the form of a traveling wave with wave number kz. The dependent
variables in 2D are rewritten as
p = p  x y e
– ik z z
u = u  x y e
– ik z z
T = T  x y e
– ik z z
and in 1D axisymmetric as
p = p  r e
– ik z z
u = u  r e
– ik z z
T = T  r e
– ik z z
Using this form of the dependent variables, differentiation with respect to z reduces to
a multiplication with – ik z . The propagating modes are determined by solving an
eigenvalue problem in the variable  = – ik z .
The expression for the pressure may now be written retaining the harmonic time
dependency, as
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CHAPTER 7: THE THERMOACOUSTICS BRANCH
k z =  + i
p = p  x e
– ik z z it
e
= p  x e
z i  t – z 
(7-8)
e
where x is the in plane coordinate(s). The axial wave number is split into a real and an
imaginary part. The imaginary part  of the wave number describes how fast the
propagating modes decay along the waveguide, it is often referred to as the attenuation
coefficient. The real part is related to the phase speed cph of the propagating mode
by cph = /. In thermoacoustics the obtained wave numbers always have an
imaginary part as the modeled system always includes losses. The relation between the
angular frequency  and the axial wave number kz is called the dispersion relation.
• Mode Analysis Study
• Mode Analysis in the COMSOL Multiphysics Reference Manual
References for the Thermoacoustics, Frequency Domain User Interface
1. W. M. Beltman, P. J. M. van der Hoogt, R. M. E. J. Spiering, and H. Tijdeman,
“Implementation and Experimental Validation of a New Viscothermal Acoustic Finite
Element for Acousto-Elastic Problems,” Journal of Sound and Vibration, vol. 216,
no. 1, pp. 159–185, 1998.
2. M. Malinen, M. Lyly, and others, “A Finite Element Method for the Modeling of
Thermo-Viscous Effects in Acoustics,” Proc. ECCOMAS 2004, Jyväskylä, 2004.
3. D. T. Blackstock, Fundamentals of Physical Acoustics, John Wiley and Sons, Inc.,
2000.
4. H. Bruus, Theoretical Microfluidics, Oxford University Press, 2010.
5. G. K. Bachelor, An Introduction to Fluid Dynamics, Cambridge University Press,
2000.
6. B. Lautrup, Physics of Continuous Matter, Exotic and Every Day Phenomena in
the Macroscopic World, Second Edition, CRC Press, 2011.
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8
The Structural Mechanics Branch
This chapter describes the enhanced Solid Mechanics interface, which is found
) in the Model Wizard and included with
under the Structural Mechanics branch (
the Acoustics Module. This version of the interface simplifies multiphysics
modeling for the acoustics-structure interaction.
In this chapter:
• The Solid Mechanics User Interface
• Theory for the Solid Mechanics User Interface
• Calculating Reaction Forces
• Geometric Nonlinearity, Frames, and the ALE Method
• Springs and Dampers
• Damping and Loss
See also The Piezoelectric Devices User Interface described in another chapter.
293
The Solid Mechanics User Interface
The Solid Mechanics (solid) user interface (
), found under the Structural Mechanics
branch (
) in the Model Wizard, has the equations and functionality for stress analysis
and general linear solid mechanics, solving for the displacements. The Linear Elastic
Material is the default material, which adds a linear elastic equation for the
displacements and has a settings window to define the elastic material properties.
When this physics user interface is added, these default nodes are also added to the
Model Builder— Linear Elastic Material, Free (a boundary condition where boundaries
are free, with no loads or constraints), and Initial Values. Right-click the Solid Mechanics
node to add nodes that implement other solid mechanics material models, boundary
conditions, and loads.
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first user interface in the model) is solid.
DOMAIN SELECTION
The default setting is to include All domains in the model to define the displacements
and the equations that describe the solid mechanics. To choose specific domains, select
Manual from the Selection list.
2D APPROXIMATION
From the 2D approximation list select Plane stress or Plane strain (the
default). For more information see the theory section.
When modeling using plane stress, the Solid Mechanics interface solves
w
for the out-of-plane strain displacement derivative, ------- , in addition to the
Z
displacement field u.
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
THICKNESS
For 2D models, enter a value or expression for the Thickness d (SI unit:
m). The default value of 1 m is suitable for plane strain models, where it
represents a a unit-depth slice, for example. For plane stress models, enter
the actual thickness, which should be small compared to the size of the
plate for the plane stress assumption to be valid.
Use a Change Thickness node to change thickness in parts of the
geometry if necessary.
S T R U C T U R A L TR A N S I E N T B E H AV I O R
From the Structural transient behavior list, select Include inertial terms (the default) or
Quasi-static. Use Quasi-static to treat the elastic behavior as quasi-static (with no mass
effects; that is, no second-order time derivatives). Selecting this option will give a more
efficient solution for problems where the variation in time is slow when compared to
the natural frequencies of the system. The default solver for the time stepping is
changed from Generalized alpha to BDF when Quasi-static is selected.
REFERENCE POINT FOR MOMENT COMPUTATION
Enter the coordinates for the Reference point for moment computation xref (SI unit: m;
variable refpnt). The resulting moments (applied or as reactions) are then computed
relative to this reference point. During the results and analysis stage, the coordinates
can be changed in the Parameters section in the result nodes.
TY P I C A L W AV E S P E E D
The typical wave speed cref is a parameter for the perfectly matched layers (PMLs) if
used in a solid wave propagation model. The default value is solid.cp, the
pressure-wave speed. To use another wave speed, enter a value or expression in the
Typical wave speed for perfectly matched layers field.
DEPENDENT VA RIA BLES
The interface uses the global spatial components of the Displacement field u as
dependent variables. You can change both the field name and the individual
component names. If a new field name coincides with the name of another
displacement field, the two fields (and the interfaces which define them) will share
degrees of freedom and dependent variable component names. You can use this
behavior to connect a Solid Mechanics user interface to a Shell directly attached to the
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boundaries of the solid domain, or to another Solid Mechanics user interface sharing
a common boundary.
A new field name must not coincide with the name of a field of another type, or with
a component name belonging to some other field. Component names must be unique
within a model except when two interfaces share a common field name.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select a
Displacement field—Linear, Quadratic (the default), Cubic, Quartic, or Quintic. Specify the
Value type when using splitting of complex variables—Real or Complex (the default). The
Frame type in the Solid Mechanics interface is always Material.
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics
• About the Body, Boundary, Edge, and Point Loads
• Theory for the Solid Mechanics User Interface
• Stresses in a Pulley: Model Library path COMSOL_Multiphysics/
Structural_Mechanics/stresses_in_pulley
• Eigenvalue Analysis of a Crankshaft: Model Library path
COMSOL_Multiphysics/Structural_Mechanics/crankshaft
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Domain, Boundary, Edge, Point, and Pair Nodes for Solid Mechanics
The Solid Mechanics User Interface has these domain, boundary, edge, point, and pair
nodes listed in alphabetical order. The list also includes subnodes.
• Added Mass
• Low-Reflecting Boundary
• Antisymmetry
• Periodic Condition
• Body Load
• Phase
• Boundary Load
• Point Load
• Change Thickness
• Pre-Deformation
• Damping
• Prescribed Acceleration
• Edge Load
• Prescribed Displacement
• Fixed Constraint
• Prescribed Velocity
• Free
• Roller
• Initial Stress and Strain
• Spring Foundation
• Initial Values
• Symmetry
• Linear Elastic Material
• Thin Elastic Layer
If there are subsequent constraints specified on the same geometrical
entity, the last one takes precedence.
For information about the Perfectly Matched Layers feature, see Infinite
Element Domains and Perfectly Matched Layers in the COMSOL
Multiphysics Reference Manual.
For 2D axisymmetric models, COMSOL Multiphysics takes the axial
symmetry boundaries (at r = 0) into account and automatically adds an
Axial Symmetry node to the model that is valid on the axial symmetry
boundaries only.
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In the COMSOL Multiphysics Reference Manual:
• Harmonic Perturbation—Exclusive and Contributing Nodes
• Continuity on Interior Boundaries
• Identity and Contact Pairs
To locate and search all the documentation, in COMSOL, select
Help>Documentation from the main menu and either enter a search term
or look under a specific module in the documentation tree.
Linear Elastic Material
The Linear Elastic Material node adds the equations for a linear elastic solid and an
interface for defining the elastic material properties. Right-click to add a Damping
subnode.
Also right-click to add an Initial Stress and Strain subnode.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains to define a linear elastic solid and compute
the displacements, stresses, and strains, or select All domains as required.
MODEL INPUTS
Define model inputs, for example, the temperature field of the material uses a
temperature-dependent material property. If no model inputs are required, this section
is empty.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes (except boundary
coordinate systems). The coordinate system is used for interpreting directions of
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orthotropic and anisotropic material data and when stresses or strains are presented in
a local system.
LINEAR ELASTIC MATERIAL
Define the Solid model and the linear elastic material properties.
Solid Model
Select a linear elastic Solid model—Isotropic (the default), Orthotropic, or Anisotropic.
Select:
• Isotropic for a linear elastic material that has the same properties in all directions.
• Orthotropic for a linear elastic material that has different material properties in
orthogonal directions, so that its stiffness depends on the properties Ei, ij, and Gij.
• Anisotropic for a linear elastic material that has different material properties in
different directions, and the stiffness comes from the symmetric elasticity matrix, D.
• Theory for the Solid Mechanics User Interface
• Orthotropic Material
• Anisotropic Material
Density
The default Density  (SI unit: kg/m3) uses values From material. If User defined is
selected, enter another value or expression.
Specification of Elastic Properties for Isotropic Materials
For an Isotropic Solid model, from the Specify list select a pair of elastic properties for
an isotropic material—Young’s modulus and Poisson’s ratio, Young’s modulus and shear
modulus, Bulk modulus and shear modulus, Lamé parameters, or Pressure-wave and
shear-wave speeds. For each pair of properties, select from the applicable list to use the
value From material or enter a User defined value or expression.
Each of these pairs define the elastic properties and it is possible to convert
from one set of properties to another (see Table 8-6).
The individual property parameters are:
• Young’s modulus (elastic modulus) E (SI unit: Pa). The default is 0 Pa.
• Poisson’s ratio  (dimensionless). The default is 0.
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• Shear modulus G (SI unit: N/m2. The default is 0 N/m2.
• Bulk modulus K (SI unit: N/m2). The default is 0 N/m2.
• Lamé parameter  (SI unit: N/m2) and Lamé parameter (SI unit: N/m2). The
defaults are 0 N/m2.
• Pressure-wave speed (longitudinal wave speed) cp (SI unit: m/s). The default
is 0 m/s.
• Shear-wave speed (transverse wave speed) cs (SI unit: m/s). The default is 0 m/s.
This is the wave speed for a solid continuum. In plane stress, for example,
the actual speed with which a longitudinal wave travels is lower than the
value given.
Specification of Elastic Properties for Orthotropic Materials
When Orthotropic is selected from the Solid model list, the material properties vary in
orthogonal directions only. The Material data ordering can be specified in either
Standard or Voigt notation. When User defined is selected in 3D, enter three values in
the fields for Young’s modulus E, Poisson’s ratio , and the Shear modulus G. This
defines the relationship between engineering shear strain and shear stress. It is
applicable only to an orthotropic material and follows the equation
 ij
 ij = -------G ij
ij is defined differently depending on the application field. It is easy to
transform among definitions, but check which one the material uses.
Specification of Elastic Properties for Anisotropic Materials
When Anisotropic is selected from the Solid model list, the material properties vary in
all directions, and the stiffness comes from the symmetric Elasticity matrix, D
(SI unit: Pa). The Material data ordering can be specified in either Standard or Voigt
notation. When User defined is selected, a 6-by-6 symmetric matrix is displayed.
GEOMETRIC NONLINEARITY
In this section there is always one check box, either Force linear strains or Include
geometric nonlinearity.
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Force Linear Strains
If a study step is geometrically nonlinear, the default behavior is to use a large strain
formulation in all domains. There are, however, some cases when the use of a small
strain formulation for a certain domain is needed. In those cases, select the Force linear
strains check box. When selected, a small strain formulation is always used,
independently of the setting in the study step. The default value is that the check box
is cleared (except when opening a model created in a COMSOL Multiphysics version
prior to 4.2a). In this case the state is chosen so that the properties of the model are
conserved.
Include Geometric Nonlinearity
The Include geometric nonlinearity check box is displayed only if the model was created
in a version prior to 4.2a, and geometric nonlinearity was originally used for the
selected domains. It is then selected and forces the Include geometric nonlinearity check
box in the study step to be selected. If the check box is cleared, the check box is
permanently removed and the study step assumes control over the selection of
geometric nonlinearity.
• Geometric Nonlinearity for the Piezoelectric Devices User Interface
• Studies and Solvers in the COMSOL Multiphysics Reference Manual
Change Thickness
The Change Thickness node is available in 2D.
Use the Change Thickness node to model domains with a thickness other than the
overall thickness defined in the physics interface’s Thickness section.
DOMAIN SELECTION
From the Selection list, choose the domains to use a different thickness.
CHANGE THICKNESS
Enter a value for the Thickness d (SI unit: m). This value replaces the overall thickness
for the domains selected above.
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Damping
Right-click the Linear Elastic Material node to add a Damping subnode, which is used
in time-dependent, eigenfrequency, and frequency domain studies to model damped
problems. The node adds Rayleigh damping by default.
The time-stepping algorithms also add numerical damping, which is
independent of any explicit damping added.
For the generalized alpha time-stepping algorithm it is possible to control
the amount of numerical damping added.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node. Or select
Manual from the Selection list to choose specific domains or select All domains as
required.
DAMPING SETTINGS
Select a Damping type—Rayleigh damping (the default), Isotropic loss factor, or
Anisotropic loss factor. If Orthotropic is selected as the Linear Elastic Material Solid
model, Orthotropic loss factor is also available.
Rayleigh Damping
Enter the Mass damping parameter dM (SI unit: 1/s) and the Stiffness damping
parameter dK (SI unit: s). The default values are 0 (no damping).
In this damping model, the damping parameter  is expressed in terms of the mass m
and the stiffness k as
 = dM m +  dK k
That is, Rayleigh damping is proportional to a linear combination of the stiffness and
mass; there is no direct physical interpretation of the mass damping parameter dM
and the stiffness damping parameter dM.
Loss Factor Damping
The loss factor is a measure of the inherent damping in a material when it is
dynamically loaded. It is typically defined as the ratio of energy dissipated in unit
volume per radian of oscillation to the maximum strain energy per unit volume. Loss
factor damping is sometimes referred to as material or structural damping.
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The use of loss factor damping traditionally refers to a scalar-valued loss factor s. But
there is no reasonthat s must be scalar. Because the loss factor is a value deduced from
true complex-valued material data, it can be represented by a matrix of the same
dimensions as the anisotropic stiffness matrix. Especially for orthotropic material, there
should be a set of loss factors of all normal and shear elasticity modulus components.
The following loss-elasticity combinations are available:
• Isotropic Loss Factor Damping
• Anisotropic Loss Factor Damping
• Orthotropic Loss Factor Damping
Isotropic Loss Factor Damping
An isotropic material is described by the different isotropic material constants. Is likely
to only have isotropic loss, described by the isotropic loss factor s. When Isotropic loss
factor is selected as the Damping type, from the Isotropic structural loss factor list, the
default s (dimensionless) uses values From material. If User defined is selected, enter
another value or expression. The default value is 0.
Anisotropic Loss Factor Damping
A symmetric anisotropic material is described by a symmetric 6-by-6 elasticity matrix
D, and the loss can be isotropic or symmetric anisotropic. The loss is described by the
isotropic loss factor s or by a symmetric anisotropic 6-by-6 loss factor matrix D.
Loss factor damping applies to frequency domain studies (that is,
frequency response and damped eigenfrequency studies).
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When Anisotropic loss factor is selected as Damping type from the Loss factor for
elasticity matrix D list, the default D or DVo (dimensionless) uses values From
material. If User defined is selected, choose:
• Isotropic (the default) to enter a single scalar loss factor.
• Symmetric to enter the components of D in the upper-triangular part of a
symmetric 6-by-6 matrix.
The values for the loss factors are ordered in two ways, consistent with the
selection of either Standard (XX, YY, ZZ, XY, YZ, XZ) or Voigt (XX, YY,
ZZ, YZ, XZ, XY) notation in the corresponding Linear Elastic Model. The
default values are 0. If the values are taken from the material, these loss
factors are found in the Anisotropic or Anisotropic, Voigt notation property
group for the material.
Orthotropic Loss Factor Damping
This option is available when Orthotropic is selected as the Linear Elastic
Material Solid model.
An orthotropic material is described by three normal Young’s modulus components
(Ex, Ey, and Ez) and three shear modulus components (Gxy, Gyz, and Gxz. The loss
can be isotropic (described by the isotropic loss factor s) or orthotropic (described by
three plus three orthotropic loss factors corresponding to the elastic moduli
components for an orthotropic material).
When Orthotropic loss factor is selected as the Damping type from the Loss factor for
orthotropic Young’s modulus list E (dimensionless), the default uses values From
material. If User defined is selected, enter another value or expression (defaults are 0).
From the Loss factor for orthotropic shear modulus list, the default G or GVo
(dimensionless) use values From material. If User defined is selected, enter other values
or expressions. The defaults are 0.
The values for the shear modulus loss factors are ordered in two ways, consistent with
the selection of either Standard (XX, YY, ZZ, XY, YZ, XZ) or Voigt (XX, YY, ZZ, YZ, XZ,
XY) notation in the corresponding Linear Elastic Model. The default values are 0. If
the values are taken from the material, these loss factors are found in the Orthotropic
or Orthotropic, Voigt notation property group for the material.
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Initial Values
The Initial Values node adds initial values for the displacement field and structural
velocity field that can serve as an initial condition for a transient simulation or as an
initial guess for a nonlinear analysis. Right-click to add additional Initial Values nodes.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INIT IA L VA LUES
Enter values or expressions for the initial values of the Displacement field u (SI unit: m)
(the displacement components u, v, and w in 3D) (the default is 0 m), and the
Structural velocity field ut (SI unit: m/s) (the default is 0 m/s)).
About the Body, Boundary, Edge, and Point Loads
Add force loads acting on all levels of the geometry to The Solid Mechanics User
Interface. Add a:
• Body Load to domains (to model gravity effects, for example).
• Boundary Load to boundaries (a pressure acting on a boundary, for example).
• Edge Load to edges in 3D (a force distributed along an edge, for example).
• Point Load to points (concentrated forces at points).
For all of these loads, right-click and choose Phase to add a phase for
harmonic loads in frequency-domain computations.
Body Load
Add a Body Load to domains for modeling gravity or centrifugal loads, for example.
Right-click to add a Phase for harmonic loads in frequency-domain computations or to
add Harmonic Perturbation.
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DOMAIN SELECTION
From the Selection list, choose the domains to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
FORCE
Select a Load type—Load defined as force per unit volume (the default) or Total force.
For 2D models, Load defined as force per unit area is also an option.
Then enter values or expressions for the components in the matrix based on the
selection and the space dimension:
• Body load FV (SI unit: N/m3)
• Total force Ftot (SI unit: N). For total force, COMSOL Multiphysics divides the
total force by the volume of the domains where the load is active.
• For 2D models: Load FA (SI unit: N/m2). The body load as force per unit volume
is then the value of F divided by the thickness.
Boundary Load
Add a Boundary Load to boundaries for a pressure acting on a boundary, for example.
Right-click to add a Phase for harmonic loads in frequency-domain computations or to
add Harmonic Perturbation.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
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FORCE
Select a Load type—Load defined as force per unit area (the default), Pressure, or Total
force. For 2D models, Load defined as force per unit length is also an option.
After selecting a Load type, the Load list normally only contains User
defined. When combining the Solid Mechanics interface with another
physics interface, it is also possible to choose a predefined load from this
list.
Then enter values or expressions for the components in the matrix based on the
selection and the space dimension:
• Load FA (SI unit: N/m2). The body load as force per unit volume is then the value
of F divided by the thickness.
• For 2D models: Load FL (SI unit: N/m).
• Total force Ftot (SI unit: N). For total force, COMSOL Multiphysics then divides
the total force by the area of the surfaces where the load is active.
• Pressure p (SI unit: Pa), which can represent a pressure or another external pressure.
The pressure is positive when directed toward the solid.
Edge Load
Add an Edge Load to 3D models for a force distributed along an edge, for example.
right-click to add a Phase for harmonic loads in frequency-domain computations or to
add Harmonic Perturbation.
EDGE SELECTION
From the Selection list, choose the edges to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
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FORCE
Select a Load type—Load defined as force per unit area (the default) or Total force. Then
enter values or expressions for the components in the matrix based on the selection:
• Load FL (SI unit: N/m). When combining the Solid Mechanics interface with, for
example, film damping, it is also possible to choose a predefined load from this list.
• Total force Ftot (SI unit: N). COMSOL Multiphysics then divides the total force by
the volume where the load is active.
Point Load
Add a Point Load to points for concentrated forces at points. Right-click to add a Phase
for harmonic loads in frequency-domain computations or to add Harmonic
Perturbation.
POINT SELECTION
From the Selection list, choose the points to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
FORCE
Enter values or expressions for the components of the Point load Fp (SI unit: N).
Fixed Constraint
The Fixed Constraint node adds a condition that makes the geometric entity fixed (fully
constrained); that is, the displacements are zero in all directions. For domains, this
condition is selected from the More submenu.
D O M A I N , B O U N D A R Y, E D G E , O R PO I N T S E L E C T I O N
From the Selection list, choose, the geometric entity (domains, boundaries, edges, or
points) to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair use. An identity pair has
to be created first.
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CONSTRAINT SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
To Apply reaction terms on all dependent variables, select All physics (symmetric).
Otherwise, select Current physics (internally symmetric) or Individual dependent
variables to restrict the reaction terms as required. Select the Use weak constraints check
box to replace the standard constraints with a weak implementation.
Using Weak Constraints to Evaluate Reaction Forces
Boundary Conditions in the COMSOL Multiphysics Reference Manual
Prescribed Displacement
The Prescribed Displacement node adds a condition where the displacements are
prescribed in one or more directions to the geometric entity (domain, boundary, edge,
or point). For domains, this condition is selected from the More submenu.
If a displacement is prescribed in one direction, this leaves the solid free to deform in
the other directions.
You can also define more general displacements as a linear combination of the
displacements in each direction.
• If a prescribed displacement is not activated in any direction, this is the
same as a Free constraint.
• If a zero displacement is applied in all directions, this is the same as a
Fixed Constraint.
D O M A I N , B O U N D A R Y, E D G E , O R P O I N T S E L E C T I O N
From the Selection list, choose the geometric entity (domains, boundaries, edges, or
points) to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
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COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. If you choose another, local
coordinate system, the displacement components change accordingly.
PRESCRIBED DISPLACEMENT
Define the prescribed displacements using a Standard notation (the default) or a General
notation.
Standard Notation
To define the displacements individually, click the Standard notation button.
Select one or all of the Prescribed in x direction, Prescribed in y direction, and for 3D
models, Prescribed in z direction check boxes. Then enter a value or expression for u0,
v0, and for 3D models, w0 (SI unit: m). For 2D axisymmetric models, select one or
both of the Prescribed in r direction and Prescribed in z direction check boxes. Then
enter a value or expression for u0 and w0 (SI unit: m).
General Notation
Click the General notation to specify the displacements using a general notation that
includes any linear combination of displacement components. For example, for 2D
models, use the relationship
H u = R
v
For H matrix H (dimensionless) select Isotropic, Diagonal, Symmetric, or Anisotropic and
then enter values as required in the field or matrix. Enter values or expressions for the
R vector R (SI unit: m)
For example, to achieve the condition u = v, use the settings
H = 1 –1 
0 0
R = 0
0
which force the domain to move only diagonally in the xy-plane.
CONSTRAINT SETTINGS
See Fixed Constraint for these settings.
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Free
The Free node is the default boundary condition. It means that there are no constraints
and no loads acting on the boundary.
BOUNDARY SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific boundaries or select All boundaries as required.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
Symmetry
The Symmetry node adds a boundary condition that represents symmetry in the
geometry and in the loads. A symmetry condition is free in the plane and fixed in the
out-of-plane direction.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
See Fixed Constraint for these settings.
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Antisymmetry
The Antisymmetry node adds a boundary condition for an antisymmetry boundary,
which must exist in both the geometry and in the loads. An antisymmetry condition is
fixed in the plane and free in the out-of-plane direction.
In a geometrically nonlinear analysis, large rotations must not occur at the
antisymmetry plane because this causes artificial straining.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect.
CONSTRAINT SETTINGS
See Fixed Constraint for these settings.
Roller
The Roller node adds a roller constraint as the boundary condition; that is, the
displacement is zero in the direction perpendicular (normal) to the boundary, but the
boundary is free to move in the tangential direction. See Fixed Constraint for all the
settings.
CONSTRAINT SETTINGS
See Fixed Constraint for these settings.
Periodic Condition
The Periodic Condition node adds a periodic boundary condition. This periodicity
makes uix0uix1 for a displacement ui. You can control the direction that the
periodic condition applies to. If the source and destination boundaries are rotated with
respect to each other, this transformation is automatically performed, so that
corresponding displacement components are connected.
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BOUNDARY SELECTION
From the Selection list, choose the boundaries to define. The software automatically
identifies the boundaries as either source boundaries or destination boundaries.
This works fine for cases like opposing parallel boundaries. In other cases
right-click the Periodic Condition node to add a Destination Selection
subnode to control the destination. By default it contains the selection
that COMSOL Multiphysics has identified.
In cases where the periodic boundary is split into several boundaries
within the geometry, it may be necessary to apply separate periodic
conditions to each pair of geometry boundaries.
PERIODICITY SETTINGS
Select a Type of periodicity—Continuity (the default), Antiperiodicity, Floquet periodicity,
Cyclic symmetry, or User defined. If User defined is selected, select the Periodic in u,
Periodic in v (for 3D and 2D models), and Periodic in w (for 3D and 2D axisymmetric
models) check boxes as required. Then for each selection, choose the Type of
periodicity—Continuity (the default) or Antiperiodicity.
• If Floquet periodicity is selected, enter a k-vector for Floquet periodicity kF (SI unit:
rad/m) for the X, Y, and Z coordinates (3D models), or the R and Z coordinates
(2D axisymmetric models), or X and Y coordinates (2D models).
• If Cyclic symmetry is selected, select a Sector angle—Automatic (the default), or User
defined. If User defined is selected, enter a value for S (SI unit: rad; default value: 0).
For any selection, also enter a Azimuthal node number m (dimensionless; default
value: 0).
CONSTRAINT SETTINGS
See Fixed Constraint for these settings.
• Cyclic Symmetry and Floquet Periodic Conditions
In the COMSOL Multiphysics Reference Manual:
• Periodic Condition and Destination Selection
• Periodic Boundary Conditions
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Initial Stress and Strain
A solid mechanics model can include the Initial Stress and Strain node, which is the
stress-strain state in the structure before applying any constraint or load. Initial strain
can, for example, describe moisture-induced swelling, and initial stress can describe
stresses from heating. Think of initial stress and strain as different ways to express the
same thing. Right-click to add this node to a Linear Elastic Material.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains to define, or select All domains as required.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. The given initial stresses
and strains are interpreted in this system.
INITIAL STRESS AND STRAIN
Enter values or expressions for the Initial stress S0 (SI unit: N/m2) and Initial strain 0
(dimensionless). The default values are 0, which is no initial stress or strain. For both,
enter the diagonal and off-diagonal components (based on space dimension):
• For a 3D Initial stress model, diagonal components S0x, S0y, and S0z and
off-diagonal components S0xy, S0yz, and S0xz, for example.
• For a 3D Initial strain model, diagonal components 0x, 0y, and 0z and off-diagonal
components 0xy, 0yz, and 0xz, for example.
Phase
Add a Phase node to a Body Load, Boundary Load, Edge Load, or Point Load. For
modeling the frequency response the physics interface splits the harmonic load into
two parameters:
• The amplitude, F, which is specified in the feature node for the load.
• The phase (FPh).
Together these define a harmonic load, for which the amplitude and phase shift can
vary with the excitation frequency, f:
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F freq = F  f   cos  2f + F Ph  f  
D O M A I N , B O U N D A R Y, E D G E , O R P O I N T S E L E C T I O N
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains, boundaries, edges, or points to define, or
select All domains, All boundaries, All edges, or All points, as required.
PHASE
Enter the components of Load phase  in radians (for a pressure the load phase  is a
scalar value). Add [deg] to a phase value to specify it using degrees.
Typically the load magnitude is a real scalar value. If the load specified in
the parent feature contains a phase (using a complex-valued expression),
the software adds the phase from the Phase node to the phase already
included in the load.
Prescribed Velocity
The Prescribed Velocity node adds a boundary or domain condition where the velocity
is prescribed in one or more directions. The prescribed velocity condition is applicable
for time-dependent and frequency-domain studies. With this boundary or domain
condition it is possible to prescribe a velocity in one direction, leaving the solid free in
the other directions. For domains, this condition is selected from the More submenu.
The Prescribed Acceleration node is a constraint, and will override any other constraint
on the same selection.
DOMAIN OR BOUNDARY SELECTION
From the Selection list, choose the geometric entity (domains or boundaries) to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. Coordinate systems with
directions that change with time should not be used. If you choose another, local
coordinate system, the velocity components change accordingly.
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P R E S C R I B E D VE L O C I T Y
Select one or all of the Prescribed in x direction, Prescribed in y direction, and for 3D
models, Prescribed in z direction check boxes. Then enter a value or expression for vx,
vy, and for 3D models, vz (SI unit: m/s). For 2D axisymmetric models, select one or
both of the Prescribed in r direction and Prescribed in z direction check boxes. Then
enter a value or expression for vr and vz (SI unit: m/s).
Prescribed Acceleration
The Prescribed Acceleration node adds a boundary or domain condition, where the
acceleration is prescribed in one or more directions. The prescribed acceleration
condition is applicable for time-dependent and frequency-domain studies. With this
boundary condition, it is possible to prescribe a acceleration in one direction, leaving
the solid free in the other directions. For domains, this condition is selected from the
More submenu.
The Prescribed Acceleration node is a constraint, and will override any other constraint
on the same selection.
DOMAIN OR BOUNDARY SELECTION
From the Selection list, choose the geometric entity (domains or boundaries) to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. Coordinate systems with
directions that change with time should not be used. If you choose another, local
coordinate system, the acceleration components change accordingly.
PRESCRIBED ACCELERATION
Select one or all of the Prescribed in x direction, Prescribed in y direction, and for 3D
models, Prescribed in z direction check boxes. Then enter a value or expression for ax,
ay, and for 3D models, az (SI unit: m/s2). For 2D axisymmetric models, select one or
both of the Prescribed in r direction and Prescribed in z direction check boxes. Then
enter a value or expression for ar and az (SI unit: m/s2).
Spring Foundation
The Spring Foundation node has elastic and damping boundary conditions for domains,
boundaries, edges, and points. To select this node for the domains, it is selected from
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the More submenu. Also right-click to add a Pre-Deformation subnode.
The Spring Foundation and Thin Elastic Layer nodes are similar, with the difference
that a Spring Foundation connects the structural part on which it is acting to a fixed
“ground,” while a Thin Elastic Layer acts between two parts, either on an interior
boundary or on a pair boundary.
D O M A I N , B O U N D A R Y, E D G E , O R P O I N T S E L E C T I O N
From the Selection list, choose the geometric entity (domains, boundaries, edges, or
points) to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. The spring and damping
constants are given with respect to the selected coordinate system.
SPRING
Select the Spring type and its associated spring constant of force using Table 8-1 as a
guide. The default option is the spring type for the type of geometric entity and space
dimension, and there are different combinations available based on this.
TABLE 8-1: SPRING TYPES FOR THE SPRING FOUNDATION FEATURE
SPRING TYPE
VARIABLE
SI UNITS
GEOMETRIC ENTITY
LEVEL
SPACE DIMENSION
Spring constant per
unit volume
kV
N/(mm3)
domains
3D, 2D, and 2D
axisymmetric
Total spring constant
ktot
N/m
domains, edges
3D, 2D, and 2D
axisymmetric
Spring constant per
unit area
kA
N/(mm)2
domains,
boundaries
3D, 2D
Spring constant per
unit length
kL
N/(mm)
edges, boundaries
(2D)
3D, 2D
Spring constant
kP
N/m
points
3D, 2D, and 2D
axisymmetric
Force per volume as
function of extension
FV
N/m3
domains
3D, 2D, and 2D
axisymmetric
Total force as function
of extension
Ftot
N
domains,
boundaries, edges
3D, 2D, and 2D
axisymmetric
Force per area as
function of extension
FA
N/m2
domains,
boundaries
3D, 2D
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TABLE 8-1: SPRING TYPES FOR THE SPRING FOUNDATION FEATURE
SPRING TYPE
VARIABLE
SI UNITS
GEOMETRIC ENTITY
LEVEL
SPACE DIMENSION
Force per length as
function of extension
FL
N/m
edges
3D
Force as function of
extension
FP
N
points
3D, 2D, and 2D
axisymmetric
LOSS FACTOR DAMPING
Enter values or expressions in the table for each coordinate based on space dimension
for the Loss factor for spring k. The loss factors act on the corresponding components
of the spring stiffness. All defaults are 0.
VISCOUS DAMPING
Select the Damping type using Table 8-2 as a guide. The default option is the default
damping type for the type of geometric entity and space dimension, and there are
different combinations available based on this.
TABLE 8-2: DAMPING TYPES FOR THE SPRING FOUNDATION FEATURE
DAMPING TYPE
VARIABLE
SI UNITS
GEOMETRIC ENTITY
LEVEL
SPACE DIMENSION
Viscous force per unit
volume
dV
Ns/(mm3)
domains,
boundaries (2D)
3D, 2D
Viscous force per unit
area
dA
Ns/(mm2)
domains,
boundaries
3D, 2D, and 2D
axisymmetric
Total viscous force
dtot
Ns/m
domains,
boundaries,
edges, points
3D, 2D, and 2D
axisymmetric
Viscous force per unit
length
dL
Ns/(mm)
edges
3D
•
Springs and Dampers
• About Spring Foundations and Thin Elastic Layers
Pre-Deformation
Right-click the Spring Foundation or Thin Elastic Layer nodes to add a
Pre-Deformation feature as a subnode and define the coordinates. By including a
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pre-deformation, you can model cases where the unstressed state of the spring is in
another configuration than the one modeled.
D O M A I N , B O U N D A R Y, E D G E , O R P O I N T S E L E C T I O N
From the Selection list, choose the geometric entity (domains, boundaries, edges, or
points) to define.
SPRING PRE-DEFORMATION
Based on space dimension, enter the coordinates for the Spring Pre-Deformation u0
(SI unit: m). The defaults are 0 m.
Thin Elastic Layer
The Thin Elastic Layer node has elastic and damping boundary conditions for
boundaries and acts between two parts, either on an interior boundary or on a pair
boundary. Also right-click to add a Pre-Deformation subnode.
The Thin Elastic Layer and Spring Foundation nodes are similar, with the difference that
a Spring Foundation connects the structural part on which it is acting to a fixed
“ground”.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
PAIR SELECTION
If this node is selected from the Pairs menu, choose the pair to define. An identity pair
has to be created first. Ctrl-click to deselect. A default Free node is added when a Thin
Elastic Layer pair node is added.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. The spring and damping
constants are given with respect to the selected coordinate system.
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SPRING
Select the Spring type and its associated spring constant of force using Table 8-3 as a
guide. The default option is the spring type for the space dimension.
TABLE 8-3: SPRING TYPES FOR THE THIN ELASTIC LAYER FEATURE
SPRING TYPE
VARIABLE
SI UNITS
SPACE DIMENSION
Total spring constant
ktot
N/m
3D, 2D, and 2D
axisymmetric
Spring constant per unit area
kA
N/(mm)2
3D, 2D, and 2D
axisymmetric
Spring constant per unit length
kL
N/(mm)
2D
Total force as function of extension
Ftot
N
3D, 2D, and 2D
axisymmetric
Force per area as function of
extension
FA
N/m2
3D, 2D, and 2D
axisymmetric
Force per length as function of
extension
FL
N/m
2D
LOSS FACTOR DAMPING
Enter values or expressions in the table for each coordinate based on space dimension
for the Loss factor for spring k. The loss factors act on the corresponding components
of the spring stiffness. All defaults are 0.
VISCOUS DAMPING
Select the Damping type using Table 8-4 as a guide. The default option is the default
damping type for the space dimension.
TABLE 8-4: DAMPING TYPES FOR THE THIN ELASTIC LAYER FEATURE
DAMPING TYPE
VARIABLE
SI UNITS
SPACE DIMENSION
2
Viscous force per unit area
dA
Ns/(mm )
3D, 2D, and 2D
axisymmetric
Total viscous force
dtot
Ns/m
3D, 2D, and 2D
axisymmetric
Viscous force per unit length
dL
Ns/(mm)
2D
•
Springs and Dampers
• About Spring Foundations and Thin Elastic Layers
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Added Mass
The Added Mass node is available on domains, boundaries, and edges and can be used
to supply inertia, which is not part of the material itself. Such inertia does not need to
be isotropic, in the sense that the inertial effects are not the same in all directions. To
select this node for the domains, it is selected from the More submenu.
To include the added mass as a static self weight, separate load physics
need to be added for the domains, boundaries, or edges. The Added Mass
node only contributes to the inertia in the dynamic sense.
D O M A I N , B O U N D A R Y, O R E D G E S E L E C T I O N
From the Selection list, choose the geometric entity (domains, boundaries, or edges)
to define.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes. The added mass values are
given with respect to the selected coordinate directions.
M A S S TY P E
Select a Mass type using Table 8-5 as a guide. The default option is the type for the
geometric entity. Then enter values or expressions into the table for each coordinate
based on space dimension. All defaults are 0.
TABLE 8-5: AVAILABLE MASS TYPES BASED ON GEOMETRIC ENTITY
MASS TYPE
VARIABLE
SI UNITS
3
GEOMETRIC ENTITY LEVEL
Mass per unit volume
pV
kg/m
domains
Mass per unit area
pA
kg/m2
domains, boundaries
Mass per unit length
pL
kg/m
edges
Total mass
m
kg
domains, boundaries, edges
About Added Mass
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321
Low-Reflecting Boundary
Use the Low-Reflecting Boundary node to let waves pass out from the model without
reflection in time-dependent analysis. As a default, it takes material data from the
domain in an attempt to create a perfect impedance match for both pressure waves and
shear waves. It may be sensitive to the direction of the incoming wave.
BOUNDARY SELECTION
From the Selection list, choose the boundaries to define.
COORDINATE SYSTEM SELECTION
The Boundary System is selected by default. The Coordinate system list contains any
additional coordinate systems that the model includes.
DAMPING
Select a Damping type—P and S waves (the default) or User defined. If User defined is
selected, enter values or expressions for the Mechanical impedance di (SI unit: Pas/m).
The defaults for all values are 0.5*solid.rho*(solid.cp+solid.cs).
About the Low-Reflecting Boundary Condition
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Theory for the Solid Mechanics User
Interface
The Solid Mechanics User Interface theory is described in this section:
• Material and Spatial Coordinates
• Damping Models
• Coordinate Systems
• Initial Stresses and Strains
• Lagrangian Formulation
• About Linear Elastic Materials
• About Spring Foundations and
Thin Elastic Layers
• Strain-Displacement Relationship
• About Added Mass
• Stress-Strain Relationship
• Geometric Nonlinearity Theory for
the Solid Mechanics User Interface
• Plane Strain and Plane Stress Cases
• About the Low-Reflecting
Boundary Condition
• Axial Symmetry
• Loads
• Cyclic Symmetry and Floquet
Periodic Conditions
• Pressure Loads
• Equation Implementation
• Setting up Equations for Different
Studies
Material and Spatial Coordinates
The Solid Mechanics interface, through its equations, describes the motion and
deformation of solid objects in a 2- or 3-dimensional space. In COMSOL Multiphysics
terminology, this physical space is known as the spatial frame and positions in the
physical space are identified by lowercase spatial coordinate variables x, y, and z (or r,
 , and z in axisymmetric models).
Continuum mechanics theory also makes use of a second set of coordinates, known as
material (or reference) coordinates. These are normally denoted by uppercase
variables X, Y, and Z (or R, , and Z) and are used to label material particles. Any
material particle is uniquely identified by its position in some given initial or reference
configuration. As long as the solid stays in this configuration, material and spatial
coordinates of every particle coincide and displacements are zero by definition.
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323
When the solid objects deform due to external or internal forces and constraints, each
material particle keeps its material coordinates X (bold font is used to denote
coordinate vectors), while its spatial coordinates change with time and applied forces
such that it follows a path
x = x  X t  = X + u  X t 
(8-1)
in space. Because the material coordinates are constant, the current spatial position is
uniquely determined by the displacement vector u, pointing from the reference
position to the current position. The global Cartesian components of this displacement
vector in the spatial frame, by default called u, v, and w, are the primary dependent
variables in the Solid Mechanics interface.
By default, the Solid Mechanics interface uses the calculated displacement and
Equation 8-1 to define the difference between spatial coordinates x and material
coordinates X. This means the material coordinates relate to the original geometry,
while the spatial coordinates are solution dependent.
Material coordinate variables X, Y, and Z must be used in coordinate-dependent
expressions that refer to positions in the original geometry, for example, for material
properties that are supposed to follow the material during deformation. On the other
hand, quantities that have a coordinate dependence in physical space, for example, a
spatially varying electromagnetic field acting as a force on the solid, must be described
using spatial coordinate variables x, y, and z. Any use of the spatial variables will be a
source of nonlinearity if a geometrically nonlinear study is performed.
Coordinate Systems
Force vectors, stress and strain tensors, as well as various material tensors are
represented by their components in a specified coordinate system. By default, material
properties use the canonical system in the material frame. This is the system whose
basis vectors coincide with the X, Y, and Z axes. When the solid deforms, these vectors
rotate with the material.
Loads and constraints, on the other hand, are applied in spatial directions, by default
in the canonical spatial coordinate system. This system has basis vectors in the x, y, and
z directions, which are forever fixed in space. Both the material and spatial default
coordinate system are referred to as the global coordinate system in the user interface.
Vector and tensor quantities defined in the global coordinate system on either frame
use the frame’s coordinate variable names as indices in the tensor component variable
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
names. For example, SXY is the material frame XY-plane shear stress, also known as a
second Piola-Kirchhoff stress, while sxy is the corresponding spatial frame stress, or
Cauchy stress. There are also a few mixed tensors, most notably the deformation
gradient FdxY, which has one spatial and one material index because it is used in
converting quantities between the material and spatial frames.
It is possible to define any number of user coordinate systems on the material and
spatial frames. Most types of coordinate systems are specified only as a rotation of the
basis with respect to the canonical basis in an underlying frame. This means that they
can be used both in contexts requiring a material system and in contexts requiring a
spatial one.
The coordinate system can be selected separately for each added material model, load,
and constraint. This is convenient if, for example, an anisotropic material with different
orientation in different domains is required. The currently selected coordinate system
is known as the local coordinate system.
Lagrangian Formulation
The formulation used for structural analysis in COMSOL Multiphysics for both small
and finite deformations is total Lagrangian. This means that the computed stress and
deformation state is always referred to the material configuration, rather than to
current position in space.
Likewise, material properties are always given for material particles and with tensor
components referring to a coordinate system based on the material frame. This has the
obvious advantage that spatially varying material properties can be evaluated just once
for the initial material configuration and do not change as the solid deforms and
rotates.
The gradient of the displacement, which occurs frequently in the following theory, is
always computed with respect to material coordinates. In 3D:
u u u
X Y Z
u = v v v
X Y Z
w w w
X Y Z
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325
The displacement is considered as a function of the material coordinates (X, Y, Z), but
it is not explicitly a function of the spatial coordinates (x, y, z). It is thus only possible
to compute derivatives with respect to the material coordinates.
About Linear Elastic Materials
The total strain tensor is written in terms of the displacement gradient
1
T
 = ---  u + u 
2
or in components as
1  u m u n 
 mn = --- 
+

2   x n  x m
(8-2)
The Duhamel-Hooke’s law relates the stress tensor to the strain tensor and
temperature:
s = s 0 + C    –  0 –  
where C is the 4th order elasticity tensor, “:” stands for the double-dot tensor product
(or double contraction), s0 and 0 are initial stresses and strains, TTref, and  is
the thermal expansion tensor.
The elastic energy is
1
W s = ---   –  0 –   C    –  0 –  
2
(8-3)
or using the tensor components:
Ws =

i j m n
ijmn
1
0
0
--- C
  ij –  ij –  ij     mn –  mn –  mn  
2
TE N S O R V S . M A T R I X F O R M U L A T I O N S
Because of the symmetry, the strain tensor can be written as the following matrix:
 x  xy  xz
 xy  y  yz
 xz  yz  z
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
Similar representation applies to the stress and the thermal expansion tensors:
 x  xy  xz
s x s xy s xz
s xy s y s yz   xy  y  yz
 xz  yz  z
s xz s yz s z
Due to the symmetry, the elasticity tensor can be completely represented by a
symmetric 6-by-6 matrix as:
D =
D 11 D 12 D 13 D 14 D 15 D 16
C
D 12 D 22 D 23 D 24 D 25 D 26
C
D 13 D 23 D 33 D 34 D 35 D 36
= C
D 14 D 24 D 34 D 44 D 45 D 46
C
D 15 D 25 D 35 D 45 D 55 D 56
C
D 16 D 26 D 36 D 46 D 56 D 66
C
1111
1122
1133
1112
1123
1113
C
C
C
C
C
C
1122
2222
2233
2212
2223
2213
C
C
C
C
C
C
1133
2233
3333
3312
3323
3313
C
C
C
C
C
C
1112
2212
3312
1212
1223
1213
C
C
C
C
C
C
1123
2223
3323
1223
2323
2313
C
C
C
C
C
C
1113
2213
3313
1213
2313
1313
which is the elasticity matrix.
ISOTROPIC MATERIAL AND ELASTIC MODULI
In this case, the elasticity matrix becomes
1– 

 1– 

 1–
E
D = -------------------------------------- 1 +    1 – 2 
0
0
0
1
– 2--------------2
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
– 2--------------2
0
0
0
0
0
0
1
– 2--------------2
and the thermal expansion matrix is:
 0 0
0  0
0 0 
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Different pairs of elastic moduli can be used, and as long as two moduli are defined,
the others can be computed according to Table 8-6.
TABLE 8-6: EXPRESSIONS FOR THE ELASTIC MODULI.
DE
DKG
D
9KG -----------------3K + G
3 + 2
 -------------------+
1
3G -
---  1 – -----------------3K + G
2
 -------------------2 + 
DESCRIPTION
VARIABLE
Young’s modulus
E
Poisson’s ratio

Bulk modulus
K
E ----------------------3  1 – 2 
2
 + ------3
Shear modulus
G
E -------------------21 + 

Lamé parameter


E
------------------------------------- 1 +    1 – 2 
2G
K – -------3
Lamé parameter


E -------------------21 + 
G
Pressure-wave
speed
cp
K
+ 4G  3------------------------
Shear-wave speed
cs
G
According to Table 8-6, the elasticity matrix D for isotropic materials is written in
terms of Lamé parameters  and ,
 + 2



 + 2



 + 2
D =
0
0
0
0
0
0
0
0
0
0
0
0

0
0
or in terms of the bulk modulus K and shear modulus G:
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
0
0
0
0

0
0
0
0
0
0

4G
2G
2G
K + -------- K – -------- K – -------- 0 0 0
3
3
3
2G
4G
2G
K – -------- K + -------- K – -------- 0 0 0
3
3
3
D =
2G
2G
4G
K – -------- K – -------- K + -------- 0 0 0
3
3
3
0
0
0
G 0 0
0
0
0
0 G 0
0
0
0
0 0 G
ORTHOTROPIC AND ANISOTROPIC MATERIALS
There are two different ways to represent orthotropic or anisotropic data. The
Standard (XX, YY, ZZ, XY, YZ, XZ) material data ordering converts the indices as:
11
1
x
2
y
22
33  3  z
12 21
4
xy
23 32
5
yz
13 31
6
xz
thus, the Hooke’s law is presented in the form involving the elasticity matrix D and the
following vectors:
sx
sx
sy
sy
sz
s xy
=
sz
s xy
s yz
s yz
s xz
s xz
 x
x
x

 y
y
y

 z
z
z
–
–
+ D
 2 xy
2 xy
2 xy

2 yz
2 yz
 2 yz

2
2
2 xz

xz
xz 0
0











COMSOL Multiphysics uses the complete tensor representation internally to perform
the coordinate system transformations correctly.
Beside the Standard (XX, YY, ZZ, XY, YZ, XZ) Material data ordering, the elasticity
coefficients are entered following the Voigt notation. In the Voigt (XX, YY, ZZ, YZ, XZ,
XY) Material data ordering, the sorting of indices is:
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11
1
x
22
2
y
33  3  z
23 32
4
yz
13 31
5
xz
12 21
6
xy
thus the last three rows and columns in the elasticity matrix D are swapped.
ORTHOTROPIC MATERIAL
The elasticity matrix for orthotropic material in the Standard (XX, YY, ZZ, XY, YZ, XZ)
Material data ordering has the following structure:
D =
D 11 D 12 D 13 0
0
0
D 12 D 22 D 23 0
0
0
D 13 D 23 D 33 0
0
0
0
0
0 D 44 0
0
0
0
0
0 D 55 0
0
0
0
0
0 D 66
where the components are as follows:
2
2
E x  E z  yz – E y 
D 11 = ---------------------------------------- ,
D denom
E x E y  E z  yz  xz + E y  xy 
D 12 = – ----------------------------------------------------------------D denom
E x E y E y   xy  yz +  xz 
D 13 = – ---------------------------------------------------------- ,
D denom
E y E z  E y  xy  xz + E x  yz 
D 23 = – ----------------------------------------------------------------- ,
D denom
D 44 = G xy ,
2
2
E y  E z  xz – E x 
D 22 = ---------------------------------------D denom
2
E y E z  E y  xy – E x 
D 33 = ----------------------------------------------D denom
D 55 = G yz , and D 66 = G xz
where
2
2
2 2
D denom = E y E z  xz – E x E y + 2 xy  yz  xz E y E z + E x E z  yz + E y  xy
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
The values of Ex, Ey, Ez, xy, yz, xz, Gxy, Gyz, and Gxz are supplied in designated
fields in the user interface. COMSOL deduces the remaining components—yx, zx,
and zy—using the fact that the matrices D and D1 are symmetric. The compliance
matrix has the following form:
D
–1
 yx  zx
1
-----– -------- – -------- 0
Ex Ey Ez
0
0
 zy
 xy 1
– -------- ------ – -------- 0
Ex Ey Ez
0
0
0
0
1
--------0
G xy
0
 xz  yz 1
– -------- – -------- -----Ex Ey Ez
=
0
0
0
0
0
0
0
1
0 --------- 0
G yz
0
0
0
0
1
0 --------G xz
The thermal expansion matrix is diagonal:
x 0 0
0 y 0
0 0 z
The elasticity matrix in the Voigt (XX, YY, ZZ, YZ, XZ, XY) Material data ordering changes
the sorting of the last three elements in the elasticity matrix:
D 44 = G yz ,
D 55 = G xz , and D 66 = G xy
ANISOTROPIC MATERIAL
In the general case of fully anisotropic material, provide explicitly 21 components of
the symmetric elasticity matrix D, in either Standard (XX, YY, ZZ, XY, YZ, XZ) or Voigt
(XX, YY, ZZ, YZ, XZ, XY) Material data ordering, and 6 components of the symmetric
thermal expansion matrix.
ENTROPY AND THERMOELASTICITY
The free energy for the linear thermoelastic material can be written as
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F = f 0  T  + W s   T 
where WsT is given by Equation 8-3. Hence, the stress can be found as
W
F
s =   =   = C    –  0 –  
  T
  T
and the entropy per unit volume can be calculated as
F
–   = C p log  T  T 0  + S elast
  T 
where T0 is a reference temperature, the volumetric heat capacity CP can be assumed
independent of the temperature (Dulong-Petit law), and
S elast = s
For an isotropic material, it simplifies into
S elast =   s x + s y + s z 
The heat balance equation can be written as
C p

T
+ T S elast =    k T  + Q h
t
t
where k are the thermal conductivity matrix, and
·
Q h = 
where · is the strain-rate tensor and the tensor  represents all possible inelastic stresses
(for example, a viscous stress).
Using the tensor components, the heat balance can be rewritten as:
C p
T
+
t

 Tmn  t smn
=    k T  + Q h
(8-4)
m n
In many cases, the second term can be neglected in the left-hand side of Equation 8-4
because all Tmn are small. The resulting approximation is often called uncoupled
thermoelasticity.
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
Strain-Displacement Relationship
The strain conditions at a point are completely defined by the deformation
components—u, v, and w in 3D—and their derivatives. The precise relation between
strain and deformation depends on the relative magnitude of the displacement.
SMALL DISPLACEMENTS
Under the assumption of small displacements, the normal strain components and the
shear strain components are related to the deformation as follows:
x =
u
x
y =
v
y
z =
w
z
 xy 1 u v
 xy = ------- = ---  + 
2
2   y  x
 yz 1 v w
 yz = ------- = ---  + 
2 z y 
2
 xz 1 u w
 xz = ------- = ---  +  .
2 z x 
2
(8-5)
To express the shear strain, use either the tensor form, xy, yz, xz, or the engineering
form, xy, yz, xz.
The symmetric strain tensor  consists of both normal and shear strain components:
 x  xy  xz
 =  xy  y  yz
 xz  yz  z
The strain-displacement relationships for the axial symmetry case for small
displacements are
r =
u
,
r
u
  = --- ,
r
z =
w
, and
z
 rz =
u w
+
z r
LARGE DEFORMATIONS
As a start, consider a certain physical particle, initially located at the coordinate X.
During deformation, this particle follows a path
x = x  X t 
For simplicity, assume that undeformed and deformed positions are measured in the
same coordinate system. Using the displacement u, it is then possible to write
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x = X+u
When studying how an infinitesimal line element dX is mapped to the corresponding
deformed line element dx, the deformation gradient, F, defined by
x
dx = ------- dX = F dX
X
is used.
The deformation gradient contains the complete information about the local
straining and rotation of the material. It is a positive definite matrix, as long as material
cannot be annihilated. The ratio between current and original volume (or mass
density) is
0
dV---------= ------ = det  F  = J
dV 0

A deformation state where J = 1 is often called incompressible. From the deformation
gradient, it is possible to define the right Cauchy-Green tensor as
T
C = F F
As can be shown by simple insertion, a finite rigid body rotation causes nonzero values
of the engineering strain defined by Equation 8-5. This is not in correspondence with
the intuitive concept of strain, and it is certainly not useful in a constitutive law. There
are several alternative strain definitions in use that do have the desired properties. The
Green-Lagrange strains, , is defined as
1
1 T
 = ---  C – I  = ---  F F – I 
2
2
Using the displacements, they be also written as
1 u i u j u k u k
 ij = ---  -------- + -------- + ---------  ---------
2 X j X i X i X j
(8-6)
The Green-Lagrange strains are defined with reference to an undeformed geometry.
Hence, they represent a Lagrangian description.
The deformation state characterized by finite (or large displacements) but small to
moderate strains is sometimes referred to as geometric nonlinearity or nonlinear
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
geometry. This typically occurs when the main part of the deformations presents a finite
rigid body rotation
STRAIN RATE AND SPIN
The spatial velocity gradient is defined in components as
L kl =
 v
 r t 
 xl k
where v k  r t  is the spatial velocity field. It can be shown that L can be computed in
terms of the deformation gradient as
L =
dF – 1
F
dt
where the material time derivative is used.
The velocity gradient can be decomposed into symmetric and skew-symmetric parts
L = Ld + Lw
where
1
T
L d = ---  L + L 
2
is called the rate of strain tensor, and
1
T
L w = ---  L – L 
2
is called the spin tensor. Both tensors are defined on the spatial frame.
It can be shown that the material time derivative of the Green-Lagrange strain tensor
can be related to the rate of strain tensor as
T
d
= F Ld F
dt
The spin tensor Lw(x,t) accounts for an instantaneous local rigid-body rotation about
an axis passing through the point x.
Components of both Ld and Lw are available as results and analysis variables under the
Solid Mechanics interface.
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Stress-Strain Relationship
The symmetric stress tensor describes stress in a material:
 x  xy  xz
 =  yx  y  yz
 xy =  yx
 xz =  zx
 yz =  zy
 zx  zy  z
This tensor consists of three normal stresses (x, y, z) and six (or, if symmetry is used,
three) shear stresses (xy, yz, xz).
For large deformations there are more than one stress measure:
• Cauchy stress (the components are denoted sx, … in COMSOL Multiphysics)
defined as force/deformed area in fixed directions not following the body.
Symmetric tensor.
• First Piola-Kirchhoff stress P (the components are denoted Px, … in COMSOL
Multiphysics). This is an unsymmetric two-point tensor.
• Second Piola-Kirchhoff stress S (the components are denoted Sx, … in COMSOL
Multiphysics). This is a symmetric tensor, for small strains same as Cauchy stress
tensor but in directions following the body.
The stresses relate to each other as
–1
S = F P
–1
 = J PF
T
–1
= J FSF
T
Plane Strain and Plane Stress Cases
For a general anisotropic linear elastic material in case of plane stress, COMSOL
Multiphysics solves three equations si30 for i3 with i = 1, 2, 3, and uses the solution
instead of Equation 8-2 for these three strain components. Thus, three components
i3 are treated as extra degrees of freedom. For isotropy and orthotropy, only with an
extra degree of freedom, 33,is used since all out of plane shear components of both
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
stress and strain are zero. The remaining three strain components are computed as in
3D case according to Equation 8-2.
For an isotropic material, only the normal out-of-plane component 33
needs to be solved for.
In case of plane strain, set i3 for i1, 2, 3. The out-of-plane stress components
si3 are results and analysis variables.
Axial Symmetry
The axially symmetric geometry uses a cylindrical coordinate system. Such a coordinate
system is orthogonal but curvilinear, and one can choose between a covariant basis
e1, e2, e3 and a contravariant basis e1, e2, e3.
The metric tensor is
1 0 0
 g ij  = 0 r 2 0
0 0 1
in the coordinate system given by e1, e2, e3, and
g
ij
1 0 0
= 0 r–2 0
0 0 1
in e1, e2, e3.
The metric tensor plays the role of a unit tensor for a curvilinear coordinate system.
For any vector or tensor A, the metric tensor can be used for conversion between
covariant, contravariant, and mixed components:
j
Ai =
  Aim g
mj

m
A
ij
=
  Anm g
ni mj
g

m n
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In both covariant and contravariant basis, the base vector in the azimuthal direction
has a nonunit length. To cope with this issue, the so called physical basis vectors of unit
length are introduced. These are
1
1
3
e r = e 1 = e  e  = --- e = re 2 e z = e 3 = e
r 2
The corresponding components for any vector or tensor are called physical.
For any tensor, the physical components are defined as
phys
A ij
g ii g jj A
=
ij
where no summation is done over repeated indices.
MIXED COMPONENTS AND PRINCIPAL INVARIANTS
The mixed strain components are given by
 g
i
Aj =
im
A mj 
m
The principal invariants are
i
I 1  A  = trace  A i  =
 Ai
i
i
1
= A 11 + A 22 ----2- + A 33
r
2
1
I 2  A  = ---   I 1  A   –
2
j
 Aj Ai
i
i j
i
I 3  A  = det  A i 
DISPLACEMENTS AND AXIAL SYMMETRY ASSUMPTIONS
The axial symmetry implementation in COMSOL Multiphysics assumes independence
of the angle, and also that the torsional component of the displacement is identically
zero. The physical components of the radial and axial displacement, u and w, are used
as dependent variables for the axially symmetric geometry.
For the linear elastic material, the stress components in coordinate system are
s
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ij
ij
= s0 + C
CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
ijkl
  kl –  kl  –  0kl 
where TTref.
For anisotropic and orthotropic materials, the 4th-order elasticity tensor is defined
from D matrix according to:
sr
sr
s
s
s z
s r
=
s z
s r
s z
s z
s rz
s rz





+ D





0
r
r


 z
2 r
–
 z
2 r
2 z
2 z
2 rz
2 rz
r 

 

 z 
–

2 r 

2 z 

2 rz 
0
The user input D matrix always contains the physical components of the elasticity
tensor
phys
C ijkl
and the corresponding tensor components are computed internally according to:
C
ijkl
C
phys
ijkl
= ----------------------------------------------g ii g jj g kk g ll
For an isotropic material:
C
ijkl
ij kl
= g g
ik jl
il jk
+ g g + g g 
where and  are the first and second Lamé elastic parameters.
Loads
Specify loads as
• Distributed loads. The load is a distributed force in a volume, on a face, or along an
edge.
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• Total force. The specification of the load is as the total force. The software then
divides this value with the area or the volume where the force acts.
• Pressure (boundaries only).
For 2D models choose how to specify the distributed boundary load as a
load defined as force per unit area or a load defined as force per unit length
acting on boundaries.
In the same way, choose between defining the load as force per unit
volume or force per unit area for body loads acting in a domain. Also
define a total force (SI unit: N) as required.
For 2D and axisymmetric models, the boundary loads apply on edges
(boundaries).
For 2D axisymmetric models, the boundary loads apply on edges
(boundaries).
For 3D solids, the boundary loads apply on faces (boundaries).
Table 8-7 shows how to define distributed loads on different geometric entity levels;
the entries show the SI unit in parentheses.
TABLE 8-7: DISTRIBUTED LOADS
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GEOMETRIC
ENTITY
POINT
EDGE
FACE
DOMAIN
2D
force (N)
force/area (N/m2) or
force/length (N/m)
Not
available
force/volume (N/m3)
or force/area (N/m2)
Axial
symmetry
total force
along the
circumferential
(N)
force/area (N/m2)
Not
available
force/volume (N/m3)
3D
force (N)
force/length (N/m)
force/area
(N/m2)
force/volume (N/m3)
CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
Pressure Loads
A pressure load is directed inward along the normal of boundary on which it is acting.
This load type acts as a source of nonlinearity, since its direction depends on the current
direction of the boundary normal. In a linearized context, for example in the frequency
domain, the pressure is equivalent to a specified normal stress.
For general cases, if the problem is linear in all other respects, the solution
can be made more efficient by forcing the solver to treat the problem as
linear. See Stationary Solver in the COMSOL Multiphysics Reference
Manual.
Equation Implementation
The COMSOL Multiphysics implementation of the equations in the Solid Mechanics
interface is based on the principle of virtual work.
The principle of virtual work states that the sum of virtual work from internal strains
is equal to work from external loads.
The total stored energy, W, for a linear material from external and internal strains and
loads equals:
W =
  – s + u  FV dv
V


S
L
+  u  F S  ds +  u  F L  dl +
 U
t
 Fp 
p
The principle of virtual work states that W0 which leads to
  –test s + utest  FV – utest  utt dv
V


S
L
+  u test  F S  ds +  u test  F L  dl +
  Utest  Fp 
t
p
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Setting up Equations for Different Studies
The Solid Mechanics interface supports stationary (static), eigenfrequency,
time-dependent (transient), frequency domain, and modal solver study types.
STATIONARY STUDIES
COMSOL Multiphysics uses an implementation based on the stress and strain
variables. The normal and shear strain variables depend on the displacement
derivatives.
Using the tensor strain, stress, and displacement variables, the principle of virtual work
is expressed as:
W =
  –test s + utest  FV dv
V


S
L
+  u test  F S  ds +  u test  F L  dl +
  Utest  Fp 
t
p
TIME-DEPENDENT STUDIES
  –test   s + dM st  + utest  FV – utest  utt – dM utest  ut dv
V
(8-7)


S
L
+  u test  F S  ds +  u test  F L  dl +
  Utest  Fp 
t
p
where the terms proportional to dM and dK appear if the Rayleigh damping is used
For more information about the equation form in case of geometric
nonlinearity see Geometric Nonlinearity Theory for the Solid Mechanics
User Interface
FREQUENCY-DOMAIN STUDIES
In the frequency domain the frequency response is studied when applying harmonic
loads. Harmonic loads are specified using two components:
• The amplitude value, Fx
• The phase, FxPh
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
To derive the equations for the linear response from harmonic excitation loads

F xfreq = F x  f  cos  t + F xPh  f  ----------
180
F xfreq
F freq = F yfreq
F zfreq
assume a harmonic response with the same angular frequency as the excitation load
u = u amp cos  t +  u 
u
u= v
w
Also describe this relationship using complex notation
u = Re  u amp e
j u jt
e
jt
j
 = Re  u˜ e  where u˜ = u amp e u
˜ e jt 
u = Re  u

jF xPh  f  ---------
180 jt
jt
F xfreq = Re  F x   e
e  = Re  F˜x e 


where
F˜x = F x  f e

jF xPh  f  ---------180
F˜x
˜
F = F˜
y
F˜z
EIGENFREQUENCY STUDIES
The eigenfrequency equations are derived by assuming a harmonic displacement field,
similar as for the frequency response formulation. The difference is that this study type
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343
uses a new variable j explicitly expressed in the eigenvalue jThe
eigenfrequency f is then derived from j as
 j f = Im
-----------------2
Damped eigenfrequencies can be studied by adding viscous damping terms to the
equation. In addition to the eigenfrequency the quality factor, Q, and decay factor, 
for the model can be examined:
Im   
Q = ------------------2Re   
 = Re   
Damping Models
The Solid Mechanics interface offers two predefined damping models: Rayleigh
damping and loss factor damping.
RAYLEIGH DAMPING
To model damping effects within the material, COMSOL Multiphysics uses Rayleigh
damping, where two damping coefficients are specified.
The weak contribution due to the alpha-damping is always accounted for as shown in
Equation 8-2. The contribution from the beta-damping that shown in Equation 8-7
corresponds to the case of small strains. In case of geometric nonlinearity, it becomes
  –dM utest Pt  dv
V
where P is the first Piola-Kirchhoff stress tensor.
Geometric Nonlinearity Theory for the Solid Mechanics User Interface
To further clarify the use of the Rayleigh damping, consider a system with a single
degree of freedom. The equation of motion for such a system with viscous damping is
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
2
du
d u
m ---------2- + c ------- + ku = f  t 
dt
dt
In the Rayleigh damping model the damping coefficient c can be expressed in terms
of the mass m and the stiffness k as
c =  dM m +  dK k
The Rayleigh damping proportional to mass and stiffness is added to the static weak
term.
A complication with the Rayleigh damping model is to obtain good values for the
damping parameters. A more physical damping measure is the relative damping, the
ratio between actual and critical damping, often expressed as a percentage of the critical
damping. Commonly used values of relative damping can be found in the literature.
It is possible to transform relative damping to Rayleigh damping parameters. The
relative damping, , for a specified pair of Rayleigh parameters, dM and dK, at a
frequency, f, is
1  dM
 = ---  ----------- +  dK 2f
2 2f
Using this relationship at two frequencies, f1 and f2, with different relative damping,
1 and 2, results in an equation system that can be solved for dM and dK:
1 ----------f
4f 1 1  dM
1  dK
----------f
4f 2 2
=
1
2
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Relative damping
Using the same relative damping, 1 = 2, does not result in a constant damping factor
inside the interval f1  f  f2. It can be shown that the damping factor is lower inside
the interval, as Figure 8-1 shows.
Rayleigh damping
Specified damping
f1
f2
f
Figure 8-1: An example of Rayleigh damping.
LOSS FACTOR DAMPING
Loss factor damping (sometimes referred to as material or structural damping) can be
applied in the frequency domain.
In COMSOL Multiphysics, the loss information appears as a multiplier of the elastic
stress in the stress-strain relationship:
s = s 0 +  1 + j s  C    –  0 –  
where s is the loss factor, and j is the imaginary unit.
Choose between these loss damping types:
• Isotropic loss damping s
• Orthotropic loss damping with components of E, the loss factor for an orthotropic
Young’s modulus, and G, the loss factor for an orthotropic shear modulus.
• Anisotropic loss damping with an isotropic or symmetric loss damping D for the
elasticity matrix.
If modeling the damping in the structural analysis via the loss factor, use the following
definition for the elastic part of the entropy:
S elast =   s – j s  C   
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This is because the entropy is a function of state and thus independent of the strain
rate, while the damping represents the rate-dependent effects in the material (for
example, viscous or viscoelastic effects). The internal work of such inelastic forces
averaged over the time period 2 can be computed as:
1
Q h = ---  s Real  Conj  C   
2
(8-8)
Equation 8-8 can be used as a heat source for modeling of the heat generation in
vibrating structures, when coupled with the frequency-domain analysis for the stresses
and strains.
Initial Stresses and Strains
Initial stress refers to the stress before the system applies any loads, displacements, or
initial strains. The initial strain is the one before the system has applied any loads,
displacements, or initial stresses.
Both the initial stress and strains are tensor variables defined via components on the
local coordinate system for each domain. Input these as the following matrices:
 0x  0xy  0xz
s 0x s 0xy s 0xz
 0xy  0y  0yz  s 0xy s 0y s 0yz
 0xz  0yz  0z
s 0xz s 0yz s 0z
In case of nearly incompressible material (mixed formulation), the components of the
total initial stress (that is, without volumetric-deviatoric split) are still input. The initial
pressure in the equation for the pressure help variable pw is computed as
1
p 0 = – --- I 1  s 0 
3
The initial stresses and strains are available with the Linear Elastic Material.
AXIAL SYMMETRY
User inputs the physical components of 0 and s0:
 0r  0r  0rz
s 0r s 0r s 0rz
 0r  0  0z  s 0r s 0 s 0z
 0rz  0z  0z
s 0rz s 0z s 0z
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OTHER POSSIBLE USES OF INITIAL STRAINS AND STRESSES
Many inelastic effects in solids and structure (creep, plasticity, damping, viscoelasticity,
poroelasticity, and so on) are additive contributions to either the total strain or total
stress. Then the initial value input fields can be used for coupling the elastic equations
(solid physics) to the constitutive equations (usually General Form PDEs) modeling
such extra effects.
About Spring Foundations and Thin Elastic Layers
In this section, the equations for the spring type physics nodes are developed using
boundaries, but the generalizations to geometrical objects of other dimensions are
obvious.
SPRING FOUNDATION
A spring gives a force that depends on the displacement and acts in the opposite
direction (in the case of a force that is proportional to the displacement, this is called
Hooke’s law). In a suitable coordinate system, a spring condition can be represented as
fs = –K   u – u0 
where fs is a force/unit area, u is the displacement, and K is a diagonal stiffness matrix.
u0 is an optional pre-deformation. If the spring properties are not constant, it is, in
general, easier to directly describe the force as a function of the displacement, so that
fs = f  u – u0 
In the same way, a viscous damping can be described as a force proportional to the
velocity
·
f v = – Du
Structural (“loss factor)” damping is only relevant for frequency domain analysis and
is defined as
f l = – iK   u – u 0 
where  is the loss factor and i is the imaginary unit (in this case, a constant or a
diagonal matrix). If the elastic part of the spring definition is given as a force versus
displacement relation, the stiffness K is taken as the stiffness at the linearization point
at which the frequency response analysis is performed. Since the loss factor force is
proportional to the elastic force, the equation can be written as
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f sl = f s + f l =  1 + i f s
The contribution to the virtual work is

T
W =  u  f sl + f v  dA
A
T H I N E L A S T I C L A Y E R B E T W E E N TW O P A R T S
A spring or damper can also act between two boundaries of an identity pair. The spring
force then depends on the difference in displacement between the surfaces.
f sD = – f sS = – K  u D – u S – u 0 
The uppercase indices refer to “source” and “destination.” When a force versus
displacement description is used,
f sD = – f sS = f  u – u 0 
u = uD – uS
The viscous and structural damping forces have analogous properties,
·
·
f vD = – f vS = – D  u D – u S 
f lD = – f lS = – iK  u D – u S – u 0 
so that
f slD = f sD + f lD =  1 + i f sD
The virtual work expression is formulated on the destination side of the pair as
W = 
  uD – uS 
T
 f slD + f vD  dA D
AD
Here the displacements from the source side are obtained using the src2dst operator
of the identity pair.
THIN ELASTIC LAYER ON INTERIOR BOUNDARIES
On an interior boundary, the Thin Elastic Layer decouples the displacements between
two sides if the boundary. The two boundaries are then connected by elastic and
viscous forces with equal size but opposite directions, proportional to the relative
displacements and velocities. The spring force can be written as
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f sd = – f su = – K  u d – u u – u 0 
or
f sd = – f su = f  u – u 0 
u = ud – uu
The viscous force is
·
·
f vd = – f vu = – D  u d – u u 
and the structural damping force is
f ld = – f lu = – iK  u d – u u – u 0 
f sld = f sd + f ld =  1 + i f sd
The subscripts u and d denote the “up” and “down” sides of the interior boundary.
The virtual work expression is formulated as
W = 
  ud – us 
T
 f sld + f vd  dA D
AD
About Added Mass
The Added Mass node can be used for supplying inertia that is not part of the material
itself. Such inertia does not need to be isotropic, in the sense that the inertial effects
are not the same in all directions. This is, for example, the case when a structure
immersed in a fluid vibrates. The fluid is added to the inertia for acceleration in the
direction normal to the boundary, but not tangential to it.
Other uses for added mass are when sheets or strips of a material that is heavy, but
having a comparatively low stiffness, are added to a structure. The data for the base
material can then be kept unaltered, while the added material is represented purely as
added mass.
The value of an added mass can also be negative. You can use such a negative value for
adjusting the mass when a part imported from a CAD system does not get exactly the
correct total mass due so simplifications of the geometry.
Added mass is an extra mass distribution that can be anisotropic. It can exist on
domains, boundaries, and edges. The inertial forces from added mass can be written as
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2
fm = –M
 u
t
2
where M is a diagonal mass distribution matrix. The contribution to the virtual work is

T
W =  u f m dA
A
for added mass on a boundary, and similarly for objects of other dimensions.
Geometric Nonlinearity Theory for the Solid Mechanics User Interface
Geometric nonlinearity formulation is suitable for any material, and it is always used
for hyperelastic materials and for large strain plasticity.
The Hyperelastic Material node is available with the Nonlinear Structural
Materials Module.
For other materials, it can be activated via the solver setting. Note however that even
together with the geometric nonlinearity, the validity of any linear material model is
usually limited to the situation of possibly large displacements but small to moderate
strains. A typical example of use is to model large rigid body rotations. The
implementation is similar to that for the geometrically linear elastic material, but with
the strain tensor replaced with the Green-Lagrange strain tensor, and the stress tensor
replaced with the second Piola-Kirchhoff stress tensor, defined as:
0
0
i
S = S + C    – I –  –  
where TTref, and i represents all possible inelastic strains (such as plastic or
creep strains). In components, it is written as:
0
0
i
S ij = S ij + C ijkl   kl –  kl  –  kl –  kl 
where the elasticity tensor C ijkl is defined from the D matrix (user input). The 2nd
Piola-Kirchhoff stress is a symmetric tensor.
The strain energy function is computed as
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1
0 i
W s = --- S   – I –  –  
2
which is a variable defined in the physics interface. Other stress variables are defined as
follows.
The first Piola-Kirchhoff stress P is calculated from the second Piola-Kirchhoff stress
as P  FS. The first Piola-Kirchhoff stress relates forces in the present configuration
with areas in the reference configuration, and it is sometimes called the nominal stress.
Using the 1st Piola-Kirchhoff stress tensor, the equation of motion can be written in
the following form:
2
0
 u
t
2
= FV – X  P
(8-9)
where the density corresponds to the material density in the initial undeformed state,
the volume force vector FV has components in the actual configuration but given with
respect to the undeformed volume, and the tensor divergence operator is computed
with respect to the coordinates on the material frame. Equation 8-9 is the strong form
that corresponds to the weak form equations solved in case of geometric nonlinearity
within the Solid Mechanics interface (and many related multiphysics interfaces) in
COMSOL Multiphysics. Using vector and tensor components, the equation can be
written as
2
0
 ux
t
2
2
0
 uy
t
2
P xX P xY P xZ
= F Vx – 
+
+
X
Y
Z 
P yX P yY P yZ
= F Vy – 
+
+
X
Y
Z 
2
0
 uz
t
2
= F Vz – 
P zX P zY P zZ
+
+
X
Y
Z 
The components of 1st Piola-Kirchhoff stress tensor are non symmetric in the general
case, thus
P iJ  P Ij
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because the component indexes correspond to different frames. Such tensors are called
two-point tensors.
The boundary load vector FA in case of geometric nonlinearity can be related to the
1st Piola-Kirchhoff stress tensor via the following formula:
FA = P  n0
where the normal n0 corresponds to the undeformed surface element. Such a force
vector is often referred to as the nominal traction. In components, it can be written as
F Ax = P xX n X + P xY n Y + P xZ n Z
F Ay = P yX n X + P yY n Y + P yZ n Z
F Az = P zX n X + P zY n Y + P zZ n Z
The Cauchy stress, s,can be calculated as
–1
s = J PF
T
–1
= J FSF
T
The Cauchy stress is a true stress that relates forces in the present configuration (spatial
frame) to areas in the present configuration, and it is a symmetric tensor. Equation 8-9
can be rewritten in terms of the Cauchy stress as
2

 u
t
2
= fV – x  s
where the density corresponds to the density in the actual deformed state, the volume
force vector fV has components in the actual configuration (spatial frame) given with
respect to the deformed volume, and the divergence operator is computed with respect
to the spatial coordinates.
The pressure is computed as
1
p = – --- trace  s 
3
which corresponds to the volumetric part of the Cauchy stress. The deviatoric part is
defined as
s d = s + pI
The second invariant of the deviatoric stress
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1
J 2  s  = --- s d :s d
2
is used for the computation of von Mises (effective) stress
s mises =
3J 2  s 
NEARLY INCOMPRESSIBLE MATERIALS
Nearly incompressible materials can cause numerical problems if only displacements
are used in the interpolating functions. Small errors in the evaluation of volumetric
strain, due to the finite resolution of the discrete model, are exaggerated by the high
bulk modulus. This leads to an unstable representation of stresses, and in general to
underestimation of the deformation because spurious volumetric stresses might
balance also applied shear and bending loads.
In such cases a mixed formulation can be used that represents the pressure as a
dependent variable in addition to the displacement components. This formulation
removes the effect of volumetric strain from the original stress tensor and replaces it
with an interpolated pressure, pw. A separate equation constrains the interpolated
pressure to make it equal (in a finite-element sense) to the original pressure calculated
from the strains.
For an isotropic linear elastic material, the second Piola-Kirchhoff stress tensor S,
computed directly from the strains, is replaced by a modified version:
s˜ = s +  p – p w I
where I is the unit tensor and the pressure p is calculated from the stress tensor
1
p = – --- trace  s 
3
The auxiliary dependent variable pw is set equal to p using the equation
pw p
------- – --- = 0
 
(8-10)
where  is the bulk modulus.
The modified stress tensor s˜ is used then in calculations of the energy variation.
For orthotropic and anisotropic materials, the auxiliary pressure equation is scaled to
make the stiffness matrix symmetric. Note, however, that the stiffness matrix in this
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formulation is not positive definite and even contains a zero block on the diagonal in
the incompressible limit. This limits the possible choices of direct and iterative linear
solver.
In case of linear elastic materials without geometric nonlinearity (and also
for hyperelastic materials), the stress tensor s in the above equations is
replaced by the 2nd Piola-Kirchhoff stress tensor S, and the pressure p
with:
1
p p = – --- trace  S 
3
About the Low-Reflecting Boundary Condition
The low-reflecting boundary condition is mainly intended for letting waves pass out
from the model domain without reflection in time-dependent analysis. It is also
available in the frequency domain, but then adding a perfectly matched layer (PML) is
usually a better option.
Infinite Element Domains and Perfectly Matched Layers in the COMSOL
Multiphysics Reference Manual
As a default, the low-reflecting boundary condition takes the material data from the
adjacent domain in an attempt to create a perfect impedance match for both pressure
waves and shear waves, so that
u
u
  n = – c p   n n – c s   t t
t
t
where n and t are the unit normal and tangential vectors at the boundary, respectively,
and cp and cs are the speeds of the pressure and shear waves in the material. This
approach works best when the wave direction in close to the normal at the wall.
In the general case, you can use
  n = – d i   c p c s 
u
t
where the mechanical impedance di is a diagonal matrix available as the user input, and
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by default it is set to
cp + cs
d i =  ---------------- I
2
More information about modeling using low-reflecting boundary
conditions can be found in M. Cohen and P.C. Jennings, “Silent
Boundary Methods for Transient Analysis,” Computational Methods for
Transient Analysis, vol 1 (editors T. Belytschko and T.J.R. Hughes),
Nort-Holland, 1983.
Cyclic Symmetry and Floquet Periodic Conditions
These boundary conditions are based on the Floquet theory which can be applied to
the problem of small-amplitude vibrations of spatially periodic structures.
If the problem is to determine the frequency response to a small-amplitude
time-periodic excitation that also possesses spatial periodicity, the theory states that the
solution can be sought in the form of a product of two functions. One follows the
periodicity of the structure, while the other one follows the periodicity of the
excitation. The problem can be solved on a unit cell of periodicity by applying the
corresponding periodicity conditions to each of the two components in the product.
The problem can be modeled using the full solution without applying the above
described multiplicative decomposition. For such a solution, the Floquet periodicity
conditions at the corresponding boundaries of the periodicity cell are expressed as
u destination = exp  – ik F  r destination – r source   u source
where u is a vector of dependent variables, and vector kF represents the spatial
periodicity of the excitation.
The cyclic symmetry boundary condition presents a special but important case of
Floquet periodicity, for which the unit periodicity cell is a sector of a structure that
possesses rotational symmetry. The frequency response problem can be solved then in
one sector of periodicity by applying the periodicity condition. The situation is often
referred to as dynamic cyclic symmetry.
For an eigenfrequency study, all the eigenmodes of the full problem can be found by
performing the analysis on one sector of symmetry only and imposing the cyclic
symmetry of the eigenmodes with an angle of periodicity  = m , where the cyclic
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symmetry mode number m can vary from 0 to N, with N being the total number of
sectors so that 2N.
The Floquet periodicity conditions at the sides of the sector of symmetry can be
expressed as
u destination = e
– i
T
R  u source
where the u represents the displacement vectors with the components given in the
default Cartesian coordinates. Multiplication by the rotation matrix given by
R =
cos    – sin    0
sin    cos    0
0
0
1
makes the corresponding displacement components in the cylindrical coordinate
system differ by the factor exp  – i  only. For scalar dependent variables, a similar
condition applies, for which the rotation matrix is replaced by a unit matrix.
The angle  represents either the periodicity of the eigenmode for an eigenfrequency
analysis or the periodicity of the excitation signal in case of a frequency-response
analysis. In the latter case, the excitation is typically given as a load vector
F = – F 0 exp  – im atan  Y  X  
when modeled using the Cartesian coordinates; parameter m is often referred to as the
azimuthal wave-number.
More information about cyclic symmetry conditions can be found in B.
Lalanne and M. Touratier, “Aeroelastic Vibrations and Stability in Cyclic
Symmetric Domains,” The International Journal of Rotating
Machinery, vol. 6, no. 6, pp 445–452, 2000.
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Calculating Reaction Forces
There are different ways to evaluate reaction forces and these are discussed in this
section.
• Using Predefined Variables to Evaluate Reaction Forces
• Using Weak Constraints to Evaluate Reaction Forces
• Using Surface Traction to Evaluate Reaction Forces
• Using Surface Traction to Evaluate Reaction Forces
The following sections describe the merits and costs of these methods.
Using Predefined Variables to Evaluate Reaction Forces
The results analysis capabilities include easy access to the reaction forces and moments.
They are available as predefined variables. The reaction force variables are available
only at the nodes, and not as a continuous field, so they are not suitable for graphic
presentation.
To compute the sum of the reaction forces over a region, use Volume
Integration, Surface Integration, or Line Integration under Derived Values.
The integration method will discover that the reaction forces are discrete
values, and apply a summation instead of an integration.
Reaction forces are computed as the sum of the nodal values over the selected volume,
face, or edge. Reaction moments are calculated as the sum of the moment from the
reaction forces with respect to a reference point, and any explicit reaction moments (if
there are rotational degrees of freedom).
Specify the default coordinates of the Reference Point for Moment Computation at the
top level of the physics interface main node’s settings window. After editing the
reference point coordinates, you need to right-click the study node and select Update
Solution for the change to take effect on the reaction moment calculation. During
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postprocessing, you can modify the coordinates of the reference point in the
Parameters section of a result feature..
Reaction forces are not available for eigenfrequency analysis or when weak
constraints are used.
If reaction forces are summed independently for two adjacent boundaries,
the total sum will not be the same as if the reaction forces were summed
for both boundaries in one operation. The values of the nodes at the
common edge always contain contributions from the elements at both
sides of the edge.
Derived Values and Tables in the COMSOL Multiphysics Reference
Manual
Using Weak Constraints to Evaluate Reaction Forces
Select the Use weak constraints check box to get accurate distributed reactions. Extra
variables that correspond to the reaction traction distribution are automatically added
to the solution components.
With weak constraints activated, COMSOL Multiphysics adds the reaction forces to
the solution components. The variables are denoted X_lm, where X is the name of the
constrained degree of freedom (as, for example, u_lm and v_lm). The extension lm
stands for Lagrange multipliers. It is only possible to evaluate reaction forces on
constrained boundaries in the directions of the constraints.
To compute the total reaction force on a boundary, integrate one of the
variables X_lm using Volume Integration, Surface Integration, or Line
Integration under Derived Values.
If the constraint is defined in a local coordinate system, the degrees of
freedom for the weak constraint variables are defined along the directions
of that system.
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359
Since the reaction force variables are added to the solution components, the number
of DOFs for the model increases slightly, depending on the mesh size for the
boundaries in question. Boundaries that are adjacent to each other must have the same
constraint settings. The reason for this is that adjacent boundaries share a common
node.
Using weak constraints will affect the structure of the equation system to be solved,
and is not suitable for all types of equation solvers.
In the COMSOL Multiphysics Reference Manual:
• Derived Values and Tables
• Bidirectional and Unidirectional Constraints
Using Surface Traction to Evaluate Reaction Forces
As an alternative method, you can obtain values of the reaction forces on constrained
boundaries by using boundary integration of the relevant components of the surface
traction vector.
For 2D models, multiply the surface traction by the cross section thickness
before integrating to calculate the total reaction force.
Two different types of surface traction results can be computed in COMSOL
Multiphysics:
The first type, contained in the variables interface.Tax, is computed from the
stresses. It is always available. Since the surface traction vector is based on computed
stress results, this method is less accurate for computing reactions than the other
methods.
The second type, contained in the variables interface.Tracx, is computed using a
method similar to the weak constraints, but without introducing the Lagrange
multipliers as extra degrees of freedom. The accuracy is high, but there is an extra
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computational cost. These traction variables are computed only if the check box
Compute boundary fluxes in the Discretization section of the physics interface is selected.
In case of geometric nonlinearity, the two types of traction variables are
interpreted differently. The interface.Tax variables are based on
Cauchy stress, and contains a force per current area. If you integrate them
you must use the spatial frame. The interface.Tracx variables are based
on First Piola-Kirchhoff stresses and contains a force per undeformed
area. An integration must then be done on the material frame.
Evaluating Surface Traction Forces on Internal Boundaries
As opposed to the other methods for reaction force computation, the boundary flux
based tractions are computed not only on external boundaries, but also on internal
boundaries. On internal boundaries, there are then two traction fields: One acting
from each of the domains sharing the boundary. These internal traction fields are
contained in the variables interface.iTracux and interface.iTracdx. The letters
u and d in the variable names indicate the up and down side of the boundary
respectively. If you need the value of the total force acting on an internal section
through your model, these variable can be integrated. The interface.iTracux
andinterface.iTracdx variables will only be available if the check box Compute
boundary fluxes in the Discretization section of the physics interface is selected and there
are internal boundaries in your model.
Computing Accurate Fluxes in the COMSOL Multiphysics Reference
Manual
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Geometric Nonlinearity, Frames, and
the ALE Method
Consider the bending of a beam in the general case of a large deformation (see
Figure 8-2). In this case the deformation of the beam introduces an effect known as
geometric nonlinearity into the equations of solid mechanics.
Figure 8-2 shows that as the beam deforms, the shape, orientation, and position of the
element at its tip changes significantly. Each edge of the infinitesimal cube undergoes
both a change in length and a rotation that depends on position. Additionally the three
edges of the cube are no longer orthogonal in the deformed configuration (although
typically for practical strains the effect of the non-orthogonality can be neglected in
comparison to the rotation).
From a simulation perspective it is desirable to solve the equations of solid mechanics
on a fixed domain, rather than on a domain that changes continuously with the
deformation. In COMSOL Multiphysics this is achieved by defining a displacement
field for every point in the solid, usually with components u, v, and w. At a given
coordinate (X, Y, Z) in the reference configuration (on the left of Figure 8-2) the
value of u describes the displacement of the point relative to its original position.
Taking derivatives of the displacement with respect to X, Y, and Z enables the
definition of a strain tensor, known as the Green-Lagrange strain (or material strain).
This strain is defined in the reference or Lagrangian frame, with X, Y, and Z
representing the coordinates in this frame. The displacement is considered as a
function of the material coordinates (X, Y, Z), but it is not explicitly a function of the
spatial coordinates (x, y, z). It is thus only possible to compute derivatives with respect
to the material coordinates.
In the Solid Mechanics interface, the Lagrangian frame is equivalent to the material
frame. An element at point (X, Y, Z) specified in this frame moves with a single piece
of material throughout a solid mechanics simulation. It is often convenient to define
material properties in the material frame as these properties move and rotate naturally
together with the volume element at the point at which they are defined as the
simulation progresses. In Figure 8-2 this point is illustrated by the small cube
highlighted at the end of the beam, which is stretched, translated, and rotated as the
beam deforms. The three mutually perpendicular faces of the cube in the Lagrange
frame are no longer perpendicular in the deformed frame. The deformed frame is
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
called the Eulerian or (in COMSOL Multiphysics) the spatial frame. Coordinates in
this frame are denoted (x, y, z) in COMSOL.
Figure 8-2: An example of the deformation of a beam showing the undeformed state (left)
and the deformed state (right) of the beam itself with an element near its tip highlighted
(top), of the element (center) and of lines parallel to the x-axis in the undeformed state
(bottom).
It is important to note that, as the solid deforms, the Lagrangian frame
becomes a non-orthogonal curvilinear coordinate system (see the lower
part of Figure 8-2 to see the deformation of the X-axis). Particular care is
therefore required when defining physics in this coordinate system.
For example, in the Eulerian system it is easy to define forces per unit area (known as
tractions) that act within the solid, and to define a stress tensor that represents all of
these forces that act on a volume element. Such forces could be physically measured,
for example using an implanted piezoresistor. The stress tensor in the Eulerian frame
is called the Cauchy or true stress tensor (in COMSOL this is referred to as the spatial
stress tensor). To construct the stress tensor in the Lagrangian frame a tensor
transformation must be performed on the Cauchy stress. This produces the second
Piola-Kirchhoff (or material) stress, which can be used with the Lagrange or material
G E O M E T R I C N O N L I N E A R I T Y, F R A M E S , A N D T H E A L E M E T H O D
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strain to solve the solid mechanics problem in the (fixed) Lagrangian frame. This is
how the Solid Mechanics interface works when geometric nonlinearities are enabled.
If the strains are small (significantly less than 10 percent), and there are no
significant rotations involved with the deformation (significantly less than
10 degrees), geometric nonlinearity can be disabled, resulting in a linear
equation system which solves more quickly (Ref. 1). This is often the case
for many practical MEMS structures.
Geometric nonlinearity can be enabled or disabled within a given model
by changing the Include geometric nonlinearity setting in the relevant
solver step.
Geometric Nonlinearity Theory for the Solid Mechanics User Interface
In the case of solid mechanics, the material and spatial frames are associated directly
with the Lagrangian and Eulerian frames, respectively. In a more general case (for
example, when tracking the deformation of a fluid, such as a volume of air surrounding
a moving structure) tying the Lagrangian frame to the material frame becomes less
desirable. Fluid must be able to flow both into and out of the computational domain,
without taking the mesh with it. The Arbitrary Lagrangian-Eulerian (ALE)
method allows the material frame to be defined with a more general mapping to the
spatial or Eulerian frame. In COMSOL, a separate equation is solved to produce this
mapping—defined by the mesh smoothing method (Laplacian, Winslow, or
hyperelastic) with boundary conditions that determine the limits of deformation (these
are usually determined by the physics of the system, whilst the domain level equations
are typically being defined for numerical convenience). The ALE method offers
significant advantages since the physical equations describing the system can be solved
in a moving domain.
Reference for Geometric Nonlinearity
1. A. F. Bower, Applied Mechanics of Solids, CRC Press, Boca Raton, FL (http://
www.solidmechanics.org), 2010.
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S pr i ng s a nd D amp ers
The Spring Foundation and Thin Elastic Layer physics nodes are available with the Solid
Mechanics interface and supply elastic and damping boundary conditions for domains,
boundaries, edges, and points. The Spring Foundation node is available also in the other
structural mechanics interfaces.
The features are completely analogous, with the difference that a Spring Foundation
node connects the structural part on which it is acting to a fixed “ground,” while the
Thin Elastic Layer acts between two parts, either on an internal boundary or on a pair.
The following types of data are defined by these nodes:
• Spring Data
• Loss Factor Damping
• Viscous Damping
SPRING DATA
The elastic properties can be defined either by a spring constant or by a force as
function of displacement. The force as a function of displacement may be more
convenient for nonlinear springs. Each spring feature has three displacement variables
defined, which can be used to describe the dependency on deformation. These
variables are named uspring1_tag, uspring2_tag, and uspring3_tag for the three
directions given by the local coordinate system. In the variable names, tag represents
the tag of the feature defining the variable The tag could for example be spf1 or tel1
for a Spring Foundation or a Thin Elastic Layer respectively. These variables measure the
relative extension of the spring after subtraction of any pre-deformation.
LOSS FACTOR DAMPING
The loss factor damping adds a loss factor to the spring data above, so that the total
force exerted by the spring with loss is
f sl =  1 + i f s
where fs is the elastic spring force, and  is the loss factor.
Loss factor damping is only applicable in for eigenfrequency and frequency domain
analysis. In time dependent analysis the loss factor is ignored.
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365
VISCOUS DAMPING
It is also possible to add viscous damping to the Spring Foundation and Thin Elastic
Layer features. The viscous damping adds a force proportional to the velocity (or in
the case of Thin Elastic Layer: the relative velocity between the two boundaries). The
viscosity constant of the feature can be made dependent on the velocity by using the
variables named vdamper1__tag, vdamper2__tag, and vdamper3__tag, which
contain the velocities in the three local directions.
The Spring Foundation feature is most commonly used for simulating boundary
conditions with a certain flexibility, such as the soil surrounding a construction. An
other important use is for stabilizing parts that would otherwise have a rigid body
singularity. This is a common problem in contact modeling before an assembly has
actually settled. In this case a Spring Foundation acting on the entire domain is useful
because it avoids the introduction of local forces.
A Thin Elastic Layer between used as a pair condition can be used to simulate thin
layers with material properties which differ significantly from the surrounding
domains. Common applications are gaskets and adhesives.
When a Thin Elastic Layer is applied on an internal boundary, it usually simulates a
local flexibility, such as a fracture zone in a geological model.
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Damping and Loss
In this section:
• Overview of Damping and Loss
• Linear Viscoelastic Materials
• Rayleigh Damping
• Equivalent Viscous Damping
• Loss Factor Damping
• Explicit Damping
Overview of Damping and Loss
In some cases damping is included implicitly in the material model. This is the case for
Linear Viscoelastic Materials, for which damping operates on the shear components of
stress and strain. Damping must be added explicitly as a subnode of the material node
for material models that do not include damping, such as linear elastic materials.
Phenomenological damping models are typically invoked to model the intrinsic
frictional damping present in most materials (material damping). These models are
easiest to understand in the context of a system with a single degree of freedom. The
following equation of motion describes the dynamics of such a system with viscous
damping:
2
d u
du
m ---------- + c ------- + ku = f  t 
dt
dt
(8-11)
In this equation u is the displacement of the degree of freedom, m is its mass, c is the
damping parameter, and k is the stiffness of the system. The time (t) dependent forcing
term is f(t). This equation is often written in the form:
2
du
2
ft
d u--------+ 2 0 ------- +  0 u = --------dt
m
dt
(8-12)
where c2m0 and 02km. In this case  is the damping ratio (1 for critical
damping) and 0 is the resonant frequency of the system. In the literature it is more
common to give values of  than c.  can also be readily related to many of the various
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367
measures of damping employed in different disciplines. These are summarized in
Table 8-8.
TABLE 8-8: RELATIONSHIPS BETWEEN MEASURES OF DAMPING
DAMPING
PARAMETER
DEFINITION
RELATION TO
DAMPING RATIO
Damping ratio
 = c  c critical
–
Logarithmic
decrement
u  t0 
 d = ln  ----------------------
 u  t 0 +  
 d  2
 « 1
where t0 is a reference time and  is the
period of vibration for a decaying, unforced
degree of freedom.
Quality factor
Loss factor
Q =   
Q  1   2 
where  is the bandwidth of the amplitude
resonance measured at 1  2 of its peak.
 « 1
1 Qh
 = ------  --------
2 W h
At the resonant
frequency:
  2
where Qh is the energy lost per cycle and Wh
is the maximum potential energy stored in the
cycle. The variables Qh and Wh are available
as solid.Qh and solid.Wh.
 « 1
In the frequency domain, the time dependence of the force and the displacement can
be represented by introducing a complex force term and assuming a similar time
dependence for the displacement. The equations
f  t  = Re  Fe
jt
 and u  t  = Re  Ue
jt

are written where  is the angular frequency and the amplitude terms U and F can in
general be complex (the arguments provide information on the relative phase of
signals). Usually the real part is taken as implicit and is subsequently dropped.
Equation 8-11 takes the following form in the frequency domain:
2
–  mU + jcU + kU = F
(8-13)
where the time dependence has canceled out on both sides. Alternatively this equation
can be written as:
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
2
2
F
–  U + 2j 0 U +  0 U = ----m
(8-14)
There are two basic damping models available—Rayleigh damping and models based
on introducing complex quantities into the equation system.
Rayleigh Damping introduces damping in a form based on Equation 8-11. This means
that the method can be applied generally in either the time or frequency domain. The
parameter c in Equation 8-11 is defined as a fraction of the mass and the stiffness using
two parameters, dM and dK, such that
c =  dM m +  dK k
(8-15)
Although this approach seems cumbersome with a one degree of freedom system,
when there are many degrees of freedom m, k, and c become matrices and the
technique can be generalized. Substituting this relationship into Equation 8-11 and
rearranging into the form of Equation 8-12 gives:
2
ft
d
u2 du
2
--------+   dM +  dK  0  ------- +  0 u = --------dt
m
dt
Rayleigh damping can therefore be identified as equivalent to a damping factor at
resonance of:
1  dM
 = ---  ----------- +  dK  0

2  0
(8-16)
Note that Equation 8-16 holds separately for each vibrational mode in the system at
its resonant frequency. In the frequency domain it is possible to use frequency
dependent values of dM and dK. For example setting dM0 and dK2/0
produces a Equivalent Viscous Damping model at the resonant frequency.
While Rayleigh damping is numerically convenient, the model does not agree with
experimental results for the frequency dependence of material damping over an
extended range of frequencies. This is because the material damping forces behave
more like frictional forces (which are frequency independent) than viscous damping
forces (which increase linearly with frequency as implied by Equation 8-13). In the
frequency domain it is possible to introduce loss factor damping, which has the desired
property of frequency independence.
Loss Factor Dampingintroduces complex material properties to add damping to the
model. As a result of this it can only be used in the frequency domain (for
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369
eigenfrequency, frequency domain, or time harmonic studies). In the single degree of
freedom case this corresponds to a complex value for the spring constant k. Setting
c=0, but modifying the spring constant of the material to take a value k1j where
 is the loss factor, modifies the form of Equation 8-13 to:
2
–  mU + jkU + kU = F
(8-17)
Alternatively writing this in the form of Equation 8-14 gives:
F
2
2
2
–  U + j 0 U +  0 U = ----m
Comparing these equations with Equation 8-13 and Equation 8-14 shows that the
loss factor  is related to  and c by:


 = 2 ------  = ---- c
k
0
Equation 8-17 shows that the loss factor has the desired property of frequency
independence. However it is clear that this type of damping cannot be applied in the
time domain. In addition to using loss factor damping the material properties can be
entered directly as complex values in COMSOL Multiphysics, which results in Explicit
Damping
are more complex and include coupling and electrical losses in addition
to the material terms.
For piezoelectric materials, dK is only used as a multiplier of the
structural contribution to the stiffness matrix when building-up the
damping matrix as given by Equation 8-15. In the frequency domain
studies, you can use the coupling and dielectric loss factors equal to dK
to effectively achieve the Rayleigh damping involving the whole stiffness
matrix.
• Linear Viscoelastic Materials
• Rayleigh Damping
• Equivalent Viscous Damping
• Loss Factor Damping
• Explicit Damping
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
Linear Viscoelastic Materials
If Linear Viscoelastic Material is selected for the Solid Mechanics interface, the
viscoelastic branches include damping automatically and no more damping is required.
In the frequency domain the damping using a viscoelastic material corresponds to loss
factor damping applied to the shear components of the material properties.
Rayleigh Damping
As discussed for a model with a single degree of freedom, the Rayleigh damping model
defines the damping parameter c in terms of the mass m and the stiffness k as
c =  dM m +  dK k
where dM and dK are the mass and stiffness damping parameters, respectively. At
any resonant frequency, f, this corresponds to a damping factor,  given by:
1  dM
 = ---  ----------- +  dK 2f

2  2f
(8-18)
Using this relationship at two resonant frequencies f1 and f2 with different damping
factors 1 and 2 results in an equation system
1 ----------f
4f 1 1  dM
1  dK
----------f
4f 2 2
=
1
2
As a result of its non-physical nature, the Rayleigh damping model can only be tuned
to give the correct damping at two independent resonant frequencies or to give an
approximately frequency independent damping response (which is physically what is
usually observed) over a limited range of frequencies.
Using the same damping factors 1 and 2 at frequencies f1 and f2 does
not result in the same damping factor in the interval. It can be shown that
the damping parameters have the same damping at the two frequencies
and less damping in between (see Figure 8-3).
Care must therefore be taken when specifying the model to ensure the
desired behavior is obtained.
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371
Damping factor
Rayleigh damping
Specified damping
f1
f2
f
Figure 8-3: An example of Rayleigh damping.
For many applications it is sufficient to leave dM as zero (the default value) and to
define damping only using the dK coefficient. Then according to Equation 8-18
linearly increasing damping is obtained. If the damping ratio f0 or loss factor f0
is known at a given frequency f0, the appropriate value for dK is:
 dK =    f 0  =    2f 0 
This model results in a well-defined, linearly increasing, damping term that has the
defined value at the given frequency.
All physics interfaces under the Structural Mechanics branch use zero
default values (that is, no damping) for dM and dK. These default values
must be changed to meet the specific modeling situation.
Equivalent Viscous Damping
Although equivalent viscous damping is independent of frequency, it is only possible
to use it in a frequency response analysis. Equivalent viscous damping also uses a loss
factor as the damping parameter, and can be implemented using the Rayleigh damping
feature, by setting the stiffness damping parameter dK, to the loss factor, , divided
by the excitation frequency:


 dK = --------- = ---2f

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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
The mass damping factor, dM, should be set to zero.
Loss Factor Damping
Loss factor damping (sometimes referred to as material or structural damping) takes
place when viscoelasticity is modeled in the frequency domain. The complex modulus
G*() is the frequency-domain representation of the stress relaxation function of
viscoelastic material. It is defined as
G = G + jG =  1 + j s G
where G' is the storage modulus, G'' is the loss modulus, and their ratio sG''G' is
the loss factor. The term G' defines the amount of stored energy for the applied strain,
whereas G'' defines the amount of energy dissipated as heat; G', G'', and s can all be
frequency dependent.
In COMSOL Multiphysics the loss information appears as a multiplier of the total
strain in the stress-strain relationship:
 = D   1 + j s  –  th –  0  +  0 .
For hyperelastic materials the loss information appears as a multiplier in strain energy
density, and thus in the second Piola-Kirchhoff stress, S:
W s
S =  1 + j s  ---------E
Loss factor damping is available for frequency response analysis and damped
eigenfrequency analysis in all interfaces,
The Hyperelastic Material node are available with the Nonlinear
Structural Materials Module.
Explicit Damping
It is possible to define damping by modeling the dissipative behavior of the material
using complex-valued material properties. In COMSOL Multiphysics, you can enter
the complex-valued data directly, using i or sqrt(-1) for the imaginary unit.
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CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH
9
The Piezoelectric Devices User Interface
This chapter describes the background theory for Piezoelectric Devices interface,
which is found under the Structural Mechanics branch (
) in the Model Wizard.
• The Piezoelectric Devices User Interface
• Theory for the Piezoelectric Devices User Interface
• Piezoelectric Damping
375
The Piezoelectric Devices User
Interface
The Piezoelectric Devices (pzd) user interface (
), found under the Structural
) in the Model Wizard, combines Solid Mechanics and
Electrostatics for modeling of piezoelectric devices, for which all or some of the
domains contain a piezoelectric material. The interface has the equations and features
for modeling piezoelectric devices, solving for the displacements and the electric
potential.
Mechanics branch (
The piezoelectric coupling can be presented in stress-charge or strain-charge form. All
solid mechanics and electrostatics functionality for modeling is also accessible to
include surrounding linear elastic solids or air domains. For example, add any solid
mechanics material for other solid domain, a dielectric model for air, or a combination.
When this interface is added, these default nodes are also added to the Model Builder—
Piezoelectric Material, Free (for the solid mechanics and default boundary conditions),
Zero Charge (for the electric potential), and Initial Values. Right-click the Piezoelectric
Devices node to add other features that implement, for example, loads, constraints, and
solid mechanics and electric materials. In 2D and 2D axial symmetry, adding a
Piezoelectric Devices interface also adds predefined base-vector coordinate systems for
the material’s (in the plane 2D case) XY-, YZ-, ZX-, YX-, XZ-, and XY-planes. These
additional coordinate systems are useful for simplifying the material orientation for the
piezoelectric material.
All functionality from the Solid Mechanics and Electric Current interfaces
is accessible for modeling the solid and electric properties and
non-piezoelectric domains. Only the features unique to this interface are
described in this section. For details about the shared features see:
• The Solid Mechanics User Interface in this guide
• The Electrostatics User Interface in the COMSOL Multiphysics
Reference Manual
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
INTERFACE IDENTIFIER
The interface identifier is used primarily as a scope prefix for variables defined by the
physics user interface. Refer to such interface variables in expressions using the pattern
<identifier>.<variable_name>. In order to distinguish between variables
belonging to different physics user interfaces, the identifier string must be unique.
Only letters, numbers and underscores (_) are permitted in the Identifier field. The first
character must be a letter.
The default identifier (for the first interface in the model) is pzd.
DOMAIN SELECTION
The default setting is to include All domains in the model to define the dependent
variables and the equations. To choose specific domains, select Manual from the
Selection list.
2D APPROXIMATION
From the 2D approximation list select Plane stress or Plane strain (the
default). When modeling using plane stress, the Piezoelectric Devices
interface solves for the out-of-plane strain components in addition to the
displacement field u.
THICKNESS
Enter a value or expression for the Thickness d (SI unit: m). The default
value of 1 m is suitable for plane strain models, where it represents a a
unit-depth slice, for example. For plane stress models, enter the actual
thickness, which should be small compared to the size of the plate for the
plane stress assumption to be valid. In rare cases, use a Change Thickness
node to change thickness in parts of the geometry.
S T R U C T U R A L TR A N S I E N T B E H AV I O R
From the Structural transient behavior list, select Include inertial terms (the default) or
Quasi-static. Use Quasi-static to treat the elastic behavior as quasi-static (with no mass
effects; that is, no second-order time derivatives). Selecting this option will give a more
efficient solution for problems where the variation in time is slow when compared to
the natural frequencies of the system. The default solver for the time stepping is
changed from Generalized alpha to BDF when Quasi-static is selected.
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377
REFERE NCE PO IN T FO R MOMENT COMPUTAT IO N
Enter the coordinates for the Reference point for moment computation xref (SI unit: m).
The resulting moments (applied or as reactions) are then computed relative to this
reference point. During the results and analysis stage, the coordinates can be changed
in the Parameters section in the result nodes.
DEPENDENT VARIABLES
This interface defines these dependent variables (fields): the Displacement field u (and
its components) and the Electric potential V. The names can be changed but the names
of fields and dependent variables must be unique within a model.
ADVANCED SETTINGS
To display this section, click the Show button (
) and select Advanced Physics Options.
Normally these settings do not need to be changed.
DISCRETIZATION
To display this section, click the Show button (
) and select Discretization. Select
Linear, Quadratic (the default), Cubic, Quartic, or (in 2D) Quintic for the Displacement
field and Electric potential. Specify the Value type when using splitting of complex
variables—Real or Complex (the default).
• Show More Physics Options
• Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric
Devices User Interface
• Theory for the Piezoelectric Devices User Interface
Domain, Boundary, Edge, Point, and Pair Nodes for the Piezoelectric
Devices User Interface
Because The Piezoelectric Devices User Interface is a multiphysics interface, many
physics nodes are shared with, and described for, other interfaces. Below are links to
the domain, boundary, edge, point, and pair nodes as indicated.
To locate and search all the documentation, in COMSOL Multiphysics,
select Help>Documentation from the main menu and either enter a search
term or look under a specific module in the documentation tree.
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
These nodes are described in this section:
• Damping and Loss
• Initial Values
• Dielectric Loss
• Periodic Condition
• Electrical Conductivity
(Time-Harmonic)
• Piezoelectric Material
• Remanent Electric Displacement
• Electrical Material Model
These nodes are described for the Solid Mechanics interface (listed in alphabetical
order):
• Added Mass
• Point Load
• Antisymmetry
• Pre-Deformation
• Body Load
• Prescribed Acceleration
• Boundary Load
• Prescribed Displacement
• Edge Load
• Prescribed Velocity
• Fixed Constraint
• Roller
• Free
• Spring Foundation
• Initial Stress and Strain
• Symmetry
• Linear Elastic Material
• Thin Elastic Layer
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|
379
These nodes are described for the Electrostatics interface in the COMSOL
Multiphysics Reference Manual:
• Electric Displacement Field
• Point Charge
• Electric Potential
• Point Charge (on Axis)
• Ground
• Space Charge Density
• Line Charge
• Surface Charge Density
• Line Charge (on Axis)
• Thin Low Permittivity Gap
• Line Charge (Out-of-Plane)
• Zero Charge
The links to the nodes described in the COMSOL Multiphysics Reference
Manual do not work in the PDF, only from the on line help in COMSOL
Multiphysics.
Piezoelectric Material
Use the Piezoelectric Material node to define the piezoelectric material properties on
stress-charge form using the elasticity matrix and the coupling matrix or on
strain-charge form using the compliance matrix and the coupling matrix. The default
settings is to use material data defined for the material in the domain. Right-click
Piezoelectric Material to add Electrical Conductivity (Time-Harmonic), Initial Stress
and Strain, and Damping and Loss nodes as required.
For entering these matrices, the ordering is different from the standard
ordering used in COMSOL Multiphysics. Instead, use the following
order (Voigt notation), which is the common convention for piezoelectric
materials: xx, yy, zz, yz, xz, zy.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
MODEL INPUTS
This section has field variables that appear as model inputs, if the current settings
include such model inputs. By default, this section is empty.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
PIEZOELECTRIC MATERIAL PROPERTIES
Select a Constitutive relation—Stress-charge form or Strain-charge form. For each of the
following, the default uses values From material. Select User defined to enter other
values in the matrix or field as required.
• For Stress-charge form, select an Elasticity matrix (ordering: xx, yy, zz, yz, xz, xy) (cE)
(SI unit: 1/Pa).
• For a Strain-charge form, select a Compliance matrix (ordering: xx, yy, zz, yz, xz, xy)
(sE) (SI unit: 1/Pa).
• Select a Coupling matrix (ordering: xx, yy, zz, yz, xz, xy) (d) (SI unit: C/m2 or C/N).
• Select a Relative permittivity (erS or erT) (dimensionless).
• Select a Density (p) (SI unit: kg/m3).
GEOMETRIC NONLINEARITY
If a study step is geometrically nonlinear, the default behavior is to use a large strain
formulation in all domains. There are however some cases when you would still want
to use a small strain formulation for a certain domain. In those cases, select the Force
linear strains check box. When selected, a small strain formulation is always used,
independently of the setting in the study step.
• Geometric Nonlinearity for the Piezoelectric Devices User Interface
• The Solid Mechanics User Interface
Electrical Material Model
The Electrical Material Model adds an electric field to domains in a piezoelectric device
model that only includes the electric field. Right-click the node to add Electrical
Conductivity (Time-Harmonic) and Dielectric Loss nodes as required.
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DOMAIN SELECTION
From the Selection list, choose the domains to define.
MODEL INPUTS
This section contains field variables that appear as model inputs, if the current settings
include such model inputs. By default, this section is empty.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
ELECTRIC FIELD
Select a Constitutive relation—Relative permittivity, Polarization, or Remanent
displacement.
• If Relative permittivity is selected, also choose a Relative permittivity (r)
(dimensionless). The default uses values From material. If User defined is selected,
choose Isotropic, Diagonal, Symmetric, or Anisotropic and enter values in the matrix
or field.
• If Polarization is selected, enter the Polarization P (SI unit: C/m2) coordinates.
• If Remanent displacement is selected, select a Relative permittivity (r). The default
uses values From material. If User defined is selected, choose Isotropic, Diagonal,
Symmetric, or Anisotropic and enter values in the matrix or field. Then enter the
Remanent displacement (Dr) (SI unit: C/m2) coordinates.
GEOMETRIC NONLINEARITY
If a study step is geometrically nonlinear, the default behavior is to use a large strain
formulation in all domains. There are however some cases when you would still want
to use a small strain formulation for a certain domain. In those cases, select the Force
linear strains check box. When selected, a small strain formulation is always used,
independently of the setting in the study step.
• Geometric Nonlinearity for the Piezoelectric Devices User Interface
• See The Solid Mechanics User Interface for details about this section.
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Electrical Conductivity (Time-Harmonic)
Right-click the Piezoelectric Material node or the Electrical Material Model to add an
Electrical Conductivity (Time-Harmonic) node. This subnode adds ohmic conductivity to
the material. For example, if the model has metal electrodes, or if the piezoelectric
material might not be a perfect insulator but has some electrical conductivity. Because
The Piezoelectric Devices User Interface solves for the charge balance equation (that
is, electrostatics) this conductivity would lead to a time integral of the ohmic current
in the equation. This feature can therefore only operate in a time-harmonic study (as
pointed out in the name), and the “equivalent electric displacement” Jij appears in
the equation.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
COORDINATE SYSTEM SELECTION
The Global coordinate system is selected by default. The Coordinate system list contains
any additional coordinate systems that the model includes.
CONDUCTION CURRENT
Select an Electrical conductivity  (SI unit: S/m). Select:
• From material to use the conductivity value from the domain material.
• Linearized resistivity to define the electric resistivity (and conductivity) as a linear
function of temperature.
• User defined to enter a value (SI unit: S/m) or expressions for an isotropic or
anisotropic conductivity. Select Isotropic, Diagonal, Symmetric, or Anisotropic from
the list based on the properties of the conductive media.
If Linearized resistivity is selected, each default setting in the corresponding Reference
temperature (Tref), Resistivity temperature coefficient (), and Reference resistivity (0)
lists is From material, which means that the values are taken from the domain material.
To specify other values for these properties, select User defined from the corresponding
list and then enter a value or expression in the applicable field.
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Damping and Loss
Right-click the Piezoelectric Material node to add a Damping and Loss subnode, which
adds damping (Rayleigh damping or loss damping), coupling losses, and dielectric
losses to the piezoelectric material.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
DAMPING SETTINGS
Select a Damping type—Rayleigh damping, Loss factor for cE, Loss factor for sE, No
damping, or Isotropic loss factor:
• No damping
• For Rayleigh damping, enter the Mass damping parameter dM and the Stiffness
damping parameter in the dM corresponding fields. The default values are 0, which
means no damping.
• For Loss factor for cE, select From material (the default) from the Loss factor for
elasticity matrix cE list to use the value from the material or select User defined to
enter values or expressions for the loss factor in the associated fields. Select
Symmetric to enter the components of cE in the upper-triangular part of a
symmetric 6-by-6 matrix or select Isotropic to enter a single scalar loss factor. The
default values are 0.
• For Loss factor for sE, from the Loss factor for compliance matrix sE list, select From
material (the default) to use the value from the material or select User defined to
enter values or expressions for the loss factor in the associated fields. Select
Symmetric to enter the components of sE in the upper-triangular part of a
symmetric 6-by-6 matrix or select Isotropic to enter a single scalar loss factor. The
default values are 0.
• For an Isotropic loss factor s, select From material (the default) from the Isotropic
structural loss factor list to take the value from the material or select User defined to
enter a value or expression for the isotropic loss factor in the field. The default value
is 0.
COUPLING LOSS SETTINGS
Select a Coupling loss—No loss, Loss factor for e, or Loss factor for d.
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For Loss factor for e and Loss factor for d, select a Loss factor for coupling matrix e or d
from the list. Select User defined to enter values or expressions for the loss factor in the
associated fields. Select Symmetric to enter the components of e or d in the
upper-triangular part of a symmetric 6-by-6 matrix or select Isotropic to enter a single
scalar loss factor. The default values are 0.
DIELECTRIC LOSS SETTINGS
From the Dielectric loss list, select Loss factor for S, Loss factor for T, or No loss.
For Loss factor for S and Loss factor for T, select a Loss factor for permittivity. Select
From material (the default) to use the value from the material or select User defined to
enter values or expressions for the loss factor in the associated fields. Select Symmetric
to enter the components of eS or eT in the upper-triangular part of a symmetric
6-by-6 matrix, select Isotropic to enter a single scalar loss factor, or select Diagonal. The
default values are 0.
Remanent Electric Displacement
Right-click the Piezoelectric Material node to add a Remanent Electric Displacement
subnode to include a remanent electric displacement vector Dr (the displacement
when no electric field is present).
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
REMANENT ELECTRIC DISPLACEMENT
Enter the components of the remanent electric displacement Dr (SI unit: C/m2) in
the Remanent displacement fields (the default values are 0 C/m2).
Dielectric Loss
Right-click the Electrical Material Model node to add a Dielectric Loss subnode to
include a dielectric loss using a dielectric loss factor.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
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interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
DIELECTRIC LOSS SETTINGS
The default Dielectric loss factor uses values From material. If User defined is selected,
then also select Isotropic, Diagonal, Symmetric, or Anisotropic and enter one or more
components in the field or matrix. The default values are 0.
Initial Values
The Initial Values node adds an initial value for the displacement field and the electric
potential. Right-click to add additional Initial Values nodes.
DOMAIN SELECTION
For a default node, the setting inherits the selection from the parent node, and cannot
be edited; that is, the selection is automatically selected and is the same as for the
interface. When nodes are added from the context menu, you can select Manual from
the Selection list to choose specific domains or select All domains as required.
INITIAL VALUES
Enter the initial values as values or expressions for the Displacement field u (SI unit: m)
and the Electric potential V (SI unit: V).
Periodic Condition
The Periodic Condition node adds a periodic boundary condition. This periodicity make
uix0uix1 for a displacement component ui or similarly for the electric potential.
You can control the direction that the periodic condition applies to and if it applies to
the electric potential. Right-click the Periodic Condition node to add a Destination
Selection subnode if required. If the source and destination boundaries are rotated with
respect to each other, this transformation is automatically performed, so that
corresponding displacement components are connected.
This feature works well for cases like opposing parallel boundaries. In
other cases use a Destination Selection subnode to control the destination.
By default it contains the selection that COMSOL Multiphysics identifies.
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PERIODICITY SETTINGS
Select a Type of periodicity—Continuity (the default), Antiperiodicity, Floquet periodicity,
Cyclic symmetry, or User defined.
• If Floquet periodicity is selected, enter a k-vector for Floquet periodicity kF (SI unit:
rad/m) for the X, Y, and Z coordinates (3D models), or the R and Z coordinates
(2D axisymmetric models), or X and Y coordinates (2D models).
• If Cyclic symmetry is selected, select a Sector angle—Automatic (the default), or User
defined. If User defined is selected, enter a value for S (SI unit: rad; default value: 0).
For any selection, also enter a Azimuthal mode number m (dimensionless; default
value: 0).
• If User defined is selected, select the Periodic in u, Periodic in v (for 3D and 2D
models), and Periodic in w (for 3D and 2D axisymmetric models) check boxes as
required. For all dimensions the Periodic in V check box is also available. Then for
each selection, choose the Type of periodicity—Continuity (the default) or
Antiperiodicity.
In the COMSOL Multiphysics Reference Manual:
• Periodic Condition and Destination Selection
• Periodic Boundary Conditions
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T he o r y f o r the Pi ez oel ect ri c D ev i c es
User Interface
The Piezoelectric Devices User Interface theory is described in this section:
• The Piezoelectric Effect
• Piezoelectric Constitutive Relations
• Piezoelectric Material
• Piezoelectric Dissipation
• Initial Stress, Strain, and Electric Displacement
• Geometric Nonlinearity for the Piezoelectric Devices User Interface
• Damping and Losses Theory
• References for the Piezoelectric Devices User Interface
The Piezoelectric Effect
The piezoelectric effect manifests itself as a transfer of electric to mechanical energy
and vice versa. It is present in many crystalline materials, while some materials such as
quartz, Rochelle salt, and lead titanate zirconate ceramics display the phenomenon
strongly enough for it to be of practical use.
The direct piezoelectric effect consists of an electric polarization in a fixed direction
when the piezoelectric crystal is deformed. The polarization is proportional to the
deformation and causes an electric potential difference over the crystal.
The inverse piezoelectric effect, on the other hand, constitutes the opposite of the
direct effect. This means that an applied potential difference induces a deformation of
the crystal.
PIEZOELECTRICITY CONVENTIONS
The documentation and the Piezoelectric Devices interface use piezoelectricity
conventions as much as possible. These conventions differ from those used in other
structural mechanics interfaces. For instance, the numbering of the shear components
in the stress-strain relation differs, as the following section describes. However, the
names of the stress and strain components remain the same as in the other structural
mechanics interfaces.
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Piezoelectric Constitutive Relations
It is possible to express the relation between the stress, strain, electric field, and electric
displacement field in either a stress-charge form or strain-charge form:
STRESS-CHARGE
T
T = cE S – e E
D = eS +  S E
STRAIN-CHARGE
T
S = sE T + d E
D = dT +  T E
The naming convention differs in piezoelectricity theory compared to structural
mechanics theory, but the Piezoelectric Devices interface uses the structural mechanics
nomenclature. The strain is named  instead of S, and the stress is named  instead of
T. This makes the names consistent with those used in the other structural mechanics
interfaces.
The numbering of the strain and stress components is also different in piezoelectricity
theory and structural mechanics theory, and it is quite important to keep track of this
aspect in order to provide material data in the correct order. In structural mechanics
the following is the most common numbering convention, and it is also the one used
as default in the structural mechanics interfaces:
 =
 xx
 xx
 xx
 yy
 yy
 yy
 zx
 zz
 xy
 =
 xy
=
 zz
2 xy
 yz
 yz
2 yz
 xz
 xz
2 xz
In contrast, textbooks on piezoelectric effects and the IEEE standard on piezoelectric
effects use the following numbering convention (also called Voigt notation):
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 =
 xx
 xx
 xx
 yy
 yy
 yy
 zz
 yz
 zz
 =
 yz
=
 zz
2 yz
 xz
 xz
2 xz
 xy
 xy
2 xy
The Piezoelectric Devices interface uses the immediately preceding piezo numbering
convention (Voigt notation) to make it easier to work with material data and to avoid
mistakes.
The constitutive relation using COMSOL Multiphysics symbols for the different
constitutive forms are thus:
STRESS-CHARGE
T
 = cE  – e E
D = e +  0  rS E
STRAIN-CHARGE
T
 = sE + d E
D = d +  0  rT E
Most material data appears in the strain-charge form, and it can be easily transformed
into the stress-charge form. In COMSOL Multiphysics both constitutive forms can be
used; simply select one, and the software makes any necessary transformations. The
following equations transform strain-charge material data to stress-charge data:
–1
cE = sE
–1
e = d sE
–1
 S =  0  rT – d s E d
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T
Piezoelectric Material
The Piezoelectric Devices interface also has different materials for easier modeling of
piezo components. This means that the material for each domain can be defined as:
• Piezoelectric material (the default material)
• Purely solid as a linear elastic or nonlinear material
• Purely dielectric using an electrical material (to model surrounding air, for example)
The piezoelectric material operates as described above, whereas using the two other
materials, structural and electrical problems can be modeled, together or either of
them independently.
Piezoelectric Dissipation
In order to define dissipation in the piezoelectric material for a time-harmonic analysis,
all material properties in the constitutive relations can be complex-valued matrices
where the imaginary part defines the dissipative function of the material.
As described in Damping and Losses Theory complex-valued data can be defined
directly in the fields for the material properties, or a real-valued material X and a set of
loss factors X can be defined, which together form the complex-valued material data
˜
X = X  1  j X 
See also the same references for an explanation of the sign convention. It is also
possible to define the electrical conductivity of the piezoelectric material: S or T
depending on the constitutive relation. Electrical conductivity does not appear directly
in the constitutive equation, but it appears as an additional term in the variational
formulation (weak equation).
The conductivity does not change during transformation between the
formulations. S and T are used to get fully-defined materials in each
formulation.
Initial Stress, Strain, and Electric Displacement
Using the piezoelectrical physics interfaces initial stress (0), initial strain (0), and
initial electric displacement (D0) can be defined for models. In the constitutive relation
for piezoelectric material these additions appear in the stress-charge formulation:
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T
 = cE   – 0  – e E + 0
D = e   –  0  +  0  rS E + D 0
When solving the model, these program does not interpret these fields as a constant
initial state, but they operate as additional fields that are continuously evaluated. Thus
use these initial field to add, for example, thermal expansion or pyroelectric effects to
models.
Geometric Nonlinearity for the Piezoelectric Devices User Interface
PIEZOELECTRIC MATERIALS WITH LARGE DEFORMATIONS
The linear piezoelectric equations as presented in Piezoelectric Constitutive Relations
with engineering strains are valid if the model undergoes only relatively small
deformations. As soon as the model contains larger displacements or rotations, these
equations produce spurious strains that result in an incorrect solution. To overcome
this problem, so-called large deformation piezoelectrical equations are required.
The Piezoelectric Devices interface implements the large deformation piezoelectrical
equations according to Yang (Ref. 8). Key items of this formulation are:
• The strains are calculated as the Green-Lagrange strains, ij:
1 u i u j u k u k
 ij = ---  -------- + -------- + ---------  ---------
2  X j X i X i X j 
(9-1)
Green-Lagrange strains are defined with reference to an undeformed geometry.
Hence, they represent a Lagrangian description. In a small-strain, large rotational
analysis, the Green-Lagrange strain corresponds to the engineering strain in
directions that follow the deformed body.
• Electrical field variables are calculated in the material directions, and the electric
displacement relation is replaced by an expression that produce electric polarization
in the material orientation of the solid.
• In the variational formulation, the electrical energy is split into two parts: The
polarization energy within the solid and the electric energy of free space occupied
by the deformed solid.
The first two items above result in another set of constitutive equations for large
deformation piezoelectricity:
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T
S = cE  – e Em
P m = e +   0  rS –  0 I E m
where S is the second Piola-Kirchhoff stress;  is the Green-Lagrange strain, Em and
Pm are the electric field and electric polarization in the material orientation; I is the
identity matrix; and cE, e, and rS are the piezoelectric material constants. The
expression within parentheses equals the dielectric susceptibility of the solid:
 =   0  rS –  0 I 
The electric displacement field in the material orientation results from the following
relation
–1
D m = P m +  0 JC E m
where C is the right Cauchy-Green tensor
T
C = F F
Fields in the global orientation result from the following transformation rules:
E = F
–T
Em
–1
P = J FP m
–1
(9-2)
D = J FD m
v = V J
–1
where F is the deformation gradient; J is the determinant of F; and v and V are the
volume charge density in spatial and material coordinates respectively. The
deformation gradient is defined as the gradient of the present position of a material
point xX + u:
x
F = ------X
DECOUPLED MATERIALS WITH LARGE DEFORMATIONS
The large deformation formulation described in the previous section applies directly to
non-piezoelectric materials if the coupling term is set to zero: e0. In that case, the
structural part corresponds to the large deformation formulation described for the
solid mechanics interfaces.
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The electrical part separates into two different cases: For solid domains the electric
energy consists of polarization energy within the solid and the electric energy of free
space occupied by the deformed solid—the same as for the piezoelectric materials. For
nonsolid domains this separation does not occur, and the electric displacement in these
domains directly results from the electric field—the electric displacement relation:
Dm = 0 r Em
On nonsolid domains the global orientation of the fields is not known
unless the ALE method is used.
LARGE DEFORMATION AND DEFORMED MESH
The Piezoelectric Devices interface can be coupled with the Moving Mesh (ALE)
physics interface in a way so that the electrical degrees of freedom are solved in an ALE
frame. This feature is intended to be used in applications where a model contains
nonsolid domains, such as modeling of electrostatically actuated structures. This
functionality is not required for modeling of piezoelectric or other solid materials.
The use of ALE has impacts on the formulation of the electrical large deformation
equations. The first impact is that with ALE, the gradient of electric potential directly
results in the electric field in the global orientation, and the material electric field
results after transformation.
The most visible impact is on the boundary conditions. With ALE any surface charge
density or electric displacement is defined per the present deformed boundary area,
whereas for the case without ALE they are defined per the undeformed reference area.
Damping and Losses Theory
HYSTERETIC LOSS
The equivalent viscous and loss factor damping are special cases of a more general way
of defining damping: hysteretic loss. Generally, and independently of the microscopic
origin of the loss, the dissipative behavior of the material can be modeled using
complex-valued material properties. For the case of piezoelectric materials, this means
that the constitutive equations are written as follows:
For the stress-charge formulation
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T
 = c˜ E  – e˜ E
D = e˜  + ̃ E
S
and for the strain-charge formulation
T
 = s˜ E  + d˜ E
D = d˜  + ̃ E
T
where c˜ E , s˜ E , ˜S , ˜T , and d˜ are complex-valued matrices, where the imaginary part
defines the dissipative function of the material.
Similarly to the real-valued material data, it is not possible to freely define the
complex-valued data. Instead the data must fulfill certain requirement to represent
physically proper materials. A key requirement is that the dissipation density is positive;
that is, there is no power gain from the passive material. This requirement sets rules for
the relative magnitudes for all material parameters. This is important to be aware of,
especially when defining the coupling losses.
In COMSOL Multiphysics the complex-valued data can be entered directly, or the
concept of loss factors can be used. Similarly to the loss factor damping, the complex
˜
data X is represented as pairs of a real-valued parameter
˜
X = real  X 
and a loss factor
˜
˜
 X = imag  X   real  X 
the ratio of the imaginary and real part, and the complex data is then
˜
X = X  1  j X 
where the sign depends on the material property used. The loss factors are specific to
the material property, and thus they are named according to the property they refer to,
for example, cE. For a structural material without coupling, simply use s, the
structural loss factor.
Depending on the field, different terminology is in use. For example, the loss tangent
tan might be referred to when working with electrical applications. The loss tangent
has the same meaning as the loss factor. Often the quality factor Qm is defined for a
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material. The quality factor Qm and the loss factor i are inversely related: i1 Qm,
where i is the loss factor for cE, sE, or the structural loss factor depending on the
material.
The Piezoelectric Devices interface uses a formulation that assumes that a positive loss
factor corresponds to a positive loss. The complex-valued data is then based on sign
rules. For piezoelectric materials, the following equations apply (m and n refer to
elements of each matrix):
m n
m n
m n
c˜ E = c E  1 + j cE 
m n
m n
m n
e˜
=e
 1 – j e 
m n
̃ S
m n
= S
m n
 1 – j S 
m n
m n
m n
s˜ E = s E  1 – j sE 
(9-3)
m n
m n
m n
d˜
=d
 1 – j d 
m n
̃ T
m n
= T
m n
 1 – j T 
The losses for the non-coupled material models are more straightforward to define.
Using the complex stiffness and permittivity, the following equations describe the lossy
material:
m n
˜ m n
m n
D
=  1 + j
D
̃ e
m n
=  1 – j e
m n
 e
m n
(9-4)
Often fully defined complex-valued data is not accessible. In the Piezoelectric Devices
interface the loss factors can be defined as full matrices or as scalar isotropic loss factors
independently of the material and the other coefficients.
For more information about hysteretic losses, see Ref. 1 to Ref. 4.
T H E L O S S F A C T O R U S I N G D I F F E R E N T D A M P I N G TY P E S
The following damping types use an isotropic loss factor s:
• Loss factor damping
• Equivalent viscous damping
• Isotropic loss
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In each case the meaning of the loss factor is the same: the fractional loss of energy per
cycle.
The difference between these damping types is how the loss enters the equations.
Using the isotropic loss, s is used to build complex-valued material properties,
whereas when using the loss factor damping, s appears in a complex-valued multiplier
in the stress-strain relation. In the equivalent viscous damping, s appears in a
complex-valued and frequency-dependent expression for dK of the Rayleigh damping
model.
ELECTRICAL CONDUCTIVITY AND DIELECTRIC LOSSES
For frequency response and damped eigenfrequency analyses, the electrical
conductivity of the piezoelectric and decoupled material (see Ref. 2, Ref. 5, and Ref.
6) can be defined. Depending on the formulation of the electrical equation, the
electrical conductivity appears in the variational formulation (the weak equation) either
as an effective electric displacement
Jp
˜
D =  r  0 E – j -----
(the actual displacement variables do not contain any conductivity effects) or in the
total current expression
J = Jd + Jp
where Jp = eE is the conductivity current and Jd is the electric displacement current.
Both a dielectric loss factor (Equation 9-3 and Equation 9-4) and the electrical
conductivity can be defined at the same time. In this case, ensure that the loss factor
refers to the alternating current loss tangent, which dominates at high frequency,
where the effect of ohmic conductivity vanishes (Ref. 7).
References for the Piezoelectric Devices User Interface
1. R. Holland and E. P. EerNisse, Design of Resonant Piezoelectric Devices, Research
Monograph No. 56, The M.I.T. Press, 1969.
2. T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, 1990.
3. A.V. Mezheritsky, “Elastic, Dielectric, and Piezoelectric Losses in Piezoceramics:
How it Works all Together,” IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control, vol. 51, no. 6, 2004.
THEORY FOR THE PIEZOELECTRIC DEVICES USER INTERFACE
|
397
4. K. Uchino and S. Hirose, “Loss Mechanisms in Piezoelectrics: How to Measure
Different Losses Separately,” IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control, vol. 48, no. 1, pp. 307–321, 2001.
5. P.C.Y. Lee, N.H. Liu, and A. Ballato, “Thickness Vibrations of a Piezoelectric Plate
with Dissipation,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency
Control, vol. 51, no. 1, 2004.
6. P.C.Y. Lee and N.H. Liu, “Plane Harmonic Waves in an Infinite Piezoelectric Plate
with Dissipation,” Frequency Control Symposium and PDA Exhibition, IEEE
International, pp. 162–169, 2002.
7. C. A. Balanis, “Electrical Properties of Matter,” Advanced Engineering
Electromagnetics, John Wiley & Sons, 1989.
8. J. Yang, An Introduction to the Theory of Piezoelectricity, Springer Science and
Business Media, N.Y., 2005.
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Piezoelectric Damping
In this section:
• About Piezoelectric Materials
• Piezoelectric Material Orientation
• Piezoelectric Losses
• References for Piezoelectric Damping
About Piezoelectric Materials
Piezoelectric materials become electrically polarized when strained. From a
microscopic perspective, the displacement of atoms within the unit cell (when the solid
is deformed) results in electric dipoles within the medium. In certain crystal structures
this combines to give an average macroscopic dipole moment or electric polarization.
This effect, known as the direct piezoelectric effect, is always accompanied by the
converse piezoelectric effect, in which the solid becomes strained when placed in an
electric field.
Within a piezoelectric there is a coupling between the strain and the electric field,
which is determined by the constitutive relation:
T
S = sE T + d E
D = dT +  T E
(9-5)
Here, S is the strain, T is the stress, E is the electric field, and D is the electric
displacement field. The material parameters sE, d, and T, correspond to the material
compliance, the coupling properties and the permittivity. These quantities are tensors
of rank 4, 3, and 2 respectively, but, since the tensors are highly symmetric for physical
reasons, they can be represented as matrices within an abbreviated subscript notation,
which is usually more convenient. In the Piezoelectric Devices interface, the Voigt
notation is used, which is standard in the literature for piezoelectricity but which differs
from the defaults in the Solid Mechanics interface. Equation 9-5 is known as the
strain-charge form of the constitutive relations. The equation can be re-arranged into
the stress-charge form, which relates the material stresses to the electric field:
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T
T = cE S – e E
D = eS +  S E
(9-6)
The material properties, cE, e, and S are related to sE, d, and T. Note that it is
possible to use either form of the constitutive relations. In addition to Equation 9-5 or
Equation 9-6, the equations of solid mechanics and electrostatics must also be solved
within the material.
• The Piezoelectric Devices User Interface
• Theory for the Piezoelectric Devices User Interface
Piezoelectric Material Orientation
The orientation of a piezoelectric crystal cut is frequently defined by the system
introduced by the I.R.E. standard of 1949 (Ref. 8). This standard has undergone a
number of subsequent revisions, with the final revision being the IEEE standard of
1989 (Ref. 9). Unfortunately the more recent versions of the standard have not been
universally adopted, and significant differences exist between the 1949 and the 1987
standards. The 1987 standard was ultimately withdrawn by the IEEE. COMSOL
Multiphysics follows the conventions used in the book by Auld (Ref. 10) and defined
by the 1987 standard. While these conventions are often used for many piezoelectric
materials, unfortunately practitioners in the quartz industry usually adhere to the older
1947 standard, which results in different definitions of crystal cuts and of material
properties.
The stiffness, compliance, coupling, and dielectric material property matrices are
defined with the crystal axes aligned with the local coordinate axes. In the absence of
a user defined coordinate system, the local system corresponds to the global X, Y, and
Z coordinate axes.
The material properties are defined in the material frame, so that if the
solid rotates during deformation the material properties rotate with the
solid. See Geometric Nonlinearity, Frames, and the ALE Method.
The crystal axes used to define material properties correspond to the 1987 IEEE
standard. All piezoelectric material properties are defined using the Voigt form of the
abbreviated subscript notation, which is almost universally employed in the literature
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
(this differs from the standard notation used for the Solid Mechanics interface material
properties). To define a particular crystal cut, a local set of rotated coordinates must
be defined; this local system then corresponds to the orientation of the crystal axes
within the model.
For some materials, the crystal X, Y, and Z axes are defined differently
between the 1987 IEEE standard and the 1949 I.R.E. standard.
Figure 9-1 shows the case of right-handed quartz (which is included in
the COMSOL material library as quartz; see Piezoelectric Materials
Database in the COMSOL Multiphysics Reference Manual), which has
different axes defined within the two standards.
The different axes sets result in different material properties so, for
example, the elasticity or stiffness matrix component cE14 of quartz takes
the value 18 GPa in the 1987 standard and 18 GPa in the 1949
standard.
The crystal cuts are also defined differently within the 1949 and 1987 standards. Both
standards use a notation that defines the orientation of a virtual slice (the plate)
through the crystal. The crystal axes are denoted X, Y, and Z and the plate, which is
usually rectangular, is defined as having sides l, w, and t (length, width, and thickness).
Initially the plate is aligned with respect to the crystal axes and then up to three
rotations are defined, using a right-handed convention about axes embedded along the
l, w, and t sides of the plate. Taking AT cut quartz as an example, the 1987 standard
defines the cut as: (YXl) 35.25°. The first two letters in the bracketed expression
always refer to the initial orientation of the thickness and the length of the plate.
Subsequent bracketed letters then define up to three rotational axes, which move with
the plate as it is rotated. Angles of rotation about these axes are specified after the
bracketed expression in the order of the letters, using a right-handed convention. For
AT cut quartz only one rotation, about the l axis, is required. This is illustrated in
Figure 9-2. Note that within the 1949 convention AT cut quartz is denoted as: (YXl)
35.25°, since the X-axis rotated by 180° in this convention and positive angles
therefore correspond to the opposite direction of rotation (see Figure 9-1).
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Figure 9-1: Crystallographic axes defined for right-handed quartz in COMSOL and the
1987 IEEE standard (color). The 1949 standard axes are shown for comparison (gray).
Figure 9-1 is reproduced with permission from: IEEE Std 176-1987 IEEE Standard on Piezoelectricity, reprinted with permission from
IEEE, 3 Park Avenue, New York, NY 10016-5997 USA, copyright 1987,
by IEEE. This figure may not be reprinted or further distributed without
prior written permission from the IEEE.
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Because COMSOL Multiphysics allows user-defined material parameters,
it is possible to add a user-defined material defined within the 1949
standard if the use of the 1987 standard is inconvenient. In any case,
significant care must be taken when entering material properties and
when defining the rotated coordinate system for a given cut. In the
literature, the particular standard being employed to define material
properties and cuts is rarely cited.
Figure 9-2: Definition of the AT cut of quartz within the IEEE 1987 standard. The AT
cut is defined as: (YXl) 35.25°. The first two bracketed letters specify the initial
orientation of the plate, with the thickness direction, t, along the crystal Y axis and the
length direction, l, along the X axis. Then up to three rotations about axes that move with
the plate are specified by the corresponding bracketed letters and the subsequent angles. In
this case only one rotation is required about the l axis, of 35.25° (in a right-handed
sense).
When defining material properties it is necessary to consider the orientation of the
plate with respect to the global coordinate system in addition to the orientation of the
plate with respect to the crystallographic axes. Consider once again the example of AT
cut quartz in Figure 9-2. The definition of the appropriate local coordinate system
depends on the desired final orientation of the plate in the global coordinate system.
One way to set up the plate is to orientate its normal parallel to the Y axis in the global
coordinate system. Figure 9-3 shows how to define the local coordinate system in this
case. Figure 9-4 shows how to define the local system such that the plate has its normal
parallel to the global Z axis.
PIEZOELECTRIC DAMPING
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403
In both cases it is critical to keep track of the orientation of the local
system with respect to the global system, which is defined depending on
the desired orientation of the plate in the model.
There are also a number of methods to define the local coordinate system
with respect to the global system.
Usually it is most convenient to define the local coordinates with a Rotated System
node, which defines three Euler angles according to the ZXZ convention (rotation
about Z, then X, then Z again). Note that these Euler angles define the local (crystal)
axes with respect to the global axes—this is distinct from the approach of defining the
cut (global) axes with respect to the crystal (local) axes.
Figure 9-3: Defining an AT cut crystal plate within COMSOL, with normal in the global
Y-direction. Within the 1987 IEEE standard the AT cut is defined as (YXl) -35.25°. Start
with the plate normal or thickness in the Ycr direction (a) and rotate the plate 35.25°
about the l axis (b). The global coordinate system rotates with the plate. Finally rotate the
entire system so that the global coordinate system is orientated as it appears in COMSOL
(c). The local coordinate system should be defined with the Euler angles (ZXZ - 0, 35.25°,
0).(d) shows a coordinate system for this system in COMSOL.
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
Figure 9-4: Defining an AT cut crystal plate within COMSOL, with normal in the global
Z-direction. Within the 1987 IEEE standard the AT cut is defined as (YXl) 35.25°.
Begin with the plate normal in the Zcr-direction, so the crystal and global systems are
coincident. Rotate the plate so that its thickness points in the Ycr-direction (the starting
point for the IEEE definition), the global system rotates with the plate (b). Rotate the plate
35.25° about the l axis (d). Finally rotate the entire system so that the global coordinate
system is orientated as it appears in COMSOL (d). The local coordinate system should be
defined with the Euler angles (ZXZ: 0, -54.75°, 0). (e) shows a coordinate system for this
system in COMSOL.
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Piezoelectric Losses
Losses in piezoelectric materials can be generated both mechanically and electrically.
In the frequency domain these can be represented by introducing complex material
properties in the elasticity and permittivity matrices, respectively. Taking the
mechanical case as an example, this introduces a phase lag between the stress and the
strain, which corresponds to a Hysteretic Loss. These losses can be added to the
Piezoelectric Materialby a Damping and Losssubnode, and are typically defined as a
loss factor (see below). For the case of electrical losses, hysteretic electrical losses are
usually used to represent high frequency electrical losses that occur as a result of
friction impeding the rotation of the microscopic dipoles that produce the material
permittivity. Low frequency losses, corresponding to a finite material conductivity, can
be added to the model through an Electrical Conductivity (Time Harmonic) node.
This feature also operates in the frequency domain. Note that the option to add
Rayleigh damping, or explicit damping (which is a particular case of Rayleigh damping
in the frequency domain), is also available in the Damping and Loss node for the
frequency domain.
In the time domain, material damping can be added using the Rayleigh Damping
option in the Damping and Loss node. Electrical damping is currently not available in
the time domain.
• Rayleigh Damping
• Explicit Damping
HYSTERETIC LOSS
In the frequency domain, the dissipative behavior of the material can be modeled using
complex-valued material properties, irrespective of the loss mechanism. Such hysteretic
losses can be applied to model both electrical and mechanical losses. For the case of
piezoelectric materials, this means that the constitutive equations are written as
follows:
For the stress-charge formulation
T
 = c˜ E  – e˜ E
D = e˜  + ̃ E
S
and for the strain-charge formulation
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
T
 = s˜ E  + d˜ E
D = d˜  + ̃ E
T
where c˜ E , d˜ , and  are complex-valued matrices, where the imaginary part defines the
dissipative function of the material.
Both the real and complex parts of the material data must be defined so as to respect
the symmetry properties of the material being modeled and with restrictions imposed
by the laws of physics.
A key requirement is that the dissipation density is positive; that is, there
is no power gain from the passive material. This requirement sets rules for
the relative magnitudes for all material parameters. This is important
when defining the coupling losses.
In COMSOL Multiphysics the complex-valued data can be entered directly, or by
˜
means of loss factors. When loss factors are used, the complex data X is represented as
pairs of a real-valued parameter
˜
X = real  X 
and a loss factor
˜
˜
 X = imag  X   real  X 
the ratio of the imaginary and real part, and the complex data is then:
˜
X = X  1  j X 
where the sign depends on the material property used. The loss factors are specific to
the material property, and thus they are named according to the property they refer to,
for example, cE. For a structural material without coupling, simply use s, the
structural loss factor.
The Piezoelectric Devices interface defines the loss factors such that a positive loss
factor corresponds to a positive loss. The complex-valued data is then based on sign
rules. For piezoelectric materials, the following equations apply (m and n refer to
elements of each matrix):
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407
m n
m n
m n
c˜ E = c E  1 + j cE 
m n
m n
m n
e˜
=e
 1 – j e 
m n
̃ S
m n
= S
m n
 1 – j S 
m n
m n
m n
s˜ E = s E  1 – j sE 
(9-7)
m n
m n
m n
d˜
=d
 1 – j d 
m n
̃ T
m n
= T
m n
 1 – j T 
The losses for non-piezoelectric materials are easier to define. Again, using the
complex stiffness and permittivity, the following equations describe the material:
m n
˜ m n
m n
D
=  1 + j
D
̃ e
m n
=  1 – j e
m n
 e
m n
(9-8)
Often there is no access to fully defined complex-valued data. The Piezoelectric
Devices interface defines the loss factors as full matrices or as scalar isotropic loss factors
independently of the material and the other coefficients. For more information about
hysteretic losses, see Ref. 1 to Ref. 4.
ELECTRICAL CONDUCTIVITY (TIME HARMONIC)
For frequency domain analyses the electrical conductivity of the piezoelectric and
decoupled material (see Ref. 2, Ref. 5, and Ref. 6) can be defined. Depending on the
formulation of the electrical equation, the electrical conductivity appears in the
variational formulation (the weak equation) either as an effective electric displacement
Jp
˜
D =  r  0 E – j -----
(the actual displacement variables do not contain any conductivity effects) or in the
total current expression J = Jd  Jp where Jp = eE is the conductivity current and
Jd is the electric displacement current.
Both a dielectric loss factor (Equation 9-3 and Equation 9-4) and the electrical
conductivity can be defined at the same time. In this case, ensure that the loss factor
refers to the alternating current loss tangent, which dominates at high frequencies,
where the effect of ohmic conductivity vanishes (Ref. 7).
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
The use of electrical conductivity in a damped eigenfrequency analysis leads to a
nonlinear eigenvalue problem, which must be solved iteratively. To compute the
correct eigenfrequency, run the eigenvalue solver once for a single mode. Then set the
computed solution to be the linearization point for the eigenvalue solver, defined in
the settings window for the Eigenvalue Solver node. Re-run the eigenvalue solver
repeatedly until the solution no longer changes. This process must be repeated for each
mode separately.
In the COMSOL Multiphysics Reference Manual:
• Selecting a Stationary, Time-Dependent, or Eigenvalue Solver
• Eigenvalue Solver
References for Piezoelectric Damping
1. R. Holland and E.P. EerNisse, Design of Resonant Piezoelectric Devices, Research
Monograph No. 56, The M.I.T. Press, 1969.
2. T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, 1990.
3. A.V. Mezheritsky, “Elastic, Dielectric, and Piezoelectric Losses in Piezoceramics:
How it Works all Together,” IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control, vol. 51, no. 6, 2004.
4. K. Uchino and S. Hirose, “Loss Mechanisms in Piezoelectrics: How to Measure
Different Losses Separately,” IEEE Transactions on Ultrasonics, Ferroelectrics, and
Frequency Control, vol. 48, no. 1, pp. 307–321, 2001.
5. P.C.Y. Lee, N.H. Liu, and A. Ballato, “Thickness Vibrations of a Piezoelectric Plate
With Dissipation,” IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency
Control, vol. 51, no. 1, 2004.
6. P.C.Y. Lee and N.H. Liu, “Plane Harmonic Waves in an Infinite Piezoelectric Plate
With Dissipation,” Frequency Control Symposium and PDA Exhibition, pp. 162–
169, IEEE International, 2002.
7. C.A. Balanis, “Electrical Properties of Matter,” Advanced Engineering
Electromagnetics, John Wiley & Sons, chapter 2, 1989.
8. “Standards on Piezoelectric Crystals, 1949”, Proceedings of the I. R. E.,vol. 37,
no.12, pp. 1378 - 1395, 1949.
PIEZOELECTRIC DAMPING
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9. IEEE Standard on Piezoelectricity, ANSI/IEEE Standard 176-1987, 1987.
10. B. A. Auld, Acoustic Fields and Waves in Solids, Krieger Publishing Company,
1990.
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CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE
10
Glossary
This Glossary of Terms contains finite element modeling terms in an acoustics
context. For mathematical terms as well as geometry and CAD terms specific to the
COMSOL Multiphysics® software and documentation, see the glossary in the
COMSOL Multiphysics Reference Manual. For references to more information
about a term, see the index.
411
Glossary of Terms
acoustic impedance At a specified surface, the complex quotient of acoustic pressure
by normal fluid velocity. SI unit: Pa/(m/s) .
acoustic reactance The imaginary part of the acoustic impedance.
acoustic resistance The real part of the acoustic impedance.
acoustic-structure interaction A multiphysics phenomenon where the fluid’s pressure
causes a fluid load on the solid domain, and the structural acceleration affects the fluid
domain as a normal acceleration across the fluid-solid boundary.
adiabatic bulk modulus One over the compressibility s measured at constant entropy.
The adiabatic bulk modulus is denoted Ks and gives a measure of the compressibility
of the fluid and is directly related to the speed of sound cs in the fluid. SI unit: Pa.
2
1
K s = ----- =  0 c s
s
admittance The reciprocal of impedance.
aeroacoustics The scientific field of study used to couple acoustics and fluid dynamics.
In COMSOL Multiphysics the formulation based on the potential field for the particle
velocity and Bernoulli’s equation.
anisotropy Variation of material properties with direction.
arbitrary Lagrangian-Eulerian (ALE) method A technique to formulate equations in a
mixed kinematical description. An ALE referential coordinate system is typically a mix
between the material (Lagrangian) and spatial (Eulerian) coordinate systems.
Bernoulli equation An integrated form of Euler’s momentum equation along a line of
flow. The equation gives an expression for an invariant quantity in an inviscid fluid. A
decrease in the speed of the fluid translates to an increase in the fluid pressure and/or
potential energy.
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CHAPTER 10: GLOSSARY
bulk modulus One over the compressibility. It gives a measure of the compressibility
of the fluid and is related to the speed of sound in the fluid. SI unit: Pa. See also
adiabatic bulk modulus.
characteristic impedance The product of the equilibrium density and the speed of
sound in a medium. SI unit: Pa/(m/s).
compliance Reciprocal of stiffness.
compliance matrix The inverse of the elasticity matrix. See elasticity matrix.
Cauchy stress The most fundamental stress measure defined as force/deformed area
in fixed directions not following the body.
constitutive equations Equations that relate two physical quantities. In
thermoacoustics both the stress tensor (relating velocity to stress) and Fourier’s law of
heat conduction (relating heat conduction to temperature) are constitutive relations.
In structural mechanics this is the equation formulating the stress-strain relationship
of a material. Constitutive equations are supplemented by equilibrium equations
(mass, momentum, and energy) and an equation of state to provide a full physical
description.
creep Time-dependent material nonlinearity that usually occurs in metals at high
temperatures in which the effect of the variation of stress and strain with time is of
interest.
damping Dissipation of energy in the fluid or a vibrating structure. The damping is
typically due to viscous losses or thermal conduction. In acoustics this happens in
structures with small geometrical dimensions, for example, small pipes or porous
materials. In structures a common assumption is viscous damping where the damping
is proportional to the velocity. See also Rayleigh damping.
decibel (dB) Logarithmic unit that indicates the ratio of a physical quantity relative to
a reference value.
dipole source An acoustic source that behaves as a translational oscillating sphere.
Doppler effect Change in the observed frequency of a wave caused by a time rate of
change in the effective length of the path of travel between the source and the
observation point.
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|
413
effective sound pressure RMS instantaneous sound pressure at a point during a time
interval, T, long enough that the measured value is effectively independent of small
changes in T. SI unit: Pa = N/m2.
equation of state The thermodynamic relation between three independent
thermodynamic variables. Typically in acoustics it is the density = (p,s) given as
function of the entropy s and the pressure p.
eigenmode A possible propagating mode of an acoustic wave.
elasticity matrix The matrix D relating strain to stresses:
 = D
Eulerian Model described and solved in a coordinate system that is fixed (spatial). See
also Lagrangian and arbitrary Lagrangian-Eulerian method.
Green-Lagrange strain Nonlinear strain measure used in large-deformation analysis.
In a small strain, large rotation analysis, the Green-Lagrange strain corresponds to
the engineering strain, with the strain values interpreted in the original directions. The
Green-Lagrange strain is a natural choice when formulating a problem in the
undeformed state. The conjugate stress is the second Piola-Kirchhoff stress.
impedance At a specified frequency, the quotient of a dynamic field quantity (such as
force, sound, pressure) by a a kinematic field quantity (such as vibration velocity,
particle velocity).
instantaneous sound pressure Total instantaneous pressure at a point in a medium
minus the static pressure at the same point. SI unit: Pa = N/m2.
irrotational background velocity field A velocity field u that has the property of having
rotation   u = 0 everywhere, where the first term is the vorticity of the fluid. In
such a fluid the viscous stress does not contribute to the acceleration of the fluid. The
mean pressure in this fluid is described by Bernoulli’s equation.
Lagrangian Model described and solved in a coordinate system that moves with the
material. See also Eulerian and arbitrary Lagrangian-Eulerian method.
monopole source An acoustic source that behaves as a radially oscillating sphere.
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CHAPTER 10: GLOSSARY
particle velocity In a sound field, the velocity caused by a sound wave of a given
infinitesimal part of the medium relative to the medium as a whole.
PML (perfectly matched layer) Domain adjoined at a system boundary designed to
emulate a non-reflecting boundary condition independently of the shape and
frequency of the incident wave front.
principle of virtual work States that the variation in internal strain energy is equal to
the work done by external forces.
propagating acoustic modes The acoustic modes or wave shapes that propagate with
no significant damping for a given frequency in a duct of a given cross-section.
Rayleigh damping A viscous damping model where the damping is proportional to the
mass and stiffness through the mass and stiffness damping parameters.
reference sound pressure See definition in the entry for sound pressure level.
resonance frequency A frequency at which the system has the tendency to oscillate at
a greater amplitude than at non-resonance frequencies. At the resonance frequencies
the system can easily transfer energy from the actuation to the vibrating structure or
acoustic wave.
RMS value Root-mean-square value; for the (complex) sound pressure, p(t), over the
time interval T1 < t < T2 defined as
p RMS =
1
------------------T2 – T1
T Re  p  t  
T2
2
dt
1
it
For a harmonic pressure wave, p  t  = p 0 e , the time interval is taken to be a
complete period, resulting in pRMS  p0 /2.
second Piola-Kirchhoff stress Conjugate stress to Green-Lagrange strain used in
large deformation analysis.
sound energy Total energy in a given part of a medium minus the energy that would
exist at the same part in the absence of sound waves. SI unit: J.
sound-energy flux density See sound intensity.
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|
415
sound intensity Average rate of sound energy transmitted in a specified direction at a
2
point through a unit area normal to this direction. SI unit: W/m .
sound pressure See effective sound pressure.
sound pressure amplitude Absolute instantaneous sound pressure in any given cycle
2
of a sound wave at some specified time. SI unit: W/m .
sound power density See sound intensity.
sound pressure level Ten times the logarithm (to the base ten) of the ratio of the
time-mean-square pressure of a sound, in a stated frequency band, to the square of a
reference sound pressure, pref. For gases, pref = 20  Pa, for other media (unless
otherwise specified) pref = 1  Pa. Unit: dB (decibel).
sound source strength Maximum instantaneous rate of volume displacement
3
produced by a source when emitting a harmonic sound wave. SI unit: m /s.
specific acoustic impedance At a point in a sound field, the quotient of sound pressure
by particle velocity. SI unit: Pa/(m/s).
speed of sound The rate of change of particle displacement with distance for a sound
wave. SI unit: m/s.
spin tensor The skew-symmetric part of the velocity gradient tensor.
static pressure Pressure that would exist at a point in the absence of a sound wave.
stiffness Ratio of change of force (or torque) to the corresponding change in
translational (or rotational) displacement of an elastic element.
thermoacoustics The interaction between thermodynamic and acoustic phenomena,
which takes into account the temperature oscillations that accompany the acoustic
pressure oscillations. The combination of these oscillations produces thermoacoustic
effects. Thermoacoustic phenomena are modeled by solving the full linearized
Navier-Stokes equation (momentum equation), the continuity equation, and the
energy equation. Thermoacoustics is also known as thermo-viscous or visco-thermal
acoustics.
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CHAPTER 10: GLOSSARY
velocity potential When a flow is irrotational   u = 0 the vector field (velocity
field) can always be derived from a scalar potential (x) as u =  , where  is the
velocity potential. See also irrotational background velocity field.
waveguide structures Structures that have the property of guiding sound waves. See
also propagating acoustic modes.
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|
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CHAPTER 10: GLOSSARY
I n d e x
A
acoustic intensity vector 43
anisotropic materials
acoustic-piezoelectric interaction, fre-
defining 331
quency domain (acpz) interface 144
elastic properties 300
acoustic-piezoelectric interaction, tran-
loss factor damping and 303, 346
sient (acpztd) interface 149
antisymmetry (node)
acoustic-shell interaction, frequency do-
solid mechanics 312
main (acsh) interface 178
Arbitrary Lagrangian-Eulerian (ALE)
acoustic-shell interaction, transient
method 364
(acshtd) interface 186
artificial boundary conditions 29
acoustic-solid interaction, frequency do-
attenuation coefficient 291
main (acsl) interface 130
axial symmetry
acoustic-solid interaction, transient
initial stress and strain 347
(actd) interface 142
axial symmetry (node) 70, 221
acoustic-structure boundary (node) 134
acoustic-thermoacoustic boundary
(node) 260
added mass (node) 321
added mass, theory 350
adiabatic (node) 264
adiabatic bulk modulus 104
advanced settings 6
aeroacoustics model (node) 213
aeroacoustics with flow model (node)
237
aeroacoustics with flow, frequency domain (aepf) interface 234
aeroacoustics with flow, transient (atpf)
interface 239
aeroacoustics, frequency domain (ae) interface 210
theory 245
aeroacoustics, transient (aetd) interface
224
theory 246
ALE method 394
analysis. see study types.
azimuthal wave-number 357
B
background pressure field (node) 78
background pressure wave 52
barotropic fluids 243
Bernoulli equation 244
Biot equivalent fluid model 27
Biot equivalents fluid model 73, 123
Biot theory 170
Biot’s high frequency range 171
body load (node) 305
boundary load (node) 306
boundary loads theory 340
boundary mode acoustics (acbm) interface 99
theory 109
boundary mode aeroacoustics (aebm)
theory 247
boundary mode aeroacoustics (aebm) interface 226
boundary nodes
acbm interface 101
acpr interface 57
acpz and acpztd interfaces 146
INDEX|
i
acsh and acshtd interfaces 181
(node) 231
acsl and astd interfaces 132
consistent stabilization settings 6
actd interface 96
constraint settings 6
ae interface 212
continuity (node)
aebm interface 228
acpr interface 70
aepf and atpf interfaces 236
ae interface 220
aetd interface 225
elw interface 167
cpf interface 230
tas interface 270
pafd interface 190
converse piezoelectric effect 399
patd interface 190
coordinate system selection 7
piezoelectric devices 378
coordinate systems
solid mechanics 297
ta interface 254
Courant number 207
tas interface 268
crystal cut standards 400
tash interface 276
cyclic symmetry, settings 313
boundary selection 7
cyclic symmetry, theory 356
boundary-layer absorption 125
cylindrical wave radiation (node) 65
boundary-layer absorption fluid model
73
bulk modulus 201
elastic moduli 328
pressure acoustics 24
C
D
damped eigenfrequency study 373
damping
boundaries 28
equation of motion and 367
linear viscoelastic material and 371
canonical systems 324
loss factors 302
Cauchy stress 336
piezoelectric devices 384, 394
Cauchy stress tensor 363
solid mechanics 302
Cauchy-Green tensors 334
types 396
CFL condition 206
damping (node) 302
change thickness (node) 301
damping and loss (node) 384
circular source (node) 93
damping models 344, 346
circumferential wave number 107, 110
defining
closed (node) 194
anisotropic materials 331
complex conjugate operator 47
isotropic materials 327
complex impedance 27
orthotropic materials 330
complex modulus 373
thermoelastic materials 331
complex wave numbers 27
deformation gradient 334
compressible potential flow (cpf) inter-
degrees of freedom 25
face 229
compressible potential flow model
ii | I N D E X
solid mechanics theory 324
dielectric loss 397
dielectric loss (node) 385
dielectric loss factor 408
acsh and acshtd interfaces 181
dipole source (node) 59
acsl and astd interfaces 132
dipole sources 24, 103
actd interface 96
direct piezoelectric effect 388, 399
ae interface 212
discretization 6
aebm interface 228
dispersion curves 40
aepf and atpf interfaces 236
dispersion relation 291
aetd interface 225
displacement field, defining 362
pafd interface 190
displacement gradients 325
patd interface 190
dissipation, piezoelectric materials 391
piezoelectric devices 378
distributed loads, theory 339
solid mechanics 297
documentation 8
ta interface 254
domain nodes
tas interface 268
acbm interface 101
edge selection 7
acpz and acpztd interfaces 146
eigenfrequency study 38
acsh and acshtd interfaces 181
acsl and astd interfaces 132
pressure acoustics 108
solid mechanics 343
actd interface 96
elastic material properties 299–300
ae interface 212
elastic moduli 327
aepf and atpf interfaces 236
elastic waves (elw) interface 152
aetd interface 225
theory 169
cpf interface 230
elasticity matrix 327
piezoelectric devices 378
electrical conductivity (time-harmonic)
solid mechanics 297
E
tash interface 276
acpr interface 57
(node) 383
ta interface 254
electrical material model (node) 381
tas interface 268
elkernel element 116
tash interface 276
emailing COMSOL 10
domain selection 7
empirical porous model 27
double dot operator 47
empty study 41
Duhamel-Hooke’s law 326
end impedance (node) 196
Dulong-Petit law 332
equation of motion, damping and 367
dynamic cyclic symmetry 356
equation view 6
edge load (node) 307
edge nodes
acbm interface 101
acpr interface 57
acpz and acpztd interfaces 146
equivalent fluid model 72
equivalent viscous damping 372
Euler equations 243
Eulerian frame 363
evanescent modes 39
INDEX|
iii
evanescent wave components 31, 113
excitation frequency 372
Green-Lagrange strains 333
H
expanding sections 6
harmonic loads 342
explicit damping 373
harmonic time dependence 44
exterior shell (node)
heat dissipation 373
acsh interface 183
heat source (node) 259
tash interface 280
F
hide button 6
far-field calculation (node) 80
Higdon conditions 111
far-field limits 32, 115
high frequency range 171
far-field regions 31, 113
hysteretic loss 394, 406
first Piola-Kirchhoff stress 336
fixed constraint (node) 308
Floquet periodicity, settings 313
Floquet periodicity, theory 356
flow line source on axis (node) 135
fluid models, pressure acoustics 72
fluid properties (node) 192
force linear strains (check box) 301
Fraunhofer diffraction 31, 114
free (node) 311
frequency domain modal study 41
frequency domain study 37
ae interface 245
solid mechanics 342
frequency response study 372–373
Fresnel numbers 31, 114
geometric entity selection 7
geometric nonlinearity 334
I
I.R.E. standard, for material orientation
400
ideal barotropic fluids 243
ideal gas fluid model 73
IEEE standard, for material orientation
400
IEEE standard, piezoelectric materials
389
impedance (node)
acpr interface 62
ae interface 219
impedance, complex 27
incident pressure field (node) 66
incident velocity potential (node) 217
include geometric nonlinearity (check
box) 301
inconsistent stabilization settings 6
initial stress and strain 391
theory 347
micromechanics and 362, 364
initial stress and strain (node) 314
piezoelectric devices 392
initial values (node)
solid mechanics theory 351
iv | I N D E X
Helmholtz-Kirchhoff integral 32, 114
far field variables 33
far-field variables 32, 116
G
Hankel function 115
acbm interface 100
global coordinate systems 324
acpr interface 59
gradient displacements 325
acpz interface 148
Green strains 333
acsh interface 182–183
Green’s function 115
acsl interface 135
Green-Lagrange strain 362
ae interface 214
aepf interface 237
loss factor damping and 303
astd interface 143
atpf interface 240
isotropic meshes 25
K
cpf interface 232
knowledge base, COMSOL 10
Korteweg formula 201
elw interface 157
pafd interface 191
patd interface 191
piezoelectric devices 386
solid mechanics 305
ta interface 259
tas interface 270
tash interface 279
insertion loss curves 43
intensity edge source (node) 138
intensity line source on axis (node) 136
intensity point source (node) 139
intensity sources, pressure acoustics
L
Lagrange elements 26
Lagrangian formulations 325
Lagrangian frame 362
Lamé parameters 328
large acoustics problems 33
large deformations 333
piezoelectric materials 392
limp porous matrix model 124
line source (node) 85
line source on axis (node) 88
line sources
pressure acoustics, intensity, on axis
136
line, on axis 136
intensity variables 43–44
interior impedance (node) 83
interior normal acceleration (node) 82
interior perforated plate (node) 84
interior shell (node)
acsh interface 184
tash interface 280
interior sound hard boundary (wall)
(node)
acpr interface 69
ae interface 220
Internet resources 8
inverse piezoelectric effect 388
irrotational velocity fields 242
isentropic speed of sound 46
isothermal (node) 260
isotropic materials
defining 327
elastic properties 299
loss damping and 346
linear elastic attenuation fluid model 119
linear elastic fluid model 118
linear elastic material (node) 298
linear elastic materials 326
linear elastic with attenuation 73
linear viscoelastic material, damping and
371
liquids and gases materials 35
loads
solid mechanics theory 339
local coordinate systems 325
logarithmic decrement 368
loss factor damping
modeling 373
solid mechanics and 302
solid mechanics theory 346
springs and 365
loss modulus 373
loss tangents 395
low-reflecting boundary (node) 322
low-reflecting boundary, theory 355
INDEX|
v
M
macroscopic empirical porous model 73,
eigenfrequency study 108
121
far field calculation 82
mass damping parameter 371
far field plots 33
mass flow (node) 233
flow point source 91
mass flow circular source (node) 223
Gaussian pulse point source 98
mass flow edge source (node) 222
interior normal acceleration 83
mass flow line source on axis (node) 222
macroscopic empirical porous fluid
mass flow point source (node) 222
model 121
mass flows 216
mass flow point source 223
matched boundary (node) 79
mode analysis study 40, 110
matched boundary conditions 29
pipe acoustics, transient 207
material coordinates 323
plane wave radiation 64
material frame 362
poroelastic waves 155
materials
power point source 91, 141
linear elastic 326
pressure acoustics, frequency domain
nearly incompressible 354
59
piezoelectric 389
radiation boundary condition 30
porous absorbing 27
solid mechanics 296
materials, piezoelectric devices 391
spherical wave radiation 65
mixed formulations 354
ta interface 253
modal reduced order model study 41
moment computations 358
mode analysis study 39
monopole point source (node) 89
boundary mode acoustics 110
monopole source (node) 58
theory 290
monopole sources 24, 103
Model Library 9
moving mesh interface, piezoelectric de-
Model Library examples
MPH-files 9
acpz interface 146
multigrid solvers 33
acsh interface 180
acsl interface 132
actd interface 96
ae interface 212
aebm interface 227
background pressure field 79
Biot equivalent fluid model 125
cpf interface 230
cylindrical wave radiation 66
damping 28
vi | I N D E X
vices and 394
acpr interface 56, 59
N
narrow ducts 127
near-field regions 31, 113
nearly incompressible materials 347, 354
no stress (node) 263
no-flow conditions 214, 220
nominal stress 352
nonlinear geometry 334, 351
nonreflecting boundary conditions 29
normal acceleration (node) 60
normal flow (node) 232
normal impedance (node) 264
particle velocity 44
normal mass flow (node) 216
perfectly matched layers (node) 30
normal stress 336
perforated plate (node) 84
normal stress (node) 263
periodic boundary conditions 312
normal velocity (node) 218
periodic condition (node) 67
NRBC. see nonreflecting boundary condition. 79
Nyquist criterion 34
O
orientation, piezoelectric material 400
orthotropic materials
defining 330
elastic properties 300
loss damping and 346
loss factor damping and 304
out-of-plane wave number 106, 245
override and contribution 6
piezoelectric devices 386
solid mechanics 312
periodic conditions, theory 356
phase (node) 314
phase factors, pressure acoustics 107
phase variables 38
piezoelectric coupling 376
piezoelectric crystal cut 400
piezoelectric devices interface 376
theory 388
piezoelectric losses 406
piezoelectric material (node) 380
P
pair impedance (node)
acpr interface 83
ae interface 219
pair nodes
acpr interface 57
acpz and acpztd interfaces 146
acsh and acshtd interfaces 181
acsl and astd interfaces 132
ae interface 212
aebm interface 228
aepf and atpf interfaces 236
aetd interface 225
cpf interface 230
pafd interface 190
patd interface 190
piezoelectric devices 378
solid mechanics 297
ta interface 254
tas interface 268
tash interface 276
pair perforated plate (node) 85
pipe acoustics, frequency domain (pafd)
interface 188
pipe acoustics, transient (patd) interface
189
theory 199
Pipe Flow Module 17
pipe properties (node) 193
plane stress and strain 294
plane wave radiation (node)
acpr interface 63
ae interface 216
point load (node) 308
point nodes
acbm interface 101
acpr interface 57
acpz and acpztd interfaces 146
acsh and acshtd interfaces 181
acsl and astd interfaces 132
actd interface 96
ae interface 212
aebm interface 228
pair selection 7
INDEX|
vii
aepf and atpf interfaces 236
patd interface 194
aetd interface 225
pipe acoustics 194
pafd interface 190
pressure acoustics interfaces 102
patd interface 190
pressure acoustics model (node) 71
piezoelectric devices 378
pressure acoustics model, fluid models
solid mechanics 297
119
ta interface 254
pressure acoustics models, theory 118
tas interface 268
pressure acoustics theory 102
tash interface 276
pressure acoustics, frequency domain
point selection 7
(acpr) interface 52
point source (node) 91
pressure acoustics, transient (actd) in-
Poisson’s ratio 328
terface 95
poroelastic material (node) 157
pressure loads 341
poroelastic septum boundary load
pressure, adiabatic (node) 261
pressure-wave speeds 328
(node) 166
poroelastic waves (elw) interface 154
principle of virtual work 341
theory 169
propagating acoustic modes 290
porous absorbing materials 27
porous, fixed constraint (node) 161
propagating modes 39
Q
quality factors and losses 395
R
radiation boundary conditions 29
porous, free (node) 161
porous, prescribed acceleration (node)
rate of strain tensor 335
165
Rayleigh damping 344, 369, 371
porous, prescribed displacement (node)
reference coordinates 323
162
reference point for moment computa-
porous, prescribed velocity (node) 164
tion 358
porous, pressure (node) 162
refpnt variable 295
porous, roller (node) 166
remanent electric displacement (node)
power dissipation variables 45
385
power edge source (node) 138
resonance frequency 38
power line source on axis (node) 137
resonant frequency 369
power point source (node) 140
results analysis and variables 43
ppr() operator 81
rigid porous matrix model 124
Prandtl number 251
RMS 43
pre-deformation (node) 319
roller (node) 312
prescribed acceleration (node) 316
rotated coordinate system 404
prescribed displacement (node) 309
prescribed velocity (node) 315
pressure (node) 62
viii | I N D E X
S
second Piola-Kirchhoff stress 336, 363
settings windows 6
shear modulus expression 328
axial symmetry 333
shear strain 333
engineering form 333
shear stress 336
shear 333
shear-wave speeds 328
tensor form 333
show button 6
strain tensor 333
skew-symmetric part 335
strain-displacement relation 333
slip (node) 262
large displacement 333
slip velocity (node) 232
small displacement 333
solid mechanics
stress 336
damping 302
Cachy 336
prescribed acceleration 316
first Piola-Kirchhoff 336
prescribed displacement 309
normal 336
prescribed velocity 315
second Piola-Kirchhoff 336
solid mechanics interface 294
shear 336
theory 323
stress (node) 263
solving large problems 33
stress and strain, piezoelectric devices
sound hard boundary (wall) (node)
389
acpr interface 60
stress-strain relation 336
ae interface 214
structural damping 373
sound hard wall (node) 259
study types
sound soft boundary (node)
acbm interface 110
acpr interface 61
actd interface 109
ae interface 218
ae interface 245
source term 45
aetd interface 246
spatial coordinates 323
eigenfrequency 343
spatial frame 363
frequency domain, solid mechanics 342
spatial stress tensor 363
stationary, solid mechanics 342
spherical wave radiation (node) 64
surface traction and reaction forces 360
spin tensor 335
swirl-correction factor 199
spring constant 365
symmetry (node)
spring foundation (node) 317
solid mechanics 311
spring foundation, solid mechanics 365
ta interface 261
spring foundation, theory 348
tas interface 271
stabilization settings 6
stationary study 36, 245
stiffness damping parameter 371
storage modulus 373
strain 333, 336
symmetry planes, far-field analysis 81
T
Taylor series 200
technical support, COMSOL 10
temperature variation (node) 265
theory 102
INDEX|
ix
acbm interface 109
transient pressure acoustics model
ae interface 245
(node) 96
aebm interface 247
transmission loss 43
aetd interface 246
true stress tensor 363
elw interface 169
U
patd interface 199
uncoupled shell (node)
acsh interface 185
piezoelectric devices 388
tash interface 281
pressure acoustics models 118
U-P formulation 171
solid mechanics 323
user community, COMSOL 10
ta interface 282
using
tas interface 282
coordinate systems 324
thermal dissipation 45
predefined variables 358
thermally conducting and viscous fluid
spatial and material coordinates 323
model 123
weak constraints 359
thermally conducting and viscous fluid
uspring variable 365
model, defining 75
thermally conducting fluid model 73, 122
thermoacoustics model (node) 255
thermoacoustics, frequency domain (ta)
interface 250
theory 282
thermoacoustic-shell interaction, frequency domain (tash) interface 273
thermoacoustic-solid interaction, frequency domain (tas) interface 266
theory 282
thermoelastic materials, defining 331
thin elastic layer (node) 319
thin elastic layer, solid mechanics 365
thin elastic layer, theory 349
time dependent study 40
pressure acoustics, transient 109
transient aeroacoustics 246
time-dependent modal study 41
time-harmonic study
aeroacoustics 245
total force loads 340
tractions 363
x | INDEX
V
variables
for far fields 33
intensity 43
material and spatial coordinates 324
phase 38
power dissipation 45
predefined 358
refpnt 295
results analysis 43
vdamper variable 366
velocity (node) 195, 262
pipe acoustics 195
velocity potential (node) 215
velocity potential, compressible potential flow, and 244
viscous damping 366, 373
viscous dissipation 45
viscous fluid model 73, 121
Voigt form 400
Voigt notation 304, 329, 389
volume ratio 334
vortex sheet (node) 219
W wall (node) 261
wave numbers 27, 106, 245
wave radiation equations 111
wave speeds 300
waveguide 39
waveguide structures 290
weak constraint settings 6
weak constraints, using 359
web sites, COMSOL 10
wide ducts 126
Y
Young’s modulus expression 328
INDEX|
xi
xii | I N D E X