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Acoustics Module User´s Guide VERSION 4.3b Acoustics Module User’s Guide © 1998–2013 COMSOL Protected by U.S. Patents 7,519,518; 7,596,474; and 7,623,991. Patents pending. This Documentation and the Programs described herein are furnished under the COMSOL Software License Agreement (www.comsol.com/sla) and may be used or copied only under the terms of the license agreement. COMSOL, COMSOL Multiphysics, Capture the Concept, COMSOL Desktop, and LiveLink are either registered trademarks or trademarks of COMSOL AB. All other trademarks are the property of their respective owners, and COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by those trademark owners. For a list of such trademark owners, see www.comsol.com/tm. Version: May 2013 COMSOL 4.3b Contact Information Visit the Contact Us page at www.comsol.com/contact to submit general inquiries, contact Technical Support, or search for an address and phone number. You can also visit the Worldwide Sales Offices page at www.comsol.com/contact/offices for address and contact information. If you need to contact Support, an online request form is located at the COMSOL Access page at www.comsol.com/support/case. Other useful links include: • Support Center: www.comsol.com/support • Download COMSOL: www.comsol.com/support/download • Product Updates: www.comsol.com/support/updates • COMSOL Community: www.comsol.com/community • Events: www.comsol.com/events • COMSOL Video Center: www.comsol.com/video • Support Knowledge Base: www.comsol.com/support/knowledgebase 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 [email protected] WHERE DO I ACCESS THE DOCUMENTATION AND MODEL LIBRARY? | 9 CONT ACT ING COMSOL BY EMAIL For general product information, contact COMSOL at [email protected] To receive technical support from COMSOL for the COMSOL products, please contact your local COMSOL representative or send your questions to [email protected] 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. ACOUSTICS MODULE CAPABILITIES | 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 20 | 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 | 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. 22 | 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 | 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 it where 2f (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 24 | 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 cf. 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 FUNDAMENTALS OF ACOUSTICS MODELING | 25 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. 26 | 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 cck and a complex density as ckZ. 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 | 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. 28 | 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 | 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. 30 | 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, Fa2/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. FUNDAMENTALS OF ACOUSTICS MODELING | 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). 32 | CHAPTER 3: MODELING WITH THE ACOUSTICS MODULE 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 | 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. 34 | 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 36 | 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 it 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 2f 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 T H E A C O U S T I C S M O D U L E S T U D Y TY P E S | 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. 38 | 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 = --------2i 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 T H E A C O U S T I C S M O D U L E S T U D Y TY P E S | 39 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. 40 | 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 T H E A C O U S T I C S M O D U L E S T U D Y TY P E S | 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 42 | 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 S P E C I A L VA R I A B L E S I N T HE A C O U S T I C S M O DU L E | 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 S P E C I A L VA R I A B L E S I N T HE A C O U S T I C S M O DU L E | 45 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 46 | 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 it Re Be it 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 S P E C I A L VA R I A B L E S I N T HE A C O U S T I C S M O DU L E | 47 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 = iv 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. S P E C I A L VA R I A B L E S I N T HE A C O U S T I C S M O DU L E | 49 50 | 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. 52 | 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. 54 | 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 (20·106 Pa). • Use reference pressure for water to use a reference pressure of 1 Pa (1·106 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 THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 55 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 56 | 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 57 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. 58 | 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 59 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 60 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 61 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: ip 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 62 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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 Zi0, 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 THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 63 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 64 | 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 65 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 re – 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. 66 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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 THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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. 68 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 69 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. 70 | 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 71 • 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 xand c = cx 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. 72 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 73 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 coefficientin 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) 74 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH • 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)) THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 75 • 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)) 76 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH • 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 77 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. 78 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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. THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 79 Properly set up, the matched boundary condition allows one mode with wave number k1 (set k2k1), 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 12f1, 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. 80 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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 THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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 82 | 1 = 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 THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 83 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 Z0, 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 8k eq Z---------= --- --------------- 1 + ------ + f + i -------- t p + h d c cc c h c c 84 | 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 1.8·105 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 (103 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: THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 85 2 k eq p t 1 1 3 – ----- p t – q d – ------------- = ----- 4S 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 rr0 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 rr0 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 86 | 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 rr0 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 rr0 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: THE PRESSURE ACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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 – ------------- = ----- 4S 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 zz0 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. 88 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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 – ------------- = ----- 4S 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 rr0. 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 rr0 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 | 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 90 | 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 – ------------- = ----- 4S 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) = rr0. 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 | 91 Flow Select Flow to add an monopole point source located at rr0 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 92 | 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 – ------------- = ----- 4S r – r 0 rd c c c 3 where r – r 0 is the delta function that adds the source at the point where rr0 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. 94 | 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 T H E P R E S S U R E A C O U S T I C S , TR A N S I E N T U S E R I N T E R F A C E | 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 96 | 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. T H E P R E S S U R E A C O U S T I C S , TR A N S I E N T U S E R I N T E R F A C E | 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 98 | 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. 100 | 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 | 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 102 | 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 = 2f (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 104 | 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 | 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)eit (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 . 106 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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: THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH | 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 i2fi. 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 22. 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. 108 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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 px, 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. THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH | 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 r1rr2. 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 110 | CHAPTER 4: THE PRESSURE ACOUSTICS BRANCH 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 )1r 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 THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH | 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 112 | 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 THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH | 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, Fa2/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: pR = G R r p r – G R r p r n dS S 114 | 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 THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH | 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 rR 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: rR 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 m0, 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 116 | 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. THEORY BACKGROUND FOR THE PRESSURE ACOUSTICS BRANCH | 117 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. 118 | 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 | 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 = 120 | 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: THEORY FOR THE PRESSURE ACOUSTICS FLUID MODELS | 121 ib –1 c = 1 + ---------2- c 1 --- ib 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: ib –1 c = 1 + ---------2- c 1 --- ib 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 122 | 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: ib – 1 c = 1 + ---------2- c 1 --- ib 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, THEORY FOR THE PRESSURE ACOUSTICS FLUID MODELS | 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 iL th Pr f 8 --------------------------= --------- – – 1 1 + --------------------------+ 1 2 p 16 iL 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 124 | 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 | 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 126 | 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 THEORY FOR THE PRESSURE ACOUSTICS FLUID MODELS | 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. 128 | 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. 130 | 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. 132 | 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 | 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) Fpn 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: ann · utt. 134 | 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: THE ACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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 136 | 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. THE ACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 137 Intensity Edge Source Use an Intensity Edge Source node to add an edge source located at rr0 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 rr0 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. 138 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 rr0. 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 4d 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). THE ACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 139 Power Point Source For 3D and 2D models, use a Power Point Source node to add a point source located at rr0. 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: 2P 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). 140 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 THE ACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 141 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. 142 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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. T H E A C O U S T I C - S O L I D I N T E R A C T I O N , TR A N S I E N T U S E R I N T E R F A C E | 143 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. 144 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 THE ACOUSTIC-PIEZOELECTRIC INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 145 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. 146 | 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. 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 THE ACOUSTIC-PIEZOELECTRIC INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 147 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). 148 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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. T H E A C O U S T I C - P I E Z O E L E C T R I C I N T E R A C T I O N , TR A N S I E N T U S E R I N T E R F A C E | 149 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 WtotQh,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, 150 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 T H E A C O U S T I C - P I E Z O E L E C T R I C I N T E R A C T I O N , TR A N S I E N T U S E R I N T E R F A C E | 151 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. 152 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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). T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 153 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 154 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 155 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 156 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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. T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 157 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 Ewhere 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. 158 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 . T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 159 • 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 160 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION ̃ f = f F ---f- fc f f c = -----------------2 2a 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 T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 161 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. 162 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 163 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). 164 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 165 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 166 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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) Fpn 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: ann · -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 T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 167 boundary equal to the acceleration based on the second derivatives of the structural displacements u with respect to time: ann · -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. 168 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 it with f (SI unit: Hz) denoting the frequency and 2f (SI unit: rad/s) the angular frequency. Assuming the same time-harmonic dependency for the source terms s0 and T H E O R Y FO R T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 169 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 + pf. Assuming a time-harmonic dependency for the variables, u(x,t) = u(x)eit, w(x,t) = w(x)eit, 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 = 0 170 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION (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 T H E O R Y FO R T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 171 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 drc: 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 172 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 KsatKdr 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) T H E O R Y FO R T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 173 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 pf0. 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 Bp, the load is equivalent to 174 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION n dr u = n p – 1 p 0 and for a soft porous matrix B1, 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 T H E O R Y FO R T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 175 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 nu = 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 FSFAsep2u 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. 176 | 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. T H E O R Y FO R T H E E L AS T I C WAVE S A N D PO RO EL A S T I C WAVES U S E R I N T E R F A C E S | 177 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. 178 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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. THE ACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 179 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 180 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 THE ACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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 182 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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: Fpn p. THE ACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 183 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. 184 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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. THE ACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 185 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. 186 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 T H E A C O U S T I C - S H E L L I N T E R A C T I O N , TR A N S I E N T U S E R I N T E R F A C E | 187 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. 188 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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. THE PIPE ACOUSTICS USER INTERFACES | 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. 190 | 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). THE PIPE ACOUSTICS USER INTERFACES | 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. 192 | 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. THE PIPE ACOUSTICS USER INTERFACES | 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. 194 | 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. THE PIPE ACOUSTICS USER INTERFACES | 195 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. 196 | 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: Pas/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 . THE PIPE ACOUSTICS USER INTERFACES | 197 • If User defined is selected, enter an End impedance Zend (SI unit: Pas/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. 198 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION 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 + Au = 0 t (5-12) and the corresponding momentum balance equation is ------------------ Au - 2 + Au = – Ap + 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 THEORY FOR THE PIPE ACOUSTICS USER INTERFACES | 199 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 200 | 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 = ------ ----20 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. THEORY FOR THE PIPE ACOUSTICS USER INTERFACES | 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 + 2Au 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 + 2Au 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 it p = p˜ x e it 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 202 | CHAPTER 5: ACOUSTIC-STRUCTURE INTERACTION u0 A i ----2- p + t A u + --------2- p e t = 0 c c 2 u0 u0 iAu + t A -----2- p + 2Au 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 = 2f 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) THEORY FOR THE PIPE ACOUSTICS USER INTERFACES | 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. 204 | 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 THEORY FOR THE PIPE ACOUSTICS USER INTERFACES | 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 12 –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 206 | 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 = --------------2a 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 | 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. 208 | 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. 210 | 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 | 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. 212 | 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 | 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. 214 | 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. THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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 nt 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 216 | 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. 218 | 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, Zip(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): 220 | 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. THE AEROACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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. 222 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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 rr0. – 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 | 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. 224 | CHAPTER 6: THE AEROACOUSTICS BRANCH • 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. T H E A E R O A C O U S T I C S , TR A N S I E N T U S E R I N T E R F A C E | 225 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. 226 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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 THE BOUNDARY MODE AEROACOUSTICS USER INTERFACE | 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. 228 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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. THE COMPRESSIBLE POTENTIAL FLOW USER INTERFACE | 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 230 | CHAPTER 6: THE AEROACOUSTICS BRANCH • 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. THE COMPRESSIBLE POTENTIAL FLOW USER INTERFACE | 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 232 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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. THE COMPRESSIBLE POTENTIAL FLOW USER INTERFACE | 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. 234 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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. T H E A E RO AC O U S T I C S W I T H F L OW, F RE Q U E N C Y D O M A I N U SE R I N T E R F A C E | 235 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 236 | CHAPTER 6: THE AEROACOUSTICS BRANCH • 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. T H E A E RO AC O U S T I C S W I T H F L OW, F RE Q U E N C Y D O M A I N U SE R I N T E R F A C E | 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). 238 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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. T H E A E R O A C O U S T I C S W I T H F L O W , TR A N S I E N T U S E R I N T E R F A C E | 239 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 phit (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. 240 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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 | 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 ar, tpr, tcmf2 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 242 | 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 | 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 pp. 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 244 | CHAPTER 6: THE AEROACOUSTICS BRANCH 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 it 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 | 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 246 | 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 | 247 248 | 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 250 | 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 THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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. 252 | 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 | 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. 254 | 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 | 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). 256 | 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. THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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. 258 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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. THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 259 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. 260 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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. THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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 262 | CHAPTER 7: THE THERMOACOUSTICS BRANCH nu = 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: THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 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: Pas/m). The default is 0 Pas/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. 264 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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 it 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. THE THERMOACOUSTICS, FREQUENCY DOMAIN USER INTERFACE | 265 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). 266 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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. THE THERMOACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 267 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 268 | 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 THE THERMOACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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 270 | 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 = iu 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 THE THERMOACOUSTIC-SOLID INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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. 272 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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. THE THERMOACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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). 274 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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. THE THERMOACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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. 276 | 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 THE THERMOACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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. 278 | 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). THE THERMOACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 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 = iu 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. 280 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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 = iu 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: Fpn 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. THE THERMOACOUSTIC-SHELL INTERACTION, FREQUENCY DOMAIN USER INTERFACE | 281 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 282 | 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 1500 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 | 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, TIn 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 0 284 | 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 it p = p 0 + p'e it T = T 0 + T'e it = 0 + 'e it F = F 0 + F'e it Q = Q 0 + Q'e it 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 + ipT 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 | 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 + ipT 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. 286 | 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 + ip + 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 THEORY BACKGROUND FOR THE THERMOACOUSTICS BRANCH | 287 c = p --------00 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, KTthe isothermal bulk modulus, and 0the 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 288 | 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 it = + 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 | 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 290 | CHAPTER 7: THE THERMOACOUSTICS BRANCH k z = + i p = p x e – ik z z it 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. THEORY BACKGROUND FOR THE THERMOACOUSTICS BRANCH | 291 292 | CHAPTER 7: THE THERMOACOUSTICS BRANCH 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. 294 | 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 THE SOLID MECHANICS USER INTERFACE | 295 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 296 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 297 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 298 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 299 • 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. 300 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 301 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. 302 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH The use of loss factor damping traditionally refers to a scalar-valued loss factor s. But there is no reasonthat 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). THE SOLID MECHANICS USER INTERFACE | 303 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. 304 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 ut (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. THE SOLID MECHANICS USER INTERFACE | 305 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. 306 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 307 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. 308 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 309 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. 310 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 311 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 uix0uix1 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. 312 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 THE SOLID MECHANICS USER INTERFACE | 313 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: 314 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH F freq = F f cos 2f + 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. THE SOLID MECHANICS USER INTERFACE | 315 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 316 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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/(mm3) 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/(mm)2 domains, boundaries 3D, 2D Spring constant per unit length kL N/(mm) 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 THE SOLID MECHANICS USER INTERFACE | 317 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 Ns/(mm3) domains, boundaries (2D) 3D, 2D Viscous force per unit area dA Ns/(mm2) domains, boundaries 3D, 2D, and 2D axisymmetric Total viscous force dtot Ns/m domains, boundaries, edges, points 3D, 2D, and 2D axisymmetric Viscous force per unit length dL Ns/(mm) 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 318 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THE SOLID MECHANICS USER INTERFACE | 319 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/(mm)2 3D, 2D, and 2D axisymmetric Spring constant per unit length kL N/(mm) 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 Ns/(mm ) 3D, 2D, and 2D axisymmetric Total viscous force dtot Ns/m 3D, 2D, and 2D axisymmetric Viscous force per unit length dL Ns/(mm) 2D • Springs and Dampers • About Spring Foundations and Thin Elastic Layers 320 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 THE SOLID MECHANICS USER INTERFACE | 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: Pas/m). The defaults for all values are 0.5*solid.rho*(solid.cp+solid.cs). About the Low-Reflecting Boundary Condition 322 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. THEORY FOR THE SOLID MECHANICS USER INTERFACE | 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 324 | 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 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, TTref, 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 326 | 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 327 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. DE DKG 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 -------------------21 + Lamé parameter E ------------------------------------- 1 + 1 – 2 2G K – -------3 Lamé parameter E -------------------21 + 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: 328 | 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: THEORY FOR THE SOLID MECHANICS USER INTERFACE | 329 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 330 | 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 D1 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 331 F = f 0 T + W s T where WsT 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 Tmn 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 Tmn are small. The resulting approximation is often called uncoupled thermoelasticity. 332 | 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 333 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 334 | 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. THEORY FOR THE SOLID MECHANICS USER INTERFACE | 335 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 si30 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 336 | 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 i1, 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 337 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 338 | ij ij = s0 + C CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH ijkl kl – kl – 0kl where TTref. 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. THEORY FOR THE SOLID MECHANICS USER INTERFACE | 339 • 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 340 | 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 W0 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 341 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 342 | 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 jt e jt j = Re u˜ e where u˜ = u amp e u ˜ e jt u = Re u jF xPh f --------- 180 jt jt 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 343 uses a new variable j explicitly expressed in the eigenvalue jThe 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 344 | 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 2f 2 2f 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 4f 1 1 dM 1 dK ----------f 4f 2 2 = 1 2 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 345 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 346 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 0z s 0r s 0 s 0z 0rz 0z 0z s 0rz s 0z s 0z THEORY FOR THE SOLID MECHANICS USER INTERFACE | 347 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 = – iK 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 348 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 = – iK 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 349 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 = – iK 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 350 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 TTref, 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 351 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 352 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 353 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 354 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 THEORY FOR THE SOLID MECHANICS USER INTERFACE | 355 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 356 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH symmetry mode number m can vary from 0 to N, with N being the total number of sectors so that 2N. 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. THEORY FOR THE SOLID MECHANICS USER INTERFACE | 357 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 358 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. CALCULATING REACTION FORCES | 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 360 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 CALCULATING REACTION FORCES | 361 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 362 | 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 | 363 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. 364 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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. SPRINGS AND DAMPERS | 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. 366 | CHAPTER 8: THE STRUCTURAL MECHANICS BRANCH 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 ft d u--------+ 2 0 ------- + 0 u = --------dt m dt (8-12) where c2m0 and 02km. 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 DAMPING AND LOSS | 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 jt and u t = Re Ue jt 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 + jcU + kU = F (8-13) where the time dependence has canceled out on both sides. Alternatively this equation can be written as: 368 | 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 ft 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 dM0 and dK2/0 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 DAMPING AND LOSS | 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 k1j where is the loss factor, modifies the form of Equation 8-13 to: 2 – mU + jkU + 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 370 | 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 2f 2 2f (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 4f 1 1 dM 1 dK ----------f 4f 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. DAMPING AND LOSS | 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 = 2f 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 = --------- = ---2f 372 | 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 sG''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. DAMPING AND LOSS | 373 374 | 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 376 | 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. THE PIEZOELECTRIC DEVICES USER INTERFACE | 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. 378 | 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 THE PIEZOELECTRIC DEVICES USER INTERFACE | 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. 380 | 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. THE PIEZOELECTRIC DEVICES USER INTERFACE | 381 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. 382 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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” Jij 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. THE PIEZOELECTRIC DEVICES USER INTERFACE | 383 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. 384 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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 THE PIEZOELECTRIC DEVICES USER INTERFACE | 385 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 uix0uix1 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. 386 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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 THE PIEZOELECTRIC DEVICES USER INTERFACE | 387 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. 388 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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): THEORY FOR THE PIEZOELECTRIC DEVICES USER INTERFACE | 389 = 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 390 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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: THEORY FOR THE PIEZOELECTRIC DEVICES USER INTERFACE | 391 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: 392 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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 xX + 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: e0. In that case, the structural part corresponds to the large deformation formulation described for the solid mechanics interfaces. THEORY FOR THE PIEZOELECTRIC DEVICES USER INTERFACE | 393 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 394 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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 THEORY FOR THE PIEZOELECTRIC DEVICES USER INTERFACE | 395 material. The quality factor Qm and the loss factor i are inversely related: i1 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 396 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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. 398 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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: PIEZOELECTRIC DAMPING | 399 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 400 | 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). PIEZOELECTRIC DAMPING | 401 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. 402 | CHAPTER 9: THE PIEZOELECTRIC DEVICES USER INTERFACE 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 | 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. 404 | 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. PIEZOELECTRIC DAMPING | 405 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 406 | 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): PIEZOELECTRIC DAMPING | 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). 408 | 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 | 409 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. 410 | 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. 412 | 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. G L O S S A R Y O F TE R M S | 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. 414 | 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 it 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. G L O S S A R Y O F TE R M S | 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. 416 | 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. G L O S S A R Y O F TE R M S | 417 418 | 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