Introduction to COMSOL Multiphysics
San Diego, CA
September 20, 2005
Mina Sierou, Ph.D.
Comsol Inc.
Contents
Morning Session
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Introduction
Demonstration of the modeling
procedure
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Workshop: Electro-thermal
analysis of semiconductor device
– 3D stationary
– 3D parametric
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Workshop: Mesh Control
Afternoon Session
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Hands-on modeling workshops:
– Flow over a Backstep
– Electric Impedance Center
– Thermal Stresses in a Layered
Plate
– MEMS Thermal Bilayer Valve
– Using Interpolation Function
– Using Mapped Meshes
COMSOL – The company
• Founded 1986
• Development of FEMLAB®
/COMSOL Multiphysics (in
Sweden)
• Business: software, support,
courses, consulting
• Today, 120 employees worldwide.
Offices in US (Boston, L.A.), UK,
Germany, France, Sweden, Finland,
Norway, and Denmark
• Distributor network covering the rest
of the world
Highlights of COMSOL Multiphysics
• General purpose Multiphysics FEA code
• MATLAB/COMSOL Script integration
– COMSOL Multiphysics can be run stand-alone
– or with MATLAB for richer set of functions
– Can use MATLAB or COMSOL Script as a scripting language
• Easy to learn and use
• Extremely adaptable and extensible
The COMSOL Multiphysics Product Line
And introducing…
COMSOL Script
CAD import module + Mesh import
• In 3.2 we can import IGES,
STEP, SAT, X_T, Pro/E, CATIA,
Inventor, VDA files with:
– More than 1000 faces
– Sliver faces, spikes, short edges
and other errors
COMSOL Multiphysics Users
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Rice, Texas A & M, UH
UT Austin, UT Arlington
Stanford, Caltech, JPL
UC’s, UW
MIT, Harvard, Princeton…
Y NL
– Y=LA,LL,LB,PN, Sandia
• NASA research centers
• NIST, NREL, USGS, SWRI
• NIH
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Shell, Exxon Mobil
Schlumberger, Dow Chemicals
Northrop-Grumman, Raytheon
Applied Materials, Agilent
Boeing, Lockheed-Martin
GE, 3M
Merck, Roche
Procter and Gamble, Gillette
Energizer, Eveready
Hewlett-Packard, Microsoft, Intel
Nissan, Sony, Toshiba
ABB, Volkswagen, GlaxoSmithKline
Philip-Morris
Mathematical Modeling
• Mathematical description of physical phenomena translates
into equations
• Description of changes in space and time results in Partial
Differential Equations (PDE’s)
• Complex geometries and phenomena require modeling with
complex equations and boundary conditions
• Resulting PDEs rarely have analytical solutions
Numerical tools are necessary
Material Balances
Material balances are usually described by an equation of the form
u
   j  F
t
where j is the flux vector and F a source term
jx j y jz
u



F
t
x y z
jx  x  jx j y  y  j y jz  z  jz
u



F
t
x
y
z
xyz
u
 yz ( j x  x  j x )  xz ( j y  y  j y ) 
t
 xy ( j z  z  j z )  xyzF
General PDE Form
u
da
   F
t
 R  
n  G    
 u  

0R

inside domain
T
on domain boundary
Example: For Poisson’s equation, the corresponding general form implies
   ux  uy
R  u.
All other coefficients are 0. (For later, note:
F 1
  n   )
Coefficient Form PDE
If equation is linear, the general form can be expanded into a coefficient form:
u
da
t
 
 F
transforms into
u
da
  (cu   u  )    u  au  f
t
n  (cu   u   )  qu  g  hT  

hu  r

Example: Poisson’s equation
  u  1
u 0
inside subdomain
on boundary
inside subdomain
on subdomain boundary
(Implies c=f=h=1 and all other coefficients are 0.)
Multiphysics Capabilities
• Very different physical phenomena can be described with the
same general equations
• Coupling of different physical formulations (multiphysics) is
thus straightforward in COMSOL Multiphysics
• Resulting systems of equations can be solved sequentially or in
a fully-coupled formulation
• Extended Multiphysics: Physics in different geometries can be
easily combined
• Coupling variables can also be used to link different physics or
geometries
Support & Knowledgebase
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support@comsol.com
www.comsol.com/support/knowledgebase
Worked Example – A Simple Fin
Purpose of the model
• Explain the modeling procedure in COMSOL Multiphysics
• Show the use of pre-defined application modes in physics
mode
• Introduce some very useful features for control of modeling
results
Problem definition
• Heat transfer by conduction (Heat Conduction application
mode)
• Linear equation, stationary solution
• Different thermal conductivities can be defined in different
subdomains
Problem Definition
T1
Step 1
k1
symmetry
T2
 kT  n  0
Step 2
k2  10  k1
   k T   0
k1
k2
Modeling, Simulation and Analysis
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Draw geometry
Define Boundary conditions
PDE specification/material parameters
Generate mesh
Solve (initial conditions, solver parameters)
Visualize solution, animation
Parametric analysis
Optional COMSOL Scripts/MATLAB interface (Optimization,
postprocessing, batch jobs etc)
Results
Example:
Thermal effects in an electric conductor
Introduction
Purpose: Introduce you to the general concepts of multiphysics
modeling methodology in COMSOL Multiphysics
• The phenomena in this example involve the coupling of
thermal and electronic current balances.
• The ohmic losses due to the device’s limited conductivity
generate heat, which increases the conductor’s temperature and
thus also changes the material’s conductivity. This implies that
a 2-way multiphysics coupling is in play.
• Parameterization to study temperature as function of different
electrode potentials
Problem definition
Conductive film
Copper conductor
Solder joints
Problem definition
Heat balance:
  (k T )  Q
QV
2
Current balance:
-·(  V) = 0
1

 0 1   T  T0 
Second Example: Mesh Control
• Automated mesh generator
– Suitable for some problems
– But not always optimal
• How can you to create a non-uniform mesh?
– Mesh parameters menu
– Mesh Quality
• Thin Geometries
– Scale problem / mesh / Unscale
• Workshop Exercise: Mesh Control
– Mesh parameters menu
– Displaying Mesh Quality
Lunch Break
• Back in 1 hour
Interpolation functions
• Interpolation of measured data is commonly necessary when
analytical expressions for material properties are not available
• New feature in 3.1
• You can use interpolation function directly in the GUI (without
the need for MATLAB)
• Data can be entered from a table (for 1D interpolation) or from
a text file (for multidimensional interpolation)
• Example: Thermal conductivity as a function of temperature
Example:
Flow over a backstep
Thermal Flow over a Backstep
• Single physics
– Fluid dynamics
• Multiphysics
– You could add heat transfer and establish temperature profiles
• Aim of the model
– To give an overview of the modeling process in COMSOL Multiphysics
– Standard CFD benchmark
– Use both regular triangular and mapped meshes and compare the
solution
for various mesh densities
Problem definition
Step 1: Flow field (note form of input velocity)
Step 2: Heat transfer
300
330
Results: Stationary velocity profile
Results: Transient temperature profile
t=5
t=10
t=15
t=20
Concluding remarks
• The model is simple to define and solve in COMSOL
Multiphysics.
• The applications can be solved simultaneusly or sequentially
and for stationary or time dependent problems.
• Different Reynolds numbers can be easily sampled
Example:
Electric Impedance Sensor
Introduction
• Electric impedance measurement techniques are used for
imaging and detection
– Geophysical imaging
– Non-destructive testing
– Medical imaging (Electrical Impedance Tomography)
• Applying voltage to an object or a matrix containing different
materials and measuring the resulting potentials or current
densities
• Frequency range: 1Hz < f < 1GHz
Main points
• Use of Electromagnetics Module, Small Currents Application
Mode
• Different Subdomains with different physical properties
• Logical expressions can be used to modify the geometry
Geometry
Electrode
Air
Ground
Conductive medium
• We will study how the lateral position of the air-filled cavity affects
the measured impedance
Domain Equation
• Modelled with Small In-Plane Currents application in COMSOL Multiphysics
– Valid for AC problems where inductive effects are negligible
– The skin depth must be large compared to the object size
• Equation of continuity
     j r  0 V   0
• Electric field
• Displacement
Ε    V
D   0 r E
Equations and boundary conditions
n  J  J n  1A
     j r  0 V   0
nJ  0
V 0
Results: Current distribution [on a dB scale]
Impedance defined as
Cavity position
voltage
total current at electrode
Results: Impedance phase angle
Cavity position
Example:
Thermal Stresses in a Layered Plate
Geometry
coating
substrate
carrier
1. The coating is deposited on the substrate, at 800 °C
2. The temperature is lowered to 150 °C -> thermal stresses
in the coating/substrate assembly.
3. The coating/substrate assembly is epoxied to a carrier
plate.
4. The temperature is lowered down to 20 °C.
Model Definition
• No motion in the z-direction (2D Plane Strain application)
• Thermal loads are introduced according to:
Constitutive relations
  D el   0  D(   th   0 )   0 ;
 th
 

 y  z  xy  yz  xz th   vec (T  Tref )
T
x
Results: First step
• Result after depositing the
coating on the substrate and
lowering the temperature to
150 °C
•
The figure shows the stress in
the x direction
Results: Final step
The stress in the x direction
after attaching the carrier
and lowering the
temperature to 20 °C
Example:
Rapid thermal annealing
Rapid Thermal Anneal –the device
• Important process step in semiconductor processing
• Rapidly heat up Silicon wafer to 1000 degrees C for 10
seconds
Rapid Thermal Anneal –model geometry
• Modeling question: what is the difference in signal from an IR
detector and a thermoresist?
Silicon wafer
Detector
Heater
Rapid Thermal Anneal –COMSOL Multiphysics model
• Transient temperature behaviour is modeled with the General
Heat Transfer application mode
• Radiative Heat Transfer is determined by Surface-to-surface
radiation (included in General Heat)
 ( J 0  T 4 )
 n  (  kT ) 
(1   )
Rapid Thermal Anneal –results
T-lamp
T-wafer
IR detector signal
Thermoresist
signal
=> IR detector gives better signal!
Example:
MEMs Thermal Bilayer Valve
Thermal Bilayer Valve
• Layered material with
different thermal expansion
coefficients
• Layers undergoing different
expansion induces
curvatures which can be
used to close a switch,
operate a valve, etc.
Thermal Bilayer Valve
• Structural deformation from
thermal expansion
• Structural buckling
• Thermal conduction
• Heat source: Joule heating
• Current from DC conductive
– Axisymmetric
• Meshing Thin Layers
Using Mapped Meshes
• Feature introduced in FEMLAB 3.1
• 2D quadrilateral elements can be generated by using a mapping
technique (defined on a unit square)
• Best suited for fairly regular domains (connected, at least four
boundary segments, no isolated vertices) but irregular
geometry can also often be modified/divided in smaller regular
ones
• 2D mesh can then extruded/revolved to generate 3D brick
elements
Example:
Printed Circuit Board
Printed Circuit Board
• Two 3D geometries, one
for the board and one for
the circuits
• Geometries are meshed
and extruded separately
• Identity coupling
variables are used to link
the two geometries back
together