Equation

Based Modeling
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Multiphysics and Single

Physics Simulation Platform
•
Mechanical
, Fluid, Electrical,
and Chemical Simulations
•
Multiphysics

Coupled P
henomena
–
Two or more physics phenomena that affect each other with
no limitation
on
•
which combinations
•
h
ow many combinations
•
Single Physics
–
One integrated environment
–
different physics and
applications
–
One day you work on Heat Transfer, next day Structural
Analysis, then Fluid Flow, and so on
–
Same workflow for any type of modeling
•
Enables cross

disciplinary product development and a
unified simulation platform
Highly
Customizable
and
Adaptable
•
Create your own multiphysics couplings
•
Customize material properties and boundary
conditions
–
Type in mathematical
expressions, combine with look

up
tables and function calls
•
User

interfaces for differential and algebraic equations
•
Parameterize on material properties, boundary
conditions, geometric dimensions, and more
•
High

Performance Computing (HPC)
–
Multicore & Multiprocessor: Included with any license
type
–
Clusters & Cloud: With floating network licenses
The COMSOL
Multiphysics
®
4.3b Product Suite
Equation

Based Modeling
•
Partial Differential Equations (PDEs)
–
PDE Interfaces
–
Boundary Conditions
•
Ordinary Differential Equations (ODEs)
–
Global ODEs
–
Distributed ODEs
•
Algebraic Equations
–
Global algebraic equations
–
Distributed algebraic equations
When is Equation

Based Modeling Needed?
•
Try to avoid equation

based modeling if possible!
–
Using built

in physics interfaces enables ready

made
postprocessing
variables
and other tools for faster model setup with much lower risk of human error
•
Applications that previously required equation

based
modeling but now has a dedicated physics interface:
–
Fluid

Structure Interaction (Structural Mechanics Module, MEMS Module)
–
Surface adsorption and reactions (Chemical Reaction Engineering Module,
Plasma Module)
–
Shell

Acoustics and
Piezo

Acoustics (Acoustics Module)
–
Thermoacoustics
(Acoustics Module)
–
and many more…
When is Equation

Based Modeling Needed?
•
Try to avoid equation

based modeling if possible!
•
But: we don’t have every imaginable physics equation built

into COMSOL (yet!). So there is sometimes a need for custom
modeling.
Custom

Modeling in COMSOL
•
COMSOL
Multiphysics
allows you to model with PDEs or
ODEs directly:
–
Use one of the equation

template user interfaces
•
You do *not* need to write “user

subroutines” in COMSOL
to implement your own equation!
–
Benefit: COMSOL’s nonlinear solver gets all the nonlinear info with
gradients and all. Faster and more robust convergence.
Customization Approaches
•
Four modeling approaches:
1.
Ready

made physics interfaces
2.
First principles with the equation templates
3.
Start with ready

made physics interface and add additional terms.
4.
Start with a ready

made physics interface and add your own
separate equation (PDE,ODE) to represent physics that is not already
available as a ready

made application mode
•
Also:
–
The Physics Builder lets you create your own user interfaces that hides the
mathematics for your colleagues and customers
PDEs
Linear Model Problems: Fundamental Phenomena
•
Laplace’s equation
•
Heat equation
•
Wave equation
•
Helmholtz equation
•
Convective Transport equation
COMSOL PDE Modes: Graphical User Interfaces
•
Coefficient form
•
General form
•
Weak form
•
All these can
be used for scalar equations or systems
•
Which to use?
–
Whichever is more convenient for you and your simulation needs
Coefficient Form
•
Coefficient Matching Example: Poisson’s equation
inside
domain
on boundary
inside subdomain
on subdomain boundary
Implies c=f=h=1 and all other coefficients are 0.
Demo:
Block: 10x1x1
PDE: default Poisson’s
equation with unknown
u
.
Dirichlet
boundary
condition everywhere:
u
=0
d(
u,x
) with no Recover smoothing
d(
u,x
) with Recover smoothing
The Recover feature applies “polynomial

preserving recovery” on the partial
derivatives (gradients).
Higher

order approximation of the
solution on a patch of mesh elements
around each mesh
vertex.
Also available as
ppr
operator.
Coefficient Form, Interpretations
mass
damping/
mass
diffusion
convection
source
convection
absorption
source
mass
damped
mass
elastic stress
initial/thermal
stress
body force
(gravitation)
Coefficient Form, Structural Analysis Wave Equation
density
damping coefficient
stress,
u
= displacement vector
stiffness, “spring constant”
accumulation/storage
diffusion
convection
source
convection
absorption
source
Coefficient Form, Transport Diffusion Equation
Coefficient Form, Steady

State Equation
diffusion
Helmholtz term
source
Helmholtz equation:
Coefficient Form, Frequency

Response Wave Equation
Wave number
Wave length
Demo:
lambda=2.5
k=2*pi/lambda
a=

k^2
f=0
u=1 one end
u=0 other end
Complex Arithmetics
•
Can compute:
real(w)
imag(w)
abs(w)
arg(w)
conj(w)
General Form
–
A more compact formulation
•
inside domain
•
on domain boundary
•
For Poisson’s equation, the corresponding general form implies
•
All other coefficients are 0
Weak Form
•
Think of the weak form as a generalization of
the principal
of
virtual
work (for
those familiar with
that) with virtual
displacement
d
u
–
The test function
n ~ d
u
•
Convection

diffusion equation:
•
Multiply by test function
n
and integrate:
•
Integrate by parts and use boundary conditions:
•
In COMSOL you can type the integrands of this integral expression:
Weak Form PDE
Typing the Weak Form
c*grad(u)
∙grad(test(u))=
c*grad(u
)
∙test(grad(u))=
c*(
ux
*test(
ux
)+
uy
*test(
uy
)+
uz
*test(
uz
))
Note: COMSOL convention has the integral
in the right

hand side so additional
negative sign needed in the GUI
accumulation/storage
diffusion
source
Transient Diffusion Equation ~ Heat Equation
“Heat Source” f=1
“Cooling” u=0 at ends
Demo:
c=1
da=1
f=0
Transient 0

>100 s
PDEs+ODEs
Transient Diffusion Equation + ODE
What if we wish to measure the global accumulation of “heat” over time?
Transient Diffusion Equation + ODE
=> This is a
Global ODE
in
the
global
state variable
w
What if we wish to measure the global accumulation of “heat” over time?
Demo:
Global Equation ODE
Same time

dependent problem as
earlier
Time

dependent 0

100
Volume integration of u
ODE:
wt

U
PDEs + Distributed ODEs
Transient Diffusion Equation + Distributed ODE
What if we get “
damage”
from local accumulation of “heat”.
Example of real application:
bioheating
We want to visualize the
P

field to assess local damage.
Let’s assume damage happens where P>20.
Disclaimer:
There is no need to use equation

based modeling for
bioheating
in COMSOL. You
are better off using the preset user interface
options of the Heat Transfer Module.
Transient Diffusion Equation + Distributed ODE
What if we get “
damage”
from local accumulation of “heat”.
Example of real application: bioheating
Transient Diffusion Equation + Distributed ODE
But this can be seen as a PDE with no spatial derivatives =
=
Distributed ODE
Use coefficient form with unknown field P, c = 0, f = u,
da
=1
Let all other coefficients be
zero
Or use Domain ODEs and DAEs interface
Volume where P>20 and
we get damage
Demo:
Distributed ODE
Same time

dependent problem as
earlier
Time

dependent 0

100
Volume integration of u
ODE:
wt

U
Distributed Algebraic Equations
Example: Ideal gas law
•
Assume
u
=(u,v,w)
and
p
given by Navier

Stokes
•
Want to solve Convection

Conduction in gas:
•
given by ideal gas law:
•
Easy

analytical
Example: Non

ideal gas law
•
Assume
u
=(
u,v,w
)
and
p
given by
Navier

Stokes
•
Want to solve Convection

Conduction in gas:
•
given by non

ideal gas law:
•
Needed for high molecular weight at very high pressures
•
Difficult
–
implicit equation
•
How to proceed?
Example: Non

ideal gas law
•
How to solve:
•
Third order equation in
•
Pressure
p
is function of space
•
So: this is an algebraic equation at each point in space!
Distributed Algebraic Equation
•
So: this is an algebraic equation at each point in space
•
See as PDE with no space or time derivatives!
–
A*(
p+B
*u^2)*(1

C*u)

D*u
–
Here we let: A=1,B=2,C=3, D=4, p=x*y
•
How: Put the entire equation in the source (f) term and zero
out the rest
•
Or, use user interface for Domain ODEs and DAEs
The solution u corresponding to the
equation A*(
p+B
*u^2)*(1

C*u)

D*u,
where p=x*y is spatially varying.
Here the equation is solved for each
point within the unit square.
Distributed Algebraic Equation
•
What about nonlinear equations with multiple solutions?
•
Which solution do you get?
•
For simplicity, consider the equation (u

2)^2

p=0, where p is a constant
•
This can be entered as earlier with an f=
(u

2)^
2

p
•
The solution is easy to get analytically
:
u=2
±
sqrt(p)
•
The solution you get will depend on the Initial Guess given by the PDE Physics
Interface
•
If we let p=x*y and let our modeling region be the unit square, then at (
x,y
)=(0,0)
we should get the unique solution u=2 but at (
x,y
)=(1,1) we get 1 or 3 depending
on our starting guess (and also the convergence region of the solver). See next
slide.
Distributed Algebraic Equation
u=3
u=2
u=1
u=2
End of Presentation
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