COMSOL_Equation_Based_Modeling_4.3bx - COMSOL.com

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Oct 24, 2013 (3 years and 7 months ago)

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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