comsol equation based modeling 4.3b

Equation-Based Modeling

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Multiphysics and Single-Physics Simulation Platform

• Mechanical, Fluid, Electrical, and Chemical Simulations

• Multiphysics - Coupled Phenomena– Two or more physics phenomena that affect each other with no

limitation on• which combinations• how 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 interfaces2. First principles with the equation templates3. 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

0 u

0)( ukut

0)( uutt

uu )(

0 xt buu

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

rhuhgquuuc T )(n

inside domain

on boundary

1 u0u

inside subdomain

on subdomain boundary

Implies c=f=h=1 and all other coefficients are 0.

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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

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mass damped mass

elastic stress initial/thermal stress

body force (gravitation)

a

aed

cc

u

2

2 ( )a ae d c a ft t

u u u u u u

Coefficient Form, Structural Analysis Wave Equation

density

damping coefficient

stress, u= displacement vectorstiffness, “spring constant”

accumulation/storage

diffusion

convection

source

convection

absorption

source

Coefficient Form, Transport Diffusion Equation

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Coefficient Form, Steady-State Equation

diffusion Helmholtz term

source2

2

( )

2

c u k u f

a k

k

Helmholtz equation:

Coefficient Form, Frequency-Response Wave Equation

Wave number

Wave length

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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

RuRG

T

0

n

uyux .uR

1F

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Weak Form• Think of the weak form as a generalization of the principal of virtual

work (for those familiar with that) with virtual displacement du– The test function n ~ du

• 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

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accumulation/storage

diffusion

source

Transient Diffusion Equation ~ Heat Equation

sourceheat volume

fkcCda

“Heat Source” f=1

“Cooling” u=0 at ends

Demo:c=1

da=1

f=0

Transient 0->100 s

PDEs+ODEs

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Transient Diffusion Equation + ODE

integral volumeof integral e tim

solution of integral volume

dtUw

dVuU

t

V

What if we wish to measure the global accumulation of “heat” over time?

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Transient Diffusion Equation + ODE

0],[ 10

Uwt

Udtdw

dtUw

dVuU

ttt

V

=> 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

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Transient Diffusion Equation + Distributed ODE

solution of integral timelocal ),,( ),,( dtzyxuzyxPt

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

solution of integral timelocal ),,( ),,( dtzyxuzyxPt

What if we get “damage” from local accumulation of “heat”.

Example of real application: bioheating

spacein point each at udtdP

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Transient Diffusion Equation + Distributed ODE

solution of integral timelocal udtdP

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

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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

0u)( TCTk

RTpM

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?

0)1)((* 2 DCBpA

0u)( TCTk

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!

0)1)((* 2 DCBpA

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=1u=2

End of Presentation