comsol

R. White, Comsol Acoustics

Introduction, 2/25/08



Finite Element Analysis (FEA / FEM) –

Numerical Solution of Partial Differential Equations (PDEs).

1. PDE representing the physics.

2. Geometry on which to solve the problem.

3. Boundary conditions (for static or steady state problems) and initial

conditions (for transient problems).

Ω − domain

Γ- boundary

(or δΩ)

Unknowns – e.g. u(x,y,z)

x

y

The Mathematical Problem:

Independent

Variables –

space and time

(x,y,z,t)

Dependent

Variables –

unknown field

(such as u)




Boundary Conditions. On each boundary you must specify either:

1) The dependent variable itself (e.g. u) – “Essential Boundary

Condition” or “Dirichlet Boundary Condition”

2) The derivative of the variable itself (e.g. du/dn) – “Natural Boundary

Condition” or “Neumann Boundary Condition”

3) The relationship between the dependent variable and its normal

derivative (e.g. du/dn=(1/z)·u)).

Ω − domain

Γ- boundary

(or δΩ)

Unknowns – e.g. u(x,y,z,t)

x

y

The Mathematical Problem:

Independent

Variables –

space and time

(x,y,z,t)

Dependent

Variables –

unknown field

(such as u)




1) Discretization of the space into pieces (the elements) – this is called

the Mesh.

2) Choice of element type - shape (triangle, quadrilateral, etc.),

number of nodes (3, 4, 5, 8, etc.) and shape function (linear,

quadratic, etc.).

3) Choice of solver (direct, iterative, preconditioning).

4) Post-processing – looking at the solution in various ways.

The Finite Element Part:

The shape is

now “meshed”

with triangle

elements.



So, this is always the sequence for any FEA problem:

1. Decide on the representative physics (choose the PDE).

2. Define the geometry on which to solve the problem.

3. Set the “material properties”… that is, all the constants that appear

in the PDE.

4. Set the boundary conditions (for static or steady state problems)

and initial conditions (for transient problems).

5. Choose an element type and mesh the geometry.

6. Choose a solver and solve for the unknowns.

7. Post-process the results to find the information you want.



Finite Element Packages - Here are some of the common ones



Comsol Multiphysics (a.k.a. FemLab)

- More recent than Ansys,

Nastran, Abaqus.

- Integrates well with Matlab

(uses Matlab syntax too).

- Focuses on “Multiphysics” –

coupling different physics

together (e.g. acoustics and

solid mechanics).

- Highly flexible… allows you to

program in your own

differential equations if they

are not already impelemented.





COMSOL – Here we go!!

I will focus on acoustics as an application, but the steps are very similar

for other kinds of physics.

Choose how many dimensions

to work in. Warning: 3D is

usually a large computational

problem, avoid if at all

possible!!

Choose your type of physics.





I have selected 2D (will solve

the problem in assuming no

variation in the z-direction).

I have selected time-harmonic

acoustics… time-harmonic means

single frequency… we are

assuming time dependence ejωt.

0

2

2 =

+∇ p

cp

ω0

112

00

=

+

∇

⋅∇ p

cp

ω

ρρ

Constant

density




I drew three rectangles using the

“Draw|Specify Objects…” Tool

Draw menu : draw

points, curves, and

2D objects (in 3D you

will have 3D tools)

Boolean operations

let you subtract,

add, etc.

Default units are mks units (SI units).

You can change units under “Physics

| Model Settings”



GO BUILD YOURSELF SOME GEOMETRY! DON’T MAKE IT TOO

COMPLEX… BE REASONABLE…




For more

complex

geometry,

you can

import CAD

data from file

(DXF format

works well)

… under

“File” menu

When you import DXF data, it will come

in as a curve, not a solid. For instance, I

drew this blob in Solidworks DWG editor,

and exported it as “Autocad 2004 DXF –

Ascii”. Once it is imported into Comsol, I

go to the Draw Menu and say “Coerce

To Solid” and it turns the curve into a 2D

solid.



3. Set the “material properties”… that is, all the constants that appear in the PDE.

Go to “subdomain settings” under the Physics menu

Each 2D

object is a

different

subdomain

Set the properties

(density and

wavespeed). They can

be functions of x and y

(and p if you want to

make the problem

nolinear)! Just use

Matlab syntax.



3. Set the “material properties”… that is, all the constants that appear in the PDE.

Go to “scalar variables” under the Physics menu

Set any global

variables… in this case,

we can set the frequency

in Hz we are solving at,

and the reference

pressure used for

displaying dB SPL

(default is 20e-6 Pa).



4. Set the boundary conditions

Go to “Boundary Settings” under the Physics menu

For each boundary,

choose a boundary

condition.

Internal Boundaries (boundaries between subdomains)

are grayed out… continuity of pressure and velocity will

be enforced at the internal boundary.




Choices of boundary conditions (acoustics mode):

1. Sound Hard Boundary – Neumann condition; dp/dn = 0 (normal velocity = 0)

2. Sound Soft Boundary – Dirichlet condition; p = 0 (pressure release)

3. Pressure – Dirichlet condition; p=p0 (sets acoustic pressure amplitude)

4. Normal Acceleration – Neumann condition since Euler says dp/dn=-ρ0an

5. Impedance Condition – set Z at the boundary (Z=p/un=-ρ0jω·p/(dp/dn))

6. Radiation Condition – set a boundary that will not reflect normally incident plane

waves…. This is how you try to approximate an infinite space; only perfect if the

incident wave is a perfect plane wave. You can include a source term in this

condition to send in a plane wave at the boundary.

For acoustics mode, we will be solving for

the complex pressure p as a function of x,

y. You can therefore use complex

numbers for any of your pressure or

velocity boundary conditions; these specify

magnitude and phase.




Notes on Radiation Condition:

- You want to use this if you are thinking of your problem extending off to infinity,

but you don’t want to mesh the problem.

- If you are working in axisymmetric 2D mode or 3D mode you will have

additional choices at the boundary to match spherical and cylindrical waves.

- If you work on our research license (which includes the acoustics module), you

have another choice for boundary conditions to simulate infinite spaces… the

Perfectly Matched Layer “PLM”.

Notes on Symmetry:

• If you are working in one of the axisymmetric modes, you may specify a

boundary as an axis of symmetry; the solution is revolved about the axis.

• If you are working in Cartesian coordinates and have a symmetric system, you

can model only part of it (this can save a lot of computation time)… often the

symmetry boundary will act like a rigid wall; the derivative of pressure will be

zero on the symmetry boundary.



SET UP PHYSICS AND BOUNDARY CONDITIONS ON YOUR

GEOMETRY…



5. Choose an element type and mesh the geometry.

These little triangle icons let you “initialize the mesh”

(first one) “refine the mesh” (second one) and

“refine a portion of the mesh” (third on). There are a

lot more meshing tools under the “Mesh” menu.

This is where you can change to quadrilaterals or

bricks if you want. Triangles and tetrahedra are

easier to mesh with, although sometimes bricks and

quads may give better results.

Keep in mind: 1. You need at least 5 elements per

wavelength… λ=c/f, so as your frequency goes up, you will need more elements! 2. If there are places in the

model where you expect complex behavior, use a

denser mesh in that region.

You can set the

element shape

function order in

“Physics |

Subdomain

Settings”.

Default is

quadratic which

should be fine.




Go to Solve | Solver

Parameters… to set

up the solution.

You have many

solver

choices… the

Direct solvers

are usually

more robust,

but require

more memory

and may not

work for large

problems.




The parametric solver is useful… you can

have Comsol solve the problem a number

of times, each time varying a parameter.

For acoustics, this might often be

frequency. You can build up a frequency

response function this way by solving the

time harmonic problem multiple times.

Adaptive mesh refinement is

available… the computer will

try to change the mesh to

reduce errors. I have not

played with this.




Click the equal sign

to solve and hope

for the best!!!!




Please remember… just

because you get pretty

colors does not mean

the solution is correct!

Be careful, please, when

you are building the

device which is

supposed to save my

life.




The Postprocessing |

Plot Parameters…

menu let’s you look at

the results. If I did a

parametric

solve, I select

which solution I

want to view

here.

I have lots of

choices for the

kind of 2D plot

to use.




These are the surface

plot options.

These are all the things it

knows how to plot. I can

also have it plot any

computatoin involving these

variables.



Convergence - Mesh Refinement

Here is what happens if I don’t

have enough elements to

capture my high-frequency

(short wavelength) solution –

the solution is not

converged!!!!!!

Watch out for this… it is an

easy mistake to make. Always

do a convergence study, solve,

then increase your mesh

density, solve again, and make

sure the solution does not

change much.



Convergence - Mesh Refinement

Much better.




The Postprocessing |

Domain Plot

Parameters… menu

let’s you look at the a

result as a function of

your parameterized

variable.

I am going to

look at a the

solution at a

point for this

range of values

of freq_aco.



I have defined the dB SPL here

using a Matlab expression

involving the solved-for complex

pressure, p.


I am looking at the

solution at two points

that I created using the

“Draw” menu.



You can see the dB SPL on the other side

of the barrier (green curve) is significantly

below that on the upstream side. The

difference is not TL or IL, since the

upstream side includes both the incident

and reflected waves!


Click here to send this

data out to a text file so

I load into Matlab or

Excel.




I ran the Comsol computation

twice, once with the

intermediate layer having the

same properties as the two

other layers (this is the “before”

insertion case) and then with

the intermediate layer as a

different layer (1cm thick

plastic).

I sent the results for the SPL at

the two points in the two cases

out to a text file, and loaded into

Matlab. The Insertion Loss is

the difference between the dB

SPL on the far side of the

barrier before and after it was

inserted.

102

103

104

20

25

30

35

40

45

50

55

60

65

Frequency (Hz)

IL (

dB

)

Comsol Result

Comsol - IL

Analytic - TL




If you want to get a

cross-section plot at a

line through your 2D

model (or a plane

through your 3D model)

that can be achieved

here.



WHENEVER YOU USE A NEW FEA

SOLVER, SOLVE A PROBLEM YOU

KNOW THE SOLUTION TO FIRST, TO

MAKE SURE YOU ARE USING IT

CORRECTLY!!!!!!!!!!



SOLVE AND PLAY WITH THE RESULTS

comsol

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