comsol acoustics

R. White, Comsol Acoustics

Introduction, © 2012

Tutorial Created in Comsol 4.3 (2012)



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).

W - domain

- boundary

(or dW)

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)).

W - domain

- boundary

(or dW)

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

- 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

implemented.



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

COMSOL – Here we go!!

I will focus on acoustics as an application, but the steps are similar for other physics.

Choose how many dimensions to work in. Warning: 3D is

usually a large computational problem, avoid if at all

possible!! Make use of symmetries to get to 2D or 2D

axisymmetric. Choose your type of physics.

You may select more than one

if you want coupling.

Tutorial Created in Comsol 4.3 (2012)

“Pressure Acoustics” is what we have been

doing in ME139 – this solves the Helmholtz

equation for the complex acoustic pressure.

Here you are choosing what kinds of solutions you want at the end of the study.

You can always add other kinds later.



• Frequency Domain : This is what

we have been doing for the most

part in ME139. The Helmholtz

equation … you are solving for

steady state pressure at a single

frequency.

• Eigenfrequency : This will allow

you to find the acoustic modes of a

domain. These are frequencies

and corresponding pressure fields

where the Helmholtz equation and

boundary conditions can be

satisfied with no external drive

(Homogeneous Solutions)

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

Type of “Study”.



1. Decide on the representative physics (choose the PDE). Complete. At this point

I have chosen my PDE and number of spatial dimensions. For pressure acoustics,

my PDE is the Hemholtz equation … but I can allow r0 and c to vary in space if

desired.

Remember, since I have chosen “Pressure Acoustics”, I have selected time-

harmonic acoustics… time-harmonic means single frequency… we are assuming

time dependence ejwt. The pressure I solve for will be the complex pressure,

2

2 0p pc

w

2

0 0

1 10p p

c

w

r r

Constant

density

( , , )p x y z Solved for in Comsol

10

( , , , ) Re ( , , )

1( , , ) ( , , )

2

( , , )( , , ) 20log

j t

rms

rms

ref

p x y z t p x y z e

p x y z p x y z

p x y zSPL x y z

p

w

Can be easily computed

in post processing using

post processing tools

inside Comsol.



Comsol 4.3 Graphical User Interface

Model tree shows all

parts of the model …

geometry, boundary

conditions, materials,

types of study to run,

results. Right click

on things to interact.

Geometry and various

results plots will be

shown in the main

Graphics window.

Various useful tools,

depending on what you

have selected in the

model tree.

Tools related to

zooming, viewing,

saving graphics objects.




Draw the geometry of the acoustic domain (the

domain over which you want to solve the PDE)

by right clicking on geometry and using various

tools (tools also appear in the toolbar at the top

when geometry is selected in the model tree).

If you select some objects in the graphics

window then Boolean tools (like subtract,

intersect, union) will also appear under

geometry.

Default units are mks units (SI units).

You can change units by selecting the

(root) object in the model tree (the

very very top object).

If you change or delete geometry

objects, sometimes you may need

to ask Comsol to Build All the

geometry again to get it to refresh.

For complicated

geometry you

may choose to

import it from a

CAD program.




Objects appear in here.

Here are all the geometry

objects I defined.

Edit them by clicking and/or right

clicking them in the model tree.

Drawing tools. For axisymmetric, the

axis of symmetry will

be r=0 and be drawn as

a line in the graphics

window.



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

For pressure acoustics, all that matters is r0 and c. (And frequency… although

that is set under “Study | Frequency Domain”, not under materials.)

Under

“materials”

select “open

material

browser” or “+

material”.

Find the material you want in the

browser, or create your own material

with “+ Material”.

After you find it, right click and

“Add Material to Model”

For pressure acoustics, the only

properties that matter are density and

speed of sound.

If you did a coupled thermal problem

these could be functions of

temperature … etc … or you can just

enter them as constants.



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

Once the materials you want are

added, assign them to whichever

geometric objects you want to have

those properties.

Also set up frequency for the

problem … I am thinking of this as

a global material property … it

appears as a constant in the

Helmholtz equation.



4. Set up the boundary conditions.

Right click on the “Pressure

Acoustics” physics in the model to

bring up options for boundary

conditions for this physics.

We have a lot of choices… for each

kind of boundary condition you will

want to add it to the model, and

then apply it to the boundaries you

want it applied to.



4. Set the boundary conditions

Choices of boundary conditions (pressure 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… remember,

everything is oscillating as ejwt)

4. Normal Acceleration – Neumann condition since Euler says dp/dn=-r0an (sets

normal acceleration amplitude… remember, everything is oscillating as ejwt)

5. Impedance Condition – set normal specific acoustic impedance zn at the boundary

(zn=p/un→ (dp/dn)=-r0jw·p/zn). This is how you could approximate an absorbing

panel or something like that … set zn to get the desired NRC.

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

waves or cylindrical waves or spherical waves…. This is how you try to

approximate an infinite space; only perfect if the incident wave is a perfect plane

wave (or cylindrical/spherical wave). You can include a source term in this

condition to send in a wave at the boundary. A preferred method is to use a

“Perfectly Matched Layer” … but radiation condition should be sufficient for

our purposes.

For pressure acoustics mode, we will be solving for the complex pressure p. You

can therefore use complex numbers for any of your pressure or velocity boundary

conditions; these specify magnitude and phase.



4. Set the boundary conditions.

Note that boundary conditions can be functions of space …

use Matlab syntax … so here I have set an = 1 mm/s2 for

r<0.1 meters, but zero for r>0.1 meters. Remember the

Matlab expression r<0.1 evaluates to 1 when true, 0 when

false. The names of the spatial coordinate variables can

be found by clicking on the Model.

Select the part of the boundary you

want to apply that condition to and

add it to the boundary selection.




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 “PML”.

Notes on Symmetry:

• If you are working in one of the axisymmetric modes, you will have 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.




Notes on Sources

- One way to create an acoustic source is to use an acceleration boundary condition which will

produce sound through the vibration of the boundary. You could set this to be a small

boundary and set up its normal acceleration to get a desired simple source strength, or make

a baffled piston, etc.

- Another way to create an acoustic source is to send in a plane wave from a plane wave

radiation boundary condition, or an incoming spherical wave from a spherical radiation

condition … it is as if a source at infinity where sending in these incoming plane or incoming

spherical waves, which will then superimpose with whatever you get from reflections,

defraction, and refraction with your geometry.

- A third way to create an acoustic source is to add a point source from under the “pressure

acoustics|points” menu. A “volume flow source” has a strength Q in m3/s that is exactly the

way we have been doing simple sources. Note that if you put this point source right on top of

a hard boundary you will get an image source … so strength will double to 2Q.

- A final way to create an acoustic source is to add a monopole distribution. You will see this

under the options when you right click on the “Pressure Acoustics” item in the model tree.

The monopole source will be applied over a volume … it is as if that volume is packed with

monopole sources with the source strength that you set. The Comsol monopole distribution

strength Q is in units of 1/s2 … so the Comsol monopole strength is volume acceleration/unit

volume = volume velocity/(jw)/unit volume. This has been checked.



5. Choose an element type and mesh the geometry.

Right click on Mesh1 and

select a type of mesh (Free

Triangular, etc). You can

modify how the mesh is

layed out by adding size,

scale, and distribution

controls.

Keep in mind: 1. You need at least 5 elements per wavelength… l=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.

IT IS CRITICAL TO HAVE A DENSE ENOUGH MESH. IF YOU DON’T HAVE

MULTIPLE ELEMENTS PER WAVELENGTH YOUR SOLUTION WILL NOT BE

CONVERGED … IT WILL NOT BE A GOOD APPROXIMATION TO THE PDE

SOLUTION!! THE PROBLEM WILL STILL RUN BUT YOUR RESULTS WILL BE

BAD.

It would be a good idea to do a simple convergence study. After solving, make

the mesh finer, re-solve, and make sure the solution didn’t change much.



6. Choose a solver and solve for the unknowns.

Finally, go to Study and select “Compute”. You can change the type of

solver under the various steps (in our case there is only one step … a

Frequency Domain pressure acoustics step). There are a variety of

direct (full matrix inversion) and iterative solvers.

Direct solvers tend to be more robust (they are most likely to converge)

but require more memory.

This is a large topic … which solvers are best … for a linear pressure

acoustics problem it should not be bad, most solvers should work. I

think.

Just make sure you mesh is fine enough to represent the wavelength of

sound at the frequency you are working at !!

Parametric sweeps are useful … especially in pressure acoustics you

might want to solve the same geometry for a range of frequencies … set

up a frequency range under “Frequency Domain” and you will get a

series of solutions at different frequencies. Just remember to keep your

mesh fine enough to represent the short wavelengths at high

frequencies!!


Introduction, 11/1/10

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

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.

There are lots of things you can plot

… these are all derived from the

solution for the complex pressure,

which is all it solved for. Once it has

that it can compute velocity,

intensity, SPL, etc. just as we have

been doing.

Keep in mind … “instantaneous” quanities are the real part…

the pressure or intensity or whatever at t=0 … this can go

positive and negative. Total or magnitude are the magnitudes

of the oscillating quantities and RMS is RMS.



WHENEVER YOU USE A NEW FEA

SOLVER, SOLVE A PROBLEM YOU

KNOW THE SOLUTION TO FIRST, TO

MAKE SURE YOU ARE USING IT

CORRECTLY!!!!!!!!!!

comsol acoustics

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