Finite Elemen Elementt Analysis Analysis (FEA (FEA / FEM) FEM) – Numerical Solution of Partial Differential Equations (PDEs). The Mathematical Problem: 1. PD PDE E repr repres esen entin ting g the the phy physi sics cs.. 2. Geomet Geometry ry on whi which ch to solv solve e the the prob problem lem.. 3. Boundary Boundary condition conditions s (for static static or steady steady state proble problems) ms) and and initial initial conditions (for transient problems). Γ -

boun unda dary ry (or δΩ)

x

Ω − domain y

e.g.. u(x, u(x,y,z y,z)) Unknowns – e.g

R. White, White, Comsol Comsol Acoustic Acoustics s Introduction, 2/25/08

Independent Variables – space and time (x,y,z,t ) Dependent Variables – unknown field (such as u )

Finite Elemen Elementt Analysis Analysis (FEA (FEA / FEM) FEM) – Numerical Solution of Partial Differential Equations (PDEs). The Mathematical Problem: 1. PD PDE E repr repres esen entin ting g the the phy physi sics cs.. 2. Geomet Geometry ry on whi which ch to solv solve e the the prob problem lem.. 3. Boundary Boundary condition conditions s (for static static or steady steady state proble problems) ms) and and initial initial conditions (for transient problems). Γ -

boun unda dary ry (or δΩ)

x

Ω − domain y

e.g.. u(x, u(x,y,z y,z)) Unknowns – e.g

R. White, White, Comsol Comsol Acoustic Acoustics s Introduction, 2/25/08

Independent Variables – space and time (x,y,z,t ) Dependent Variables – unknown field (such as u )

Finite Elemen Elementt Analysis Analysis (FEA (FEA / FEM) FEM) – The Mathematical Problem: Boundary Conditions. On each boundary you must specify either: 1) The de depe pend nden entt vari variab able le its itsel elff (e.g. u ) – “Essen “Essentia tiall Boundar Boundary y Condit Con dition ion”” or “Dirichl “Dirichlet et Bounda Boundary ry Conditi Condition” on” 2) The deriva derivativ tive e of the variab variable le itse itself lf (e.g. du/dn) – “Natural Boundary Condition Condition”” or “Neumann “Neumann Boundary Boundary Condition” Condition” 3) The relati relationsh onship ip between between the depend dependent ent variab variable le and and its normal normal derivative (e.g. du/dn=(1/z)·u)). Γ -

boun unda dary ry (or δΩ)

x

Ω − domain y

Unknowns – e.g. u(x,y,z,t)

R. White, White, Comsol Comsol Acoustic Acoustics s Introduction, 2/25/08

Independent Variables – space and time (x,y,z,t ) Dependent Variables – unknown field (such as u )

Finite Element Analysis (FEA / FEM) – The Finite Element Part: 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 shape is now “meshed” with triangle elements.

R. White, Comsol Acoustics Introduction, 2/25/08

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.

R. White, Comsol Acoustics Introduction, 2/25/08

Finite Element Packages - Here are some of the common ones

R. White, Comsol Acoustics Introduction, 2/25/08

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.

R. White, Comsol Acoustics Introduction, 2/25/08

COMSOL – Here we go!! I will focus on acoustics as an application, but the steps are very similar for other kinds of physics.

1.

Decide on the representative physics (choose the PDE).

2.

Define the geometry on which to solve the problem.

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.

R. White, Comsol Acoustics Introduction, 2/25/08

1.

Decide on the representative physics (choose the PDE).

2.

Define the geometry on which to solve the problem.

1 1 ω 2 ∇ p + p = 0 ∇ ⋅ ρ 0 ρ 0 c

Constant density

ω c

∇ p + 2

2

p=0

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 e j ωt . R. White, Comsol Acoustics Introduction, 2/25/08

2. Define the geometry on which to solve the problem. Default units are mks units (SI units). You can change units under “Physics | Model Settings” Draw menu : draw points, curves, and 2D objects (in 3D you will have 3D tools) Boolean operations let you subtract, add, etc.

I drew three rectangles using the “Draw|Specify Objects…” Tool

R. White, Comsol Acoustics Introduction, 2/25/08

GO BUILD YOURSELF SOME GEOMETRY! DON’T MAKE IT TOO COMPLEX… BE REASONABLE…

R. White, Comsol Acoustics Introduction, 2/25/08

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

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.

R. White, Comsol Acoustics Introduction, 2/25/08

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

Go to “subdomain settings” under the Physics menu

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.

Each 2D object is a different subdomain

R. White, Comsol Acoustics Introduction, 2/25/08

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 ).

R. White, Comsol Acoustics Introduction, 2/25/08

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.

R. White, Comsol Acoustics Introduction, 2/25/08

4. Set the boundary conditions 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.

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 =- ρ 0 a n 5. Impedance Condition – set Z at the boundary (Z=p/u n =- ρ 0 j ω· 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. R. White, Comsol Acoustics Introduction, 2/25/08

4. Set the boundary conditions 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. R. White, Comsol Acoustics Introduction, 2/25/08

SET UP PHYSICS AND BOUNDARY CONDITIONS ON YOUR GEOMETRY…

R. White, Comsol Acoustics Introduction, 2/25/08

5. Choose an element type and mesh the geometry.

You can set the element shape function order in “Physics | Subdomain Settings”. Default is quadratic which should be fine.

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. R. White, Comsol Acoustics Introduction, 2/25/08

6. Choose a solver and solve for the unknowns.

You have many solver choices… the Direct solvers are usually more robust, but require more memory and may not work for large problems.

Go to Solve | Solver Parameters… to set up the solution.

R. White, Comsol Acoustics Introduction, 2/25/08

6. Choose a solver and solve for the unknowns.

Adaptive mesh refinement is available… the computer will try to change the mesh to reduce errors. I have not played with this.

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.

R. White, Comsol Acoustics Introduction, 2/25/08

6. Choose a solver and solve for the unknowns.

Click the equal sign to solve and hope for the best!!!!

R. White, Comsol Acoustics Introduction, 2/25/08

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.

R. White, Comsol Acoustics Introduction, 2/25/08

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

The Postprocessing | Plot Parameters… menu let’s you look at the results. I have lots of choices for the kind of 2D plot to use.

R. White, Comsol Acoustics Introduction, 2/25/08

If I did a parametric solve, I select which solution I want to view here.

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

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.

R. White, Comsol Acoustics Introduction, 2/25/08

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.

R. White, Comsol Acoustics Introduction, 2/25/08

Convergence - Mesh Refinement

Much better.

R. White, Comsol Acoustics Introduction, 2/25/08

☺

7. Post-process the results to find the information you want. I am going to look at a the solution at a point for this range of values of freq_aco.

The Postprocessing | Domain Plot Parameters… menu let’s you look at the a result as a function of your parameterized variable.

R. White, Comsol Acoustics Introduction, 2/25/08

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

R. White, Comsol Acoustics Introduction, 2/25/08

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

Click here to send this data out to a text file so I load into Matlab or Excel.

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!

R. White, Comsol Acoustics Introduction, 2/25/08

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

Comsol Result 65 60

Comsol - IL Analytic - TL

55 50

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.

) 45 B d ( L I 40

35 30 25 20 2 10

3

10 Frequency (Hz)

R. White, Comsol Acoustics Introduction, 2/25/08

10

4

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

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.

R. White, Comsol Acoustics Introduction, 2/25/08

WHENEVER YOU USE A NEW FEA SOLVER, SOLVE A PROBLEM YOU KNOW THE SOLUTION TO FIRST, TO MAKE SURE YOU ARE USING IT CORRECTLY!!!!!!!!!!

R. White, Comsol Acoustics Introduction, 2/25/08

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