Tutorial Created in Comsol 4.3 (2012)

Transcription

1 Tutorial Created in Comsol 4.3 (2012)

2 Finite Element Analysis (FEA / FEM) Numerical Solution of Partial Differential Equations (PDEs). The Mathematical Problem: 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). x y W - domain - boundary (or dw) Unknowns e.g. u(x,y,z) Independent Variables space and time (x,y,z,t) Dependent Variables unknown field (such as u)

3 Finite Element Analysis (FEA / FEM) The Mathematical Problem: 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)). x y W - domain - boundary (or dw) Unknowns e.g. u(x,y,z,t) Independent Variables space and time (x,y,z,t) Dependent Variables unknown field (such as u)

4 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.

5 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.

6 Finite Element Packages - Here are some of the common ones

7 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.

8 COMSOL Here we go!! Tutorial Created in Comsol 4.3 (2012) I will focus on acoustics as an application, but the steps are similar for other physics. 1. Decide on the representative physics (choose the PDE). 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. Pressure Acoustics is what we have been doing in ME139 this solves the Helmholtz equation for the complex acoustic pressure.

9 1. Decide on the representative physics (choose the PDE) : Choose Type of Study. 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)

10 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 r 0 and c to vary in space if desired w p p 0 r 0 r0 c Constant density Remember, since I have chosen Pressure Acoustics, I have selected timeharmonic acoustics time-harmonic means single frequency we are assuming time dependence e jwt. The pressure I solve for will be the complex pressure, p( x, y, z) p( x, y, z, t) Re p( x, y, z) e 1 prms ( x, y, z) p( x, y, z) 2 Solved for in Comsol jwt prms ( x, y, z) SPL( x, y, z) 20log10 p ref 2 2 w p p c Can be easily computed in post processing using post processing tools inside Comsol. 0

11 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. Various useful tools, depending on what you have selected in the model tree. Tools related to zooming, viewing, saving graphics objects. Geometry and various results plots will be shown in the main Graphics window.

12 2. Define the geometry on which to solve the problem. 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). 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). For complicated geometry you may choose to import it from a CAD program. If you select some objects in the graphics window then Boolean tools (like subtract, intersect, union) will also appear under geometry. 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.

13 2. Define the geometry on which to solve the problem. Drawing tools. Here are all the geometry objects I defined. Edit them by clicking and/or right clicking them in the model tree. For axisymmetric, the axis of symmetry will be r=0 and be drawn as a line in the graphics window. Objects appear in here.

14 3. Set the material properties that is, all the constants that appear in the PDE. For pressure acoustics, all that matters is r 0 and c. (And frequency although that is set under Study Frequency Domain, not under materials.) 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 Under materials select open material browser or + material. 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.

15 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.

16 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.

17 4. Set the boundary conditions 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. 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=p 0 (sets acoustic pressure amplitude remember, everything is oscillating as e jwt ) 4. Normal Acceleration Neumann condition since Euler says dp/dn=-r 0 a n (sets normal acceleration amplitude remember, everything is oscillating as e jwt ) 5. Impedance Condition set normal specific acoustic impedance z n at the boundary (z n =p/u n (dp/dn)=-r 0 jw p/z n ). This is how you could approximate an absorbing panel or something like that set z n 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.

18 4. Set the boundary conditions. Select the part of the boundary you want to apply that condition to and add it to the boundary selection. Note that boundary conditions can be functions of space use Matlab syntax so here I have set a n = 1 mm/s 2 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.

19 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 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.

20 4. Set the boundary conditions 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 m 3 /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/s 2 so the Comsol monopole strength is volume acceleration/unit volume = volume velocity/(jw)/unit volume. This has been checked.

21 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 R. White, convergence Comsol Acoustics study. After solving, make the mesh finer, re-solve, and make sure Introduction, the solution 2012 didn t change much.

22 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!!

23 7. Post-process the results to find the information you want. 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. 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. 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 Introduction, save my life. 11/1/10

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