Post on 06-Feb-2018
R. White, Comsol Acoustics
Introduction, 2/25/08
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).
domain
- boundary
(or )
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)
R. White, Comsol Acoustics
Introduction, 2/25/08
Finite Element Analysis (FEA / FEM)
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)).
domain
- boundary
(or )
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)
R. White, Comsol Acoustics
Introduction, 2/25/08
Finite Element Analysis (FEA / FEM)
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.
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
1. Decide on the representative physics (choose the PDE).
2. Define the geometry on which to solve the problem.
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.
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.
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 ejt.
0
2
2 =
+ p
cp
0
112
00
=
+
p
cp
Constant
density
R. White, Comsol Acoustics
Introduction, 2/25/08
2. Define the geometry on which to solve the problem.
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
R. White, Comsol Acoustics
Introduction, 2/25/08
GO BUILD YOURSELF SOME GEOMETRY! DONT 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
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.
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
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=-0an5. Impedance Condition set Z at the boundary (Z=p/un=-0jp/(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.
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 dont 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.
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 =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.
R. White, Comsol Acoustics
Introduction, 2/25/08
6. Choose a solver and solve for the unknowns.
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.
R. White, Comsol Acoustics
Introduction, 2/25/08
6. Choose a solver and solve for the unknowns.
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.
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 lets 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.
R. White, Comsol Acoustics
Introduction, 2/25/08
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 dont
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.
The Postprocessing |
Domain Plot
Parameters menu
lets 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.
R. White, Comsol Acoustics
Introduction, 2/25/08
I have defined the dB SPL here
using a Matlab expression
involving the solved-for complex
pressure, p.
7. Post-process the results to find the information you want.
I am looking at the
solution at two points
that I created using the
Draw menu.
R. White, Comsol Acoustics
Introduction, 2/25/08
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!
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.
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).
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
R. White, Comsol Acoustics
Introduction, 2/25/08
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
SOLVE AND PLAY WITH THE RESULTS