MA557/MA578/CS557
Lecture 3
Spring 2003
Prof. Tim Warburton
timwar@math.unm.edu
1
Week 1 (01/22/03, 01/24/03)
• Introduction to partial differential equations and their use.
• Examples of some applications for PDEs (acoustics,
electromagnetics, fluid dynamics ….. )
• Review of some basic notation and definitions for multivariate
calculus.
• Inner-products, norms, Sobolev spaces….
2
PDE’s – Why Do We Care ?
1)
Money:
a) If you can modify a vehicle’s geometry to significantly reduce turbulent drag (race
car, commercial airplane…)
b) Modeling financial instruments (derivatives…)
2)
Scientific curiosity:
a) Model’s of poorly understood physical phenomena (turbulence…)
b) Astrophysical models, solar models…
3)
Engineering Applications:
a) Structural modeling
b) Electromagnetics, acoustics, fluid dynamics…
4)
Environment:
a) Modeling environmental impact of those pesky greenhouse gases
b) Modeling weather to avoid damage or to predict crop performance
c) Predicting earthquakes, volcanic eruptions, tsunami (all belong in the “Money”
section too?.
5)
Defense:
a) Designing materials and profiles for stealth aircraft
b) Nuclear weapon stockpile stewardship
6)
Discussion…. what else comes to mind – also how would you rank the relevant
importance of the above (and how well do you think each area is funded) ?.
3
Some Time Dependent PDE
•
A typical PDE which is first order in time, and possibly higher order in
space will have the general form:

q
q q q  2q 
 F  x, y, z, t , q, , , , 2 ,... 
t
x y z x


q  u, F  a
u
u
u
gives us
a
0
x
t
x
•
Example:
•
We will see where these come from next lecture.
4
Commonly Used Numerical Methods
•
•
•
•
•
Finite difference
Finite volume
Finite element
hp-finite element
Spectral methods
•
•
•
•
Boundary elements
Numerical Greens function methods
Fast multipole methods
Meshfree methods
Each has its own practical range of operation….
5
Industry Solvers
• The state of the art in industrial solvers has evolved PDE
solvers into word processor like technology (to some degree).
• It is now possible to apply some of the previous methods to
PDEs entered with math formulae (i.e. not computer code).
• A few clicks will now allow an engineer to solve extremely
complex problems
• But…..
6
Your Turn To Solve a PDE
•
Download:
– http://www.useme.org/WUM_v5.zip
– Or
– http://www.math.unm.edu/~timwar/WUM_v5.zip
– Or – grab a spare cd-rom and copy the WUM_v5.zip file
– Save it to the desktop and double click on it.
– When you have unzip’d the file indicate that you are done.
– We will now go through an insane sequence to simulate
Maxwell’s equations in a two-dimensional domain 
7
2D Transverse Magnetic
Mode Maxwell’s Equations
•
•
We are going to solve the following equations to obtain
Hx,Hy,Ez as coupled functions of time and space.
We will specify that:
Hx(t=0,x,y)=Hy(t=0,x,y)=Ez(t=0,x,y)=0
•
•
We also specify that no electric or magnetic fields travel
inwards from the limit of large (x,y)
All boundaries we create will be perfectly electrically
conducting (superconducting) where Ez=0 and (Hx,Hy) is
tangential to the boundary.
•
We will specify epsilon (whereas mu=1 by default)
•
We have now specified the PDEs, the initial conditions and
sufficient boundary conditions to allow us to solve for
{Hx(t,x,y),Hy(t,x,y),Ez(t,x,y), t>=0}
Hx 
t
H y 
t
   Ez 
t


Hx 
x
Ez

y

Ez
x
H y
x

H x
y
H y 
y
8
0
Windows USEMe
USEMe solvers by Tim Warburton
USEMe gui by Nigel Nunn
9
Starting Up
• Click on the WinUSEMe application
10
First screen
11
Click on Ellipse
12
First we build a circular far field
(must be unit radius for the Hagstrom
boundary conditions – current implementation)
13
Note the 32 node circle
14
Zoom in using right mouse
and moving mouse
15
Next make a rectangle
4) Here it is
1) Click on Rect
2) Fill in rectangle details
3) Press Apply
16
Make the rectangle a hole
-- press Hole
17
Left mouse click inside
the Rect
18
Now build a rectangle
which has no associated
boundary conditions
19
Maxwell’s Hagstrom Module
• This module is able to simulate variable
epsilon Maxwell’s…
• We need to click on each region and
specify the epsilon for that region
• The region including the far field should
be set to material parameter=1
20
Next click on region
so we can set the region
material properties
21
1) Pin the regions dialogue
2) Click in each material region
22
Edit the first region selected to
set epsilon=9
23
Save the geometry by clicking “save as poly”
24
Click on Generate to make mesh
25
Save mesh by clicking on “write as neu”
26
Click on the “Solve” tab
27
Set the run directory by clicking on “Find”
28
Locate a .neu file in the
run directory and click on it
29
Locate .neu file saved previously
on pull-down menu and click on “Load”
30
Ready to set simulation parameters
31
Choose simulation type
32
Choose order of scheme
33
Click “Run” to start simulation
34
Field 0 (Hx) after a few time steps
35
Click on “Viz” tab
36
Change the number of nodes
used for plotting
37
Click “Apply” to
set resolution
38
Note nice and smooth fields
39
Choose “Colormap” to
change contour ranges
40
Using left mouse can
change viewpoint
1) Click on “Auto Z-scale”
2) Increasing Surface scale raises surface
41
Note RCS in
right window
Click on Window/Tile Vertical
42
Homework. Due on 01/27/03
1) Master the WUM code – so that you are able to build
a mesh with:
a) a plus sign shaped PEC hole
b) far field is far type unit circle (see next slide)
c) Make sure the Region is set to one
2) Run the code for 15 units and print out a snap shot of the
results (use alt-print scrn and paste into Powerpoint). Repeat
this for different orders. Generally experiment.
3) Read chapters 1 and 2 of Leveque
4) In a few weeks you will be able to code up the Maxwell’s
solver yourself and prove it converges 
43
44