MA557/MA578/CS557 Lecture 3 Spring 2003 Prof. Tim Warburton

[email protected]

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

1) PDE’s – Why Do We Care ?

Money: a) b) If you can modify a vehicle’s geometry to significantly reduce turbulent drag (race car, commercial airplane…) 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) b) Structural modeling Electromagnetics, acoustics, fluid dynamics… 4) Environment: a) Modeling environmental impact of those pesky greenhouse gases b) c) Modeling weather to avoid damage or to predict crop performance Predicting earthquakes, volcanic eruptions, tsunami (all belong in the “Money” section too?.

5) 6) Defense: a) b) Designing materials and profiles for stealth aircraft Nuclear weapon stockpile stewardship 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

t

F



q q

,

q

y

,

q

z

, 2

q

x

2 ,...

  • Example:

q



u

,

F



a



u



x

gives us 

u



t



a



u



x

 0 • 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….

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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}        

t



t H x H y

  

t E z

    

E z



y

   

E



x z



H y



x

 

H



y x

  

H x



x

    

H y



y

  0 8

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 1) Click on Rect 4) Here it is 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

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