Exploring Partial Differential Equations: Applications, Methods, and Solvers

1. MA557/MA578/CS557Lecture 3 Spring 2003 Prof. Tim Warburton timwar@math.unm.edu

2. 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….

3. PDE’s – Why Do We Care ? • Money: • If you can modify a vehicle’s geometry to significantly reduce turbulent drag (race car, commercial airplane…) • Modeling financial instruments (derivatives…) • Scientific curiosity: • Model’s of poorly understood physical phenomena (turbulence…) • Astrophysical models, solar models… • Engineering Applications: • Structural modeling • Electromagnetics, acoustics, fluid dynamics… • Environment: • Modeling environmental impact of those pesky greenhouse gases • Modeling weather to avoid damage or to predict crop performance • Predicting earthquakes, volcanic eruptions, tsunami (all belong in the “Money” section too?. • Defense: • 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) ?.

4. Some Time Dependent PDE • A typical PDE which is first order in time, and possibly higher order in space will have the general form: • Example: • We will see where these come from next lecture.

5. 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….

6. 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…..

7. 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 

8. 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}

9. Windows USEMe USEMe solvers by Tim Warburton USEMe gui by Nigel Nunn

10. Starting Up • Click on the WinUSEMe application

11. First screen

12. Click on Ellipse

13. First we build a circular far field (must be unit radius for the Hagstrom boundary conditions – current implementation)

14. Note the 32 node circle

15. Zoom in using right mouseand moving mouse

16. Next make a rectangle 4) Here it is 1) Click on Rect 2) Fill in rectangle details3) Press Apply

17. Make the rectangle a hole-- press Hole

18. Left mouse click inside the Rect

19. Now build a rectanglewhich has no associatedboundary conditions

20. 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 shouldbe set to material parameter=1

21. Next click on regionso we can set the regionmaterial properties

22. Pin the regions dialogue • Click in each material region

23. Edit the first region selected toset epsilon=9

24. Save the geometry by clicking “save as poly”

25. Click on Generate to make mesh

26. Save mesh by clicking on “write as neu”

27. Click on the “Solve” tab

28. Set the run directory by clicking on “Find”

29. Locate a .neu file in the run directory and click on it

30. Locate .neu file saved previouslyon pull-down menu and click on “Load”

31. Ready to set simulation parameters

32. Choose simulation type

33. Choose order of scheme

34. Click “Run” to start simulation

35. Field 0 (Hx) after a few time steps

36. Click on “Viz” tab

37. Change the number of nodes used for plotting

38. Click “Apply” to set resolution

39. Note nice and smooth fields

40. Choose “Colormap” to change contour ranges

41. Using left mouse can change viewpoint • Click on “Auto Z-scale” • Increasing Surface scale raises surface

42. Note RCS in right window Click on Window/Tile Vertical

43. Homework. Due on 01/27/03 • Master the WUM code – so that you are able to build a mesh with: • a) a plus sign shaped PECholeb) far field is far type unit circle (see next slide) • c) Make sure the Region is set to one • 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. • Read chapters 1 and 2 of Leveque • In a few weeks you will be able to code up the Maxwell’s solver yourself and prove it converges 