Solving Maxwells Equations FAST

1
Solving Maxwells Equations FAST!

• Capt. Michael A. Saville
• Center for Computational Electromagnetics and
  Electromagnetics LaboratoryUniversity of
  Illinois at Urbana-Champaign
• CS 598 Calculus on MeshesOctober 20, 2005

Special Acknowledgements to Professors W. C. Chew
and J. Jin of the CCEML and their associated
graduate students for images and slides
throughout this presentation. The views expressed
in this presentation are those of the author and
do not reflect the official policy or position of
the United States Air Force, Department of
Defense, or the U.S. Government
2
Overview

• Computational Electromagnetics
• Method of Moments
• Fast Algorithms
• Basis Functions and Integral Equation Operators
• Summary

Oil Well Industry Borehole ProblemComplex,
Layered-Media
3
Notes on Introduction to CEM Courtesy of Prof.
Jian-Ming Jin (ECE 540)
4
(No Transcript)
5
DoD Applications
6
Radar Cross Section (RCS)
7
Stealth Technology
8
Directed EM Radiation
9
EM Interference and Compatibility
10
Imaging
SAR Synthetic Aperture Radar
11
Biomedical Imaging
MRI Magnetic Resonance Imaging
12
EM Simulation
13
Numerical Techniques in CEM
14
CEM Recipe
15
Cross Disciplinary
16
Maxwells Equations

• Time Domain
• Vector Wave Equation (in time-harmonic form)

17
Overview

• Computational Electromagnetics
• Method of Moments
• Fast Algorithms
• Basis Functions and Integral Equation Operators
• Summary

18
Method of Moments

• Integral equation solver
• Also known as
• Boundary element method
• Method of weighted residuals
• Four basic steps
• 1. Formulate Problem as integral equation
• 2. Expand unknown in a basis
• 3. Test, or weight, the residual
• 4. Solve the ensuing matrix equation

19
Whats an integral equation?

• Integral operator acts on the unknown
• Greens function can be difficult to derive, but
  is a powerful tool

Integral equation of the 1st kind
Integral equation of the 2nd kind
20
Greens Functions

• Consider Inhomogeneous scalar Helmholtz equation
• Linear superposition (Maxwells eqs. are linear)
• When
• Solution is the Greens function

21
Formulate Problem (1/3)

• 2D Helmholtz equation
• Radiation condition (b.c.)
• Greens function is well known

22
Formulate Problem (2/3)

• Integrate exterior region
• Apply second scalar Greens Theorem

23
Formulate Problem (3/3)

• Select appropriate boundary condition on
• Consider Dirichlet boundary condition
  (impenetrable)

24
Expand Unknown

• First stage of discretization basis is tied to
  geometry
• Select vn(r) according to physics of the problem
• Continuity of current, frequency, simplicity of
  implementation

Current distribution
Cone with a groove
Wire Antenna
25
Choice of Basis Functions (1/2)
26
Choice of Basis Functions (2/2)
Vector Basis Functions
27
Weight Residual

• Idea is to minimize residual error hence the
  termweighted residual method

28
Solve System

• Dense matrix system
• Simple solution
• Iterative solver

Still too costly!
29
MoM Example

• Integral Equation Solution to
• EM scattering from an object

Einc, Hinc fields incident on object
Escat, Hscat fieldsscattered off object
Radar Cross Section
30
Method of Moments (1/3)
N107
31
Overview

• Computational Electromagnetics
• Method of Moments
• Fast Algorithms
• Basis Functions and Integral Equation Operators
• Summary

32
Why Fast Solvers?--Computational Complexity
Computation time vs. Number of unknowns (CPU
100M Flops)
Ref 2 J. Jin
33
Memory Complexity
Memory requirements vs. Number of unknowns
Ref 2 J. Jin
34
Bottleneck in MoM
Only matrix-vector product matters If we have
the matrix-vector product, matrix is not
necessary
35
Multi-Level Fast Multipole Algorithm (MLFMA)

• Accelerate Matrix-Vector Product
• Greens function describes interaction
• Bottleneck is iterative solver

Ref 2,3 W. Chew, V. Rokhlin
36
Discretize Object

• Recursively divide object
• Group bases bybox in a tree-likestructure

2-D Quad Tree
3-D Oct Tree
37
Decompose Interaction

• Recall 3D Greens function in free space
• Decompose r-r
• Factored Greens Function

Rokhlins Multipole Translator, 3
38
Translate and Disaggregate Translator

• To the board

39
Overview

• Computational Electromagnetics
• Method of Moments
• Fast Algorithms
• Basis Functions and Integral Equation Operators
• Summary

40
Summary

• Linear equations facilitate Greens function
  solution
• Practical applications require fast algorithms
  (MLFMA)
• Judicious choice of basis function may be
  derivable from discrete exterior calculus
• Discrete exterior calculus may enable error
  analysis of integro-differential operators (EFIE,
  MFIE)

41
References

• 1 J. Jin,, Finite Element Method in
  Computational Electromagnetics, John Wiley
  Sons, Inc., 1999.
• 2 W.C. Chew, J. Jin, E. Michielssen, and J.
  Song, Fast and Efficient Algorithms in
  Computational Electromagnetics, Artech House,
  Inc., 2000.
• 3 V. Rokhlin, Rapid solution of integral
  equations of scattering theory in two
  dimensions, J. Comput. Phys., vol. 86, no. 2,
  pp. 414-439, Feb 1990.
• Additional References
• R. Coifman, V. Rokhlin, and S. Wandzura, The
  fast multipole method for the wave equation A
  pedestrian prescription, IEEE Ant. Prop. Mag.,
  vol. 35, pp. 7-12, June 1993.
• C.C. Lu and W.C. Chew, "A multilevel algorithm
  for solving boundary integral equation of
  scattering," Micro. Opt. Tech. Lett., vol. 7,
  no. 10, pp.466-470, July 1994.
• J.M. Song and W.C. Chew, "Multilevel fast
  multipole algorithm for solving combined field
  integral equation of electromagnetic scattering,"
  Mico. Opt. Tech. Lett., vol. 10, no. 1, pp 14-19,
  September 1995.
• J.M. Song, C.-C. Lu, and W.C. Chew, "MLFMA for
  electromagnetic scatteringby large complex
  objects," IEEE Trans. Ant. Propag., vol. 45, no.
  10, pp. 1488-1493, October 1997.