Title: Solving Maxwells Equations FAST
1Solving 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
2Overview
- Computational Electromagnetics
- Method of Moments
- Fast Algorithms
- Basis Functions and Integral Equation Operators
- Summary
Oil Well Industry Borehole ProblemComplex,
Layered-Media
3Notes on Introduction to CEM Courtesy of Prof.
Jian-Ming Jin (ECE 540)
4(No Transcript)
5DoD Applications
6Radar Cross Section (RCS)
7Stealth Technology
8Directed EM Radiation
9EM Interference and Compatibility
10Imaging
SAR Synthetic Aperture Radar
11Biomedical Imaging
MRI Magnetic Resonance Imaging
12EM Simulation
13Numerical Techniques in CEM
14CEM Recipe
15Cross Disciplinary
16Maxwells Equations
- Time Domain
- Vector Wave Equation (in time-harmonic form)
17Overview
- Computational Electromagnetics
- Method of Moments
- Fast Algorithms
- Basis Functions and Integral Equation Operators
- Summary
18Method 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
19Whats 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
20Greens Functions
- Consider Inhomogeneous scalar Helmholtz equation
- Linear superposition (Maxwells eqs. are linear)
- When
- Solution is the Greens function
21Formulate Problem (1/3)
- 2D Helmholtz equation
- Radiation condition (b.c.)
- Greens function is well known
22Formulate Problem (2/3)
- Integrate exterior region
- Apply second scalar Greens Theorem
23Formulate Problem (3/3)
- Select appropriate boundary condition on
- Consider Dirichlet boundary condition
(impenetrable)
24Expand 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
25Choice of Basis Functions (1/2)
26Choice of Basis Functions (2/2)
Vector Basis Functions
27Weight Residual
- Idea is to minimize residual error hence the
termweighted residual method
28Solve System
- Dense matrix system
- Simple solution
- Iterative solver
Still too costly!
29MoM Example
- Integral Equation Solution to
- EM scattering from an object
Einc, Hinc fields incident on object
Escat, Hscat fieldsscattered off object
Radar Cross Section
30Method of Moments (1/3)
N107
31Overview
- Computational Electromagnetics
- Method of Moments
- Fast Algorithms
- Basis Functions and Integral Equation Operators
- Summary
32Why Fast Solvers?--Computational Complexity
Computation time vs. Number of unknowns (CPU
100M Flops)
Ref 2 J. Jin
33Memory Complexity
Memory requirements vs. Number of unknowns
Ref 2 J. Jin
34Bottleneck in MoM
Only matrix-vector product matters If we have
the matrix-vector product, matrix is not
necessary
35Multi-Level Fast Multipole Algorithm (MLFMA)
- Accelerate Matrix-Vector Product
- Greens function describes interaction
- Bottleneck is iterative solver
Ref 2,3 W. Chew, V. Rokhlin
36Discretize Object
- Recursively divide object
- Group bases bybox in a tree-likestructure
2-D Quad Tree
3-D Oct Tree
37Decompose Interaction
- Recall 3D Greens function in free space
- Decompose r-r
- Factored Greens Function
Rokhlins Multipole Translator, 3
38Translate and Disaggregate Translator
39Overview
- Computational Electromagnetics
- Method of Moments
- Fast Algorithms
- Basis Functions and Integral Equation Operators
- Summary
40Summary
- 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)
41References
- 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.