Numerical Analysis for Electromagnetic Intergral Equations

by Karl F. Warnick

Overview

Improve EM simulation efforts fast with this applications-focused resource. This unique volume is the first book on integral equation-based methods that combines quantitative formulas for predicting numerical simulation accuracy together with rigorous error estimates and results for dozens of actual electromagnetics and wave propagation problems. You get the latest insights on accuracy-improving methods like regularization and error-increasing effects such as edge singularities and resonance, along with full details on how to determine mesh density, choice of basis functions, and other parameters needed to optimize any numerical simulation.

Bridging the gap between abstract academic treatments and the real-world needs of engineers, this timely work introduces various surface integral equation formulations, approaches to discretizing the integral equations, and measures of solution accuracy. It gives you numerical methods for 2D radiation and scattering problems, emphasizing concrete solution error bounds with exactly given constants. Moreover, the book provides techniques for higher order basis functions and 3D problems, focusing on smooth scatterers and edge singularity effects. This informative reference also explores problems involving resonant cavities and structures, and features a comprehensive treatment of resonant scatterers. The final chapter covers the convergence of the fast multipole method with iterative linear system solvers, complete with practical methods for improving the efficiency of iterative solutions.

Product Details

ISBN-13:	9781596933330
Publisher:	Artech House, Incorporated
Publication date:	09/30/2008
Series:	Artech House Electromagnetic Analysis
Pages:	219
Product dimensions:	6.10(w) x 9.20(h) x 0.70(d)

About the Author

Karl F. Warnick is an associate professor in the Department of Electrical Engineering, Brigham Young University in Utah, where he earned his Ph.D. in electrical engineering. He is also co-author of Problem Solving in Electromagnetics, Microwave Circuit, and Antenna Design for Communications Engineering (Artech House, 2006).

Weng Cho Chew is a professor in the Electrical and Computer Engineering Department at the University of Illinois at Urbana-Champaign, and serves as the dean of faculty of engineering at the University of Hong Kong. He earned his Ph.D. in electrical engineering at the Massachusetts Institute of Technology. He is the coauthor of two books, over 300 journal articles, and more than 400 conference papers.

Table of Contents

Preface xi

Chapter 1 Introduction 1

1.1 Approaches to Error Analysis 2

1.2 Empirical Methods 2

1.3 Sobolev Spaces and Asymptotic Error Estimates 3

1.3.1 Sobolev Norms 4

1.3.2 Sobolev Norms and the Scattering Amplitude 5

1.3.3 Static Limit 5

1.3.4 Quasioptimality and Approximation Error 6

1.3.5 Convergence Theorems 6

1.3.6 Limitations of Existing Error Estimates 7

1.4 Spectral Convergence Theory 8

1.4.1 Normal Operator Decomposition 8

1.4.2 Spectral Error 8

1.4.3 Error Contributions 10

1.4.4 Canonical Scattering Problems 11

References 12

Chapter 2 Surface Integral Equation Formulations and the Method of Moments 15

2.1 Electric Field Integral Equation 16

2.1.1 2D Scattering Problems 17

2.2 Magnetic Field Integral Equation 18

2.2.1 2D Scattering Problems 19

2.3 Combined Field Integral Equation 19

2.3.1 2D Scattering Problems 20

2.4 Method of Moments 20

2.4.1 Vector Basis Functions 22

2.5 Number of Unknowns 23

2.6 Scattering Amplitude, Scattering Width, and Radar Cross-section 24

2.7 Error Measures 26

2.8 Basic Concepts of Modal Error Analysis 28

2.8.1 Interpolation Error 28

2.8.2 Mesh Nyquist Frequency 30

2.8.3 Projection Error 30

References 32

Chapter 3 Error Analysis of the EFIE with W.C. Chew 33

3.1 TM-EFIE with Ideal Discretizations 34

3.1.1 Discretized Operator Spectrum 36

3.1.2 Comparison of the Discretized and Exact Operator Spectra 38

3.1.3 Spectral Error 40

3.1.4 Spectral Error for Low-Order Basis Functions 43

3.1.5 Current Solution Error 46

3.1.6 Scattering Amplitude Error 49

3.2 Variational Principles, the Moment Method, and Superconvergence 51

3.2.1 Superconvergence 54

3.2.2 IdealizingAssumptions 55

3.3 TM-EFIE with Nonideal Discretizations 56

3.3.1 Quadrature Error 56

3.3.2 Reducing Quadrature Error 58

3.3.3 Geometrical Discretization Error 59

3.4 TE-EFIE 62

3.4.1 Spectral Error 62

3.4.2 Spectral Error for Low-Order Basis Functions 63

3.4.3 Quadrature Error 65

3.4.4 Geometrical Discretization Error 66

3.5 Solution Error for Other Smooth Scatterers 67

3.6 Summary 69

References 70

Chapter 4 Error Analysis of the MFIE and CFIE with C. P. Davis 73

4.1 TM-MFIE with Ideal Discretizations 74

4.1.1 Operator Smoothing Properties 74

4.1.2 Discretized Operator Spectrum 75

4.1.3 Spectral Error 76

4.1.4 Current Solution Error 77

4.1.5 Current Error for the Point/Pulse Discretization 79

4.1.6 Scattering Amplitude Error 80

4.1.7 Scattering Amplitude Error for the Point/Pulse Discretization 82

4.2 Nonideal Discretizations 83

4.2.1 Quadrature Error 83

4.2.2 Single Integration Point 84

4.2.3 Geometrical Discretization Error 85

4.3 CFIE 86

4.4 Solution Error for Other Smooth Scatterers 87

4.5 Superconvergence and Regularization 88

4.5.1 Convergence Rates for EFIE and MFIE 89

4.5.2 Nonsuperconvergent Cases 93

4.5.3 First- and Second-Kind Operators 95

4.5.4 Higher-Order Basis Functions 96

4.5.5 High-Order Convergence with Low-Order Basis Functions 96

4.6 Summary 99

References 100

Chapter 5 Geometrical Singularities and the Flat Strip 101

5.1 Flat Strip Interior Error, TM-EFIE 101

5.1.1 Normal Operator Approximation 102

5.1.2 Discretized Operator Spectrum 104

5.1.3 Spectral Error 107

5.1.4 Spectral Error for Low-Order Basis Functions 108

5.1.5 The Magic "1/3" Discretization 109

5.1.6 Quadrature Error 110

5.1.7 Current Solution Error 110

5.1.8 Scattering Amplitude Error 114

5.2 Flat Strip Interior Error, TE-EFIE 114

5.2.1 Discretized Operator Spectrum 116

5.2.2 Spectral Error 117

5.2.3 Current Solution Error 118

5.2.4 Quadrature Error 119

5.3 Edge Error Analysis 121

5.4 Wedges 124

5.5 Summary 125

References 127

Chapter 6 Resonant Structures 129

6.1 Resonance and the EFIE Operator Spectrum 130

6.1.1 Quasi-Resonant Modes 131

6.1.2 Resonance and the Method of Moments 132

6.2 Internal Resonance 132

6.3 Cavities 134

6.3.1 Resonant Case 137

6.3.2 Near-Resonant Case 140

6.3.3 Spectral Error 141

6.3.4 Scattering Amplitude Error 143

6.4 Summary 145

References 145

Chapter 7 Error Analysis for 3D Problems 147

7.1 Flat Plate 148

7.1.1 Moment Matrix Spectrum 149

7.1.2 Rooftop Basis Functions 150

7.2 RWG Basis Functions 155

7.2.1 Nonideal Discretizations 158

7.3 Summary 160

References 160

Chapter 8 Higher-Order Basis Functions with A. F. Peterson 161

8.1 Higher-Order Basis Functions for 2D Problems 162

8.2 Interpolatory Polynomials 164

8.2.1 Discretized Operator Spectrum 164

8.2.2 Interpolation Transfer Function 166

8.2.3 Spectral Error 169

8.2.4 Current Solution Error 169

8.2.5 Scattering Amplitude Error 172

8.3 Orthogonal Polynomials 174

8.3.1 Discretized Operator Spectrum 174

8.3.2 Projection Transfer Function 175

8.3.3 Spectral Error 176

8.3.4 Current Solution Error 176

8.3.5 Scattering Amplitude Error 178

8.4 3D Problems 179

8.4.1 Numerical Results 181

8.5 Summary 184

References 185

Chapter 9 Operator Spectra and Iterative Solution Methods 187

9.1 Krylov Subspace Algorithms 188

9.1.1 CG Algorithm 189

9.1.2 CGNE and CGNR 189

9.1.3 Residual Error 190

9.1.4 Condition Number 191

9.1.5 Other Krylov Subspace Methods 192

9.2 Iteration Count Estimates 192

9.3 Condition Number Estimates 193

9.3.1 Circular Cylinder, TM-EFIE 193

9.3.2 Circular Cylinder, TE-EFIE 196

9.3.3 Flat Strip, TM-EFIE 196

9.3.4 Flat Strip, TE-EFIE 198

9.3.5 Parallel Strip Resonator 198

9.3.6 Higher-Order Basis Functions 201

9.3.7 Flat Plate-3D 203

9.4 Low-Frequency Breakdown 205

9.4.1 Helmholtz Decomposition 207

9.5 Preconditioners 208

9.6 Summary 210

References 210

About the Author 213

Index 215