Preface ........................................................ ix
Acknowledgments .............................................. xiii
About the Author ............................................... xv
Chapter 1 Computational Electromagnetics ........................ 1
1.1 Computational Electromagnetics Algorithms .................. 1
1.1.1 Low-Frequency Methods ............................... 2
1.1.2 High-Frequency Methods .............................. 2
References ................................................. 4
Chapter 2 A Brief Review of Electromagnetics .................... 5
2.1 Maxwell's Equations ........................................ 5
2.2 Electromagnetic Boundary Conditions ........................ 6
2.3 Formulations for Radiation ................................. 6
2.3.1 Three-Dimensional Green's Function .................. 8
2.3.2 Two-Dimensional Green's Function .................... 9
2.4 Vector Potentials ......................................... 10
2.4.1 Magnetic Vector Potential .......................... 11
2.4.2 Electric Vector Potential .......................... 12
2.4.3 Comparison of Radiation Formulas ................... 13
2.5 Near and Far Fields ....................................... 14
2.5.1 Near Field ......................................... 15
2.5.2 Far Field .......................................... 16
2.6 Equivalent Problems ....................................... 18
2.6.1 Surface Equivalent ................................. 18
2.6.2 Physical Equivalent ................................ 20
2.7 Surface Integral Equations ................................ 25
2.7.1 Electric Field Integral Equation ................... 25
2.7.2 Magnetic Field Integral Equation ................... 26
2.7.3 Combined Field Integral Equation ................... 28
2.7.1 References ......................................... 30
Chapter 3 The Method of Moments ............................... 33
3.1 Electrostatic Problems .................................... 33
3.1.1 Charged Wire ....................................... 34
3.1.2 Charged Plate ...................................... 39
3.2 The Method of Moments ..................................... 43
3.2.1 Point Matching ..................................... 44
3.2.2 Galerkin's Method .................................. 44
3.3 Common Two-Dimensional Basis Functions .................... 45
3.3.1 Pulse Functions .................................... 45
3.3.2 Piecewise Triangular Functions ..................... 45
3.3.3 Piecewise Sinusoidal Functions ..................... 46
3.3.4 Entire-Domain Functions ............................ 47
3.3.5 Number of Basis Functions .......................... 47
3.4 Solution of Matrix Equations .............................. 48
3.4.1 Gaussian Elimination ............................... 48
3.4.2 LU Decompositon .................................... 50
3.4.3 Condition Number ................................... 52
3.4.4 Iterative Methods .................................. 53
3.4.5 Examples ........................................... 57
3.4.6 Commonly Used Matrix Algebra Software .............. 58
References ................................................ 61
Chapter 4 Thin Wires .......................................... 63
4.1 Thin Wire Approximation ................................... 63
4.2 Thin Wire Excitations ..................................... 65
4.2.1 Delta-Gap Source ................................... 65
4.2.2 Magnetic Frill ..................................... 66
4.2.3 Plane Wave ......................................... 67
4.3 Solving Hallen's Equation ................................. 68
4.3.1 Symmetric Problems ................................. 69
4.3.2 Asymmetric Problems ................................ 71
4.4 Solving Pocklington's Equation ............................ 72
4.4.1 Solution by Pulse Functions and Point Matching ..... 73
4.5 Thin Wires of Arbitrary Shape ............................. 73
4.5.1 Redistribution of EFIE Differential Operators ...... 74
4.5.2 Solution Using Triangle Basis and Testing
Functions .......................................... 75
4.5.3 Solution Using Sinusoidal Basis and Testing
Functions .......................................... 77
4.5.4 Lumped and Distributed Impedances .................. 78
4.6 Examples .................................................. 79
4.6.1 Comparison of Thin Wire Models ..................... 79
4.6.2 Circular Loop Antenna .............................. 83
4.6.3 Folded Dipole Antenna .............................. 86
4.6.4 Two-Wire Transmission Line ......................... 87
4.6.5 Matching a Yagi Antenna ............................ 89
References ................................................ 94
Chapter 5 Two-Dimensional Problems ............................ 95
5.1 Two-Dimensional EFIE ...................................... 95
5.1.1 EFIE for a Strip: TM Polarization .................. 95
5.1.2 Generalized EFIE: TM Polarization ................. 100
5.1.3 EFIE for a Strip: ТЕ Polariation .................. 102
5.1.4 Generalized EFIE: ТЕ Polarization ................. 107
5.2 Two-Dimensional MFIE ..................................... 109
5.2.1 MFIE: TM Polarization ............................. 109
5.2.2 MFIE: ТЕ Polarization ............................. 111
5.3 Examples ................................................. 113
5.3.1 Scattering by an Infinite Cylinder: TM
Polarization ...................................... 113
5.3.2 Scattering by an Infinite Cylinder: ТЕ
Polarization ...................................... 115
References ............................................... 124
Chapter 6 Bodies of Revolution ............................... 125
6.1 BOR Surface Description .................................. 125
6.2 Surface Current Expansion on a BOR ....................... 126
6.3 EFIE for a Conducting BOR ................................ 127
6.3.1 EFIE Matrix Elements .............................. 127
6.3.2 Excitation ........................................ 130
6.3.3 Scattered Field ................................... 134
6.4 MFIE for a Conducting BOR ................................ 136
6.4.1 MFIE Matrix Elements .............................. 137
6.4.2 Excitation ........................................ 140
6.4.3 Scattered Field ................................... 141
6.5 Notes on Software Implementation ......................... 141
6.5.1 Parallelization ................................... 141
6.5.2 Convergence ....................................... 142
6.6 Examples ................................................. 142
6.6.1 Galaxy ............................................ 142
6.6.2 Conducting Sphere ................................. 142
6.6.3 EMCC Benchmark Targets ............................ 145
6.6.4 Biconic Reentry Vehicle ........................... 152
6.6.5 Summary of Examples ............................... 159
References ............................................... 159
Chapter 7 Three-Dimensional Problems ......................... 161
7.1 Representation of Three-Dimensional Surfaces ............. 161
7.2 Surface Currents on a Triangle ........................... 164
7.2.1 Edge Finding Algorithm ............................ 165
7.3 EFIE for Three-Dimensional Conducting Surfaces ........... 167
7.3.1 EFIE Matrix Elements .............................. 167
7.3.2 Singular Matrix Element Evaluation ................ 168
7.3.3 EFIE Excitation Vector Elements ................... 176
7.3.4 Radiated Field .................................... 178
7.4 MFIE for Three-Dimensional Conducting Surfaces ........... 179
7.4.1 MFIE Matrix Elements .............................. 179
7.4.2 MFIE Excitation Vector Elements ................... 184
7.4.3 Radiated Field .................................... 184
7.4.4 Accuracy of RWG Functions in MFIE ................. 184
7.5 Notes on Software Implementation ......................... 185
7.5.1 Memory Management ................................. 185
7.5.2 Parallelization ................................... 185
7.6 Considerations for Modeling with Triangles ............... 187
7.6.1 Triangle Aspect Ratios ............................ 187
7.6.2 Watertight Meshes and T-Junctions ................. 188
7.7 Examples ................................................. 188
7.7.1 Serenity .......................................... 189
7.7.2 RCS of a Sphere ................................... 189
7.7.3 EMCC Plate Benchmark Targets ...................... 189
7.7.4 Strip Dipole Antenna .............................. 198
7.7.5 Bowtie Antenna .................................... 199
7.7.6 Archimedean Spiral Antenna ........................ 201
7.7.7 Summary of Examples ............................... 204
References ............................................... 205
Chapter 8 The Fast Multipole Method .......................... 209
8.1 The Matrix-Vector Product ................................ 210
8.2 Addition Theorem ......................................... 210
8.2.1 Wave Translation .................................. 212
8.3 FMM Matrix Elements ...................................... 213
8.3.1 EFIE Matrix Elements .............................. 213
8.3.2 MFIE Matrix Elements .............................. 214
8.3.3 CFIE Matrix Elements .............................. 215
8.3.4 Matrix Transpose .................................. 215
8.4 One-Level Fast Multipole Algorithm ....................... 215
8.4.1 Grouping of Basis Functions ....................... 215
8.4.2 Near and Far Groups ............................... 216
8.4.3 Number of Multipoles .............................. 216
8.4.4 Sampling Rates and Integration .................... 218
8.4.5 Transfer Functions ................................ 219
8.4.6 Radiation and Receive Functions ................... 220
8.4.7 Near-Matrix Elements .............................. 220
8.4.8 Matrix-Vector Product ............................. 221
8.4.9 Computational Complexity .......................... 222
8.5 Multi-Level Fast Multipole Algorithm (MLFMA) ............. 222
8.5.1 Grouping via Octree ............................... 222
8.5.2 Matrix-Vector Product ............................. 223
8.5.3 Interpolation Algorithms .......................... 227
8.5.4 Transfer Functions ................................ 229
8.5.5 Radiation and Receive Functions ................... 230
8.5.6 Interpolation Steps in MLFMA ...................... 230
8.5.7 Computational Complexity .......................... 231
8.6 Notes on Software Implementation ......................... 231
8.6.1 Initial Guess in Iterative Solution ............... 231
8.6.2 Memory Management ................................. 232
8.6.3 Parallelization ................................... 234
8.6.4 Vectorization ..................................... 234
8.7 Preconditioning .......................................... 235
8.7.1 Diagonal Preconditioner ........................... 235
8.7.2 Block Diagonal Preconditioner ..................... 236
8.7.3 Inverse LU Preconditioner ......................... 236
8.7.4 Sparse Approximate Inverse ........................ 237
8.8 Examples ................................................. 240
8.8.1 Bistatic RCS of a Sphere .......................... 240
8.8.2 EMCC Benchmark Targets ............................ 240
8.8.3 Summary of Examples ............................... 245
References ............................................... 252
Chapter 9 Integration ........................................ 255
9.1 One-Dimensional Integration .............................. 255
9.1.1 Centroidal Approximation .......................... 255
9.1.2 Trapezoidal Rule .................................. 256
9.1.3 Simpson's Rule .................................... 258
9.1.4 One-Dimensional Gaussian Quadrature ............... 259
9.2 Integration over Triangles ............................... 260
9.2.1 Simplex Coordinates ............................... 260
9.2.2 Radiation Integrals with a Constant Source ........ 262
9.2.3 Radiation Integrals with a Linear Source .......... 265
9.2.4 Gaussian Quadrature on Triangles .................. 267
References ............................................... 269
Index ......................................................... 271
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