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The Method of Moments in Electromagnetics Walton C. Gibson http://www.tripointindustries.com kalla@tripoint.org

Chapman & Hall/CRC Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487‑2742 © 2008 by Taylor & Francis Group, LLC Chapman & Hall/CRC is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid‑free paper 10 9 8 7 6 5 4 3 2 1 International Standard Book Number‑13: 978‑1‑4200‑6145‑1 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the conse‑ quences of their use. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www. copyright.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978‑750‑8400. CCC is a not‑for‑profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Gibson, Walton C. p. cm. The method of moments in electromagnetics / Walton C. Gibson. Includes bibliographical references and index. ISBN 978‑1‑4200‑6145‑1 (alk. paper) 1. Electromagnetism‑‑Data processing. 2. Electromagnetic fields‑‑Mathematical models. 3. Moments method (Statistics) 4. Electromagnetic theory‑‑Data processing. 5. Integral equations‑‑Numerical solutions. I. Title. QC665.E4.G43 2008 530.14’1015118‑‑dc22 2007037311 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com C614X_FM.indd 2 10/23/07 9:35:42 AM

Contents Preface Acknowledgments About the Author Chapter 1 Computational Electromagnetics 1.1 Computational Electromagnetics Algorithms 1.1.1 Low-Frequency Methods 1.1.2 High-Frequency Methods References Chapter 2 A Brief Review of Electromagnetics 2.1 Maxwell’s Equations 2.2 Electromagnetic Boundary Conditions 2.3 Formulations for Radiation 2.3.1 Three-Dimensional Green’s Function 2.3.2 Two-Dimensional Green’s Function 2.4 Vector Potentials 2.4.1 Magnetic Vector Potential 2.4.2 Electric Vector Potential 2.4.3 Comparison of Radiation Formulas 2.5 Near and Far Fields 2.5.1 Near Field 2.5.2 Far Field 2.6 Equivalent Problems 2.6.1 2.6.2 Surface Equivalent Physical Equivalent 2.7 Surface Integral Equations 2.7.1 Electric Field Integral Equation 2.7.2 Magnetic Field Integral Equation 2.7.3 Combined Field Integral Equation iii ix xiii xv 1 1 2 2 4 5 5 6 6 8 9 10 11 12 13 14 15 16 18 18 20 25 25 26 28

iv Contents References Chapter 3 The Method of Moments 3.1 Electrostatic Problems 3.1.1 Charged Wire 3.1.2 Charged Plate 3.2 The Method of Moments Point Matching 3.2.1 3.2.2 Galerkin’s Method 3.3 Common Two-Dimensional Basis Functions Pulse Functions Piecewise Triangular Functions Piecewise Sinusoidal Functions 3.3.1 3.3.2 3.3.3 3.3.4 Entire-Domain Functions 3.3.5 Number of Basis Functions 3.4 Solution of Matrix Equations 3.4.1 Gaussian Elimination 3.4.2 LU Decompositon 3.4.3 Condition Number 3.4.4 Iterative Methods 3.4.5 Examples 3.4.6 Commonly Used Matrix Algebra Software References 30 33 33 34 39 43 44 44 45 45 45 46 47 47 48 48 50 52 53 57 58 61 Chapter 4 Thin Wires 4.6 Examples 4.1 Thin Wire Approximation 4.2 Thin Wire Excitations 4.2.1 Delta-Gap Source 4.2.2 Magnetic Frill 4.2.3 Plane Wave 4.3 Solving Hall´en’s Equation 4.3.1 Symmetric Problems 4.3.2 Asymmetric Problems 4.4 Solving Pocklington’s Equation 63 63 65 65 66 67 68 69 71 72 73 73 74 Solution Using Triangle Basis and Testing Functions 75 Solution Using Sinusoidal Basis and Testing Functions 77 78 79 79 83 86 87 Solution by Pulse Functions and Point Matching 4.5.1 Redistribution of EFIE Differential Operators 4.5.2 4.5.3 4.5.4 Lumped and Distributed Impedances 4.6.1 Comparison of Thin Wire Models 4.6.2 Circular Loop Antenna 4.6.3 Folded Dipole Antenna 4.6.4 Two-Wire Transmission Line 4.4.1 4.5 Thin Wires of Arbitrary Shape

Contents 4.6.5 Matching a Yagi Antenna References Chapter 5 Two-Dimensional Problems 5.1 Two-Dimensional EFIE 5.1.1 EFIE for a Strip: TM Polarization 5.1.2 Generalized EFIE: TM Polarization 5.1.3 EFIE for a Strip: TE Polariation 5.1.4 Generalized EFIE: TE Polarization 5.2 Two-Dimensional MFIE 5.2.1 MFIE: TM Polarization 5.2.2 MFIE: TE Polarization 5.3 Examples 5.3.1 5.3.2 References Scattering by an Inﬁnite Cylinder: TM Polarization Scattering by an Inﬁnite Cylinder: TE Polarization Chapter 6 Bodies of Revolution 6.1 BOR Surface Description 6.2 Surface Current Expansion on a BOR 6.3 EFIE for a Conducting BOR 6.3.1 EFIE Matrix Elements 6.3.2 Excitation 6.3.3 Scattered Field 6.4 MFIE for a Conducting BOR 6.4.1 MFIE Matrix Elements 6.4.2 Excitation 6.4.3 Scattered Field 6.5 Notes on Software Implementation 6.5.1 Parallelization 6.5.2 Convergence 6.6 Examples 6.6.1 Galaxy 6.6.2 Conducting Sphere 6.6.3 EMCC Benchmark Targets 6.6.4 Biconic Reentry Vehicle 6.6.5 References Summary of Examples Chapter 7 Three-Dimensional Problems 7.1 Representation of Three-Dimensional Surfaces 7.2 Surface Currents on a Triangle 7.2.1 Edge Finding Algorithm 7.3 EFIE for Three-Dimensional Conducting Surfaces 7.3.1 EFIE Matrix Elements v 89 94 95 95 95 100 102 107 109 109 111 113 113 115 124 125 125 126 127 127 130 134 136 137 140 141 141 141 142 142 142 142 145 152 159 159 161 161 164 165 167 167

vi Contents Singular Matrix Element Evaluation 7.3.2 7.3.3 EFIE Excitation Vector Elements 7.3.4 Radiated Field 7.4 MFIE for Three-Dimensional Conducting Surfaces 7.4.1 MFIE Matrix Elements 7.4.2 MFIE Excitation Vector Elements 7.4.3 Radiated Field 7.4.4 Accuracy of RWG Functions in MFIE 7.5 Notes on Software Implementation 7.5.1 Memory Management 7.5.2 Parallelization 7.6 Considerations for Modeling with Triangles 7.6.1 Triangle Aspect Ratios 7.6.2 Watertight Meshes and T-Junctions 7.7 Examples Serenity 7.7.1 7.7.2 RCS of a Sphere 7.7.3 EMCC Plate Benchmark Targets 7.7.4 7.7.5 Bowtie Antenna 7.7.6 Archimedean Spiral Antenna 7.7.7 References Summary of Examples Strip Dipole Antenna Chapter 8 The Fast Multipole Method 8.1 The Matrix-Vector Product 8.2 Addition Theorem 8.2.1 Wave Translation 8.3 FMM Matrix Elements 8.3.1 EFIE Matrix Elements 8.3.2 MFIE Matrix Elements 8.3.3 CFIE Matrix Elements 8.3.4 Matrix Transpose 8.4 One-Level Fast Multipole Algorithm 8.4.1 Grouping of Basis Functions 8.4.2 Near and Far Groups 8.4.3 Number of Multipoles 8.4.4 8.4.5 Transfer Functions 8.4.6 Radiation and Receive Functions 8.4.7 Near-Matrix Elements 8.4.8 Matrix-Vector Product 8.4.9 Computational Complexity Sampling Rates and Integration 8.5 Multi-Level Fast Multipole Algorithm (MLFMA) 8.5.1 Grouping via Octree 168 176 178 179 179 184 184 184 185 185 185 187 187 188 188 189 189 189 198 199 201 204 205 209 210 210 212 213 213 214 215 215 215 215 216 216 218 219 220 220 221 222 222 222

Contents Interpolation Algorithms 8.5.2 Matrix-Vector Product 8.5.3 8.5.4 Transfer Functions 8.5.5 Radiation and Receive Functions 8.5.6 8.5.7 Computational Complexity 8.6 Notes on Software Implementation Interpolation Steps in MLFMA Initial Guess in Iterative Solution 8.6.1 8.6.2 Memory Management 8.6.3 Parallelization 8.6.4 Vectorization 8.7 Preconditioning 8.7.1 Diagonal Preconditioner 8.7.2 Block Diagonal Preconditioner 8.7.3 8.7.4 Inverse LU Preconditioner Sparse Approximate Inverse 8.8 Examples 8.8.1 Bistatic RCS of a Sphere 8.8.2 EMCC Benchmark Targets 8.8.3 References Summary of Examples Chapter 9 Integration 9.1 One-Dimensional Integration 9.2 9.1.1 Centroidal Approximation 9.1.2 Trapezoidal Rule 9.1.3 Simpson’s Rule 9.1.4 One-Dimensional Gaussian Quadrature Integration over Triangles 9.2.1 9.2.2 Radiation Integrals with a Constant Source 9.2.3 Radiation Integrals with a Linear Source 9.2.4 Gaussian Quadrature on Triangles Simplex Coordinates References Index vii 223 227 229 230 230 231 231 231 232 234 234 235 235 236 236 237 240 240 240 245 252 255 255 255 256 258 259 260 260 262 265 267 269 271

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