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The Nystrom Method in Electromagnetics

Tong, Mei Song / Chew, Weng Cho

Wiley - IEEE

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1. Auflage Oktober 2020
528 Seiten, Hardcover
Praktikerbuch

ISBN: 978-1-119-28484-0
John Wiley & Sons

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A comprehensive, step-by-step reference to the Nyström Method for solving Electromagnetic problems using integral equations

Computational electromagnetics studies the numerical methods or techniques that solve electromagnetic problems by computer programming. Currently, there are mainly three numerical methods for electromagnetic problems: the finite-difference time-domain (FDTD), finite element method (FEM), and integral equation methods (IEMs). In the IEMs, the method of moments (MoM) is the most widely used method, but much attention is being paid to the Nyström method as another IEM, because it possesses some unique merits which the MoM lacks. This book focuses on that method--providing information on everything that students and professionals working in the field need to know.

Written by the top researchers in electromagnetics, this complete reference book is a consolidation of advances made in the use of the Nyström method for solving electromagnetic integral equations. It begins by introducing the fundamentals of the electromagnetic theory and computational electromagnetics, before proceeding to illustrate the advantages unique to the Nyström method through rigorous worked out examples and equations. Key topics include quadrature rules, singularity treatment techniques, applications to conducting and penetrable media, multiphysics electromagnetic problems, time-domain integral equations, inverse scattering problems and incorporation with multilevel fast multiple algorithm.
* Systematically introduces the fundamental principles, equations, and advantages of the Nyström method for solving electromagnetic problems
* Features the unique benefits of using the Nyström method through numerical comparisons with other numerical and analytical methods
* Covers a broad range of application examples that will point the way for future research

The Nystrom Method in Electromagnetics is ideal for graduate students, senior undergraduates, and researchers studying engineering electromagnetics, computational methods, and applied mathematics. Practicing engineers and other industry professionals working in engineering electromagnetics and engineering mathematics will also find it to be incredibly helpful.

About the Authors xiii

Preface xv

Acknowledgment xxi

1 Electromagnetics, Physics, and Mathematics 1

1.1 A Brief History of Electromagnetics 1

1.2 Enduring Legacy of Electromagnetic Theory-Why? 3

1.3 The Rise of Quantum Optics and Electromagnetics 4

1.3.1 Connection of Quantum Electromagnetics to Classical Electromagnetics 5

1.4 The Early Days - Descendent from Fluid Physics 6

1.5 The Complete Development of Maxwell's Equations 7

1.5.1 Derivation of Wave Equation 9

1.6 Circuit Physics,Wave Physics, Ray Physics, and Plasmonic Resonances 10

1.6.1 Circuit Physics 10

1.6.2 Wave Physics 14

1.6.3 Ray Physics 15

1.6.4 Plasmonic Resonance 17

1.7 The Age of Closed Form Solutions 20

1.7.1 Separable Coordinate Systems 20

1.7.2 Integral Transform Solution 21

1.8 The Age of Approximations 23

1.8.1 Asymptotic Expansions 23

1.8.2 Matched Asymptotic Expansions 24

1.8.3 Ansatz-Based Approximations 27

1.9 The Age of Computations 28

1.9.1 Computations and Mathematics 30

1.9.2 Sobolev Space and Dual Space 33

1.10 Fast Algorithms 35

1.10.1 Cruelty of Computational Complexity 36

1.10.2 Curse of Dimensionality 38

1.10.3 Multiscale Problems 38

1.10.4 Fast Algorithm for Multiscale Problems 39

1.10.5 Domain Decomposition Methods 40

1.11 High Frequency Solutions 41

1.12 Inverse Problems 41

1.12.1 Distorted Born Iterative Method 42

1.12.2 Super-Resolution Reconstruction 43

1.12.3 Super-Resolution and the Weyl-Sommerfeld Identity 43

1.13 Metamaterials 46

1.14 Small Antennas 47

1.15 Conclusions 48

Bibliography 49

2 Computational Electromagnetics 75

2.1 Introduction 75

2.2 Analytical Methods 77

2.3 Numerical Methods 82

2.3.1 The Finite-Difference Time-Domain (FDTD)Method 83

2.3.2 The Finite Element Method (FEM) 83

2.3.3 The Method of Moments (MoM) 84

2.4 Electromagnetic Integral Equations 87

2.4.1 Surface Integral Equations (SIEs) 88

2.4.2 Volume Integral Equations (VIEs) 91

2.4.3 Volume-Surface Integral Equations (VSIEs) 93

2.5 Summary 95

Bibliography 95

3 The Nyström Method 99

3.1 Introduction 99

3.2 Basic Principle 100

3.3 Singularity Treatment 101

3.4 Higher-Order Scheme 102

3.5 Comparison to the Method of Moments 103

3.6 Comparison to the Point-Matching Method 104

3.7 Summary 105

Bibliography 106

4 Numerical Quadrature Rules 107

4.1 Introduction 107

4.2 Definition and Design 108

4.3 Quadrature Rules for a Segmental Mesh 108

4.4 Quadrature Rules for a Surface Mesh 109

4.4.1 Quadrature Rules for a Triangular Patch 109

4.4.2 Quadrature Rules for a Square Patch 112

4.5 Quadrature Rules for a Volumetric Mesh 116

4.5.1 Quadrature Rules for a Tetrahedral Element 116

4.5.2 Quadrature Rules for a Cuboid Element 121

4.6 Summary 122

Bibliography 123

5 Singularity Treatment 125

5.1 Introduction 125

5.2 Singularity Subtraction 126

5.2.1 Basic Principle 126

5.2.2 Subtraction for the Kernel of Operator 127

5.2.3 Subtraction for the Kernel of Operator 130

5.2.4 Subtraction for the Kernels of VIEs 132

5.3 Singularity Cancellation 133

5.3.1 Surface Integral Equation 134

5.3.2 Evaluation of the Weakly-Singular Integrals 135

5.3.3 Numerical Examples 138

5.4 Evaluation of Hypersingular and Weakly-Singular Integrals over Triangular Patches 143

5.4.1 Hypersingular Integrals 144

5.4.2 Weakly-Singular Integrals 149

5.4.3 Non-Singular Integrals 152

5.4.4 Numerical Examples 154

5.5 Different Scheme for Evaluating Strongly-Singular and Hypersingular Integrals Over Triangular Patches 154

5.5.1 Strongly-Singular and Hypersingular Integrals 157

5.5.2 Stokes' Theorem 159

5.5.3 Derivation of New Formulas for HSIs and SSIs 160

5.5.4 Numerical Tests 164

5.5.5 Numerical Examples 164

5.6 Evaluation of Singular Integrals Over Volume Domains 167

5.6.1 Representation of Volume Current Density 168

5.6.2 Evaluation of Singular Integrals 169

5.6.3 Numerical Examples 172

5.7 Evaluation of Near-Singular Integrals 176

5.7.1 Integral Equations and Near-Singular Integrals 177

5.7.2 Evaluation 179

5.7.3 Numerical Examples 185

5.8 Summary 187

Bibliography 188

6 Application to Conducting Media 193

6.1 Introduction 193

6.2 Solution for 2D Structures 193

6.2.1 General 2D Structures 194

6.2.2 2D Open Structures with Edge Conditions 196

6.2.3 Evaluation of Singular and Near-Singular Integrations 199

6.2.4 Numerical Examples 204

6.3 Solution for Body-of-Revolution (BOR) Structures 211

6.3.1 2D Integral Equations 212

6.3.2 Evaluation of Singular Fourier Expansion Coefficients 215

6.3.3 Numerical Examples 219

6.4 Solutions of the Electric Field Integral Equation 221

6.4.1 Higher-order Nyström method 222

6.4.2 Numerical Examples 225

6.5 Solutions of the Magnetic Field Integral Equation 228

6.5.1 Integral Equations 229

6.5.2 Singularity and Near-Singularity Treatment 230

6.5.3 Numerical Examples 233

6.6 Solutions of the Combined Field Integral Equation 238

6.6.1 Integral Equations 239

6.6.2 Quality of Triangular Patches 240

6.6.3 Nyström Discretization 241

6.6.4 Numerical Examples 242

6.7 Summary 245

Bibliography 246

7 Application to Penetrable Media 253

7.1 Introduction 253

7.2 Surface Integral Equations for Homogeneous and Isotropic Media 254

7.2.1 Surface Integral Equations 254

7.2.2 Nyström Discretization 259

7.2.3 Numerical Examples 260

7.3 Volume Integral Equations for Homogeneous and Isotropic Media 266

7.3.1 Volume Integral Equations 268

7.3.2 Nyström Discretization 268

7.3.3 Local Correction Scheme 271

7.3.4 Numerical Examples 274

7.4 Volume Integral Equations for Inhomogeneous or/and Anisotropic Media 279

7.4.1 Volume Integral Equations 280

7.4.2 Inconvenience of the Method of Moments 282

7.4.3 Nyström Discretization 283

7.4.4 Numerical Examples 284

7.5 Volume Integral Equations for Conductive Media 287

7.5.1 Volume Integral Equations 289

7.5.2 Nyström Discretization 290

7.5.3 Numerical Examples 291

7.6 Volume-Surface Integral Equations for Mixed Media 296

7.6.1 Volume-Surface Integral Equations 298

7.6.2 Nyström-Based Mixed Scheme for Solving the VSIEs 299

7.6.3 Numerical Examples 301

7.7 Summary 306

Bibliography 309

8 Incorporation with Multilevel Fast Multipole Algorithm 317

8.1 Introduction 317

8.2 Multilevel Fast Multipole Algorithm 318

8.3 Surface Integral Equations for Conducting Objects 320

8.3.1 Integral Equations 321

8.3.2 Nyström Discretization and MLFMA Acceleration 321

8.3.3 Numerical Examples 323

8.4 Surface Integral Equations for Penetrable Objects 325

8.4.1 Integral Equations 327

8.4.2 MLFMA Acceleration 329

8.4.3 Numerical Examples 331

8.5 Volume Integral Equations for Conductive Media 335

8.5.1 Integral Equations 336

8.5.2 Nyström Discretization 337

8.5.3 Incorporation with the MLFMA 338

8.5.4 Numerical Examples 338

8.6 Volume-Surface Integral Equations for Conducting-Anisotropic Media 342

8.6.1 Integral Equations for Anisotropic Objects 343

8.6.2 Nyström Discretization 344

8.6.3 MLFMA Acceleration 345

8.6.4 Numerical Examples 347

8.7 Summary 352

Bibliography 353

9 Application to Solve Multiphysics Problems 357

9.1 Introduction 357

9.2 Solution of Elastic Wave Problems 359

9.2.1 Boundary Integral Equations 359

9.2.2 Singularity Treatment 362

9.2.3 Numerical Examples 364

9.3 MLFMA Acceleration for Solve Large Elastic Wave Problems 369

9.3.1 Formulations 370

9.3.2 Reformulation of Near Terms 375

9.3.3 Reduction of Number of Patterns 377

9.3.4 Numerical Examples 379

9.4 Solution of Acoustic Wave Problems with MLFMA Acceleration 383

9.4.1 Implementation of the MLFMA for the Acoustic BIE 383

Acoustic BIE 384

Radiation and Receiving Patterns 384

Near Terms 385

9.4.2 Numerical Examples 388

9.5 Unified Boundary Integral Equations for Elastic Wave and Acoustic Wave 395

9.5.1 Elastic Wave BIEs 397

9.5.2 Limit of Dyadic Green's Function 398

9.5.3 Vector BIE for Acoustic Wave 399

9.5.4 Method of Moments (MoM) Solutions 401

9.5.5 Numerical Examples 403

9.6 Coupled Integral Equations for Electromagnetic Wave and Elastic Wave 411

9.6.1 EM Wave Integral Equations 412

9.6.2 Elastic Wave Integral Equations 415

9.6.3 Coupled Integral Equations 418

9.6.4 Solving Method 420

9.6.5 Numerical Examples 421

9.7 Summary 425

Bibliography 429

10 Application to Solve Time Domain Integral Equations 437

10.1 Introduction 437

10.2 Time Domain Surface Integral Equations for Conducting Media 438

10.2.1 Time Domain Electric Field Integral Equation 438

Formulations 439

Numerical Solution 440

Numerical Examples 442

10.2.2 Time Domain Magnetic Field Integral Equation 446

Formulations 447

Numerical Solution 447

Numerical Examples 449

10.3 Time Domain Surface Integral Equations for Penetrable Media 454

10.3.1 Formulations 455

10.3.2 Numerical Solution 456

10.3.3 Numerical Examples 459

10.4 Time Domain Volume Integral Equations for Penetrable Media 465

10.4.1 Formulations 466

10.4.2 Numerical Solution 467

10.4.3 Numerical Examples 470

10.5 Time Domain Combined Field Integral Equations for Mixed Media 476

10.5.1 Formulations 476

10.5.2 Numerical Solution 479

10.5.3 Numerical Examples 484

10.6 Summary 488

Bibliography 488

Index 493
Mei Song Tong, PhD, is currently a Distinguished Professor, Chair of the Department of Electronic Science and Technology, and Vice-Dean of the College of Microelectronics, Tongji University, Shanghai, China.

Weng Cho Chew, PhD, is a Distinguished Professor at the School of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana, USA, and a Fellow of IEEE, OSA, IOP, and HKIE.

W. C. Chew, University of Illinois, Urbana-Champaign