The Nystrom Method in Electromagnetics

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.

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.

1 Electromagnetics, Physics, and Mathematics 1

1.1 A Brief History of Electromagnetics 1

1.2 Enduring Legacy of Electromagnetic Theory Why? 2

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 10

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

1.6.1 Circuit Physics 11

1.6.2 Wave Physics 14

1.6.3 Ray Physics 16

1.6.4 Plasmonic Resonance 18

1.7 The Age of Closed Form Solutions 21

1.7.1 Separable Coordinate Systems 21

1.7.2 Integral Transform Solution 22

1.8 The Age of Approximations 23

1.8.1 Asymptotic Expansions 24

1.8.2 Matched Asymptotic Expansions 25

1.8.3 Ansatz-based Approximations 28

1.9 The Age of Computations 29

1.9.1 Computations and Mathematics 31

1.9.2 Sobolev Space and Dual Space 35

1.10 Fast Algorithms 37

1.10.1 Cruelty of Computational Complexity 38

1.10.2 Curse of Dimensionality 40

1.10.3 Multiscale Problems 40

1.10.4 Fast Algorithm for Multiscale Problems 41

1.10.5 Domain Decomposition Methods 42

1.11 High Frequency Solutions 43

1.12 Inverse Problems 43

1.12.1 Distorted Born Iterative Method 44

1.12.2 Super-Resolution Reconstruction 45

Nystrom Method

1.12.3 Super-resolution and Weyl-Sommerfeld Identity 46

1.13 Metamaterials 48

1.14 Small Antennas 49

1.15 Conclusions 51

Chapter 2 Computational Electromagnetics

2.1 Analytical Methods

2.2 Numerical Methods

2.3 Matrix Equation Solvers

2.4 Electromagnetic Integral Equations

2.4.1 Surface Integral Equations (SIEs)

2.4.2 Volume Integral Equations (VIEs)

2.4.3 Volume-Surface Integral Equations (VSIEs)

2.5 Summary

Bibliography

Chapter 3 Nystrom Method

3.1 Basic Principle

3.2 Singularity Treatment

3.3 Higher-Order Scheme

3.3.1 Geometry mapping for Curvilinear Meshes

3.3.2 Singularity Treatment in Curvilinear Meshes

3.4 Comparison to the Method of Moments

3.5 Summary

Bibliography

Chapter 4 Numerical Quadrature Rules

4.1 Definition and Design

4.2 Quadrature Rules for a Segmental Mesh

4.2 Quadrature Rules for a Segmental Mesh

4.3.1 Quadrature Rules for a Triangular Patch

4.3.2 Quadrature Rules for a Square Patch

4.4 Quadrature Rules for a Volumetric Mesh

4.4.1 Quadrature Rules for a Tetrahedral Element

4.4.2 Quadrature Rules for a Cuboid Element

4.5 Summary

Bibliography

Chapter 5 Singularity Treatment

5.1 Singularity Subtraction

5.1.1 Process

5.1.2 Subtraction for the Kernel of EFIE of SIEs

5.1.3 Subtraction for the Kernel of MFIE of SIEs

5.1.4 Subtraction for the Kernels of VIEs

5.2 Singularity Cancellation

5.3 Surface Integral Equations

5.4 Novel Approach for Evaluating the Weakly Singular Integrals

5.5 Numerical Examples

5.6 Evaluation of Hypersingular Integrals over Triangular Patches

5.6.1 Numerical Examples

5.7 Di_erent Scheme for Evaluating Strongly-Singular and Hypersingular Integrals over Triangular Patches

5.7.1 SIEs and Singularity Subtraction

5.7.2 Stokes' Theorem

5.7.3 Derivation of New Formulas for HSIs and SSIs

5.7.4 Numerical Tests

5.7.5 Numerical Examples

5.8 Evaluation of Singular Integrals over Volume Do-Mains

5.8.1 Representation of Volume Current Density

5.8.2 Evaluation of Singular Integrals

5.8.3 Numerical Examples

5.9 Evaluation of Near-Singular Integrals

5.9.1 Integral Equations and Singularity Subtraction

5.9.2 Evaluation of NI Elements

5.9.3 Numerical Examples

5.10 Summary

Bibliography

Chapter 6 Application to Conducting Media

6.1 Solution for 2D Structures

6.1.1 2D Concave Structures

6.1.2 2D Open Structures with Edge Conditions

6.1.3 Evaluation of singular and near-singular integrations

6.1.4 Numerical Examples

6.2 Solution for Body-of-Revolution Structures

6.2.1 2D Integral Equation

6.2.2 Evaluation of Singular Fourier Coe_cients

6.2.3 Numerical Examples

6.3 Solution of Electric Field Integral Equation

6.3.1 Higher-order Nystrom method

6.3.2 Numerical Examples

6.4 Solution of Magnetic Field Integral Equation

6.4.1 Integral Equations

6.4.2 Near-Singularity Treatment

6.4.3 Numerical Examples

6.5 Solution of Combined Field Integral Equation

6.5.2 Quality of Triangular Patches

6.5.3 Nystrom Solutions

6.5.4 Numerical Examples

6.6 Summary

Bibliography

Chapter 7 Application to Penetrable Media

7.1. SURFACE INTEGRAL EQUATIONS FOR HOMOGENEOUS AND ISOTROPIC MEDIA261

7.1.2 Nystrom Discretization

7.1.3 Numerical Examples

7.2 Volume Integral Equations for Homogeneous and Isotropic Media

7.2.1 Volume Integral Equations

7.2.2 Nystrom Discretization

7.2.3 Local Correction Scheme

7.2.4 Numerical Examples

7.3 Volume Integral Equations for Inhomogeneous or/and Anisotropic Media

7.3.1 Volume Integral Equations

7.3.2 Inconvenience of the Method of Moments

7.3.3 Nystrom Discretization

7.3.4 Numerical Examples

7.4 Volume Integral Equations for Conductive Media

7.4.1 Volume Integral Equations

7.4.2 The Nystrom Method

7.4.3 Numerical Examples

7.5 Volume-Surface Integral Equations for Mixed Media

7.5.1 Volume-Surface Integral Equations (VSIEs)

7.5.2 Nystrom-Based Mixed Scheme for Solving the VSIEs

7.5.3 Numerical Examples

7.6 Summary

Bibliography

Chapter 8 Incorporation with Multilevel Fast Multipole Algorithm

8.1 Multilevel Fast Multipole Algorithm

8.2 Surface Integral Equations for Conducting Objects

8.2.1 Integral Equations

8.2.2 Nystrom Discretization and MLFMA Acceleration

8.2.3 Numerical Examples

8.3 Surface Integral Equations for Penetrable Objects

8.3.1 Integral Equations

8.3.2 MLFMA Acceleration

8.3.3 Numerical Examples

8.4 Volume Integral Equations for Conductive Media

8.4.1 Integral Equations

8.4.2 Nystrom Discretization

8.4.3 Incorporation with the MLFMA

8.4.4 Numerical Examples

8.5 Volume-Surface Integral Equations for Conducting-Anisotropic Media

8.5.1 Integral Equations for Anisotropic Objects

8.5.2 Nystrom Discretization

8.5.3 MLFMA Acceleration

8.5.4 Numerical Examples

8.6 Summary

Bibliography

Chapter 9 Application to Solve Multiphysics Problems

9.1 Solution of Elastic Wave Problems

9.1.1 Boundary Integral Equations

9.1.2 Singularity Treatment

9.1.3 Numerical Examples

9.2 MLFMA Acceleration for Solving Large ElasticWave

Problems

9.2.1 Formulations

9.2.2 Reformulation of Near Terms

9.2.3 Reduction of Number of Patterns

9.2.4 Numerical Examples

9.3 Solution of Acoustic Wave Problems with an MLFMA Acceleration

9.4 Implementation of MLFMA for Acoustic BIE

9.4.1 Acoustic BIE

9.4.2 Radiation and Receiving Patterns

9.4.3 Nystrom Method

9.4.4 Near Terms

9.5 Numerical Results

9.6 Uni_ed Boundary Integral Equation for ElasticWave and Acoustic Wave

9.6.1 Elastic Wave BIE

9.6.2 Limit of Dyadic Green's Function

9.6.3 Vector BIE for Acoustic Wave

9.6.4 Method of Moments (MoM)

9.6.5 Numerical Results

9.7 Coupled Integral Equations for MicrowaveWave and Elastic Wave

9.7.1 EM Wave Integral Equation

9.7.2 Elastic Wave Integral Equation

9.7.3 Coupled Integral Equation

9.7.4 Solving Method

9.7.5 Numerical Examples

9.8 Summary

Bibliography

Chapter 10 Application to Solve Time-Domain Integral Equations

10.1 Time-Domain Surface Integral Equations for Conducting Media

10.1.1 Time-Domain Electric Field Integral Equation

10.1.2 Time-Domain Magnetic Field Integral Equation

10.2 Time-Domain Surface Integral Equations for Penetrable Media

10.2.1 Formulations

10.2.2 Numerical Solution

10.2.3 Numerical Examples

10.3 Time-Domain Volume Integral Equations for Penetrable Media

10.3.1 Formulations

10.3.2 Numerical Solution

10.3.3 Numerical Examples

10.4 Time-Domain Combined Field Integral Equations for Mixed Media

10.4.1 Formulations

10.4.2 Numerical Solution

10.4.3 Numerical Examples

10.5 Summary

Bibliography

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