Nystrom Method in Electromagnetics

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.

• About the Authors xiiiPreface xvAcknowledgment xxi1 Electromagnetics, Physics, and Mathematics 11.1 A Brief History of Electromagnetics 11.2 Enduring Legacy of Electromagnetic Theory–Why? 31.3 The Rise of Quantum Optics and Electromagnetics 41.3.1 Connection of Quantum Electromagnetics to Classical Electromagnetics 51.4 The Early Days – Descendent from Fluid Physics 61.5 The Complete Development of Maxwell’s Equations 71.5.1 Derivation of Wave Equation 91.6 Circuit Physics,Wave Physics, Ray Physics, and Plasmonic Resonances 101.6.1 Circuit Physics 101.6.2 Wave Physics 141.6.3 Ray Physics 151.6.4 Plasmonic Resonance 171.7 The Age of Closed Form Solutions 201.7.1 Separable Coordinate Systems 201.7.2 Integral Transform Solution 211.8 The Age of Approximations 231.8.1 Asymptotic Expansions 231.8.2 Matched Asymptotic Expansions 241.8.3 Ansatz-Based Approximations 271.9 The Age of Computations 281.9.1 Computations and Mathematics 301.9.2 Sobolev Space and Dual Space 331.10 Fast Algorithms 351.10.1 Cruelty of Computational Complexity 361.10.2 Curse of Dimensionality 381.10.3 Multiscale Problems 381.10.4 Fast Algorithm for Multiscale Problems 391.10.5 Domain Decomposition Methods 401.11 High Frequency Solutions 411.12 Inverse Problems 411.12.1 Distorted Born Iterative Method 421.12.2 Super-Resolution Reconstruction 431.12.3 Super-Resolution and the Weyl-Sommerfeld Identity 431.13 Metamaterials 461.14 Small Antennas 471.15 Conclusions 48Bibliography 492 Computational Electromagnetics 752.1 Introduction 752.2 Analytical Methods 772.3 Numerical Methods 822.3.1 The Finite-Difference Time-Domain (FDTD)Method 832.3.2 The Finite Element Method (FEM) 832.3.3 The Method of Moments (MoM) 842.4 Electromagnetic Integral Equations 872.4.1 Surface Integral Equations (SIEs) 882.4.2 Volume Integral Equations (VIEs) 912.4.3 Volume-Surface Integral Equations (VSIEs) 932.5 Summary 95Bibliography 953 The Nyström Method 993.1 Introduction 993.2 Basic Principle 1003.3 Singularity Treatment 1013.4 Higher-Order Scheme 1023.5 Comparison to the Method of Moments 1033.6 Comparison to the Point-Matching Method 1043.7 Summary 105Bibliography 1064 Numerical Quadrature Rules 1074.1 Introduction 1074.2 Definition and Design 1084.3 Quadrature Rules for a Segmental Mesh 1084.4 Quadrature Rules for a Surface Mesh 1094.4.1 Quadrature Rules for a Triangular Patch 1094.4.2 Quadrature Rules for a Square Patch 1124.5 Quadrature Rules for a Volumetric Mesh 1164.5.1 Quadrature Rules for a Tetrahedral Element 1164.5.2 Quadrature Rules for a Cuboid Element 1214.6 Summary 122Bibliography 1235 Singularity Treatment 1255.1 Introduction 1255.2 Singularity Subtraction 1265.2.1 Basic Principle 1265.2.2 Subtraction for the Kernel of ; Operator 1275.2.3 Subtraction for the Kernel of ; Operator 1305.2.4 Subtraction for the Kernels of VIEs 1325.3 Singularity Cancellation 1335.3.1 Surface Integral Equation 1345.3.2 Evaluation of the Weakly-Singular Integrals 1355.3.3 Numerical Examples 1385.4 Evaluation of Hypersingular and Weakly-Singular Integrals over Triangular Patches 1435.4.1 Hypersingular Integrals 1445.4.2 Weakly-Singular Integrals 1495.4.3 Non-Singular Integrals 1525.4.4 Numerical Examples 1545.5 Different Scheme for Evaluating Strongly-Singular and Hypersingular Integrals Over Triangular Patches 1545.5.1 Strongly-Singular and Hypersingular Integrals 1575.5.2 Stokes’ Theorem 1595.5.3 Derivation of New Formulas for HSIs and SSIs 1605.5.4 Numerical Tests 1645.5.5 Numerical Examples 1645.6 Evaluation of Singular Integrals Over Volume Domains 1675.6.1 Representation of Volume Current Density 1685.6.2 Evaluation of Singular Integrals 1695.6.3 Numerical Examples 1725.7 Evaluation of Near-Singular Integrals 1765.7.1 Integral Equations and Near-Singular Integrals 1775.7.2 Evaluation 1795.7.3 Numerical Examples 1855.8 Summary 187Bibliography 1886 Application to Conducting Media 1936.1 Introduction 1936.2 Solution for 2D Structures 1936.2.1 General 2D Structures 1946.2.2 2D Open Structures with Edge Conditions 1966.2.3 Evaluation of Singular and Near-Singular Integrations 1996.2.4 Numerical Examples 2046.3 Solution for Body-of-Revolution (BOR) Structures 2116.3.1 2D Integral Equations 2126.3.2 Evaluation of Singular Fourier Expansion Coefficients 2156.3.3 Numerical Examples 2196.4 Solutions of the Electric Field Integral Equation 2216.4.1 Higher-order Nyström method 2226.4.2 Numerical Examples 2256.5 Solutions of the Magnetic Field Integral Equation 2286.5.1 Integral Equations 2296.5.2 Singularity and Near-Singularity Treatment 2306.5.3 Numerical Examples 2336.6 Solutions of the Combined Field Integral Equation 2386.6.1 Integral Equations 2396.6.2 Quality of Triangular Patches 2406.6.3 Nyström Discretization 2416.6.4 Numerical Examples 2426.7 Summary 245Bibliography 2467 Application to Penetrable Media 2537.1 Introduction 2537.2 Surface Integral Equations for Homogeneous and Isotropic Media 2547.2.1 Surface Integral Equations 2547.2.2 Nyström Discretization 2597.2.3 Numerical Examples 2607.3 Volume Integral Equations for Homogeneous and Isotropic Media 2667.3.1 Volume Integral Equations 2687.3.2 Nyström Discretization 2687.3.3 Local Correction Scheme 2717.3.4 Numerical Examples 2747.4 Volume Integral Equations for Inhomogeneous or/and Anisotropic Media 2797.4.1 Volume Integral Equations 2807.4.2 Inconvenience of the Method of Moments 2827.4.3 Nyström Discretization 2837.4.4 Numerical Examples 2847.5 Volume Integral Equations for Conductive Media 2877.5.1 Volume Integral Equations 2897.5.2 Nyström Discretization 2907.5.3 Numerical Examples 2917.6 Volume-Surface Integral Equations for Mixed Media 2967.6.1 Volume-Surface Integral Equations 2987.6.2 Nyström-Based Mixed Scheme for Solving the VSIEs 2997.6.3 Numerical Examples 3017.7 Summary 306Bibliography 3098 Incorporation with Multilevel Fast Multipole Algorithm 3178.1 Introduction 3178.2 Multilevel Fast Multipole Algorithm 3188.3 Surface Integral Equations for Conducting Objects 3208.3.1 Integral Equations 3218.3.2 Nyström Discretization and MLFMA Acceleration 3218.3.3 Numerical Examples 3238.4 Surface Integral Equations for Penetrable Objects 3258.4.1 Integral Equations 3278.4.2 MLFMA Acceleration 3298.4.3 Numerical Examples 3318.5 Volume Integral Equations for Conductive Media 3358.5.1 Integral Equations 3368.5.2 Nyström Discretization 3378.5.3 Incorporation with the MLFMA 3388.5.4 Numerical Examples 3388.6 Volume-Surface Integral Equations for Conducting-Anisotropic Media 3428.6.1 Integral Equations for Anisotropic Objects 3438.6.2 Nyström Discretization 3448.6.3 MLFMA Acceleration 3458.6.4 Numerical Examples 3478.7 Summary 352Bibliography 3539 Application to Solve Multiphysics Problems 3579.1 Introduction 3579.2 Solution of Elastic Wave Problems 3599.2.1 Boundary Integral Equations 3599.2.2 Singularity Treatment 3629.2.3 Numerical Examples 3649.3 MLFMA Acceleration for Solve Large Elastic Wave Problems 3699.3.1 Formulations 3709.3.2 Reformulation of Near Terms 3759.3.3 Reduction of Number of Patterns 3779.3.4 Numerical Examples 3799.4 Solution of Acoustic Wave Problems with MLFMA Acceleration 3839.4.1 Implementation of the MLFMA for the Acoustic BIE 383Acoustic BIE 384Radiation and Receiving Patterns 384Near Terms 3859.4.2 Numerical Examples 3889.5 Unified Boundary Integral Equations for Elastic Wave and Acoustic Wave 3959.5.1 Elastic Wave BIEs 3979.5.2 Limit of Dyadic Green’s Function 3989.5.3 Vector BIE for Acoustic Wave 3999.5.4 Method of Moments (MoM) Solutions 4019.5.5 Numerical Examples 4039.6 Coupled Integral Equations for Electromagnetic Wave and Elastic Wave 4119.6.1 EM Wave Integral Equations 4129.6.2 Elastic Wave Integral Equations 4159.6.3 Coupled Integral Equations 4189.6.4 Solving Method 4209.6.5 Numerical Examples 4219.7 Summary 425Bibliography 42910 Application to Solve Time Domain Integral Equations 43710.1 Introduction 43710.2 Time Domain Surface Integral Equations for Conducting Media 43810.2.1 Time Domain Electric Field Integral Equation 438Formulations 439Numerical Solution 440Numerical Examples 44210.2.2 Time Domain Magnetic Field Integral Equation 446Formulations 447Numerical Solution 447Numerical Examples 44910.3 Time Domain Surface Integral Equations for Penetrable Media 45410.3.1 Formulations 45510.3.2 Numerical Solution 45610.3.3 Numerical Examples 45910.4 Time Domain Volume Integral Equations for Penetrable Media 46510.4.1 Formulations 46610.4.2 Numerical Solution 46710.4.3 Numerical Examples 47010.5 Time Domain Combined Field Integral Equations for Mixed Media 47610.5.1 Formulations 47610.5.2 Numerical Solution 47910.5.3 Numerical Examples 48410.6 Summary 488Bibliography 488Index 493

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