Computational Electrodynamics
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1 Computational Electrodynamics The Finite-Difference Time-Domain Method 1 Allen Taflove Artech House Boston London
2 Contents Preface Chapter 1 Reinventing Electromagnetics 1.1 Background The Heritage of the 1980s The Rise of Partial Differential Equation Methods Interdisciplinary Impact of Emerging Time-Domain PDE Solvers 1.2 History of Space-Grid Time-Domain Techniques for Maxwell's Equations 1.3 General Characteristics of Space-Grid Time-Domain Approaches Classes of FD-TD and FV-TD Algorithms Predictive Dynamic Range 1.4 Scaling to Very Large Problem Sizes Algorithm Scaling Factors Computer Architecture Scaling Factors 1.5 Defense Applications RCS Modeling of Entire Aircraft to 1 GHz Desirable Additional Modeling Capabilities for Defense Applications 1.6 Dual-Use Electromagnetics Technology Antenna Design Bioelectromagnetic Systems 1: Hyperthermia Treatment of Cancer Bioelectromagnetic Systems 2: Physics of Human Vision Digital Circuit Packaging and Interconnects Incorporation of Models of Active Circuit Devices Subpicosecond Photonic Devices 1.7 Conclusions Supplemental Reading References Chapter 2 The One-Dimensional Scalar Wave Equation 2.1 Introduction 2.2 Propagating-Wave Solutions 2.3 Finite Differences 2.4 Finite-Difference Approximation of the Scalar Wave Equation 2.5 Dispersion Relations for the One-Dimensional Wave Equation 2.6 Numerical Phase Velocity Case 1: Very Fine Mesh Case 2: Magic Time Step Case 3: Dispersive Wave Propagation 2.7 Numerical Group Velocity Case 1: Very Fine Mesh Case 2: Magic Time Step 2.8 Numerical Stability The Time Eigenvalue Problem The Space Eigenvalue Problem Enforcement of Stability Problems vi l
3 Chapter 3 Introduction to Maxwell's Equations and the Yee Algorithm Introduction Maxwell's Equations in Three Dimensions Reduction to Two Dimensions TMMode ТЕ Mode Reduction to One Dimension TMMode ТЕ Mode Equivalence to the Wave Equation in One Dimension The Yee Algorithm Basic Ideas Finite Differences and Notation Finite-Difference Expressions for Maxwell's Equations in Three Dimensions Space Region with a Continuous Variation of Material Properties Space Region with a Finite Number of Distinct Media Space Region with Nonpermeable Media Reduction to the Two-Dimensional TM and ТЕ Modes Interpretation as Faraday's and Ampere's Laws in Integral Form Divergence-Free Nature Exponential Time-Stepping 77 References 79 Problems 79 Chapter 4 Numerical Stability Introduction Basic Stability Analysis Procedure, TM Mode Time Eigenvalue Problem Space Eigenvalue Problem Enforcement of Stability Extension to the Full Three-Dimensional Yee Algorithm Generalized Stability Problem Boundary Conditions Variable and Unstructured Meshing Lossy, Dispersive, Nonlinear, and Gain Materials 91 Reference 92 Bibliography on Stability of Finite-Difference Methods 92 Problems 92 Chapter 5 Numerical Dispersion Introduction Basic Procedure Substitution of Traveling-Wave Trial Solutions Extension to the Full Three-Dimensional Yee Algorithm Comparison with the Ideal Dispersion Case Reduction to the Ideal Dispersion Case for Special Grid Conditions Dispersion-Optimized Basic Yee Algorithm Dispersion-Optimized Yee Algorithm wim Fourth-Order Accurate Spatial Central Differences Formulation 102 vni
4 5.8.2 Example Pros and Cons 104 Problems 105 Chapter 6 Incident Wave Source Conditions for Free Space and Waveguides Introduction Requirements for the Plane Wave Source Condition Insertion of the Incident Wave As an Initial Condition The Hard Source Total-Field/Scattered-Field Formulation Ideas One-Dimensional Example Two-Dimensional Example Calculation of the Incident Field to Implement the Connecting Condition Extension to Three Dimensions Pure Scattered-Field Formulation Application to PEC Structures Application to Lossy Dielectric Structures Choice of Incident Plane Wave Formulation Waveguide Source Conditions Pulsed Electric Field Hard Source Total-Field/Reflected-Field Formulation Resistive Source and Load Conditions 141 References 142 Problems 142 Chapter 7 Absorbing Boundary Conditions for Free Space and Waveguides Introduction Bayliss-Turkel Scattered-Wave Annihilating Operators Spherical Coordinates Cylindrical Coordinates Engquist-Majda One-Way Wave Equations One-Term and Two-Term Taylor Series Approximations Mur Finite-Difference Scheme Trefethen-Halpern Generalized and Higher-Order ABCs Theoretical Reflection Coefficient Analysis Numerical Experiments Higdon Operator Liao Extrapolation Mei-Fang Superabsorption Berenger Perfectly Matched Layer Theory, Two-Dimensional ТЕ Case Theory, Two-Dimensional TM Case Theory, Full-Vector Three-Dimensional Case Numerical Experiments Verifying the Effectiveness of PML ABC Absorbing Boundary Conditions for Waveguides 195 References 198 Problems 201 ix
5 Chapter 8 Near-to-Far-Field Transformation Introduction Two-Dimensional Transformation, Phasor Domain Application of Green's Theorem Far-Field Limit Reduction to Standard Form Obtaining Phasor Quantities Via Discrete Fourier Transformation Surface Equivalence Theorem Extension to Three Dimensions, Phasor Domain Time-Domain Near-to-Far-Field Transformation 219 References 224 Problems 225 Chapter 9 Dispersive, Nonlinear, and Gain Materials Introduction Recursive Convolution Method, Linear Isotropic Case Total-Field Formulation Scattered-Field Formulation Discussion Recursive Convolution Method, Linear Gyrotropic Case Time-Domain Susceptibility Functions Time-Stepping Algorithm Discussion Auxiliary Differential Equation Method, Linear Isotropic Case Introduction: The Single-Relaxation Dispersion, Initial Formulation Multiple Lorentzian Relaxations, Refined Formulation Comparison with the Recursive Convolution Method Auxiliary Differential Equation Method, Nonlinear Lorentz Media (Nonlinear Optics) Governing Equations Equivalent System of Ordinary Differential Equations Algorithm Results for Temporal Solitons Results for Spatial Solitons Discussion Auxiliary Differential Equation Method, Lorentz Gain Media (Active Lasing Materials) Theory and Numerical Implementation Generic Validations Discussion 274 References 274 Problems 278 Chapter 10 Local Subcell Models of Fine Geometrical Features Introduction Basis of Contour-Path FD-TD Modeling The Simplest Contour-Path Subcell Models Diagonal Split-Cell Model for PEC Surfaces Average Properties Model for Material Surfaces The Contour-Path Model of the Narrow Slot The Thin Wire 288
6 10.6 Conformal Modeling of Curved Surfaces Two-Dimensional PEC Structures, ТЕ Case Two-Dimensional PEC Structures, TM Case Illustrative Results, Aluminum Winglike Object Three-Dimensional PEC Structures Two-Dimensional Material Structures The Thin Material Sheet Basis Illustrative Results Dispersive Surface Impedance Maloney-Smith Method Beggs et al. Method Lee et al. Method Relativistic Motion of PEC Boundaries Basis Illustrative Results Numerical Stability 338 References 338 Additional Bibliography 341 Problems 342 Chapter 11 Explicit Time-Domain Solution of Maxwell's Equations Using Nonorthogonal and Unstructured Grids Introduction Nonuniform Orthogonal Grids Locally Conformal Grids, Globally Orthogonal Global Curvilinear Coordinates Nonorthogonal Curvilinear FD-TD Algorithm Stability Criterion Irregular Nonorthogonal Structured Grids Irregular Nonorthogonal Unstructured Grids Generalized Yee Algorithm Inhomogeneous Media Practical Implementation of the Generalized Yee Algorithm A Planar Generalized Yee Algorithm Time-Stepping Expressions Projection Operators Efficient Time-Stepping Implementation Analysis of Printed Circuit Devices Using the Planar Generalized Yee Algorithm 382 References 390 Problems 393 Chapter 12 The Body of Revolution FD-TD Algorithm Introduction Field Expansion Difference Equations for Off-Axis Cells The e r Patch Integral The e. Patch Integral The e z Patch Integral Difference Equations 407 xi
7 Surface-Conforming Patch Integrals Difference Equations for On-Axis Cells The е г Patch Integral Near the Axis The e f Patch Integral Near the Axis The ft r Patch Integral Near the Axis Numerical Stability PML Absorbing Boundary Condition BOR FD-TD Background Extension of PML to the BOR Waveguide Case Examples Application to Particle Accelerator Physics Definitions and Concepts Examples 426 References 429 Problems 430 Chapter 13 Modeling of Electromagnetic Fields in High-Speed Electronic Circuits Introduction Basic Circuit Parameters Transmission Line Parameters Impedance Parameters Differential Capacitance Calculation Differential Inductance Calculation Lumped Inductance Due to a Discontinuity Flux/Current Definition Fitting Z(a) or S mn (co) to an Equivalent Circuit Discussion: Choice of Methods Parallel Coplanar Microstrips Multilayered Interconnect Modeling Example Digital Signal Processing and Spectrum Estimation Techniques Prony's Method Autoregressive Models System Identification Modeling of Lumped Circuit Elements FD-TD Formulation Extended to Circuit Elements The Resistor The Resistive Voltage Source The Capacitor The Inductor TheDiode The Bipolar Junction Transistor Direct Linking of FD-TD and SPICE 465 References 470 Additional Bibliography 472 Problems 473
8 Chapter 14 FD-TD and Antenna Analysis 14.1 Introduction 14.2 Antenna Characteristics Antenna Fields and Radiation Patterns Antenna Impedance 14.3 Motivation for Using FD-TD in Antenna Design 14.4 The Monopole Over a PEC Ground Plane Modeling Considerations Results Discussion 14.5 Waveguide and Horn Antennas Two-Dimensional Horn Antenna Three-Dimensional Waveguide Radiator Discussion 14.6 The Vivaldi Slotline Array Background The Planar Element The Two-Element Vivaldi Pair The Quad Element The Linear Phased Array Active Impedance of the Phased Array 14.7 Linear Superposition References Bibliography Projects Chapter 15 Electromagnetic Wave Scattering, Penetration, and Coupling for Complex Structures 15.1 Introduction 15.2 Scattering and Radar Cross Section Small PEC Cube, Broadside Incidence Nine-Wavelength T-Shaped Conducting Target Generic Curved-Surface Targets, Conformally Modeled Full-Scale Military Fighter Aircraft 15.3 Penetration and Coupling Empty Cylindrical PEC Cavity Loaded Missile Guidance Section Spatial Decomposition via a Schelkunoff Equivalence Principle Cylindrical PEC Cavity Loaded by a Wire Bundle 15.4 Biological Tissue Structures The First FD-TD Biological Tissue Model: The Human Eye Patient-Specific Electromagnetic Hyperthermia Models Derived from Computed Tomography Imaging and Analysis Visible Light Interactions with the Vertebrate Retinal Rod 15.5 Microlaser Cavities References Bibliography
9 Chapter 16 Efficient FD-TD Algorithms for Vector and Multiprocessor Computers Introduction Pipelined RISC and Vector Processors Multiprocessor Computers Introduction to Parallelism Parallel FD-TD Algorithm Numerical Benchmarks Parallel Nonorthogonal FD-TD Algorithms Parallel Planar Generalized Yee Algorithm The Parallel Algorithm Spatial Decomposition Algorithms Porting FDTD3D to the Cray T3D Background T3D Programming Models Program Design Issues Single-Processor Optimization Performance 579 References 581 Problems 583 Author and Citation Index 585 Subject Index 591 About the Chapter Authors 597
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