The Method of Moments in Electromagnetics

Cover
Half Title
Title Page
Copyright Page
Contents
Preface to the Third Edition
Preface to the Second Edition
Preface
Acknowledgments
About the Author
1. Computational Electromagnetics
1.1. CEM Algorithms
1.1.1. Low-Frequency Methods
1.1.1.1. Finite Difference Time Domain Method
1.1.1.2. Finite Element Method
1.1.1.3. Method of Moments
1.1.2. High-Frequency Methods
1.1.2.1. Geometrical Theory of Diffraction
1.1.2.2. Physical Optics
1.1.2.3. Physical Theory of Diffraction
1.1.2.4. Shooting and Bouncing Rays
References
2. The Method of Moments
2.1. Electrostatic Problems
2.1.1. Charged Wire
2.1.1.1. Matrix Element Evaluation
2.1.1.2. Solution
2.1.2. Charged Plate
2.1.2.1. Matrix Element Evaluation
2.1.2.2. Solution
2.2. The Method of Moments
2.2.1. Point Matching
2.2.2. Galerkin’s Method
2.3. Common One-Dimensional Basis Functions
2.3.1. Pulse Functions
2.3.2. Piecewise Triangular Functions
2.3.3. Piecewise Sinusoidal Functions
2.3.4. Entire-Domain Functions
2.3.5. Number of Basis Functions
References
3. Radiation and Scattering
3.1. Maxwell’s Equations
3.2. Electromagnetic Boundary Conditions
3.3. Formulations for Radiation
3.3.1. Three-Dimensional Green’s Function
3.3.2. Two-Dimensional Green’s Function
3.4. Vector Potentials
3.4.1. Magnetic Vector Potential
3.4.1.1. Three-Dimensional Magnetic Vector Potential
3.4.1.2. Two-Dimensional Magnetic Vector Potential
3.4.2. Electric Vector Potential
3.4.2.1. Three-Dimensional Electric Vector Potential
3.4.2.2. Two-Dimensional Electric Vector Potential
3.4.3. Total Fields
3.4.4. Comparison of Radiation Formulas
3.5. Near and Far Field
3.5.1. Three-Dimensional Near Field
3.5.2. Two-Dimensional Near Field
3.5.3. Three-Dimensional Far Field
3.5.4. Two-Dimensional Far Field
3.6. Formulations for Scattering
3.6.1. Surface Equivalent
3.6.2. Surface Integral Equations
3.6.2.1. Interior Resonance Problem
3.6.2.2. Discretization and Testing
3.6.2.3. Modification of Matrix Elements
3.6.3. Enforcement of Boundary Conditions
3.6.3.1. EFIE-CFIE-PMCHWT Approach
3.6.4. Physical Optics Equivalent
References
4. Solution of Matrix Equations
4.1. Direct Methods
4.1.1. Gaussian Elimination
4.1.1.1. Pivoting
4.1.2. LU Factorization
4.1.3. Block LU Factorization
4.1.4. Condition Number
4.2. Iterative Methods
4.2.1. Conjugate Gradient
4.2.2. Biconjugate Gradient
4.2.3. Conjugate Gradient Squared
4.2.4. Biconjugate Gradient Stabilized
4.2.5. GMRES
4.2.6. Stopping Criteria
4.2.7. Preconditioning
4.3. Software for Linear Systems
4.3.1. BLAS
4.3.2. LAPACK
4.3.3. MATLAB
References
5. Thin Wires
5.1. Thin Wire Approximation
5.2. Thin Wire Excitations
5.2.1. Delta-Gap Source
5.2.2. Magnetic Frill
5.2.3. Plane Wave
5.3. Hallén’s Equation
5.3.1. Symmetric Problems
5.3.1.1. Solution Using Pulse Functions and Point Matching
5.3.2 Asymmetric Problems
5.3.2.1 Solution Using Pulse Functions and Point Matching
5.4. Pocklington’s Equation
5.4.1. Solution Using Pulse Functions and Point Matching
5.5. Thin Wires of Arbitrary Shape
5.5.1. Method of Moments Discretization
5.5.2. Solution Using Triangle Basis and Testing Functions
5.5.2.1. Non-Self Terms
5.5.2.2. Self Terms
5.5.3. Solution Using Sinusoidal Basis and Testing Functions
5.5.3.1. Self Terms
5.5.4. Lumped and Distributed Impedances
5.6. Examples
5.6.1. Comparison of Thin Wire Models
5.6.1.1. Input Impedance
5.6.1.2. Induced Current Distribution
5.6.2. Half-Wavelength Dipole
5.6.3. Circular Loop Antenna
5.6.4. Folded Dipole Antenna
5.6.5. Two-Wire Transmission Line
5.6.6. Yagi Antenna for 146 MHz
References
6. Two-Dimensional Problems
6.1. Conducting Objects
6.1.1. EFIE: TM Polarization
6.1.1.1. Solution Using Pulse Functions
6.1.1.2. Solution Using Triangle Functions
6.1.2. Generalized EFIE: TM Polarization
6.1.2.1. MoM Discretization
6.1.2.2. Solution Using Triangle Functions
6.1.3. EFIE: TE Polarization
6.1.3.1. Pulse Function Solution
6.1.4. Generalized EFIE: TE Polarization
6.1.4.1. MoM Discretization
6.1.4.2. Solution Using Triangle Functions
6.1.5. nMFIE: TM Polarization
6.1.5.1. Solution Using Triangle Functions
6.1.6. nMFIE: TE Polarization
6.1.6.1. Solution Using Triangle Functions
6.1.7. Examples
6.1.7.1. Conducting Cylinder: TM Polarization
6.1.7.2. Conducting Cylinder: TE Polarization
6.2. Dielectric and Composite Objects
6.2.1. Basis Function Orientation
6.2.2. EFIE: TM Polarization
6.2.2.1. MoM Discretization
6.2.3. MFIE: TM Polarization
6.2.3.1. MoM Discretization
6.2.4. nMFIE: TM Polarization
6.2.4.1. MoM Discretization
6.2.5. EFIE: TE Polarization
6.2.5.1. MoM Discretization
6.2.6. MFIE: TE Polarization
6.2.6.1. MoM Discretization
6.2.7. nMFIE: TE Polarization
6.2.7.1. MoM Discretization
6.2.8. Numerical Stability
6.2.9. Examples
6.2.9.1. Dielectric Cylinder
6.2.9.2. Dielectric Cylinder: TM Polarization
6.2.9.3. Dielectric Cylinder: TE Polarization
6.2.9.4. Coated Cylinder
6.2.9.5. Coated Cylinder: TM Polarization
6.2.9.6. Coated Cylinder: TE Polarization
6.2.9.7. Effect of Number of Segments per Wavelength on Accuracy
References
7. Bodies of Revolution
7.1. BoR Surface Description
7.2. Expansion of Surface Currents
7.3. EFIE
7.3.1. L Operator
7.3.1.1. L Matrix Elements
7.3.2. K Operator
7.3.2.1. K Matrix Elements
7.3.3. Excitation
7.3.3.1. Plane Wave Excitation
7.4. MFIE
7.4.1. Excitation
7.4.1.1. Plane Wave Excitation
7.5. Solution
7.5.1. Plane Wave Solution
7.5.1.1. Currents
7.5.2. Scattered Field
7.5.2.1. Scattered Far Fields
7.6. nMFIE
7.6.1. n × L Operator
7.6.1.1. nL Matrix Elements
7.6.2. n × K Operator
7.6.2.1. nK Matrix Elements
7.6.3. Excitation
7.6.3.1. Plane Wave Excitation
7.6.3.2. Plane Wave Solution
7.7. Numerical Discretization
7.8. Notes on Software Implementation
7.8.1. Geometry Processing and Basis Function Assignment
7.8.2. Parallelization
7.8.3. Convergence
7.9. Examples
7.9.1. Spheres
7.9.1.1. Conducting Sphere
7.9.1.2. Stratified Sphere
7.9.1.3. Dielectric Sphere
7.9.1.4. Coated Sphere
7.9.2. EMCC Benchmark Targets
7.9.2.1. EMCC Ogive
7.9.2.2. EMCC Double Ogive
7.9.2.3. EMCC Cone-Sphere
7.9.2.4. EMCC Cone-Sphere with Gap
7.9.3. Biconic Reentry Vehicle
7.10. Treatment of Junctions
7.10.1. Orientation of Basis Functions
7.10.1.1. Longitudinal Basis Vectors
7.10.1.2. Azimuthal Basis Vectors
7.10.2. Examples with Junctions
7.10.2.1. Dielectric Sphere with Septum
7.10.2.2. Coated Sphere with Septum
7.10.2.3. Stratified Sphere with Septum
7.10.2.4. Monoconic Reentry Vehicle with Dielectric Nose
References
8. Three-Dimensional Problems
8.1. Modeling of Three-Dimensional Surfaces
8.1.1. Facet File
8.1.2. Edge-Finding Algorithm
8.1.2.1. Shared Nodes
8.2. Expansion of Surface Currents
8.2.1. Divergence of the RWG Function
8.2.2. Assignment and Orientation of Basis Functions
8.3. EFIE
8.3.1. L Operator
8.3.1.1. Non-Near Terms
8.3.1.2. Near and Self Terms
8.3.2. K Operator
8.3.2.1. Non-Near Terms
8.3.2.2. Near Terms
8.3.3. Excitation
8.3.3.1. Plane Wave Excitation
8.3.3.2. Planar Antenna Excitation
8.4. MFIE
8.4.1. Excitation
8.4.1.1. Plane Wave Excitation
8.5. nMFIE
8.5.1. n × K Operator
8.5.1.1. Non-Near Terms
8.5.1.2. Near Terms
8.5.2. n × L Operator
8.5.2.1. Non-Near Terms
8.5.2.2. Near and Self Terms
8.5.3. Excitation
8.5.3.1. Plane Wave Excitation
8.6. Enforcement of Boundary Conditions
8.6.1. Classification of Edges and Junctions
8.6.1.1. Dielectric Edges and Junctions
8.6.1.2. Conducting Edges and Junctions
8.6.1.3. Composite Conducting-Dielectric Junctions
8.6.2. Reducing the Overdetermined System
8.6.2.1. PMCHWT at Dielectric Edges and Junctions
8.6.2.2. EFIE and CFIE at Conducting Edges and Junctions
8.6.2.3. EFIE and CFIE at Composite Conducting-Dielectric Junctions
8.7. Software Implementation Notes
8.7.1. Pre-Processing and Bookkeeping
8.7.1.1. Region and Interface Assignments
8.7.1.2. Geometry Processing
8.7.1.3. Assignment and Orientation of Basis Functions
8.7.2. Matrix and Right-Hand Side Fill
8.7.3. Parallelization
8.7.3.1. Shared Memory Systems
8.7.3.2. Distributed Memory Systems
8.7.4. Triangle Mesh Considerations
8.7.4.1. Aspect Ratio
8.7.4.2. T-Junctions
8.8. Numerical Examples
8.8.1. Serenity
8.8.2. Compute Platform
8.8.3. Spheres
8.8.3.1. Conducting Sphere
8.8.3.2. Dielectric Sphere
8.8.3.3. Coated Sphere
8.8.4. EMCC Plate Benchmark Targets
8.8.4.1. Wedge Cylinder
8.8.4.2. Wedge-Plate Cylinder
8.8.4.3. Plate Cylinder
8.8.4.4. Business Card
8.8.5. Strip Dipole Antenna
8.8.6. Bowtie Antenna
8.8.7. Archimedean Spiral Antenna
8.8.8. Monoconic Reentry Vehicle with Dielectric Nose
8.8.9. Summary of Examples
References
9. Adaptive Cross Approximation
9.1. Rank Deficiency
9.1.1. Limitations of Using SVD For Compression
9.2. Adaptive Cross Approximation
9.2.1. Modifications
9.2.1.1. Initialization
9.2.1.2. Early Termination
9.2.1.3. Pathological Failure Case
9.2.2. QR/SVD Recompression
9.3. Clustering Techniques
9.3.1. Target Group Size For ACA
9.4. LU Factorization of ACA-Compressed Matrix
9.4.1. ACA-Compressed Block LU Factorization
9.4.1.1. Compressibility of the LU Matrix
9.5. Solution of the ACA-Compressed Matrix System
9.6. Software Implementation Notes
9.6.1. Software Class Support
9.6.1.1. Element Engine Class
9.6.1.2. Matrix Classes
9.6.2. Shared Memory Processing
9.6.2.1. ACA CPU Thread Class
9.6.2.2. ACA GPU Thread Class
9.6.3. Distributed Memory Processing
9.6.3.1. Parallelization Strategy
9.6.3.2. Block LU Factorization Using MPI
9.6.3.3. Block-RHS Solution Using MPI
9.7. Numerical Examples
9.7.1. Compute Platform
9.7.2. Adaptive ACA Tolerance
9.7.3. Spheres
9.7.3.1. Conducting Sphere
9.7.3.2. Dielectric Sphere
9.7.3.3. Coated Sphere
9.7.4. EMCC Benchmark Targets
9.7.4.1. EMCC Ogive
9.7.4.2. EMCC Double Ogive
9.7.4.3. EMCC Cone-Sphere
9.7.4.4. EMCC Cone-Sphere with Gap
9.7.4.5. NASA Almond
9.7.4.6. EMCC Cube
9.7.4.7. EMCC Prism
9.7.5. Dielectric Cube and Ogive
9.7.5.1. Small Polyethylene Cube
9.7.5.2. Polyethylene Ogive
9.7.6. UT Austin Benchmark Targets
9.7.6.1. PEC Almond
9.7.6.2. Solid Resin Almond
9.7.6.3. Closed-Tail Almond
9.7.6.4. Open-Tail Almond
9.7.6.5. EXPEDITE-RCS Aircraft
9.7.7. Monoconic Reentry Vehicle
9.7.7.1. Conducting RV
9.7.7.2. RV with Dielectric Nose
9.7.8. Summary of Examples
References
10. Multi-Level Adaptive Cross Approximation
10.1. MLACA Compression of Matrix Blocks
10.1.1. MLACA Fundamentals
10.1.1.1. SVD-Based Compression on Higher Levels
10.1.2. Hierarchical Clustering of Sub-Groups
10.1.3. Compression of Diagonal Blocks
10.2. Direct Solution of MLACA-Compressed Matrix System
10.2.1. MLACA Block Reconstruction
10.2.2. Matrix Product and V-Type MLACA
10.2.2.1. Top-Level Matrix Product
10.2.2.2. Bottom-Level Matrix Product
10.2.3. MLACA Block-RHS Solution
10.3. Software Implementation Notes
10.3.1. Software Class Support
10.3.1.1. Abstract Matrix Class
10.3.1.2. Element Engine Class
10.3.1.3. MLACA Translator Class
10.3.1.4. MLACAMatrix and MLACANode Classes
10.3.1.5. HMatrix Class
10.3.2. Shared Memory Processing
10.3.2.1. Reconstruction of Blocks and Intermediate Products
10.3.2.2. MLACA CPU Thread Class
10.3.2.3. MLACA GPU Thread Class
10.3.3. Distributed Memory Processing
10.4. Numerical Examples
10.4.1. Compute Platform
10.4.2. Conducting Spheres
10.4.2.1. Variation of Target Group Size
10.4.3. Polyethylene Cone-Sphere
10.4.4. Monoconic Reentry Vehicle
10.4.4.1. EMCC Prism
References
11. The Fast Multipole Method
11.1. The N-Body Problem
11.2. Matrix-Vector Product
11.2.1. Addition Theorem
11.2.2. Wave Translation
11.2.2.1. Complex Wavenumbers
11.2.3. Far Matrix Elements
11.2.3.1. EFIE
11.2.3.2. MFIE
11.2.3.3. nMFIE
11.2.4. Unit Sphere Decomposition
11.3. One-Level Fast Multipole Algorithm
11.3.1. Clustering of Basis Functions
11.3.1.1. Classification of Near and Far Groups
11.3.2 Near Matrix
11.3.2.1 Compression of Near Matrix
11.3.3. Number of Multipoles
11.3.3.1. Limiting L for Transfer Functions
11.3.3.2. L for Complex Wavenumbers
11.3.4. Integration on the Sphere
11.3.4.1. Spherical Harmonic Representation
11.3.4.2. Total Bandwidth
11.3.4.3. Computation and Storage of Transfer Functions
11.3.4.4. Computation of Radiation and Receive Functions
11.3.4.5. Compression of Radiation and Receive Functions
11.3.5. Matrix-Vector Product
11.3.5.1. Near Product
11.3.5.2. Far Product
11.4. Multi-Level Fast Multipole Algorithm (MLFMA)
11.4.1. MLFMA/SVD
11.4.2. Spatial Subdivision and Clustering via Octree
11.4.3. Near Matrix and Near Product
11.4.4. Unit Sphere Sampling Rates
11.4.5. Far Product
11.4.5.1. Upward Pass (Aggregation)
11.4.5.2. Downward Pass (Disaggregation)
11.4.6. Interpolation Algorithms
11.4.6.1. Statement of the Problem
11.4.6.2. Global Interpolation by Spherical Harmonics
11.4.6.3. Local Interpolation by Lagrange Polynomials
11.5. Preconditioners
11.5.1. Information Content
11.5.2. Diagonal Preconditioner
11.5.3. Incomplete Block LU (ILU) Preconditioners
11.5.3.1. Block Diagonal
11.5.3.2. Block ILU with Zero Fill-In (ILU(0))
11.5.3.3. Block ILU with Threshold (ILUT)
11.5.4. Sparse Approximate Inverse (SAI)
11.5.4.1. Dense QR Factorization
11.6. Software Implementation Notes
11.6.1. Software Class Support
11.6.1.1. Element Engine Class
11.6.1.2. Sparse Block Matrix Class
11.6.1.3. FMM Region Class
11.6.1.4. FMM Octree Class
11.6.2. Shared Memory Processing
11.6.2.1. FMM CPU Thread Class
11.7. Numerical Examples
11.7.1. Compute Platform
11.7.2. Run Parameters
11.7.3. Spheres
11.7.3.1. Conducting Sphere
11.7.3.2. Dielectric Sphere
11.7.3.3. Coated Sphere
11.7.4. Thin Square Plate
11.7.5. Monoconic Reentry Vehicle
11.7.6. Business Jet
11.7.7. Summary of Examples
11.7.8. Preconditioner Performance
11.7.8.1. Compressed versus Uncompressed Preconditioners
11.7.8.2. Performance versus Incident Angle
11.7.8.3. Performance versus Cube Size
11.7.9. Initial Guess in Iterative Solution
References
12. Integration
12.1. One-Dimensional Integration
12.1.1. Centroidal Approximation
12.1.2. Rectangular Rule
12.1.3. Trapezoidal Rule
12.1.3.1. Romberg Integration
12.1.4. Simpson’s Rule
12.1.4.1. Adaptive Simpson’s Rule
12.1.5. One-Dimensional Gaussian Quadrature
12.2. Integration over Triangles
12.2.1. Simplex Coordinates
12.2.2. Radiation Integrals with a Constant Source
12.2.2.1. Special Cases
12.2.3. Radiation Integrals with a Linear Source
12.2.3.1. General Case
12.2.3.2. Special Cases
12.2.4. Gaussian Quadrature on Triangles
12.2.4.1. Comparison with Analytic Solution
References
A. Scattering Using Physical Optics
A.1. Field Scattered at a Conducting Interface
A.2. Plane Wave Decomposition at a Planar Interface
A.3. Field Scattered at a Dielectric Interface
A.4. Layered Dielectrics over Conductor
References
Index