book

The Method of Moments in Electromagnetics, 3rd Edition

by Walton C. Gibson

September 2021

Intermediate to advanced

510 pages

10h 39m

English

Chapman and Hall/CRC

Read now

Unlock full access

1.1 CEM Algorithms1.1.1 Low-Frequency Methods1.1.1.1 Finite Difference Time Domain Method1.1.1.2 Finite Element Method1.1.1.3 Method of Moments1.1.2 High-Frequency Methods1.1.2.1 Geometrical Theory of Diffraction1.1.2.2 Physical Optics1.1.2.3 Physical Theory of Diffraction1.1.2.4 Shooting and Bouncing RaysReferences
2.1 Electrostatic Problems2.1.1 Charged Wire2.1.1.1 Matrix Element Evaluation2.1.1.2 Solution2.1.2 Charged Plate2.1.2.1 Matrix Element Evaluation2.1.2.2 Solution2.2 The Method of Moments2.2.1 Point Matching2.2.2 Galerkin's Method2.3 Common One-Dimensional Basis Functions2.3.1 Pulse Functions2.3.2 Piecewise Triangular Functions2.3.3 Piecewise Sinusoidal Functions2.3.4 Entire-Domain Functions2.3.5 Number of Basis FunctionsReferences
3.1 Maxwell's Equations3.2 Electromagnetic Boundary Conditions3.3 Formulations for Radiation3.3.1 Three-Dimensional Green's Function3.3.2 Two-Dimensional Green's Function3.4 Vector Potentials3.4.1 Magnetic Vector Potential3.4.1.1 Three-Dimensional Magnetic Vector Potential3.4.1.2 Two-Dimensional Magnetic Vector Potential3.4.2 Electric Vector Potential3.4.2.1 Three-Dimensional Electric Vector Potential3.4.2.2 Two-Dimensional Electric Vector Potential3.4.3 Total Fields3.4.4 Comparison of Radiation Formulas3.5 Near and Far Field3.5.1 Three-Dimensional Near Field3.5.2 Two-Dimensional Near Field3.5.3 Three-Dimensional Far Field3.5.4 Two-Dimensional Far Field3.6 Formulations for Scattering3.6.1 Surface Equivalent3.6.2 Surface Integral Equations3.6.2.1 Interior Resonance Problem3.6.2.2 Discretization and Testing3.6.2.3 Modification of Matrix Elements3.6.3 Enforcement of Boundary Conditions3.6.3.1 EFIE-CFIE-PMCHWT Approach3.6.4 Physical Optics EquivalentReferences
4.1 Direct Methods4.1.1 Gaussian Elimination4.1.1.1 Pivoting4.1.2 LU Factorization4.1.3 Block LU Factorization4.1.4 Condition Number4.2 Iterative Methods4.2.1 Conjugate Gradient4.2.2 Biconjugate Gradient4.2.3 Conjugate Gradient Squared4.2.4 Biconjugate Gradient Stabilized4.2.5 GMRES4.2.6 Stopping Criteria4.2.7 Preconditioning4.3 Software for Linear Systems4.3.1 BLAS4.3.2 LAPACK4.3.3 MATLABReferences
5.1 Thin Wire Approximation5.2 Thin Wire Excitations5.2.1 Delta-Gap Source5.2.2 Magnetic Frill5.2.3 Plane Wave5.3 Hallén's Equation5.3.1 Symmetric Problems5.3.1.1 Solution Using Pulse Functions and Point Matching5.3.2 Asymmetric Problems5.3.2.1 Solution Using Pulse Functions and Point Matching5.4 Pocklington's Equation5.4.1 Solution Using Pulse Functions and Point Matching5.5 Thin Wires of Arbitrary Shape5.5.1 Method of Moments Discretization5.5.2 Solution Using Triangle Basis and Testing Functions5.5.2.1 Non-Self Terms5.5.2.2 Self Terms5.5.3 Solution Using Sinusoidal Basis and Testing Functions5.5.3.1 Self Terms5.5.4 Lumped and Distributed Impedances5.6 Examples5.6.1 Comparison of Thin Wire Models5.6.1.1 Input Impedance5.6.1.2 Induced Current Distribution5.6.2 Half-Wavelength Dipole5.6.3 Circular Loop Antenna5.6.4 Folded Dipole Antenna5.6.5 Two-Wire Transmission Line5.6.6 Yagi Antenna for 146 MHzReferences
6.1 Conducting Objects6.1.1 EFIE: TM Polarization6.1.1.1 Solution Using Pulse Functions6.1.1.2 Solution Using Triangle Functions6.1.2 Generalized EFIE: TM Polarization6.1.2.1 MoM Discretization6.1.2.2 Solution Using Triangle Functions6.1.3 EFIE: TE Polarization6.1.3.1 Pulse Function Solution6.1.4 Generalized EFIE: TE Polarization6.1.4.1 MoM Discretization6.1.4.2 Solution Using Triangle Functions6.1.5 nMFIE: TM Polarization6.1.5.1 Solution Using Triangle Functions6.1.6 nMFIE: TE Polarization6.1.6.1 Solution Using Triangle Functions6.1.7 Examples6.1.7.1 Conducting Cylinder: TM Polarization6.1.7.2 Conducting Cylinder: TE Polarization6.2 Dielectric and Composite Objects6.2.1 Basis Function Orientation6.2.2 EFIE: TM Polarization6.2.2.1 MoM Discretization6.2.3 MFIE: TM Polarization6.2.3.1 MoM Discretization6.2.4 nMFIE: TM Polarization6.2.4.1 MoM Discretization6.2.5 EFIE: TE Polarization6.2.5.1 MoM Discretization6.2.6 MFIE: TE Polarization6.2.6.1 MoM Discretization6.2.7 nMFIE: TE Polarization6.2.7.1 MoM Discretization6.2.8 Numerical Stability6.2.9 Examples6.2.9.1 Dielectric Cylinder6.2.9.2 Dielectric Cylinder: TM Polarization6.2.9.3 Dielectric Cylinder: TE Polarization6.2.9.4 Coated Cylinder6.2.9.5 Coated Cylinder: TM Polarization6.2.9.6 Coated Cylinder: TE Polarization6.2.9.7 Effect of Number of Segments per Wavelength on AccuracyReferences
7.1 BoR Surface Description7.2 Expansion of Surface Currents7.3 EFIE7.3.1 ℒ Operator7.3.1.1 L Matrix Elements7.3.2 κ Operator7.3.2.1 K Matrix Elements7.3.3 Excitation7.3.3.1 Plane Wave Excitation7.4 MFIE7.4.1 Excitation7.4.1.1 Plane Wave Excitation7.5 Solution7.5.1 Plane Wave Solution7.5.1.1 Currents7.5.2 Scattered Field7.5.2.1 Scattered Far Fields7.6 nMFIE7.6.1 n^ × ℒ Operator7.6.1.1 nL Matrix Elements7.6.2 n^ × κ Operator7.6.2.1 nK Matrix Elements7.6.3 Excitation7.6.3.1 Plane Wave Excitation7.6.3.2 Plane Wave Solution7.7 Numerical Discretization7.8 Notes on Software Implementation7.8.1 Geometry Processing and Basis Function Assignment7.8.2 Parallelization7.8.3 Convergence7.9 Examples7.9.1 Spheres7.9.1.1 Conducting Sphere7.9.1.2 Stratified Sphere7.9.1.3 Dielectric Sphere7.9.1.4 Coated Sphere7.9.2 EMCC Benchmark Targets7.9.2.1 EMCC Ogive7.9.2.2 EMCC Double Ogive7.9.2.3 EMCC Cone-Sphere7.9.2.4 EMCC Cone-Sphere with Gap7.9.3 Biconic Reentry Vehicle7.10 Treatment of Junctions7.10.1 Orientation of Basis Functions7.10.1.1 Longitudinal Basis Vectors7.10.1.2 Azimuthal Basis Vectors7.10.2 Examples with Junctions7.10.2.1 Dielectric Sphere with Septum7.10.2.2 Coated Sphere with Septum7.10.2.3 Stratified Sphere with Septum7.10.2.4 Monoconic Reentry Vehicle with Dielectric NoseReferences
8.1 Modeling of Three-Dimensional Surfaces8.1.1 Facet File8.1.2 Edge-Finding Algorithm8.1.2.1 Shared Nodes8.2 Expansion of Surface Currents8.2.1 Divergence of the RWG Function8.2.2 Assignment and Orientation of Basis Functions8.3 EFIE8.3.1 ℒ Operator8.3.1.1 Non-Near Terms8.3.1.2 Near and Self Terms8.3.2 κ Operator8.3.2.1 Non-Near Terms8.3.2.2 Near Terms8.3.3 Excitation8.3.3.1 Plane Wave Excitation8.3.3.2 Planar Antenna Excitation8.4 MFIE8.4.1 Excitation8.4.1.1 Plane Wave Excitation8.5 nMFIE8.5.1 n^ × κ Operator8.5.1.1 Non-Near Terms8.5.1.2 Near Terms8.5.2 n^ × ℒ Operator8.5.2.1 Non-Near Terms8.5.2.2 Near and Self Terms8.5.3 Excitation8.5.3.1 Plane Wave Excitation8.6 Enforcement of Boundary Conditions8.6.1 Classification of Edges and Junctions8.6.1.1 Dielectric Edges and Junctions8.6.1.2 Conducting Edges and Junctions8.6.1.3 Composite Conducting-Dielectric Junctions8.6.2 Reducing the Overdetermined System8.6.2.1 PMCHWT at Dielectric Edges and Junctions8.6.2.2 EFIE and CFIE at Conducting Edges and Junctions8.6.2.3 EFIE and CFIE at Composite Conducting-Dielectric Junctions8.7 Software Implementation Notes8.7.1 Pre-Processing and Bookkeeping8.7.1.1 Region and Interface Assignments8.7.1.2 Geometry Processing8.7.1.3 Assignment and Orientation of Basis Functions8.7.2 Matrix and Right-Hand Side Fill8.7.3 Parallelization8.7.3.1 Shared Memory Systems8.7.3.2 Distributed Memory Systems8.7.4 Triangle Mesh Considerations8.7.4.1 Aspect Ratio8.7.4.2 T-Junctions8.8 Numerical Examples8.8.1 Serenity8.8.2 Compute Platform8.8.3 Spheres8.8.3.1 Conducting Sphere8.8.3.2 Dielectric Sphere8.8.3.3 Coated Sphere8.8.4 EMCC Plate Benchmark Targets8.8.4.1 Wedge Cylinder8.8.4.2 Wedge-Plate Cylinder8.8.4.3 Plate Cylinder8.8.4.4 Business Card8.8.5 Strip Dipole Antenna8.8.6 Bowtie Antenna8.8.7 Archimedean Spiral Antenna8.8.8 Monoconic Reentry Vehicle with Dielectric Nose8.8.9 Summary of ExamplesReferences
9.1 Rank Deficiency9.1.1 Limitations of Using SVD For Compression9.2 Adaptive Cross Approximation9.2.1 Modifications9.2.1.1 Initialization9.2.1.2 Early Termination9.2.1.3 Pathological Failure Case9.2.2 QR/SVD Recompression9.3 Clustering Techniques9.3.1 Target Group Size For ACA9.4 LU Factorization of ACA-Compressed Matrix9.4.1 ACA-Compressed Block LU Factorization9.4.1.1 Compressibility of the LU Matrix9.5 Solution of the ACA-Compressed Matrix System9.6 Software Implementation Notes9.6.1 Software Class Support9.6.1.1 Element Engine Class9.6.1.2 Matrix Classes9.6.2 Shared Memory Processing9.6.2.1 ACA CPU Thread Class9.6.2.2 ACA GPU Thread Class9.6.3 Distributed Memory Processing9.6.3.1 Parallelization Strategy9.6.3.2 Block LU Factorization Using MPI9.6.3.3 Block-RHS Solution Using MPI9.7 Numerical Examples9.7.1 Compute Platform9.7.2 Adaptive ACA Tolerance9.7.3 Spheres9.7.3.1 Conducting Sphere9.7.3.2 Dielectric Sphere9.7.3.3 Coated Sphere9.7.4 EMCC Benchmark Targets9.7.4.1 EMCC Ogive9.7.4.2 EMCC Double Ogive9.7.4.3 EMCC Cone-Sphere9.7.4.4 EMCC Cone-Sphere with Gap9.7.4.5 NASA Almond9.7.4.6 EMCC Cube9.7.4.7 EMCC Prism9.7.5 Dielectric Cube and Ogive9.7.5.1 Small Polyethylene Cube9.7.5.2 Polyethylene Ogive9.7.6 UT Austin Benchmark Targets9.7.6.1 PEC Almond9.7.6.2 Solid Resin Almond9.7.6.3 Closed-Tail Almond9.7.6.4 Open-Tail Almond9.7.6.5 EXPEDITE-RCS Aircraft9.7.7 Monoconic Reentry Vehicle9.7.7.1 Conducting RV9.7.7.2 RV with Dielectric Nose9.7.8 Summary of ExamplesReferences
10.1 MLACA Compression of Matrix Blocks10.1.1 MLACA Fundamentals10.1.1.1 SVD-Based Compression on Higher Levels10.1.2 Hierarchical Clustering of Sub-Groups10.1.3 Compression of Diagonal Blocks10.2 Direct Solution of MLACA-Compressed Matrix System10.2.1 MLACA Block Reconstruction10.2.2 Matrix Product and V-Type MLACA10.2.2.1 Top-Level Matrix Product10.2.2.2 Bottom-Level Matrix Product10.2.3 MLACA Block-RHS Solution10.3 Software Implementation Notes10.3.1 Software Class Support10.3.1.1 Abstract Matrix Class10.3.1.2 Element Engine Class10.3.1.3 MLACA Translator Class10.3.1.4 MLACAMatrix and MLACANode Classes10.3.1.5 HMatrix Class10.3.2 Shared Memory Processing10.3.2.1 Reconstruction of Blocks and Intermediate Products10.3.2.2 MLACA CPU Thread Class10.3.2.3 MLACA GPU Thread Class10.3.3 Distributed Memory Processing10.4 Numerical Examples10.4.1 Compute Platform10.4.2 Conducting Spheres10.4.2.1 Variation of Target Group Size10.4.3 Polyethylene Cone-Sphere10.4.4 Monoconic Reentry Vehicle10.4.4.1 EMCC PrismReferences
11.1 The N-Body Problem11.2 Matrix-Vector Product11.2.1 Addition Theorem11.2.2 Wave Translation11.2.2.1 Complex Wavenumbers11.2.3 Far Matrix Elements11.2.3.1 EFIE11.2.3.2 MFIE11.2.3.3 nMFIE11.2.4 Unit Sphere Decomposition11.3 One-Level Fast Multipole Algorithm11.3.1 Clustering of Basis Functions11.3.1.1 Classification of Near and Far Groups11.3.2 Near Matrix11.3.2.1 Compression of Near Matrix11.3.3 Number of Multipoles11.3.3.1 Limiting L for Transfer Functions11.3.3.2 L for Complex Wavenumbers11.3.4 Integration on the Sphere11.3.4.1 Spherical Harmonic Representation11.3.4.2 Total Bandwidth11.3.4.3 Computation and Storage of Transfer Functions11.3.4.4 Computation of Radiation and Receive Functions11.3.4.5 Compression of Radiation and Receive Functions11.3.5 Matrix-Vector Product11.3.5.1 Near Product11.3.5.2 Far Product11.4 Multi-Level Fast Multipole Algorithm (MLFMA)11.4.1 MLFMA/SVD11.4.2 Spatial Subdivision and Clustering via Octree11.4.3 Near Matrix and Near Product11.4.4 Unit Sphere Sampling Rates11.4.5 Far Product11.4.5.1 Upward Pass (Aggregation)11.4.5.2 Downward Pass (Disaggregation)11.4.6 Interpolation Algorithms11.4.6.1 Statement of the Problem11.4.6.2 Global Interpolation by Spherical Harmonics11.4.6.3 Local Interpolation by Lagrange Polynomials11.5 Preconditioners11.5.1 Information Content11.5.2 Diagonal Preconditioner11.5.3 Incomplete Block LU (ILU) Preconditioners11.5.3.1 Block Diagonal11.5.3.2 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 Factorization11.6 Software Implementation Notes11.6.1 Software Class Support11.6.1.1 Element Engine Class11.6.1.2 Sparse Block Matrix Class11.6.1.3 FMM Region Class11.6.1.4 FMM Octree Class11.6.2 Shared Memory Processing11.6.2.1 FMM CPU Thread Class11.7 Numerical Examples11.7.1 Compute Platform11.7.2 Run Parameters11.7.3 Spheres11.7.3.1 Conducting Sphere11.7.3.2 Dielectric Sphere11.7.3.3 Coated Sphere11.7.4 Thin Square Plate11.7.5 Monoconic Reentry Vehicle11.7.6 Business Jet11.7.7 Summary of Examples11.7.8 Preconditioner Performance11.7.8.1 Compressed versus Uncompressed Preconditioners11.7.8.2 Performance versus Incident Angle11.7.8.3 Performance versus Cube Size11.7.9 Initial Guess in Iterative SolutionReferences
12.1 One-Dimensional Integration12.1.1 Centroidal Approximation12.1.2 Rectangular Rule12.1.3 Trapezoidal Rule12.1.3.1 Romberg Integration12.1.4 Simpson's Rule12.1.4.1 Adaptive Simpson's Rule12.1.5 One-Dimensional Gaussian Quadrature12.2 Integration over Triangles12.2.1 Simplex Coordinates12.2.2 Radiation Integrals with a Constant Source12.2.2.1 Special Cases12.2.3 Radiation Integrals with a Linear Source12.2.3.1 General Case12.2.3.2 Special Cases12.2.4 Gaussian Quadrature on Triangles12.2.4.1 Comparison with Analytic SolutionReferences
A.1 Field Scattered at a Conducting InterfaceA.2 Plane Wave Decomposition at a Planar InterfaceA.3 Field Scattered at a Dielectric InterfaceA.4 Layered Dielectrics over ConductorReferences

Overview

This book is intended for students, researchers and industry experts working in area of computational electromagnetics and the MoM. Providing a bridge between theory and software implementation, the book incorporates significant background material, while presenting practical, nuts-and-bolts implementation details.