Integral Equations for Real-Life Multiscale Electromagnetic Problems (Electromagnetic Waves)

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Contents
About the editors
1 Introduction
	References
2 Surface integral equation formulations
	2.1 Maxwell’s equations
		2.1.1 Integral form of Maxwell’s equations
		2.1.2 Point or differential form of Maxwell’s equations
		2.1.3 Boundary form of Maxwell’s equations
		2.1.4 The Helmholtz equations and potential representations
		2.1.5 Far fields and far potentials
		2.1.6 The duality principle
		2.1.7 Uniqueness theorem
	2.2 Equivalence principles
		2.2.1 The volumetric equivalence principle
		2.2.2 The surface equivalence principle
	2.3 Boundary field representations
		2.3.1 The Calderón identities
	2.4 The Lorentz reciprocity theorem
	2.5 Surface integral equation formulations and solutions by moment methods
		2.5.1 Surface representation by triangulation
		2.5.2 Defining electromagnetic quantities on a mesh
		2.5.3 The electric field integral equation (EFIE)
		2.5.4 Fill and assembly of element and system matrices and column excitation vectors
		2.5.5 The magnetic field integral equation (MFIE)
		2.5.6 Conducting sheets and the EFIE and MFIE
		2.5.7 Internal resonances and the CFIE
		2.5.8 Integral equation formulations for dielectrics
	2.6 Surface integral equation challenges
		2.6.1 Vector norms, matrix norms, and condition number
		2.6.2 The EFIE and L operator
		2.6.3 The MFIE and K operator
		2.6.4 Mixed operator integral equations
	References
3 Kernel-based fast factorization techniques
	3.1 Introduction
	3.2 Multilevel fast multipole algorithm
		3.2.1 Conventional MLFMA based on plane waves
		3.2.2 Low-frequency and broadband MLFMA implementations
	3.3 Large-scale simulations and parallel computing
	3.4 Material modeling
		3.4.1 Material simulations with the conventional MLFMA
		3.4.2 Simulations of plasmonic structures
		3.4.3 Simulations of near-zero-index (NZI) structures
	3.5 Problems with dense discretizations
	3.6 Problems with non-uniform discretizations
	3.7 Conclusions and new trends
	Acknowledgments
	References
4 Kernel-independent fast factorization methods for multiscale electromagnetic problems
	4.1 Introduction
	4.2 Adaptive cross approximation (ACA) method
	4.3 Multilevel matrix compression method for multiscale problems
		4.3.1 Background and theory
		4.3.2 Accuracy validation
		4.3.3 Computational complexity analysis
		4.3.4 Numerical evaluation of the induced fields in a real-life aircraft
	4.4 Nested equivalence source approximation for low-frequency multiscale problems
		4.4.1 Equivalent source distributions for field representation
		4.4.2 Field representation via equivalent RWG basis functions
		4.4.3 Single-level nested matrix compression approximation algorithm
		4.4.4 Multilevel NESA
		4.4.5 Matrix–vector product and computation complexity
		4.4.6 Numerical results
	4.5 Wideband nested equivalence source approximation for multiscale problems
		4.5.1 Far-field factorization admissibility conditions
		4.5.2 High-frequency-nested approximation in directions
		4.5.3 Multilevel WNESA
		4.5.4 MVP and computation complexity
		4.5.5 Numerical results
	4.6 Mixed-form nested equivalence source approximation for multiscale problems
		4.6.1 Multiscale sampling for skeletons
		4.6.2 Mixed-form wideband-nested approximation
		4.6.3 Numerical results
	4.7 Conclusion and prospect
	Acknowledgments
	References
5 Domain decomposition method (DDM)
	5.1 Discontinuous Galerkin DD method for PEC objects
		5.1.1 Introduction to discontinuous Galerkin method
		5.1.2 SIE formulation
		5.1.3 Domain partitioning and basis function space
		5.1.4 Interior penalty formulation
		5.1.5 Matrix equation and preconditioner
		5.1.6 Iterative solution of preconditioned matrix equation
		5.1.7 Numerical experiments
	5.2 DG DD method for penetrable objects
		5.2.1 DG-DDM-SIE for homogeneous objects
		5.2.2 DG-DDM-SIE for piecewise homogeneous objects
	5.3 Tear-and-interconnect DDM
		5.3.1 Preconditioner formulation
		5.3.2 A note on parallelization
		5.3.3 Numerical examples
	References
6 Multi-resolution preconditioner
	6.1 Preliminaries
		6.1.1 Introduction and scope
		6.1.2 Basis functions
		6.1.3 MoM linear system
		6.1.4 Multi-resolution strategy
	6.2 Basis functions generation
		6.2.1 Generalized basis functions
		6.2.2 Multi-resolution basis functions
		6.2.3 PEC ground plane handling
		6.2.4 Basis for electrical sizes beyond the resonance region
		6.2.5 Algorithm flow chart and computational complexity
	6.3 Generation of a hierarchical family of meshes
		6.3.1 Cells grouping strategy
		6.3.2 Cells ranking and aggregation
		6.3.3 Cells grouping refinement
		6.3.4 Maximum cell size grouping limiting
		6.3.5 Computational complexity
	6.4 Application to MoM
		6.4.1 Change-of-basis matrix memory allocation
		6.4.2 Direct solution
		6.4.3 Application to iterative solvers
		6.4.4 Application to electrically large multi-scale structures
		6.4.5 Low-frequency matrix entries evaluation
	6.5 Numerical results
		6.5.1 Ferrari Testarossa test case
		6.5.2 Realistic vessel test case
	6.6 Conclusion and perspectives
	Acknowledgments
	References
7 Calderón preconditioners for electromagnetic integral equations
	7.1 Introduction
	7.2 Background and notations
	7.3 Calderón identities
	7.4 Discretization
	7.5 Electric field IE
		7.5.1 The original equation
		7.5.2 The preconditioned equation
	7.6 Combined field IE
		7.6.1 The original equation
		7.6.2 The preconditioned equation
	7.7 PMCHWT
		7.7.1 The original equation
		7.7.2 The preconditioned equation
		7.7.3 Different solution strategies
	7.8 Conclusions
	References
8 Decoupled potential integral equation
	8.1 Scattering problem and boundary conditions
	8.2 Low-frequency limit boundary value problems
	8.3 Stabilizing conditions
	8.4 Decoupled potentials and different Lorenz gauge fixings
	8.5 Incoming potentials in a low-frequency stable Lorenz gauge
	8.6 Decoupled potential boundary value problems
	8.7 Second-kind integral equation
	8.8 Discretization of an integral equation of the second kind
		8.8.1 High-order accurate self-interaction integral
	8.9 Near interaction quadrature
	Appendix A: Differential geometry of surfaces
	Appendix B: Numerical integration and interpolation in 1D
	Appendix C: Numerical integration and interpolation in 2D
	Appendix D: Generalized Gaussian quadrature for arbitrary non-smooth functions
	Appendix E: Function spaces
	References
9 Conclusion and perspectives
	References
Index
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