Multigrid Finite Element Methods for Electromagnetic Field Modeling: Multigrid Finite Element Methods for Electromagnetic Field Modeling .part1.rar

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ISBN-1 3 978-0-471-741 10-7
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10 9 8 7 6 5 4 3 2 1

CONTENTS
List of Figures
List of Tables
Preface
Acknowledgments
1 Introduction
1.1 Statement of the Boundary Value Problem
1.2 Ritz Finite Element Method
1.3 Petrov-Galerkin’s Finite Element Method
1.4 Time-Harmonic Maxwell’s Equations and Boundary Conditions
1.4.1 Boundary conditions at material interfaces
1.4.2 Boundary conditions at the enclosing boundary
1.4.3 Uniqueness in the presence of impedance boundaries
1.5 Present and Future Challenges in Finite Element Modeling
2 Hierarchical Basis Functions for Triangles and Tetrahedra
2.1 The Importance of Proper Choice of Finite Element Bases
2.2 Two-Dimensional Finite Element Spaces
2.2.1 Two-dimensional potential space
2.2.2 Two-dimensional field space
viii CONTENTS
2.2.3 Two-dimensional flux space
2.2.4 Two-dimensional charge space
2.3 Relationship Among 2D Finite Element Spaces
2.4 Gradient, Curl and Divergence Matrices for 2D Finite Element Spaces
2.5 Three-Dimensional Finite Element Spaces
2.5.1 Three-dimensional potential space
2.5.2 Three-dimensional field space
2.5.3 Three-dimensional flux space
2.5.4 Three-dimensional charge space
Relationship Among 3D Finite Element Spaces
Gradient, Curl and Divergence Matrices for 3D Finite Element Spaces
The Spaces ‘HP(curZ) and ‘HP(div)
The Issue of Orthogonality in Hierarchical Bases
3 Finite Element Formulations of Electromagnetic BVPs
3.1 Electrostatic Boundary Value Problems

Governing equations and boundary conditions
Weak statement of the electrostatic BVP
The case of unbounded domains
3.2 Magnetostatic Boundary Value Problems
3.2.1
3.2.2
3.2.3 Existence of solution (solvability)
3.2.4 Uniqueness of solution
3.3 Magneto-Quasi-Static (Eddy-Current) Problems
3.3.1
3.3.2 Electric field formulation
3.3.3 Potential formulation
3.4 Full-Wave Boundary Value Problems
3.4.1
3.4.2 Electric field formulation
3.4.3 Potential formulation
3.4.4 Field-flux formulation
Partial Element Equivalent Circuit Model
3.5.1 Electric field integral equation
3.5.2 Development of the PEEC model
3.5.3 The case of surface current flow
3.5.4 Low-frequency numerical instability
Governing equations and boundary conditions
Weak statement of the magnetostatic BVP
Governing equations and boundary conditions
Governing equations and boundary conditions
4 Iterative Methods, Preconditioners, and Multigrid
4.1 Definitions
111
112
CONTENTS ix
4.1.1 Vector space, inner product, and norm 112
4.1.2 Matrix eigenvalues and eigenvectors 113
4.1.3 Properties of Hermitian matrices 114
4.1.4 Positive definite matrices 115
4.1.5 Independence, invariant subspaces, and similarity transformations 115
4.2 Iterative Methods for the Solution of Large Matrices
4.2.1 Stationary methods
4.2.2 Convergence of iterative methods
4.2.3 Non-stationary methods
4.3 Generalized Minimum Residual Method
4.4 Conjugate Gradient Method
4.5 The Preconditioner Matrix
4.5.1 The Jacobi preconditioner
4.5.2 The symmetric Gauss-Seidel preconditioner
4.5.3 Incomplete LU factorization
Multigrid Process and Its Use as a Preconditioner
4.6.1 Motivation for multigrid
4.6.2 The two-grid process
4.6.3 The multigrid process
4.6
5 Nested Multigrid Preconditioner
5.1 Weak Statement of the Two-Dimensional Helmholtz Equation
5.1.1 Total field formulation
5.1.2 Scattered field formulation
5.2 Development of the Finite Element System
5.3 Nested Multigrid Preconditioner
5.4 Intergrid Transfer Operators
5.5 Applications
5.5.1
5.5.2
5.5.3
Plane wave scattering by a PEC cylinder
Plane wave scattering by dielectric cylinder
Plane wave scattering by electrically large cylinders
6 Nested Multigrid Vector and Scalar Potential Preconditioner
6.1 'rho-Dimensional Electromagnetic Scattering
6.1.1
6.1.2
6.1.3 'rho-dimensional potential formulation
6.1.4 Nested multigrid potential preconditioner
6.1.5 lko-dimensional intergrid transfer operators
6.1.6 Applications
lko-dimensional field formulation - TE, case
The finite element matrix and its properties
6.2 Three-Dimensional Electromagnetic Scattering
X CONTENTS
6.2.1 Three-dimensional field formulation
6.2.2 Three-dimensional potential formulation
6.2.3 Nested multigrid potential preconditioner
6.2.4 Three-dimensional intergrid transfer operators
6.2.5
6.2.6
6.2.7 Applications
Finite Element Modeling of Passive Microwave Components
6.3.1
6.3.2 Transfinite-element boundary truncation
6.3.3
6.3.4 Applications
Symmetry of the Nested Multigrid Potential Preconditioner
6.4.1 Potential smoothing operators
6.4.2 Symmetric nested two-grid potential preconditioner
Grid truncation via a boundary integral operator
Approximate boundary integral equation preconditioner
6.3
Electromagnetic ports and associated boundary condition
Nested multigrid potential TFE preconditioner
6.4
7 Hierarchical Multilevel and Hybrid Potential Preconditioners
7.1 Higher-Order Field Formulation
7.2 Higher-Order Potential Formulation
7.3 Hierarchical Multilevel Potential Preconditioner
7.4 Hybrid MultileveVMultigrid Potential Preconditioner
7.5 Numerical Experiments
7.6 Symmetric Hierarchical Multilevel Potential Preconditioner
7.6.1 Potential smoothing operations
7.6.2
Key Attributes of Multigrid and Multilevel Potential Preconditioners
Symmetric hierarchical two-level potential preconditioner
7.7
8 Krylov-Subspace Based Eigenvalue Analysis
8.1
8.2
8.3
8.4
8.5
Subspace Iteration
Methods Based on Krylov Subspace Projection
8.2.1 Arnoldi algorithm
8.2.2 Lanczos algorithm
Deflation Techniques
8.3.1
8.3.2
Non-standard Eigenvalue Problems
8.4.1 Generalized eigenvalue problems
8.4.2 Quadratic eigenvalue problems
Shift-and-Invert Preconditioner
Deflation techniques for symmetric matrices
Deflation techniques for non-symmetric matrices
CONTENTS
9 Two-Dimensional Eigenvalue Analysis of Waveguides
FEM Formulations of the -0-Dimensional Eigenvalue Problem
9.1.1
Transverse-Field Methods
Transverse-Longitudinal-Field Methods
9.3.1 Field TLF formulation
9.3.2 Potential TLF formulation
9.3.3
9.3.4 Numerical examples
Transverse-Transverse-Field Method
9.4.1 Finite element formulation
9.4.2 Algorithms
9.4.3 Applications
Equivalent Transmission-Line Formalism for Planar Waveguides
9.5.1 Multi-conductor transmission line theory
9.5.2 S-parameter representation of a section of an MTL
Mathematical statement of the 2D vector eigenvalue problem
Computer algorithms for eigenvalue calculation
10 Three-Dimensional Eigenvalue Analysis of Resonators
10.1 FEM Formulation of the Three-Dimensional Electromagnetic Eigenvalue
Problem
10.1.1 Finite element approximation: The case of symmetric material
tensors
10.1.2 Finite element approximation: The case of Hermitian material
tensors
10.1.3 Lumped parallel resonant circuit
10.2.1 Elimination of spurious DC modes
10.2.2 Extraction of multiple modes
10.3.1 Elimination of spurious DC modes
10.3.2 Extraction of multiple modes
10.4 Multigrid/hlultilevel Eigenvalue Analysis
10.5 Numerical Validation
10.2 Eigensolver for Lossless Media
10.3 Eigensolver for Lossy Media
11 Model Order Reduction of Electromagnetic Systems
11.1 Asymptotic Waveform Evaluation
11.2 Krylov Subspace-Based Model Order Reduction
1 1.2.1 Pad6 Via Lanczos process
11.2.2 Arnoldi process: SISO system
1 1.2.3 Arnoldi process: MIMO system
xi
1 1.3 Passive Reduced-Order Interconnect Macromodeling Algorithm (PRIMA) 335
Xii CONTENTS
1 1.3.1 Preservation of moments in PRIMA
11.3.2 Preservation of passivity
11.3.3 Error estimate in model order reduction
11.3.4 Pole-residue representation of the reduced order model
1 1.4 Model Order Reduction of Electric Field-Based Finite Element Models
11.4.1 Adaptive Lanczos-Pad6 sweep
11.5 Maxwell’s Curl Equations-Based Model Order Reduction
11.5.1 Passivity of discrete model
11.5.2 Incorporation of lumped elements
11.5.3 PRIMA-based model order reduction
1 1.6 Applications
12 Finite Element Analysis of Periodic Structures
12.1 Finite Element Formulation of the Scattering and Radiation Problem
12.2 Computational Domain Truncation Schemes
12.2.1 Space harmonics expansion of EM fields
12.2.2 Grid truncation using transfinite flements
12.2.3 Grid truncation using anisotropic perfectly matched layers
12.3 Finite Element Approximation Inside the Unit Cell
12.4 Periodic Boundary Condition
12.5 MultileveVMultigrid Preconditioner
12.6 Applications
12.7 Finite Element Modeling of Periodic Waveguides
12.8 Application
Appendix A: Identities and Theorems from Vector Calculus

Index 404

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