Numerical Methods in Electromagnetism

Numerical Methods in Electromagnetism......Page 4
Copyright Page......Page 5
Contents......Page 6
Foreword......Page 12
Preface......Page 14
1.2 Static Electric Fields......Page 16
1.3 The Electric Potential......Page 18
1.4 Electric Fields and Materials......Page 26
1.5 Interface Conditions on the Electric Field......Page 28
1.6 Laplace's and Poisson's Equations......Page 30
1.7 Static Magnetic Fields......Page 39
1.8 Energy in the Magnetic Field......Page 51
1.9 Quasi-statics: Eddy Currents and Diffusion......Page 54
1.10 The Wave Equation......Page 57
1.11 Discussion of Choice of Variables......Page 59
1.12 Classification of Differential Equations......Page 75
2.1 Introduction and Historical Background......Page 78
2.2 Graphical Methods......Page 80
2.4 Experimental Methods......Page 83
2.5 ElectroConducting Analog......Page 84
2.6 Resistive Analog......Page 85
2.7 Closed Form Analytical Methods......Page 86
2.8 Discrete Analytical Methods......Page 89
2.9 Transformation Methods for Nonlinear Problems......Page 90
2.10 Nonlinear Magnetic Circuit Analysis......Page 94
2.11 Finite Difference Method......Page 95
2.12 Integral Equation Method......Page 99
2.13 The Finite Element Method......Page 111
3.1 Introduction......Page 120
3.2 Difference Equations......Page 121
3.3 Laplace's and Poisson's Equations......Page 123
3.4 Interfaces Between Materials......Page 124
3.5 Neumann Boundary Conditions......Page 126
3.6 Treatment of Irregular Boundaries......Page 129
3.7 Equivalent Circuit Representation......Page 130
3.8 Formulas For High-Order Schemes......Page 132
3.9 Finite Differences With Symbolic Operators......Page 137
3.10 Diffusion Equation......Page 141
3.11 Conclusions......Page 156
4.1 Introduction......Page 158
4.2 The Variational Method......Page 159
4.3 The Functional and its Extremum......Page 160
4.4 Functional in more than one space variable and its extremum......Page 168
4.5 Derivation of the Energy-Related Functional......Page 172
4.6 Ritz's method......Page 185
4.7 The Wave Equation......Page 191
4.8 Variational Method for Integral Equations......Page 194
4.9 Introduction to The Galerkin Method......Page 197
4.10 Example of the Galerkin Method......Page 198
5.1 Introduction......Page 204
5.2 Polynomial Interpolation......Page 216
5.3 Deriving Shape Functions......Page 222
5.4 Lagrangian Interpolation......Page 226
5.5 Two-Dimensional Elements......Page 229
5.6 High-Order Triangular Interpolation Functions......Page 237
5.7 Rectangular Elements......Page 242
5.8 Derivation of Shape Functions for Serendipity Elements......Page 249
5.9 Three-Dimensional Finite Elements......Page 255
5.10 Orthogonal Basis Functions......Page 293
6.1 Introduction......Page 298
6.2 Functional minimization and global assembly......Page 310
6.3 Solution to the nonlinear magnetostatic problem with first-order triangular finite elements......Page 317
6.4 Application of the Newton–Raphson Method to a First-Order Element......Page 321
6.5 Discretization of Time by the Finite Element Method......Page 325
6.6 Axisymmetric Formulation for the Eddy Current Problem Using Vector Potential......Page 328
6.7 Finite Difference and First-Order Finite Elements......Page 335
6.8 Galerkin Finite Elements......Page 337
6.9 Three-Element Magnetostatic Problem......Page 341
6.10 Permanent Magnets......Page 353
6.11 Numerical Example of Matrix Formation for Isoparametric Elements......Page 357
6.12 Edge Elements......Page 368
7.2 Basic Integral Equations......Page 374
7.3 Method of Moments......Page 377
7.4 The Charge Simulation Method......Page 385
7.5 Boundary Element Equations for Poisson's Equation in Two Dimensions......Page 389
7.6 Example of BEM Solution of a Two-Dimensional Potential Problem......Page 396
7.7 Axisymmetric Integral Equations for Magnetic Vector Potential......Page 404
7.8 Two-Dimensional Eddy Currents With T – Ω......Page 408
7.9 BEM Formulation of The Scalar Poisson Equation in Three Dimensions......Page 418
7.10 Green's functions for some typical electromagnetics applications......Page 424
8.2 Hybrid Harmonic Finite Element Method......Page 428
8.3 Infinite Elements......Page 432
8.4 Ballooning......Page 442
8.5 Infinitesimal Scaling......Page 448
8.6 Hybrid Finite Element–Boundary Element Method......Page 452
9.1 Introduction......Page 466
9.2 Finite Element Formulation in Two Dimensions......Page 467
9.3 Boundary Element Formulation......Page 474
9.4 Implementation of the Hybrid Method (HEM)......Page 482
9.5 Evaluation of the Far-Field......Page 486
9.6 Scattering Problems......Page 496
9.7 Numerical Examples......Page 503
9.8 Three Dimensional FEM Formulation for the Electric Field......Page 509
9.9 Example......Page 532
10.1 Time Domain Modeling of Electromechanical Devices......Page 534
10.2 Modeling of Flow Electrification in Insulating Tubes......Page 565
10.3 Coupled Finite Element and Fourier Transform Method for Transient Scalar Field Problems......Page 576
10.4 Axiperiodic Analysis......Page 585
11.1 Introduction......Page 606
11.2 Direct Methods......Page 610
11.3 LU Decomposition......Page 613
11.4 Cholesky Decomposition......Page 620
11.5 Sparse Matrix Techniques......Page 622
11.6 The Preconditioned Conjugate Gradient Method......Page 642
11.7 GMRES......Page 660
11.8 Solution of Nonlinear Equations......Page 673
APPENDIX A. VECTOR OPERATORS......Page 722
APPENDIX B. TRIANGLE AREA IN TERMS OF VERTEX COORDINATES......Page 724
APPENDIX C. FOURIER TRANSFORM METHOD......Page 726
C.1 Computation of Element Coefficient Matrices and Forcing Functions......Page 729
APPENDIX D. INTEGRALS OF AREA COORDINATES......Page 734
APPENDIX E. INTEGRALS OF VOLUME COORDINATES......Page 736
APPENDIX F. GAUSS–LEGENDRE QUADRATURE FORMULAE, ABSCISSAE, AND WEIGHT COEFFICIENTS......Page 738
APPENDIX G. SHAPE FUNCTIONS FOR 1D FINITE ELEMENTS......Page 740
APPENDIX H. SHAPE FUNCTIONS FOR 2D FINITE ELEMENTS......Page 742
APPENDIX I. SHAPE FUNCTIONS FOR 3D FINITE ELEMENTS......Page 750
REFERENCES......Page 764
INDEX......Page 774