Finite Element Methods For Electromagnetics
Stanley Humphries
Contents
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 1
1.2 Summary of material . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 2
1.3 Some precautions . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 6
2 Finite-element Electrostatic Solutions 9
2.1 Coulomb’s law . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 10
2.2 Gauss’ law and charge density . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 12
2.3 Differential equations for electrostatic fields . . . .
. . . . . . . . . . . . . . . . . 13
2.4 Charge density distributions and dielectric materials .
. . . . . . . . . . . . . . . 17
2.5 Finite elements . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 21
2.6 Coordinate relationships for triangles . . . . . . . . .
. . . . . . . . . . . . . . . 23
2.7 Gauss’s law for elements at a vertex point . . . . . . .
. . . . . . . . . . . . . . 26
2.8 Solution procedure and boundary conditions . . . . . . .
. . . . . . . . . . . . . 30
2.9 Electrostatic equations in cylindrical coordinates . . .
. . . . . . . . . . . . . . . 32
3 Minimum-energy Principles in Electrostatics 37
3.1 Electrostatic field energy . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 37
3.2 Elements of the calculus of variations . . . . . . . . .
. . . . . . . . . . . . . . . 40
3.3 Poisson equation as a condition of minimum energy . . .
. . . . . . . . . . . . . 42
3.4 Finite-element equations for two-dimensional
electrostatics . . . . . . . . . . . . 43
3.5 Three-dimensional finite-element electrostatics on
arbitrary meshes . . . . . . . 45
3.6 High-order finite-element formulations . . . . . . . . .
. . . . . . . . . . . . . . 48
4 Finite-difference Solutions and Regular Meshes 51
4.1 Difference operators . . . . . . . . . . . . . . . . ..
. . . . . . 52
4.2 Initial value solutions of ordinary differential
equations . . . . . . . . . . . . . . 58
4.3 One-dimensional Poisson equation . . . . . . . . . . . .
. . . . . . . . . . . . . . 61
4.4 Solving the Poisson equation by back-substitution . . .
. . . . . . . . . . . . . . 63
4.5 Two dimensional electrostatic solutions on a regular
mesh . . . . . . . . . . . . 64
4.6 Three-dimensional electrostatic solutions on a regular
mesh . . . . . . . . . . . . 68
5 Techniques for Numerical Field Solutions 73
5.1 Regular meshes in three dimensions . . . . . . . . . . .
. . . . . . . . . . . . . . 74
5.2 Two-dimensional conformal triangular meshes . . . . . .
. . . . . . . . . . . . . 77
5.3 Fitting triangular elements to physical boundaries . . .
. . . . . . . . . . . . . . 83
i
5.4 Neumann boundaries in resistive media . . . . . . . . .
. . . . . . . . . . . . . . 87
5.5 The method of successive over-relaxation . . . . . . . .
. . . . . . . . . . . . . . 89
6 Matrix Methods for Field Solutions 93
6.1 Gauss-Jordan elimination . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 94
6.2 Solving tridiagonal matrices . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 96
6.3 Matrix solutions for one-dimensional electrostatics . .
. . . . . . . . . . . . . . . 98
6.4 Matrices for two-dimensional finite-element solutions .
. . . . . . . . . . . . . . 101
6.5 Solving Tridiagonal Block Matrix Problems . . . . . . .
. . . . . . . . . . . . . . 103
7 Analyzing Numerical Solutions 107
7.1 Locating elements . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 107
7.2 Generalized least-squares fits . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 110
7.3 Field calculations on a two-dimensional triangular mesh
. . . . . . . . . . . . . . 113
7.4 Mesh and boundary plots . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . 116
7.5 Contour, element, elevation and field line plots . . . .
. . . . . . . . . . . . . . . 119
8 Non-linear and Anisotropic Materials 131
8.1 Iterative solutions to boundary value problems . . . . .
. . . . . . . . . . . . . . 131
8.2 Numerical data for material properties . . . . . . . . .
. . . . . . . . . . . . . . 134
8.3 Finite-element equations for anisotropic materials . . .
. . . . . . . . . . . . . . 140
9 Finite-element Magnetostatic Solutions 145
9.1 Differential and integral magnetostatic equations . . .
. . . . . . . . . . . . . . . 146
9.2 Vector potential and field equations in two dimensions .
. . . . . . . . . . . . . 152
9.3 Isotropic magnetic materials . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 155
9.4 Finite-element magnetostatic equations . . . . . . . . .
. . . . . . . . . . . . . . 160
9.5 Magnetic field solutions . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 163
9.6 Properties of permanent magnet materials . . . . . . . .
. . . . . . . . . . . . . 165
9.7 Magnetostatic solutions with permanent magnets . . . . .
. . . . . . . . . . . . 169
10 Static Field Analysis and Applications 177
10.1 Volume and surface integrals on a finite-element mesh .
. . . . . . . . . . . . . . 177
10.2 Electric and magnetic field energy . . . . . . . . . .
. . . . . . . . . . . . . . . . 178
10.3 Capacitance calculations . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 180
10.4 Inductance calculations . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 182
10.5 Electric and magnetic forces on materials . . . . . . .
. . . . . . . . . . . . . . 185
10.6 Charged particle orbits . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . 188
10.7 Electron and ion guns . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 190
10.8 Generalized Neumann boundaries - Hall effect devices .
. . . . . . . . . . . . . . 197
11 Low-frequency Electric and Magnetic Fields 207
11.1 Maxwell equations . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 208
11.2 Complex numbers for harmonic quantities . . . . . . . .
. . . . . . . . . . . . . 211
11.3 Electric field equations in resistive media . . . . . .
. . . . . . . . . . . . . . . . 213
11.4 Electric field solutions with complex number potentials
. . . . . . . . . . . . . . 215
11.5 Magnetic fields with eddy currents . . . . . . . . . .
. . . . . . . . . . . . . . . 218
ii
12 Thermal Transport and Magnetic Field Diffusion 225
12.1 Thermal transport equation . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 226
12.2 Finite-difference solution of the diffusion equation .
. . . . . . . . . . . . . . . . 228
12.3 Finite-element diffusion solutions . . . . . . . . . .
. . . . . . . . . . . . . . . . 231
12.4 Instabilities in finite-element diffusion solutions . .
. . . . . . . . . . . . . . . . 234
12.5 Magnetic field diffusion . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 237
13 Electromagnetic Fields in One Dimension 245
13.1 Planar electromagnetic waves . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 246
13.2 Time-domain electromagnetism in one dimension . . . . .
. . . . . . . . . . . . 250
13.3 Electromagnetic pulse solutions . . . . . . . . . . . .
. . . . . . . . . . . . . . . 253
13.4 Frequency-domain equations . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 259
13.5 Scattering solutions . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . 262
13.6 One-dimensional resonant modes . . . . . . . . . . . .
. . . . . . . . . . . . . . 265
14 Two and Three-dimensional Electromagnetic Simulations 273
14.1 Time-domain equations on a conformal mesh . . . . . . .
. . . . . . . . . . . . . 274
14.2 Electromagnetic pulse solutions . . . . . . . . . . . .
. . . . . . . . . . . . . . . 277
14.3 Frequency-domain equations . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 283
14.4 Methods for scattering solutions . . . . . . . . . . .
. . . . . . . . . . . . . . . . 285
14.5 Waveguides and resonant cavities . . . . . . . . . . .
. . . . . . . . . . . . . . . 289
14.6 Power losses and Q factors . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . 291
14.7 Finite-difference time-domain method in three
dimensions . . . . . . . . . . . . 295
14.8 Three-dimensional element-based time-domain equations .
. . . . . . . . . . . . 299
