Field Solutions on Computers

Field Solutions on Computers covers a broad range of practical applications involving electric and magnetic fields. The text emphasizes finite-element techniques to solve real-world problems in research and industry. After introducing numerical methods with a thorough treatment of electrostatics, the book moves in a structured sequence to advanced topics. These include magnetostatics with non-linear materials, permanent magnet devices, RF heating, eddy current analysis, electromagnetic pulses, microwave structures, and wave scattering. The mathematical derivations are supplemented with chapter exercises and comprehensive reviews of the underlying physics. The book also covers essential supporting techniques such as mesh generation, interpolation, sparse matrix inversions, and advanced plotting routines.

Chapter 1 Introduction

1

(9)

1.1 Overview

1

(1)

1.2 The Load Ahead

2

(5)

1.3 Some Precautions

7

(3)

Chapter 2 Finite-Element Electrostatic Solutions

10

(34)

2.1 Coulomb's Law

10

(2)

2.2 Gauss' Law and Charge Density

12

(2)

2.3 Differential Equations for Electrostatic Fields

14

(5)

2.4 Charge Density Distributions and Dielectric Materials

19

(4)

2.5 Finite Elements

23

(2)

2.6 Coordinate Relationships for Triangles

25

(3)

2.7 Gauss' Law for Elements at a Vertex Point

28

(5)

2.8 Solution Procedure and Boundary Conditions

33

(2)

2.9 Electrostatic Equations in Cylindrical Coordinates

35

(3)

Exercises

38

(6)

Chapter 3 Minimum-Energy Principles in Electrostatics

44

(18)

3.1 Electrostatic Field Energy

44

(3)

3.2 Elements of the Calculus of Variations

47

(2)

3.3 Poisson Equation as a Condition of Minimum Energy

49

(1)

3.4 Finite-Element Equations for Two-Dimensional Electrostatics

50

(2)

3.5 Three-Dimensional Finite-Element Electrostatics on Arbitrary Meshes

52

(4)

3.6 High-Order Finite-Element Formulations

56

(2)

Exercises

58

(4)

Chapter 4 Finite-Difference Solutions and Regular Meshes

62

(26)

4.1 Difference Operators

62

(7)

4.2 Initial Value Solutions of Ordinary Differential Equations

69

(4)

4.3 One-Dimensional Poisson Equation

73

(2)

4.4 Solution of the Poisson Equation by Back-Substitution

75

(2)

4.5 Two-Dimensional Electrostatic Solutions on a Regular Mesh

77

(4)

4.6 Three-Dimensional Electrostatic Solutions on a Regular Mesh

81

(2)

Exercises

83

(5)

Chapter 5 Techniques for Numerical Field Solutions

88

(24)

5.1 Regular Meshes in Three Dimensions

88

(4)

5.2 Two-Dimensional Conformal Triangular Meshes

92

(4)

5.3 Fitting Triangular Elements to Physical Boundaries

96

(6)

5.4 Neumann Boundaries in Resistive Media

102

(2)

5.5 Boundary Value Solutions by Successive Over-Relaxation

104

(3)

Exercises

107

(5)

Chapter 6 Matrix Methods for Field Solutions

112

(22)

6.1 Gauss-Jordan Elimination

112

(3)

6.2 Solving Tridiagonal Matrices

115

(4)

6.3 Matrix Solutions for One-Dimensional Electrostatics

119

(4)

6.4 Matrices for Two-Dimensional Finite-Element Solutions

123

(4)

6.5 Solving Tridiagonal Block Matrix Problems

127

(3)

Exercises

130

(4)

Chapter 7 Analyzing Numerical Solutions

134

(26)

7.1 Locating Elements

134

(3)

7.2 Generalized Least-Squares Fits

137

(4)

7.3 Field Calculations on a Two-Dimensional Triangular Mesh

141

(4)

7.4 Mesh and Boundary Plots

145

(3)

7.5 Contour, Element, Elevation, and Field Line Plots

148

(6)

Exercises

154

(6)

Chapter 8 Nonlinear and Anisotropic Materials

160

(22)

8.1 Iterative Solutions to Boundary Value Problems

160

(3)

8.2 Numerical Data for Material Properties

163

(6)

8.3 Finite-Element Equations for Anisotropic Materials

169

(4)

8.4 Steady-State Gas Flow

173

(6)

Exercises

179

(3)

Chapter 9 Finite-Element Magnetostatic Solutions

182

(36)

9.1 Differential and Integral Magnetostatic Equations

182

(7)

9.2 Vector Potential and Field Equations in Two Dimensions

189

(4)

9.3 Isotropic Magnetic Materials

193

(5)

9.4 Finite-Element Magnetostatic Equations

198

(3)

9.5 Magnetic Field Solutions

201

(3)

9.6 Properties of Permanent Magnet Materials

204

(4)

9.7 Magnetostatic Solutions with Permanent Magnets

208

(4)

Exercises

212

(6)

Chapter 10 Static Field Analysis and Applications

218

(34)

10.1 Volume and Surface Integrals on a Finite-Element Mesh

218

(1)

10.2 Electric and Magnetic Field Energy

218

(3)

10.3 Capacitance Calculations

221

(2)

10.4 Inductance Calculations

223

(3)

10.5 Electric and Magnetic Forces on Materials

226

(3)

10.6 Charged-Particle Orbits

229

(4)

10.7 Electron and Ion Guns

233

(6)

10.8 Generalized Neumann Boundaries -- Hall-Effect Devices

239

(5)

Exercises

244

(8)

Chapter 11 Low-Frequency Electric and Magnetic Fields

252

(22)

11.1 Maxwell Equations

252

(3)

11.2 Complex Numbers for Harmonic Quantities

255

(3)

11.3 Electric Field Equations in Resistive Media

258

(2)

11.4 Electric Field Solutions with Complex Number Potentials

260

(3)

11.5 Magnetic Fields with Eddy Currents

263

(4)

Exercises

267

(7)

Chapter 12 Thermal Transport and Magnetic Field Diffusion

274

(24)

12.1 Thermal Transport Equation

274

(3)

12.2 Finite-Difference Solution of the Diffusion Equation

277

(3)

12.3 Finite-Element Diffusion Solutions

280

(2)

12.4 Instabilities in Finite-Element Diffusion Solutions

282

(5)

12.5 Magnetic Field Diffusion

287

(6)

Exercises

293

(5)

Chapter 13 Electromagnetic Fields in One Dimension

298

(32)

13.1 Planar Electromagnetic Waves

298

(5)

13.2 Time-Domain Electromagnetism in One Dimension

303

(4)

13.3 Electromagnetic Pulse Solutions

307

(7)

13.4 Frequency-Domain Equations

314

(3)

13.5 Scattering Solutions

317

(2)

13.6 One-Dimensional Resonant Modes

319

(6)

Exercises

325

(5)

Chapter 14 Two- and Three-Dimensional Electromagnetic Simulations

330

(41)

14.1 Time-Domain Equations on a Conformal Mesh

330

(4)

14.2 Electromagnetic Pulse Solutions

334

(6)

14.3 Frequency-Domain Equations

340

(3)

14.4 Methods for Scattering Solutions

343

(5)

14.5 Waveguides and Resonant Cavities

348

(2)

14.6 Power Losses and Q Factors

350

(6)

14.7 Finite-Difference Time-Domain Method in Three Dimensions

356

(4)

14.8 Three-Dimensional Element-Based Time-Domain Equations

360

(6)

Exercises

366

(5)

Bibliography

371

(4)

Index

375

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Field Solutions on Computers

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