Finite-element Methods for Electromagnetics

Preface to the electronic edition

There were many texts available on finite-element methods for electromagnetic fields when the

print version of this book was released in 1997, and there are even more now. Most of them are

mathematically oriented, presenting a generalized, sophisticated theoretical framework. This

book differs in two respects:

• It emphasizes the physical properties of electric and magnetic fields and how they can be

translated to a numerical representation.

• It shows how to do useful things, concentrating on the nuts-and-bolts of numerical solu-

tions.

Mexico. I would like to thank Ashley Gasque of Taylor and Francis for arranging permission

to create and to distribute this electronic text. The PDF file was generated from my original

manuscript and illustrations using Latex. The conversion effort was supported by Field Preci-

sion LLC. Please note that the PDF file and the included material are copyrighted by Stanley

Humphries (2010). You have permission to download the file and to produce a single printed

copy. The following activities are illegal without the copyright holder’s permission:

• Distributing the PDF file or any included material in electronic or printed form.

• Posting the PDF file or any portion of it on the Internet for download.

• Adding any portion of the text or illustrations to another work, either published or un-

published.

http://www.fieldp.com/sate.html

A three-dimensional electromagnetic code that employs the FETD methods covered in Sect.

14.8 is available at:

http://www.fieldp.com/emp3.html

Finally, please contact me if you find errors in the text. The Latex project can easily be

modified and regenerated.

-Stanley Humphries, Jr., Albuquer que, New Mexico, U.S.A., January 2010

Contents

1 Introduction 1

1.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

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

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 th ree dimensions . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2 Two-dimensional conformal triangular meshes . . . . . . . . . . . . . . . . . . . 77

5.3 Fitting triangular elements to physical boundaries . . . . . . . . . . . . . . . . . 83

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5.4 Neumann boundaries in resistive media . . . . . . . . . . . . . . . . . . . . . . . 87

6.5 Solving Tridiagonal Block Matrix Problems . . . . . . . . . . . . . . . . . . . . . 103

7 Analyzing Numerical Solutions 109

7.1 Locating elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

7.2 Generalized least-squares fits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

7.3 Field calculations on a two-dimensional triangular mesh . . . . . . . . . . . . . . 115

7.4 Mesh and boundary plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

7.5 Contour, element, elevation and field line plots . . . . . . . . . . . . . . . . . . . 121

8 Non-linear and Anisotropic Materials 131

8.1 Iterative solutions to boundary value pr oblems . . . . . . . . . . . . . . . . . . . 131

8.2 Numerical data for material properties . . . . . . . . . . . . . . . . . . . . . . . 134

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

11.4 Electric field solutions with complex number potentials . . . . . . . . . . . . . . 215

11.5 Magnetic fields with eddy currents . . . . . . . . . . . . . . . . . . . . . . . . . 218

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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

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

References 309

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