The Finite Element Method in Electromagnetics

The finite element method (FEM) is a powerful simulation technique used to solve boundary-value problems in a variety of engineering circumstances. It has been widely used for analysis of electromagnetic fields in antennas, radar scattering, RF and microwave engineering, high-speed/high-frequency circuits, wireless communication, electromagnetic compatibility, photonics, remote sensing, biomedical engineering, and space exploration.

The Finite Element Method in Electromagnetics, Third Edition explains the method’s processes and techniques in careful, meticulous prose and covers not only essential finite element method theory, but also its latest developments and applications—giving engineers a methodical way to quickly master this very powerful numerical technique for solving practical, often complicated, electromagnetic problems.

Featuring over thirty percent new material, the third edition of this essential and comprehensive text now includes:

• A wider range of applications, including antennas, phased arrays, electric machines, high-frequency circuits, and crystal photonics
• The finite element analysis of wave propagation, scattering, and radiation in periodic structures
• The time-domain finite element method for analysis of wideband antennas and transient electromagnetic phenomena
• Novel domain decomposition techniques for parallel computation and efficient simulation of large-scale problems, such as phased-array antennas and photonic crystals

Along with a great many examples, The Finite Element Method in Electromagnetics is an ideal book for engineering students as well as for professionals in the field.

JIAN-MING JIN, PhD, is Y. T. Lo Chair Professor in Electrical and Computer Engineering and Director of the Electromagnetics Laboratory and Center for Computational Electromagnetics at the University of Illinois at Urbana-Champaign. He authored Theory and Computation of Electromagnetic Fields (Wiley) and Electromagnetic Analysis and Design in Magnetic Resonance Imaging, and coauthored Computation of Special Functions (Wiley) and Finite Element Analysis of Antennas and Arrays (Wiley). A Fellow of the IEEE, he is listed by ISI among the world’s most cited authors.

Preface to the First Edition xxiii

Preface to the Second Edition xxvii

1 Basic Electromagnetic Theory 1

1.1 Brief Review of Vector Analysis 2

1.2 Maxwell's Equations 4

1.3 Scalar and Vector Potentials 6

1.4 Wave Equations 7

1.5 Boundary Conditions 8

1.6 Radiation Conditions 11

1.7 Fields in an Infinite Homogeneous Medium 11

1.8 Huygen's Principle 13

1.9 Radar Cross Sections 14

1.10 Summary 15

2 Introduction to the Finite Element Method 17

2.1 Classical Methods for Boundary-Value Problems 17

2.2 Simple Example 21

2.3 Basic Steps of the Finite Element Method 27

2.4 Alternative Presentation of the Finite Element Formulation 34

2.5 Summary 36

3 One-Dimensional Finite Element Analysis 39

3.1 Boundary-Value Problem 39

3.2 Variational Formulation 40

3.3 Finite Element Analysis 42

3.4 Plane-Wave Reflection by a Metal-Backed Dielectric Slab 53

3.5 Scattering by a Smooth, Convex Impedance Cylinder 59

3.6 Higher-Order Elements 62

3.7 Summary 74

4 Two-Dimensional Finite Element Analysis 77

4.1 Boundary-Value Problem 77

4.2 Variational Formulation 79

4.3 Finite Element Analysis 81

4.4 Application to Electrostatic Problems 98

4.5 Application to Magnetostatic Problems 103

4.6 Application to Quasistatic Problems: Analysis of Multiconductor Transmission Lines 105

4.7 Application to Time-Harmonic Problems 109

4.8 Higher-Order Elements 128

4.9 Isoparametric Elements 144

4.10 Summary 149

5 Three-Dimensional Finite Element Analysis 151

5.1 Boundary-Value Problem 151

5.2 Variational Formulation 152

5.3 Finite Element Analysis 153

5.4 Higher-Order Elements 160

5.5 Isoparametric Elements 162

5.6 Application to Electrostatic Problems 168

5.7 Application to Magnetostatic Problems 169

5.8 Application to Time-Harmonic Field Problems 176

5.9 Summary 188

6 Variational Principles for Electromagnetics 191

6.1 Standard Variational Principle 192

6.2 Modified Variational Principle 197

6.3 Generalized Variational Principle 201

6.4 Variational Principle for Anisotrpic Medium 203

6.5 Variational Principle for Resistive Sheets 207

6.6 Concluding Remarks 209

7 Eigenvalue Problems: Waveguides and Cavities 211

7.1 Scalar Formulations for Closed Waveguides 212

7.2 Vector Formulations for Closed Waveguides 225

7.3 Open Waveguides 235

7.4 Three-Dimensional Cavities 238

7.5 Summary 239

8 Vector Finite Elements 243

8.1 Two-Dimensional Edge Elements 244

8.2 Waveguide Problem Revisited 256

8.3 Three-Dimensional Edge Elements 259

8.4 Cavity Problem Revisited 270

8.5 Waveguide Discontinuities 274

8.6 Higher-Order Interpolatory Vector Elements 278

8.7 Higher-Order Hierarchical Vector Elements 293

8.8 Computational Issues 305

8.9 Summary 309

9 Absorbing Boundary Conditions 315

9.1 Two-Dimensional Absorbing Boundary Conditions 316

9.2 Three-Dimensional Absorbing Boundary Conditions 323

9.3 Scattering Analysis Using Absorbing Boundary Conditons 328

9.4 Adaptive Absorbing Boundary Conditons 339

9.5 Fictitious Absorbers 348

9.6 Perfectly Matched Layers 350

9.7 Application of PML to Body-of-Revolutions Problems 368

9.8 Summary 371

10 Finite Element-Boundary Integral Methods 379

10.1 Scattering by Two-Dimensional Cavity-Backed Apertures 381

10.2 Scattering by Two-Dimensional Cylindrical Structures 399

10.3 Scattering by Three-Dimensional Cavity-Backed Apertures 411

10.4 Radiation by Microstrip Patch Antennas in a Cavity 425

10.5 Scattering by General Three-Dimensional Bodies 430

10.6 Solution of the Finite Element-Boundary Integral System 436

10.7 Symmetric Finite Element-Boundary Integral Formulations 447

10.8 Summary 462

11 Finite Element-Eigenfunction Expansion Methods 469

11.1 Waveguide Port Boundary Conditions 470

11.2 Open-Region Scattering 487

11.3 Coupled Basis Functions: The Unimoment Method 494

11.4 Finite Element-Extended Boundary Condition Method 502

11.5 Summary 509

12 Finite Element Analysis in the Time Domain 513

12.1 Finite Element Formulation and Temporal Excitation 514

12.2 Time-Domain Discretization 518

12.3 Stability Analysis 523

12.4 Modeling of Dispersive Media 529

12.5 Truncation via Absorbing Boundary Conditions 538

12.6 Truncation via Perfectly Matched Layers 541

12.7 Truncation via Boundary Integral Equations 551

12.8 Time-Domain Wqaveguide Port Boundary Conditions 562

12.9 Hybrid Field-Circuit Analysis 569

12.10 Dual-Field Domain Decomposition and Element-Level Methods 587

12.11 Discontinuous Galerkin Time-Domain Methods 605

12.12 Summary 625

13 Finite Element Analysis of Periodic Structures 637

13.1 Finite Element Formulation for a Unit Cell 638

13.2 Scattering by One-Dimensional Periodic Structures: Frequency-Domain Analysis 651

13.3 Scattering by One-Dimensional Periodic Structures: Time-Domain Analysis 656

13.4 Scattering by Two-Dimensional Periodic Structures: Frequency-Domain Analysis 663

13.5 Scattering by Two-Dimensonal Periodic Structures: Time-Domain Analysis 670

13.6 Analysis of Angular Periodic Strctures 678

13.7 Summary 682

14 Domain Decompsition for Large-Scale Analysis 687

14.1 Schwarz Methods 688

14.2 Schur Complement Methods 693

14.3 FETI-DP Method for Low-Frequency Problems 705

14.4 FETI-DP Method for High-Frequency Problems 728

14.5 Noncomformal FETI-DP Method Based on Cement Elements 743

14.6 Application of Second-Order Transmission Conditions 753

14.7 Summary 760

15 Solution of Finite Element Equations 767

15.1 Decomposition Methods 769

15.2 Conjugate Gradient Methods 778

15.3 Solution of Eigenvalue Problems 791

15.4 Fast Frequency-Sweep Computation 797

15.5 Summary 803

Appendix A: Basic Vector Identities and Integral Theorems 809

Appendix B: The Ritz Procedure for Complex-Valued Problems 813

Appendix C: Green's Functions 817

Appendix D: Singular Integral Evaluation 825

Appendix E: Some Special Functions 829

Index 837

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