Finite Element Method Electromagnetics

Employed in a large number of commercial electromagnetic simulation packages, the finite element method is one of the most popular and well-established numerical techniques in engineering. This book covers the theory, development, implementation, and application of the finite element method and its hybrid versions to electromagnetics. FINITE ELEMENT METHOD FOR ELECTROMAGNETICS begins with a step-by-step textbook presentation of the finite method and its variations then goes on to provide up-to-date coverage of three dimensional formulations and modern applications to open and closed domain problems. Worked out examples are included to aid the reader with the fine features of the method and the implementation of its hybridization with other techniques for a robust simulation of large scale radiation and scattering. The crucial treatment of local boundary conditions is carefully worked out in several stages in the book. Sponsored by: IEEE Antennas and Propagation Society.

About the Authors John L. Volakis is professor at the Department of Electrical Engineering and Computer Science at the University of Michigan. He has published more than 140 refereed journal articles and more than 140 conference papers on numerical and analytical techniques in electromagnetics. Dr. Volakis is also coauthor of Approximate Boundary Conditions in Electromagnetics (IEE Press, 1995) and several book chapters.

PREFACE

xiii

(2)

ACKNOWLEDGMENTS

xv

CHAPTER 1 FUNDAMENTAL CONCEPTS

1

(36)

1.1 Time-Harmonic Maxwell's Equations

1

(4)

1.2 Wave Equation

5

(1)

1.3 Electrostatics and Magnetostatics

6

(3)

1.3.1 Electrostatics

6

(3)

1.3.2 Magnetostatics

9

(1)

1.4 Surface Equivalence

9

(5)

1.5 Natural Boundary Conditions

14

(3)

1.6 Approximate Boundary Conditions

17

(3)

1.6.1 Impedance Boundary Conditions

17

(2)

1.6.2 Sheet Transition Conditions

19

(1)

1.7 Poynting's Theorem

20

(2)

1.8 Uniqueness Theorem

22

(1)

1.9 Superposition Theorem

23

(1)

1.10 Duality Theorem

23

(1)

1.11 Numerical Techniques

24

(13)

1.11.1 The Ritz Method

24

(3)

1.11.2 Functionals for Anisotropic Media

27

(1)

1.11.3 Method of Weighted Residuals

28

(1)

1.11.4 Vector and Matrix Norms in Linear Space

29

(2)

1.11.5 Some Matrix Definitions

31

(1)

1.11.6 Comparison of Solution Methods and Their Convergence

32

(2)

1.11.7 Field Formulation Issues

34

(3)

CHAPTER 2 SHAPE FUNCTIONS FOR SCALAR AND VECTOR FINITE ELEMENTS

37

(28)

2.1 Introduction

37

(1)

2.2 Features of Finite Element Shape Functions

38

(1)

2.2.1 Spatial Locality

38

(1)

2.2.2 Approximation Order

38

(1)

2.2.3 Continuity

38

(1)

2.3 Node-Based Elements

39

(9)

2.3.1 One-Dimensional Basis Functions

39

(1)

2.3.2 Two-Dimensional Basis Functions

40

(5)

2.3.3 Three-Dimensional Basis Functions

45

(3)

2.4 Edge-Based Elements

48

(17)

2.4.1 Two-Dimensional Basis Functions

49

(4)

2.4.2 Three-Dimensional Basis Functions

53

(12)

CHAPTER 3 OVERVIEW OF THE FINITE ELEMENT METHOD: ONE-DIMENSIONAL EXAMPLES

65

(28)

3.1 Introduction

65

(1)

3.2 Overview of the Finite Element Method

66

(3)

3.3 Examples of One-Dimensional Problems in Electromagnetics

69

(2)

3.4 The Weighted Residual Method

71

(2)

3.5 Discretization of the "Weak" Differential Equation

73

(3)

3.6 Assembly of the Element Equations

76

(3)

3.7 Enforcement of Boundary Conditions

79

(4)

3.7.1 Neumann Boundary Conditions (Homogeneous)

80

(1)

3.7.2 Dirichlet Boundary Conditions (Homogeneous)

80

(1)

3.7.3 Nonzero Boundary Constraints (Inhomogeneous)

81

(1)

3.7.4 Impedance Boundary Conditions

82

(1)

3.8 Examples

83

(6)

Appendix 1: Sample One-Dimensional MATLAB FEM Analysis Program

89

(2)

Appendix 2: Useful Integration Formulae for One-Dimensional FEM Analysis

91

(2)

CHAPTER 4 TWO-DIMENSIONAL APPLICATIONS

93

(64)

4.1 Introduction

93

(1)

4.2 Two-Dimensional Wave Equations

94

(6)

4.2.1 Transmission Lines

94

(1)

4.2.2 Two-Dimensional Scattering

95

(2)

4.2.3 Waveguide Propagation (Homogeneous Cross Section)

97

(1)

4.2.4 Waveguide Propagation (Inhomogeneous Cross Section)

98

(2)

4.3 Discretization of the Two-Dimensional Wave Equation

100

(18)

4.3.1 Weak Form of the Wave Equation

101

(1)

4.3.2 Discretization of the Weak Wave Equation

102

(3)

4.3.3 Assembly of Element Equations

105

(3)

4.3.4 Assembly Example: Waveguide Eigenvalues

108

(10)

4.4 Two-Dimensional Scattering

118

(19)

4.4.1 Treatment of Metallic Boundaries

119

(2)

4.4.2 Absorbing Boundary Conditions

121

(3)

4.4.3 Scattered Field Computation

124

(3)

4.4.4 Scattering Example Using ABCs

127

(3)

4.4.5 Artificial Absorbers for Mesh Truncation

130

(4)

4.4.6 Boundary Integral Mesh Truncation

134

(3)

4.5 Edge Elements

137

(12)

4.5.1 Example 1: Propagation Constants of a Homogeneously Filled Waveguide

144

(1)

4.5.2 Example 2: Scattering by a Square-Shaped Material Coated Cylinder

145

(4)

Appendix 1: Element Matrix for Node-Based Bilinear Rectangles

149

(1)

Appendix 2: Sample MATLAB Code for Implementing the Matrix Assembly

150

(7)

CHAPTER 5 THREE-DIMENSIONAL PROBLEMS: CLOSED DOMAIN

157

(26)

5.1 Introduction

157

(1)

5.2 Formulation

158

(5)

5.2.1 Field Formulation

159

(3)

5.2.2 Potential Formulation

162

(1)

5.3 Origin of Spurious Solutions

163

(1)

5.4 Matrix Generation and Assembly

164

(4)

5.5 Source Modeling

168

(3)

5.6 Applications

171

(5)

5.6.1 Cavity Resonators

171

(2)

5.6.2 Circuit Applications

173

(3)

Appendix: Edge-Based Right Triangular Prisms

176

(7)

CHAPTER 6 THREE-DIMENSIONAL PROBLEMS: RADIATION AND SCATTERING

183

(44)

6.1 Introduction

183

(1)

6.2 Survey of Vector ABCs

184

(17)

6.2.1 Three-Dimensional Vector ABCs

184

(10)

6.2.2 Artificial Absorbers

194

(7)

6.3 Formulation

201

(3)

6.3.1 Scattered and Total Field Formulations

201

(3)

6.4 Applications

204

(17)

6.4.1 Scattering Examples

205

(11)

6.4.2 Antenna and Circuit Examples

216

(5)

Appendix: Derivation of Some Vector Identities

221

(6)

CHAPTER 7 THREE-DIMENSIONAL FE-BI METHOD

227

(50)

7.1 Introduction

227

(1)

7.2 General Formulation

228

(10)

7.2.1 Derivation of the FE-BI Equations

229

(4)

7.2.2 Solution of the FE-BI Equations

233

(3)

7.2.3 Comments on the General FE-BI Formulation

236

(2)

7.3 Excitation and Feed Modeling

238

(7)

7.3.1 Plane Wave

238

(1)

7.3.2 Probe Feed

239

(1)

7.3.3 Voltage Gap Feed

240

(1)

7.3.4 Coaxial Cable Feed

241

(1)

7.3.5 Aperture-Coupled Microstrip Line

242

(1)

7.3.6 Mode Matched Feed

243

(2)

7.4 Cavity Recessed in a Ground Plane

245

(12)

7.4.1 Formulation

246

(1)

7.4.2 Solution Using Brick Elements

247

(2)

7.4.3 FFT-Based Matrix-Vector Multiply Scheme

249

(3)

7.4.4 Examples

252

(3)

7.4.5 Aperture in a Thick Metallic Plane

255

(2)

7.5 Cavity-Backed Antennas on a Circular Cylinder

257

(5)

7.5.1 Examples

260

(2)

7.6 Recent Advances in the FE-BI Method

262

(5)

7.6.1 Finite Element-Periodic Method of Moments

262

(2)

7.6.2 Finite Element-Surface of Revolution Method

264

(2)

7.6.3 Fast Integral Solution Methods

266

(1)

Appendix 1: Explicit Formulas for Brick Elements

267

(5)

Appendix 2: Brick Finite Element-Boundary Integral Computer Program

272

(5)

CHAPTER 8 FAST INTEGRAL METHODS

277

(22)

S. Bindiganavale

J. L. Volakis

8.1 The Adaptive Integral Method

277

(2)

8.2 Fast Multipole Method

279

(8)

8.2.1 Boundary Integral Equation

279

(1)

8.2.2 Exact FMM

280

(3)

8.2.3 Windowed FMM

283

(1)

8.2.4 Fast Far Field Algorithm

284

(3)

8.3 Logic Flow

287

(7)

8.4 Results

294

(5)

CHAPTER 9 NUMERICAL ISSUES

299

(38)

9.1 Introduction

299

(1)

9.2 Sparse Storage Schemes

300

(3)

9.3 Direct Equation Solver

303

(4)

9.3.1 Factorization Schemes

303

(1)

9.3.2 Error Control

304

(1)

9.3.3 Matrix Ordering Strategies

305

(2)

9.4 Iterative Equation Solvers

307

(6)

9.5 Preconditioning

313

(7)

9.5.1 Diagonal Preconditioner

313

(2)

9.5.2 Incomplete LU (ILU) Preconditioner

315

(3)

9.5.3 Approximate Inverse Preconditioner

318

(2)

9.5.4 Flexible GMRES with Preconditioning

320

(1)

9.6 Eigenanalysis

320

(7)

9.6.1 Direct and Inverse Iteration

322

(2)

9.6.2 Simultaneous Iteration

324

(1)

9.6.3 Lanczos Algorithm

325

(2)

9.7 Parallelization

327

(10)

9.7.1 Analysis of Communication

330

(7)

INDEX

337

(6)

ABOUT THE AUTHORS

343

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Finite Element Method Electromagnetics Antennas, Microwave Circuits, and Scattering Applications

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