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.
PREFACE xiii(2) ACKNOWLEDGMENTS xv CHAPTER 1 FUNDAMENTAL CONCEPTS 1(36) 1.1 Time-Harmonic Maxwells 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 Poyntings 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
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. Arindam Chaterjee has developed three-dimensional computer simulation of electromagnetic fields for scattering and microwave circuits, and is currently a member of the finite element development group for the HFSS finite element commercial package at Hewlett-Packard. Leo C. Kempel developed three-dimensional antenna simulation packages using the finite element-boundary integral method and has extensive experience with all popular numerical techniques in electromagnetics. He is currently at Mission Research Corporation, Florida, conducting research and development on all aspects of electromagnetics.
