Pei-bai Zhou Numerical Analysis of Electromagnetic Fields With 157 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Contents Part 1 Universal Concepts for Numerical Analysis of Electromagnetic Field Problems Chapter 1 Fundamental Concepts of Electromagnetic Field Theory... 3 1.1 Maxwell's equations and boundary value problems 3 1.1.1 Potential equations in different frequency ranges 4 1.1.2 Boundary conditions of the interface 8 1.1.3 Boundary value problems 10 1.2 Green's theorem, Green's functions and fundamental Solutions 11 1.2.1 Green's theorem 12 1.2.2 Vector analogue of Green's theorem 14 1.2.3 Green's function 15 1.2.3.1 Dirac-delta function 15 1.2.3.2 Green's function 16 1.2.4 Fundamental Solutions 18 1.3 Equivalent sources 19 1.3.1 Single layer charge distribution 20 1.3.2 Double layer source distributions 22 1.3.3 Equivalent polarization Charge and magnetization current.. 26 1.4 Integral equations of electromagnetic fields 28 1.4.1 Integral form of Poisson's equation 28 1.4.2 Integral equation for the exterior region 29 1.5 Summary 30 References 31 Appendix 1.1 The integral equation of 3-D magnetic fields 31 Chapter 2 General Outline of Numerical Methods 35 2.1 Introduction 35 2.2 Operator equations 37 2.2.1 Hilbert Space 38
XIV Contents 2.2.2 Definition and properties of Operators 40 2.2.3 The relationship between the properties of the Operators and the Solution of Operator equations 42 2.2.4 Operator equations of electromagnetic fields 43 2.3 Principles of error minimization 46 2.3.1 Principle of weighted residuals 47 2.3.2 Orthogonal projection principle 48 2.3.2.1 Projection Operator 49 2.3.2.2 Orthogonal projection 49 2.3.2.3 Orthogonal projection methods 51 2.3.2.4 Non-orthogonal projection methods 51 2.3.3 Variational principle 51 2.4 Categories of various numerical methods 52 2.4.1 Methods of weighted residuals 53 2.4.1.1 Method of moments 53 2.4.1.2 Galerkin's finite element method 54 2.4.1.3 Collocation methods 55 2.4.1.4 Boundary element methods 55 2.4.2 Variational approach 56 2.5 Summary 58 References 59 Part 2 Domain Methods Cbapter 3 Finite Difference Method (FDM) 63 3.1 Introduction 63 3.2 Difference formulation of Poisson's equation 64 3.2.1 Discretization mode for 2-D problems 64 3.2.2 Difference equations in 2-D Cartesian coordinates 64 3.2.3 Discretization equation in polar coordinates 69 3.2.4 Discretization formula of axisymmetric fields 72 3.2.5 Discretization formula of the non-linear magnetic fields... 73 3.2.6 Difference equations for time-dependent problems 73 3.3 Solution methods for difference equations 76 3.3.1 Properties of simultaneous equations 76 3.3.2 Successive over-relaxation (SOR) method 77 3.3.3 Convergence criterion 80 3.4. Difference formulations of arbitrary boundaries and interfacial boundaries between different materials 80 3.4.1 Difference formulations on the lines of symmetry 81 3.4.2 Difference equation of a curved boundary 81 3.4.3 Difference formulations for the interface of different materials 82
Contents XV 3.5 Examples 84 3.6 Further discussions about the finite difference method... 87 3.6.1 Physical explanation of the finite difference method 87 3.6.2 The error analysis of the finite difference method 88 3.6.3 Difference equation and the principle of weighted residuals. 90 3.6.4 Difference equation and the variational principle 91 3.7 Summary 92 References 93 Chapter 4 Fundamentals of Finite Element Method (FFM) 95 4.1 Introduction 95 4.2 General procedures of the finite dement method 96 4.2.1 Domain discretization and shape functions 97 4.2.2 Method using Galerkin residuals 100 4.2.2.1 Element matrix equations 101 4.2.2.2 System matrix equation 106 4.2.2.3 Storage of the system matrix 109 4.2.2.4 Treatment of the Dirichlet boundary condition 111 4.3 Solution methods of finite element equations 113 4.3.1 Direct methods 113 4.3.1.1 Gaussian elimination method 113 4.3.1.2 Cholesky's decomposition (triangulär decomposition) 115 4.3.2 Iterative methods 117 4.3.2.1 Method of over-relaxation iteration 118 4.3.2.2 Conjugate-gradient method (CGM) 119 4.4 Mesh generation 120 4.4.1 Mesh generation of a triangulär element 121 4.4.2 Automatic mesh generation 123 4.5 Examples 124 4.6 Summary 127 References 128 Chapter 5 Variational Finite Element Method 130 5.1 Introduction 130 5.2 Basic concepts of the functional and its variations 131 5.2.1 Definition of the functional and its variations 132 5.2.1.1 The functional 132 5.2.1.2 The differentiation and Variation of a function 134 5.2.1.3 Variation of the functional 135
XVI Contents 5.2.2 Calculus of variations and Euler's equation 136 5.2.2.1 Euler's equation 137 5.2.2.2 Euler's equation for multivariable functions 138 5.2.2.3 The shortest length of a curve 140 5.2.3 Relationship between the Operator equation and the functional. 141 5.3 Variational expressions for electromagnetic field problems.. 142 5.3.1 Variational expression for Poisson's equation 143 5.3.1.1 Mathematical manipulation 143 5.3.1.2 Physical manipulation 146 5.3.2 Variational expressions for Poisson's equations in piece-wise homogeneous materials 147 5.3.3 Variational expression for the scalar Helmholtz equation... 148 5.3.4 Variational expression for the magnetic field in a non-linear medium 150 5.4 Variational finite element method 153 5.4.1 Ritz method 153 5.4.2 Finite element method (FEM) 155 5.4.2.1 Domain discretization 155 5.4.2.2 Finite element equation of a Laplacian problem 156 5.4.2.3 Finite element equation for 2-D magnetic fields 161 5.4.2.4 Finite element equation for non-linear magnetic fields... 163 5.4.2.5 Finite element equation for Helmholtz's equation (2-D-case) 164 5.5 Special problems using the finite element method 166 5.5.1 Approaching floating electrodes by the variational finite element method 166 5.5.2 Open boundary problems 167 5.5.2.1 Introduction 167 5.5.2.2 Ballooning method 168 5.6 Summary 170 References 171 Chapter 6 Elements and Shape Functions 172 6.1 Introduction 172 6.2 Types and requirements of the approximating functions... 173 6.2.1 Lagrange and Hermite shape functions 173 6.2.2 Requirements of the approximating functions 174 6.3 Global, natural, and local coordinates 176 6.3.1 Natural coordinates 176 6.3.2 Local coordinates 182 6.4 Lagrange shape function 183 6.4.1 Triangulär elements 184 6.4.2 Quadrilateral elements 186 6.4.3 Tetrahedral and hexahedral elements 188
Contents XVII 6.5 Parametric elements 189 6.6 Element matrix equation 191 6.6.1 Coordinate transformations, Jacobian matrix 192 6.6.2 Evaluation of the Lagrangian element matrix 193 6.6.3 Universal matrix 195 6.7 Hermite shape function 199 6.7.1 One dimensional Hermite shape function 199 6.7.2 Triangulär Hermite shape functions 200 6.7.3 Evaluation of a Hermite element matrix 202 6.8 Application discussions 204 6.9 Summary 206 References 206 Appendix 6.1 Langrangian shape functions for 2-D cases 207 Appendix 6.2 Commonly used shape functions for 3-D cases 208 Appendix 6.3 The universal matrix of axisymmetric fields 209 Part 3 Boundary Methods Chapter 7 Charge Simulation Method (CSM) 215 7.1 Introduction 215 7.2 Matrix equations of simulated charges 216 7.2.1 Matrix equation in homogeneous dielectrics 216 7.2.1.1 Governing equation subject to Dirichlet boundary conditions 216 7.2.1.2 Governing equation subject to Neumann boundary conditions. 218 7.2.1.3 Mixed boundary conditions and free potential conductors.. 218 7.2.1.4 Matrix form of Poisson's equation 219 7.2.2 Matrix equation in piece-wise homogeneous dielectrics... 220 7.3 Commonly used simulated charges 221 7.3.1 Point Charge 222 7.3.2 Line Charge 223 7.3.3 Ring Charge 223 7.3.4 Charged elliptic cylinder, 225 7.4 Applications of the Charge Simulation method 226 7.5 Coordinate transformations 232 7.5.1 Transformation matrix 233 7.5.2 Inverse transformation of the field strength 234 7.6 Optimized charge Simulation method (OCSM) 235 7.6.1 Objective function 235 7.6.2 Transformation of constrained conditions 237 7.6.3 Examples 237 7.7 Error analysis in the charge Simulation method 241 7.7.1 Properties of the errors 241
XVIII Contents 7.7.2 Error distribution pattern along the electrode contour... 242 7.7.3 Factors influencing the errors 243 7.8 Summary 246 References 247 Appendix 7.1 Formulations for a point Charge 248 Appendix 7.2 Formulations for a line Charge 248 Appendix 7.3 Formulations for a ring Charge 249 Appendix 7.4 Formulations for a charged elliptic cylinder 249 Appendix 7.5 Approximate formulations for calculating K(k) and E(k) 250 Chapter 8 Surface Charge Simulation Method (SSM) 251 8.1 Introduction 251 8.1.1 Example 252 8.2 Surface integral equations 254 8.2.1 Single layer or double layer integral equations 254 8.2.2 Integral equations of the interfacial surface 255 8.3 Types of surface boundary elements and surface Charge densities 257 8.3.1 Representations of boundary and Charge density 257 8.3.2 Potential and field strength coefficients for 2-D and axisymmetrical problems 258 8.3.2.1 Planar element with constant or linear Charge density 259 8.3.2.2 Arced element with constant or linear Charge density 262 8.3.2.3 Ring element with linear Charge density 264 8.3.3 Elements for 3-D problems 267 8.3.3.1 Planar triangulär element 267 8.3.3.2 Cylindrical tetragonal bilinear element 267 8.3.3.3 Isoparametric high order element 268 8.3.3.4 Spline function element 269 8.4 Magnetic surface Charge Simulation method 270 8.5 Evaluation of Singular integrals 272 8.5.1 The semi-analytical technique 273 8.5.2 Method using coordinate transformations 274 8.5.3 Numerical techniques 275 8.5.4 Combine the analytical integral and Gaussian quadrature.. 275 8.6 Applications 275 8.7 Summary 280 References 280 Appendix 8.1 Potential and field strength coefficients of 2-D planar elements with constant and linear charge density... 281 Appendix 8.2 Potential and field strength coefficients of 2-D arced elements with constant and linear charge density... 283 Appendix 8.3 Coefficients of ring elements with linear charge density 285
Contents XIX Chapter 9 Boundary Element Method (BEM) 287 9.1 Introduction 287 9.2 Boundary element equations 288 9.2.1 Method of weighted residuals 288 9.2.2 Green's theorem 291 9.2.3 Variational principle 291 9.2.4 Boundary integral equation 292 9.2.5 Indirect boundary integral equation 295 9.3 Matrix formulations of the boundary integral equation... 295 9.3.1 Discretization and shape functions 296 9.3.2 Matrix equation of a 2-dimensional constant element 298 9.3.2.1 Evaluation of H {j and G {j 300 9.3.2.2 Evaluation of H ü and G 301 9.3.3 Matrix equation of 2-D linear elements 302 9.3.4 Matrix form of Poisson's equation 304 9.3.5 Matrix equation of a piecewise homogeneous domain 305 9.3.6 Matrix equation of axisymmetric problems 306 9.3.7 Discretization of 3-dimensional problems 307 9.3.8 Use of symmetry 309 9.4 Eddy current problems 310 9.4.1 Eddy current equations 310 9.4.1.1 A-cp formulations 311 9.4.1.2 T-Q formulations 311 9.4.2 One-dimensional Solution of an Eddy current problem... 312 9.4.3 BEM for solving Eddy current problems 313 9.4.4 Surface impedance boundary conditions 316 9.5 Non-linear and time-dependent problems 317 9.5.1 BEM for non-linear problems 317 9.5.2 Time-dependent problems 320 9.6 Summary 321 References 322 Appendix 9.1 Bessel function 323 Chapter 10 Moment Methods 327 10.1 Introduction 327 10.2 Basis functions and weighting functions 330 10.2.1 Galerkin's methods 332 10.2.2 Point matching method 332 10.2.3 Sub-regions and sub-sectional basis 334 10.3 Interpretation using variations 335
XX Contents 10.4 Moment methods for solving static field problems 336 10.4.1 Charge distribution of an isolated plate 336 10.4.2 Charge distribution of a charged cylinder 338 10.5 Moment methods for solving eddy current problems 341 10.5.1 Integral equation of a 2-D eddy current problem 341 10.5.2 Sub-sectional basis method 342 10.6 Moment methods to solve the current distribution of a line antenna 343 10.6.1 Integral equation of a line antenna 343 10.6.2 Solution of Hallen's equation 345 10.7 Summary 347 References 348 Part 4 Optimization Methods of Electromagnetic Field Problems Chapter 11 Methods of Applied Optimization 351 11.1 Introduction 351 11.2 Fundamental concepts 351 11.2.1 Necessary and sufficient conditions for the local minimum.. 352 11.2.2 Geometrical Interpretation of the minimizer 354 11.2.3 Quadratic functions 355 11.2.4 Basic method for solving unconstrained non-linear optimization problems 356 11.2.5 Stability and convergence 357 11.3 Linear search and single variable optimization 358 11.3.1 Golden section method 358 11.3.2 Methods of polynomial interpolation 360 11.4 Analytic methods of unconstrained optimization problems.. 362 11.4.1 The method of steepest descent 363 11.4.2 Conjugate gradient method 364 11.4.2.1 Conjugate direction 364 11.4.2.2 Quadratic convergence 364 11.4.2.3 Selection of conjugate directions 365 11.4.3 Quasi-Newton's methods 367 11.4.3.1 Davidon-Fletcher-Powell (DFP) method 368 11.4.3.2 BFGS formulation 370 11.4.3.3 B matrix formulae 371 11.4.3.4 Cholesky factorization of the Hessian matrix 371 11.4.4 Method of non-linear least Squares 372 11.4.4.1 Gauss-Newton method 372 11.4.4.2 Levenberg-Marquardt method 373 11.5 Function comparison methods 374
Contents XXI 11.5.1 Polytype method 374 11.5.2 Powell's method of quadratic convergence 375 11.6 Constrained optimization methods 376 11.6.1 Basic concepts of constrained optimization 377 11.6.2 Kuhn-Tucker conditions 378 11.6.2.1 Lagrange multiplier method 378 11.6.2.2 Necessary condition of the first order 379 11.6.2.3 Necessary and sufficient conditions of the second order... 380 11.6.3 Penalty and barrier function methods 381 11.6.4 Sequential unconstrained minimization technique 384 11.7 Summary 385 References 386 Chapter 12 Optimizing Electromagnetic Devices 387 12.1 Introduction 387 12.2 General concepts of Optimum design 388 12.2.1 Objective function 388 12.2.2 Mathematical expressions of the boundary value problem.. 389 12.2.3 Optimization methods 390 12.2.4 Categories of optimization 391 12.3 Contour optimization 391 12.3.1 Method of curvature adjustment 391 12.3.2 Method of charge redistribution 393 12.3.3 Contour optimization by using non-linear programming... 395 12.4 Problems of domain optimization 396 12.4.1 Field synthesis by using Fredholm's integral equation 396 12.4.2 Domain optimization by using non-linear programming... 398 12.5 Summary 400 References 400 Subject Index 403