Numerical Analysis of Electromagnetic Fields

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1 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.

2 Contents Part 1 Universal Concepts for Numerical Analysis of Electromagnetic Field Problems Chapter 1 Fundamental Concepts of Electromagnetic Field Theory Maxwell's equations and boundary value problems Potential equations in different frequency ranges Boundary conditions of the interface Boundary value problems Green's theorem, Green's functions and fundamental Solutions Green's theorem Vector analogue of Green's theorem Green's function Dirac-delta function Green's function Fundamental Solutions Equivalent sources Single layer charge distribution Double layer source distributions Equivalent polarization Charge and magnetization current Integral equations of electromagnetic fields Integral form of Poisson's equation Integral equation for the exterior region Summary 30 References 31 Appendix 1.1 The integral equation of 3-D magnetic fields 31 Chapter 2 General Outline of Numerical Methods Introduction Operator equations 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 XIV Contents Definition and properties of Operators The relationship between the properties of the Operators and the Solution of Operator equations Operator equations of electromagnetic fields Principles of error minimization Principle of weighted residuals Orthogonal projection principle Projection Operator Orthogonal projection Orthogonal projection methods Non-orthogonal projection methods Variational principle Categories of various numerical methods Methods of weighted residuals Method of moments Galerkin's finite element method Collocation methods Boundary element methods Variational approach Summary 58 References 59 Part 2 Domain Methods Cbapter 3 Finite Difference Method (FDM) Introduction Difference formulation of Poisson's equation Discretization mode for 2-D problems Difference equations in 2-D Cartesian coordinates Discretization equation in polar coordinates Discretization formula of axisymmetric fields Discretization formula of the non-linear magnetic fields Difference equations for time-dependent problems Solution methods for difference equations Properties of simultaneous equations Successive over-relaxation (SOR) method Convergence criterion Difference formulations of arbitrary boundaries and interfacial boundaries between different materials Difference formulations on the lines of symmetry Difference equation of a curved boundary 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.

4 Contents XV 3.5 Examples Further discussions about the finite difference method Physical explanation of the finite difference method The error analysis of the finite difference method Difference equation and the principle of weighted residuals Difference equation and the variational principle Summary 92 References 93 Chapter 4 Fundamentals of Finite Element Method (FFM) Introduction General procedures of the finite dement method Domain discretization and shape functions Method using Galerkin residuals Element matrix equations System matrix equation Storage of the system matrix Treatment of the Dirichlet boundary condition Solution methods of finite element equations Direct methods Gaussian elimination method Cholesky's decomposition (triangulär decomposition) Iterative methods Method of over-relaxation iteration Conjugate-gradient method (CGM) Mesh generation Mesh generation of a triangulär element Automatic mesh generation Examples Summary 127 References 128 Chapter 5 Variational Finite Element Method Introduction Basic concepts of the functional and its variations Definition of the functional and its variations The functional The differentiation and Variation of a function 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.

5 XVI Contents Calculus of variations and Euler's equation Euler's equation Euler's equation for multivariable functions The shortest length of a curve Relationship between the Operator equation and the functional Variational expressions for electromagnetic field problems Variational expression for Poisson's equation Mathematical manipulation Physical manipulation Variational expressions for Poisson's equations in piece-wise homogeneous materials Variational expression for the scalar Helmholtz equation Variational expression for the magnetic field in a non-linear medium Variational finite element method Ritz method Finite element method (FEM) Domain discretization Finite element equation of a Laplacian problem Finite element equation for 2-D magnetic fields Finite element equation for non-linear magnetic fields Finite element equation for Helmholtz's equation (2-D-case) Special problems using the finite element method Approaching floating electrodes by the variational finite element method Open boundary problems Introduction Ballooning method Summary 170 References 171 Chapter 6 Elements and Shape Functions Introduction Types and requirements of the approximating functions Lagrange and Hermite shape functions Requirements of the approximating functions Global, natural, and local coordinates Natural coordinates Local coordinates Lagrange shape function Triangulär elements Quadrilateral elements 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.

6 Contents XVII 6.5 Parametric elements Element matrix equation Coordinate transformations, Jacobian matrix Evaluation of the Lagrangian element matrix Universal matrix Hermite shape function One dimensional Hermite shape function Triangulär Hermite shape functions Evaluation of a Hermite element matrix Application discussions 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) Introduction Matrix equations of simulated charges Matrix equation in homogeneous dielectrics Governing equation subject to Dirichlet boundary conditions Governing equation subject to Neumann boundary conditions Mixed boundary conditions and free potential conductors Matrix form of Poisson's equation Matrix equation in piece-wise homogeneous dielectrics Commonly used simulated charges Point Charge Line Charge Ring Charge Charged elliptic cylinder, Applications of the Charge Simulation method Coordinate transformations Transformation matrix Inverse transformation of the field strength Optimized charge Simulation method (OCSM) Objective function Transformation of constrained conditions Examples Error analysis in the charge Simulation method 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.

7 XVIII Contents Error distribution pattern along the electrode contour Factors influencing the errors 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) Introduction Example Surface integral equations Single layer or double layer integral equations Integral equations of the interfacial surface Types of surface boundary elements and surface Charge densities Representations of boundary and Charge density Potential and field strength coefficients for 2-D and axisymmetrical problems Planar element with constant or linear Charge density Arced element with constant or linear Charge density Ring element with linear Charge density Elements for 3-D problems Planar triangulär element Cylindrical tetragonal bilinear element Isoparametric high order element Spline function element Magnetic surface Charge Simulation method Evaluation of Singular integrals The semi-analytical technique Method using coordinate transformations Numerical techniques Combine the analytical integral and Gaussian quadrature Applications Summary 280 References 280 Appendix 8.1 Potential and field strength coefficients of 2-D planar elements with constant and linear charge density Appendix 8.2 Potential and field strength coefficients of 2-D arced elements with constant and linear charge density 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.

8 Contents XIX Chapter 9 Boundary Element Method (BEM) Introduction Boundary element equations Method of weighted residuals Green's theorem Variational principle Boundary integral equation Indirect boundary integral equation Matrix formulations of the boundary integral equation Discretization and shape functions Matrix equation of a 2-dimensional constant element Evaluation of H {j and G {j Evaluation of H ü and G Matrix equation of 2-D linear elements Matrix form of Poisson's equation Matrix equation of a piecewise homogeneous domain Matrix equation of axisymmetric problems Discretization of 3-dimensional problems Use of symmetry Eddy current problems Eddy current equations A-cp formulations T-Q formulations One-dimensional Solution of an Eddy current problem BEM for solving Eddy current problems Surface impedance boundary conditions Non-linear and time-dependent problems BEM for non-linear problems Time-dependent problems Summary 321 References 322 Appendix 9.1 Bessel function 323 Chapter 10 Moment Methods Introduction Basis functions and weighting functions Galerkin's methods Point matching method Sub-regions and sub-sectional basis 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.

9 XX Contents 10.4 Moment methods for solving static field problems Charge distribution of an isolated plate Charge distribution of a charged cylinder Moment methods for solving eddy current problems Integral equation of a 2-D eddy current problem Sub-sectional basis method Moment methods to solve the current distribution of a line antenna Integral equation of a line antenna Solution of Hallen's equation Summary 347 References 348 Part 4 Optimization Methods of Electromagnetic Field Problems Chapter 11 Methods of Applied Optimization Introduction Fundamental concepts Necessary and sufficient conditions for the local minimum Geometrical Interpretation of the minimizer Quadratic functions Basic method for solving unconstrained non-linear optimization problems Stability and convergence Linear search and single variable optimization Golden section method Methods of polynomial interpolation Analytic methods of unconstrained optimization problems The method of steepest descent Conjugate gradient method Conjugate direction Quadratic convergence Selection of conjugate directions Quasi-Newton's methods Davidon-Fletcher-Powell (DFP) method BFGS formulation B matrix formulae Cholesky factorization of the Hessian matrix Method of non-linear least Squares Gauss-Newton method Levenberg-Marquardt method 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.

10 Contents XXI Polytype method Powell's method of quadratic convergence Constrained optimization methods Basic concepts of constrained optimization Kuhn-Tucker conditions Lagrange multiplier method Necessary condition of the first order Necessary and sufficient conditions of the second order Penalty and barrier function methods Sequential unconstrained minimization technique Summary 385 References 386 Chapter 12 Optimizing Electromagnetic Devices Introduction General concepts of Optimum design Objective function Mathematical expressions of the boundary value problem Optimization methods Categories of optimization Contour optimization Method of curvature adjustment Method of charge redistribution Contour optimization by using non-linear programming Problems of domain optimization Field synthesis by using Fredholm's integral equation Domain optimization by using non-linear programming Summary 400 References 400 Subject Index 403

Inverse Problems and Optimal Design in Electricity and Magnetism

Inverse Problems and Optimal Design in Electricity and Magnetism P. Neittaanmäki Department of Mathematics, University of Jyväskylä M. Rudnicki Institute of Electrical Engineering, Warsaw and A. Savini