Numerical Analysis of Electromagnetic Fields

1 Universal Concepts for Numerical Analysis of Electromagnetic Field Problems.- 1 Fundamental Concepts of Electromagnetic Field Theory.- 1.1 Maxwell's equations and boundary value problems.- 1.1.1 Potential equations in different frequency ranges.- 1.1.2 Boundary conditions of the interface.- 1.1.3 Boundary value problems.- 1.2 Green's theorem, Green's functions and fundamental solutions.- 1.2.1 Green's theorem.- 1.2.2 Vector analogue of Green's theorem.- 1.2.3 Green's function.- 1.2.3.1 Dirac-delta function.- 1.2.3.2 Green's function.- 1.2.4 Fundamental solutions.- 1.3 Equivalent sources.- 1.3.1 Single layer charge distribution.- 1.3.2 Double layer source distributions.- 1.3.3 Equivalent polarization charge and magnetization current.- 1.4 Integral equations of electromagnetic fields.- 1.4.1 Integral form of Poisson's equation.- 1.4.2 Integral equation for the exterior region.- 1.5 Summary.- References.- Appendix 1.1 The integral equation of 3-D magnetic fields.- 2 General Outline of Numerical Methods.- 2.1 Introduction.- 2.2 Operator equations.- 2.2.1 Hubert space.- 2.2.2 Definition and properties of operators.- 2.2.3 The relationship between the properties of the operators and the solution of operator equations.- 2.2.4 Operator equations of electromagnetic fields.- 2.3 Principles of error minimization.- 2.3.1 Principle of weighted residuals.- 2.3.2 Orthogonal projection principle.- 2.3.2.1 Projection operator.- 2.3.2.2 Orthogonal projection.- 2.3.2.3 Orthogonal projection methods.- 2.3.2.4 Non-orthogonal projection methods.- 2.3.3 Variational principle.- 2.4 Categories of various numerical methods.- 2.4.1 Methods of weighted residuals.- 2.4.1.1 Method of moments.- 2.4.1.2 Galerkin's finite element method.- 2.4.1.3 Collocation methods.- 2.4.1.4 Boundary element methods.- 2.4.2 Variational approach.- 2.5 Summary.- References.- 2 Domain Methods.- 3 Finite Difference Method (FDM).- 3.1 Introduction.- 3.2 Difference formulation of Poisson's equation.- 3.2.1 Discretization mode for 2-D problems.- 3.2.2 Difference equations in 2-D Cartesian coordinates.- 3.2.3 Discretization equation in polar coordinates.- 3.2.4 Discretization formula of axisymmetric fields.- 3.2.5 Discretization formula of the non-linear magnetic fields.- 3.2.6 Difference equations for time-dependent problems.- 3.3 Solution methods for difference equations.- 3.3.1 Properties of simultaneous equations.- 3.3.2 Successive over-relaxation (SOR) method.- 3.3.3 Convergence criterion.- 3.4. Difference formulations of arbitrary boundaries and interfacial boundaries between different materials.- 3.4.1 Difference formulations on the lines of symmetry.- 3.4.2 Difference equation of a curved boundary.- 3.4.3 Difference formulations for the interface of different materials.- 3.5 Examples.- 3.6 Further discussions about the finite difference method.- 3.6.1 Physical explanation of the finite difference method.- 3.6.2 The error analysis of the finite difference method.- 3.6.3 Difference equation and the principle of weighted residuals.- 3.6.4 Difference equation and the variational principle.- 3.7 Summary.- References.- 4 Fundamentals of Finite Element Method (FFM).- 4.1 Introduction.- 4.2 General procedures of the finite element method.- 4.2.1 Domain discretization and shape functions.- 4.2.2 Method using Galerkin residuals.- 4.2.2.1 Element matrix equations.- 4.2.2.2 System matrix equation.- 4.2.2.3 Storage of the system matrix.- 4.2.2.4 Treatment of the Dirichlet boundary condition.- 4.3 Solution methods of finite element equations.- 4.3.1 Direct methods.- 4.3.1.1 Gaussian elimination method.- 4.3.1.2 Cholesky's decomposition (triangular decomposition).- 4.3.2 Iterative methods.- 4.3.2.1 Method of over-relaxation iteration.- 4.3.2.2 Conjugate-gradient method (CGM).- 4.4 Mesh generation.- 4.4.1 Mesh generation of a triangular element.- 4.4.2 Automatic mesh generation.- 4.5 Examples.- 4.6 Summary.- References.- 5 Variational Finite Element Method.- 5.1 Introduction.- 5.2 Basic concepts of the functional and its variations.- 5.2.1 Definition of the functional and its variations.- 5.2.1.1 The functional.- 5.2.1.2 The differentiation and variation of a function.- 5.2.1.3 Variation of the functional.- 5.2.2 Calculus of variations and Euler's equation.- 5.2.2.1 Euler's equation.- 5.2.2.2 Euler's equation for multivariable functions.- 5.2.2.3 The shortest length of a curve.- 5.2.3 Relationship between the operator equation and the functional.- 5.3 Variational expressions for electromagnetic field problems.- 5.3.1 Variational expression for Poisson's equation.- 5.3.1.1 Mathematical manipulation.- 5.3.1.2 Physical manipulation.- 5.3.2 Variational expressions for Poisson's equations in piece-wise homogeneous materials.- 5.3.3 Variational expression for the scalar Helmholtz equation.- 5.3.4 Variational expression for the magnetic field in a non-linear medium.- 5.4 Variational finite element method.- 5.4.1 Ritz method.- 5.4.2 Finite element method (FEM).- 5.4.2.1 Domain discretization.- 5.4.2.2 Finite element equation of a Laplacian problem.- 5.4.2.3 Finite element equation for 2-D magnetic fields.- 5.4.2.4 Finite element equation for non-linear magnetic fields.- 5.4.2.5 Finite element equation for Helmholtz's equation (2-D-case).- 5.5 Special problems using the finite element method.- 5.5.1 Approaching floating electrodes by the variational finite element method.- 5.5.2 Open boundary problems.- 5.5.2.1 Introduction.- 5.5.2.2 Ballooning method.- 5.6 Summary.- References.- 6 Elements and Shape Functions.- 6.1 Introduction.- 6.2 Types and requirements of the approximating functions.- 6.2.1 Lagrange and Hermite shape functions.- 6.2.2 Requirements of the approximating functions.- 6.3 Global, natural, and local coordinates.- 6.3.1 Natural coordinates.- 6.3.2 Local coordinates.- 6.4 Lagrange shape function.- 6.4.1 Triangular elements.- 6.4.2 Quadrilateral elements.- 6.4.3 Tetrahedral and hexahedral elements.- 6.5 Parametric elements.- 6.6 Element matrix equation.- 6.6.1 Coordinate transformations, Jacobian matrix.- 6.6.2 Evaluation of the Lagrangian element matrix.- 6.6.3 Universal matrix.- 6.7 Hermite shape function.- 6.7.1 One dimensional Hermite shape function.- 6.7.2 Triangular Hermite shape functions.- 6.7.3 Evaluation of a Hermite element matrix.- 6.8 Application discussions.- 6.9 Summary.- References.- Appendix 6.1 Langrangian shape functions for 2-D cases.- Appendix 6.2 Commonly used shape functions for 3-D cases.- Appendix 6.3 The universal matrix of axisymmetric fields.- 3 Boundary Methods.- 7 Charge Simulation Method (CSM).- 7.1 Introduction.- 7.2 Matrix equations of simulated charges.- 7.2.1 Matrix equation in homogeneous dielectrics.- 7.2.1.1 Governing equation subject to Dirichlet boundary conditions.- 7.2.1.2 Governing equation subject to Neumann boundary conditions.- 7.2.1.3 Mixed boundary conditions and free potential conductors.- 7.2.1.4 Matrix form of Poisson's equation.- 7.2.2 Matrix equation in piece-wise homogeneous dielectrics.- 7.3 Commonly used simulated charges.- 7.3.1 Point charge.- 7.3.2 Line charge.- 7.3.3 Ring charge.- 7.3.4 Charged elliptic cylinder.- 7.4 Applications of the charge simulation method.- 7.5 Coordinate transformations.- 7.5.1 Transformation matrix.- 7.5.2 Inverse transformation of the field strength.- 7.6 Optimized charge simulation method (OCSM).- 7.6.1 Objective function.- 7.6.2 Transformation of constrained conditions.- 7.6.3 Examples.- 7.7 Error analysis in the charge simulation method.- 7.7.1 Properties of the errors.- 7.7.2 Error distribution pattern along the electrode contour.- 7.7.3 Factors influencing the errors.- 7.8 Summary.- References.- Appendix 7.1 Formulations for a point charge.- Appendix 7.2 Formulations for a line charge.- Appendix 7.3 Formulations for a ring charge.- Appendix 7.4 Formulations for a charged elliptic cylinder.- Appendix 7.5 Approximate formulations for calculating K(k) and E(k).- 8 Surface Charge Simulation Method (SSM).- 8.1 Introduction.- 8.1.1 Example.- 8.2 Surface integral equations.- 8.2.1 Single layer or double layer integral equations.- 8.2.2 Integral equations of the interfacial surface.- 8.3 Types of surface boundary elements and surface charge densities.- 8.3.1 Representations of boundary and charge density.- 8.3.2 Potential and field strength coefficients for 2-Dand axisymmetrical problems.- 8.3.2.1 Planar element with constant or linear charge density.- 8.3.2.2 Arced element with constant or linear charge density.- 8.3.2.3 Ring element with linear charge density.- 8.3.3 Elements for 3-D problems.- 8.3.3.1 Planar triangular element.- 8.3.3.2 Cylindrical tetragonal bilinear element.- 8.3.3.3 Isoparametric high order element.- 8.3.3.4 Spline function element.- 8.4 Magnetic surface charge simulation method.- 8.5 Evaluation of singular integrals.- 8.5.1 The semi-analytical technique.- 8.5.2 Method using coordinate transformations.- 8.5.3 Numerical techniques.- 8.5.4 Combine the analytical integral and Gaussian quadrature.- 8.6 Applications.- 8.7 Summary.- References.- 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.- 9 Boundary Element Method (BEM).- 9.1 Introduction.- 9.2 Boundary element equations.- 9.2.1 Method of weighted residuals.- 9.2.2 Green's theorem.- 9.2.3 Variational principle.- 9.2.4 Boundary integral equation.- 9.2.5 Indirect boundary integral equation.- 9.3 Matrix formulations of the boundary integral equation.- 9.3.1 Discretization and shape functions.- 9.3.2 Matrix equation of a 2-dimensional constant element.- 9.3.2.1 Evaluation of Hij and Gij.- 9.3.2.2 Evaluation of Hii and Gii.- 9.3.3 Matrix equation of 2-D linear elements.- 9.3.4 Matrix form of Poisson's equation.- 9.3.5 Matrix equation of a piecewise homogeneous domain.- 9.3.6 Matrix equation of axisymmetric problems.- 9.3.7 Discretization of 3-dimensional problems.- 9.3.8 Use of symmetry.- 9.4 Eddy current problems.- 9.4.1 Eddy current equations.- 9.4.1.1 A-? formulations.- 9.4.1.2 T-? formulations.- 9.4.2 One-dimensional solution of an Eddy current problem.- 9.4.3 BEM for solving Eddy current problems.- 9.4.4 Surface impedance boundary conditions.- 9.5 Non-linear and time-dependent problems.- 9.5.1 BEM for non-linear problems.- 9.5.2 Time-dependent problems.- 9.6 Summary.- References.- Appendix 9.1 Bessel function.- 10 Moment Methods.- 10.1 Introduction.- 10.2 Basis functions and weighting functions.- 10.2.1 Galerkin's methods.- 10.2.2 Point matching method.- 10.2.3 Sub-regions and sub-sectional basis.- 10.3 Interpretation using variations.- 10.4 Moment methods for solving static field problems.- 10.4.1 Charge distribution of an isolated plate.- 10.4.2 Charge distribution of a charged cylinder.- 10.5 Moment methods for solving eddy current problems.- 10.5.1 Integral equation of a 2-D eddy current problem.- 10.5.2 Sub-sectional basis method.- 10.6 Moment methods to solve the current distribution of a line antenna.- 10.6.1 Integral equation of a line antenna.- 10.6.2 Solution of Hallen's equation.- 10.7 Summary.- References.- 4 Optimization Methods of Electromagnetic Field Problems.- 11 Methods of Applied Optimization.- 11.1 Introduction.- 11.2 Fundamental concepts.- 11.2.1 Necessary and sufficient conditions for the local minimum.- 11.2.2 Geometrical interpretation of the minimizer.- 11.2.3 Quadratic functions.- 11.2.4 Basic method for solving unconstrained non-linear optimization problems.- 11.2.5 Stability and convergence.- 11.3 Linear search and single variable optimization.- 11.3.1 Golden section method.- 11.3.2 Methods of polynomial interpolation.- 11.4 Analytic methods of unconstrained optimization problems.- 11.4.1 The method of steepest descent.- 11.4.2 Conjugate gradient method.- 11.4.2.1 Conjugate direction.- 11.4.2.2 Quadratic convergence.- 11.4.2.3 Selection of conjugate directions.- 11.4.3 Quasi-Newton's methods.- 11.4.3.1 Davidon-Fletcher-Powell (DFP) method.- 11.4.3.2 BFGS formulation.- 11.4.3.3 B matrix formulae.- 11.4.3.4 Cholesky factorization of the Hessian matrix.- 11.4.4 Method of non-linear least squares.- 11.4.4.1 Gauss-Newton method.- 11.4.4.2 Levenberg-Marquardt method.- 11.5 Function comparison methods.- 11.5.1 Polytype method.- 11.5.2 Powell's method of quadratic convergence.- 11.6 Constrained optimization methods.- 11.6.1 Basic concepts of constrained optimization.- 11.6.2 Kuhn-Tucker conditions.- 11.6.2.1 Lagrange multiplier method.- 11.6.2.2 Necessary condition of the first order.- 11.6.2.3 Necessary and sufficient conditions of the second order.- 11.6.3 Penalty and barrier function methods.- 11.6.4 Sequential unconstrained minimization technique.- 11.7 Summary.- References.- 12 Optimizing Electromagnetic Devices.- 12.1 Introduction.- 12.2 General concepts of optimum design.- 12.2.1 Objective function.- 12.2.2 Mathematical expressions of the boundary value problem.- 12.2.3 Optimization methods.- 12.2.4 Categories of optimization.- 12.3 Contour optimization.- 12.3.1 Method of curvature adjustment.- 12.3.2 Method of charge redistribution.- 12.3.3 Contour optimization by using non-linear programming.- 12.4 Problems of domain optimization.- 12.4.1 Field synthesis by using Fredholm's integral equation.- 12.4.2 Domain optimization by using non-linear programming.- 12.5 Summary.- References.