Numerical Analysis of Electromagnetic Fields

Numerical methods for solving boundary value problems have developed rapidly. Knowledge of these methods is important both for engineers and scientists. There are many books published that deal with various approximate methods such as the finite element method, the boundary element method and so on. However, there is no textbook that includes all of these methods. This book is intended to fill this gap. The book is designed to be suitable for graduate students in engineering science, for senior undergraduate students as well as for scientists and engineers who are interested in electromagnetic fields. Objective Numerical calculation is the combination of mathematical methods and field theory. A great number of mathematical concepts, principles and techniques are discussed and many computational techniques are considered in dealing with practical problems. The purpose of this book is to provide students with a solid background in numerical analysis of the field problems. The book emphasizes the basic theories and universal principles of different numerical methods and describes why and how different methods work. Readers will then understand any methods which have not been introduced and will be able to develop their own new methods. Organization Many of the most important numerical methods are covered in this book. All of these are discussed and compared with each other so that the reader has a clear picture of their particular advantage, disadvantage and the relation between each of them. The book is divided into four parts and twelve chapters.

Table of Contents:
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 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.- 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'#8217;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 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.