* Comprehensive descriptions of numerical methods
* In-depth introduction to finite differences, finite elements, and integral equations
* Illustrations and applications of linear and nonlinear solutions for multi-dimensional analysis
* Numerical examples to facilitate understanding of the methods
* Appendices for quick reference of mathematical and numerical methods employed
Electromagnetics is the foundation of our electric technology. It describes the fundamental principles upon which electricity is generated and used. This includes electric machines, high voltage transmission, telecommunication, radar, and recording and digital computing. Numerical Methods in Electromagnetism will serve both as an introductory text for graduate students and as a reference book for professional engineers and researchers. This book leads the uninitiated into the realm of numerical methods for solving electromagnetic field problems by examples and illustrations. Detailed descriptions of advanced techniques are also included for the benefit of working engineers and research students. Comprehensive descriptions of numerical methods In-depth introduction to finite differences, finite elements, and integral equations Illustrations and applications of linear and nonlinear solutions for multi-dimensional analysis Numerical examples to facilitate understanding of the methods Appendices for quick reference of mathematical and numerical methods employed
Cover 1
Contents 6
Foreword 12
Preface 14
CHAPTER 1. BASIC PRINCIPLES OF ELECTROMAGNETIC FIELDS 16
1.1 Introduction 16
1.2 Static Electric Fields 16
1.3 The Electric Potential 18
1.4 Electric Fields and Materials 26
1.5 Interface Conditions on the Electric Field 28
1.6 Laplace's and Poisson's Equations 30
1.7 Static Magnetic Fields 39
1.8 Energy in the Magnetic Field 51
1.9 Quasi-statics: Eddy Currents and Diffusion 54
1.10 The Wave Equation 57
1.11 Discussion of Choice of Variables 59
1.12 Classification of Differential Equations 75
CHAPTER 2. OVERVIEW OF COMPUTATIONAL METHODS IN ELECTROMAGNETICS 78
2.1 Introduction and Historical Background 78
2.2 Graphical Methods 80
2.3 Conformal Mapping 83
2.4 Experimental Methods 83
2.5 ElectroConducting Analog 84
2.6 Resistive Analog 85
2.7 Closed Form Analytical Methods 86
2.8 Discrete Analytical Methods 89
2.9 Transformation Methods for Nonlinear Problems 90
2.10 Nonlinear Magnetic Circuit Analysis 94
2.11 Finite Difference Method 95
2.12 Integral Equation Method 99
2.13 The Finite Element Method 111
CHAPTER 3. THE FINITE DIFFERENCE METHOD 120
3.1 Introduction 120
3.2 Difference Equations 121
3.3 Laplace's and Poisson's Equations 123
3.4 Interfaces Between Materials 124
3.5 Neumann Boundary Conditions 126
3.6 Treatment of Irregular Boundaries 129
3.7 Equivalent Circuit Representation 130
3.8 Formulas For High-Order Schemes 132
3.9 Finite Differences With Symbolic Operators 137
3.10 Diffusion Equation 141
3.11 Conclusions 156
CHAPTER 4. VARIATIONAL AND GALERKIN METHODS 158
4.1 Introduction 158
4.2 The Variational Method 159
4.3 The Functional and its Extremum 160
4.4 Functional in more than one space variable and its extremum 168
4.5 Derivation of the Energy-Related Functional 172
4.6 Ritz's method 185
4.7 The Wave Equation 191
4.8 Variational Method for Integral Equations 194
4.9 Introduction to The Galerkin Method 197
4.10 Example of the Galerkin Method 198
CHAPTER 5. SHAPE FUNCTIONS 204
5.1 Introduction 204
5.2 Polynomial Interpolation 216
5.3 Deriving Shape Functions 222
5.4 Lagrangian Interpolation 226
5.5 Two-Dimensional Elements 229
5.6 High-Order Triangular Interpolation Functions 237
5.7 Rectangular Elements 242
5.8 Derivation of Shape Functions for Serendipity Elements 249
5.9 Three-Dimensional Finite Elements 255
5.10 Orthogonal Basis Functions 293
CHAPTER 6. THE FINITE ELEMENT METHOD 298
6.1 Introduction 298
6.2 Functional minimization and global assembly 310
6.3 Solution to the nonlinear magnetostatic problem with first-order triangular finite elements 317
6.4 Application of the Newton–Raphson Method to a First-Order Element 321
6.5 Discretization of Time by the Finite Element Method 325
6.6 Axisymmetric Formulation for the Eddy Current Problem Using Vector Potential 328
6.7 Finite Difference and First-Order Finite Elements 335
6.8 Galerkin Finite Elements 337
6.9 Three-Element Magnetostatic Problem 341
6.10 Permanent Magnets 353
6.11 Numerical Example of Matrix Formation for Isoparametric Elements 357
6.12 Edge Elements 368
CHAPTER 7. INTEGRAL EQUATIONS 374
7.1 Introduction 374
7.2 Basic Integral Equations 374
7.3 Method of Moments 377
7.4 The Charge Simulation Method 385
7.5 Boundary Element Equations for Poisson's Equation in Two Dimensions 389
7.6 Example of BEM Solution of a Two-Dimensional Potential Problem 396
7.7 Axisymmetric Integral Equations for Magnetic Vector Potential 404
7.8 Two-Dimensional Eddy Currents With T–O 408
7.9 BEM Formulation of The Scalar Poisson Equation in Three Dimensions 418
7.10 Green's functions for some typical electromagnetics applications 424
CHAPTER 8. OPEN BOUNDARY PROBLEMS 428
8.1 Introduction 428
8.2 Hybrid Harmonic Finite Element Method 428
8.3 Infinite Elements 432
8.4 Ballooning 442
8.5 Infinitesimal Scaling 448
8.6 Hybrid Finite Element–Boundary Element Method 452
CHAPTER 9. HIGH-FREQUENCY PROBLEMS WITH FINITE ELEMENTS 466
9.1 Introduction 466
9.2 Finite Element Formulation in Two Dimensions 467
9.3 Boundary Element Formulation 474
9.4 Implementation of the Hybrid Method (HEM) 482
9.5 Evaluation of the Far-Field 486
9.6 Scattering Problems 496
9.7 Numerical Examples 503
9.8 Three Dimensional FEM Formulation for the Electric Field 509
9.9 Example 532
CHAPTER 10. LOW-FREQUENCY APPLICATIONS 534
10.1 Time Domain Modeling of Electromechanical Devices 534
10.2 Modeling of Flow Electrification in Insulating Tubes 565
10.3 Coupled Finite Element and Fourier Transform Method for Transient Scalar Field Problems 576
10.4 Axiperiodic Analysis 585
CHAPTER 11. SOLUTION OF EQUATIONS 606
11.1 Introduction 606
11.2 Direct Methods 610
11.3 LU Decomposition 613
11.4 Cholesky Decomposition 620
11.5 Sparse Matrix Techniques 622
11.6 The Preconditioned Conjugate Gradient Method 642
11.7 GMRES 660
11.8 Solution of Nonlinear Equations 673
APPENDIX A. VECTOR OPERATORS 722
APPENDIX B. TRIANGLE AREA IN TERMS OF VERTEX COORDINATES 724
APPENDIX C. FOURIER TRANSFORM METHOD 726
C.1 Computation of Element Coefficient Matrices and Forcing Functions 729
APPENDIX D. INTEGRALS OF AREA COORDINATES 734
APPENDIX E. INTEGRALS OF VOLUME COORDINATES 736
APPENDIX F. GAUSS–LEGENDRE QUADRATURE FORMULAE, ABSCISSAE, AND WEIGHT COEFFICIENTS 738
APPENDIX G. SHAPE FUNCTIONS FOR 1D FINITE ELEMENTS 740
APPENDIX H. SHAPE FUNCTIONS FOR 2D FINITE ELEMENTS 742
APPENDIX I. SHAPE FUNCTIONS FOR 3D FINITE ELEMENTS 750
REFERENCES 764
INDEX 774
BASIC PRINCIPLES OF ELECTROMAGNETIC FIELDS
1.1 INTRODUCTION
This chapter is intended to introduce and to discuss the basic principles and concepts of electromagnetic fields, which will be dealt with in detail in succeeding chapters. The concept of flux and potential is discussed and Maxwell’s set of equations is presented. The description of electric and magnetic materials then follows. We then discuss the set of differential equations that are commonly used as a starting point for the numerical methods we will study. An explanation of some of the more popular choices of state variables (scalar and vector potential or direct field variables) is then presented.
1.2 STATIC ELECTRIC FIELDS
The electric charge is the source of all electromagnetic phenomena. Electrostatics is the study of electric fields due to charges at rest. In SI units the electric charge has the units of coulombs after Charles Augustin de Coulomb (1736–1806). The charge of one electron is 1.602 × 10−19 coulombs. In rationalized MKS units an electric charge of Q coulombs produces an electric flux of Q Coulombs. In Figure 1.1 we see the electric charge emitting flux uniformly in the radial direction in a spherical coordinate system with origin at the charge. If we consider a closed surface around the charge (for example, the sphere shown in Figure 1.1), then all of the flux must pass through the surface. This would be true for any number of charges in the volume enclosed by the surface. Using the principle of superposition, we conclude that the total flux passing through any closed surface is equal to the total charge enclosed by that surface. This is Gauss’ law and one of Maxwell’s equations. If we define the electric flux density as the local value of the flux per unit area then we may write
Figure 1.1 Point Charge and Flux
(1.1)
where ρ is the charge density. This is the integral form of Gauss’ law. From the symmetry of Figure 1.1 we find immediately that for a point charge
(1.2)
where r is the distance from the charge (source point) to the location where we compute the field quantity (field point).
The inverse square law of equation (1.2) is one of the most important results of electromagnetism. There are two consequences of the inverse square law that we shall encounter in subsequent chapters. One is that there is a second-order singularity which occurs at r = 0. The second is that the electric flux density goes to zero as r → ∞.
Coulomb, in a famous experiment, found that there is a force between two electric charges acting along the line connecting them that varies as . This force is proportional to the magnitude of the charges and depends on the medium. Coulomb’s law states that for two charges Q1 and Q2 in vacuum,
(1.3)
Here ∈0 is the permittivity of free space, farads/meter. The force on a unit charge is called the electric field and is denoted as . The units of electric field are volts/meter. From (1.2) and (1.3) we see that
(1.4)
1.3 THE ELECTRIC POTENTIAL
For computational purposes it is often difficult to compute the field quantities and directly. One of the reasons is that we need to find three components at each point and all of the algebra is vectorial. For these reasons we find it simpler to compute scalar variables and then compute the field from these values. In the case of the electric field we use the electric potential, V. Recalling that the electric field is the force on a unit charge, we define the electric potential at a point in space as the work expended in moving a unit charge from infinity to that point. Consider Figure 1.2, where we are moving a charge a distance Δ at an angle θ with respect to the electric field. We see that we do work only when moving the charge in the direction of the electric field. The work done to move the charge in Figure 1.2 a distance Δ is E Δ cos θ, because no work is required to move the distance Δ sin θ perpendicular to the field.
Figure 1.2 Work and Electric Field
We write
(1.5)
The minus sign is a convention and means that we do work moving a positive charge toward a field produced by a positive charge. The work done moving a unit charge a distance Δ in the presence of an electric field is ΔW = − · Δ so that
(1.6)
Therefore, if we can find V, a scalar, we can compute the electric field by taking the gradient. An important property of the scalar potential is that the work done to move a charge between points P1 and P2 is independent of the path taken by the charge. Fields having this property are called conservative. This is typical of lossless systems in which we have thermodynamically reversible processes. Systems with losses do not have this property, and this will have implications later in our discussion of variational principles. A corollary of this principle is that it takes no work to move a charge around a closed path in a static electric field. This is a very important property of the potential. Without this, the potential would not be uniquely defined at a point and the path would have to be specified. We will see in the next section that the magnetostatic field with current sources does not have this property and the scalar potential is of more limited use. We have therefore
(1.7)
another of Maxwell’s equations.
A few examples of the electric field will be of interest to us.
Potential Due to a Point Charge
The electric field due to a point charge of magnitude Q is, from (1.3) and (1.4),
(1.8)
We find the potential by integrating equation (1.8) to obtain
(1.9)
We make note of some properties that will be of interest in later chapters. First, there is a singularity at r = 0, but it is of first order and not of second order as in the case of the field. Second, the potential goes to zero as r approaches infinity.
The Logarithmic Potential
If we consider a long line of charge density ρ coulombs/meter, then the application of Gauss’ law in Figure 1.3 gives a flux density of
Figure 1.3 Logarithmic Potential
(1.10)
and an electric field of
(1.11)
Integrating from infinity to a point P gives the potential
(1.12)
This logarithmic potential is frequently found in two-dimensional problems and problems in cylindrical coordinates. The potential has a singularity at r = 0 but unfortunately does not go to zero as r goes to infinity. In fact, the potential slowly approaches infinity as r increases. The potential vanishes at r = 1.
Potential of a Dipole
The field and potential due to an electric dipole will be important in our discussion of integral equations. Consider two charges +Q and –Q separated by a distance d. We can find the potential by superposition. We will be interested in the potential at a distance from the charges much greater than their separation. The potential due to a single point charge is
(1.13)
For both charges (see Figure 1.4) we have
Figure 1.4 Dipole Field and Potential
(1.14)
We see that
(1.15)
We make the approximation that for r > > d
(1.16)
so
(1.17)
This potential drops off faster than the potential of a point charge and the singularity at r = 0 is of second order.
Potential Due to a Ring of Charge
Another configuration that will be useful is the potential due to a uniform ring of charge with charge per unit length of ρ [1]. Consider the ring with center at the origin of a cylindrical coordinate system illustrated in Figure 1.5.
Figure 1.5 Potential of Ring of Charge
The potential is
(1.18)
where
(1.19)
and
(1.20)
This is equivalent to
(1.21)
where
(1.22)
and
(1.23)
equation (1.21) is an elliptic integral of the first kind. Results for this are tabulated in many standard references [2].
1.3.1 Potential Energy and Energy Density
Using our definition...
| Erscheint lt. Verlag | 10.11.1999 |
|---|---|
| Sprache | englisch |
| Themenwelt | Geisteswissenschaften |
| Mathematik / Informatik ► Informatik | |
| Mathematik / Informatik ► Mathematik ► Angewandte Mathematik | |
| Naturwissenschaften ► Physik / Astronomie ► Elektrodynamik | |
| Sozialwissenschaften ► Pädagogik | |
| Technik ► Bauwesen | |
| Technik ► Elektrotechnik / Energietechnik | |
| Technik ► Maschinenbau | |
| ISBN-10 | 0-08-051289-5 / 0080512895 |
| ISBN-13 | 978-0-08-051289-1 / 9780080512891 |
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