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## Über dieses Buch

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.

## Inhaltsverzeichnis

### Chapter 1. Fundamental Concepts of Electromagnetic Field Theory

Abstract
The solution of many practical electromagnetic field problems can only be undertaken by applying numerical methods. Before such a solution can be undertaken, it is important that a correct mathematical model be established for the problem considered. Maxwell’s equations and the associated boundary conditions provide the necessary basis for the modelling of practical electromagnetic problems which are reviewed in this chapter. Further, as Green’s theorem, fundamental solutions, and equivalent sources are the basic tools used in some numerical techniques, they are also presented here.
Pei-bai Zhou

### Chapter 2. General Outline of Numerical Methods

Abstract
According to Maxwell’s equations, all electromagnetic field problems can be expressed in partial differential equations which are subject to specific boundary conditions. By using Green’s function, the partial differential equations can be transformed into integral equations or differential-integral equations. The analytical solution of these equations can only be obtained in very simple cases. Therefore numerical methods are significant for the solution of practical problems. In numerical solutions the following aspects have to be considered.
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### Chapter 3. Finite Difference Method

Abstract
The finite difference method (FDM) is an approximate method for solving partial differential equations. It has been used to solve a wide range of problems. These include linear and non-linear, time independent and dependent problems. This method can be applied to problems with different boundary shapes, different kinds of boundary conditions, and for a region containing a number of different materials. Even though the method was known by such workers as Gauss and Boltzmann, it was not widely used to solve engineering problems until the 1940s. The mathematical basis of the method was already known to Richardson in 1910 [1] and many mathematical books such as references [2 and 3] were published which discussed the finite difference method. Specific reference concerning the treatment of electric and magnetic field problems is made in [4]. The application of FDM is not difficult as it involves only simple arithmetic in the derivation of the discretization equations and in writing the corresponding programs. During 1950–1970 FDM was the most important numerical method used to solve practical problems ([5–7]). With the development of high speed computers having large scale storage capability many numerical solution techniques appeared for solving partial differential equations. However, due to the ease of application of the finite difference method it is still a valuable means of solving these problems ([8–11]).
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### Chapter 4. Fundamentals of Finite Element Method (FEM)

Abstract
The main idea behind FDM is the approximation of the derivative operations ∂u/∂x and 2 u/∂x 2 by the difference quotients Δu(x)/Δx and Δ 2 u(x)/Δx 2 , which reduces the partial differential equation to a set of algebraic equations. The application of FDM has two serious limitations. First, the regular steps of h x , h y , h z which construct an array of grid nodes in the x, y, z directions are not suitable for a field with a rapidly changing gradient or for problems having a curved boundary. Second, different formulae must be derived for specific interfaces between different media and for the various shapes of boundaries.
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### Chapter 5. Variational Finite Element Method

Abstract
In general, most of the problems in engineering and science can be described by variational principles. For instance, the principle of least action exists in mechanics and electrodynamics [1]. In electrostatic fields, Thomson’s theory [2] states that the electric energy is minimum if the system is in equilibrium. In classical thermodynamics, the entropy remains at maximum for any equilibrated isolated system.
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### Chapter 6. Elements and Shape Functions

Abstract
Domain discretization is one of the most important steps in many numerical methods to solve boundary value problems. In finite element method (FEM), the whole domain is discretized by elements. In boundary element method (BEM) the boundary of the domain is discretized by elements. The choice of the geometry of the element and the form of the approximating function to represent the behaviour of field variables (such as the potential φ, field strength E H and so on) within each element are both extremely important. They strongly influence the accuracy of the results, the computing time and the software engineering of computer programs. Hence the problem of element discretization is a generic problem and is common to element approximate methods as well as FEM. Hence element discretization techniques are discussed in one chapter.
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### Chapter 7. Charge Simulation Method

Abstract
According to the uniqueness theorem of electromagnetic fields, if a solution satisfies Laplace’s equation or Poissons equation and all the corresponding boundary conditions, no matter how that solution is obtained — even if guessed — it is the only solution of the specified boundary value problem. For example, the field distribution of an isolated charged spherical conductor equals the field distribution of a point charge if it is located at the centre of the sphere and its charge equals the total amount of surface charge of the sphere. This point charge is called the equivalent charge or simulated charge of the original charged conductor. Thus the distributed charge on the conductor surface is replaced by a lumped fictitious point charge. It should be noted that the region of interest is now the region outside the sphere. In other words, the fictitious simulated charges must be placed outside the space in which the field is under consideration.
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### Chapter 8. Surface Charge Simulation Method (SSM)

Abstract
The surface charge simulation method (SSM) is similar to the charge simulation method (CSM). In these methods, the real source distribution is simulated by a great number of accumulated or surface charges. They are both convenient for solving open boundary and 3-D problems. SSM can solve field problems that cannot be solved by CSM and also those that can. Using SSM one can obtain solutions to problems containing a number of dielectric materials and problems having thin electrodes. Hence SSM is a more general method than CSM. One could also consider it to be a kind of boundary element method, or a method of equivalent source.
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### Chapter 9. Boundary Element Method

Abstract
It is difficult to say who was the pioneer of the boundary element method (BEM). In Brebbia’s opinion [1], the work started in 1960s. The first book entitled Boundary Elements was published in 1978 [2]. After that BEM developed rapidly. It has been expanded so as to include time-dependent and non-linear problems [3,4]. During this time many papers [5,6], theses [7–9] and books [10–12] have been published. The method is now regarded as important as FEM. An international conference to discuss BEM is held every year and the edited proceedings are valuable references.
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### Chapter 10. Moment Methods

Abstract
As outlined in Chapter 2, the method of moments is a generalized method based on the principle of weighted residuals. It covers the many specific methods discussed such as the charge simulation method, the surface charge simulation method, boundary element method and even the finite element method which is regarded as one of the special cases of the method of moments. The name ‘moment’ is understood here as the product of an appropriate weighting function with an approximate solution. Any method whereby an operator equation is reduced to a matrix equation can be interpreted as a method of moments. It is also considered as the unified treatment of a matrix method (R.F. Harrington [1]).
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### Chapter 11. Methods of Applied Optimization

Abstract
The advanced purpose of the analysis of electromagnetic fields is to determine a better design of a practical problem. Design optimization (sometimes called the inverse problem) deals with the problem of finding the source distribution or the dimensions of devices for a specific purpose. Most inverse problems involve numerical optimization. This chapter provides the mathematical methods used for the optimum design.
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### Chapter 12. Optimizing Electromagnetic Devices

Abstract
The problem of optimum design has been studied for a long time. One of the earliest example was to find a 2-D shape which occupied the maximum area with its circumference as a given constant. Before the 1950’s classical mathematical methods such as the differential and variational methods were used to solve these problems. With digital computers and using numerical methods, the methods of optimal design developed rapidly.
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### Backmatter

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