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## Über dieses Buch

The Complex Variable Boundary Element Method or CVBEM is a generalization of the Cauchy integral formula into a boundary integral equation method or BIEM. This generalization allows an immediate and extremely valuable transfer of the modeling techniques used in real variable boundary integral equation methods (or boundary element methods) to the CVBEM. Consequently, modeling techniques for dissimilar materials, anisotropic materials, and time advancement, can be directly applied without modification to the CVBEM. An extremely useful feature offered by the CVBEM is that the pro­ duced approximation functions are analytic within the domain enclosed by the problem boundary and, therefore, exactly satisfy the two-dimensional Laplace equation throughout the problem domain. Another feature of the CVBEM is the integrations of the boundary integrals along each boundary element are solved exactly without the need for numerical integration. Additionally, the error analysis of the CVBEM approximation functions is workable by the easy-to-understand concept of relative error. A sophistication of the relative error analysis is the generation of an approximative boundary upon which the CVBEM approximation function exactly solves the boundary conditions of the boundary value problem' (of the Laplace equation), and the goodness of approximation is easily seen as a closeness-of-fit between the approximative and true problem boundaries.

## Inhaltsverzeichnis

### Chapter 1. Flow Processes and Mathematical Models

Abstract
In the following discussions, several mathematical models of flow processes will be developed. Each model is an attempt to precisely describe some physical process by using experimentally calibrated relationships assumed by man. Generally, the assumed relationship (or physical law) expresses a rate of flow or flux of some type of specie as a function of a linear gradient of some type of defined potential.
$$\rm q_x=-kA\frac{\partial \phi}{\partial x}$$
(1.1)

### Chapter 2. A Review of Complex Variable Theory

Abstract
Before developing the mathematical foundations of the Complex Variable Boundary Element Method (CVBEM), the basic tools needed for the method’s development need to be reviewed. In this chapter, a brief summary of the prerequisite complex variable theory is presented. The basic definitions of analytic function theory are reviewed, and the necessary line integral theory including the Cauchy theorems will be addressed.

### Chapter 3. Mathematical Development of the Complex Variable Boundary Element Method

Abstract
In this chapter, the CVBEM will be rigorously developed with special attention given to the often used linear trial function model. Basic theorems will be proven which address convergence of the approximation function to an analytic function which satisfies the boundary conditions continuously.

### Chapter 4. The Complex Variable Boundary Element Method

Abstract
In this chapter, the Complex Variable Boundary Element Method will be developed for use in approximating solutions to potential problems defined on a simply connected domain. The strategy of the method is to (1) define the problem domain Ω and boundary Γ such that there are no singularities of the state variable φ(x, y) and conjugate stream function Ψ(x, y) for z = x + iy and z ε Ω UΓ; (2) develop an approximation function which is analytic in Q and continuous on r and satisfies specified boundary conditions values on Γ; and (3) determine a relative error distribution between the approximation function and the analytic solution of the boundary value problem to be used for error analysis and subsequent modeling refinement. Steps (1) and (2) will be discussed in this chapter, with the error analysis and subsequent modeling refinement methods developed in Chapter 5.

### Chapter 5. Reducing CVBEM Approximation Relative Error

Abstract
In the previous chapters, the complex variable boundary element method (CVBEM) is used to develop an approximation function $$\hat{\omega}(\rm z)$$ which is analytic in the interior of the domain Ω UΓ ε P, where the boundary Γ is a simply connected contour. The function $$\hat{\omega}(\rm z)$$, therefore, exactly satisfies the Laplace equation in Ω and generally approximates the boundary conditions on Γ. Let ω(z) be the solution of the boundary value problem (Laplace equation) on Ω UΓ. Then a relative error function is defined on Ω UΓ by $$\rm e(z)=\omega(z)-\hat{\omega}(z)$$. Should e(z) = 0 on Γ, then $$\hat{\omega}(\rm z)=\omega(\rm z)$$ on Ω UΓ.