**Preface**

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 produced 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.

Due to the convenient approximation error evaluation afforded by the CVBEM, the modeling approach is extremely useful in developing highly accurate approximations for two-dimensional potential problems. This numerical approach can potential problems which occur in engineering applications, or to aid in numerically calibrating (e.g. finite element or fir diffusion type problems.

This numerical approach can then be used to develop solutions for potential problems which occur in engineering applications, or to aid in numerically calibrating and verifying domain method numerical models

(e.g. finite element or finite difference methods) of steady state diffusion type problems.

Because of the direct link between the real variable BIEM models and the CVBEM, detailed discussions of accomodating anisotropic and dissimilar materials and time advancement techniques are not presented. Rather, the real variable BIEM literature should be consulted such as Brebbia (1978, 1980). The main objective of this book is to present the detailed mathematics which are associated to the CVBEM, and the simplifications which allow the interpretation of the CVBEM approximation error. Computer programs are presented which allow an immediate use of the CVBEM in solving two-dimensional problems. Several application problems are solved which demonstrate the use of the CVBEM and the interpretation of the produced approximation error for subsequent reduction.

The mathematics literature has shown some recent attention to the applications and solutions of Cauchy singular integral equations although the overall thrust has been more focused towards real variable applications. In a Symposium on the Application and Numerical Solution of Integral Equations (1978), two papers were presented which address the use of the Cauchy singular integral equation for the solution of potential problems.

A review of the engineering literature indicates that the use of analytic function theory in developing BIEM models is sparse. Possibly one of the most significant citations is the Analytic Function Method (AFM) presented in van der Veer (1978). This comprehensive study develops an analytic approximation function as a sum of simple products of complex linear polynomials and complex logarithm functions. The work includes a comprehensive review of the keystone numerical modeling Chapter 1: Flow Processes literature for solution of two-dimensional potential problems by both real and complex variable methods, and boundary and domain numerical methods. It will be shown in Chapter 6 that a linear trial function CVBEM model results in the AFM. The AFM serves as the starting point for the generalization of the CVBEM theory which was developed during a research engagement (1979 through 1981) at the University of California, Irvine.

Citations of engineering applications include Hunt and Isaacs (1981) who use a BIEM model based on the Cauchy integral to approximate groundwater flow problems. Hromadka and Guymon (1982) use a similar BIEM model to approximate the temporal and spatial evolution of a slow moving freezing front in freezing and thawing soils. In another application, the time derivative was also included in the work of Brevig et al (1982) where a Cauchy integral model is used to approximate the time evolution of two-dimensional seawater waves and associated forces.

Because the CVBEM is a linear combination of real variable functions, much of the real variable BIEM theory is applicable to the CVBEM. Consequently, the extensive BIEM literature requires citation. The bibliography includes a brief list of the real variable BIEM literature.

**Table Of Contents****Chapter 1** – Flow Processes and Mathematical Models

1.0 Introduction

1.1 Ideal Fluid Flow

1.2 Steady State Heat Flow

1.3 Saturated Groundwater Flow

1.4 Steady State Fickian Diffusion

1.5 Use of the Laplace Equation**Chapter 2** – A Review of Complex Variable Theory

2.0 Introduction

2.1 Preliminary Definitions

2.2 Polar Forms-of Complex Numbers

2.3 Limits and Continuity

2.4 Derivatives

2.5 The Cauchy-Riemann Equations and Harmonic Functions.

2.6 Complex Line Integration

2.7 Cauchy’s Integral Theorem

2.8 The Cauchy Integral Formula

2.9 Taylor Series

2.10 Program 1: A Complex Polynomial Approximation Method

2.11 Potential Theory and Analytic Functions**Chapter 3** – Mathematical Development of the Complex

3.0 Introduction

3.1 Basic Definitions

3.2 Linear Global Trial Function Characteristics

3.3 The Hi Approximation Function

3.4 Higher Order H Approximation Functions

3.5 Engineering Applications**Chapter 4** – The Complex Variable Boundary Element Method

4.0 Introduction

4.1 A Complex Variable Boundary Element Approximation Model

4.2 The Analytic Function Defined by the Approximator _(z)

4.3 Program 2: A Linear Basis Function Approximator _(z).

4.4 A Constant Boundary Element Method

4.5 The Complex Variable Boundary Element Method (CVBEM)**Chapter 5** – Reducing CVBEM Approximation Relative Error

5.0 Introduction

5.1 Application of the CVBEM to the Unit Circle

5.2 Approximation Error from the CVBEM

5.3 A CVBEM Modeling Strategy to Reduce Approximation Error

5.4 A Modified CVBEM Numerical Model

5.5 Program 3: A Modified CVBEM Numerical Model

5.6 Determining some Useful Relative Error Bounds for the CVBEM**Chapter 6** – Advanced Topics

6.0 Introduction

6.1 Expansion of the H k Approximation Function

6.2 Upper Half Plane Boundary Value Problems

6.3 Sources and Sinks

6.4 The Approximative Boundary for Error Analysis

Estimating Boundary Spatial Coordinates