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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.
List of contents
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.- 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.- 3: Mathematical Development of the Complex Variable Boundary Element Method.- 3.0 Introduction.- 3.1 Basic Definitions.- 3.2 Linear Global Trial Function Characteristics.- 3.3 The H1 Approximation Function.- 3.4 Higher Order Hk Approximation Functions.- 3.5 Engineering Applications.- 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 $$rm hat{omega }$$(z).- 4.3 Program 2: A Linear Basis Function Approximator $$rm hat{omega }$$(z).- 4.4 A Constant Boundary Element Method.- 4.5 The Complex Variable Boundary Element Method (CVBEM).- 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.- 6: Advanced Topics.- 6.0 Introduction.- 6.1 Expansion of the HkApproximation Function.- 6.2 Upper Half Plane Boundary Value Problems.- 6.3 Sources and Sinks.- 6.4 The Approximative Boundary for Error Analysis.- 6.5 Estimating Boundary Spatial Coordinates.
