The Study Of The Charge Simulation Method For Numerical Conformal Mapping

Conformal transformation is also called conformal mapping, it is a basic problem of complex function. In terms of physics, especially in the theory of elasticity, electromagnetic theory, fluid dynamics, heat transfer, and other fields has been widely used. Due to the numerical calculation method of conformal mapping in practical engineering problems has a unique advantage in the application, and therefore its researchers have done a lot of research work.The calculation of conformal mapping is a very difficult problem, in general, its main method for solving can be divided into two major categories of analytical method and numerical methods. Based on the complexity of the actual engineering application of the analytical method, and only a handful of cases to the analytic function with elementary function, and therefore is not commonly used; However, for numerical calculations, the conformal mapping is the stereotype transformation can be proved by the Riemann existence and uniqueness theorems, and therefore usually seen only know the transform function is exists but can not obtained, in most cases, this conformal mapping function can only be approximate solved by numerical methods with the satisfied a given condition.Amano.K and other scholars work done a lot of research on charges simulation method and numerical conformal mapping since the 1980s, and puts forward the numerical conformal mapping based on charge simulation method (Amano.K method). Compared with the traditional numerical conformal mapping method, such as the integral equation method, variation method, series expansion method algorithm, the numerical conformal mapping based on charge simulation method Required calculation time is shorter, it has a higher accuracy and calculation more simple, Numerical integral calculation is avoided, the computer program are more easier, and error evaluation is more simple.The main work in this paper is as follows:1. We review the development situation of conformal mapping with other countries. In this paper, we introduce the basic concepts and methods of conformal mapping in detail.2. We introduce the charge simulation method for the numerical conformal mapping of exterior and interior domains in detail. In order to improve the accuracy of conformal mapping, the constraint equations are solved by using QR decomposition method, LU decomposition method and BiCR iterative method respectively.3. For the charge simulation method for the numerical conformal mapping of exterior and interior domains, we study how to obtain the charges of the typical domains with smooth boundary and angle boundary.4. In numerical experiments, we use the ellipse for typical smooth boundary and the square for typical angle boundary respectively. We do the experiments of errors and conformal mappings for interior domains and exterior domains.In this paper, we consider improved methods of numerical conformal mapping based on charge simulation method by solving linear equations using QR method, LU method and BiCR method. The efficiency of the proposed methods is illustrated by some numerical examples.