Numerical Applications of Conformal Mapping with Explicit Formulae Based on the Power Series Method

Abstract

A general optimization method, based on the power series method, is presented for computing the conformal mappings with explicit expressions from: (a) the unit disc onto an infinite domain exterior of a closed Jordan curve, (b) the circular annulus domain onto a finite doubly-connected domain bounded by two closed Jordan curves, (c) the infinite domain bounded by two circular curves onto an infinite domain bounded by two non-circular closed Jordan curves. The unknown mapping functions are approximated by the power series method. The problem of solving the mapping function coefficients is transformed into the problem of determining the image points on the image plane by means of the least square method. Different from most of the previous optimization methods, the angles are set as the design variables rather than the mapping function coefficients in the paper. The influence of the terms of the series on the calculation accuracy is investigated. The successful applications of the proposed method are shown by a large number of numerical examples.