Exact Boundary Derivative Formulation for Numerical Conformal Mapping Method

Abstract

Conformal mapping is a useful technique for handling irregular geometries when applying the finite difference method to solve partial differential equations. When the mapping is from a hyperrectangular region onto a rectangular region, a specific length-to-width ratio of the rectangular region that fitted the Cauchy-Riemann equations must be satisfied. In this research, a numerical integral method is proposed to find the specific length-to-width ratio. It is conventional to employ the boundary integral method (BIEM) to perform the conformal mapping. However, due to the singularity produced by the BIEM in seeking the derivatives on the boundaries, the transformation Jacobian determinants on the boundaries have to be evaluated at inner points instead of directly on the boundaries. This approximation is a source of numerical error. In this study, the transformed rectangular property and the Cauchy-Riemann equations are successfully applied to derive reduced formulations of the derivatives on the boundaries for the BIEM. With these boundary derivative formulations, the Jacobian determinants can be evaluated directly on the boundaries. Furthermore, the results obtained are more accurate than those of the earlier mapping method.

Improving the Accuracy of the Charge Simulation Method for Numerical Conformal Mapping

“…Symm [3,4] proposed the integral equation method for the conformal mapping of a simply connected domain bounded by a closed Jordan curve onto the unit disk, of its exterior onto the exterior of the unit disk and of a doubly connected domain onto a concentric circular annulus. In [5], a numerical integral method is proposed to find the specific length-to-width ratio, which is able to improve the accuracy of the numerical conformal mapping on the boundaries and makes the mapping method more reliable when solving problems that are sensitive to accuracy. Nasser [6][7][8][9] proposed a method which can be used to compute the mapping function onto the five canonical regions in a unified way.…”