An improved exponential transformation for accurate evaluation of nearly singular boundary integrals in 3D BEM

Abstract

Accurate evaluation of nearly singular integrals is an important issue for the successful implementation of the boundary element method (BEM), and the exponential transformation has been proved to be feasible in dealing with nearly weak and strong singular integrals for 2D and 3D boundary elements with regular shape. In this paper, in order to eliminate the shape effect from distorted elements, an improved exponential transformation is proposed by combination of the conformal mapping and sigmoidal transformation in angular direction. Besides, the nearly hyper-singular integrals have been solved readily with a simple geometrical manipulation to convert them into line integrals. The performance of the conformal mapping based exponential transformation, including sensitivity to the relative distance, location of projection point and number of Gauss points used, is tested in detail through a distorted curved element extracted from cylinder surface. Finally, an alternative form of exponential transformation and comparisons with the distance transformation are discussed.