Efficient evaluation of weakly singular integrals with Duffy-distance transformation in 3D BEM

Abstract

Weakly singular integrals are commonly encountered in the application of BIEs. In general, the 3D integral boundary element is divided into several triangle patches using the image of singular point in the local coordinate system, which transforms the integral into vertex singularity problem over each triangle patch. In this paper, firstly the classic Duffy transformation used for vertex singularity problem are modified into an optimized form termed as ‘Duffy-distance transformation’, taking into account the near singularity caused by the integral patch shape. Besides, the previously proposed conformal mapping is coupled with the Duffy-distance transformation to eliminate the near singularity caused by element shape distortion. Planar quadrilateral elements with different inclined angles and aspect ratios are given to verify the accuracy and efficiency in detail, and another two curved elements extracted from cylinder and sphere surfaces are presented to demonstrate the applicability for higher-ordered elements.