An adaptive element subdivision method based on the affine transformations and partitioning techniques for evaluating the weakly singular integrals

Abstract

In this paper, an adaptive element Subdivision Method based on the Affine transformations and Partitioning techniques (APSM) is presented for evaluating the weakly singular integrals with arbitrary shape of elements. The basic idea consists in partitioning the input surface element via affine transformation and then in generating a set of high-quality patches by adaptive binary-tree subdivision. The proposed method has some advantages over the conventional methods based on elements subdivision, i.e. the adaptive element subdivision, an improvement of the accuracy and a straight-forward implementation. By using the domain partitioning technique, the surface element can be divided into several element projection and subdivision regions under affine transformations. It is far more efficient to separately perform the element subdivision for different regions where the desirable patches are required. The ultimate patches in the vicinity of the singular point are calculated numerically by the coordinate transformation technique, while the remaining patches are evaluated accurately by the standard Gaussian quadrature. Several numerical examples are presented in order to validate the accuracy, efficiency and availability of the proposed method.