A general algorithm for numerical integration of three-dimensional crack singularities in PU-based numerical methods

Abstract

With the development of PU-based numerical methods for crack problems, the evaluation of various orders of vertex/edge singularity has been one of the most critical issues, which restrains the computational efficiency of PU-based methods, especially for 3D crack problems. In this paper, based on the conventional Duffy transformation, a general algorithm for numerical integration of three-dimensional crack singularities is proposed for the vertex/edge singularity problems, which takes the integration cell shape into full consideration. Besides, the corresponding 3D conformal preconditioning strategy is constructed to fully eliminate the shape influence of tetrahedron elements. Extensive numerical examples, including ill-shaped integration cells and crack-front tetrahedron elements with parallel/nonparallel crack front, are given to validate the feasibility and accuracy of the proposed method. As a result, for each crack-front element, several hundreds of Gauss points are sufficient to achieve the precision of 10−6 for both kernels 1∕r and 1∕r, in sharp contrast with around ten thousands of Gauss points using the conventional Duffy transformation.