The Distance-Sinh combined transformation for near-singularity cancelation based on the generalized Duffy normalization

Abstract

The singularity cancelation is usually of great importance in the calculation of nearly singular integral that is involved in the boundary element analysis of thin structures and in the extended finite element method for crack problems. The well-known distance transformation can be performed for various kinds of singular kernels to compute the near singular integral. However, in the case of that the source point is located nearby the boundary of the integral element, the result is usually very sensitive to the shape of the sub-triangles. A Distance-Sinh combined transformation method based on the generalized Duffy normalization is proposed in this paper to improve the accuracy of the nearly singular integral calculation. Furthermore, a two-level transformation method is proposed to cancel the singularity of the integrand. In first level of the method, the generalized Duffy transformation is implemented. In the transformed integral space (u, v), the singularity of the integrand along u-direction is eliminated by the distance transformation. The singularity of the integrand along the other direction is canceled by the Sinh transformation. Numerical examples are presented to verify the proposed method.