A series of Duffy-distance transformation for integrating 2D and 3D vertex singularities

With the development of the generalized/extended finite element method for fracture problems, the accurate and efficient integration of singular enrichment functions has been an open issue, especially for the 3D case. In this paper, we reveal the near singularities caused by distorted integral patch/cell shape numerically and theoretically during the implementation of generalized Duffy transformation, and the Duffy-distance transformation is developed step by step for the 2D and 3D vertex singularities. Meanwhile, the 3D conformal preconditioning strategy is constructed to eliminate the near singularity caused by element shape distortion during the iso-parametric transformation, which enables the Duffy-distance transformation to be applicable for arbitrary shaped tetrahedral elements. As a result, the near singularities can be fully or partly canceled depending on the order of singularity. The implementation of the proposed scheme in existing codes is straightforward. Numerous numerical examples for arbitrary shaped triangles and tetrahedrons are presented to demonstrate its robustness and efficiency, along with comparisons to the generalized Duffy transformation.