Abstract
New integration schemes are presented for integrands with singularity of 1/r. We partition the element with a singular center into several triangles sharing the center. Then, a transformation between a standard square and each of the triangles is conducted. We prove such a transformation itself brings about the Jacobian with the factor r, leading to no need to introduce any other transformation. Both two-dimensional and three-dimensional cases are considered. Compared to the Duffy transformation, the proposed methods enjoy more excellent numerical properties. Numerical examples in elastic fracture are also presented to illustrate the performance of the new integration techniques.
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