New Numerical Quadrature of Integrand with Singularity of 1/r and its Application

A new quadrature method is proposed for numerical integration of integrands with the singularity of 1/r occurring at the computation of stiffness matrix when a singular physical cover is introduced to the numerical manifold method (NMM) for linear fracture problems. The detailed proof is presented, which shows the Jacobian has a factor of r that can be used to eliminate the singularity. Compared with the Duffy transformation, it proves more simple and easier to implement while owning the same precision. A numerical example in elastic fracture by the NMM is presented to illustrate the performance of the proposed method. The result has a good agreement with the reference solution.

Simple but Efficient Method for Integrating 1/r Singularities

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.