Spherical polar coordinate transformation for integration of singular functions on tetrahedra

TL;DR

This paper introduces a spherical polar coordinate transformation method for accurately evaluating integrals with singularities on tetrahedra, improving convergence and accuracy especially for poorly conditioned geometries.

Contribution

The paper presents a novel coordinate transformation technique that explicitly removes singularities in tetrahedral integrals and implements an adaptive algorithm for high-precision results.

Findings

High accuracy compared to analytical solutions

Effective for poorly conditioned tetrahedra

Demonstrates good convergence with adaptive method

Abstract

A method is presented for the evaluation of integrals on tetrahedra where the integrand has an integrable singularity at one vertex. The approach uses a transformation to spherical polar coordinates which explicitly eliminates the singularity and facilitates the evaluation of integration limits. The method is also implemented in an adaptive form which gives convergence to a required tolerance. Results from the method are compared to the output from an exact analytical method for one tetrahedron and show high accuracy. In particular, when the adaptive algorithm is used, highly accurate results are found for poorly conditioned tetrahedra which normally present difficulties for numerical quadrature techniques. The approach is also demonstrated for evaluation of the Biot-Savart integral on an unstructured mesh in combination with a fixed node quadrature rule and demonstrates good convergence and accuracy.