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.

翻译：这种方法还以适应性形式实施,使所需的容度趋同;该方法的结果与一四面体精确分析方法的输出相比较,并显示高准确性;特别是,在使用适应性算法时,发现条件差的四面体的极性结果非常准确,这些四面体通常会给数字二次曲线技术带来困难;该方法还演示了在未结构的网格上对生物-沙瓦集成物与固定的结晶体规则相结合的评估方法,并展示了良好的趋同性和准确性。