Computing singular and near-singular integrals over curved boundary elements: The strongly singular case

TL;DR

This paper introduces advanced algorithms for accurately computing strongly singular and near-singular integrals over curved boundary elements, improving boundary element methods for 3D scattering problems.

Contribution

The paper develops novel algorithms combining singularity subtraction, continuation, and transplanted Gauss quadrature for curved elements, enhancing accuracy and efficiency.

Findings

Algorithms achieve high accuracy for quadratic basis functions.

Curved boundary elements improve computational efficiency.

Method successfully applied to 3D scattering problems.

Abstract

We present algorithms for computing strongly singular and near-singular surface integrals over curved triangular patches, based on singularity subtraction, the continuation approach, and transplanted Gauss quadrature. We demonstrate the accuracy and robustness of our method for quadratic basis functions and quadratic triangles by integrating it into a boundary element code and solving several scattering problems in 3D. We also give numerical evidence that the utilization of curved boundary elements enhances computational efficiency compared to conventional planar elements.