Numerical quadratures for near-singular and near-hypersingular integrals in boundary element methods

TL;DR

This paper introduces a new quadrature rule derivation method for efficiently integrating near-singular and hypersingular functions in boundary element methods, avoiding explicit singularity analysis.

Contribution

It extends previous techniques to handle more complex integrals without explicit singularity separation, improving accuracy and efficiency.

Findings

Achieves near machine-precision accuracy in tests

Handles complex near-singular integrals without explicit singularity analysis

Extends previous methods for boundary element integral evaluation

Abstract

A method of deriving quadrature rules has been developed which gives nodes and weights for a Gaussian-type rule which integrates functions of the form: f(x,y,t) = a(x,y,t)/((x-t)^2+y^2) + b(x,y,t)/([(x-t)^2+y^2]^{1/2}) + c(x,y,t)\log[(x-t)^2+y^2]^{1/2} + d(x,y,t), without having to explicitly analyze the singularities of $f(x,y,t)$ or separate it into its components. The method extends previous work on a similar technique for the evaluation of Cauchy principal value or Hadamard finite part integrals, in the case when $y\equiv0$. The method is tested by evaluating standard reference integrals and its error is found to be comparable to machine precision in the best case.