Quadrature Rules for Functions with a Mid-point Logarithmic Singularity in the Boundary Element Method based on the x=tp Substitution

Abstract

Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric threedimensional problems are considered. This paper focuses on the numerical integration of the functions on the standard domain [-1,1], with a logarithmic singularity at the centre. The subtitution x=tp, where p(>3) is an odd integer is given particular attention, since this returns a regular integral and the domain unchanged. Gauss-Legendre quadrature rules are applied to the transformed integrals for a number of values of p. It is shown that a high values for p typically gives more accurate results.