Quadrature Rules for Functions with a Mid-Point Logarithmic Singularity in the Boundary Element Method Based on the <i>x = t<sup>p</sup></i> Substitution

Abstract

Quadrature rules for evaluating singular integrals that typically occur in the boundary element method (BEM) for two-dimensional and axisymmetric three-dimensional 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 substitutionis an odd integer is given particular attention, as 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 value for p typically gives more accurate results.

Tensor-Train Numerical Integration of Multivariate Functions with Singularities

“…There are several approaches to solve this problem (see e.g. [24]). One of the most powerful is the variable substitution.…”

“…Other useful substitutions include tanh-sinh [25] (x = tanh( 1 2 π sinh t)) and erf [24] (x = 1 2 (1 − erf(t))) ones. The substitution approach can be easily adopted for TT integration technique if there is only one singularity in the multivariate integration domain [0, 1] d .…”