Isoparametric singularity extraction technique for 3D potential problems in BEM

TL;DR

This paper introduces an isoparametric singularity extraction method for 3D potential problems in boundary element methods, improving the accuracy of integral evaluations on smooth curved geometries by series expansion and analytical integration.

Contribution

It develops a novel series expansion approach for singular kernel regularization in 3D BEM using isoparametric mappings, enhancing integral accuracy on curved geometries.

Findings

Series expansion increases kernel smoothness at source points.

Analytical formulas for integral terms are derived using recurrence relations.

Numerical tests show improved accuracy in singular integral evaluation.

Abstract

To solve boundary integral equations for potential problems using collocation Boundary Element Method (BEM) on smooth curved 3D geometries, an analytical singularity extraction technique is employed. By adopting the isoparametric approach, curved geometries that are represented by mapped rectangles or triangles from the parametric domain are considered. The singularity extraction on the governing singular integrals can be performed either as an operation of subtraction or division, each having some advantages. A particular series expansion of a singular kernel about a source point is investigated. The series in the intrinsic coordinates consists of functions of a type $R^p x^q y^r$, where $R$ is a square root of a quadratic bivariate homogeneous polynomial, corresponding to the first fundamental form of a smooth surface, and $p,q,r$ are integers, satisfying $p\leq -1$ and $q,r \geq 0$. By extracting more terms from the series expansion of the singular kernel, the smoothness of the regularized kernel at the source point can be increased. Analytical formulae for integrals of such terms are obtained from antiderivatives of $R^p x^q y^r$, using recurrence formulae, and by evaluating them at the edges of rectangular or triangular parametric domains. Numerical tests demonstrate that the singularity extraction technique can be a useful prerequisite for a numerical quadrature scheme to obtain accurate evaluations of the governing singular integrals in 3D collocation BEM.