Isoparametric singularity extraction technique for 3D potential problems in BEM

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 ≤ −1 and q, r ≥ 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.

IgA-BEM for 3D Helmholtz problems on multi-patch domains using B-spline tailored numerical integration

“…where U m s is an approximation of U(s, •), obtained by truncating a particular series expansion of U about the singular point t = s after the m-th term; see [12] for the first application of these expansions in the simplest form and [15] for detailed construction and analysis. Note that the singularity subtraction technique that splits an integral into two ones is applied only to a smaller portion of integrals.…”

“…Owing to the division step, the regularized kernel becomes singular if U m s has roots inside the integration domain. For additional analysis of the latter regularization technique see [15].…”

“…The continuity of the integrand in the regularized integrals is equal to C m−2 at t = s for the chosen m ≥ 1 (with C −1 we denote functions that are integrable in the standard sense but have discontinuities of the first kind at t = s), if the geometry map F is a smooth enough function [15]. By defining local coordinates z := s − t, the kernel reads…”

“…For this aim we have been relying on the analytical expressions derived in [15] and computed with a help of Wolfram Mathematica software.…”