Analytical integrations in 3D BEM: preliminaries

Abstract

This work provides a preliminary contribution in the context of analytical integrations of strongly and hyper singular kernels in boundary element methods (BEMs) in 3D. It concerns the integral of 1=r 3 over a triangle in R 3 , that plays a fundamental role in BEMs in 3D, especially for the Galerkin implementation. Because the existence of the aforementioned integral depends on the position of the source point, all significant instances of the position of the source point will be considered and detailed. For its interest in the context of BEM, the integral is also considered in the more general sense of finite part of Hadamard.

Analytical integrations in 3D BEM for elliptic problems: Evaluation and implementation

“…A preliminary work [26] concerned the analytical integration of function r −3 over triangle T j , as the sum of two factors I r −3 (x, d 2 ):…”

“…In all the aforementioned papers, analytical integrations are provided for all singular integrals, whereas standard quadrature formulae are used for non-singular integrals. In the third scheme [26,[35][36][37], the complete analytical integration has been provided, also in time-dependent problems [38], evaluating HFP and CPV directly as well as by means of a limit to the boundary process. The present note falls into this class.…”

A study of three-dimensional edge and corner problems using the neBEM solver

“…The numerical implementation requires considerable care [1] because it involves evaluation of singular (weak, strong and hyper) integrals. Some of the notable two-dimensional (while all the devices are 3D by definition, useful insight is often obtained by performing a 2D analysis) and three-dimensional approaches used to evaluate the singular integrals are discussed in [1,2] and [3,4,5,6,7] and the references in these papers. It is wellunderstood that many of the difficulties in the available BEM solvers stem from the assumption of nodal concentration of singularities which leads to various mathematical difficulties and to the infamous numerical boundary layers [8,9,33] when the source is placed very close to the field point ( [2] and references [4][5][6] therein).…”

Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity: Displacement Equation

“…A general introduction to the Galerkin method can be found in [5], while [21,34] are two basic references for elasticity. For a Galerkin approximation in three dimensions, a number of singular integration methods have proved successful in handling the hypersingular kernel: transformation of the integral using Stokes' Theorem [8,9,10,18], and in particular for anisotropic elasticity [4]; numerical methods [33] based upon the Duffy transformation [24]; and analytic integration approaches utilizing either Hadamard Finite Part [1,6,30,31,33] or limit definitions [12,13]. While the direct limit analysis is convenient, in that it does not require a reformulation of the integral equations, it does require the ability to integrate analytically and to manipulate the integral.…”