Symmetric Galerkin boundary formulations employing curved elements

Abstract

SUMMARYAccounts of the symmetric Galerkin approach to boundary element analysis (BEA) have recently been published. This paper attempts to add to the understanding of this method by addressing a series of fundamental issues associated with its potential computational efficiency. A new symmetric Galerkin theoretical formulation for both the (harmonic) heat conduction and the (biharmonic) elasticity problem that employs regularized singular and hypersingular boundary integral equations (BIEs) is presented. The novel use of regularized BIEs in the Galerkin context is shown to allow straightforward incorporation of curved, isoparametric elements. A symmetric reusable intrinsic sample point (REP) numerical integration algorithm is shown to produce a Galerkin (i.e. double) integration strategy that is competitive with its counterpart (i.e. singular) integration procedure in the collocation BEA approach when the time saved in the symmetric equation solution phase is also taken into account. This new formulation is shown to be capable of employing hypersingular BIEs while obviating the requirement of C1 continuity, a fact that allows the employment of the popular continuous element technology. The behaviour of the symmetric Galerkin BEA method with regard to both direct and iterative equation solution operations is also addressed. A series of example problems are presented to quantify the performance of this symmetric approach, relative to the more conventional unsymmetric BEA, in terms of both accuracy and efficiency. It is concluded that appropriate implementations of the symmetric Galerkin approach to BEA indeed have the potential to be competitive with, if not superior to, collocation-based BEA, for large-scale problems.

Regularized direct and indirect symmetric variational BIE formulations for three-dimensional elasticity

“…It is also similar in nature to the weighted residual statement developed in [11], from which it differs mainly by the kind of regularization used for the originally hypersingular kernel.…”

Galerkin BEM with Direct Evaluation of Hypersingular Integrals

“…They have never been detected before because in no other paper a limit process based on the vanishing neighbourhood approach has been used in conjunction with SGBEM. All formulations of SGBEM based on regularization of the kernels before performing the limit, like in [11,12,13,14,15], do not provide free terms at all (and this is indeed one of their advantages). The same is true for the limit to the boundary method [16,17,18].…”

“…In [11] simple solutions were employed, whereas in [12] a procedure based on Stokes theorem (or integration by parts in 2D, as in [13,14]) was developed. Regularization via integration by parts has been also presented in [15] for Kirchhoff plates.…”

On the extraction technique in boundary integral equations

“…The evaluation of layer potentials and also their derivatives near the supporting surface Γ ⊂ R d , d = 2 or 3, has attracted attention in boundary element computations recently (see e.g., [8,10,11,12,20]). If the point of interest x is sufficiently distant from Γ, the evaluation does not cause any difficulties due to the smoothness of the kernels.…”