Symmetric Galerkin BEM for multi‐connected bodies

Abstract

SUMMARYIn this paper, it is shown that the symmetric Galerkin boundary element formulation cannot be used in its standard form for multiple connected bodies. This is because the traction integral equation used for boundaries with Neuman boundary condition give non-unique solutions. While this fact is well known from the classical theory of integral equations, the problem has not been fully addressed in the literature related to symmetric Galerkin formulations. In this paper, the problem is reviewed and a general way to deal with it is proposed. The details of the numerical implementation are discussed and an example is solved to demonstrate the e ectiveness of the proposed solution.

Development and implementation of some BEM variants—A critical review

“…In an attempt to expand the SGBEM applicability, the SGBEM was then extended by Pé rez-Gavilá n and Aliabadi [196] in which the modified method can be demonstrated for multi-connected bodies. However, in their following reports [193,194], they also pointed out that the applications of symmetric Galerkin boundary element method to multiple connected regions, where at least one closed boundary is subjected to Neumann boundary condition only, will result in non-uniqueness. To overcome the non-uniqueness problem, they proposed a new approach which consists of partitioning the domain in two simply connected subregions to deal with the nonuniqueness in symmetric Galerkin scheme.…”

On the Identities for Elastostatic Fundamental Solution and Nonuniqueness of the Traction BIE Solution for Multiconnected Domains

“…The defect related to the traction or hypersingular BIEs for multiconnected domains has been reported in Refs, [9,10] for elasticity problems and in Refs, [11,12] for Stokes flow problems, although it has not been proved in a general setting. In Ref, [9], only the constant displacement term is dealt with and an integral representation for the domain enclosed by the hole is applied to determine the unknown constant displacement in the solution on the edge of a hole for the original traction BIE for multidomain problems, A few selected constraints on the edge of the hole are also introduced in order to remove the rigid-body motion in the solution of the traction BIE, In Ref, [10], both the constant and linear terms are discussed, however, only for 2D cases as the integral identities applied are derived only for such cases.…”

“…[10] for the 2D case based on a physical argument (equilibrium of moments of the traction) and with yj = 0. Taking the derivative of the new identity, Eq.…”

“…This defect in the traction BIEs for multiconnected domains was reported in Refs. [9,10] for elasticity problems, and in Refs. [11,12] for Stokes fiow problems.…”

“…Special care or techniques were also proposed in Refs. [9,10] to remedy this situation with the traction BIE for multiconnected domain elasticity problems. For Stokes flow problems, a dual BIE formulation using a linear combination of the singular and hypersingular BIEs is suggested in Refs.…”