On the validity of conforming BEM algorithms for hypersingular boundary integral equations

Abstract

The widely held notion that the use of standard conforming isoparametric boundary elements may not be used in the solution of hypersingular integral equations is investigated. It is demonstrated that for points on the boundary where the underlying ®eld is C 1Ya continuous, a class of rigorous nonsingular conforming BEM algorithms may be applied. The justi®cation for this class of algorithms is interpreted in terms of some recent criticism. It is shown that the numerical integration in these conforming BEM algorithms using relaxed regularization represents a ®nite approximation to the standard twosided Hadamard ®nite part interpretation of hypersingular integrals. It is also shown that the integration schemes in this class of algorithms are not based upon one-sided ®nite part interpretations. As a result, the attendant ambiguities associated with the incorrect use of the one-sided interpretations in boundary integral equations pose no problem for this class of algorithms. Additionally, the distinction is made between the analytic discontinuities in the ®eld which place limitations on the applicability of the conforming BEM and the discontinuities resulting from the use of piece-wise C 1Ya interpolations. IntroductionHypersingular integral equations are formed from differentiation of singular integral equations and as a result, the order of the singularities in the kernels is increased in these equations. The higher order singularities in the kernels requires more smoothness of the densities for the given integral to be ®nite. A suf®cient condition for the existence of hypersingular integrals is C 1Ya continuity of the density at the source point. This is a condition which is not met by the use of standard isoparametric boundary elements which only provide a piece-wise C 1Ya interpolation. For this reason, approximate solution techniques for solving hypersingular integral equations using the boundary element method (BEM) require special consideration.In general, this special consideration takes one of several forms. The hypersigularity may be reduced through integration by parts (Guidera and Lardner 1975;Giroire and Ne Âde Âlec 1995). In addition to being rather mathematically tedious, approaches of this form have yet to produce numerical results of the same accuracy as have been obtained with the more conventional approaches.Each of several remaining approaches seeks to impose C 1Ya continuity at the source point on the particular density in the hypersingular integral which is used in the numerical integration, as is required mathematically. The ®rst class of algorithms is based on interpolation schemes which are C 1Ya continuous. One way in which this is done is to introduce nodal displacement gradients in the interpolation scheme. Polch et al. (Polch et al. 1987) used a standard isoparametric representation for displacement gradients. This continuous displacement gradient ®eld was then coupled to the standard interpolation for displacements through a least-squares method. Young (Young 1996) has used a hybrid in...

Evaluation of non-singular BEM algorithms for potential problems

“…In spite of the very successful numerical results reported by Richardson et al (1997), Richardson and Cruse (1999), Chien et al (1991), Huang and Cruse (1994) using various forms of these relaxed algorithms combined with piece-wise C 1,α interpolations, Martin and Rizzo (1996), Krishnasamy et al (1992) have concluded that these algorithms could not be theoretically justified. This means that, from a strictly mathematical point of view, only boundary element implementations that ensure C 0,α or C 1,α continuity at each collocation point can be applied in the discretizations of the standard, or the hypersingular boundary integral equations, respectively.…”

“…The next section discusses the discretization of these integral equations, where the boundary is discretized into standard isoparametric elements, which satisfy all the continuity requirements for the potential-BIE, but do not satisfy the continuity requirements for the flux-BIE at a finite number of inter-element nodes in case standard continuous isoparametric boundary elements are used. Since these continuity requirements refer only to collocation points, a 'relaxed continuity' approach is introduced for the case of collocation at an inter-element node according to Huang and Cruse (1994), Cruse and Richardson (1996), Richardson et al (1997), Richardson and Cruse (1999), and different alternatives are discussed, which either allow or avoid the collocation at such nodes. Numerical results obtained from these alternatives, as well as from the self-regular potential BIE are presented.…”

“…(9)). However, in Richardson et al (1997) and Richardson and Cruse (1999) the authors claim that this BEM algorithm matches the bounded nature of the BIE in a two-sided sense without invalidating the underlying continuity requirements for the flux formulation, which is the essence of relaxed continuity approach (piecewise smoothness). Unlike the potential-BIE, on the flux-BIE all integrals in Eq.…”

“…Self-regular forms of the Somigliana displacement identity (SDI) and the Somigliana stress identity (SSI) have been presented by Cruse and Richardson (1996) and Richardson et al (1997). For 3-D potential theory, Cruse and Richardson (2000) have presented two self-regular formulations of the BIE, while Jorge et al (2001) in a similar fashion have developed the correspondent self-regular forms for 2-D problems.…”

Smoothness-relaxation strategies for singular and hypersingular integral equations

“…However, the third author has written several papers with Huang and Richardson [2][3][4] in which they do so collocate. In fact, they report good numerical computations (see also References 5 and 6), using regularized integral equations.…”