Smoothness-relaxation strategies for singular and hypersingular integral equations

Abstract

Three stages are involved in the formulation of a typical direct boundary element method: derivation of an integral representation; taking a Limit To the Boundary (LTB) so as to obtain an integral equation; and discretization. We examine the second and third stages, focussing on strategies that are intended to permit the relaxation of standard smoothness assumptions. Two such strategies are indicated. The ÿrst is the introduction of various apparent or 'pseudo-LTBs'. The second is 'relaxed regularization', in which a regularized integral equation, derived rigorously under certain smoothness assumptions, is used when less smoothness is available. Both strategies are shown to be based on inconsistent reasoning. Nevertheless, reasons are o ered for having some conÿdence in numerical results obtained with certain strategies. Our work is done in two physical contexts, namely two-dimensional potential theory (using functions of a complex variable) and three-dimensional elastostatics. ? 1998 John Wiley & Sons, Ltd.

Two-dimensional numerical approach for the vibration isolation analysis of thin walled wave barriers in poroelastic soils

“…The treatment of those integrals for this problem is analogous to that of acoustics [11] and elastodynamics [33,15] problems. The treatment of the HBIE is based on a regularization process that requires the integrands (excluding the term r −2 ) belong to the Hölder function space C 1,α [30]. To do so, the collocation point must be in a boundary point where the primary variables are differentiable, i.e.…”

New identities for fundamental solutions and their applications to non-singular boundary element formulations

“…Relaxation of this smoothness requirement for the hypersingular BIEs, in either stronglysingular or weakly-singular forms and with conforming C 0 elements, have been attempted by several authors (see, e.g., (Chien, Rajiyah et al 1990;Wu, Seybert et al 1991;Liu and Rizzo 1992;Cruse and Suwito 1993;Huang and Cruse 1994;Cruse and Richardson 1996)). The validation of this relaxation has also been provided (Richardson, Cruse et al 1997;Liu and Chen 1999;Martin, Rizzo et al 1998) due to the strong need to do this in the BEM community. It has been postulated in Ref.…”

“…The validation of this relaxation received renewed attention recently (Richardson, Cruse et al 1997;Liu and Chen 1999;Martin, Rizzo et al 1998) due to a strong desire to do so in the BEM community.…”

Evaluation of non-singular BEM algorithms for potential problems

“…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. Martin et al (1998) renewed the discussion between the theoretical continuity requirements and the good numerical results reported by Cruse and his co-workers. Jorge et al (2003) and Porto et al (2005) presented a non-symmetric variational approach to enforce C 1,α continuity requirement at inter-element nodes in the self-regular traction-BIE and flux-BIE, discretized using relaxed continuity approach with Lagrangian C 0 elements.…”