On the non‐singular traction‐BIE in elasticity

Abstract

SUMMARYThe work reported herein develops a generalized traction-BIE formulation which involves only weakly singular integrals (in the three-dimensional problem) or totally regular integrals (in the two-dimensional problem). The first step deals with the terms in the Somigliana displacement identity, and then the derivatives of these terms. The only conditions required for the existence of the traction-BIE and the related Somigliana stress identity are weak continuity of the in-plane derivatives of the surface displacements and of the surface tractions. It is shown that the Cauchy Principal Value (CPV) interpretations so commonly used in BIE developments are unnecessary. The formulation is established not only at a smooth boundary point, but also at a comer point. The extension of the non-singular formulation to discontinuous boundary tractions and tangential derivatives of the boundary displacements applicable to a generalized problem statement as well as the usual BEM implementations is also shown. In the demonstrated formulation, the source points are located directly at the boundary nodes and non-conformal elements are not needed.

The complex one-sided integrals of Cauchy and Hadamard and application to boundary element method

“…If unknown functions are approximated by the polynomials or the polynomials multiplied by a weight function (for the tip elements) all the integrals (hypersingular, singular and regular) involved in this equation can be evaluated in a closed form. The expressions to evaluate the integrals involved in (26) are given in Reference 27.…”

“…It can be written as (26) where¸is the totality of external and internal boundaries; u"u>!u\ is the displacement discontinuity (DD); u> and u\ are the limit values of the displacements if we approach the contour from its left or right side, respectively; " L #i O ; L is a normal and O is a shear traction on¸; the unit normal n is directed to the right of the direction of travel; the direction of travel is arbitrary for contact boundaries between regions; it is counterclockwise for the external boundary; a bar over a symbol denotes complex conjugation; the origin of the co-ordinate system is assumed to be inside some region; i"(!1;…”

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.…”

Hypersingular Integrals: How Smooth Must the Density Be?

“…For simplicity, let us take x = 0 and define T(y) = T(0, y). Then, we have wheref, is defined by (9).…”