Analytical integrations in 2D BEM elasticity

Abstract

SUMMARYIn the context of two-dimensional linear elasticity, this paper presents the closed form of the integrals that arise from both the standard (collocation) boundary element method and the symmetric Galerkin boundary element method. Adopting polynomial shape functions of arbitrary degree on straight elements, ÿnite part of Hadamard, Cauchy principal values and Lebesgue integrals are computed analytically, working in a local coordinate system. For the symmetric Galerkin boundary element method, a study on the singularity of the external integral is conducted and the outer weakly singular integral is analytically performed. Numerical tests are presented as a validation of the obtained results.

Hypersingular formulation for boundary stress evaluation revisited. Part 2: Corners.

“…Such an integral equation is however due to a property of the global elastic integral and not of each singular operator. Only in this view equation (2) still makes sense at a corner.…”

“…For all fields that do not fulfill the constitutive law, all aforesaid properties do not seem to hold and equation (2) does not seem to make sense at a corner.…”

“…(5) and of the hyper singularity of the kernel G pp , the function G pp (x − y; n(x); l(y)) [u(y) − u(x 0 ) − (x 0 ) (y − x 0 )] is Lebesgue integrable on Γ also when x ∈ Ω moves to x 0 ∈ Γ ε . Free terms arise therefore from the remaining terms, which will be dealt with in Section 2 by means of recently proposed formulae [2]. As a main result, it will be shown that the presence of a corner causes the non-existence of limit (7): it turns out to consist of the finite part of Hadamard (HFP), of a bounded term independent on the size of the support (expected to be the free term contribution D pp (x, n(x))), and of an unbounded logarithmic term.…”

“…In dealing with smooth boundaries, p(x) in equation (2) denotes the traction at x ∈ Γ referring to the outward normal l(x), that is p(x) stands for p(x, l(x)). Proposition 4.1 in reference [1] states that equation (2) holds even with respect to any direction n(x). The present work aims at extending such a result in the presence of corners, when normal at the boundary is not uniquely defined.…”

“…x ∈ Ω has been solved in [2] by means of the variable change r + = x + − y + ; matrices L pu , A pu , and S pu follow in the local reference L + :…”