A galerkin symmetric boundary‐element method in elasticity: Formulation and implementation

Abstract

SUMMARYStatic discontinuities (i.e. distributions of forces along a line or a surface, implying a jump of tractions across it) and kinematic (displacement) discontinuities are considered simultaneously as sources acting on the unbounded elastic space R, along the boundary r of a homogeneous elastic body R embedded in R,. The auxiliary elastic state thus generated in the body is associated with the actual elastic state by a Betti reciprocity equation. Using suitable discretizations of actual and fictitious boundary variables, a symmetric Galerkin formulation of the direct boundary element method is generated.The foliowing topics are addressed: reciprocity relations among kernels with particular attention to the role of singularities; conditions to be satisfied by the boundary field modelling in order to achieve the symmetry of the coefficient matrix; variational properties of the solution.With reference to two-dimensional problems, a technique based on a complex-variable formalism is proposed to perform the double integrations involved in this apprgach. An implementation of this technique for elastic analysis is described assuming straight elements, with continuous linear displacements and piecewise-constant tractions; all the double integrations are carried out analytically.Comparisons, from the computational standpoint, with the traditional non-symmetric method based on collocation and single integration, demonstrate the effectiveness of the present approach.

Accelerating Galerkin BEM for linear elasticity using adaptive cross approximation

“…The problem (4.1) admits the following symmetric boundary integral formulation (for details we refer to Sirtori [25] and Steinbach [26]):…”

Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity: Displacement Equation

“…The Galerkin approximation is advantageous for dealing with the hypersingular equation, and applications in anisotropic elasticity have been reported [4,23]. A general introduction to the Galerkin method can be found in [5], while [21,34] are two basic references for elasticity. For a Galerkin approximation in three dimensions, a number of singular integration methods have proved successful in handling the hypersingular kernel: transformation of the integral using Stokes' Theorem [8,9,10,18], and in particular for anisotropic elasticity [4]; numerical methods [33] based upon the Duffy transformation [24]; and analytic integration approaches utilizing either Hadamard Finite Part [1,6,30,31,33] or limit definitions [12,13].…”

Analytical integrations in 2D BEM elasticity

“…Since all the above-introduced kernels are inÿnitely smooth in their domain, which is the whole space R 2 with exception of the origin (that is x = y), the traction operator can be applied to Somigliana's identity, thus obtaining the boundary integral representation of tractions on a surface of normal n(x) in the interior of the domain [11]. Such representation formula (by some authors named 'hypersingular identity' [12]) involves Green's functions (collected in matrices G pu and G pp ) which describe components (p i ) of the traction vector p on a surface of normal n(x) due to: (i) a unit force concentrated in space (point y) and acting on the unbounded elastic space ∞ in direction j; (ii) a unit relative displacement concentrated in space (at a point y), crossing a surface with normal l(y) and acting on the unbounded elastic space ∞ (in direction j).…”