Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

Bécache, Éliane; Nédélec, Jean-Claude; Nishimura, Naoshi

doi:10.1007/bf00041622

Cited by 25 publications

(29 citation statements)

References 6 publications

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“…The new kernels B ikqs (x, x) and A ik (x, x), given by Eqs. (88), (89), are associated with the decomposition (87) of the kernel D ik (x, x) (given in [38] and generalized to anisotropic elasticity in [3]), by virtue of which the Stokes formula (13) could be applied. An identity similar to (51), with S, u, v replaced with Γ, φ, ψ, has been used to obtain (41).…”

Section: Symmetric Galerkin Bie Formulationmentioning

confidence: 99%

Boundary element based formulations for crack shape sensitivity analysis

Bonnet

2001

Engineering Analysis with Boundary Elements

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The present paper addresses several BIE-based or BIE-oriented formulations for sensitivity analysis of integral functionals with respect to the geometrical shape of a crack. Functionals defined in terms of integrals over the external boundary of a cracked body and involving the solution of a frequency-domain boundary-value elastodynamic problem are considered, but the ideas presented in this paper are applicable, with the appropriate modifications, to other kinds of linear field equations as well. Both direct differentiation and adjoint problem techniques are addressed, with recourse to either collocation or symmetric Galerkin BIE formulations. After a review of some basic concepts about shape sensitivity and material differentiation, the derivative integral equations for the elastodynamic crack problem are discussed in connection with both collocation and symmetric Galerkin BIE formulations. Building upon these results, the direct differentiation and the adjoint solution approaches are then developed. In particular, the adjoint solution approach is presented in three different forms compatible with boundary element method (BEM) analysis of crack problems, based on the discretized collocation BEM equations, the symmetric Galerkin BEM equations and the direct and adjoint stress intensity factors, respectively. The paper closes with a few comments

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Section: Symmetric Galerkin Bie Formulationmentioning

confidence: 99%

Boundary element based formulations for crack shape sensitivity analysis

Bonnet

2001

Engineering Analysis with Boundary Elements

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“…The regularization procedure yielding (7)- (9) involves the Stokes theorem together with indirect regularization. The surface curl operator R arising as a result of this manipulation, introduced e.g in [31,4], is defined by [Ru] ks (x) = e jrs n j u k,r (y) (where e jrs denotes the permutation symbol), while the weakly singular fourth-order tensor B ikqs is given by …”

Section: Introductionmentioning

confidence: 99%

Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Pham¹,

Mouhoubi²,

Bonnet³

et al. 2012

Engineering Analysis with Boundary Elements

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…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]. While the direct limit analysis is convenient, in that it does not require a reformulation of the integral equations, it does require the ability to integrate analytically and to manipulate the integral.…”

Section: Introductionmentioning

confidence: 99%

Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity: Displacement Equation

Gray

Griffith

Johnson

et al. 2007

EJBE

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Algorithms for the direct evaluation of singular Galerkin boundary integrals for three-dimensional anisotropic elasticity are presented. The integral of the traction kernel is defined as a boundary limit, and (partial) analytic evaluation is employed to compute the limit. The spherical angle components of the Green's function and its derivatives are not known in closed form, and thus the analytic integration requires a splitting of the kernel, into 'singular' and 'non-singular' terms. For the coincident singular integral, a single analytic evaluation suffices to isolate the potentially divergent term, and to show that this term self-cancels. The implementation for a linear element is considered in detail, and the extension to higher order curved interpolation is also discussed. Results from test calculations establish that the algorithms are successful.

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Regularization in 3D for anisotropic elastodynamic crack and obstacle problems

Cited by 25 publications

References 6 publications

Boundary element based formulations for crack shape sensitivity analysis

Boundary element based formulations for crack shape sensitivity analysis

Fast multipole method applied to Symmetric Galerkin boundary element method for 3D elasticity and fracture problems

Evaluation of Galerkin Singular Integrals for Anisotropic Elasticity: Displacement Equation

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