Mixed Finite Element Methods for Smooth Domain Formulation of Crack Problems

Belhachmi, Zakaria; Sac-Epee, Jean-Marc; Sokołowski, Jan

doi:10.1137/s0036142903429729

Cited by 14 publications

(13 citation statements)

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“…Therefore, there is a unique weak solution to the associated variational inequality. We introduce also the so-called smooth domain formulation [15] which have some implications in numerical analysis, for the related results in the case of a scalar problem of an elastic membrane with a cut we refere the reader to [1]. The smooth domain formulation allows obtain variational solutions to the crack problem in the geometrical domain without any cut, the crack is present only in the subset of admissible functions for the variational solution, i.e., some inequality constraints are imposed for the admissible functions over the crack Γ c .…”

Section: 1)mentioning

confidence: 99%

“…Numerical aspects for the problems like (1.1)-(1.5) can be found, for example, in [1], [17]. In particular, in [1] the convergence of the finite element approximation is proved for a scalar problem, and some error estimates are derived.…”

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confidence: 99%

“…In particular, in [1] the convergence of the finite element approximation is proved for a scalar problem, and some error estimates are derived.…”

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confidence: 99%

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Shape and topological sensitivity analysis in domains with cracks

2010

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Abstract. Framework for shape and topology sensitivity analysis in geometrical domains with cracks is established for elastic bodies in two spatial dimensions. Equilibrium problem for elastic body with cracks is considered. Inequality type boundary conditions are prescribed at the crack faces providing a non-penetration between the crack faces. Modelling of such problems in two spatial dimensions is presented with all necessary details for further applications in shape optimization in structural mechanics. In the paper, general results on the shape and topology sensitivity analysis of this problem are provided. The results are interesting on its own. In particular, the existence of the shape and topological derivatives of the energy functional is obtained. It is shown, in fact, that the level set type method [4] can be applied to shape and topology opimization of the related variational inequalities for elasticity problems in domains with cracks, with the nonpenetration condition prescribed on the crack faces. The results presented in the paper can be used for numerical solution of shape optimization and inverse problems in structural mechanics.Key words. Crack with non-penetration, shape sensitivity, derivative of energy functional, topological derivative AMS subject classifications. Primary 35J85, 74K20 Secondary 35J25, 74M151. Introduction. Shape optimization requires few mathematical results, in the framework of modelling and numerical solution, for any specific class of problems governed by partial differential equations of mathematical physics. Usually, we need to show the well posedness of the specific problem, and also we can propose a numerical method for the effective solution procedure. Hence, in order to solve a shape optimization problem we are obliged to have the results on• the existence and continuous dependence with respect to the shape of solutions to the model, which may result in the existence of optimal shapes, • the differentiability of solutions with respect to the boundary variations, which imply the existence of shape gradients and leads to some necessary conditions for optimality, of the first order and possibly of the second order which leads to the Newton method of shape optimization, • and in addition, perform the asymptotic analysis of the related boundary value problem in singularly perturbed geometrical domains and derive the form of the topological derivative for the shape functional of interest, which allows for the topology changes in the process of numerical optimization, if necessary, • and finally, we may device a numerical method and show its efficiency in numerical examples, and its convergence form the mathematical point of view.

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Shape and topological sensitivity analysis in domains with cracks

2010

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“…Both their mathematical analysis [15,25] and their numerical analysis [19,18,17,24,31] and more recently [5,7,3,6,21,14,26,27,22] have been widely studied but still remain a source of many challenging problems. The unilateral crack problems are a particular class where the Signorini conditions are prescribed at the crack faces [23,4,29,30].…”

Section: Introductionmentioning

confidence: 99%

“…The mixed variational formulation to problem (1.1)-(1.6) in displacement-stress fields could be extended to the whole domain including the crack [23,4,30]. Therefore, with this extended formulation, we work in the Lipschitzian domain and the difficulties due to the presence of the crack as geometric object are reduced.…”

Section: Introductionmentioning

confidence: 99%

Locking-Free Finite Elements for Unilateral Crack Problems in Elasticity

Belhachmi¹,

Sac-Epee²,

Tahir³

2009

Math. Model. Nat. Phenom.

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Abstract. We consider mixed and hybrid variational formulations to the linearized elasticity system in domains with cracks. Inequality type conditions are prescribed at the crack faces which results in unilateral contact problems. The variational formulations are extended to the whole domain including the cracks which yields, for each problem, a smooth domain formulation. Mixed finite element methods such as PEERS or BDM methods are designed to avoid locking for nearly incompressible materials in plane elasticity. We study and implement discretizations based on such mixed finite element methods for the smooth domain formulations to the unilateral crack problems. We obtain convergence rates and optimal error estimates and we present some numerical experiments in agreement with the theoretical results.

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