Zakaria Belhachmi

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We study the Neumann-Laplacian eigenvalue problem in domains with multiple cracks. We derive a mixed variational formulation which holds on the whole geometric domain (including the cracks) and implement efficient finite element discretizations for the computation of eigenvalues. Optimal error estimates are given and several numerical examples are(More)
We introduce and discuss shape based models for finding the best interpolation data when reconstructing missing regions in images by means of solving the Laplace equation. The shape analysis is done in the framework of Γ-convergence, from two different points of view. First, we propose a continuous PDE model and get pointwise information on the " importance(More)
This paper is devoted to an elementary introduction to the homogenization method applied to topology and shape optimization of elastic structures under single and multiple external loads. The single load case, in the context of minimum compliance and weight design of elastic structures, has been fully described in its theoretical as well as its numerical(More)
We consider a general formulation for shape optimization problems involving the eigenvalues of the Laplace operator. Both the cases of Dirichlet and Neumann conditions on the free boundary are studied. We survey the most recent results concerning the existence of optimal domains, and list some conjectures and open problems. Some open problems are supported(More)
The paper deals with the identifiability of non-smooth defects by boundary measurements , and the stability of their detection. We introduce and analyse a new pointwise regularity concept at the boundary of an open set which turns out to play a crucial role in the identifiabilty of defects by two boundary measurements. As a consequence, we prove the unique(More)
Applying high order finite elements to unilateral contact varia-tional inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the(More)