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Optimal Shape for Elliptic Problems with Random Perturbations

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      Abstract

      In this paper we analyze the relaxed form of a shape optimization problem with state equation \(\{{array}{ll} -div \big(a(x)Du\big)=f\qquad\hbox{in}D \hbox{boundary conditions on}\partial D. {array}.\) The new fact is that the term \(f\) is only known up to a random perturbation \(\xi(x,\omega)\). The goal is to find an optimal coefficient \(a(x)\), fulfilling the usual constraints \(\alpha\le a\le\beta\) and \(\displaystyle\int_D a(x) dx\le m\), which minimizes a cost function of the form \[\int_\Omega\int_Dj\big(x,\omega,u_a(x,\omega)\big) dx dP(\omega).\] Some numerical examples are shown in the last section, to stress the difference with respect to the case with no perturbation.

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      Exact estimates of conductivity of composites formed by two isotropically conducting media taken in prescribed proportion

       A. Cherkaev,  K. Lurie (1984)
      This paper is a sequel to [1-4]. We consider the problem of G-closure, i.e. the description of the setGUof effective tensors of conductivity for all possible mixtures assembled from a number of initially given components belonging to some fixed setU. Effective tensors are determined here in a sense of G-convergence relative to the operator ∇·D· ∇, of the elementsDeU∈ [5, 6].
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        H-Convergence

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          Variational Methods for Structural Optimization

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            Author and article information

            Journal
            1002.2770

            Analysis, Numerical methods

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