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Reports Year : 2002

Asymptotic analysis of shape functionals


A family of boundary value problems is considered in domains $\Omega(\varepsilon)=\Omega\setminus\overline\omega_\varepsilon \subset\mathbb{R}^n$, $n\ge 3$, with cavities $\omega_\varepsilon$ depending on a small parameter $\varepsilon\in (0,\varepsilon_0]$. An approximation ${\mathcal U}(\varepsilon, x)$, $x\in\Omega(\varepsilon)$, of the solution $u(\varepsilon, x)$, $x\in\Omega(\varepsilon)$, to the boundary value problem is obtained by an application of the methods of matched and compound asymptotic expansions. The asymptotic expansion is constructed with precise a priori estimates for solutions and remainders in Hölder spaces, i.e., pointwise estimates are established as well. The asymptotic solution ${\mathcal U}(\varepsilon, x)$ is used in order to derive the first term of the asymptotic expansion with respect to $\varepsilon$ for the shape functional ${\mathcal J}(\Xi(\varepsilon))={\mathbb{J}}_\varepsilon(u) \cong{\mathbb{J}}_\varepsilon({\mathcal U})$. In particular, we obtain the {\it topological derivative} $\cT(x)$ of the shape functional $\cJ(\Xi)$ at a point $x\in\O$. Volume and surface functionals are considered in the paper.


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Dates and versions

inria-00071952 , version 1 (23-05-2006)


  • HAL Id : inria-00071952 , version 1


Serguei Nazarov, Jan Sokolowski. Asymptotic analysis of shape functionals. RR-4633, INRIA. 2002. ⟨inria-00071952⟩
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