Crack localization problem

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Crack localization problem

Let $\Omega$ be a bounded open set of $\mathbb{R}^2$ . We assume in this section that $\Omega$ contains a perfectly insulating crack $\sigma^\ast$ . We impose a flux $\phi\in H^{-1/2}(\Gamma)$ on the boundary $\Gamma$ of $\Omega$ , and we want to find $\sigma\subset\Omega$ such that the solution $u\in H^1(\Omega\backslash\sigma)$ of

$\displaystyle \left\{ \begin{array}{l} \Delta u = 0 \quad in \ \Omega\backslash... ... \phi\quad on\ \Gamma, \\ \partial_n u = 0\quad on\ \sigma, \end{array} \right.$

(2.7)

satisfies $u\vert _\Gamma = T$ , where $T\in H^{1/2}(\Gamma)$ is a given function. We also assume some compatibility conditions in order to have a well-posed direct problem.

A topological gradient approach has been introduced in [8], and consists of defining a Dirichlet and a Neumann problem, as we have an over-determination in the boundary conditions:

$\displaystyle u_D\in H^1(\Omega\backslash\sigma) \textrm{ such that } \ \left\{... ...= T\quad on\ \Gamma, \\ \partial_n u_D = 0\quad on\ \sigma, \end{array} \right.$

(2.8)

$\displaystyle u_N\in H^1(\Omega\backslash\sigma) \textrm{ such that } \ \left\{... ...phi\quad on\ \Gamma, \\ \partial_n u_N = 0\quad on\ \sigma. \end{array} \right.$

(2.9)

It is clear that for the actual crack $\sigma^\ast$ , the two solution and are equal. The idea is then to consider and minimize the following cost function

$\displaystyle J(\sigma) = \frac{1}{2} \Vert u_D-u_N\Vert^2_{L^2(\Omega)}.$

(2.10)

The topological asymptotic expansion of this cost function is detailed in [8].

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