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Presentation of the method

Let $ \Omega$ be a regular open bounded domain of $ \mathbb{R}^2$ (or $ \mathbb{R}^3$ ). Let us consider a Partial Differential Equation (PDE) problem defined in $ \Omega$ , written in its variational formulation:

$\displaystyle \textrm{find } u\in\mathcal{V} \textrm{ such that } a(u,w) = l(w), \forall w\in \mathcal{V},$ (2.1)

where $ \mathcal{V}$ is a Hilbert space on $ \Omega$ , usually $ H^1(\Omega)$ , $ a$ is a bilinear continuous and coercive form defined on $ \mathcal{V}$ , and $ l$ is a linear continuous form on $ \mathcal{V}$ . We finally consider a cost function $ J(\Omega,u)$ to be minimized, where $ u$ is the solution of equation (2.1).

We now consider a small perturbation of the domain, e.g. by the insertion of a crack $ \sigma_\rho = x_0+\rho \sigma(n)$ , where $ x_0\in\Omega$ represents the point where the crack is inserted, $ \sigma(n)$ is a straight crack containing the origin of the domain, and $ n$ is a unit vector normal to the crack. Finally, $ \rho>0$ represents the size of the perturbation, assumed to be small. Let $ \Omega_\rho = \Omega\backslash\sigma_\rho$ be the perturbed domain. We can consider the same PDE problem as before, but on the perturbed domain:

$\displaystyle \textrm{find } u_\rho\in\mathcal{V_\rho} \textrm{ such that } a_\rho(u_\rho,w) = l_\rho(w), \forall w\in \mathcal{V_\rho},$ (2.2)

where $ \mathcal{V}_\rho$ , $ a_\rho$ and $ l_\rho$ represent the restriction of the Hilbert space $ \mathcal{V}$ to $ \Omega_\rho$ , and the perturbed bilinear and linear forms respectively.

We can rewrite the cost function $ J$ as a function of $ \rho$ by considering the following map:

% latex2html id marker 5604 $\displaystyle j: \rho \mapsto \Omega_\rho \mapsto ... ...solution of (\ref{eq:inpainting:eq2}) } \mapsto j(\rho):=J(\Omega_\rho,u_\rho).$ (2.3)

The topological sensitivity theory provides an asymptotic expansion of $ j$ when $ \rho$ tends to zero. It takes the general form:

$\displaystyle j(\rho) - j(0) = f(\rho) G(x_0) + o(f(\rho)),$ (2.4)

where $ f(\rho)$ is an explicit positive function going to zero with $ \rho$ , and $ G(x_0)$ is called the topological gradient at point $ x_0$ [83].

Then to minimize the criterion $ j$ , one has to insert small holes (or cracks) at points where the topological gradient $ G$ is the most negative, in order to make the cost function $ j$ decrease quickly (see the asymptotic expansion (2.4)).

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