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Asymptotic expansion

The cost function remains unchanged, and is still defined by (2.10), as the idea is to find some cracks $ \sigma\subset\omega$ that minimize the difference between the two solutions $ u_N$ and $ u_D$ . We assume that the crack $ \sigma$ is equal to $ x+\rho\overline{\sigma}$ , where $ x$ is the point of insertion of the crack, $ \rho$ is the size of the inserted crack (assumed to be small), and $ \overline{\sigma}$ is a reference crack, of unit normal vector $ n$ . Then, we can rewrite the cost function $ J$ defined by equation (2.10) as a function $ j(\rho)$ of $ \rho$ . The asymptotic expansion is then the following:

$\displaystyle j(\rho)-j(0) = f(\rho) g(x,n) + o(f(\rho)),$ (2.14)

where the topological gradient $ g$ is defined by

$\displaystyle g(x,n) = -\left[ (\nabla u_D(x).n)(\nabla p_D(x).n)+(\nabla u_N(x).n)(\nabla p_N(x).n)\right],$ (2.15)

where $ u_D$ and $ u_N$ are the solutions of (2.11) and (2.12) respectively, but without any inserted crack ( $ \sigma=\emptyset$ ). Also, $ p_D$ and $ p_N$ are the corresponding adjoint states, respectively solutions in $ H^1(\Omega)$ of the following equations:

$\displaystyle \left\{ \begin{array}{l} p_D = 0 \quad in \ \Omega\backslash\omeg... ...ad on\ \gamma, \\ -\Delta p_D = -(u_D-u_N)\quad in\ \omega, \end{array} \right.$ (2.16)

$\displaystyle \left\{ \begin{array}{l} p_N = 0 \quad in \ \Omega\backslash\omeg... ...ad on\ \gamma, \\ -\Delta p_N = +(u_D-u_N)\quad in\ \omega. \end{array} \right.$ (2.17)

The topological gradient defined by equation (2.15) can be rewritten in the following way:

$\displaystyle g(x,n) = n^T M(x) n,$ (2.18)

where $ M(x)$ is the $ 2\times 2$ (resp. $ 3\times 3$ in the case of 3D images, or movies) symmetric matrix defined by

$\displaystyle M(x) = -sym(\nabla u_D(x) \otimes \nabla p_D(x)+\nabla u_N(x)\otimes \nabla p_N(x)).$ (2.19)

From this equation, we can deduce that the minimum of $ g(x,n)$ is reached when $ n$ is the eigenvector associated to the lowest eigenvalue $ \lambda_{min}(M(x))$ of $ M(x)$ .

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