Topological derivative

The topological derivative is, conceptually, a derivative of a shape functional with respect to infinitesimal changes in its topology, such as adding an infinitesimal hole or crack. When used in higher dimensions than one, the term topological gradient is also used to name the first-order term of the topological asymptotic expansion, dealing only with infinitesimal singular domain perturbations. It has applications in shape optimization, topology optimization, image processing and mechanical modeling.

Let ${\displaystyle \Omega }$ be an open bounded domain of ${\displaystyle \mathbb {R} ^{d}}$, with ${\displaystyle d\geq 2}$, which is subject to a nonsmooth perturbation confined in a small region ${\displaystyle \omega _{\varepsilon }({\tilde {x}})={\tilde {x}}+\varepsilon \omega }$ of size ${\displaystyle \varepsilon }$ with ${\displaystyle {\tilde {x}}}$ an arbitrary point of ${\displaystyle \Omega }$ and ${\displaystyle \omega }$ a fixed domain of ${\displaystyle \mathbb {R} ^{d}}$. Let ${\displaystyle \Psi }$ be a characteristic function associated to the unperturbed domain and ${\displaystyle \Psi _{\varepsilon }}$ be a characteristic function associated to the perforated domain ${\displaystyle \Omega _{\varepsilon }=\Omega \backslash {\overline {\omega _{\varepsilon }}}}$. A given shape functional ${\displaystyle \Phi (\Psi _{\varepsilon }({\tilde {x}}))}$ associated to the topologically perturbed domain, admits the following topological asymptotic expansion:

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