Topological derivative

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In the field of shape optimization, a topological derivative is, conceptually, a derivative of a function of a region with respect to small changes in its topology, such as adding a small hole.

Shape optimization concerns itself with finding an optimal shape. That is, find $Ω$ to minimize some scalar-valued objective function, $J (Ω)$ . Neglecting changes in topology, an initial guess can be improved by perturbing the shape of $Ω$ by methods of calculus of variations and functional analysis. However, this does not permit a discrete operation such as making a hole in $Ω$ .