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High Quality Content by WIKIPEDIA articles! 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 to minimize some scalar-valued objective function, J( ). Neglecting changes in topology, an initial guess can be improved by perturbing the shape of by methods of calculus of variations and functional analysis. However, this does not permit a discrete operation such as making a hole in .