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Version: publishedVersion × Subject: Shape optimization ×

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    Domain expression of the shape derivative and application to electrical impedance tomography
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2013) Laurain, Antoine; Sturm, Kevin
    The well-known structure theorem of Hadamard-Zolésio states that the derivative of a shape functional is a distribution on the boundary of the domain depending only on the normal perturbations of a smooth enough boundary. However a volume representation (distributed shape derivative) is more general than the boundary form and allows to work with shapes having a lower regularity. It is customary in the shape optimization literature to assume regularity of the domains and use the boundary expression of the shape derivative for numerical algorithm. In this paper we describe the numerous advantages of the distributed shape derivative in terms of generality, easiness of computation and numerical implementation. We give several examples of numerical applications such as the inverse conductivity problem and the level set method.
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    Reproducing kernel Hilbert spaces and variable metric algorithms in PDE constrained shape optimisation
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Eigel, Martin; Sturm, Kevin
    In this paper we investigate and compare different gradient algorithms designed for the domain expression of the shape derivative. Our main focus is to examine the usefulness of kernel reproducing Hilbert spaces for PDE constrained shape optimisation problems. We show that radial kernels provide convenient formulas for the shape gradient that can be efficiently used in numerical simulations. The shape gradients associated with radial kernels depend on a so called smoothing parameter that allows a smoothness adjustment of the shape during the optimisation process. Besides, this smoothing parameter can be used to modify the movement of the shape. The theoretical findings are verified in a number of numerical experiments.