Volume 28, 2022
|Number of page(s)||29|
|Published online||11 January 2022|
A novel W1,∞ approach to shape optimisation with Lipschitz domains
Institut für Analysis und Numerik, Otto-von-Guericke-Universität Magdeburg,
2 Mathematisches Institut, Campus Koblenz, Universität Koblenz-Landau, Universitätsstr. 1, 56070 Koblenz, Germany.
* Corresponding author: Michael Hinze firstname.lastname@example.org
Accepted: 10 December 2021
This article introduces a novel method for the implementation of shape optimisation with Lipschitz domains. We propose to use the shape derivative to determine deformation fields which represent steepest descent directions of the shape functional in the W1,∞-topology. The idea of our approach is demonstrated for shape optimisation of n-dimensional star-shaped domains, which we represent as functions defined on the unit (n − 1)-sphere. In this setting we provide the specific form of the shape derivative and prove the existence of solutions to the underlying shape optimisation problem. Moreover, we show the existence of a direction of steepest descent in the W1,∞− topology. We also note that shape optimisation in this context is closely related to the ∞−Laplacian, and to optimal transport, where we highlight the latter in the numerics section. We present several numerical experiments in two dimensions illustrating that our approach seems to be superior over a widely used Hilbert space method in the considered examples, in particular in developing optimised shapes with corners.
Mathematics Subject Classification: 35Q93 / 49Q10 / 35R30 / 49K20 / 49J20
Key words: PDE constrained shape optimization / star-shaped domain / W1,∞ descent / optimal transport / ∞-Laplacian shape derivative
© The authors. Published by EDP Sciences, SMAI 2022
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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