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 W$$-topology. The idea of our approach is demonstrated for shape optimisation of n-dimensional star-shaped domains, which we represent as functions defined on the unit (n − 1)-sphere. In this setting we provide the specific form of the shape derivative and prove the existence of solutions to the underlying shape optimisation problem. Moreover, we show the existence of a direction of steepest descent in the W$$− 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.
Keywords: PDE constrained shape optimization, star-shaped domain, $$1,∞ descent, optimal transport, $$-Laplacian shape derivative
@article{COCV_2022__28_1_A2_0,
author = {Deckelnick, Klaus and Herbert, Philip J. and Hinze, Michael},
title = {A novel $W^{1 , \infty}$ approach to shape optimisation with {Lipschitz} domains},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2021108},
mrnumber = {4362195},
zbl = {1483.35299},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021108/}
}
TY - JOUR
AU - Deckelnick, Klaus
AU - Herbert, Philip J.
AU - Hinze, Michael
TI - A novel $W^{1 , \infty}$ approach to shape optimisation with Lipschitz domains
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2022
VL - 28
PB - EDP-Sciences
UR - https://www.numdam.org/articles/10.1051/cocv/2021108/
DO - 10.1051/cocv/2021108
LA - en
ID - COCV_2022__28_1_A2_0
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%A Herbert, Philip J.
%A Hinze, Michael
%T A novel $W^{1 , \infty}$ approach to shape optimisation with Lipschitz domains
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2022
%V 28
%I EDP-Sciences
%U https://www.numdam.org/articles/10.1051/cocv/2021108/
%R 10.1051/cocv/2021108
%G en
%F COCV_2022__28_1_A2_0
Deckelnick, Klaus; Herbert, Philip J.; Hinze, Michael. A novel $W^{1 , \infty}$ approach to shape optimisation with Lipschitz domains. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 2. doi: 10.1051/cocv/2021108
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