Convergence of a steepest descent algorithm in shape optimisation using $W^{1,\infty}$ functions
Year: 2023
Type: preprint
Abstract: Built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), we present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty}$ topology together with a suitable finite element discretisation. For the numerical solution of the respective discrete shape optimisation problems we propose a steepest descent minimisation algorithm with Armijo-Goldstein stepsize rule. We show that the sequence generated by this descent method globally converges, and under mild assumptions also, that every accumulation point of this sequence is a stationary point of the shape functional. Moreover, for the mesh discretisation parameter tending to zero we under mild assumptions prove convergence of the discrete stationary shapes in the Hausdorff complementary metric. To illustrate our approach we present a selection of numerical examples for PDE constrained shape optimisation problems, where we include numerical convergence studies which support our analytical findings.
Source: arXiv (Cornell University)
Language: English
Subfield: Computational Mechanics
Field: Engineering
Domain: Physical Sciences
Open Access status: green
Funder: Deutsche Forschungsgemeinschaft
Award: G816307624
OpenAlex ID: W4387929755