2023/10/23 by Klaus Deckelnick, Philip J. Herbert, Deckelnick, Klaus +3
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Analysis Techniques #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Topology Optimization in Engineering
paper · pdf · doi:10.48550/arxiv.2310.15078
openalex publication_date 2023/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 W1,∞ 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.