PDE constrained shape optimisation with first-order and Newton-type methods in the $W^{1,\infty }$ topology

Abstract

We present a general shape optimisation framework based on the method of mappings in the $W^{1,\infty }$ topology. We propose steepest descent and Newton-like minimisation algorithms for the numerical solution of the respective shape optimisation problems. Our work is built upon previous work of the authors in (Deckelnick, Herbert, and Hinze, ESAIM: COCV 28 (2022)), where a $W^{1,\infty }$ framework for star-shaped domains is proposed. To illustrate our approach we present a selection of PDE constrained shape optimisation problems and compare our findings to results from so far classical Hilbert space methods and recent $p$-approximations.

References in corpus (11)

• SciPy 1.0--Fundamental Algorithms for Scientific Computing in Python
• Efficient PDE constrained shape optimization based on Steklov-Poincaré type metrics
• Shape optimisation with nearly conformal transformations
• Algorithmic aspects of multigrid methods for optimization in shape spaces
• A phase field approach to shape optimization in Navier--Stokes flow with integral state constraint
• A novel $W^{1,\infty }$ approach to shape optimisation with Lipschitz domains
• Mesh quality preserving shape optimization using nonlinear extension operators
• A Scalable Algorithm for Shape Optimization with Geometric Constraints in Banach Spaces
• A continuous perspective on modeling of shape optimal design problems
• A formula for membrane mediated point particle interactions on near spherical biomembranes
• Higher-order moving mesh methods for PDE-constrained shape optimization

Cited by in corpus (1)

• Shape optimization in $W^{1,\infty }$ with geometric constraints: a study in distributed-memory systems