A Scalable Algorithm for Shape Optimization with Geometric Constraints in Banach Spaces

TL;DR

This paper introduces a scalable PDE-constrained shape optimization algorithm using Lipschitz transformations and the p-Laplace operator, emphasizing geometric constraints, mesh quality, and parallel computation for large-scale problems.

Contribution

It presents a novel scalable algorithm for shape optimization in Banach spaces that incorporates geometric constraints without penalty terms and leverages multigrid solvers for parallel efficiency.

Findings

Effective handling of geometric constraints without penalty terms.

Demonstrated scalability on large parallel computers.

Successful application to fluid dynamics energy minimization.

Abstract

This work develops an algorithm for PDE-constrained shape optimization based on Lipschitz transformations. Building on previous work in this field, the $p$-Laplace operator is utilized to approximate a descent method for Lipschitz shapes. In particular, it is shown how geometric constraints are algorithmically incorporated avoiding penalty terms by assigning them to the subproblem of finding a suitable descent direction. A special focus is placed on the scalability of the proposed methods for large scale parallel computers via the application of multigrid solvers. The preservation of mesh quality under large deformations, where shape singularities have to be smoothed or generated within the optimization process, is also discussed. It is shown that the interaction of hierarchically refined grids and shape optimization can be realized by the choice of appropriate descent directions. The performance of the proposed methods is demonstrated for energy dissipation minimization in fluid dynamics applications.