Fluid dynamic shape optimization using self-adapting nonlinear extension operators with multigrid preconditioners

TL;DR

This paper introduces a scalable shape optimization algorithm that combines multigrid PDE solvers with a self-adapting nonlinear extension operator, enabling efficient large-scale geometric modifications in fluid dynamics problems.

Contribution

The paper presents a novel self-adapting nonlinear extension operator within the method of mappings, improving shape optimization by identifying critical regions and adapting transformations accordingly.

Findings

Achieves weak scalability and grid-independent convergence.

Effectively identifies critical regions with geometric singularities.

Demonstrates improved drag minimization in Navier-Stokes flow.

Abstract

In this article we propose a scalable shape optimization algorithm which is tailored for large scale problems and geometries represented by hierarchically refined meshes. Weak scalability and grid independent convergence is achieved via a combination of multigrid schemes for the simulation of the PDEs and quasi Newton methods on the optimization side. For this purpose a self-adapting, nonlinear extension operator is proposed within the framework of the method of mappings. This operator is demonstrated to identify critical regions in the reference configuration where geometric singularities have to arise or vanish. Thereby the set of admissible transformations is adapted to the underlying shape optimization situation. The performance of the proposed method is demonstrated for the example of drag minimization of an obstacle within a stationary, incompressible Navier-Stokes flow.