Mesh quality preserving shape optimization using nonlinear extension operators

TL;DR

This paper introduces a nonlinear extension operator for shape optimization that preserves mesh quality during large deformations, improving shape admissibility and computational efficiency in fluid flow simulations.

Contribution

The paper presents a novel nonlinear extension operator based on the method of mappings, enhancing shape optimization by maintaining mesh quality and reducing computational complexity.

Findings

Nonlinear extension operators improve mesh quality during large deformations.

The proposed method effectively handles 2D and 3D Navier-Stokes flow shape optimization.

Numerical tests demonstrate the benefits of nonlinearity in extension operators.

Abstract

In this article, we propose a shape optimization algorithm which is able to handle large deformations while maintaining a high level of mesh quality. Based on the method of mappings we introduce a nonlinear extension operator, which links a boundary control to domain deformations, ensuring admissibility of resulting shapes. The major focus is on comparisons between well-established approaches involving linear-elliptic operators for the extension and the effect of additional nonlinear advection on the set of reachable shapes. It is moreover discussed how the computational complexity of the proposed algorithm can be reduced. The benefit of the nonlinearity in the extension operator is substantiated by several numerical test cases of stationary, incompressible Navier-Stokes flows in 2d and 3d.