A Novel $p$-Harmonic Descent Approach Applied to Fluid Dynamic Shape Optimization

TL;DR

This paper presents a new shape optimization method for fluid dynamics using the $p$-Laplace operator to improve convergence and shape quality, especially for shapes with sharp features.

Contribution

The paper introduces a $p$-harmonic descent method for shape optimization in fluid dynamics, emphasizing the use of the $W^{1, ext{infinity}}$-topology for better results.

Findings

The $p$-harmonic approach outperforms classical methods in convergence.

Optimized shapes have improved sharp corner features.

Mesh quality remains high after large deformations.

Abstract

We introduce a novel method for the implementation of shape optimziation in fluid dynamics applications, where we propose to use the shape derivative to determine deformation fields with the help of the $p-$ Laplacian for $p > 2$. This approach is closely related to the computation of steepest descent directions of the shape functional in the $W^{1,\infty}-$ topology. Our approach is demonstrated for shape optimization related to drag-minimal free floating bodies. The method is validated against existing approaches with respect to convergence of the optimization algorithm, the obtained shape, and regarding the quality of the computational grid after large deformations. Our numerical results strongly indicate that shape optimization related to the $W^{1,\infty}$-topology -- though numerically more demanding -- seems to be superior over the classical approaches invoking Hilbert space methods, concerning the convergence, the obtained shapes and the mesh quality after large deformations, in particular when the optimal shape features sharp corners.