A Shape Optimization Problem Constrained with the Stokes Equations to Address Maximization of Vortices

TL;DR

This paper formulates and solves a shape optimization problem for fluid flow governed by Stokes equations, aiming to maximize vorticity through regularized shape design, with theoretical and numerical analysis including gradient-based methods.

Contribution

It introduces a novel shape optimization framework for Stokes flow that incorporates vorticity maximization, regularization, and topological flexibility, with proven existence and practical algorithms.

Findings

Existence of an optimal obstacle shape was established.

Numerical methods successfully optimized shapes to maximize vorticity.

Two volume constraint handling approaches were demonstrated in simulations.

Abstract

We study an optimization problem that aims to determine the shape of an obstacle that is submerged in a fluid governed by the Stokes equations. The mentioned flow takes place in a channel, which motivated the imposition of a Poiseuille-like input function on one end and a do-nothing boundary condition on the other. The maximization of the vorticity is addressed by the $L^2$-norm of the curl and the {\it det-grad} measure of the fluid. We impose a Tikhonov regularization in the form of a perimeter functional and a volume constraint to address the possibility of topological change. Having been able to establish the existence of an optimal shape, the first order necessary condition was formulated by utilizing the so-called rearrangement method. Finally, numerical examples are presented by utilizing a finite element method on the governing states, and a gradient descent method for the deformation of the domain. On the said gradient descent method, we use two approaches to address the volume constraint: one is by utilizing the augmented Lagrangian method; and the other one is by utilizing a class of divergence-free deformation fields.