Open AccessA shape optimization problem constrained with the Stokes equations to address maximization of vortices
Author(s) -
John Sebastian H. Simon,
Hirofumi Notsu
Publication year - 2022
Publication title -
evolution equations and control theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.665
H-Index - 19
eISSN - 2163-2480
pISSN - 2163-2472
DOI - 10.3934/eect.2022003
Subject(s) - tikhonov regularization , mathematics , gradient descent , mathematical analysis , regularization (linguistics) , maximization , boundary (topology) , shape optimization , vorticity , stokes flow , vortex , finite element method , flow (mathematics) , physics , mathematical optimization , inverse problem , geometry , computer science , machine learning , artificial intelligence , artificial neural network , thermodynamics
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 \begin{document}$ L^2 $\end{document} -norm of the curl and the 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.