Vorticity Maximization of a Linear Fluid Flow via Volume Constrained and Perimeter Regularized Shape Optimization

by   John Sebastian H. Simon, et al.

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 det-grad measure of the fluid. Meanwhile, to ensure the existence of an optimal shape, a Tikhonov regularization in the form of a perimeter objective, and a volume constraint is imposed. 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.



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