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.
翻译:我们研究一个优化问题,目的是确定在斯托克斯方程式所调节的流体中淹没的障碍的形状。 所提到的流动发生在一个频道中,它促使将类似于Poisuille的输入功能强加在一端,而另一端则是无边界条件。 园林和流体的渐渐脱位测量以L$2$-norm 和 $it det-grad) 解决园艺的最大性能问题。 我们以周边功能和体积限制的形式对Tikhonov进行正规化,以解决地形变化的可能性。 在建立了最佳的形状之后,第一个必要状态是通过使用所谓的重新布局方法。 最后,通过在治理状态上使用一个限定元素法和一种梯度下降法对域进行变形,提出了数字实例。 关于上述梯度下降法,我们用两种方法来解决体积限制: 一种是使用扩大的拉格方法;另一种是利用一个不裂变的场。