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.
翻译:我们研究一个优化问题,目的是确定一个被斯托克斯方程式控制的流体所淹没的障碍的形状;所述流动发生在一个频道中,该通道促使将类似于Poisuille的输入功能强加在一端,而另一端则是无边界条件。园林和流体分层分层测量法以L$2美元为核心,解决园艺最大化问题。与此同时,为确保存在一个最佳形状,即以周边目标为形式的Tikhonov正规化,并施加体积限制。由于能够确立一种最佳形状的存在,第一个必要顺序是利用所谓的重新安排方法。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来提出数字实例。关于上述梯度下降法,我们用两种方法来解决体积限制:一种是利用扩大的拉格朗江法;另一种是利用一种无差异的变形场。