In this study, a shape optimization problem for the two-dimensional stationary Navier--Stokes equations with an artificial boundary condition is considered. The fluid is assumed to be flowing through a rectangular channel, and the artificial boundary condition is formulated so as to take into account the possibility of ill-posedness caused by the usual do-nothing boundary condition. The goal of the optimization problem is to maximize the vorticity of the said fluid by determining the shape of an obstacle inside the channel. Meanwhile, the shape variation is limited by a perimeter functional and a volume constraint. The perimeter functional was considered to act as a Tikhonov regularizer and the volume constraint is added to exempt us from topological changes in the domain. The shape derivative of the objective functional was formulated using the rearrangement method, and this derivative was later on used for gradient descent methods. Additionally, an augmented Lagrangian method and a class of solenoidal deformation fields were considered to take into account the goal of volume preservation. Lastly, numerical examples based on the gradient descent and the volume preservation methods are presented.

翻译：在本研究中,考虑了具有人为边界条件的两维定点纳维耶-斯托克方程式的形状优化问题,假定液体通过长方形通道流出,而人为边界条件的设定是为了考虑到通常的无边界条件可能造成的不正确状态。优化问题的目的是通过确定通道内障碍的形状形状形状的形状形状变异,同时,形状变异受周边功能和体积限制的限制。周边功能被视为一种Tikhonov定律器,而体积限制则被添加来免除我们在该域的地形变化。目标功能的形状衍生物是使用重新排列方法拟订的,这种衍生物后来用于梯度下移方法。此外,还考虑了增强拉格朗加法和一种单氮变形场,以考虑到体积保存的目标。最后,还介绍了基于梯度位位和体积保护方法的数字示例。