In this work, we study shape optimization problems in the Stokes flows. By phase-field approaches, the resulted total objective function consists of the dissipation energy of the fluids and the Ginzburg–Landau energy functional as a regularizing term for the generated diffusive interface, together with a Lagrangian multiplier for volume constraint. An efficient decoupled scheme is proposed to implement by the gradient flow approach to decrease the objective function. In each loop, we first update the velocity field by solving the Stokes equation with the phase field variable given in the previous iteration, which is followed by updating the phase field variable by solving an Allen–Cahn-type equation using a stabilized scheme. We then take the cut-off post-processing for the phase-field variable to constrain its value in $[0,1]$. In the last step of each loop, the Lagrangian parameter is updated with an appropriate artificial time step. We rigorously prove that the proposed scheme permits an unconditionally monotonic-decreasing property. To enhance the overall efficiency of the algorithm, in each loop we update the phase field variable and Lagrangian parameter several time steps but update the velocity field only one time. Numerical results for various shape optimizations are presented to validate the effectiveness of our numerical scheme.

在这项工作中，我们研究了斯托克斯流中的形状优化问题。通过相场方法，得到的总目标函数包括流体的耗散能和作为生成扩散界面的正则化项的 Ginzburg-Landau 能量泛函，以及用于体积约束的拉格朗日乘数。提出了一种有效的解耦方案，通过梯度流方法来实现，以降低目标函数。在每个循环中，我们首先更新速度场通过使用先前迭代中给出的相场变量求解斯托克斯方程，然后通过使用稳定方案求解 Allen-Cahn 型方程来更新相场变量。然后，我们对相场变量进行截止后处理，以将其值限制在$[0,1]$. 在每个循环的最后一步，使用适当的人工时间步更新拉格朗日参数。我们严格证明所提出的方案允许无条件单调递减性质。为了提高算法的整体效率，在每个循环中我们更新相位场变量和拉格朗日参数几个时间步，但只更新一次速度场。给出了各种形状优化的数值结果，以验证我们数值方案的有效性。