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 Lagrangian multiplayer 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 a 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, which allows us to use the adaptive mesh strategy. 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.
翻译:在此工作中, 我们研究 Stokes 流中的优化问题 。 通过阶段化方法, 结果的总目标功能包括流体和Ginzburg- Landau 能源功能的耗损能量和Ginzburg- Landau 能源功能, 作为生成的 diffusive 界面的常规术语, 以及 Lagrangian 多玩家 的体积限制 。 提议了一个高效的脱钩方案, 通过梯度流方法实施一个高效的脱钩方案, 以降低目标函数 。 在每一个循环中, 我们首先通过解决 Stokes 方程式的方程式和先前迭代中给出的阶段字段变量来更新速度字段。 之后, 通过使用一个稳定的方案来更新阶段域变量变量, 解决Allen- Cahn 型方程式的耗能。 然后我们用一个固定的公式进行截断断后处理, 以限制其值 $[ 10, 1, $ 。 在每一个循环的最后一步中, Lagrang 参数会用一个适当的人工时间步骤更新。 我们严格证明, 拟议的方案允许无条件的单调的单体- decreasreasreasasasasion 属性, 属性,, 允许我们使用调适合的Mine 战略。