Stokes流中形状optimization的单调递减相场算法
A provably efficient monotonic-decreasing algorithm for shape optimization in Stokes flows by phase-field approaches
摘要：
本文研究了Stokes流中的形状最优化问题。通过相场方法，得到的总目标函数由流体的耗散能和Ginzburg-Landau能量泛函组成，作为生成扩散界面的正则化项，以及体积约束的Lagrangian多层。提出了一种有效的解耦方案，通过梯度流方法来降低目标函数。在每个循环中，我们首先更新的速度场求解斯托克斯方程与相场变量在前一次迭代，其次是更新的相场变量求解Allen-卡恩型方程使用稳定的计划。然后对相位场变量进行截断后处理，以约束其在$[0,1]$中的值。在每个循环的最后一步中，使用适当的人工时间步长更新拉格朗日参数。我们严格证明，该计划允许无条件单调递减的属性，这使我们能够使用自适应网格策略。为了提高算法的整体效率，在每个循环中，我们更新相场变量和拉格朗日参数几个时间步长，但只更新一次速度场。各种形状优化的数值结果验证了我们的数值方法的有效性。
英文摘要：
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.
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