Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows
Numerical algorithms and simulations of boundary dynamic control for optimal mixing in unsteady Stokes flows
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DOI：
10.1016/j.cma.2023.116455
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发表时间：
2023-06
影响因子：
7.2
通讯作者：
Xiaoming Zheng;Weiwei Hu;Jiahong Wu
中科院分区：
文献类型：
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作者：
Xiaoming Zheng;Weiwei Hu;Jiahong Wu
本工作基于胡和吴在一系列工作中奠定的理论基础，开发了一种高效、准确的优化算法来研究非定常斯托克斯流边界控制驱动的最优混合问题。混合的标量纯粹是由流动平流输送的，控制是通过纳维滑移条件切向施加在域边界上的力。该控制设计在许多工业过程中具有潜在的应用，例如旋转壁驱动混合、声波微混合器和人工纤毛混合。由于混合问题的多尺度性质和优化要求，数值算法具有高复杂性、高精度要求和高计算费用。一个关键问题是成本函数的 Gâteaux 导数的计算。为此，建立了一种基于变分公式和有限差分的混合方法，具有高精度和高效率来处理各种类型的控制输入函数。我们尝试了各种优化算法，包括最速下降算法、共轭梯度法和两种线搜索选项（回溯和精确线搜索）。我们能够识别并实施最佳组合。数值模拟表明，当仅应用一种控制类型时，混合效果有限，但当使用更多样的控制类型和更多的时间分段时，混合效果可以得到增强。最佳混合中的混合范数呈指数衰减。这项工作的数值研究表明，单独的边界控制可能是不可压缩流混合的有效策略。
This work develops an efficient and accurate optimization algorithm to study the optimal mixing problem driven by boundary control of unsteady Stokes flows, based on the theoretical foundation laid by Hu and Wu in a series of work. The scalar being mixed is purely advected by the flow and the control is a force exerted tangentially on the domain boundary through the Navier slip conditions. The control design has potential applications in many industrial processes such as rotating wall driven mixing, mircomixers with acoustic waves, and artificial cilia mixing.The numerical algorithms have high complexity, high accuracy demand, and high computing expense, due to the multiscale nature of the mixing problem and the optimization requirements. A crucial problem is the computation of the Gâteaux derivative of the cost functional. To this end, a hybrid approach based on variational formula and finite difference is built with high accuracy and efficiency to treat various types of control input functions. We have experimented with various optimization algorithms including the steepest descent algorithm, the conjugate gradient method and two line search options (backtracking and exact line search). We are able to identify and implement the best combinations.The numerical simulations show that the mixing efficacy is limited when only one single type of control is applied, but can be enhanced when more diverse control types and more time segmentation are utilized. The mix-norm in the optimal mixings decays exponentially. The numerical study in this work demonstrates that boundary control alone could be an effective strategy for mixing in incompressible flows.