Optimal Control of Variational Inequalities of the Second Kind with Application to Yield Stress Fluids

第二类变分不等式的优化控制及其在屈服应力流体中的应用

• 批准号: 314144749
• 负责人: Professor Dr. Christian Meyer
• 金额: --
• 依托单位: Lehrstuhl I: Analysis
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2016
• 资助国家: 德国
• 起止时间: 2015-12-31 至 2019-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/314144749?language=en
• 关键词: Optimal; Control; Variational; Inequalities; Second

The project is concerned with the optimal control of variational inequalities of the second kind involving a non-differentiable norm of first derivatives as characteristic nonsmooth feature. Such kind of variational inequalities arise in the modeling of yield stress fluids as for instance Bingham fluids. Within this project we focus on Bingham fluids in their stationary form. The simulation of such nonsmooth fluid flows is frequently based on regularization techniques and the same holds for their optimization. In order to avoid an associated regularization error, the aim of our project is to optimally control yield stress fluid flows with minimal, respectively, without any regularization.The project addresses theoretical and numerical aspects. First we investigate the limited differentiability properties of the solution operator associated with the variational inequality. Based on these findings, necessary and sufficient optimality conditions are derived in the analytical part of the project. Concerning the numerical component of the project, the differentiability results are used to design a mesh-independent trust-region algorithm in function space for the solution of the optimal control problem. This algorithm will be realized with the in-house flow simulation package FEATFLOW in order to solve application-driven benchmark problems.

该项目涉及第二类变分不等式的最优控制，涉及一阶导数的不可微范数作为特征非光滑特征。这种变分不等式出现在屈服应力流体（例如宾汉流体）的建模中。在这个项目中，我们重点研究静止形式的宾汉流体。这种非光滑流体流动的模拟通常基于正则化技术，并且同样适用于它们的优化。为了避免相关的正则化误差，我们项目的目标是在不进行任何正则化的情况下分别以最小的速度最佳地控制屈服应力流体流动。该项目解决了理论和数值方面的问题。首先，我们研究与变分不等式相关的解算子的有限可微性质。基于这些发现，在项目的分析部分导出了必要和充分的最优性条件。关于该项目的数值部分，可微分结果用于设计函数空间中与网格无关的信赖域算法，以求解最优控制问题。该算法将通过内部流量仿真包 FEATFLOW 来实现，以解决应用程序驱动的基准问题。