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

该项目涉及第二类变分不等式的最优控制问题，涉及一阶导数的不可微范数作为特征的非光滑特征。在对屈服应力流体(例如宾汉流体)进行建模时，会出现这样的变分不等。在这个项目中，我们将重点放在固定形式的宾汉姆流体上。对这种非光滑流体流动的模拟通常基于正则化技术，对于它们的优化也是如此。为了避免相关的正则化误差，我们的项目的目标是在没有任何正则化的情况下，分别以最小的屈服应力流体流动来优化控制。该项目涉及理论和数值方面。首先，我们研究了与变分不等式有关的解算子的有限可微性。基于这些结果，在项目的分析部分得到了最优性的充要条件。对于工程的数值部分，利用可微性结果在函数空间设计了一个求解最优控制问题的网格无关信赖域算法。该算法将用内部流程模拟包FEATFLOW实现，以解决应用驱动的基准问题。

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.