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

该项目关注的是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的建模中，例如宾汉流体。在这个项目中，我们专注于宾汉流体在其固定的形式。这种非光滑流体流动的模拟通常基于正则化技术，并且同样适用于其优化。为了避免相关的正则化错误，我们的项目的目的是最佳地控制屈服应力流体流动，分别最小，没有任何regularization.The项目地址的理论和数值方面。首先，我们研究了与变分不等式相关的解算子的有限可微性。基于这些发现，必要和充分的最优性条件推导出的项目的分析部分。关于该项目的数值部分，可微性的结果被用来设计一个网格独立的信赖域算法在函数空间的最优控制问题的解决方案。该算法将实现与内部流模拟包FEATFLOW，以解决应用程序驱动的基准问题。

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.