Simulation and Control of a Nonsmooth Cahn-Hilliard Navier-Stokes System with Variable Fluid Densities

在这个项目中，我们研究了由非光滑能量的扩散界面模型控制的变密度多相流的最优和闭环控制问题。由此产生的问题属于函数空间中具有平衡约束的数学规划领域。研究了非光滑Cahn-Hilliard - Navier-Stokes系统的模型降阶技术及其在多相流控制中的应用。我们打算(a)研究耦合Cahn-Hilliard Navier-Stokes系统解算子的微分稳定性性质，并导出所谓的最优控制问题的强平稳条件；(b)为简化控制问题的数值解开发、分析和实施无捆绑隐式规划技术；(c)发展和分析完全综合的自适应数值方法，以控制流体密度可变的多相流问题，保证对界面上的局部过程进行局部精确的解决；(d)推导基于双加权残差的自适应网格细化误差估计，用于非光滑Cahn-Hilliard Navier-Stokes系统的有效离散化；(e)开发、实施和分析多相流的降阶模型，这些模型由具有非光滑能量的扩散界面模型控制，具有可变密度。(f)开发、实施和分析由具有非光滑能量的变密度扩散界面模型控制的多相流的模型预测反馈控制策略。

In this project we study optimal and closed-loop control problems for variable density multiphase flows governed by diffuse interface models with nonsmooth energies. The resulting problems fall into the realm of mathematical programs with equilibrium constraints (MPECs) in function space. Moreover, model order reduction (MOR) techniques for nonsmooth Cahn-Hilliard Navier-Stokes systems and their application in control of multiphase flows are considered. We intend to (a) study differential stability properties of the solution operator of the coupled Cahn-Hilliard Navier-Stokes system and derive so-called strong stationarity conditions for the optimal control problem;(b) develop, analyse and implement bundle-free implicit programming techniques for the numerical solution of the reduced control problem;(c) develop and analyse fully integrated adaptive numerical methods for the control of multiphase flow problems with variable fluid densities which guarantee a locally refined resolution of the local processes at the interface;(d) derive dual-weighted residual based error estimates for adaptive mesh refinement for the efficient discretization of the non-smooth Cahn-Hilliard Navier-Stokes system;(e) develop, implement and analyse reduced order models for multiphase flows with variable densities governed by diffuse interface models with nonsmooth energies.(f) develop, implement and analyze model-predictive feedback control strategies for multiphase flows governed by variable density diffuse interface models with nonsmooth energies.