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

在这个项目中，我们研究了由具有非光滑能量的扩散界面模型控制的变密度多相流的最优控制问题和闭环系统控制问题。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(MPEC)的领域。此外，还研究了非光滑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，得到最优控制问题的所谓强平稳性条件；(B)发展、分析和实现约化控制问题数值解的无束隐式编程技术；(C)发展和分析控制具有变流体密度的多相流问题的完全集成的自适应数值方法，保证界面上局部过程的局部精细分辨率；(D)推导基于双重加权残差的误差估计，用于对非光滑Cahn-Hilliard Navier-Stokes系统的有效离散化进行自适应网格加密；(E)开发、实施和分析由非光滑能量扩散界面模型控制的变密度多相流的降阶模型。(F)开发、实施和分析由非光滑能量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.