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

Description

In this project we study optimal and closed-loop control problems for variable density multiphase flows governed by diffuse interface models with non-smooth 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 non-smooth 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 analyze reduced order models for multiphase flows with variable densities governed by diffuse interface models with non-smooth energies. (f) develop, implement and analyze model-predictive feedback control strategies for multiphase flows governed by variable density diffuse interface models with non-smooth energies.

Publications

Carmen Gräßle, Michael Hinze: POD Reduced Order Modeling for Evolution Equations Utilizing Arbitrary Finite Element Discretizations, Advances in Computational Mathematics, 2018 (SPP1962-010).

Michael Hintermüller, Michael Hinze, Christian Kahle, Tobias Keil: A Goal-Oriented Dual-Weighted Adaptive Finite Element Approach for the Optimal Control of a Nonsmooth Cahn-Hilliard-Navier-Stokes System , Optimization and Engineering, 2018 (SPP1962-011).

Michael Hintermüller, Tobias Keil: Some Recent Developments in Optimal Control of Multiphase Flows, Shape Optimization, Homogenization and Optimal Control, 2018.

Harald Garcke, Michael Hinze, Christian Kahle: Optimal Control of Time-Discrete Two-Phase Flow Driven by a Diffuse-Interface Model, ESAIM: Control, Optimisation and Calculus of Variations , 2018 (SPP1962-058).

Harald Garcke, Michael Hinze, Christian Kahle, Kei Fong Lam: A Phase Field Approach to Shape Optimization in Navier--Stokes Flow with Integral State Constraints , Adv. Comput. Math., 2018 (SPP1962-059).

Carmen Gräßle, Michael Hinze: The Combination of POD Model Reduction with Adaptive Finite Element Methods in the Context of Phase Field Models, PAMM. Volume 17(1): 47-50 , 2018 (SPP1962-061).

Michael Hintermüller, Carlos N. Rautenberg, Simon Rösel: Density of Convex Intersections and Applications, Proc. R. Soc. A 2017 473 20160919, 2017 (SPP1962-005).

Harald Garcke, Claudia Hecht, Michael Hinze, Christian Kahle, Kei Fong Lam: Shape Optimization for Surface Functionals in Navier--Stokes Flow using a Phase Field Approach, Interfaces and Free Boundaries 18(2):219-261 , 2016 (SPP1962-060).

Preprints

Ahmad Ali, Klaus Deckelnick, Michael Hinze: Global Minima for Optimal Control of the Obstacle Problem (SPP1962-095, 10/2018, [bib])

Carmen Gräßle, Michael Hinze, Jens Lang, Sebastian Ullmann: POD Model Order Reduction with Space-adapted Snapshots for Incompressible Flows (SPP1962-096, 10/2018, [bib])

Carmen Gräßle, Michael Hinze, Nicolas Scharmacher: POD for Optimal Control of the Cahn-Hilliard System using Spatially Adapted Snapshots (SPP1962-092, 10/2018, [bib])

Harald Garcke, Michael Hinze, Christian Kahle: Optimal Control of Time-Discrete Two-Phase Flow Driven by a Diffuse-Interface Model (SPP1962-058, 06/2018, [bib])

Harald Garcke, Michael Hinze, Christian Kahle, Kei Fong Lam: A Phase Field Approach to Shape Optimization in Navier--Stokes Flow with Integral State Constraints (SPP1962-059, 06/2018, [bib])

Harald Garcke, Claudia Hecht, Michael Hinze, Christian Kahle, Kei Fong Lam: Shape Optimization for Surface Functionals in Navier--Stokes Flow using a Phase Field Approach (SPP1962-060, 06/2018, [bib])

Carmen Gräßle, Michael Hinze: The Combination of POD Model Reduction with Adaptive Finite Element Methods in the Context of Phase Field Models (SPP1962-061, 06/2018, [bib])

Carmen Gräßle, Michael Hinze: POD Reduced Order Modeling for Evolution Equations Utilizing Arbitrary Finite Element Discretizations (SPP1962-010, 03/2017, [bib])

Michael Hintermüller, Michael Hinze, Christian Kahle, Tobias Keil: A Goal-Oriented Dual-Weighted Adaptive Finite Element Approach for the Optimal Control of a Nonsmooth Cahn-Hilliard-Navier-Stokes System (SPP1962-011, 03/2017, [bib])

Michael Hintermüller, Carlos N. Rautenberg, Simon Rösel: Density of Convex Intersections and Applications (SPP1962-005, 01/2017, [bib])

Research Area

Modeling, problem analysis, algorithm design and convergence analysis

The focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.

Realization of algorithms, adaptive discretization and model reduction

As the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.

Members

  • member's portrait

    Prof. Michael Hintermüller

    Weierstraß-Institut
    Principal Investigator
  • member's portrait

    Prof. Michael Hinze

    Universität Hamburg
    Principal Investigator
  • member's portrait

    Carmen Gräßle

    Universität Hamburg
    Research Assistant
  • member's portrait

    Tobias Keil

    Humboldt-Universität zu Berlin
    Research Assistant

Project Related News

  • Oct 29, 2018 : New preprint submitted

    Michael Hinze submitted the preprint SPP1962-095, Global Minima for Optimal Control of the Obstacle Problem

  • Oct 29, 2018 : New preprint submitted

    Carmen Gräßle submitted the preprint SPP1962-096, POD Model Order Reduction with Space-adapted Snapshots for Incompressible Flows

  • Oct 26, 2018 : New preprint submitted

    Carmen Gräßle submitted the preprint SPP1962-092, POD for Optimal Control of the Cahn-Hilliard System using Spatially Adapted Snapshots

  • Oct 10, 2016 : Welcome to our new project member

    Carmen Gräßle joins project 13.