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Lothar Collatz Center for Computing in Science
Lothar Collatz Center
for Computing in Science

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Project C6 (finished)

Project C6 - Simulation and control of the Cahn-Hilliard Navier-Stokes System

Principal Investigator: Michael Hinze

Scientific Background and Motivation

Hydrodynamics of two-phase flows play an important role in many applications, such as the mixture of e.g. polymers. The mathematical modeling is based on the Cahn-Hilliard Navier-Stokes system. In this project we develop a reliable and efficient adaptive finite element scheme for solving the coupled Cahn-Hilliard/Navier-Stokes system with a double-obstacle homogeneous free energy density. Its significance is reflected in the development of a robust, highly accurate, fully automatic scheme for the numerical treatment of the hydrodynamics of two-phase flows modeled with the diffuse interface method, which then forms the basis for optimization of multi-phase flows.

Aims and Objectives

Our principal aim consists in developing a robust, reliable and fully adaptive solver for simulation and control of mutliphase flows. In our approach we develop an adaptive finite element a posteriori error estimator for the numerical solution of a coupled Cahn-Hilliard/Navier-Stokes system with a double-obstacle homogenous free (interfacial) energy density. We use the semi-implicit Euler scheme for the time-integrationwhich results in a system coupling a quasi-Stokes or Oseen-type problem for the fluid flow to a variational inequality for the concentration and the chemical potential according to the Cahn-Hilliard model. Furthermore, a Moreau-Yosida regularization is employed which relaxes the constraints contained in the variational inequality. For the finite dimensional approximation of the concentration and the chemical potential we employ piecewise linear and globally continuous finite elements, and for the numerical approximation of the fluid velocity Taylor-Hood finite elements are used. For the optimization of multiphase flows we intend to apply model predictive control strategies.

PhD Student: Christian Kahle

Christian Kahle successfully defended his PhD on 29 October 2014.

Publications

1. M. Hinze and C. Kahle: A Nonlinear Model Predictive Concept for Control of Two-Phase Flows Governed by the Cahn-Hilliard Navier-Stokes System. System Modeling and Optimization, 25th IFIP TC 7 Conference 2011, IFIP AICT 391: 348-357 (2012).

2. M. Hinze and C. Kahle: Model predictive control of variable density multiphase flows governed by diffuse interface models. Hamburger Beiträge zur Angewandten Mathematik 2013-01 (2013), accepted for publication with first IFAC Workshop on Control of Systems Modeled by Partial Differential Equations, Paris 2013.

3. M. Hintermüller, M. Hinze and C. Kahle: An adaptive finite element Moreau-Yosida-based solver for a coupled Cahn-Hilliard/Navier-Stokes system. J. Comput. Phys. 235:810-827 (2013).

4. H. Garcke, C. Hecht, M. Hinze, C. Kahle: Numerical approximation of phase field based shape and topology optimization for fluids. SIAM J. Sci. Comp. 37:1848-1871 (2015)

5. H. Garcke, M. Hinze, C. Kahle: A stable and linear time discretization for a thermodynamically consistent model for two-phase incompressible flow. Applied Numerical Mathematics 99:151-171 (2016).

6. H. Garcke, C. Hecht, M. Hinze, C. Kahle, K.F. Lam: Shape optimization for surface functionals in Navier-Stokes flow using a phase field approach. arXiv:1504.06402. Accepted for publication in Interfaces and Free Boundaries (2016).

7. Christian Kahle: A $L^\infty$ bound for the Cahn--Hilliard equation with relaxed non-smooth free energy. arXiv:1511.02618 and Hamburger Beiträge zur angewandten Mathematik 2015-40 (2015),

Further details can be found here.