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Fully Adaptive and Integrated Numerical Methods for the Simulation and Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models

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Transport Processes at Fluidic Interfaces

Part of the book series: Advances in Mathematical Fluid Mechanics ((AMFM))

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Abstract

The present work is concerned with the simulation and optimal control of two-phase flows. We provide stable time discretization schemes for the simulation based on both, smooth and non-smooth free energy densities, which we combine with a practical, reliable and efficient adaptive mesh refinement concept for the spatial variables. Furthermore, we consider optimal control problems for two-phase flows and, among other things, derive first order optimality conditions. In the presence of smooth free energies we encounter classical Karush-Kuhn-Tucker (KKT) conditions, while in the case of non-smooth free energies we can prove C(larke)-stationarity. Moreover, we propose a dual weighted residual concept for spatial mesh adaptivity which is based on the newly derived stationarity conditions. We also address future research directions, including closed-loop control concepts and model order reduction techniques for simulation and control of variable density multiphase flows.

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Acknowledgements

The authors gratefully acknowledge the support of the DFG through the priority programm 1506 “Transport processes at fluidic interfaces” under the grants HI 689_7-1 and HI 1466/2-1. This research was further supported by the Research Center MATHEON through project C-SE5 and D-OT1 funded by the Einstein Center for Mathematics Berlin. In addition, this research was partly supported by the Berlin Mathematical School.

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Authors and Affiliations

  1. Weierstraß-Institut, Mohrenstrasse 39, 10117, Berlin, Germany

    Michael Hintermüller

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Hintermüller

  3. Fachbereich Mathematik, Universität Hamburg, Bundesstrasse 55, 20146, Hamburg, Germany

    Michael Hinze

  4. Center for Mathematical Sciences, Technical University of Munich, Chair of Optimal Control, M17, Boltzmannstraße 3, 85748, Garching bei Muenchen, Germany

    Christian Kahle

  5. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Tobias Keil

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  1. Michael Hintermüller
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  4. Tobias Keil
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Correspondence to Michael Hinze .

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  1. Fachbereich Mathematik, Technische Universität Darmstadt , Darmstadt, Germany

    Dieter Bothe

  2. Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Arnold Reusken

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Hintermüller, M., Hinze, M., Kahle, C., Keil, T. (2017). Fully Adaptive and Integrated Numerical Methods for the Simulation and Control of Variable Density Multiphase Flows Governed by Diffuse Interface Models. In: Bothe, D., Reusken, A. (eds) Transport Processes at Fluidic Interfaces. Advances in Mathematical Fluid Mechanics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-56602-3_13

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