Computational Methods in Systems and Control Theory

OptConFee- StabMultiFlow

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DFG Priority Program 1253: Optimization With Partial Differential Equations

Project

Optimal Control-Based Feedback Stabilization of Multi-Field Flow Problems

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The goal of this project is to derive and investigate numerical algorithms for optimal control-based boundary feedback stabilization of multi-field flow problems.

We will follow an approach laid out during the last years in a series of papers by Barbu, Lasiecka, Triggiani, Raymond, and others. They have shown that it is possible to stabilize perturbed flows described by the Navier-Stokes equation by designing a stabilizing controller based on a corresponding linear-quadratic optimal control problem.

Until recently, the numerical solution of these linear-quadratic optimal control problem was a numerical challenge due to the complexity of existing algorithms. Employing recent advances in reducing these complexities essentially to a cost proportional to the simulation of the forward problem, we plan to apply this methodology to multi-field problems where the flow is coupled with other field equations.

We suggest three scenarios with increasing difficulty for which we want to demonstrate the applicability of the optimal control-based feedback stabilization approach.

The scenarios are the following:

• Navier-Stokes coupled with (passive) transport of some (reactive) species:

  This example may in a rather crude way model a reactor, where a chemical substance is transported by a flow field and reacts at the surface. The reaction is considered to be fast (compared to diffusion and transport), such that a homogeneous Dirichlet boundary condition can be used as simplified modell.
  The control acts by varying the inflow boundary condition. The idea behind this setting is to stabilize and control the process of reaction, which is strongly influenced by the transport of the substance from the inflow to the reacting surface.

• Phase transition liquid/solid with convection:

  Consider a hot melt that solidifies while flowing through a mould. The objective here is to control the phase boundary between the liquid part of the mould and the solid part. In addition to considering the Navier-Stokes equations in the liquid part, there is a heat equation for the temperature to be solved in the whole domain. The control is given as the temperature distribution on a part of the boundary.

• Stabilization of a flow with a free capillary surface:

  Capillary free surfaces play a decisive role in many technological applications. Thus control of the free boundary can be of paramount interest. Here, we consider a model problem, where a fluid is flowing over an obstacle and the upper boundary is a free capillary one. This boundary will be oscillatory due to the Karman vortex shedding in the wake of the obstacle. The goal is to stabilize the free boundary.

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Principal investigators: Prof. Dr. Peter Benner Prof. Dr. Eberhard Bänsch (Applied Mathematics III, FAU Erlangen)

Researcher: Anne Katrin Heubner (11/2006-09/2008) Dr. Jens Saak (10/2009-01/2011) Heiko Weichelt (since 06/2011)

Student Assistants: Martin Köhler (05/09-10/10) Heiko Weichelt (11/08-12/10)

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Simulations:

Navier-Stokes on von Kármán vortex street Example: Re=500, t_end=30 Control input over boundary after t_control=10 Openloop control with constant input Download as AVI in higher quality (ca. 12 MB) (Non optimal) feedback Download as AVI in higher quality (ca. 19 MB) Goal: Get laminar flow behind the obstacle.		Navier-Stokes coupled with transport of some species Example: Re=10, Sc=10, t_end=60 (left picture) Piecewise constant control input (right picture) 3D-Simulation of concentration Download as AVI in higher quality (ca. 60 MB) 2D-Simulation of concentration Download as AVI in higher quality (ca. 80 MB) Goal: Get a fixed rate of reaction on the obstacle.
Navier-Stokes on von Kármán vortex street Example: Re=300, t_end=40 Feedback stabilization over boundary influence after t_control=7.5 for initial and optimal feedback. Download as AVI in higher quality (ca. 69 MB) Goal: Vanish y-components of flow in grid fields 3 and 4 of the 3rd row.

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Publications:

• Eberhard Bänsch, Peter Benner, Jens Saak, und Heiko K. Weichelt; Riccati-Based Boundary Feedback Stabilization of Incompressible Navier-Stokes Flow;
  Preprint SPP1253-154, September 2013.
• Peter Benner, Jens Saak, Martin Stoll, and Heiko K. Weichelt; Efficient Solution of Large-Scale Saddle Point Systems Arising in Riccati-Based Boundary Feedback Stabilization of Incompressible Stokes Flow;
  Preprint SPP1253-130, June 2012.
• Eberhard Bänsch and Peter Benner; Stabilization of Incompressible Flow Problems by Riccati-Based Feedback;
  published in Constrained Optimization and Optimal Control for Partial Differential Equations, Birkhäuser Verlag 2012, pages 5-20.
• Peter Benner und Jens Saak; A Galerkin-Newton-ADI Method for Solving Large-Scale Algebraic Riccati Equations;
  Preprint SPP1253-090, January 2010.
• Peter Benner, Tobias Rothaug and Rene Schneider; Flow stabilisation by Dirichlet boundary control;
  Proceedings in Applied Mathematics and Mechanics, Vol. 8, No. 1, pp. 10961-10962, 2008.
• Eberhard Bänsch, Peter Benner and Anne Heubner; Riccati-Based Feedback Stabilization of Flow Problems ;
  23rd IFIP TC 7 Conference on System Modelling and Optimization Cracow, 2007.

Related Posters:

• Optimal Control-Based Feedback Stabilization of Multi-Field Flow Problems;
  Evaluation Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg;
  Magdeburg, March 14, 2012.
• DFG-Priority Program 1253: Optimization with Partial Differential Equations;
  Review meeting;
  Freising, March 29-31, 2009.
• DFG-Priority Program 1253: Optimization with Partial Differential Equations;
  Opening meeting;
  Bad Honnef, February 06-07, 2006.

Reports:

• Heiko Weichelt; Feedback-Stabilisierung von instationären, inkompressiblen Strömungen mit Riccati-Ansatz;
  diploma thesis; (German only)
  Chemnitz UT, December 21 2010.
• Heiko Weichelt; Navier-Stokes-Gleichung gekoppelt mit dem Transport von (reaktiven) Substanzen;
  final modeling seminar report; (German only)
  Chemnitz UT, April 2010.

Talks and Presentations:

• Heiko Weichelt; Riccati-Based Boundary Feedback Stabilization of Multi-Field Flow Problems;
  Workshop on Numerical Methods for Optimal Control and Inverse Problems 2013;
  Garching, March 13 2013.
• Heiko Weichelt; Optimal Control-Based Feedback Stabilization of Multi-Field Flow Problems;
  SPP1253 Final meeting;
  Kloster Banz, February 26, 2013.
• Heiko Weichelt; Riccati-Based Boundary Feedback Stabilization of Flow Problems -- A Summary --;
  Absolventen Seminar 2012, Fachgebiet Numerische Mathematik;
  TU Berlin, November 22, 2012.
• Heiko Weichelt; Riccati-Based Boundary Feedback Stabilization of Flow Problems;
  DMV-Jahrestagung 2012;
  Saarbrücken, September 19, 2012.
• Heiko Weichelt; Preconditioning of Large-Scale Saddle Point Systems Arising in Riccati Feedback Control of Flow Problems;
  XII GAMM Workshop: Angewandte und Numerische Lineare Algebra;
  Chateau Liblice (Tschechien), September 3, 2012.
• Heiko Weichelt; Efficient Solution of Large-Scale Saddle Point Systems Arising in Feedback Control of the Stokes Equations;
  Twelth Copper Mountain Conference on Iterative Methods;
  Copper Mountain, Colorado (USA), March 29, 2012.
• Heiko Weichelt; Riccati-Based Boundary Feedback Stabilization of Incompressible Flow Problems;
  Workshop on Numerical Methods for Optimal Control and Inverse Problems 2012;
  Garching, March 13, 2012.
• Heiko Weichelt; Efficient Solution of Large-Scale Saddle Point Systems Arising in Feedback Control of Flow Problems;
  CSC Seminar;
  MPI Magdeburg, Februar 14, 2012.
• Heiko Weichelt; Optimal Control-Based Feedback Stabilization of Multi-Field Flow Problems;
  SPP1253 annual meeting;
  Kloster Banz, September 26, 2011.
• Heiko Weichelt; Feedback-Stabilisierung von instationären, inkompressiblen Strömungen mit Riccati-Ansatz;
  diploma presentation; (German only)
  Chemnitz UT, December 21, 2010.
• Heiko Weichelt; Navier-Stokes-Gleichung gekoppelt mit dem Transport von (reaktiven) Substanzen -Abschlussbericht-;
  modeling seminar; (German only)
  Chemnitz UT, April 14, 2010.
• Peter Benner; ADI-based Methods for Algebraic Lyapunov and Riccati Equations;
  CICADA / MIMS workshop on Numerics for Control and Simulation;
  University of Manchaster, June 17, 2009.
• Peter Benner; Solving algebraic Riccati equations for stabilization of incompressible flows;
  Plenary Lecture at Householder Symposium XVII, ;
  Zeuthen, Germany, June 1-6, 2008.
• Peter Benner; Tobias Rothaug; Rene Schneider; Flow stabilisation by Dirichlet boundary control;
  79th GAMM annual meeting 2008;
  Universität Bremen, March 31 - April 04, 2008.
• Anne Heubner; Riccati-Based Feedback Boundary Stabilization of Flow Problems;
  79th GAMM annual meeting 2008;
  Universität Bremen, March 31 - April 04, 2008.
• Jens Saak; Efficient numerical solution of large scale matrix equations arising in LQR/LQG design for parabolic PDEs;
  Workshop PDE Constrained Optimization: Recent Challenges and Future Developments;
  University of Hamburg, Germany, March 27-29, 2008.
• Peter Benner; On the Numerical Solution of Differential Operator Riccati Equations in PDE Control;
  Workshop PDE Constrained Optimization: Recent Challenges and Future Developments;
  University of Hamburg, Germany, March 27-29, 2008.
• Anne Heubner; Riccati-Basierte Feedback-Stabilisierung von Strömungsproblemen;
  5. Elgersburg Workshop `Mathematische Systemtheorie';
  Elgersburg (Thüringen), February 11-14, 2008.
• Anne Heubner; Optimal Control-Based Feedback Stabilization in Multi-Field Flow Problems;
  First annual meeting of SPP1253;
  Bad Honnef, October 4-5, 2007.
• Anne Heubner; Riccati-Based Feedback Stabilization of Flow Problems
  23rd IFIP TC 7 Conference on System Modelling and Optimization;
  Cracow, Poland, July 23-27, 2007