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DFG Priority Program 1253:
Optimization With Partial Differential Equations
Project
Optimal Control-Based Feedback Stabilization of Multi-Field Flow Problems
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:
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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.
Simulations:
Navier-Stokes on von Kármán vortex street
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Navier-Stokes coupled with transport of some species
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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:
Reports:
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
