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Stabilization of Incompressible Flow Problems by RiccatiBased Feedback
 OF INTERNATIONAL SERIES OF NUMERICAL MATHEMATICS, BIRKHÄUSER, 2012
"... We consider optimal controlbased boundary feedback stabilization of flow problems for incompressible fluids. We follow an analytical 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 p ..."
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We consider optimal controlbased boundary feedback stabilization of flow problems for incompressible fluids. We follow an analytical 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 NavierStokes equations by designing a stabilizing controller based on a corresponding linearquadratic optimal control problem. For this purpose, algorithmic advances in solving the associated algebraic Riccati equations are needed and investigated here. The computational complexity of the new algorithms is essentailly proportional to the simulation of the forward problem.
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"... Abstract. In this article a boundary feedback stabilization approach for incompressible Navier– Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a dif ..."
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Abstract. In this article a boundary feedback stabilization approach for incompressible Navier– Stokes flows is studied. One of the main difficulties encountered is the fact that after space discretization by a mixed finite element method (because of the solenoidal condition) one ends up with a differential algebraic system of index 2. The remedy here is to use a discrete realization of the Leray projection used by Raymond [J.P. Raymond, SIAM J. Control Optim., 45 (2006), pp. 790–828] to analyze and stabilize the continuous problem. Using the discrete projection, a linear quadratic regulator (LQR) approach can be applied to stabilize the (discrete) linearized flow field with respect to small perturbations from a stationary trajectory. We provide a novel argument that the discrete Leray projector is nothing else but the numerical projection method proposed by Heinkenschloss and colleagues in [M. Heinkenschloss, D. C. Sorensen, and K. Sun, SIAM J. Sci. Comput., 30 (2008), pp. 1038–1063]. The nested iteration resulting from applying this approach within the NewtonADI method to solve the LQR algebraic Riccati equation is the key to compute a feedback matrix that in turn can be applied within a closedloop simulation. Numerical examples for various parameters influencing the different levels of the nested iteration are given. Finally, the stabilizing property of the computed feedback matrix is demonstrated using the von Kármán vortex street within a finite element based flow solver.
FRIEDRICHALEXANDERUNIVERSITÄT ERLANGENNÜRNBERG Lehrstuhl für Angewandte Mathematik III Simulation of Instabilities occurring in Liquid Jets
"... angegebenen Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form ..."
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angegebenen Quellen angefertigt habe und dass die Arbeit in gleicher oder ähnlicher Form
Numerical simulation of the free fall of a rigid body in a viscous fluid
"... Abstract. We consider the problem of the free fall of a rigid body in a viscous fluid. This problem has many important applications and the present paper is devoted to its numerical solution. We present a model of the problem, discuss various aspects of its finite element discretization and present ..."
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Abstract. We consider the problem of the free fall of a rigid body in a viscous fluid. This problem has many important applications and the present paper is devoted to its numerical solution. We present a model of the problem, discuss various aspects of its finite element discretization and present some first numerical results. 1
NavierStokes
, 2013
"... Riccatibased boundary feedback stabilization of incompressible ..."
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Lehrstuhl für Angewandte Mathematik III
, 2004
"... Computational comparison between the Taylor–Hood and the conforming Crouzeix–Raviart element ..."
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Computational comparison between the Taylor–Hood and the conforming Crouzeix–Raviart element