Optimal boundary control with critical penalization for a PDE model of fluid–solid interactions

Bucci, F.; Lasiecka, Irena

doi:10.1007/s00526-009-0259-9

Cited by 33 publications

(60 citation statements)

References 32 publications

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“…If we enter (13) in the weak formulation (9), we immediately obtain with (12) that the triplet (v,p,û) solves the weak formulation (11) with the right-hand sidesf f ,f s , and g = 0.…”

Section: Novel Symmetric Weak Formulationmentioning

confidence: 99%

“…In [12], the authors derive formally necessary optimality conditions for an optimal control problem of a nonlinear time-dependent FSI configuration using shape derivatives. Further results on optimal feedback control of FSI can be found in [13][14][15], where corresponding Riccati equations are derived. In [16], the authors apply reduced basis methods for a shape optimization problem in the context of arterial blood flow.…”

Section: Introductionmentioning

confidence: 99%

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Optimal Control of a Linear Unsteady Fluid–Structure Interaction Problem

Failer

Meidner

Vexler

2016

J Optim Theory Appl

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In this paper, we consider optimal control problems governed by linear unsteady fluid-structure interaction problems. Based on a novel symmetric monolithic formulation, we derive optimality systems and provide regularity results for optimal solutions. The proposed formulation allows for natural application of gradient-based optimization algorithms and for space-time finite element discretizations.

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“…If we enter (13) in the weak formulation (9), we immediately obtain with (12) that the triplet (v,p,û) solves the weak formulation (11) with the right-hand sidesf f ,f s , and g = 0.…”

Section: Novel Symmetric Weak Formulationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Optimal Control of a Linear Unsteady Fluid–Structure Interaction Problem

Failer

Meidner

Vexler

2016

J Optim Theory Appl

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“…Based on this, the theory of optimal control and associate Riccati equations has been constructed in [45]. It is reasonable to expect that such theory could also be developed for the mini-max game problem studied in this paper [46]. However, this theory is not expected to recover a full boundedness of the gain operator B * P(t) .…”

Section: Conclusion and Some Open Problemsmentioning

confidence: 99%

Min-max Game Theory for Elastic and Visco-Elastic Fluid Structure Interactions

Lasiecka¹

2013

TOAMJ

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Abstract:We present the salient features of a min-max game theory developed in the context of coupled PDE's with an interface. Canonical applications include linear fluid-structure interaction problem modeled by Oseen's equations coupled with elastic waves. We shall consider two models for the structures: elastic and visco-elastic. Control and disturbance are allowed to act at the interface between the two media. The sought-after saddle solutions are expressed in a pointwise feedback form, which involves a Riccati operator; that is, an operator satisfying a suitable non-standard Riccati differential equation. Motivations, applications as well as a brief historical account are also provided.

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“…A nonstationary situation assuming a rigid solid was theoretically studied in [48]. Further theoretical results for a boundary control FSI problem were established in [8].Parameter estimation to detect the stiffness of an arterial wall with a well-posedness analysis and numerical simulations was addressed in [49]. Again in blood flow simulations, a data assimilation problem was formulated in [33], in which however, the arterial walls were not considered.…”

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confidence: 99%

Optimization with nonstationary, nonlinear monolithic fluid‐structure interaction

Wick

Wollner

2020

Numerical Meth Engineering

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Within this work, we consider optimization settings for nonlinear, nonstationary fluid-structure interaction. The problem is formulated in a monolithic fashion using the arbitrary Lagrangian-Eulerian framework to set-up the fluid-structure forward problem. In the optimization approach, either optimal control or parameter estimation problems are treated. In the latter, the stiffness of the solid is estimated from given reference values. In the numerical solution, the optimization problem is solved with a gradient-based solution algorithm. The nonlinear subproblems of the FSI forward problem are solved with a Newton method including line search. Specifically, we will formally provide the backward-in-time running adjoint state used for gradient computations.Our algorithmic developments are demonstrated with some numerical examples as for instance extensions of the well-known fluid-structure benchmark settings and a flapping membrane test in a channel flow with elastic walls. arXiv:1910.03424v1 [math.NA] 8 Oct 2019 [53,57,60,12]. Here, we notice that the required adjoints are the same as used for adjoint-based error estimation; see for instance [62,51]. Nonlinear (stationary) FSI investigating various partitioned coupling techniques was recently subject in [54]. The by far more challenging situation of nonstationary settings is listed in the following. A nonstationary situation assuming a rigid solid was theoretically studied in [48]. Further theoretical results for a boundary control FSI problem were established in [8].Parameter estimation to detect the stiffness of an arterial wall with a well-posedness analysis and numerical simulations was addressed in [49]. Again in blood flow simulations, a data assimilation problem was formulated in [33], in which however, the arterial walls were not considered. A full FSI problem for data assimilation using a Kalman filter was subject in [6]. In [44], the authors used optimization techniques to formulate the FSI coupling conditions. Adjoints for 1D FSI were derived in [16,47]. Reduced basis methods for FSI-based optimization were developed in [45]. Optimal control of nonstationary FSI applied to benchmark settings was investigated in [3]. A linearized FSI optimization problem was addressed in [23] and detailed results for full-time-dependent FSI optimal control were summarized in [22]. In this respect, we also mention [24] in which the adjoints required for optimization were employed for dual-weighted residual error estimation for time adaptivity. Most recently, a uncertainty quantification framework for fluid-structure interaction with applications in aortic biomechanics was developed in [43].The significance of the current work is on the development of a robust fully monolithic formulation for gradient-based optimization for nonstationary, nonlinear FSI problems. Here, the coupled problem is prescribed in the reference configuration with the help of the ALE approach in a variationalmonolithic way. As previously summarized in our literature review, only very few results exis...

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Optimal boundary control with critical penalization for a PDE model of fluid–solid interactions

Cited by 33 publications

References 32 publications

Optimal Control of a Linear Unsteady Fluid–Structure Interaction Problem

Optimal Control of a Linear Unsteady Fluid–Structure Interaction Problem

Min-max Game Theory for Elastic and Visco-Elastic Fluid Structure Interactions

Optimization with nonstationary, nonlinear monolithic fluid‐structure interaction

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