Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions

Lasiecka, Irena; Szulc, Katarzyna; Żochowski, Antoni

doi:10.1016/j.nonrwa.2018.04.004

Cited by 5 publications

(26 citation statements)

References 17 publications

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“…We recall the problem setting from Lasiecka et al [LSZ18]. Let D ⊂ R n , n = 2, 3, be a bounded domain with piecewise regular boundary ∂D and straight corners as shown in Figure 1.…”

Section: The Domainmentioning

confidence: 99%

“…Existence follows by Banach's fixed point theorem, see [LSZ18,(68),(85)], using smallness of the data g.…”

Section: 1mentioning

confidence: 99%

“…The elastic body deforms under the flow and is modelled by linear elasticity, for the fluid we consider the steady Navier-Stokes equation with Dirichlet condition at the inlet, no-slip condition on the wall, and do-nothing condition on the outlet. The configuration is taken from Lasiecka, Szulc, and Zochoswki [LSZ18] who analyze existence of solutions to this FSI problem and existence of an optimal inflow profile, considered as a boundary control, which minimizes the drag at the interface of the elastic body and the fluid. Let g denote the Dirichlet inflow boundary values and (u, w, p) be the solution of the FSI problem after transforming the variables for the fluid to a reference domain, that means u solves the elasticity equation, (w, p) is the solution of the Navier-Stokes equation and both equations are coupled via the traction force at the interface and via coefficients in the Navier-Stokes equation.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, the linearized Navier-Stokes operator needs to be an isomorphism in suitable spaces; hence, main parts of the paper deal with the derivation of regularity results for this equation. We proceed in three steps following the procedure in [LSZ18]: (i) Derivation of a lower regularity result for the velocity pressure pair in W 1,2 × L 2 based on Lax-Milgram arguments, (ii) derivation of a higher regularity result in W 2,2 × W 1,2 which uses estimates from [LSZ18] which relies on results from the Agmon, Douglis, and Nirenberg [ADN59] theory on ellitpic systems, (iii) higher p-integrability, namely W 2,p × W 1,p on compact subsets using commutator analysis. For the analysis of linear elasticity we rely on classical theory.…”

Section: Introductionmentioning

confidence: 99%

“…On optimal control and FSI: In [LSZ18] boundary control of a FSI problem with stationary Navier-Stokes equation is considered. The authors show existence of a unqiue solution of the underlying equation under a smallness condition as well as of an optimal control.…”

Section: Introductionmentioning

confidence: 99%

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Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier-Stokes equations with mixed-boundary conditions in a channel

Hintermüller¹,

Kröner²

2021

Preprint

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In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from [Lasiecka, Szulc, and Zochoswki, Nonl. Anal.: Real World Appl., 44, 2018 ]. An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of [Wollner and Wick, J. Math. Fluid Mech., 21, 2019 ] to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners. Contents 1. Introduction 2. The domain 2.1. The flow map and some transformation rules 2.2. Transformation of integrals 2.3. Transformation of the Navier-Stokes equation 3. Existence of solutions for the considered systems 3.1. The Navier-Stokes system 3.2. The elasticity system and the traction force 3.3. The fluid-structure interation system 4. The linearized equations 4.1. Linearized state equation: Coefficients equal to one 4.2. The linearized state equation 4.3. Lower regularity 4.4. Higher regularity 4.5. Limit behaviour 5. Differentiability Appendix A. Transformation of the Navier-Stokes equation Appendix B. Transformation of the linearized Navier-Stokes equation Appendix C. Some properties References

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“…We recall the problem setting from Lasiecka et al [LSZ18]. Let D ⊂ R n , n = 2, 3, be a bounded domain with piecewise regular boundary ∂D and straight corners as shown in Figure 1.…”

Section: The Domainmentioning

confidence: 99%

“…Existence follows by Banach's fixed point theorem, see [LSZ18,(68),(85)], using smallness of the data g.…”

Section: 1mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier-Stokes equations with mixed-boundary conditions in a channel

Hintermüller¹,

Kröner²

2021

Preprint

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show abstract

Differentiability Properties for Boundary Control of Fluid-Structure Interactions of Linear Elasticity with Navier-Stokes Equations with Mixed-Boundary Conditions in a Channel

Hintermüller

Kröner

2023

Appl Math Optim

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In this paper we consider a fluid-structure interaction problem given by the steady Navier Stokes equations coupled with linear elasticity taken from (Lasiecka et al. in Nonlinear Anal 44:54–85, 2018). An elastic body surrounded by a liquid in a rectangular domain is deformed by the flow which can be controlled by the Dirichlet boundary condition at the inlet. On the walls along the channel homogeneous Dirichlet boundary conditions and on the outflow boundary do-nothing conditions are prescribed. We recall existence results for the nonlinear system from that reference and analyze the control to state mapping generalizing the results of (Wollner and Wick in J Math Fluid Mech 21:34, 2019) to the setting of the nonlinear Navier-Stokes equation for the fluid and the situation of mixed boundary conditions in a domain with corners.

show abstract

Shape Gradient Methods for Shape Optimization of an Unsteady Multiscale Fluid–Structure Interaction Model

Zhang,

Zhu,

et al. 2024

J Geom Anal

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Boundary control of small solutions to fluid–structure interactions arising in coupling of elasticity with Navier–Stokes equation under mixed boundary conditions

Cited by 5 publications

References 17 publications

Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier-Stokes equations with mixed-boundary conditions in a channel

Differentiability properties for boundary control of fluid-structure interactions of linear elasticity with Navier-Stokes equations with mixed-boundary conditions in a channel

Differentiability Properties for Boundary Control of Fluid-Structure Interactions of Linear Elasticity with Navier-Stokes Equations with Mixed-Boundary Conditions in a Channel

Shape Gradient Methods for Shape Optimization of an Unsteady Multiscale Fluid–Structure Interaction Model

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