Smoothness of weak solutions to a nonlinear fluid-structure interaction model

Barbu, Viorel; Grujić, Zoran; Lasiecka, Irena; Tuffaha, Amjad

doi:10.1512/iumj.2008.57.3284

Cited by 86 publications

(79 citation statements)

References 27 publications

(26 reference statements)

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“…FSI problems coupling the Navier-Stokes equations with linear elasticity where the coupling was calculated at a fixed fluid domain boundary, were considered in [23], and in [2,3,35] where an additional nonlinear coupling term was added at the interface. A study of well-posedness for FSI problems between an incompressible, viscous fluid and an elastic/viscoelastic structure with nonlinear coupling evaluated at a moving interface started with the result by daVeiga [4], where existence of a strong solution was obtained locally in time for an interaction between a 2D fluid and a 1D viscoelastic string, assuming periodic boundary conditions.…”

Section: A Brief Literature Reviewmentioning

confidence: 99%

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

Muha

Čanić

2013

Communications in Information and Systems

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We study a 3D fluid-structure interaction (FSI) problem between an incompressible, viscous fluid modeled by the Navier-Stokes equations, and the motion of an elastic structure, modeled by the linearly elastic cylindrical Koiter shell equations, allowing structure displacements that are not necessarily radially symmetric. The problem is set on a cylindrical domain in 3D, and is driven by the time-dependent inlet and outlet dynamic pressure data. The coupling between the fluid and the structure is fully nonlinear (2-way coupling), giving rise to a nonlinear, moving-boundary problem in 3D. We prove the existence of a weak solution to this 3D FSI problem by using an operator splitting approach in combination with the Arbitrary Lagrangian Eulerian mapping, which satisfies a geometric conservation law property. We effectively prove that the resulting computational scheme converges to a weak solution of the full, nonlinear 3D FSI problem.

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Section: A Brief Literature Reviewmentioning

confidence: 99%

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

Muha

Čanić

2013

Communications in Information and Systems

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“…The relation in (5.10) proves dissipativity of A. The proof of maximality follows now the same arguments as in [30,33].…”

Section: Generation Of the Semigroupmentioning

confidence: 71%

“…In the case when ! = 0 the proof of this result is given in [33]. Similar argument can be applied also to the case !…”

Section: Generation Of the Semigroupmentioning

confidence: 74%

“…s . The second approach, instead, is variational: it was presented in [33][34][35][36]. We shall use this approach here.…”

Section: Specialization Of Abstract Model To the Fluidstructure Intermentioning

confidence: 99%

“…The Neumann map N is linear bounded from 33,35] . We are in a position to define the dynamics of the overall coupled system.…”

Section: Specialization Of Abstract Model To the Fluidstructure Intermentioning

confidence: 99%

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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 global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate

Čhuešhov¹

2011

Math. Meth. Appl. Sci.

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We study asymptotic dynamics of a coupled system consisting of linearized 3D Navier-Stokes equations in a bounded domain and the classical (nonlinear) elastic plate equation for in-plane motions on a flexible flat part of the boundary. The main novelty of the model is the assumption that the transversal displacements of the plate are negligible relative to in-plane displacements. These kinds of models arise in the study of blood flows in large arteries. Our main result states the existence of a compact global attractor of finite dimension. Under some conditions this attractor is an exponentially attracting single point. We also show that the corresponding linearized system generates an exponentially stable C 0 -semigroup. We do not assume any kind of mechanical damping in the plate component. Thus our results mean that dissipation of the energy in the fluid because of viscosity is sufficient to stabilize the system. Copyright © 2011 John Wiley & Sons, Ltd.Keywords: fluid-structure interaction; linearized 3D Navier-Stokes equations; nonlinear plate; finite-dimensional attractorwhere >0 is the dynamical viscosity and Q G is a volume force. We supplement Equations (1)- (3) with the (non-slip) boundary conditions imposed on the velocity field v D v.x, t/:where andwithwhere D D Eh= 1 2 , E is Young's modulus, 0 < < 1=2 is Poisson's ratio, h is the thickness of the plate, is the mass density. The external (in-plane) force F 1 ; F 2 in Equations (5) and (6) consists of two parts,where f 1 .u 1 , u 2 /; f 2 .u 1 , u 2 / is a nonlinear feedback force represented by some potentialˆ(which we specify below)and .T 1 .v/; T 2 .v// is the viscous shear stress exerted by the fluid on the plate, T i .v/ D ..Tn/j , e i / R 3 . Here, T D fT ij g 3 i,jD1 is the stress tensor of the fluid, given by (in the last equality, we use the fact that v 3 .x 1 ; x 2 ; 0/ D 0 for .x 1 ; x 2 / 2 because of the second relation in Equation (4) and henceThus we arrive at the following equations for the in-plane displacement u D u 1 ; u 2 of the plate (below for some notational simplifications we assume that h D 1 and D.1 /=2 D 1):where D .1 C /.1 / 1 is a nonnegative parameter. For the displacement u D u 1 ; u 2 we impose clamped boundary conditions on D @ :Our main point of interest is the well-posedness and long-time dynamics of solutions to the coupled problem in Equations (1)- (4), (7), and (8) for the velocity v and the displacement u D .u 1 ; u 2 / with the initial dataThis problem in the case when Q G Á 0, D 0, and f i .u/ Á 0 was considered in [8] (see also [5,7]) with an additional locally distributed strong (Kelvin-Voight type) damping force applied to the interior of the plate. These papers deal with the existence and asymptotic stability of the corresponding semigroup. In contrast with [5,7,8], we do not assume the presence of mechanical damping terms in the plate component of the system but consider nonlinearly forced model.

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Smoothness of weak solutions to a nonlinear fluid-structure interaction model

Cited by 86 publications

References 27 publications

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

A nonlinear, 3D fluid-structure interaction problem driven by the time-dependent dynamic pressure data: a constructive existence proof

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

A global attractor for a fluid-plate interaction model accounting only for longitudinal deformations of the plate

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