Weak Solutions of Fluid–Solid Interaction Problems

Hsiao, George; Kleinman, R. E.; Roach, G. F.

doi:10.1002/1522-2616(200010)218:1<139::aid-mana139>3.3.co;2-j

Cited by 34 publications

(65 citation statements)

References 16 publications

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“…Hence,

ρ_{0} ω^{2} u \cdot n = \frac{\partial p}{\partial n} + \frac{\partial p^{0}}{\partial n} .

Then, the fluid–solid interaction problem can be formulated as follows (c.f. ): For a given external force f applied to the solid and an incident field p 0 ∈ C 1 that satisfies Δ p 0 + k 2 p 0 = 0 almost everywhere in

normalΩ \cup {normalΩ}^{+}

, find

boldu \in C^{2} false(normalΩ false) \cap C^{1} false(normalΩ \cup normalΓ false)

and

p \in C^{2} false({normalΩ}^{+} false) \cap C^{1} false({normalΩ}^{+} \cup normalΓ false)

satisfying

\begin{matrix} right div σ (u) + ρ ω^{2} u & left = f in Ω, & right \\ right Δ p + k^{2} p & left = 0 in Ω^{+}, \\ right σ (u) n & left = - (p + p^{0}) n on Γ, \\ right ρ_{0} ω^{2} u \cdot n & left = \frac{\partial p}{\partial n} + \frac{\partial p^{0}}{\partial n} on Γ, \\ right p satisfies the radiation condition & left (1) in Ω^{+} . \end{matrix}

Now, we consider the following reduced problem, such that we find the pressure p on the boundary Γ; that is, because p satisfies the Helmholtz equation, we can apply th...…”

Section: State Equation: Fluid‐structure Interaction Problemmentioning

confidence: 99%

“…Here, ⟨·,·⟩ indicates the dual product between H 1/2 (Γ) and H − 1/2 (Γ). Now, taking

\begin{matrix} scriptA false(boldu, γ_{0}^{+} p; boldv, q false) & : = \int_{normalΩ} σ false(boldu false) : grad \overset{v}{true} d x - ρ ω^{2} \int_{normalΩ} boldu \cdot \overset{v}{true} d x, + ⟨ γ_{0}^{+} p boldn, boldv ⟩ \\ - (⟨⟩, ((), K - \frac{I}{2}) γ_{0}^{+} p, q) + ρ_{0} ω^{2} ⟨ V false(boldu \cdot boldn false), q ⟩, \end{matrix}

and

F (f, p^{0}; v, q) : = - ⟨ f, v ⟩ - ⟨ {γ_{0}^{+} p}^{0} n, v ⟩ + ⟨ V {γ_{1}^{+} p}^{0}, q ⟩,

we can rewrite the weak formulation as follows: Find

false(boldu, γ_{0}^{+} p false) \in H^{1} false(normalΩ false) \times H^{1 false/ 2} false(normalΓ false)

such that

scriptA false(boldu, γ_{0}^{+} p; boldv, q false) = scriptF false(boldf; boldv, q false) \forall false(boldv, q false) \in H^{1} false(normalΩ false) \times H^{1 false/ 2} false(normalΓ false) .

According to …”

Section: State Equation: Fluid‐structure Interaction Problemmentioning

confidence: 99%

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Local active control for an exterior fluid‐structure interaction problem

Domínguez

Hernández

Torres

2017

Numerical Meth Engineering

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Summary In this paper, we propose a numerical method based on an active structural control to dampen the discrete critical frequencies from a fe/be coupled system, which models a time‐harmonic fluid‐structure interaction problem. The active structural acoustic control used consists of applying external forces directly on the structure, so that in the presence of critical frequencies, the linear system remains stable and, in turn, in the presence of non‐critical frequencies, the active control does not alter the system. We consider the problem in the framework of the theory of optimal control and present bi‐dimensional numerical simulations to show the behavior of the scheme in some vibro‐acoustic structures. Copyright © 2016 John Wiley & Sons, Ltd.

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“…Hence,

ρ_{0} ω^{2} u \cdot n = \frac{\partial p}{\partial n} + \frac{\partial p^{0}}{\partial n} .

normalΩ \cup {normalΩ}^{+}

, find

boldu \in C^{2} false(normalΩ false) \cap C^{1} false(normalΩ \cup normalΓ false)

and

p \in C^{2} false({normalΩ}^{+} false) \cap C^{1} false({normalΩ}^{+} \cup normalΓ false)

satisfying

\begin{matrix} right div σ (u) + ρ ω^{2} u & left = f in Ω, & right \\ right Δ p + k^{2} p & left = 0 in Ω^{+}, \\ right σ (u) n & left = - (p + p^{0}) n on Γ, \\ right ρ_{0} ω^{2} u \cdot n & left = \frac{\partial p}{\partial n} + \frac{\partial p^{0}}{\partial n} on Γ, \\ right p satisfies the radiation condition & left (1) in Ω^{+} . \end{matrix}

Now, we consider the following reduced problem, such that we find the pressure p on the boundary Γ; that is, because p satisfies the Helmholtz equation, we can apply th...…”

Section: State Equation: Fluid‐structure Interaction Problemmentioning

confidence: 99%

“…Here, ⟨·,·⟩ indicates the dual product between H 1/2 (Γ) and H − 1/2 (Γ). Now, taking

\begin{matrix} scriptA false(boldu, γ_{0}^{+} p; boldv, q false) & : = \int_{normalΩ} σ false(boldu false) : grad \overset{v}{true} d x - ρ ω^{2} \int_{normalΩ} boldu \cdot \overset{v}{true} d x, + ⟨ γ_{0}^{+} p boldn, boldv ⟩ \\ - (⟨⟩, ((), K - \frac{I}{2}) γ_{0}^{+} p, q) + ρ_{0} ω^{2} ⟨ V false(boldu \cdot boldn false), q ⟩, \end{matrix}

and

F (f, p^{0}; v, q) : = - ⟨ f, v ⟩ - ⟨ {γ_{0}^{+} p}^{0} n, v ⟩ + ⟨ V {γ_{1}^{+} p}^{0}, q ⟩,

we can rewrite the weak formulation as follows: Find

false(boldu, γ_{0}^{+} p false) \in H^{1} false(normalΩ false) \times H^{1 false/ 2} false(normalΓ false)

such that

scriptA false(boldu, γ_{0}^{+} p; boldv, q false) = scriptF false(boldf; boldv, q false) \forall false(boldv, q false) \in H^{1} false(normalΩ false) \times H^{1 false/ 2} false(normalΓ false) .

According to …”

Section: State Equation: Fluid‐structure Interaction Problemmentioning

confidence: 99%

Local active control for an exterior fluid‐structure interaction problem

Domínguez

Hernández

Torres

2017

Numerical Meth Engineering

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“…In addition, in this reference, one finds an historical description of the fluid-structure problem, along with an extensive list of references up to 2003, to which we shall defer. Some include [17], [11], [22], and yet, to quote from [16]: "Rigorous mathematical models are rare for fluid-solid interaction problems in which both the fluid and the solid occupy true spatial domains." Mathematically, [16] has two main results: (1) an existence and uniqueness result of a weak solution to a divergence-free (weak) formulation of the fluid-structure system, with (in our notation) Initial Con- …”

Section: B)mentioning

confidence: 99%

Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction

Avalos¹,

Triggiani²

2009

Discrete &Amp; Continuous Dynamical Systems - S

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This paper is focused on an established, genuinely physical fluidstructure interaction model, herein the structure is immersed in a fluid with coupling taking place at the boundary interface between the two media. Mathematically, the model is a coupled parabolic-hyperbolic system of two partial differential equations in 3-d with nonstandard coupling at the boundary interface. Fluid and structure are mathematically expressed by the (dynamic) Stokes system (parabolic) and the Lamé system (hyperbolic), respectively. The main claim presented is a contraction semigroup well-posedness result on the natural space of finite energy. There are two main features in the analysis: (i) a nonstandard elimination of the pressure term, as the boundary coupling between fluid and structure rules out application of the classical Leray/Helmoltz projection; (ii) a nonstandard usage of the Babuska-Brezzi "inf-sup" theory to assert maximal dissipativity of the candidate generator. A unified treatment includes both undamped and (perhaps, partially) damped boundary conditions at the interface. With the generator explicitly at hand, an analysis of its point spectrum on the imaginary axis is also included. In the undamped case, it depends on the geometry of the structure. In the case of full boundary damping, it implies as a by-product a strong stability result for the solutions by soft methods.2000 Mathematics Subject Classification. Primary: 35A05, 35M10; Secondary: 35P05.

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“…wave is incident upon the elastic target and the problem is to determine the scattered acoustic pressure in the fluid domain and the displacement field in the elastic structure (for the detailed physical background we refer to [26]). …”

Section: Communicated By E Meistermentioning

confidence: 99%

“…The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov-Poincaret ype operators and on the basis of the Ga s rding inequality and the Lax-Milgram theorem. In particular, it is shown that the physical fluid-solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter.wave is incident upon the elastic target and the problem is to determine the scattered acoustic pressure in the fluid domain and the displacement field in the elastic structure (for the detailed physical background we refer to [26]). …”

mentioning

confidence: 99%

Non-local approach in mathematical problems of fluid-structure interaction

Jentsch

Natroshvili

1999

Math. Meth. Appl. Sci.

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Three‐dimensional mathematical problems of interaction between elastic and scalar oscillation fields are investigated. An elastic field is to be defined in a bounded inhomogeneous anisotropic body occupying the domain Ω¯1⊂ℝ3 while a physical (acoustic) scalar field is to be defined in the exterior domain Ω¯2=ℝ3\Ω1 which is filled up also by an anisotropic (fluid) medium. These two fields satisfy the governing equations of steady‐state oscillations in the corresponding domains together with special kinematic and dynamic transmission conditions on the interface ∂Ω1. The problems are studied by the so‐called non‐local approach, which is the coupling of the boundary integral equation method (in the unbounded domain) and the functional‐variational method (in the bounded domain). The uniqueness and existence theorems are proved and the regularity of solutions are established with the help of the corresponding Steklov–Poincaré type operators and on the basis of the Gårding inequality and the Lax–Milgram theorem. In particular, it is shown that the physical fluid–solid acoustic interaction problem is solvable for arbitrary values of the frequency parameter. Copyright © 1999 John Wiley & Sons, Ltd.

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Weak Solutions of Fluid–Solid Interaction Problems

Cited by 34 publications

References 16 publications

Local active control for an exterior fluid‐structure interaction problem

Local active control for an exterior fluid‐structure interaction problem

Semigroup well-posedness in the energy space of a parabolic-hyperbolic coupled Stokes-Lamé PDE system of fluid-structure interaction

Non-local approach in mathematical problems of fluid-structure interaction

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