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Advanced Computational Dissipative Structural Acoustics and Fluid-Structure Interaction in Low-and Medium-Frequency Domains. Reduced-Order Models and Uncertainty Quantification
Advanced Computational Dissipative Structural Acoustics and Fluid-Structure Interaction in Low-and Medium-Frequency Domains. Reduced-Order Models and Uncertainty Quantification
International Journal of Aeronautical and Space Sciences. 2012. Jun, 13(2): 127-153
Copyright ©2012, The Korean Society for Aeronautical Space Science
This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.
  • Received : January 15, 2012
  • Accepted : March 15, 2012
  • Published : June 30, 2012
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About the Authors
R. Ohayon
Structural Mechanics and Coupled Systems Laboratory, Conservatoire National des Arts et Metiers (CNAM), 2 rue Conte, 75003 Paris, France
roger.ohayon@cnam.fr
C. Soize
Laboratoire Modélisation et Simulation Multi-Echelle (MSME UMR 8208 CNRS), Université Paris-Est, 5 bd Descartes, 77454 Marne-la-Vallée, France
christian.soize@univ-paris-est.fr

Abstract
This paper presents an advanced computational method for the prediction of the responses in the frequency domain of general linear dissipative structural-acoustic and fluid-structure systems, in the low-and medium-frequency domains and this includes uncertainty quantification. The system under consideration is constituted of a deformable dissipative structure that is coupled with an internal dissipative acoustic fluid. This includes wall acoustic impedances and it is surrounded by an infinite acoustic fluid. The system is submitted to given internal and external acoustic sources and to the prescribed mechanical forces. An efficient reduced-order computational model is constructed by using a finite element discretization for the structure and an internal acoustic fluid. The external acoustic fluid is treated by using an appropriate boundary element method in the frequency domain. All the required modeling aspects for the analysis of the medium-frequency domain have been introduced namely, a viscoelastic behavior for the structure, an appropriate dissipative model for the internal acoustic fluid that includes wall acoustic impedance and a model of uncertainty in particular for the modeling errors. This advanced computational formulation, corresponding to new extensions and complements with respect to the state-of-the-art are well adapted for the development of a new generation of software, in particular for parallel computers.
Keywords
Nomenclature
  • aijkh= elastic coefficients of the structure
  • bijkh= damping coefficients of the structure
  • c0= speed of sound in the internal acoustic fluid
  • cE= speed of sound in the external acoustic fluid
  • f= vector of the generalized forces for the internal acoustic fluid
  • fS= vector of the generalized forces for the structure
  • g= mechanical body force field in the structure
  • i= imaginary complex number i
  • k= wave number in the external acoustic fluid
  • n= number of internal acoustic DOF
  • ns= number of structure DOF
  • nj= component of vectorn
  • n= outward unit normal to∂Ω
  • nsj= component of vectornS
  • nS= outward unit normal to∂ΩS
  • p= internal acoustic pressure field
  • pE= external acoustic pressure field
  • pE|ΓE= value of the external acoustic pressure field on ΓE
  • pgiven= given external acoustic pressure field
  • pgiven|ΓE= value of the given external acoustic pressure field on ΓE
  • q= vector of the generalized coordinates for the internal acoustic fluid
  • qS= vector of the generalized coordinates for the structure
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  • = component of the damping stress tensor in the structure
  • t= time
  • u= structural displacement field
  • v= internal acoustic velocity field
  • xj= coordinate of pointx
  • x= generic point of R3
  • [A] = reduced dynamical matrix for the internal acoustic fluid
  • [A] = random reduced dynamical matrix for the internal acoustic fluid
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  • = dynamical matrix for the internal acoustic fluid
  • [ABEM] = reduced matrix of the impedance boundary operator for the external acoustic fluid
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  • = matrix of the impedance boundary operator for the external acoustic fluid
  • [AFSI] = reduced dynamical matrix for the fluid-structure coupled system
  • [AFSI] = random reduced dynamical matrix for the fluid-structure coupled system
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  • = dynamical matrix for the fluid-structure cou
  • [AS] = reduced dynamical matrix for the structure
  • [AS] = random reduced dynamical matrix for the structure
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  • = dynamical matrix for the structure
  • [AZ] = reduced dynamical matrix associated with the wall acoustic impedance
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  • = dynamical matrix associated with the wall acoustic impedance
  • [C] = reduced coupling matrix between the internal acoustic fluid and the structure
  • [C] = random reduced coupling matrix between the internal acoustic fluid and the structure
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  • = coupling matrix between the internal acoustic fluid and the structure
  • [D] = reduced damping matrix for the internal acoustic fluid
  • [D] = random reduced damping matrix for the internal acoustic fluid
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  • = damping matrix for the internal acoustic fluid
  • [DS] = reduced damping matrix for the structure
  • [DS] = random reduced damping matrix for the structure
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  • = damping matrix for the structure
  • DOF = degrees of freedom
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  • = vector of discretized acoustic forces
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  • = vector of discretized structural forces
  • Gijkh(0) = initial elasticity tensor for viscoelastic material
  • Gijkh(t) = relaxation functions for viscoelastic material
  • G= mechanical surface force field on∂Ωs
  • [G] = random matrix
  • [G0] = random matrix
  • [K] = reduced “stiffness” matrix for the internal acoustic fluid
  • [K] = random reduced “stiffness” matrix for the internal acoustic fluid
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  • = “stiffness” matrix for the internal acoustic fluid
  • [KS] = reduced stiffness matrix for the structure
  • [KS] = random reduced stiffness matrix for the structure
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  • = stiffness matrix for the structure
  • [M] = reduced “mass” matrix for the internal acoustic fluid
  • [M] = random reduced “mass” matrix for the internal acoustic fluid
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  • = “mass” matrix for the internal acoustic fluid
  • [MS] = reduced mass matrix for the structure
  • [MS] = random reduced mass matrix for the structure
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  • = mass matrix for the structure
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  • = internal acoustic mode
  • [P] = matrix of internal acoustic modes
  • Q= internal acoustic source density
  • QE= external acoustic source density
  • Q= random vector of the generalized coordinates for the internal acoustic fluid
  • QS= random vector of the generalized
  • P= random vector of internal acoustic pressure DOF
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  • = vector of internal acoustic pressure DOF
  • U= random vector of structural displacement DOF
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  • = vector of structural displacement DOF
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  • = elastic structural mode α
  • [u] = matrix of elastic structural modes
  • Z= wall acoustic impedance
  • ZΓE= impedance boundary operator for external acoustic fluid
  • δ = dispersion parameter
  • εkh= component of the strain tensor in the structure
  • ω = circular frequency in rad/s
  • ρ0= mass density of the internal acoustic fluid
  • ρE= mass density of the external acoustic fluid
  • ρS= mass density of the structure
  • σ = stress tensor in the structure
  • σij= component of the stress tensor in the structure
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  • = component of the elastic stress tensor in the structure
  • τ = damping coefficient for the internal acoustic fluid
  • ∂Ω= boundary of Ω
  • ∂ΩE= boundary of ΩEequal to ΓE
  • ∂ΩS= boundary of Ωs
  • Γ = coupling interface between the structure and the internal acoustic fluid
  • ΓE= coupling interface between the structure and the external acoustic fluid
  • ΓZ= coupling interface between the structure and the internal acoustic fluid with acoustical properties
  • Ω = internal acoustic fluid domain
  • Ωi=
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  • (ΩE⋃ΓE)
  • ΩE= external acoustic domain
  • ΩS= structural domain
1. Introduction
The fundamental objective of this paper is to present an advanced computational method for the prediction of the responses in the low-and medium-frequency domains of general linear dissipative structural- acoustic and fluid-structure systems. The system under consideration is constituted of a deformable dissipative structure and it is coupled with an internal dissipative acoustic fluid which includes wall acoustic impedances. The system is surrounded by an infinite acoustic fluid and it is submitted to a given internal and external acoustic sources and to the prescribed mechanical forces.
Instead of presenting an exhaustive review of such a problem in this introductory section, we have preferred to move on to the review discussions in each relevant section.
Concerning the appropriate formulations for computing the elastic, acoustic and elastoacoustic modes of the associated conservative fluid-structure system, including substructuring techniques, for the construction of the reduced-order computational models in fluid-structure interaction and for structural-acoustic systems, refer to Ref. [1 - 5] . For the dissipative complex systems, readers can find out the details of the basic formulations in Ref. [3] .
In this paper, the proposed formulation that corresponds to new extensions and complements with respect to the state-of-the-art can be used for the development of a new generation of computational software in particular to the context of parallel computers. We present here an advanced computational formulation. This is based on an efficient reduced-order model in the frequency domain and for this all the required modeling aspects for the analysis of the medium-frequency domain have been taken into account. To be more precise, we have introduced a viscoelastic modeling for the structure, an appropriate dissipative model for the internal acoustic fluid that includes wall acoustic impedance and finally, a global model of uncertainty. It should be noted that model uncertainties must be absolutely taken into account in the computational models of complex vibroacoustic systems in order to improve the prediction of responses in the medium-frequency range. The reduced-order computational model is constructed by using finite element discretization for the structure and for the internal acoustic fluid.
The external acoustic fluid is treated by using an approximate boundary element method in the frequency domain.
  • The sections of the paper are:
  • 1. Introduction
  • 2. Statement of the problem in the frequency domain
  • 3. External inviscid acoustic fluid equations
  • 4. Internal dissipative acoustic fluid equations
  • 5. Structure equations
  • 6. Boundary value problem in terms of {u, p}
  • 7. Computational model
  • 8. Reduced-order computational model
  • 9. Uncertainty quantification
  • 10. Symmetric boundary element method without spurious frequencies for the external acoustic fluid
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Configuration of the system
  • 11. Conclusion
  • References are given at the end of the paper.
2. Statement of the Problem in the Frequency Domain
We consider a mechanical system made up of a damped linear elastic free-free structure Ω S that contains a dissipative acoustic fluid (gas or liquid) which occupies a domain Ω. This system is surrounded by an infinite external inviscid acoustic fluid domain Ω E (gas or liquid) (see Fig. 2 ). A part Γ Z of the internal fluid-structure interface is assumed to be dissipative and it is modeled by a wall acoustic local impedance Z. This system is submitted to a given internal acoustic source in the acoustic cavity and to the given mechanical forces that are applied to the structure. In the infinite external acoustic fluid domain, external acoustic sources are given. It is assumed that the external forces are in equilibrium.
We are interested in the responses in the low-and medium-frequency domains for the displacement field in the structure, the pressure field in the acoustic cavity and the pressure fields on the external fluid-structure interface and also in the external acoustic fluid (near and far fields). It is now well established that the predictions in the medium-frequency domain must be improved by taking into account both the system-parameter uncertainties and the model uncertainties that are induced by modeling errors. Such aspects will be considered in the last section of the paper, which is devoted
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Configuration of the structural-acoustic system for a liquid with free surface.
to Uncertainty Quantification (UQ) in structural acoustics and in fluid-structure interaction.
- 2.1 Main notations
The physical space
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is referred to a cartesian reference system and we denote the generic point of
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by x = ( x 1 , x 2 , x 3 ). For any function f ( x ), the notation f , j denotes the partial derivative with respect to xj . We also use the classical convention for summations over repeated Latin indices but not over Greek indices. As explained earlier, we are interested in the vibration problems that are formulated in the frequency domain for structural- acoustic and fluid-structure interaction systems. Therefore, we introduce the Fourier transform for the various quantities involved. For instance, for the displacement field u , the stress tensor σ ij and the strain tensor ε ij of the structure, we will use the following simplified notation consisting in using the same symbol for a quantity and its Fourier transform. We then have,
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in which the circular frequency ω is real. Nevertheless, for other quantities some exceptions to this rule are done and in such a case, the Fourier transform of a function f will be noted
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- 2.2 Geometry -Mechanical and acoustical hypotheses Given loadings
The coupled system is assumed to be in linear vibrations around a static equilibrium state and this is taken as a natural state at rest.
Structure Ω S . In general, a complex structure is composed of a main part called the master structure . It is defined as the “primary” structure and it is accessible to conventional modeling which includes uncertainties modeling. A secondary part called as the fuzzy substructure is related to the structural complexity and it includes for example many equipment units that are attached to the master structure. In the present paper, we will not consider fuzzy substructures and this concerns the fuzzy structure theory, refer to Ref. [6 , 7] , to Chapter 15 of Ref. [3] for a synthesis, and to Ref. [8] for the extension of the theory to uncertain complex vibroacoustic system with fuzzy interface modeling. Consequently, the so-called “master structure” will be simply called here as “structure”
The structure at the equilibrium occupies the three-dimensional bounded domain Ω S with a boundary ∂Ω S . This is made up of a part Γ E which is the coupling interface between the structure and the external acoustic fluid, a part Γ which is a coupling interface between the structure and the internal acoustic fluid. Finally, the part Γ Z is another part of the coupling interface between the structure and the internal acoustic fluid with acoustical properties. The structure is assumed to be free (free-free structure), i.e. not fixed on any part of the boundary ∂Ω S . The outward unit normal to ∂Ω S is denoted as
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(see Fig. 2 ). The displacement field in Ω S is denoted by u ( x , ω) = ( u 1 ( x , ω), u 2 ( x , ω), u 3 ( x , ω)). A surface force field G ( x , ω) = ( G 1 ( x , ω), G 2 ( x , ω), G 3 ( x , ω)) is given on ∂Ω S and a body force field g ( x , ω) = ( g 1 ( x , ω), g 2 ( x , ω), g 3 ( x , ω)) is given in Ω S . The structure is a dissipative medium whose viscoelastic constitutive equation is defined in Section 5.2.
Internal dissipative acoustic fluid Ω. Let Ω be the internal bounded domain that is filled with a dissipative acoustic fluid (gas or liquid) as described in Section 4. The boundary ∂Ω of Ω is Γ⋃Γ Z . The outward unit normal to ∂Ω is denoted as n = ( n 1 , n 2 , n 3 ) and we have n = − n S on ∂Ω (see Fig. 2 ). Part Γ Z of the boundary has acoustical properties that are modeled by wall acoustic impedance Z ( x , ω)and this satisfies the hypotheses defined in Section 4.2. We denote the pressure field in Ω as p( x , ω) and the velocity field as v ( x , ω). We assume that there is no Dirichlet boundary condition on any part of ∂Ω. An acoustic source density Q ( x , ω) is given inside Ω.
External inviscid acoustic fluid Ω E . The structure is surrounded by an external inviscid acoustic fluid (gas or liquid) and it is as described in Section 10. The fluid occupies the infinite three-dimensional domain Ω E whose boundary ∂Ω E is Γ E . We introduce the bounded open domain Ω i which is defined by
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Note that in general, Ω i does not coincide with the internal acoustic cavity Ω. The boundary ∂Ω i of Ω i is then Γ E . The outward unit normal to ∂Ω i is n S and it is defined above (see Fig. 2 ). We denote the pressure field in Ω E as p E ( x , ω). We assume that there is no Dirichlet boundary condition on any part of Γ E . An acoustic source density Q E ( x , ω) is given in Ω E . This acoustic source density induces a pressure field p given (ω) on Γ E and it is defined in Section 10. For the sake of brevity, we do not consider the case of an incident plane wave here and for this case we refer the reader to Ref. [3] .
3. External Inviscid Acoustic Fluid Equations
An inviscid acoustic fluid occupies an infinite domain Ω E and it is described by the acoustic pressure field p E ( x , ω)at point x of Ω E and at circular frequency ω. Let ρ E be the constant mass density of an external acoustic fluid at equilibrium. Let, cE be the constant speed of sound in the external acoustic fluid at equilibrium and let, k = ω/c E be the wave number at frequency ω. The pressure is then the solution of the classical exterior Neumann problem that is related to the Helmholtz equation with a source term,
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with R = || x || → +∞, where ∂ / ∂ R is the derivative in the radial direction and u · n S is the normal displacement field on Γ E that is induced by the deformation of the structure. Equation (7) corresponds to the outward Sommerfeld radiation condition at infinity. In Section 10, it is proven that the value p E | Γ E of the pressure field pE on the external fluid-structure interface Γ E is related to pgiven | Γ E and to u by Eq. (141),
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in which the different quantities are defined in Section 10. This is a self-contained section that describes the computational modeling of the external inviscid acoustic fluid by an appropriate boundary element method. It should be noted that in Eq. (8), the pressure field p E | Γ E (ω) is related to the value of the normal displacement field u (ω)· n S on the external fluid-structure interface Γ E through an operator Z Γ E (ω).
4. Internal Dissipative Acoustic Fluid Equations
- 4.1 Internal dissipative acoustic fluid equations in the frequency domain
The fluid is assumed to be homogeneous, compressible and dissipative. In the reference configuration, the fluid is at rest. The fluid is either a gas or a liquid and the gravity effects are neglected (see Ref. [9] to take into account both gravity and compressibility effects for an inviscid internal fluid). Such a fluid is called as a dissipative acoustic fluid . Generally, there are two main physical dissipations. The first one is an internal acoustic dissipation inside the cavity. This is due to the viscosity and the thermal conduction of the fluid. These dissipation mechanisms are assumed to be small. In the model proposed, we consider only the dissipation that is due to the viscosity. This correction introduces an additional dissipative term in the Helmholtz equation without the modification of the conservative part. The second one is the dissipation that is generated inside the “wall viscothermal boundary layer” of the cavity and it is neglected here. We then, consider only the acoustic mode (irrotational motion) that is predominant in the volume. The vorticity and entropy modes which mainly play a role in the “wall viscothermal boundary layer” are not modeled. For additional details concerning dissipation in acoustic fluids, refer to Ref. [10 - 13] .
The dissipation due to thermal conduction is neglected and the motions are assumed to be irrotational. Let, ρ 0 be the mass density and c 0 be the constant speed of sound in the fluid at equilibrium in the reference configuration Ω. We have (see the details in Ref. [3] ),
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τ is given by,
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The constant η is the dynamic viscosity, v = η 0 is the kinematic viscosity and ζ is the second viscosity which can depend on ω. Therefore, τ can depend on the frequency ω. In order to simplify the notation, we write τ instead of τ (ω). Eliminating v between Eqs. (9) and (10), then dividing by ρ 0 , yields the Helmholtz equation with a dissipative term and a source term,
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Taking τ = 0 and Q = 0 in Eq. (12) yields the usual Helmholtz equation for wave propagation in inviscid acoustic fluid.
- 4.2 Boundary conditions in the frequency domain
(i) Neumann boundary condition on Γ. By using Eq. (10) and v · n = i ω u · n on Γ yields the following Neumann boundary condition,
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(ii) Neumann boundary condition on Γ Z with wall acoustic impedance. The part Γ Z of the boundary ∂Ω has acoustical properties that are modeled by a wall acoustic impedance Z ( x , ω) which is defined for x ∈ Γ Z , with complex values. The wall impedance boundary condition on Γ Z is written as,
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Wall acoustic impedance Z ( x , ω) must satisfy appropriate conditions in order to ensure that the problem is stated correctly (see Ref. [3] for a general formulation and see Ref. [14] for a simplified model of the Voigt type with an internal inviscid fluid). By using Eq. (10), v · n = i ω u · n and Eq. (14) on Γ, yields the following Neumann boundary condition with a wall acoustic impedance,
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- 4.3 Case of a free surface for a liquid
Cavity Ω is partially filled with a liquid (dissipative acoustic fluid) that occupies the domain Ω L . It is assumed that the complementary part Ω/Ω L is a vacuum domain. The boundary, ∂Ω L of Ω L is constituted of three boundaries namely Γ Z , Γ 0 that corresponds to the free surface of the liquid and a part Γ L of Γ. The Neumann boundary condition on Γ L is given by Eq. (13), on Γ Z which is given by Eq. (15). By neglecting the gravity effects, the following Dirichlet condition is written on the free surface,
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5. Structure Equations
- 5.1 Structure equations in the frequency domain
The equation of the structure that occupies the domain Ω S is written as,
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in which ρ s ( x ) is the mass density of the structure. The constitutive equation (linear viscoelastic model, see Section 5.2, Eq. (31)) is such that the symmetric stress tensor σ ij is written as,
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in which the symmetric strain tensor εkh ( u ) is such that
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and where the tensors aijkh (ω) and bijkh (ω) depend on ω (see Section 5.2). The boundary condition on the fluid-structure external interface Γ E is such that
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in which p E | Γ E is given by Eq. (8) and it yields
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As n S = − n , the boundary condition on Γ∪Γ Z is written as,
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in which p is the internal acoustic pressure field that is defined in Section 4.
- 5.2 Viscoelastic constitutive equation
In dynamics, the structure must always be modeled as a dissipative continuum. For the conservative part of the structure, we use the linear elasticity theory which allows the structural modes to be introduced. This was justified by the fact that in the low-frequency range, the conservative part of the structure can be modeled as an elastic continuum. In this section, we introduce damping models for the structure that is based on the general linear theory of viscoelasticity and it is presented in Ref. [15] (see also Ref. [16 , 17] ). Complementary developments are presented with respect to the viscoelastic constitutive equation detailed in Ref. [3] .
In this section, x is fixed in Ω S , and we rewrite the stress tensor σij ( x , t ) as σij ( t ), the strain tensor εij ( x , t ) as εij ( t ) and its time derivative
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as
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Constitutive equation in the time domain. The stress tensor σij ( t ) is written as,
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Where, σij ( t ) = 0 and ε ( t ) = 0 for t ≤ 0. The real functions Gijkh ( x , t ) are denoted as Gijkh ( t ) and they are called as the relaxation functions. The tensor Gijkh ( t ) (and thus
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has the usual property of symmetry and Gijkh ( 0 ), which is called as the initial elasticity tensor is positive definite. The relaxation functions are defined on [0, +∞[ and are differentiable with respect to t on ]0, +∞[. Their derivatives are denoted as
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and are assumed to be integrable on [0, +∞[. Functions Gijkh ( t ) can be written as,
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Therefore, the limit of Gijkh ( t ), denoted as Gijkh (∞), is finite as t and it tends to +∞,
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The tensor Gijkh (∞), called as the equilibrium modulus at x , is symmetric and positive definite. It corresponds to the usual elasticity coefficients of the elastic material for a static deformation. In effect, the static equilibrium state is obtained for t and it tends to infinity.
For all x that is fixed in Ω S , we introduce the real functions t → gijkh ( x , t ), denoted as gijkh ( t ), such that
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As gijkh ( t ) = 0 for t < 0, we deduce that gijkh ( t ) is a causal function.
By using Eq. (26), Eq. (23) can be rewritten as,
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It should be noted that Eq. (27) corresponds to the most general formulation in the time domain within the framework of the linear theory of viscoelasticity. The usual approach which consists in modeling the constitutive equation in time domain by a linear differential equation in σ ( t ) and ε ( t ) (see for instance Ref. [15 , 18] ) and this corresponds to a particular case which is an approximation of the general Eq. (27). An alternative approximation of Eq. (27) consists of representing the integral operator by a differential operator that acts on additional hidden variables. This type of approximation can efficiently be described by using fractional derivative operators (see for instance Ref. [19 , 20] ).
Constitutive equation in the frequency domain. The general constitutive equation in the frequency domain is written as,
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in which,
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Equation (28) can then be rewritten as,
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Tensors aijkh (ω) and bijkh (ω) must satisfy the symmetry properties
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and the positive-definiteness properties, i.e., for all the second-order real symmetric tensors Xij ,
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in which the positive constants ca (ω) and cb (ω) are such that ca (ω) ≥ c 0 > 0 and cb (ω) ≥ c 0 > 0 where c 0 is a positive real constant that is independent of ω.
As gijkh ( t ) is an integrable function on ]−∞, +∞[, its Fourier transform
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is defined by,
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and it is a complex function which is continuous on ]−∞, +∞[ and such that
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The real part
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and the imaginary part
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of
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are even and odd functions. So, it is easy to say that
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and
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We can then deduce that
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We can now take the Fourier transform of Eq. (27) and using Eq. (31) yields the relations,
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Eqs. (37), (39) and (40) yields,
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From Eqs. (31), (41) and (42), we deduce that
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Eq. (43) shows that viscoelastic materials behave elastically at high frequencies with elasticity coefficients that are defined by the initial elasticity tensor Gijkh ( 0 ) that differs from the equilibrium modulus tensor Gijkh (∞) which is written by using Eqs. (25) and (38) as,
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As pointed out before, a positive-definite tensor Gijkh (∞) corresponds to the usual elasticity coefficients of a linear elastic material for a static deformation process. More specifically for ω = 0 by using Eqs. (38) to (40) and Eq. (31) yield,
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in which σ ijkh (0) = {σ ijkh (ω)} ω=0 and ε ijkh (0) = {ε ijkh (ω)} ω=0 where,
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The reader should be aware of the fact that the constitutive equation of an elastic material in a static deformation process is defined by Gijkh (∞) and not by the initial elasticity tensor, Gijkh (0). Referring to Ref. [15 , 21] , it has been proven that Gijkh (0) − Gijkh (∞)is a positive-definite tensor. Consequently,
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is a negative- definite tensor.
As gijkh ( t ) is a causal function, the real part
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and the imaginary part
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of its Fourier transform
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are related by the following relations that involve the Hilbert transform (see Ref. [22 , 23] ),
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in which p.v denotes the Cauchy principal value which is defined as,
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The relations defined by Eqs. (47) and (48) are also called as the Kramers and Kronig relations for the function gijkh ( t ) (see Ref. [24 , 25] ).
LF-range constitutive equation approximation. In the low-frequency range and in most cases, the coefficients aijkh (ω) was given by the linear viscoelastic model. It was defined by Eq. (39) and it is almost frequency independent. In such a case, they can be approximated by aijkh ( ω )≅ aijkh (0) and this is independent of ω(but which depends on x ). It should be noted that this approximation can only be made on a finite interval that corresponds to low-frequency range and it cannot be used in the entire frequency domain as Eqs. (47) and (48) are not satisfied and the integrability property is lost.
MF range constitutive equation. In the medium-frequency range, the previous LF-range constitutive equation approximation is generally invalid and the entire linear viscoelastic theory which is defined by Eq. (31) must be used.
Bibliographical comments concerning expressions of frequency-dependent coefficients. Some algebraic representations of functions aijkh (ω) and bijkh (ω) have been proposed in literature (see for instance Refs. [3 , 15 - 16 , 18 , 20 , 26 - 30] ). Concerning linear hysteretic damping which is correctly written in the present context, refer to Refs. [31 - 32] .
6. Boundary Value Problem in Terms of {u, p}
The boundary value problem in terms of { u , p } is written as follows. For all real ω and for the given G (ω), g (ω), pgiven | Γ E (ω) and Q (ω), we calculate u (ω) and p (ω), such that
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In case of a free surface in the internal acoustic cavity (see Section 4.3), we must add the following boundary condition
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Comments.
  • We are interested in studying the linear vibrations of the coupled system that is around a static equilibrium and this is considered as a natural state at rest (then, the external solid and acoustic forces are assumed to be in equilibrium).
  • Eq. (50) corresponds to the structure equation (see Eqs. (17) and (28)), in which {divσ(u)}i=σij,j(u).
  • Eqs. (51) and (52) are the boundary conditions for the structure (see Eqs. (21) and (22)).
  • Eq. (53) corresponds to the internal dissipative acoustic fluid equation (see Eq. (12)).
  • Finally, Eqs. (54) and (55) are the boundary conditions for the acoustic cavity (see Eqs. (13) and (15)).
  • It is important to note that the external acoustic pressure fieldpEhas been eliminated as a function ofuby using the acoustic impedance boundary operatorZΓE(ω) while the internal acoustic pressure fieldpis kept.
7. Computational Model
The computational model is constructed by using the finite element discretization of the boundary value problem. We also consider a finite element mesh of structure, Ω S and a finite element mesh of internal acoustic fluid Ω. We assume that the two finite element meshes are compatible on an interface Γ⋃Γ Z . The finite element mesh of surface Γ E is the trace of the mesh of Ω S (see Fig. 3 ).
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Example of the structure and internal fluid finite element meshes.
We classically use the finite element method to construct the discretization of the variational formulation of the boundary value problem. This is defined by using Eqs. (50) to (55), with additional boundary condition that is defined by Eq. (56) in the case of a free surface for an internal liquid. For the details that concern with the practical construction of the finite element matrices, refer to Ref. [3] . Let,
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be a complex vector of the n s degrees- of-freedom (DOFs) which are the values of u (ω) at the nodes of the finite element mesh of the domain Ω S . For the internal acoustic fluid, let
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be the complex vectors of n DOFs which are the values of p (ω) at the nodes of a finite element mesh of domain Ω. The finite element method yields the following complex matrix equation,
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in which the complex matrix
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is defined by,
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In Eq. (58), the symmetric ( nS × nS ) complex matrix
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is defined by,
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where,
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and
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are symmetric ( nS × nS ) real matrices which represent the mass matrix, the damping matrix and the stiffness matrix of the structure. Matrix
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is positive and invertible (positive definite). Matrices
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and
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are positive and not invertible (positive semidefinite). This is due to the presence of six rigid body motions since the structure has been considered as a free-free structure. The symmetric ( n × n ) complex matrix
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is defined by,
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Where,
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and
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are symmetric ( n × n ) real matrices. Matrix
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is positive and invertible. Matrices
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and
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are positive and are not invertible with rank n − 1. From Eq. (53), it can easily be deduced that
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in which τ(ω) is defined by Eq. (11). The internal fluid-structure coupling matrix
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is related to the coupling between the structure and the internal fluid on an internal fluid-structure interface. This is a ( nS × n ) real matrix which is only related to the values of
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and
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on the internal fluid-structure interface. The wall acoustic impedance matrix
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is a symmetric ( n × n ) complex matrix that depends on the wall acoustic impedance Z ( x , ω) on Γ Z and this is only related to the values of
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on boundary Γ Z . The boundary element matrix
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which depends on ω/c E , is a symmetric ( nS × nS ) complex matrix and it is only related to the values of
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on the external fluid-structure interface Γ E . This matrix is written as,
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in which [ B ΓE ( ω /c E )] is a full symmetric ( nE × nE ) complex matrix which is defined in Section 10.7. Here,
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is a sparse ( nE × nS ) real matrix that is related to finite element discretization.
8. Reduced-Order Computational Model
The strategy used for the construction of the reduced-order computational model consists in using the projection basis constituted of [3] :
  • the undamped elastic structural modes of the structure in vacuo for which the constitutive equation corresponds to elastic materials (see Eq. (45)), and consequently, the stiffness matrix has to be taken for ω = 0.
  • the undamped acoustic modes of the acoustic cavity is with fixed boundary and without wall acoustic impedance. Two cases must be considered: one for which the internal pressure varies with the variation of the volume of the cavity (a cavity with a sealed wall is called as a closed cavity) and the other one for which the internal pressure does not vary along with the variation of the volume of the cavity (a cavity with a non sealed wall is called as an almost closed cavity).
- 8.1 Computation of the elastic structural modes
This step concerns with the finite element calculation of the undamped elastic structural modes of structure Ω S in vacuo for which the constitutive equation corresponds to elastic materials. By setting λ S = ω 2 , we then have the following classical ( nS × nS ) which is a generalized symmetric real eigenvalue problem
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It can be shown that there is a zero eigenvalue with multiplicity 6 (corresponding to the six rigid body motions) and that there is an increasing sequence of ns − 6 strictly positive eigenvalues (corresponding to the elastic structural modes). Each positive eigenvalue can be a multiple (case of a structure with symmetries),
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Let
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be the eigenvectors (the elastic structural modes) that is associated with
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Let 0 < NS nS − 6. We introduce ( nS × NS ) real matrix of the NS elastic structural modes
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that is associated with the first NS strictly positive eigenvalues,
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One has classical orthogonality properties,
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where, [ MS ] is a diagonal matrix of positive real numbers and [ KS (0)] is a diagonal matrix of eigenvalues such that
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(the eigenfrequencies are,
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- 8.2 Computation of the acoustic modes
This step concerns the finite element calculation of the undamped acoustic modes of a closed (sealed wall) or an almost closed (non sealed wall) acoustic cavity, Ω. By setting λ = ω 2 , we then have the following classical ( n × n ) generalized symmetric real eigenvalue problem
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It can be shown that there is a zero eigenvalue with multiplicity 1 and denoted as λ 0 (corresponding to constant eigenvector denoted as
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). Moreover, there is an increasing sequence of n − 1 strictly positive eigenvalues (corresponding to the acoustic modes) and each positive eigenvalue can be multiple (case of an acoustic cavity with symmetries),
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Let
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be the eigenvectors (the acoustic modes) that is associated with λ 1 , …, λ α , …
  • Closed (sealed wall) acoustic cavity.Let be 0
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  • and of theN− 1 acoustic modes
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  • that is associated with the firstN− 1 strictly positive eigenvalues as,
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  • Almost closed (non sealed wall) acoustic cavity.Let be 0
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  • is associated with the firstNstrictly positive eigenvalues,
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One has classical orthogonality properties,
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where, [ M ] is a diagonal matrix of positive real numbers and [ K ] is a diagonal matrix of eigenvalues such that [ K ] αβ = λ α δ αβ (for non zero eigenvalue, the eigenfrequencies are
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- 8.3 Construction of the reduced-order computational model
The reduced-order computational model, of order NS << ns and N << n , is obtained by projecting Eq. (57) as follows,
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Complex vectors q S (ω) and q (ω) of dimensions NS and N are the solution of the following equation,
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in which the complex matrix [ AFSI (ω)] is defined by,
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In Eq. (76), the symmetric ( NS × NS ) complex matrix [ AS (ω)] is defined by,
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in which [ MS ], [ DS (ω)] and [ KS (ω)] are positive-definite symmetric ( NS × NS ) real matrices such that [ DS ( ω )] = [ u ] T
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and
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The symmetric ( N × N ) complex matrix [ A ( ω )] is defined by,
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Where, [ M ], [ D ( ω )] and [ K ] are symmetric ( N × N ) real matrices. Matrix [ M ] is positive and invertible. Diagonal ( N × N ) real matrix [ D ( ω )] is written as [ D ( ω )] = τ( ω )[ K ] in which τ( ω ) is defined by, Eq. (11). For a closed (sealed wall) acoustic cavity, matrix [ K ] is positive and it is not invertible with rank N − 1, while for an almost closed (non sealed wall) acoustic cavity, matrix [ K ] is positive and invertible. The ( NS × N ) real matrix [ C ] and it is written as,
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