This paper presents an advanced computational method for the prediction of the responses in the frequency domain of general linear dissipative structuralacoustic and fluidstructure systems, in the lowand mediumfrequency 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 reducedorder 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 mediumfrequency 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 stateoftheart are well adapted for the development of a new generation of software, in particular for parallel computers.
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

pEΓE= value of the external acoustic pressure field on ΓE

pgiven= given external acoustic pressure field

pgivenΓE= 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

= 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

= dynamical matrix for the internal acoustic fluid

[ABEM] = reduced matrix of the impedance boundary operator for the external acoustic fluid

= matrix of the impedance boundary operator for the external acoustic fluid

[AFSI] = reduced dynamical matrix for the fluidstructure coupled system

[AFSI] = random reduced dynamical matrix for the fluidstructure coupled system

= dynamical matrix for the fluidstructure cou

[AS] = reduced dynamical matrix for the structure

[AS] = random reduced dynamical matrix for the structure

= dynamical matrix for the structure

[AZ] = reduced dynamical matrix associated with the wall acoustic impedance

= 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

= 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

= damping matrix for the internal acoustic fluid

[DS] = reduced damping matrix for the structure

[DS] = random reduced damping matrix for the structure

= damping matrix for the structure

DOF = degrees of freedom

= vector of discretized acoustic forces

= 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

= “stiffness” matrix for the internal acoustic fluid

[KS] = reduced stiffness matrix for the structure

[KS] = random reduced stiffness matrix for the structure

= stiffness matrix for the structure

[M] = reduced “mass” matrix for the internal acoustic fluid

[M] = random reduced “mass” matrix for the internal acoustic fluid

= “mass” matrix for the internal acoustic fluid

[MS] = reduced mass matrix for the structure

[MS] = random reduced mass matrix for the structure

= mass matrix for the structure

= 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

= vector of internal acoustic pressure DOF

U= random vector of structural displacement DOF

= vector of structural displacement DOF

= 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

= 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=

(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 lowand mediumfrequency domains of general linear dissipative structural acoustic and fluidstructure 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 fluidstructure system, including substructuring techniques, for the construction of the reducedorder computational models in fluidstructure interaction and for structuralacoustic 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 stateoftheart 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 reducedorder model in the frequency domain and for this all the required modeling aspects for the analysis of the mediumfrequency 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 mediumfrequency range. The reducedorder 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. Reducedorder computational model

9. Uncertainty quantification

10. Symmetric boundary element method without spurious frequencies for the external acoustic fluid
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 freefree 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 fluidstructure 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
lowand mediumfrequency
domains for the displacement field in the structure, the pressure field in the acoustic cavity and the pressure fields on the external fluidstructure interface and also in the external acoustic fluid (near and far fields). It is now well established that the predictions in the mediumfrequency domain must be improved by taking into account both the systemparameter 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
Configuration of the structuralacoustic system for a liquid with free surface.
to
Uncertainty Quantification
(UQ) in structural acoustics and in fluidstructure interaction.
 2.1 Main notations
The physical space
is referred to a cartesian reference system and we denote the generic point of
by
x
= (
x
_{1}
,
x
_{2}
,
x
_{3}
). For any function
f
(
x
), the notation
f
,
_{j}
denotes the partial derivative with respect to
x_{j}
. 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 fluidstructure 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,
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
 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 socalled “master structure” will be simply called here as “structure”
The structure at the equilibrium occupies the threedimensional 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 (freefree structure), i.e. not fixed on any part of the boundary ∂Ω
_{S}
. The outward unit normal to ∂Ω
_{S}
is denoted as
(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 threedimensional domain Ω
_{E}
whose boundary ∂Ω
_{E}
is Γ
_{E}
. We introduce the bounded open domain Ω
_{i}
which is defined by
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,
c_{E}
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,
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
p_{E}
on the external fluidstructure interface Γ
_{E}
is related to
p_{given}

_{Γ}
_{E}
and to
u
by Eq. (141),
in which the different quantities are defined in Section 10. This is a selfcontained 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 fluidstructure 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]
),
τ is given by,
The constant
η
is the dynamic viscosity,
v
=
η
/ρ
_{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,
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,
(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,
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,
 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,
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,
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,
in which the symmetric strain tensor
ε_{kh}
(
u
) is such that
and where the tensors
a_{ijkh}
(ω) and
b_{ijkh}
(ω) depend on ω (see Section 5.2). The boundary condition on the fluidstructure external interface Γ
_{E}
is such that
in which
p
_{E}

_{Γ}
_{E}
is given by Eq. (8) and it yields
As
n
^{S}
= −
n
, the boundary condition on Γ∪Γ
_{Z}
is written as,
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 lowfrequency 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
σ_{ij}
(
x
,
t
) as
σ_{ij}
(
t
), the strain tensor
ε_{ij}
(
x
,
t
) as
ε_{ij}
(
t
) and its time derivative
as
Constitutive equation in the time domain. The stress tensor
σ_{ij}
(
t
) is written as,
Where,
σ_{ij}
(
t
) = 0 and
ε
(
t
) = 0 for t ≤ 0. The real functions
G_{ijkh}
(
x
,
t
) are denoted as
G_{ijkh}
(
t
) and they are called as the relaxation functions. The tensor
G_{ijkh}
(
t
) (and thus
has the usual property of symmetry and
G_{ijkh}
(
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
and are assumed to be integrable on [0, +∞[. Functions
G_{ijkh}
(
t
) can be written as,
Therefore, the limit of
G_{ijkh}
(
t
), denoted as
G_{ijkh}
(∞), is finite as
t
and it tends to +∞,
The tensor
G_{ijkh}
(∞), 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 →
g_{ijkh}
(
x
,
t
), denoted as
g_{ijkh}
(
t
), such that
As
g_{ijkh}
(
t
) = 0 for
t
< 0, we deduce that
g_{ijkh}
(
t
) is a causal function.
By using Eq. (26), Eq. (23) can be rewritten as,
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,
in which,
Equation (28) can then be rewritten as,
Tensors
a_{ijkh}
(ω) and
b_{ijkh}
(ω) must satisfy the symmetry properties
and the positivedefiniteness properties, i.e., for all the secondorder real symmetric tensors
X_{ij}
,
in which the positive constants
c_{a}
(ω) and
c_{b}
(ω) are such that
c_{a}
(ω) ≥
c
_{0}
> 0 and
c_{b}
(ω) ≥
c
_{0}
> 0 where
c
_{0}
is a positive real constant that is independent of ω.
As
g_{ijkh}
(
t
) is an integrable function on ]−∞, +∞[, its Fourier transform
is defined by,
and it is a complex function which is continuous on ]−∞, +∞[ and such that
The real part
and the imaginary part
of
are even and odd functions. So, it is easy to say that
and
We can then deduce that
We can now take the Fourier transform of Eq. (27) and using Eq. (31) yields the relations,
Eqs. (37), (39) and (40) yields,
From Eqs. (31), (41) and (42), we deduce that
Eq. (43) shows that viscoelastic materials behave elastically at high frequencies with elasticity coefficients that are defined by the initial elasticity tensor
G_{ijkh}
(
0
) that differs from the equilibrium modulus tensor
G_{ijkh}
(∞) which is written by using Eqs. (25) and (38) as,
As pointed out before, a positivedefinite tensor
G_{ijkh}
(∞) 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,
in which σ
_{ijkh}
(0) = {σ
_{ijkh}
(ω)}
_{ω=0}
and ε
_{ijkh}
(0) = {ε
_{ijkh}
(ω)}
_{ω=0}
where,
The reader should be aware of the fact that the constitutive equation of an elastic material in a static deformation process is defined by
G_{ijkh}
(∞) and not by the initial elasticity tensor,
G_{ijkh}
(0). Referring to Ref.
[15
,
21]
, it has been proven that
G_{ijkh}
(0) −
G_{ijkh}
(∞)is a positivedefinite tensor. Consequently,
is a negative definite tensor.
As
g_{ijkh}
(
t
) is a causal function, the real part
and the imaginary part
of its Fourier transform
are related by the following relations that involve the Hilbert transform (see Ref.
[22
,
23]
),
in which p.v denotes the Cauchy principal value which is defined as,
The relations defined by Eqs. (47) and (48) are also called as the Kramers and Kronig relations for the function
g_{ijkh}
(
t
) (see Ref.
[24
,
25]
).
LFrange constitutive equation approximation.
In the lowfrequency range and in most cases, the coefficients
a_{ijkh}
(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
a_{ijkh}
(
ω
)≅
a_{ijkh}
(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 lowfrequency 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 mediumfrequency range, the previous LFrange 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 frequencydependent coefficients.
Some algebraic representations of functions
a_{ijkh}
(ω) and
b_{ijkh}
(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),
p_{given}

_{Γ}
_{E}
(ω) and
Q
(ω), we calculate
u
(ω) and
p
(ω), such that
In case of a free surface in the internal acoustic cavity (see Section 4.3), we must add the following boundary condition
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
).
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,
be a complex vector of the
n
_{s}
degrees offreedom (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
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,
in which the complex matrix
is defined by,
In Eq. (58), the symmetric (
n_{S}
×
n_{S}
) complex matrix
is defined by,
where,
and
are symmetric (
n_{S}
×
n_{S}
) real matrices which represent the mass matrix, the damping matrix and the stiffness matrix of the structure. Matrix
is positive and invertible (positive definite). Matrices
and
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 freefree structure. The symmetric (
n
×
n
) complex matrix
is defined by,
Where,
and
are symmetric (
n
×
n
) real matrices. Matrix
is positive and invertible. Matrices
and
are positive and are not invertible with rank
n
− 1. From Eq. (53), it can easily be deduced that
in which τ(ω) is defined by Eq. (11). The internal fluidstructure coupling matrix
is related to the coupling between the structure and the internal fluid on an internal fluidstructure interface. This is a (
n_{S}
×
n
) real matrix which is only related to the values of
and
on the internal fluidstructure interface. The wall acoustic impedance matrix
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
on boundary Γ
_{Z}
. The boundary element matrix
which depends on ω/c
_{E}
, is a symmetric (
n_{S}
×
n_{S}
) complex matrix and it is only related to the values of
on the external fluidstructure interface Γ
_{E}
. This matrix is written as,
in which [
B
_{ΓE}
(
ω
/c
_{E}
)] is a full symmetric (
n_{E}
×
n_{E}
) complex matrix which is defined in Section 10.7. Here,
is a sparse (
n_{E}
×
n_{S}
) real matrix that is related to finite element discretization.
8. ReducedOrder Computational Model
The strategy used for the construction of the reducedorder 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 (
n_{S}
×
n_{S}
) which is a generalized symmetric real eigenvalue problem
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
n_{s}
− 6 strictly positive eigenvalues (corresponding to the elastic structural modes). Each positive eigenvalue can be a multiple (case of a structure with symmetries),
Let
be the eigenvectors (the elastic structural modes) that is associated with
Let 0 <
N_{S}
≤
n_{S}
− 6. We introduce (
n_{S}
×
N_{S}
) real matrix of the
N_{S}
elastic structural modes
that is associated with the first
N_{S}
strictly positive eigenvalues,
One has classical orthogonality properties,
where, [
M^{S}
] is a diagonal matrix of positive real numbers and [
K^{S}
(0)] is a diagonal matrix of eigenvalues such that
(the eigenfrequencies are,
 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
It can be shown that there is a zero eigenvalue with multiplicity 1 and denoted as λ
_{0}
(corresponding to constant eigenvector denoted as
). 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),
Let
be the eigenvectors (the acoustic modes) that is associated with λ
_{1}
, …, λ
_{α}
, …

Closed (sealed wall) acoustic cavity.Let be 0

and of theN− 1 acoustic modes

that is associated with the firstN− 1 strictly positive eigenvalues as,

Almost closed (non sealed wall) acoustic cavity.Let be 0

is associated with the firstNstrictly positive eigenvalues,
One has classical orthogonality properties,
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
 8.3 Construction of the reducedorder computational model
The reducedorder computational model, of order
N_{S}
<<
n_{s}
and
N
<<
n
, is obtained by projecting Eq. (57) as follows,
Complex vectors
q
^{S}
(ω) and
q
(ω) of dimensions
N_{S}
and
N
are the solution of the following equation,
in which the complex matrix [
A_{FSI}
(ω)] is defined by,
In Eq. (76), the symmetric (
N_{S}
×
N_{S}
) complex matrix [
A^{S}
(ω)] is defined by,
in which [
M^{S}
], [
D^{S}
(ω)] and [
K^{S}
(ω)] are positivedefinite symmetric (
N_{S}
×
N_{S}
) real matrices such that [
D^{S}
(
ω
)] = [
u
]
^{T}
and
The symmetric (
N
×
N
) complex matrix [
A
(
ω
)] is defined by,
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 (
N_{S}
×
N
) real matrix [
C
] and it is written as,