Advanced Computational Dissipative Structural Acoustics and FluidStructure Interaction in Lowand MediumFrequency Domains. ReducedOrder Models and Uncertainty Quantification
 Author: Ohayon R., Soize C.
 Organization: Ohayon R.; Soize C.
 Publish: International Journal Aeronautical and Space Sciences Volume 13, Issue2, p127~153, 30 June 2012

ABSTRACT
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.

KEYWORD
Computational mechanics , Structural acoustics , Vibroacoustic , Fluidstructure interaction , Uncertainty quantification , Reducedorder model , Medium frequency , Low frequency , Dissipative system , Viscoelasticity , Wall acoustic impedance , Finite element discretization , Boundary element method

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 vector n
n = outward unit normal to ∂Ω
nsj = component of vector nS
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 point x
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 ΩE equal 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. [15]. 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
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 devotedto
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 functionf (x ), the notationf ,_{j} denotes the partial derivative with respect tox_{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 fieldu , 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 noted2.2 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 themaster structure . It is defined as the “primary” structure and it is accessible to conventional modeling which includesuncertainties modeling. A secondary part called as thefuzzy 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 byu (x , ω) = (u _{1}(x , ω),u _{2}(x , ω),u _{3}(x , ω)). A surface force fieldG (x , ω) = (G _{1}(x , ω),G _{2}(x , ω),G _{3}(x , ω)) is given on ∂Ω_{S} and a body force fieldg (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 asn = (n _{1},n _{2},n _{3}) and we haven = ？n ^{S} on ∂Ω (see Fig. 2). Part Γ_{Z} of the boundary has acoustical properties that are modeled by wall acoustic impedanceZ (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 asv (x , ω). We assume that there is no Dirichlet boundary condition on any part of ∂Ω. An acoustic source densityQ (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 byNote 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} isn ^{S} and it is defined above (see Fig. 2). We denote the pressure field in Ω_{E} asp _{E} (x , ω). We assume that there is no Dirichlet boundary condition on any part of Γ_{E} . An acoustic source densityQ _{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 pointx 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 andu ·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 valuep _{E} _{Γ}_{E} of the pressure fieldp_{E} on the external fluidstructure interface Γ_{E} is related top_{given} _{Γ}_{E} and tou 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 fieldu (ω)·n ^{S} on the external fluidstructure interface Γ_{E} through an operatorZ _{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. [1013].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 andζ 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 ofτ (ω). Eliminatingv 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 andQ = 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 ωu ·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 impedanceZ (x , ω) which is defined forx ∈ 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 ωu ·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 thatand where the tensors
a_{ijkh} (ω) andb_{ijkh} (ω) depend on ω (see Section 5.2). The boundary condition on the fluidstructure external interface Γ_{E} is such thatin which
p _{E} _{Γ}_{E} is given by Eq. (8) and it yieldsAs
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 tensorσ_{ij} (x ,t ) asσ_{ij} (t ), the strain tensorε_{ij} (x ,t ) asε_{ij} (t ) and its time derivativeas
Constitutive equation in the time domain. The stress tensor
σ_{ij} (t ) is written as,Where,
σ_{ij} (t ) = 0 andε (t ) = 0 for t ≤ 0. The real functionsG_{ijkh} (x ,t ) are denoted asG_{ijkh} (t ) and they are called as the relaxation functions. The tensorG_{ijkh} (t ) (and thushas 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 tot on ]0, +∞[. Their derivatives are denoted asand are assumed to be integrable on [0, +∞[. Functions
G_{ijkh} (t ) can be written as,Therefore, the limit of
G_{ijkh} (t ), denoted asG_{ijkh} (∞), is finite ast and it tends to +∞,The tensor
G_{ijkh} (∞), called as the equilibrium modulus atx , 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 fort 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 asg_{ijkh} (t ), such thatAs
g_{ijkh} (t ) = 0 fort < 0, we deduce thatg_{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 ) andε (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) andb_{ijkh} (ω) must satisfy the symmetry propertiesand the positivedefiniteness properties, i.e., for all the secondorder real symmetric tensors
X_{ij} ,in which the positive constants
c_{a} (ω) andc_{b} (ω) are such thatc_{a} (ω) ≥c _{0} > 0 andc_{b} (ω) ≥c _{0} > 0 wherec _{0} is a positive real constant that is independent of ω.As
g_{ijkh} (t ) is an integrable function on ]？∞, +∞[, its Fourier transformis 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 tensorG_{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 thatG_{ijkh} (0) ？G_{ijkh} (∞)is a positivedefinite tensor. Consequently,is a negative definite tensor.
As
g_{ijkh} (t ) is a causal function, the real partand 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 coefficientsa_{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 bya_{ijkh} (ω )？a_{ijkh} (0) and this is independent of ω(but which depends onx ). 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 functionsa_{ijkh} (ω) andb_{ijkh} (ω) have been proposed in literature (see for instance Refs. [3,1516,18,20,2630] ). Concerning linear hysteretic damping which is correctly written in the present context, refer to Refs. [3132].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 givenG (ω),g (ω),p_{given} _{Γ}_{E} (ω) andQ (ω), we calculateu (ω) andp (ω), such thatIn 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 field pE has been eliminated as a function of u by using the acoustic impedance boundary operator ZΓE(ω) while the internal acoustic pressure field p is 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).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 ofu (ω) at the nodes of the finite element mesh of the domain Ω_{S} . For the internal acoustic fluid, letbe the complex vectors of
n DOFs which are the values ofp (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 matrixis 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. Matrixis 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 matrixis defined by,
Where,
and
are symmetric (
n ×n ) real matrices. Matrixis positive and invertible. Matrices
and
are positive and are not invertible with rank
n ？ 1. From Eq. (53), it can easily be deduced thatin 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 ofand
on the internal fluidstructure interface. The wall acoustic impedance matrix
is a symmetric (
n ×n ) complex matrix that depends on the wall acoustic impedanceZ (x , ω) on Γ_{Z} and this is only related to the values ofon boundary Γ
_{Z} . The boundary element matrixwhich depends on ω/c
_{E} , is a symmetric (n_{S} ×n_{S} ) complex matrix and it is only related to the values ofon 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 problemIt 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 theN_{S} elastic structural modesthat 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 problemIt 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 < N ≤ n. We introduce the (n × N) real matrix of the constant eigenvector
and of the N ？ 1 acoustic modes
that is associated with the first N ？ 1 strictly positive eigenvalues as,
Almost closed (non sealed wall) acoustic cavity. Let be 0 < N ≤ n ？ 1. We introduce the (n × N) real matrix of N acoustic modes
is associated with the first N strictly 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 are8.3 Construction of the reducedorder computational model
The reducedorder computational model, of order
N_{S} <<n_{s} andN <<n , is obtained by projecting Eq. (57) as follows,Complex vectors
q ^{S}(ω) andq (ω) of dimensionsN_{S} andN 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 rankN ？ 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,Symmetric (
N ×N ) complex matrix [A^{Z} (ω )] is such thatand finally, the symmetric (
N_{S} ×N_{S} ) complex matrix [A_{BEM} (ω /c_{E} )] is given by,The given forces are written as
and
9. Uncertainty Quantification
9.1 Short overview on uncertainty quantification
In this section, we summarize the fundamental concepts that are related to uncertainties and their stochastic modeling in computational structuralacoustic models (extracted from Refs. [3334]).
9.1.1 Uncertainty and variability
The
designed structuralacoustic system is used to manufacture thereal system and to construct the nominal computational model (also called themean computational model or sometimes, the mean model) by using a mathematicalmechanical modeling process for which the main objective is the prediction of the responses of the real system. The real system can exhibit variability in its responses due to fluctuations in the manufacturing process. This is due to small variations of the configuration around a nominal configuration that is associated with the designed structuralacoustic system. The mean computational model which results from a mathematical mechanical modeling process of the designed structural acoustic system, has parameters (such as geometry, mechanical properties, boundary conditions) and this can be uncertain (for example, parameters related to the structure, the internal acoustic fluid, the wall acoustic impedance). In this case,there are
uncertainties on the computational model parameters . On the other hand, the modeling process induces some modeling errors that are defined as themodel uncertainties . Fig 4 summarizes two types of uncertainties in a computational model and the variabilities of a real system. It is important to take into account both the uncertainties on the computational model parameters and the model uncertainties to improve the predictions in order to use such a computational model to carry out robust optimization, robust design and robust updating with respect to uncertainties. Today, it is well understood that, as soon as the probability theory can be used, then the stochastic approach of uncertainties is the most powerful, efficient and effective tool for modeling and for solving direct problem and inverse problem related to the identification. The developments are presented below and they are carried out within the framework of the probability theory.9.1.2 Types of approach for stochastic modeling of uncertainties
The
parametric probabilistic approach consists in modeling theuncertain parameters of the computational model by random variables and then, in constructing the stochastic model of these random variables by using the available information. Such an approach is very well adapted and very efficient to take into account the uncertainties in the computational model parameters. Many works have been published and a stateoftheart can be found out, for instance, in Refs. [3540].Concerning model uncertainties that is induced by modeling errors, it is well understood that the prior and posterior probability models of the uncertain parameters of the computational model are insufficient and they do not have the capability to take into account the model uncertainties in the context of computational mechanics as explained, for instance, in Ref. [41] and in Ref. [4244]. Two main methods can be used to take into account the model uncertainties (modeling errors).
(i)
Outputpredictionerror method . It consists in introducing a stochastic model of the system output which is the difference between the real system output and the computational model output. If there are no experimental data then, this method cannot really be used as there is generally no information that concerns with the probability model of the noise and it is added to the computational model output. If the experiments are available then, the observed prediction error is then the difference between the measured real system output and the computational model output. Then, a posterior probability model can be constructed in Refs. [41,45] by using the Bayesian method [4647]. Such an approach is efficient but it requires experimental data. In this case, the posterior probability model of the uncertain parameters of the computational model strongly depends on the probability model of the noise that is added to the model output and it is often unknown. Moreover, for many problems, it can be necessary to take into account the modeling errors at the operators’ level of the mean computational model. For instance, such an approach seems to be necessary to take into account the modeling errors on the mass and the stiffness operators of a computational dynamical model in order to analyze the generalized eigenvalue problem. It is also the case for the robust design optimization that is performed with an uncertain computational model for which the design parameters of the computational model are not fixed but they vary inside an admissible set of values.(ii)
Nonparametric probabilistic approach of model uncertainties induced by modeling errors. This approach is proposed in Ref. [42] as an alternative method to the previous outputpredictionerror method. This allows modeling errors to be taken into account at the operators’ level by introducing random operators and not at the model output level by introducing an additive noise. It should be noted that this second approach allows a prior probability model of model uncertainties to be constructed even if no experimental data are available. This nonparametric probabilistic approach is based on the use of a reducedorder model and the random matrix theory. It consists in directly constructing the stochastic modeling of the operators of the mean computational model (Ref. [42]). The random matrix theory [48] and its developments in the context of dynamics, vibration and acoustics (Refs. [4244,4950]) is used to construct the prior probability distribution of the random matrices modeling the uncertain operators of the mean computational model. This prior probability distribution is constructed by using the maximum entropy principle [51], in the context of Information Theory [52] and the constraints are defined by the available information (Refs. [4244,49,5354]. Since the basic paper Ref. [42], many works have been published in order:to validate, using the experimental results, the nonparametric probabilistic approach of both the computational modelparameter uncertainties and the model uncertainties that are induced by modeling errors (Refs. [8, 44, 5560]),
to extend the applicability of the theory to other areas (Refs. [6169]),
to extend the theory to new ensembles of positivedefinite random matrices that yield a more flexible description of the dispersion levels (Ref. [70]),
to apply the theory for the analysis of complex dynamical systems in the mediumfrequency range that include structuralacoustic systems, (Refs. [8,33,55,57,5961,7176]),
to analyze nonlinear dynamical systems (i) with local nonlinear elements (Refs. [64, 7783]) and (ii) with nonlinear geometrical effects (Refs. [8485]).
Concerning the coupling of the parametric probabilistic approach of uncertain computational model parameters, with the nonparametric probabilistic approach of model uncertainties that are induced by modeling errors, a methodology has been recently proposed in Refs. 8687. This generalized probabilistic approach of uncertainties in computational dynamics uses the random matrix theory. The proposed approach allows the prior probability model of each type of uncertainties (uncertainties on the computational model parameters and model uncertainties) which are to be separately constructed and identified.
Concerning robust updation or robust design optimization consists of updating a computational model or in optimizing the design of a mechanical system with a computational model by taking into account the uncertainties in the computational model parameters and the modeling uncertainties. An overview of the computational methods in optimization that considers uncertainties can be found in Ref. [88]. Robust updating and robust design developments with uncertainties in the computational model parameters are developed in Refs. [8991] while robust updating and robust design optimization with modeling uncertainties can be found in Refs. [82, 9295].
9.2 Uncertainties and stochastic reducedorder computational structuralacoustic model
This section is devoted to the construction of the stochastic model of both computational model parameters uncertainties and modeling errors by using the nonparametric probabilistic approach and random matrix theory (for the details, see Refs. [3334, 49, 59]). We apply this methodology to the reducedorder computational structural acoustic model that is defined by using Eqs. (73) to (78). It is assumed that there is no uncertainties in the boundary element matrix [A
_{BEM} (ω/c_{E} )] and in the wall acoustic impedance matrix [A^{Z} (ω)]. Consequently, for fixed values ofN_{S} andN , the stochastic reducedorder computational structuralacoustic model of the orderN_{S} andN is written as,Where, for all fixed ω, the complex random vectors
Q ^{S} (ω) andQ (ω) of dimensionN_{S} andN are the solution of the following equation,Where, the complex random matrix [
A _{FSI}(ω)] is written as,The symmetric (
N_{S} ×N_{S} ) complex random matrix [A ^{S} (ω)] is defined by,Where, the positivedefinite symmetric (
N_{S} ×N_{S} ) real matrices [M ^{S} ], [D ^{S} (ω)] and [K ^{S} (ω)] are random matrices whose probability distributions are constructed in Sections 9.4 and 9.5. The symmetric (N ×N ) complex random matrix [A (ω)] is written as,Where, [
M ], [D (ω)] and [K ]are symmetric (N ×N ) real random matrices. Random matrix [M ] is a positive definite. The diagonal (N ×N ) real random matrix [D (ω)] is written as,in which τ(ω) is deterministic and it is defined by Eq. (11). For a closed (sealed wall) acoustic cavity, random matrix
[
K ] is positive and it is not invertible with rankN ？ 1, while for an almost closed (non sealed wall) acoustic cavity, random matrix [K ] is positive definite. The probability distributions of random matrices [M ], [K ] and of the (N_{S} ×N ) real random matrix [C ] are constructed in Sections 9.6 to 9.8.9.3 Preliminary results for the stochastic modeling of the random matrices for the stochastic reducedorder computational structuralacoustic model
In the framework of the nonparametric probabilistic approach of uncertainties the probability distributions and the generators of independent realizations of such random matrices. They are constructed by using random matrix theory [48] and the maximum entropy principle [51, 67] from Information Theory [52], in which Shannon introduced the notion of entropy as a measure of the level of uncertainties for a probability distribution. For instance, if
p_{X} (x ) is a probability density function on a real random variableX then, the entropyε (p_{X} ) ofp_{X} is defined by,The maximum entropy principle consists in maximizing the entropy, that is to say, maximizing the uncertainties, under the constraints that are defined by the available information. Consequently, it is important to define the algebraic properties of the random matrices for which the probability distributions are to be constructed. Let
E be the mathematical expectation. For instance,Consequently, we have
ε (p_{X} ) = ？E {log(p_{X} (X ))}. In order to construct the probability distributions of the random matrices that are introduced in Section 9.2, we need to define a basic ensemble of random matrices.It is well known that a real Gaussian random variable can take in negative values. Consequently, the Gaussian orthogonal ensemble (GOE) of random matrices [48] is the generalization for the matrix case of the Gaussian random variable which cannot be used when the positiveness property of the random matrix is required. Therefore, new ensembles of random matrices are required for the implementation of the nonparametric probabilistic approach of uncertainties. Below, we summarize the construction [4243] of an ensemble of positivedefinite symmetric (
m ×m ) real random matrices.9.3.1 Definition of the available information
For the probabilistic construction using the maximum entropy principle, the available information corresponds to two constraints. First is the mean value which is given and it is equal to the identity matrix. Second is an integrability condition which has to be imposed in order to ensure the decrease in the probability density function around the origin. These two constraints are written as,
Where, [
X ] is finite and [I_{m} ] is the (m ×m ) identity matrix.9.3.2 Probability density function
The value of the probability density function of the random matrix [
G _{0}] for the matrix [G ] is notedp _{[G0]}([G ]) and this satisfies the usual normalization condition,and the integration is carried out on the set of all the positivedefinite symmetric (
m ×m ) real matrices and it can be shown that the volume elementis,
Let δ be a positive real number defined by,
and this will allow the dispersion of the probability model of the random matrix [
G _{0}] that is to be controlled and ∥M ∥_{F} is the Frobenius matrix norm of the matrix ∥M ∥such thatFor δ such that 0 < δ < (m+1)^{1/2}(m+5)^{？1/2}, the use of the maximum entropy principle under the two constraints are defined by using Eq. (86) and the normalization condition is defined by Eq. (87). This yields, for all positivedefinite symmetric (
m ×m ) real matrix [G],Where, the positive constant of normalization is
c _{0}, the constantc _{1} = (m + 1)(1 ？ δ^{2})/(2δ^{2}) and the constantc _{2} = (m + 1)/(2δ^{2}) depends onm and δ.9.3.3 Generator of independent realizations
The generator of the independent realizations (which is required to solve the random equations with the Monte Carlo method) is constructed by using the following algebraic representations. By using the Cholesky decomposition, random matrix [
G _{0}] is written as [G _{0}] = [L ]^{T} [L ] in which [L ] is an upper triangular (m ×m )random matrix such that:random variables {[L]ij´, j ≤ j´} are independent;
for j < j´, the realvalued random variable [L]jj´ is written as, [L]jj´ = σmUjj´ where, σm = δ(m+1)？1/2 and Ujj´ is a realvalued Gaussian random variable with zero mean and variance equal to 1;
for j = j´, the positivevalue random variable [L]jj is written as,
in which Vj is a positivevalued Gamma random variable with probability density function Γ(aj, 1)
where,
9.3.4 Ensemble SG^{+}_{ε} of random matrices
Let 0 ≤ ε << 1 be a positive number (for instance, ε can be chosen as 10?6). We then define the ensemble
of all the random matrices such that
Where, [
G _{0}] is a random matrix whose probability density function is defined in Section 9.3.2 and whose generator of independent realizations is defined in Section 9.3.3.9.3.5 Cases of several random matrices
It can be proved (Ref. [44]) that if there are several random matrices for which there is no available information concerning their statistical dependencies then, the use of the maximum entropy principle yields the best model that maximizes the entropy (the uncertainties). This is a stochastic model for which all these random matrices are independent.
9.4 Stochastic modeling of random matrix [M^{S}]
As there is no available information that concerns to the statistical dependency of [
M ^{S} ] with the other random matrices of the problem then, the random matrix [M ^{S} ] is independent of all the other random matrices. The deterministic matrix [M^{S} ] is positive definite and consequently, it can be written as [M^{S} ] = [L_{MS} ]^{T}[L_{MS} ] where, [L_{MS} ] is an upper triangular real matrix. By using the nonparametric probabilistic approach of uncertainties, the stochastic model of the positivedefinite symmetric random matrix [M ^{S}] is then defined by,Where, [
G _{MS} ] is a (N_{S} ×N_{S} ) random matrix that belongs to ensemblewhich is defined in Section 9.3.4. Its probability distribution and generator of independent realizations depend only on the dimension
N_{S} and on the dispersion parameter, δ_{MS} .9.5 Stochastic modeling of the family of random matrices [D^{S}(ω)] and [K^{S}(ω)]
As there is no available information concerning the statistical dependency of the random matrices {[
D ^{S}(ω)], [K ^{S}(ω)]} with the other random matrices of the problem then, {[D ^{S}(ω)], [K ^{S}(ω)]} are independent of all the other random matrices. But we will see below that [D ^{S}(ω)] and [K ^{S}(ω)] are statistically dependent random matrices. For stochastic modeling of [D ^{S}(ω)] and [K ^{S}(ω)]that is related to the linear viscoelastic structure, we propose to use the new extension which is presented in Ref. [96]. This is based on the Hilbert transform [22] in the frequency domain to express the causality properties (similar to the transforms used in Section 5.2). Then, the nonparametric probabilistic approach of uncertainties consists in modeling the positivedefinite symmetric (N_{S} ×N_{S} ) real matrices [D ^{S}(ω)] and [K ^{S}(ω)] by random matrices [D ^{S}(ω)] and [K ^{S}(ω)] such thatFor ω ≥ 0, the construction of the stochastic model of the family of random matrices [
D ^{S}(ω)] and [K ^{S}(ω)] is carried out as follows,Constructing the family [DS(ω)] of random matrices such that [DS(ω)] = [LDS(ω)]T[GDS][LDS(ω)] where, [LDS(ω)] is such that [DS(ω)] = [LDS(ω)]T[LDS(ω)] and where, [GDS] is a (NS × NS) random matrix that belongs to ensemble
and it is defined in Section 9.3.4. Its probability distribution and its generator of independent realizations depend only on the dimension NS and on the dispersion parameter δDS which allows the level of uncertainties to be controlled.
Defining the family
of random matrices such that
Constructing the family
of random matrices by using the equation,
or equivalently by using the two following equations that are useful for computation:
and, for ω > 0,
Defining the family
of random matrices such that
Constructing the random matrix [KS(0)] = [LKS(0)]T[GKS(0)] [LKS(0)] where, [LKS(0)] is such that [KS(0)] = [LKS(0)]T[LKS(0)] and where [GKS(0)] is a (NS × NS) random matrix that belongs to ensemble
is defined in Section 9.3.4. Its probability distribution and generator of independent realizations depend only on the dimension NS and on the dispersion parameter δKS(0) which allows the level of uncertainties to be controlled. It should be noted that the random matrix [GKS (0)] is independent of random matrix [GDS].
Computing the random matrix
Defining the random matrix
Constructing the random matrix
and verifying that [KS(ω)] is an effectively increasing function on [0, +∞[.
9.6 Stochastic modeling of random matrix [M]
As there is no available information concerning the
statistical dependency of [
M ] with the other random matrices of the problem, the random matrix [M ] is independent of all the other random matrices. The deterministic matrix [M ], is positive definite and consequently, it can be written as [M ] = [L_{M} ]^{T} [L_{M} ] in which [L_{M} ] is an upper triangular real matrix. By using the nonparametric probabilistic approach of uncertainties, the stochastic model of the positivedefinite symmetric random matrix [M ] is then defined by,Where, [
G _{M}] is a (N ×N ) random matrix that belongs to ensembleand it is defined in Section 9.3.4. Its probability distribution and generator of independent realizations depend only on the dimension
N and on the dispersion parameter, δ_{M} .9.7 Stochastic modeling of random matrix [K]
As there is no available information concerning the
statistical dependency of [
K ] with the other random matrices of the problem, the random matrix [K ]is independent of all the other random matrices. For the stochastic modeling of [K ], two cases have to be considered.Closed (sealed wall) acoustic cavity. In such a case, the symmetric positive matrix [K] is of rank N ？ 1 and it can then be written as [K] = [LK]T [LK] where, [LK] is a rectangular (N, N ？ 1) real matrix. By using the nonparametric probabilistic approach of uncertainties, the stochastic model of the positive symmetric random matrix [K] of rank N ？ 1 is then defined [44] by,
Where, [
G _{K}] is a ((N ？ 1) × (N ？ 1)) random matrix that belongs to ensembleand it is defined in Section 9.3.4. It’s probability distribution and generator of independent realizations depend only on the dimension
N ？ 1 and on the dispersion parameter, δ_{K} .Almost closed (non sealed wall) acoustic cavity.
The matrix [
K ] is positive definite and thus it is invertible. Consequently, it can be written as [K ] = [L_{K} ]^{T} [L_{K} ] in which [L_{K} ] is an upper triangular (N ,N ) real matrix. By using the nonparametric probabilistic approach of uncertainties, the stochastic model of this positive symmetric random matrix yields,Where, [
G _{K}] is a (N ×N ) random matrix that belongs to ensembleand it is defined in Section 9.3. It’s probability distribution and generator of independent realizations depend only on the dimension
N and on the dispersion parameterδ_{K} .9.8 Stochastic modeling of random matrix [C]
As there is no available information concerning the statistical dependency of [
C ] with the other random matrices of the problem, the random matrix [C ] is independent of all the other random matrices. We use the construction that is proposed in Ref. [44]) in the context of the nonparametric probabilistic approach. Let us assume thatN_{S} ≥N and that (N_{S} ×N ) real matrix [C ] is such that [C ]q = 0 impliesq = 0. IfN ≥N_{S} then, the following construction must be applied to [C ]^{T} instead of [C ]. By using the singular value decomposition of rectangular matrix [C ], one can write [C ] = [R ][T ] in which the (N_{S} ×N ) real matrix [R ] is such that [R ]^{T}[R ]=[I_{N} ] and where, the symmetric square matrix [T ] is a positivedefinite symmetric (N ×N ) real matrix. By using, Cholesky decomposition we then have [T ] = [L_{T} ]^{T} [L_{T} ] in which [L_{T} ] is an upper triangular matrix. The (N_{S} ×N ) real random matrix [C ] is then written as,Where, [
G _{C}] is a (N ×N ) random matrix that belongs to ensembleand it is defined in Section 9.3.4. It’s probability distribution and generator of independent realizations depend only on the dimension
N_{S} ,N and on the dispersion parameter δ_{c}.9.9 Comments about the stochastic model parameters of uncertainties and the stochastic solver
The dispersion parameter δ of each random matrix [
G ] allows its level of dispersion (statistical fluctuations) to be controlled. The dispersion parameters of random matrices [G _{MS}], [G _{DS}], [G _{KS(0)}], [G _{M}], [G _{K}], and [G _{C}] are represented by a vectorδ such thatThis belongs to an admissible set
C_{δ} and it allows the level of uncertainties to be controlled for each type of operators that are introduced in the stochastic reducedorder computational structuralacoustic model. Consequently, if no experimental data are available then,δ has to be used in order to analyze the robustness of the solution of the structuralacoustic problem with respect to the uncertainties by varyingδ inC_{δ} .For a given value of
δ , there are two major classes of methods for solving the stochastic reducedorder computational structuralacoustic model and it is defined by using Eqs. (79) to (85). The first one belongs to the category of the spectral stochastic methods (see Refs. [3536,97]). The second one belongs to the class of stochastic sampling techniques for which the Monte Carlo method is the most popular. Such a method is often called nonintrusive as it offers the advantage of only requiring the availability of classical deterministic codes. It should be noted that the Monte Carlo numerical simulation method (see for instance Refs. [9899]) is a very effective and efficient method as it as the four following advantages,it is a nonintrusive method,
it is adapted to massively parallel computation without any software developments,
it is such that its convergence can be controlled during the computation,
the speed of convergence is independent on the dimension.
If the experimental data are available then, there are several possible methodologies (whose one is the maximum likelihood method) to identify the optimal values of
δ (for the sake of brevity, these aspects are not considered in this paper and we refer the reader to Ref. [33]).10. Symmetric Boundary Element Method Without Spurious Frequencies for the External Acoustic Fluid
The inviscid acoustic fluid occupies the infinite threedimensional domain Ω
_{E} whose boundary ∂Ω_{E} is Γ_{E} . This section is devoted to the construction of the frequencydependent impedance boundary operatorZ _{ΓE} (ω), for the external acoustic problem. We recall that the operatorZ _{ΓE} (ω) is such thatp _{E} _{Γ}_{E} (ω) =Z _{ΓE} (ω)υ (ω). It relates to the pressure fieldp _{E} _{Γ}_{E} (ω) that is exerted by the external fluid on Γ_{E} to the normal velocity field,υ (ω) which is induced by the deformation of this boundary Γ_{E} .Many methods can be found in literature for solving this problem: the boundary element methods, the artificial boundary conditions and the local/nonlocal nonreflecting boundary condition (NRBC) to take into account the Sommerfeld radiation condition at infinity, the DirichlettoNeumann (DtN) boundary condition are related to a nonlocal artificial boundary condition and they match the analytical and numerical solutions, the infinite element method, the doubly asymptotic approximation method, the finite element method in unbounded domain and related a
posteriori error estimation and finally, the wave based method for unbounded domain, see for instance Refs. [100107]. This section is devoted to the presentation on the boundary element methods.The frequencydependent impedance boundary operator
Z _{ΓE} (ω) can be constructed, either in the time domain and then, taking the Fourier transform, or by directly constructing in the frequency domain. One technique for the construction ofZ _{ΓE} (ω) consists of using the boundary integral formulations (Refs. [18, 108115]). In the time domain, it uses the socalled Kirchhoff retarded potential formula (see for instance Refs. [116117]). It should be noted that the formulations in the frequency domain can be easily implemented in massively parallel computers.The finite element discretization of the boundary integral equations yields the Boundary Element Method [3, 118121]. Furthermore, most of those formulations yield unsymmetric fully populated complex matrices. The computational cost can then be reduced by using fast multipole methods [122126].
A major drawback of the classical boundary integral formulations for the exterior Neumann problem related to the Helmholtz equation. This is related to the uniqueness problem although the boundary value problem has a unique solution for all real frequencies [18, 127]. Precisely, there is no unique solution of the physical problem for a sequence of real frequencies called as
spurious orirregular frequencies and they are also called asJones eigenfrequencies [112,128131]. Various methods are proposed in the literature to overcome this mathematical difficulty that arises in the boundary element method [3,129,132137].In this section, we present a method that was initially developed in Ref. [134]. This yields an appropriate symmetric boundary element method that is valid for all real values of the frequency and it is numerically stable and very efficient. This method is detailed in Ref. [3]. It does not require introduction of additional degrees of freedom in the numerical discretization for the treatment of irregular frequencies. This method has been extended to the Maxwell equations [138]. In the case of an external liquid domain with a zeropressure free surface (which is not presented here for the sake of brevity) the method presented below can be adapted by using the image method (for the details, see Ref. [3]).
10.1 Exterior Neumann problem related to the Helmholtz equation
The geometry is defined in Fig. 5. The inviscid fluid occupies the infinite domain Ω
_{E} . For practical computational considerations, the exterior Neumann problem related to the Helmholtz equation (see Eqs. (5) to (7)) is rewritten in terms of a velocity potential, ψ(x , ω). Let,v (x , ω) = ∇ψ(x , ω) be the velocity field of the fluid. The acoustic pressurep (x , ω) is related to ψ(x , ω) by using the following equation,Where, ρ
_{E} is the constant mass density of the external fluid at equilibrium. Let,c_{E} be the constant speed of sound in the external fluid at equilibrium and letk = ω/c_{E} be the wave number at frequency, ω. The exterior Neumann problem is written as,with
R = x  → +∞, where ∂/∂R is the derivative in the radial direction and whereυ (y ) is the prescribed normal velocity field on Γ_{E} . Equation (103) is the Helmholtz equation in the external acoustic fluid, Eq. (104) is the Neumann condition on external fluidstructure interface Γ_{E} and Eq. (105) corresponds to the outward Sommerfeld radiation condition at infinity.10.2 Pressure field in Ω_{E} and on Γ_{E}
For arbitrary real ω ≠ 0, it can be shown that the boundary value problem is defined by using Eqs. (103) to (105). It admits a unique solution that is denoted by ψ^{sol}. It depends linearly on the normal velocity
υ [18,127]. Letbe the value of ψ^{sol} on Γ
_{E} . For allx in Ω_{E} , let us introduce the linear operatorR (x , ω/c_{E} ) such thatWe also introduce the linear boundary operator
B _{ΓE} (ω/c_{E} ) such thatBy using Eq. (102), for all
x in Ω_{E}, the pressure fieldp (x , ω) is written as,in which
Z _{rad} (x , ω) is called as the radiation impedance operator which can then be written as,Similarly, the pressure field
p _{ΓE} (ω) on Γ_{E} is written as,where,
Z _{ΓE} (ω) is called the acoustic impedance boundary operator and this can then be written as,Note that
Z _{ΓE} (ω) is a nonlocal operator.10.3 Symmetry property of the acoustic impedance boundary operator
The transpose of the operator
B _{ΓE} (ω/c_{E} ) is denoted by^{t} B _{ΓE} (ω/c_{E} ). It can then be proven (see Ref. [3]) that the following symmetry property,and from Eq. (111), we deduce that
It should be noted that these complex operators are symmetric but not hermitian.
10.4 Positivity of the real part of the acoustic impedance boundary operator
Operator
i ωZ _{ΓE} (ω) can be written as,Where,
M _{ΓE} (ω/c_{E} ) andD _{ΓE} (ω/c_{E}) are two linear operators such thatIt can be shown that (Ref. [3]) the following positivity property of the real part
D _{ΓE} (ω/c_{E} ) of the acoustic impedance boundary operator is due to the Sommerfeld radiation condition at infinity.10.5 Construction of the acoustic impedance boundary operator for all real values of the frequency
We present here an appropriate symmetric boundary element method without spurious frequencies, for which the details can be found in Ref. [3]. This formulation simultaneously uses two boundary singular integral equations on Γ
_{E} . The first one is based on the use of a singleand doublelayer potentials on Γ_{E}. The second integral equation is obtained by a normal derivative on Γ_{E} of the first one. We then, obtained the following system that relates ψ^{sol} toυ and this then allowsB _{ΓE} (ω/c_{E} ) to be defined by using Eq. (107),The linear boundary integral operators
S _{S} (ω/c_{E} ),S _{D} (ω/c_{E} )) andS _{T} (ω/c_{E} ) are defined by,Where,
G (x ？y ) is the Green function which is written as,Where,
r = x ？y . In Eqs. (118) to (120), the brackets correspond to bilinear forms that allow the operators to be defined and the functions δυ and δψ_{ΓE} are associated to functionsυ and ψ_{ΓE}. By taking in to consideration Eq. (117), letH (ω/c_{E}) be the operator that is defined by,It can be proven that the operator
H (ω/c_{E} )has the symmetric property,^{t} H (ω/c_{E} ) =H (ω/c_{E} ). In Eq. (117), the first equation can be rewritten as,This classical boundary equation that allows the velocity potential to be calculated for a given normal velocity, has a unique solution for all real ω . It does not belong to the set of frequencies for which
S _{T} (ω/c_{E} ) has a null space which is not reduced to {0}. This set of frequencies is called as the set ofspurious orirregular frequencies. Consequently, as proven in Ref. [3], for a spurious frequency, ψ_{ΓE} is the sum of solution ψ_{ΓE} with an arbitrary element that belongs to the null space of the operatorS _{T} (ω/c_{E} ). The originality of the proposed method [3,134] (extended to the Maxwell equations in Ref. [138]), then consists in using the second equation and it is written as,This yields the solution
for all real ω as the elements that belong to the null space are filtered when ω is a spurious frequency. Concerning the practical construction of
for all real values of ω, by using Eq. (117), a particular elimination procedure will be described in Section 10.7.
10.6 Construction of the radiation impedance operator
The solution {ψ^{sol} (
x , ω),x ∈ Ω_{E}} of Eqs. (103) to (105) can be calculated by using the following integral equation,For all
x that is fixed in Ω_{E} , we define the linear integral operatorsR _{S} (x , ω/c_{E} ) andR _{D} (x , ω/c_{E} ) by,By using Eq. (107), Eq. (123) can be rewritten as
From Eq. (106), we deduce that for all
x fixed in Ω_{E} ,and the radiation impedance operator
Z _{rad}(x , ω) is calculated by using Eqs. (109) and (127),10.7 Symmetric boundary element method without spurious frequencies
We use the finite element method to discretize the boundary integral operators
S _{S} (ω/c_{E} ),S _{D}(ω/c_{E} ) andS _{T}(ω/c_{E} ) (corresponding to a boundary element method). Let us consider a finite element mesh of boundary Γ_{E}. LetV = (V _{1}, …,V_{nE} ) andΨ _{ΓE} = (Ψ_{ΓE,1}, …, Ψ_{ΓE,nE}) be the complex vectors of then_{E} degreesoffreedom that constituted of the values ofυ and Ψ_{ΓE} at the nodes of the mesh. Let S_{S}(ω/c_{E} ), S_{D}(ω/c_{E} ) and S_{T}(ω/c_{E} ) be the full complex matrices that correspond to the discretization of the operators which are defined in Eqs. (118) to (120). The complex matrices S_{S}(ω/c_{E} ) and S_{T}(ω/c_{E} ) are symmetric. The finite element discretization of Eq. (117) yields,Where, the symmetric complex matrix [
H (ω/c_{E} )] is the matrix,In Eq. (129),
is the complex vector of the nodal unknowns that correspond to the finite element discretization of
The matrix [
E ] is the nondiagonal (n_{E} ×n_{E} ) real matrix that correspond to the discretization of the identity operatorI . The elimination ofΨ _{ΓE} in Eq. (129) yields a linear equation betweenand
V that defines the symmetric (n_{E} ×n_{E} ) complex matrix [B_{ΓE} (ω/c_{E} )]. This corresponds to the finite element discretization of the boundary integral operatorB _{ΓE} (ω/c_{E} ). We then have,The particular elimination procedure discussed in Section 10.5 which avoids the spurious frequencies is defined below. Vector
Ψ _{ΓE} is eliminated by using Gauss elimination with a partial row pivoting algorithm [139]. If ω does not belong to the set of the spurious frequencies then, [S_{T} (ω/c_{E} )] is invertible and the elimination in Eq. (129) is performed up to row numbern_{E} . If ω coincides with a spurious frequency ω_{α} that is to say ω = ω_{α} then, [S_{T} (ω/c_{E} )] is not invertible and its null space is a real subspace ofof dimension
n _{α} <n_{E} . In this case, the elimination in Eq. (129) is performed up to row numbern_{E} ？n _{α}. Practically,n _{α} is unknown. During the Gauss elimination with a partial row pivoting algorithm, the elimination process is stopped when a “zero” pivot is encountered. It should be noted that when the elimination is stopped, the equations that correspond to the row numbersn_{E} ？n _{α} +1, …,n_{E} are automatically satisfied. From Eq. (111), we deduce that the (n_{E} ×n_{E} ) complex symmetric matrix [Z _{ΓE} (ω)] of operatorZ _{ΓE} (ω) is such thatFinally, the finite element discretization of the acoustic radiation impedance operator
Z _{rad}(x , ω) defined by Eq. (129) is written as,10.8. Acoustic response to prescribed wall displacement field and acoustic source density
We now consider the acoustic response of the infinite external acoustic fluid submitted to a prescribed external acoustic excitation, namely an acoustic source
Q_{E} (x , ω), and to a prescribed normal velocity field on Γ_{E}. This is written asυ =i ωu (ω)·n ^{S} where,n ^{s} is the unit normal to Γ_{E}, external to the structure Ω_{S} andu is the displacement field of the external fluidstructure interface Γ_{E}. This response is formulated by using the results that are related to the exterior Neumann problem for the Helmholtz equation that have been presented in Sections 10.1 to 10.7 and by using the linearity of the problem.Pressure in Ω_{E}. At any point
x fixed in Ω_{E}, the resultant pressurep_{E} (x , ω) is written as,where,
p_{rad} (x , ω) is the field radiated by the boundary Γ_{E} and it is submitted to the prescribed velocity fieldυ . It is written (see Eq. (108)) as,The pressure
p_{given} (x , ω) is such thatwhere,
p _{inc,Q}(x , ω) is the pressure in the free space that is induced by the acoustic sourceQ_{E} and it is written as,where, the Green function
G is defined by Eq. (121) and ∂ψ_{inc,Q}/∂n ^{S} is deduced from Eqs. (137) and (102). The second term in the righthand side of the Eq. (136) corresponds to the scattering of the incident wave (induced by the external acoustic source) by the boundary Γ_{E} is considered as rigid and fixed.Pressure on Γ_{E}. The resultant pressure on Γ_{E} is then written as,
where,
p _{rad}_{ΓE} (ω) is written as,and the pressure field
p _{given}_{Γ}_{E} (ω) on Γ_{E} is such thatBy substituting Eq. (139) in (138), it yields
For details, we refer the reader to Chapter 12 of Ref. [3] .
10.9 Asymptotic formula for the radiated pressure farfield
At point
x in the external domain Ω_{E}, the radiated pressurep (x , ω) is given (see Eq. (108)) byp (x , ω) =Z _{rad}(x , ω)υ . LetR ande be such that (see Fig. 6.)Definition of integral operators
and
For all
x =R e fixed in the external domainΩ_{E}, we define the linear integral operators
and
by,
Where,
N _{e}(y ) is defined by,Asymptotic formula for radiation impedance operator
Z _{rad}(x , ω). We have the following asymptotic formulas,From Eq. (127), we deduce the asymptotic formula for theradiation impedance operator as,
11. Conclusion
We have presented an advanced computational formulation for dissipative structuralacoustics systems and fluidstructure interaction. This is adapted for the development of a new generation of software. An efficient stochastic reducedorder model in the frequency domain is proposed to analyze low and mediumfrequency ranges. 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 which includes wall acoustic impedance and a model of uncertainty in particular for modeling errors.

[Fig. 1.] Configuration of the system

[Fig. 2.] Configuration of the structuralacoustic system for a liquid with free surface.

[Fig. 3.] Example of the structure and internal fluid finite element meshes.

[Fig. 4.] Variabilities and the types of uncertainties in computational structural acoustics and fluidstructure interaction

[Fig. 5.] Geometry of an external infinite domain.

[Fig. 6.] Geometrical configuration.