Advanced Computational Dissipative Structural Acoustics and Fluid-Structure Interaction in Low-and Medium-Frequency Domains. Reduced-Order Models and Uncertainty Quantification

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  • ABSTRACT

    This paper presents an advanced computational method for the prediction of the responses in the frequency domain of general linear dissipative structural-acoustic and fluid-structure systems, in the low-and medium-frequency domains and this includes uncertainty quantification. The system under consideration is constituted of a deformable dissipative structure that is coupled with an internal dissipative acoustic fluid. This includes wall acoustic impedances and it is surrounded by an infinite acoustic fluid. The system is submitted to given internal and external acoustic sources and to the prescribed mechanical forces. An efficient reduced-order computational model is constructed by using a finite element discretization for the structure and an internal acoustic fluid. The external acoustic fluid is treated by using an appropriate boundary element method in the frequency domain. All the required modeling aspects for the analysis of the medium-frequency domain have been introduced namely, a viscoelastic behavior for the structure, an appropriate dissipative model for the internal acoustic fluid that includes wall acoustic impedance and a model of uncertainty in particular for the modeling errors. This advanced computational formulation, corresponding to new extensions and complements with respect to the state-of-the-art are well adapted for the development of a new generation of software, in particular for parallel computers.


  • KEYWORD

    Computational mechanics , Structural acoustics , Vibroacoustic , Fluid-structure interaction , Uncertainty quantification , Reduced-order 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

    pE|ΓE = value of the external acoustic pressure field on ΓE

    pgiven = given external acoustic pressure field

    pgiven|ΓE = value of the given external acoustic pressure field on ΓE

    q = vector of the generalized coordinates for the internal acoustic fluid

    qS = vector of the generalized coordinates for the structure

    image

    = 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

    image

    = dynamical matrix for the internal acoustic fluid

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

    image

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

    [AFSI] = reduced dynamical matrix for the fluid-structure coupled system

    [AFSI] = random reduced dynamical matrix for the fluid-structure coupled system

    image

    = dynamical matrix for the fluid-structure cou

    [AS] = reduced dynamical matrix for the structure

    [AS] = random reduced dynamical matrix for the structure

    image

    = dynamical matrix for the structure

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

    image

    = 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

    image

    = 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

    image

    = damping matrix for the internal acoustic fluid

    [DS] = reduced damping matrix for the structure

    [DS] = random reduced damping matrix for the structure

    image

    = damping matrix for the structure

    DOF = degrees of freedom

    image

    = vector of discretized acoustic forces

    image

    = 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

    image

    = “stiffness” matrix for the internal acoustic fluid

    [KS] = reduced stiffness matrix for the structure

    [KS] = random reduced stiffness matrix for the structure

    image

    = stiffness matrix for the structure

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

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

    image

    = “mass” matrix for the internal acoustic fluid

    [MS] = reduced mass matrix for the structure

    [MS] = random reduced mass matrix for the structure

    image

    = mass matrix for the structure

    image

    = 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

    image

    = vector of internal acoustic pressure DOF

    U = random vector of structural displacement DOF

    image

    = vector of structural displacement DOF

    image

    = 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

    image

    = 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 =

    image

    (ΩE?ΓE)

    ΩE = external acoustic domain

    ΩS = structural domain

    1. Introduction

    The fundamental objective of this paper is to present an advanced computational method for the prediction of the responses in the low-and medium-frequency domains of general linear dissipative structural- acoustic and fluid-structure systems. The system under consideration is constituted of a deformable dissipative structure and it is coupled with an internal dissipative acoustic fluid which includes wall acoustic impedances. The system is surrounded by an infinite acoustic fluid and it is submitted to a given internal and external acoustic sources and to the prescribed mechanical forces.

    Instead of presenting an exhaustive review of such a problem in this introductory section, we have preferred to move on to the review discussions in each relevant section.

    Concerning the appropriate formulations for computing the elastic, acoustic and elastoacoustic modes of the associated conservative fluid-structure system, including substructuring techniques, for the construction of the reduced-order computational models in fluid-structure interaction and for structural-acoustic systems, refer to Ref. [1-5]. For the dissipative complex systems, readers can find out the details of the basic formulations in Ref. [3].

    In this paper, the proposed formulation that corresponds to new extensions and complements with respect to the state-of-the-art can be used for the development of a new generation of computational software in particular to the context of parallel computers. We present here an advanced computational formulation. This is based on an efficient reduced-order model in the frequency domain and for this all the required modeling aspects for the analysis of the medium-frequency domain have been taken into account. To be more precise, we have introduced a viscoelastic modeling for the structure, an appropriate dissipative model for the internal acoustic fluid that includes wall acoustic impedance and finally, a global model of uncertainty. It should be noted that model uncertainties must be absolutely taken into account in the computational models of complex vibroacoustic systems in order to improve the prediction of responses in the medium-frequency range. The reduced-order computational model is constructed by using finite element discretization for the structure and for the internal acoustic fluid.

    The external acoustic fluid is treated by using an approximate boundary element method in the frequency domain.

    The sections of the paper are:

    1. Introduction

    2. Statement of the problem in the frequency domain

    3. External inviscid acoustic fluid equations

    4. Internal dissipative acoustic fluid equations

    5. Structure equations

    6. Boundary value problem in terms of {u, p}

    7. Computational model

    8. Reduced-order computational model

    9. Uncertainty quantification

    10. Symmetric boundary element method without spurious frequencies for the external acoustic fluid

    11. Conclusion

    References are given at the end of the paper.

    2. Statement of the Problem in the Frequency Domain

    We consider a mechanical system made up of a damped linear elastic free-free structure ΩS 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 ΩE (gas or liquid) (see Fig. 2). A part ΓZ of the internal fluid-structure interface is assumed to be dissipative and it is modeled by a wall acoustic local impedance Z. This system is submitted to a given internal acoustic source in the acoustic cavity and to the given mechanical forces that are applied to the structure. In the infinite external acoustic fluid domain, external acoustic sources are given. It is assumed that the external forces are in equilibrium.

    We are interested in the responses in the low-and medium-frequency domains for the displacement field in the structure, the pressure field in the acoustic cavity and the pressure fields on the external fluid-structure interface and also in the external acoustic fluid (near and far fields). It is now well established that the predictions in the medium-frequency domain must be improved by taking into account both the system-parameter uncertainties and the model uncertainties that are induced by modeling errors. Such aspects will be considered in the last section of the paper, which is devoted

    to Uncertainty Quantification (UQ) in structural acoustics and in fluid-structure interaction.

       2.1 Main notations

    The physical space

    image

    is referred to a cartesian reference system and we denote the generic point of

    image

    by x = (x1, x2, x3). For any function f(x), the notation f, j denotes the partial derivative with respect to xj. We also use the classical convention for summations over repeated Latin indices but not over Greek indices. As explained earlier, we are interested in the vibration problems that are formulated in the frequency domain for structural- acoustic and fluid-structure interaction systems. Therefore, we introduce the Fourier transform for the various quantities involved. For instance, for the displacement field u, the stress tensor σij and the strain tensor εij 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,

    image
    image
    image

    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

    image
    image

       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 ΩS. In general, a complex structure is composed of a main part called the master structure. It is defined as the “primary” structure and it is accessible to conventional modeling which includes uncertainties modeling. A secondary part called as the fuzzy substructure is related to the structural complexity and it includes for example many equipment units that are attached to the master structure. In the present paper, we will not consider fuzzy substructures and this concerns the fuzzy structure theory, refer to Ref. [6,7], to Chapter 15 of Ref. [3] for a synthesis, and to Ref. [8] for the extension of the theory to uncertain complex vibroacoustic system with fuzzy interface modeling. Consequently, the so-called “master structure” will be simply called here as “structure”

    The structure at the equilibrium occupies the three-dimensional bounded domain ΩS with a boundary ∂ΩS. This is made up of a part ΓE 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 ΓZ is another part of the coupling interface between the structure and the internal acoustic fluid with acoustical properties. The structure is assumed to be free (free-free structure), i.e. not fixed on any part of the boundary ∂ΩS. The outward unit normal to ∂ΩS is denoted as

    image

    (see Fig. 2). The displacement field in ΩS is denoted by u(x, ω) = (u1(x, ω), u2(x, ω), u3(x, ω)). A surface force field G(x, ω) = (G1(x, ω), G2(x, ω), G3(x, ω)) is given on ∂ΩS and a body force field g(x, ω) = (g1(x, ω), g2(x, ω), g3(x, ω)) is given in ΩS. 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?ΓZ. The outward unit normal to ∂Ω is denoted as n = (n1, n2, n3) and we have n = ?nS on ∂Ω (see Fig. 2). Part ΓZ 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 ΩE. The structure is surrounded by an external inviscid acoustic fluid (gas or liquid) and it is as described in Section 10. The fluid occupies the infinite three-dimensional domain ΩE whose boundary ∂ΩE is ΓE. We introduce the bounded open domain Ωi which is defined by

    image

    Note that in general, Ωi does not coincide with the internal acoustic cavity Ω. The boundary ∂Ωi of Ωi is then ΓE. The outward unit normal to ∂Ωi is nS and it is defined above (see Fig. 2). We denote the pressure field in ΩE as pE(x, ω). We assume that there is no Dirichlet boundary condition on any part of ΓE. An acoustic source density QE(x, ω) is given in ΩE. This acoustic source density induces a pressure field pgiven(ω) on ΓE 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 ΩE and it is described by the acoustic pressure field pE(x, ω)at point x of ΩE and at circular frequency ω. Let ρE be the constant mass density of an external acoustic fluid at equilibrium. Let, cE be the constant speed of sound in the external acoustic fluid at equilibrium and let, k = ω/cE 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,

    image
    image
    image

    with R = ||x|| → +∞, where ∂ / ∂R is the derivative in the radial direction and u·nS is the normal displacement field on ΓE 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 pE|ΓE of the pressure field pE on the external fluid-structure interface ΓE is related to pgiven|ΓE and to u by Eq. (141),

    image

    in which the different quantities are defined in Section 10. This is a self-contained section that describes the computational modeling of the external inviscid acoustic fluid by an appropriate boundary element method. It should be noted that in Eq. (8), the pressure field pE|ΓE(ω) is related to the value of the normal displacement field u(ω)·nS on the external fluid-structure interface ΓE through an operator ZΓE (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, ρ0 be the mass density and c0 be the constant speed of sound in the fluid at equilibrium in the reference configuration Ω. We have (see the details in Ref. [3]),

    image
    image

    τ is given by,

    image

    The constant η is the dynamic viscosity, v = η0 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 ρ0, yields the Helmholtz equation with a dissipative term and a source term,

    image

    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 = iω u·n on Γ yields the following Neumann boundary condition,

    image

    (ii) Neumann boundary condition on ΓZ with wall acoustic impedance. The part ΓZ of the boundary ∂Ω has acoustical properties that are modeled by a wall acoustic impedance Z(x, ω) which is defined for x ∈ ΓZ, with complex values. The wall impedance boundary condition on ΓZ is written as,

    image

    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 = iω u·n and Eq. (14) on Γ, yields the following Neumann boundary condition with a wall acoustic impedance,

    image

       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 ΩL. It is assumed that the complementary part Ω/ΩL is a vacuum domain. The boundary, ∂ΩL of ΩL is constituted of three boundaries namely ΓZ, Γ0 that corresponds to the free surface of the liquid and a part ΓL of Γ. The Neumann boundary condition on ΓL is given by Eq. (13), on ΓZ which is given by Eq. (15). By neglecting the gravity effects, the following Dirichlet condition is written on the free surface,

    image

    5. Structure Equations

       5.1 Structure equations in the frequency domain

    The equation of the structure that occupies the domain ΩS is written as,

    image

    in which ρs(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 σij is written as,

    image

    in which the symmetric strain tensor εkh(u) is such that

    image

    and where the tensors aijkh(ω) and bijkh(ω) depend on ω (see Section 5.2). The boundary condition on the fluid-structure external interface ΓE is such that

    image

    in which pE|ΓE is given by Eq. (8) and it yields

    image

    As nS = ? n, the boundary condition on Γ∪ΓZ is written as,

    image

    in which p is the internal acoustic pressure field that is defined in Section 4.

       5.2 Viscoelastic constitutive equation

    In dynamics, the structure must always be modeled as a dissipative continuum. For the conservative part of the structure, we use the linear elasticity theory which allows the structural modes to be introduced. This was justified by the fact that in the low-frequency range, the conservative part of the structure can be modeled as an elastic continuum. In this section, we introduce damping models for the structure that is based on the general linear theory of viscoelasticity and it is presented in Ref. [15] (see also Ref. [16,17]). Complementary developments are presented with respect to the viscoelastic constitutive equation detailed in Ref. [3].

    In this section, x is fixed in ΩS, and we rewrite the stress tensor σij(x, t) as σij(t), the strain tensor εij(x, t) as εij(t) and its time derivative

    image

    as

    image

    Constitutive equation in the time domain. The stress tensor σij(t) is written as,

    image

    Where, σij(t) = 0 and ε(t) = 0 for t ≤ 0. The real functions Gijkh(x, t) are denoted as Gijkh(t) and they are called as the relaxation functions. The tensor Gijkh(t) (and thus

    image

    has the usual property of symmetry and Gijkh(0), which is called as the initial elasticity tensor is positive definite. The relaxation functions are defined on [0, +∞[ and are differentiable with respect to t on ]0, +∞[. Their derivatives are denoted as

    image

    and are assumed to be integrable on [0, +∞[. Functions Gijkh(t) can be written as,

    image

    Therefore, the limit of Gijkh(t), denoted as Gijkh(∞), is finite as t and it tends to +∞,

    image

    The tensor Gijkh(∞), called as the equilibrium modulus at x, is symmetric and positive definite. It corresponds to the usual elasticity coefficients of the elastic material for a static deformation. In effect, the static equilibrium state is obtained for t and it tends to infinity.

    For all x that is fixed in ΩS, we introduce the real functions t → gijkh(x, t), denoted as gijkh(t), such that

    image

    As gijkh(t) = 0 for t < 0, we deduce that gijkh(t) is a causal function.

    By using Eq. (26), Eq. (23) can be rewritten as,

    image

    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,

    image

    in which,

    image
    image

    Equation (28) can then be rewritten as,

    image

    Tensors aijkh(ω) and bijkh(ω) must satisfy the symmetry properties

    image
    image

    and the positive-definiteness properties, i.e., for all the second-order real symmetric tensors Xij,

    image
    image

    in which the positive constants ca(ω) and cb(ω) are such that ca(ω) ≥ c0 > 0 and cb(ω) ≥ c0 > 0 where c0 is a positive real constant that is independent of ω.

    As gijkh(t) is an integrable function on ]?∞, +∞[, its Fourier transform

    image

    is defined by,

    image

    and it is a complex function which is continuous on ]?∞, +∞[ and such that

    image

    The real part

    image

    and the imaginary part

    image

    of

    image

    are even and odd functions. So, it is easy to say that

    image

    and

    image

    We can then deduce that

    image

    We can now take the Fourier transform of Eq. (27) and using Eq. (31) yields the relations,

    image
    image

    Eqs. (37), (39) and (40) yields,

    image
    image

    From Eqs. (31), (41) and (42), we deduce that

    image

    Eq. (43) shows that viscoelastic materials behave elastically at high frequencies with elasticity coefficients that are defined by the initial elasticity tensor Gijkh(0) that differs from the equilibrium modulus tensor Gijkh(∞) which is written by using Eqs. (25) and (38) as,

    image

    As pointed out before, a positive-definite tensor Gijkh(∞) corresponds to the usual elasticity coefficients of a linear elastic material for a static deformation process. More specifically for ω = 0 by using Eqs. (38) to (40) and Eq. (31) yield,

    image

    in which σijkh(0) = {σijkh(ω)}ω=0 and εijkh(0) = {εijkh(ω)}ω=0 where,

    image

    The reader should be aware of the fact that the constitutive equation of an elastic material in a static deformation process is defined by Gijkh(∞) and not by the initial elasticity tensor, Gijkh(0). Referring to Ref. [15, 21], it has been proven that Gijkh(0) ? Gijkh(∞)is a positive-definite tensor. Consequently,

    image

    is a negative- definite tensor.

    As gijkh(t) is a causal function, the real part

    image

    and the imaginary part

    image

    of its Fourier transform

    image

    are related by the following relations that involve the Hilbert transform (see Ref. [22, 23]),

    image
    image

    in which p.v denotes the Cauchy principal value which is defined as,

    image

    The relations defined by Eqs. (47) and (48) are also called as the Kramers and Kronig relations for the function gijkh(t) (see Ref. [24, 25]).

    LF-range constitutive equation approximation. In the low-frequency range and in most cases, the coefficients aijkh(ω) was given by the linear viscoelastic model. It was defined by Eq. (39) and it is almost frequency independent. In such a case, they can be approximated by aijkh(ω)?aijkh(0) and this is independent of ω(but which depends on x). It should be noted that this approximation can only be made on a finite interval that corresponds to low-frequency range and it cannot be used in the entire frequency domain as Eqs. (47) and (48) are not satisfied and the integrability property is lost.

    MF range constitutive equation. In the medium-frequency range, the previous LF-range constitutive equation approximation is generally invalid and the entire linear viscoelastic theory which is defined by Eq. (31) must be used.

    Bibliographical comments concerning expressions of frequency-dependent coefficients. Some algebraic representations of functions aijkh(ω) and bijkh(ω) have been proposed in literature (see for instance Refs. [3,15-16,18,20,26-30] ). Concerning linear hysteretic damping which is correctly written in the present context, refer to Refs. [31-32].

    6. Boundary Value Problem in Terms of {u, p}

    The boundary value problem in terms of {u, p} is written as follows. For all real ω and for the given G(ω), g(ω), pgiven|ΓE (ω) and Q(ω), we calculate u(ω) and p(ω), such that

    image
    image
    image
    image
    image
    image

    In case of a free surface in the internal acoustic cavity (see Section 4.3), we must add the following boundary condition

    image

    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, ΩS and a finite element mesh of internal acoustic fluid Ω. We assume that the two finite element meshes are compatible on an interface Γ?ΓZ. The finite element mesh of surface ΓE is the trace of the mesh of ΩS (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,

    image

    be a complex vector of the ns degrees- of-freedom (DOFs) which are the values of u(ω) at the nodes of the finite element mesh of the domain ΩS. For the internal acoustic fluid, let

    image

    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,

    image

    in which the complex matrix

    image

    is defined by,

    image

    In Eq. (58), the symmetric (nS × nS) complex matrix

    image

    is defined by,

    image

    where,

    image

    and

    image

    are symmetric (nS × nS) real matrices which represent the mass matrix, the damping matrix and the stiffness matrix of the structure. Matrix

    image

    is positive and invertible (positive definite). Matrices

    image

    and

    image

    are positive and not invertible (positive semidefinite). This is due to the presence of six rigid body motions since the structure has been considered as a free-free structure. The symmetric (n × n) complex matrix

    image

    is defined by,

    image

    Where,

    image

    and

    image

    are symmetric (n × n) real matrices. Matrix

    image

    is positive and invertible. Matrices

    image

    and

    image

    are positive and are not invertible with rank n ? 1. From Eq. (53), it can easily be deduced that

    image

    in which τ(ω) is defined by Eq. (11). The internal fluid-structure coupling matrix

    image

    is related to the coupling between the structure and the internal fluid on an internal fluid-structure interface. This is a (nS × n) real matrix which is only related to the values of

    image

    and

    image

    on the internal fluid-structure interface. The wall acoustic impedance matrix

    image

    is a symmetric (n × n) complex matrix that depends on the wall acoustic impedance Z(x, ω) on ΓZ and this is only related to the values of

    image

    on boundary ΓZ. The boundary element matrix

    image

    which depends on ω/cE, is a symmetric (nS × nS) complex matrix and it is only related to the values of

    image

    on the external fluid-structure interface ΓE. This matrix is written as,

    image

    in which [BΓE (ω/cE)] is a full symmetric (nE × nE) complex matrix which is defined in Section 10.7. Here,

    image

    is a sparse (nE × nS) real matrix that is related to finite element discretization.

    8. Reduced-Order Computational Model

    The strategy used for the construction of the reduced-order computational model consists in using the projection basis constituted of [3]:

    the undamped elastic structural modes of the structure in vacuo for which the constitutive equation corresponds to elastic materials (see Eq. (45)), and consequently, the stiffness matrix has to be taken for ω = 0.

    the undamped acoustic modes of the acoustic cavity is with fixed boundary and without wall acoustic impedance. Two cases must be considered: one for which the internal pressure varies with the variation of the volume of the cavity (a cavity with a sealed wall is called as a closed cavity) and the other one for which the internal pressure does not vary along with the variation of the volume of the cavity (a cavity with a non sealed wall is called as an almost closed cavity).

       8.1 Computation of the elastic structural modes

    This step concerns with the finite element calculation of the undamped elastic structural modes of structure ΩS in vacuo for which the constitutive equation corresponds to elastic materials. By setting λS = ω2, we then have the following classical (nS × nS) which is a generalized symmetric real eigenvalue problem

    image

    It can be shown that there is a zero eigenvalue with multiplicity 6 (corresponding to the six rigid body motions) and that there is an increasing sequence of ns ? 6 strictly positive eigenvalues (corresponding to the elastic structural modes). Each positive eigenvalue can be a multiple (case of a structure with symmetries),

    image

    Let

    image

    be the eigenvectors (the elastic structural modes) that is associated with

    image

    Let 0 < NSnS ? 6. We introduce (nS × NS) real matrix of the NS elastic structural modes

    image

    that is associated with the first NS strictly positive eigenvalues,

    image

    One has classical orthogonality properties,

    image
    image

    where, [MS] is a diagonal matrix of positive real numbers and [KS (0)] is a diagonal matrix of eigenvalues such that

    image

    (the eigenfrequencies are,

    image

       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 = ω2, we then have the following classical (n × n) generalized symmetric real eigenvalue problem

    image

    It can be shown that there is a zero eigenvalue with multiplicity 1 and denoted as λ0 (corresponding to constant eigenvector denoted as

    image

    ). 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),

    image

    Let

    image

    be the eigenvectors (the acoustic modes) that is associated with λ1, …, λα, …

    Closed (sealed wall) acoustic cavity. Let be 0 < N ≤ n. We introduce the (n × N) real matrix of the constant eigenvector

    image

    and of the N ? 1 acoustic modes

    image

    that is associated with the first N ? 1 strictly positive eigenvalues as,

    image

    Almost closed (non sealed wall) acoustic cavity. Let be 0 < N ≤ n ? 1. We introduce the (n × N) real matrix of N acoustic modes

    image

    is associated with the first N strictly positive eigenvalues,

    image

    One has classical orthogonality properties,

    image
    image

    where, [M] is a diagonal matrix of positive real numbers and [K] is a diagonal matrix of eigenvalues such that [K]αβ = λα δαβ (for non zero eigenvalue, the eigenfrequencies are

    image

       8.3 Construction of the reduced-order computational model

    The reduced-order computational model, of order NS << ns and N << n, is obtained by projecting Eq. (57) as follows,

    image
    image

    Complex vectors qS(ω) and q(ω) of dimensions NS and N are the solution of the following equation,

    image

    in which the complex matrix [AFSI(ω)] is defined by,

    image

    In Eq. (76), the symmetric (NS × NS) complex matrix [AS(ω)] is defined by,

    image

    in which [MS], [DS(ω)] and [KS(ω)] are positive-definite symmetric (NS × NS) real matrices such that [DS(ω)] = [u]T

    image

    and

    image

    The symmetric (N × N) complex matrix [A(ω)] is defined by,

    image

    Where, [M], [D(ω)] and [K] are symmetric (N × N) real matrices. Matrix [M] is positive and invertible. Diagonal (N × N) real matrix [D(ω)] is written as [D(ω)] = τ(ω)[K] in which τ(ω) is defined by, Eq. (11). For a closed (sealed wall) acoustic cavity, matrix [K] is positive and it is not invertible with rank N ? 1, while for an almost closed (non sealed wall) acoustic cavity, matrix [K] is positive and invertible. The (NS × N) real matrix [C] and it is written as,

    image

    Symmetric (N × N) complex matrix [AZ(ω)] is such that

    image

    and finally, the symmetric (NS × NS) complex matrix [ABEM(ω/cE)] is given by,

    image

    The given forces are written as

    image

    and

    image

    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 structural-acoustic models (extracted from Refs. [33-34]).

    9.1.1 Uncertainty and variability

    The designed structural-acoustic system is used to manufacture the real system and to construct the nominal computational model (also called the mean computational model or sometimes, the mean model) by using a mathematical-mechanical 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 structural-acoustic 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 the model 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 the uncertain 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 state-of-the-art can be found out, for instance, in Refs. [35-40].

    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. [42-44]. Two main methods can be used to take into account the model uncertainties (modeling errors).

    (i) Output-prediction-error 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 [46-47]. 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 output-prediction-error 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 reduced-order 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. [42-44,49-50]) 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. [42-44,49,53-54]. 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 model-parameter uncertainties and the model uncertainties that are induced by modeling errors (Refs. [8, 44, 55-60]),

    to extend the applicability of the theory to other areas (Refs. [61-69]),

    to extend the theory to new ensembles of positive-definite 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 medium-frequency range that include structural-acoustic systems, (Refs. [8,33,55,57,59-61,71-76]),

    to analyze nonlinear dynamical systems (i) with local nonlinear elements (Refs. [64, 77-83]) and (ii) with nonlinear geometrical effects (Refs. [84-85]).

    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. 86-87. 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. [89-91] while robust updating and robust design optimization with modeling uncertainties can be found in Refs. [82, 92-95].

       9.2 Uncertainties and stochastic reduced-order computational structural-acoustic 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. [33-34, 49, 59]). We apply this methodology to the reduced-order 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 [ABEM(ω/cE)] and in the wall acoustic impedance matrix [AZ(ω)]. Consequently, for fixed values of NS and N, the stochastic reduced-order computational structural-acoustic model of the order NS and N is written as,

    image
    image

    Where, for all fixed ω, the complex random vectors QS(ω) and Q(ω) of dimension NS and N are the solution of the following equation,

    image

    Where, the complex random matrix [AFSI(ω)] is written as,

    image

    The symmetric (NS × NS) complex random matrix [AS(ω)] is defined by,

    image

    Where, the positive-definite symmetric (NS × NS) real matrices [MS], [DS(ω)] and [KS(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,

    image

    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,

    image

    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 rank N ? 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 (NS × 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 reduced-order computational structural-acoustic 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 pX(x) is a probability density function on a real random variable X then, the entropy ε(pX) of pX is defined by,

    image

    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,

    image

    Consequently, we have ε(pX) = ?E{log(pX(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 [42-43] of an ensemble of positive-definite 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,

    image

    Where, [X] is finite and [Im] is the (m × m) identity matrix.

    9.3.2 Probability density function

    The value of the probability density function of the random matrix [G0] for the matrix [G] is noted p[G0]([G]) and this satisfies the usual normalization condition,

    image

    and the integration is carried out on the set of all the positive-definite symmetric (m × m) real matrices and it can be shown that the volume element

    image

    is,

    image

    Let δ be a positive real number defined by,

    image

    and this will allow the dispersion of the probability model of the random matrix [G0] that is to be controlled and ∥MF is the Frobenius matrix norm of the matrix ∥M∥such that

    image

    For 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 positive-definite symmetric (m × m) real matrix [G],

    image

    Where, the positive constant of normalization is c0, the constant c1 = (m + 1)(1 ? δ2)/(2δ2) and the constant c2 = (m + 1)/(2δ2) depends on m 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 [G0] is written as [G0] = [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 real-valued random variable [L]jj´ is written as, [L]jj´ = σmUjj´ where, σm = δ(m+1)?1/2 and Ujj´ is a real-valued Gaussian random variable with zero mean and variance equal to 1;

    for j = j´, the positive-value random variable [L]jj is written as,

    image

    in which Vj is a positive-valued Gamma random variable with probability density function Γ(aj, 1)

    where,

    image

    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

    image

    of all the random matrices such that

    image

    Where, [G0] 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 [MS]

    As there is no available information that concerns to the statistical dependency of [MS] with the other random matrices of the problem then, the random matrix [MS] is independent of all the other random matrices. The deterministic matrix [MS] is positive definite and consequently, it can be written as [MS] = [LMS]T[LMS] where, [LMS] is an upper triangular real matrix. By using the nonparametric probabilistic approach of uncertainties, the stochastic model of the positive-definite symmetric random matrix [MS] is then defined by,

    image

    Where, [GMS] is a (NS × NS) random matrix that belongs to ensemble

    image

    which 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, δMS.

       9.5 Stochastic modeling of the family of random matrices [DS(ω)] and [KS(ω)]

    As there is no available information concerning the statistical dependency of the random matrices {[DS(ω)], [KS(ω)]} with the other random matrices of the problem then, {[DS(ω)], [KS(ω)]} are independent of all the other random matrices. But we will see below that [DS(ω)] and [KS(ω)] are statistically dependent random matrices. For stochastic modeling of [DS(ω)] and [KS(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 positive-definite symmetric (NS × NS) real matrices [DS(ω)] and [KS(ω)] by random matrices [DS(ω)] and [KS(ω)] such that

    image
    image

    For ω ≥ 0, the construction of the stochastic model of the family of random matrices [DS(ω)] and [KS(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

    image

    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

    image

    of random matrices such that

    image

    Constructing the family

    image

    of random matrices by using the equation,

    image

    or equivalently by using the two following equations that are useful for computation:

    image

    and, for ω > 0,

    image

    Defining the family

    image

    of random matrices such that

    image

    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

    image

    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

    image

    Defining the random matrix

    image

    Constructing the random matrix

    image

    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] = [LM]T [LM] in which [LM] is an upper triangular real matrix. By using the nonparametric probabilistic approach of uncertainties, the stochastic model of the positive-definite symmetric random matrix [M] is then defined by,

    image

    Where, [GM] is a (N × N) random matrix that belongs to ensemble

    image

    and 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, δM.

       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,

    image

    Where, [GK] is a ((N ? 1) × (N ? 1)) random matrix that belongs to ensemble

    image

    and 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, δK.

    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] = [LK]T [LK] in which [LK] 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,

    image

    Where, [GK] is a (N × N) random matrix that belongs to ensemble

    image

    and 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 parameter δK.

       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 that NSN and that (NS × N) real matrix [C] is such that [C]q = 0 implies q = 0. If NNS 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 (NS × N) real matrix [R] is such that [R]T[R]=[IN] and where, the symmetric square matrix [T] is a positive-definite symmetric (N × N) real matrix. By using, Cholesky decomposition we then have [T] = [LT]T [LT] in which [LT] is an upper triangular matrix. The (NS × N) real random matrix [C] is then written as,

    image

    Where, [GC] is a (N × N) random matrix that belongs to ensemble

    image

    and it is defined in Section 9.3.4. It’s probability distribution and generator of independent realizations depend only on the dimension NS, N and on the dispersion parameter δc.

       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 [GMS], [GDS], [GKS(0)], [GM], [GK], and [GC] are represented by a vector δ such that

    image

    This belongs to an admissible set Cδ and it allows the level of uncertainties to be controlled for each type of operators that are introduced in the stochastic reduced-order computational structural-acoustic 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 structural-acoustic problem with respect to the uncertainties by varying δ in Cδ.

    For a given value of δ, there are two major classes of methods for solving the stochastic reduced-order computational structural-acoustic 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. [35-36,97]). 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 non-intrusive 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. [98-99]) is a very effective and efficient method as it as the four following advantages,

    it is a non-intrusive 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 three-dimensional domain ΩE whose boundary ∂ΩE is ΓE. This section is devoted to the construction of the frequency-dependent impedance boundary operator ZΓE (ω), for the external acoustic problem. We recall that the operator ZΓE (ω) is such that pE|ΓE (ω) = ZΓE (ω) υ(ω). It relates to the pressure field pE|ΓE (ω) that is exerted by the external fluid on ΓE to the normal velocity field, υ(ω) which is induced by the deformation of this boundary ΓE.

    Many methods can be found in literature for solving this problem: the boundary element methods, the artificial boundary conditions and the local/nonlocal non-reflecting boundary condition (NRBC) to take into account the Sommerfeld radiation condition at infinity, the Dirichlet-to-Neumann (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. [100-107]. This section is devoted to the presentation on the boundary element methods.

    The frequency-dependent impedance boundary operator ZΓE (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 of ZΓE (ω) consists of using the boundary integral formulations (Refs. [18, 108-115]). In the time domain, it uses the so-called Kirchhoff retarded potential formula (see for instance Refs. [116-117]). 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, 118-121]. Furthermore, most of those formulations yield unsymmetric fully populated complex matrices. The computational cost can then be reduced by using fast multipole methods [122-126].

    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 or irregular frequencies and they are also called as Jones eigenfrequencies[112,128-131]. Various methods are proposed in the literature to overcome this mathematical difficulty that arises in the boundary element method [3,129,132-137].

    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 zero-pressure 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 ΩE. 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 pressure p(x, ω) is related to ψ(x, ω) by using the following equation,

    image

    Where, ρE is the constant mass density of the external fluid at equilibrium. Let, cE be the constant speed of sound in the external fluid at equilibrium and let k = ω/cE be the wave number at frequency, ω. The exterior Neumann problem is written as,

    image
    image
    image

    with R = ||x|| → +∞, where ∂/∂R is the derivative in the radial direction and where υ(y) is the prescribed normal velocity field on ΓE. Equation (103) is the Helmholtz equation in the external acoustic fluid, Eq. (104) is the Neumann condition on external fluid-structure interface ΓE and Eq. (105) corresponds to the outward Sommerfeld radiation condition at infinity.

       10.2 Pressure field in ΩE and on ΓE

    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 ψsol. It depends linearly on the normal velocity υ[18,127]. Let

    image

    be the value of ψsol on ΓE. For all x in ΩE, let us introduce the linear operator R(x, ω/cE) such that

    image

    We also introduce the linear boundary operator BΓE (ω/cE) such that

    image

    By using Eq. (102), for all x in ΩE, the pressure field p(x, ω) is written as,

    image

    in which Zrad(x, ω) is called as the radiation impedance operator which can then be written as,

    image

    Similarly, the pressure field p|ΓE (ω) on ΓE is written as,

    image

    where, ZΓE (ω) is called the acoustic impedance boundary operator and this can then be written as,

    image

    Note that ZΓE (ω) is a nonlocal operator.

       10.3 Symmetry property of the acoustic impedance boundary operator

    The transpose of the operator BΓE (ω/cE) is denoted by tBΓE (ω/cE). It can then be proven (see Ref. [3]) that the following symmetry property,

    image

    and from Eq. (111), we deduce that

    image

    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 iωZΓE (ω) can be written as,

    image

    Where, MΓE (ω/cE) and DΓE (ω/cE) are two linear operators such that

    image
    image

    It can be shown that (Ref. [3]) the following positivity property of the real part DΓE (ω/cE) 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 ΓE. The first one is based on the use of a single-and double-layer potentials on ΓE. The second integral equation is obtained by a normal derivative on ΓE of the first one. We then, obtained the following system that relates ψsol to υ and this then allows BΓE (ω/cE) to be defined by using Eq. (107),

    image

    The linear boundary integral operators SS (ω/cE), SD (ω/cE)) and ST (ω/cE) are defined by,

    image
    image
    image

    Where, G(xy) is the Green function which is written as,

    image

    Where, r = ||xy||. In Eqs. (118) to (120), the brackets correspond to bilinear forms that allow the operators to be defined and the functions δυ and δψΓE are associated to functions υ and ψΓE. By taking in to consideration Eq. (117), let H(ω/cE) be the operator that is defined by,

    image

    It can be proven that the operator H(ω/cE)has the symmetric property, tH(ω/cE) = H(ω/cE). In Eq. (117), the first equation can be rewritten as,

    image

    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 ST(ω/cE) has a null space which is not reduced to {0}. This set of frequencies is called as the set of spurious or irregular frequencies. Consequently, as proven in Ref. [3], for a spurious frequency, ψΓE is the sum of solution ψΓE with an arbitrary element that belongs to the null space of the operator ST (ω/cE). 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,

    image

    This yields the solution

    image

    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

    image

    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 {ψsol (x, ω), x ∈ ΩE} of Eqs. (103) to (105) can be calculated by using the following integral equation,

    image

    For all x that is fixed in ΩE, we define the linear integral operators RS (x, ω/cE) and RD (x, ω/cE) by,

    image
    image

    By using Eq. (107), Eq. (123) can be rewritten as

    image

    From Eq. (106), we deduce that for all x fixed in ΩE,

    image

    and the radiation impedance operator Zrad(x, ω) is calculated by using Eqs. (109) and (127),

    image

       10.7 Symmetric boundary element method without spurious frequencies

    We use the finite element method to discretize the boundary integral operators SS(ω/cE), SD(ω/cE) and ST(ω/cE) (corresponding to a boundary element method). Let us consider a finite element mesh of boundary ΓE. Let V = (V1, …, VnE) and ΨΓE = (ΨΓE,1, …, ΨΓE,nE) be the complex vectors of the nE degrees-of-freedom that constituted of the values of υ and ΨΓE at the nodes of the mesh. Let SS(ω/cE), SD(ω/cE) and ST(ω/cE) be the full complex matrices that correspond to the discretization of the operators which are defined in Eqs. (118) to (120). The complex matrices SS(ω/cE) and ST(ω/cE) are symmetric. The finite element discretization of Eq. (117) yields,

    image

    Where, the symmetric complex matrix [H(ω/cE)] is the matrix,

    image

    In Eq. (129),

    image

    is the complex vector of the nodal unknowns that correspond to the finite element discretization of

    image

    The matrix [E] is the non-diagonal (nE × nE) real matrix that correspond to the discretization of the identity operator I. The elimination of ΨΓE in Eq. (129) yields a linear equation between

    image

    and V that defines the symmetric (nE × nE) complex matrix [BΓE (ω/cE)]. This corresponds to the finite element discretization of the boundary integral operator BΓE (ω/cE). We then have,

    image

    The particular elimination procedure discussed in Section 10.5 which avoids the spurious frequencies is defined below. Vector ΨΓE 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, [ST (ω/cE)] is invertible and the elimination in Eq. (129) is performed up to row number nE. If ω coincides with a spurious frequency ωα that is to say ω = ωα then, [ST (ω/cE)] is not invertible and its null space is a real subspace of

    image

    of dimension nα < nE . In this case, the elimination in Eq. (129) is performed up to row number nEnα. Practically, nα 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 numbers nEnα +1, …, nE are automatically satisfied. From Eq. (111), we deduce that the (nE × nE) complex symmetric matrix [ZΓE (ω)] of operator ZΓE (ω) is such that

    image

    Finally, the finite element discretization of the acoustic radiation impedance operator Zrad(x, ω) defined by Eq. (129) is written as,

    image

       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 QE(x, ω), and to a prescribed normal velocity field on ΓE. This is written as υ = iωu(ω)·nS where, ns is the unit normal to ΓE, external to the structure ΩS and u is the displacement field of the external fluid-structure interface ΓE. 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 ΩE. At any point x fixed in ΩE, the resultant pressure pE(x, ω) is written as,

    image

    where, prad(x, ω) is the field radiated by the boundary ΓE and it is submitted to the prescribed velocity field υ. It is written (see Eq. (108)) as,

    image

    The pressure pgiven(x, ω) is such that

    image

    where, pinc,Q(x, ω) is the pressure in the free space that is induced by the acoustic source QE and it is written as,

    image

    where, the Green function G is defined by Eq. (121) and ∂ψinc,Q/∂nS is deduced from Eqs. (137) and (102). The second term in the right-hand side of the Eq. (136) corresponds to the scattering of the incident wave (induced by the external acoustic source) by the boundary ΓE is considered as rigid and fixed.

    Pressure on ΓE. The resultant pressure on ΓE is then written as,

    image

    where, prad|ΓE (ω) is written as,

    image

    and the pressure field pgiven|ΓE (ω) on ΓE is such that

    image

    By substituting Eq. (139) in (138), it yields

    image

    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 ΩE, the radiated pressure p(x, ω) is given (see Eq. (108)) by p(x, ω) = Zrad(x, ω) υ. Let R and e be such that (see Fig. 6.)

    image

    Definition of integral operators

    image

    and

    image

    For all x = Re fixed in the external domain

    ΩE, we define the linear integral operators

    image

    and

    image

    by,

    image

    Where, Ne(y) is defined by,

    image

    Asymptotic formula for radiation impedance operator Zrad(x, ω). We have the following asymptotic formulas,

    image
    image

    From Eq. (127), we deduce the asymptotic formula for theradiation impedance operator as,

    image

    11. Conclusion

    We have presented an advanced computational formulation for dissipative structural-acoustics systems and fluid-structure interaction. This is adapted for the development of a new generation of software. An efficient stochastic reduced-order model in the frequency domain is proposed to analyze low- and medium-frequency 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.

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  • [Fig. 1.] Configuration of the system
    Configuration of the system
  • [Fig. 2.] Configuration of the structural-acoustic system for a liquid with free surface.
    Configuration of the structural-acoustic system for a liquid with free surface.
  • [Fig. 3.] Example of the structure and internal fluid finite element meshes.
    Example of the structure and internal fluid finite element meshes.
  • [Fig. 4.] Variabilities and the types of uncertainties in computational structural acoustics and fluid-structure interaction
    Variabilities and the types of uncertainties in computational structural acoustics and fluid-structure interaction
  • [Fig. 5.] Geometry of an external infinite domain.
    Geometry of an external infinite domain.
  • [Fig. 6.] Geometrical configuration.
    Geometrical configuration.