Didier Clouteau

Christian Soize

Y Le Guennec

R Cottereau
email: regis.cottereau@ecp.fr

A coupling method for stochastic continuum models at diff erent scales

Keywords: Multiscale modeling, Coupling numerical method, Stochastic mechanics, Arlequin method

In this paper, we present a novel approach that allows to couple two stochastic continuum models describing the same random medium at different observation scales. The coupling strategy is performed in the Arlequin framework, which is based on a volume coupling and a partition of the energy. Suitable functional space and coupling operator are chosen for the weak enforcement of the continuity between the two models. This choice ensures that the resulting mixed problem is well posed. The Monte-Carlo based numerical strategy for the solution of the mixed problem is briefly outlined. An application is presented, emphasizing on the interest of the chosen coupling operator. Finally, some remarks are provided concerning a stochastic multi-model coupling.

Introduction

Classical deterministic models provide global predictions that are satisfactory for many industrial applications. However, when one is interested in a very localized behavior or quantity, or when multiscale phenomena come into play, these models may not be sufficient. For instance the limited heterogeneity of a material modeled as a continuum may not have significant influence on its behavior at a large scale while it may influence greatly a local stress factor, which would typically be a fundamental quantity of interest for structural design. Unfortunately, for these problems, the information necessary to parametrize the relevant, very complex, models is usually not available. Stochastic methods have therefore been proposed [START_REF] Matthies | Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations[END_REF][START_REF] Frauenfelder | Finite elements for elliptic problems with stochastic coefficients[END_REF][START_REF] Nouy | Recent developments in spectral stochastic methods for the numerical solution of stochastic partial differential equations[END_REF][START_REF] Xiu | Fast numerical methods for stochastic computations: a review[END_REF][START_REF] Maître | Spectral methods for uncertainty quantification: with applications to computational fluid dynamics[END_REF][START_REF] Babuška | A stochastic collocation method for elliptic partial differential equations with random input data[END_REF]] and now appear unavoidable in multiscale modeling. Although the use of stochastic models and methods has expanded rapidly in the last decades, the related numerical costs are still often prohibitive. Hence, the application of these methods in a complex or industrial context remains limited. An important field of research is therefore concerned with the general reduction of the costs associated with the use of stochastic methods, for example by using iterative methods and preconditioners specially adapted to the structure of the matrices arising in the Stochastic Finite Element (FE) method [START_REF] Ghanem | Numerical solution of spectral stochastic finite element systems[END_REF][START_REF] Pellissetti | Iterative solution of systems of linear equations arising in the context of stochastic finite elements[END_REF][START_REF] Desceliers | Polynomial chaos representation of a stochastic preconditioner[END_REF][START_REF] Ullmann | A Kronecker product preconditioner for stochastic Galerkin finite element discretizations[END_REF][START_REF] Wheeler | A multiscale preconditioner for stochastic mortar mixed finite elements[END_REF], using reduced bases for the representation of random fields [START_REF] Doostan | Stochastic model reduction for chaos representation[END_REF][START_REF] Arnst | Probabilistic equivalence and stochastic model reduction in multiscale analysis[END_REF][START_REF] Soize | Reduced chaos decomposition with random coefficients of vector-valued random variables and random fields[END_REF], domain decomposition [START_REF] Sarkar | Domain decomposition of stochastic PDEs: theoretical formulations[END_REF][START_REF] Ganis | A stochastic mortar mixed finite element method for flow in porous media with multiple rock types[END_REF], or multigrid techniques [START_REF] Maître | A multigrid solver for two-dimensional stochastic diffusion equations[END_REF].

Another approach consists, when it is possible, in trying to localize the use of stochastic models to limited regions of the computational domain, and to model the rest of the domain with a deterministic model [START_REF] Cottereau | A stochastic-deterministic coupling method for continuum mechanics[END_REF][START_REF] Chevreuil | A multiscale method with patch for the solution of stochastic partial differential equations with localized uncertainties[END_REF]. This is typically possible only when a homogenized medium can indeed be defined. The homogenized model is then used for the bulk of the domain, and the stochastic model is used close to boundaries, loadings or geometrical features that preclude the use of homogenized models, and in areas over which the quantities of interest are defined. Note that the areas over which the model should be stochastic are not necessarily obvious, although error estimation can help in that matter [START_REF] Zaccardi | Error estimation and model adaptation for stochastic-deterministic coupling in the Arlequin framework[END_REF]. When it works, this approach can be seen as a model reduction technique, as the model is simplified wherever the simplification does not impact the quality of the estimation of the quantities of interest. It is appealing in terms of numerical cost for the direct solution of the corresponding problems, and also in terms of parameterization when the identification of the parameters of the models is considered.

There are unfortunately cases when homogenization theory cannot be applied straightforwardly, such as close to boundaries. This means in particular that over elongated domains, or when parameters fluctuate slowly with respect to the size of the domain, homogenization cannot be performed in the classical sense. However, in such cases, it might still be possible to perform a similar but more limited type of model reduction, and to partially upscale the parameters fields. A stochastic parameter fluctuating rapidly would hence be replaced by a parameter fluctuating more slowly over some parts of the computational domain rather than by a deterministic fully homogenized parameter. This is the approach followed in this paper, and it requires two ingredients: (i) a recipe for upscaling partially a random medium, and (ii) a numerical method for coupling stochastic models described at two different scales. We consider that the models at the two scales are stochastic, in the sense that they are driven by partial differential equations with stochastic parameter fields. The first model describes the material heterogeneity at a micro scale, with a parameter field typically oscillating over small distances. The second model is coarser, and describes the same material at a meso-scale, typically with smaller variations in both amplitude and wavelength. The parameter of the second model may be seen as a filtered (or partially homogenized) version of the parameter of the first model. We will consider a case in which the upscaling method is known (step (i) above), so that the micro-scale and meso-scale (partially upscaled) models are given. This paper concentrates on the proposal of a coupling method for two stochastic models (step (ii) above).

Note that recent papers on stochastic multiscale mechanics have rather focused on an integrated approach to the problems (i) and (ii) seen above. Extensions of the multiscale finite element method [START_REF] Hou | A multiscale finite element method for elliptic problems in composite materials and porous media[END_REF] and variational multiscale method [START_REF] Hughes | Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods[END_REF][START_REF] Hughes | The variational multiscale method -a paradigm for computational mechanics[END_REF] have been proposed for stochastic elliptic problems [START_REF] Velamur Asokan | A stochastic variational multiscale method for diffusion in heterogeneous random media[END_REF][START_REF] Xu | A multiscale stochastic finite element method on elliptic problems involving uncertainties[END_REF][START_REF] Ganapathysubramanian | Modeling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multiscale method[END_REF][START_REF] Chen | Analysis of the multiscale finite element method for nonlinear and random homogenization problems[END_REF] and for stochastic mixed formulations [START_REF] Velamur Asokan | Variational multiscale stabilized FEM formulations for transport equations: stochastic advection-diffusion and incompressible stochastic Navier-Stokes equations[END_REF][START_REF] Ganapathysubramanian | A stochastic multiscale framework for modeling flow through random heterogeneous porous media[END_REF][START_REF] Jiang | Stochastic mixed multiscale finite element methods and their applications in random porous media[END_REF]. In these extensions, the authors discuss in particular the way to compute the meso-scale functional bases in efficient ways, using hypotheses of weak randomness [START_REF] Ginting | A novel method for solving multiscale elliptic problems with randomly perturbed data[END_REF][START_REF] Bris | Multiscale finite element approach for "weakly" random problems and related issues[END_REF], interpolation [START_REF] Aarnes | Mixed multiscale finite element methods for stochastic porous media flows[END_REF] or stochastic collocation [START_REF] Ganapathysubramanian | Modeling diffusion in random heterogeneous media: data-driven models, stochastic collocation and the variational multiscale method[END_REF][START_REF] Dostert | Multiscale finite element methods for stochastic porous media flow equations and applications to uncertainty quantification[END_REF][START_REF] Jiang | A resourceful splitting technique with application to deterministic and stochastic multiscale finite element methods[END_REF]. Extensions of the heterogeneous multiscale method were also proposed [START_REF] Ming | Analysis of the heterogeneous multiscale method for elliptic homogenization problems[END_REF][START_REF] Ma | A stochastic mixed finite element heterogeneous multiscale method for flow in porous media[END_REF], and the asymptotic behavior of the extensions of both the multiscale finite element method and heterogeneous multiscale method are discussed in [START_REF] Bal | Corrector theory for MsFEM and HMM in random media[END_REF].

Besides considering a segregated approach of (i) and (ii), the present proposal differs from the papers above in its focus. Indeed, the quantities of interest that we aim at approximating are assumed to depend on the micro-scale properties only locally. When this assumption is appropriate, the bulk of the model can be rep-resented by a meso-scale model without degrading the evaluation of the quantity of interest, and the zone where the micro-scale is considered can be seen as a zoom in a global meso-scale model. This paper therefore stands in the line of classical adaptive mesh and local enrichment techniques [START_REF] Stein | Coupled model-and solution-adaptivity in the finite-element method[END_REF][START_REF] Melenk | The partition of unity finite element method: basic theory and applications[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF][START_REF] Düster | Applying the hp-d version of the FEM to locally enhance dimensionally reduced models[END_REF], global-local iterative methods [START_REF] Gendre | A two-scale approximation of the Schur complement and its use for non-intrusive coupling[END_REF][START_REF] Lozinski | Numerical zoom for advection diffusion problems with localized multiscales[END_REF][START_REF] Hager | Solving dynamic contact problems with local refinement in space and time[END_REF] and bridging methods [START_REF] Dhia | Multiscale mechanical problems: the Arlequin method[END_REF][START_REF] Brezzi | Analysis of a Chimera method[END_REF][START_REF] Wagner | Coupling of atomistic and continuum simulations using a bridging scale decomposition[END_REF][START_REF] Xiao | A bridging domain method for coupling continua with molecular dynamics[END_REF], widely developed for deterministic applications.

In particular, we construct our coupling technique in the Arlequin framework [START_REF] Dhia | Multiscale mechanical problems: the Arlequin method[END_REF], which is a bridging method. Within this framework, the main issue of the paper lies in the choice of coupling operator and spaces that ensure that the resulting mixed formulation is well posed and that the results obtained from the coupled method are representative of the full micro-scale solution. This paper describes an extension of [START_REF] Cottereau | Localized modeling of uncertainty in the Arlequin framework[END_REF][START_REF] Cottereau | A stochastic-deterministic coupling method for continuum mechanics[END_REF], which considered the coupling of a deterministic model with a stochastic one. It bears similarities also with the paper [START_REF] Chamoin | A stochastic coupling method for atomic-to-continuum Monte-Carlo simulations[END_REF] on discrete-to-continuum coupling, but the coupling operator is there completely different.

In Section 2, the macro-scale and meso-scale models of the random medium are described. In Section 3, the Arlequin modeling is introduced in the particular case of the coupling of two stochastic models. In Section 4 the numerical implementation is considered, based on the Monte-Carlo approach. In Section 5, some examples are provided. Finally, some conclusions are drawn in Section 6 and the extension of this method for two models driven by different partial differential equations (3D-beam coupling, for instance) is outlined.

Stochastic modeling and upscaling

In this section, we introduce the two stochastic continuum models that we will consider in the paper. We start with the micro-scale one. The meso-scale model is then obtained by considering a partial homogenization of the micro-scale mechanical parameter field. The micro-scale model is the reference one, and we only expect that the meso-scale solution can reproduce some of the features of the micro-scale solution. Finally, we will discuss at the end of this section the link between the partial upscaling considered here and classical homogenization.

Micro-scale (reference) model

Let us consider a domain Ω of R d , with outgoing normal vector n and smooth boundary ∂Ω, separated into Dirichlet and Neumann boundaries Γ D and Γ N , such that Γ D ∪ Γ N = ∂Ω, Γ D ∩ Γ N = ∅, and Γ D ∅. We consider Poisson's equation, with a random fluctuating mechanical parameter k m (x). In the particular case of homogeneous Dirichlet boundary conditions, the weak formulation of the corresponding stochastic boundary value problem reads: find u m ∈ V m such that

a m (u m , v) = f (v), ∀v ∈ V m , (1)
where andE[•] = Θ • dP denotes the mathematical expectation. The mechanical parameter k m (x) is modeled as a positive, second-order, mean-square continuous stochastic field indexed on R d , and defined on a probability space (Θ, F , P). We assume that it is statistically homogeneous and bounded almost surely for any

a m : V m ×V m → R and f : V m → R are defined, respectively, by a m (u, v) = E Ω k m (x)∇u • ∇v dx , f (v) = Ω f E[v] dx + Γ N g E[v] dx,
x ∈ Ω c m < k(x) < c M , with c m > 0. The functional space is V m = L 2 (Θ, H 1 0 ), with H 1 0 = {v ∈ H 1 (Ω), v |Γ D = 0}.
Endowed with the appropriate inner product and norm, V m is a Hilbert space. Using Lax-Milgram theorem, it can be proved that the problem (1) has a unique solution u m (see for instance [START_REF] Babuška | Galerkin finite element aproximations of stochastic elliptic partial differential equations[END_REF]). An approximation of that solution can then be obtained, for example, by using a Stochastic FE method [START_REF] Ghanem | Stochastic finite elements: a spectral approach[END_REF][START_REF] Stefanou | The stochastic finite element method: past, present and future[END_REF] or a Monte Carlo approach [START_REF] Robert | Monte Carlo statistical methods[END_REF].

The field k m (x) will be called throughout micro-scale mechanical parameter field, the boundary value problem (1) will be called micro-scale mono-model problem, and its solution u m (x) will be called micro-scale monomodel solution. Note that we use the term mono-model for the problem and its solution, in order not to mistake them with the quantities relevant to the coupled system that will be introduced further down, and that will include a micro-scale component.

The micro-scale mechanical parameter field k m (x) is fluctuating, and those fluctuations are controlled by its power spectral density Rm (ζ) (PSD), which is the Fourier transform in

R d of its autocovariance R m (y) = E[(k m (x) -k m )(k m (x + y) -k m )],
where k m = E[k m (x)] is the expectation of k m (x), and we assume that the auto covariance is integrable on R d . Although the power spectral density and autocovariance in general depend on the considered direction, we assume here that they are isotropic, so that they only depend on the amplitude of the y and ζ. Furthermore, we assume that a correlation length m can be defined as

m = R |R m (y)|dy/σ 2 m , where σ 2 m = R m (0) is the variance of k m .
Examples of realizations of micro-scale parameter fields can be observed in Figures 1 (dashed

Meso-scale model

From the given PSD Rm (ζ) of the micro-scale parameter field k m (x), a hierarchy of upscaled parameter fields can be derived by trimming its tails. This trimming translates into an increase of the correlation length and a decrease of the variance. Figure 1 shows an example of such trimming on the PSD and the corresponding trajectory in 1D-model (solid lines). Figure 2 presents a similar example in 2D (lower figure). On both figures, the increase of correlation length and decrease of variance is clear. In this paper, and for a given micro-scale PSD, we refer to meso-scale mechanical parameter (or coarse mechanical parameter) as the field k M (x) corresponding to the trimmed PSD, with a correlation length

M > m , variance σ 2 M < σ 2 m
, and average k M , which is a priori different from k m (see the discussion in Section 2.3).

The construction above naturally introduces a segmentation of the probability space (Θ, F , P) into two probability spaces (Θ M , F M , P M ) and

(Θ m , F m , P m ) such that k M (x) is defined on F M ⊗ B(Ω), where B(Ω) is the Borel σ-algebra generated by the open subsets of Ω, (Θ, F ) = (Θ M × Θ m , F M ⊗ F m )
, and P is the unique product measure of P m and P M , such that

∀X m ∈ F m and X M ∈ F M , P(X m × X M ) = P m (X m )P M (X M ). We in- troduce the additional notations E M [•] = Θ M • dP M and E m [•] = Θ m • dP m .
From this definition of the meso-scale parameter field k M (x), we also define a meso-scale mono-model boundary value problem, which states: find the meso-scale mono-model solution u M (x) ∈ V M , such that

a M (u M , v) = f (v), ∀v ∈ V M , (2)
where a M :

V M × V M → R is defined by a M (u, v) = E M Ω k M (x)∇u • ∇v dx , and V M = L 2 (Θ M , H 1 0 )
. Endowed with appropriate inner product and norm, V M is a Hilbert space. As before, it can be proved that the problem (2) has a unique solution u M .

Partial upscaling and homogenization

Extending the upscaling process described above to the limit M → +∞, one obtains a deterministic homogeneous mechanical parameter k * . In general, this parameter k * is not equal to the parameter obtained from classical homogenization [START_REF] Papanicolaou | Boundary value problems with rapidly oscillating random coefficients[END_REF][START_REF] Bourgeat | Approximations of effective coefficients in stochastic homogenization[END_REF][START_REF] Tartar | The general theory of homogenization: a personalized introduction[END_REF][START_REF] Cottereau | Numerical strategy for unbiased homogenization of random media[END_REF]]. However, in the particular case of a micro-scale parameter with lognormal first-order marginal density in 2D, the homogenized tensor is equal to the geometric average of the field

k * = k m /(1 + σ 2 m /k 2 m ) 1/2 .
Hence, if we assume that both the micro-scale and meso-scale models follow a log-normal first-order marginal density, the average k M of the meso-scale model can be chosen such that the two models are coherent in the sense that they have the same homogenized coefficient

k * = k m /(1+σ 2 m /k 2 m ) 1/2 = k M /(1 + σ 2 M /k 2 M ) 1/2 .
With this hypothesis, a meso-scale model k M (x), with trimmed triangular spectrum and lognormal first-order marginal density, can be defined starting from a micro-scale model k m (x) for any choice of correlation length M > m . Using the technique described in [START_REF] Noetinger | The future of stochastic and upscaling methods in hydrogeology[END_REF], it is furthermore possible to generate realizations of the two fields k M (x) and k m (x) that correspond one to the other, realization by realization, as illustrated in Figures 1 and2.

It should be noted that if the mechanical parameter fields k m and k M follow log-normal first-order marginal densities, they are not bounded almost surely. Lax-Milgram theorem can then not be used to prove existence and uniqueness of the micro-and meso-scale mono-model solutions. Hence, we will rather use trimmed log-normal first-order marginal densities. The inferior bound b min and superior bound b max should be chosen far enough from unity: 0 < b min << 1 and 1 >> b max , so that the distribution remains close to a log-normal one, and the analytical formula can still be used (this is only for simplicity). Also, the bounds should be chosen such that b min = 1/b max , so that the first-order marginal distributions of the variables and their inverse remain equal and the formula above for the 2D homogenized coefficient k * [61, Chap. 3] remains valid.

Note that this question of upscaling of random microscale models has been widely treated in the literature [START_REF] Wen | Upscaling hydraulic conductivities in heterogeneous media: an overview[END_REF][START_REF] Renard | Calculating equivalent permeability: a review[END_REF][START_REF] Farmer | Upscaling: a review[END_REF][START_REF] Noetinger | The future of stochastic and upscaling methods in hydrogeology[END_REF]. In particular, some upscaling techniques seem to provide partial upscaling naturally [START_REF] Brewster | A multiresolution strategy for numerical homogenization[END_REF][START_REF] Dorobantu | Wavelet-based numerical homogenization[END_REF]. The choice of micro-scale and mesoscale models described in this section does not pretend to be representative of the state-of-the-art in this field. It is rather one very particular choice for which a sequence of models can be constructed analytically in a manner consistent with (classical) homogenization. Our objective in this paper does not lie in the actual process of partial upscaling and homogenization, but in the description of a coupling technique for two stochastic continuum models at two different scales.

Coupling method in a stochastic framework

This section now concentrates on this issue of coupling two stochastic continuum media. It constitutes the core of this paper. We consider again the domain Ω and now divide it into two overlapping subdomains Ω m and Ω M such that Ω m ∪ Ω M = Ω (see Figure 3). The Dirichlet and Neumann boundaries separate into corresponding overlapping boundaries Γ m D , Γ M D , Γ m N and Γ M N . Note that one of the two Dirichlet boundaries may be empty. On each subdomain, a different model will be considered, with a different mechanical parameter field, namely k m (x) on Ω m and k M (x) on Ω M . As with other similar techniques, the general idea is here to replace, where it is relevant, a fine-scale model by a coarser one in order to focus the available resources on a limited area of the domain. This section describes the coupled problem then posed on Ω. This coupling problem is developed in the Arlequin framework [START_REF] Dhia | Multiscale mechanical problems: the Arlequin method[END_REF][START_REF] Dhia | Mathematical analysis of the mixed Arlequin method[END_REF][START_REF] Dhia | The Arlequin method as a flexible engineering design tool[END_REF][START_REF] Dhia | Further insights by theoretical investigations of the multiscale Arlequin method[END_REF][START_REF] Cottereau | Localized modeling of uncertainty in the Arlequin framework[END_REF][START_REF] Cottereau | A stochastic-deterministic coupling method for continuum mechanics[END_REF][START_REF] Zaccardi | Error estimation and model adaptation for stochastic-deterministic coupling in the Arlequin framework[END_REF]. This framework is based on three ingredients: (i) splitting of the domain into overlapping subdomains to which different models are attached, (ii) introduction of weight functions to dispatch the global energy among the models, (iii) imposition of a weak compatibility constraint between the solutions of the different models. The developments proposed below for a coupling between two stochastic continuum models stand mainly in the choice of the coupling operator and space (ingredient (iii) above).

Arlequin formulation

We choose a so-called coupling volume Ω c ⊂ (Ω m ∩ Ω M ), over which the two models are assumed to exchange information. In the particular case of homogeneous Dirichlet boundary conditions (the extension to non-homogeneous boundary conditions is obvious), the mixed Arlequin problem reads: find

(w m , w M , Φ) ∈ W m × W M × W c such that              A m (w m , v) + C(Φ, v) = f m (v), ∀v ∈ W m A M (w M , v) -C(Φ, v) = f M (v), ∀v ∈ W M C(Ψ, w m -w M ) = 0, ∀Ψ ∈ W c , (3)
where the bilinear forms

A m : W m × W m → R, A M : W M × W M → R, and C : W c × W c → R are defined by A m (w, v) = E Ω m α m k m (x)∇w • ∇v dx , A M (w, v) = E M Ω M α M k M (x)∇w • ∇v dx , and
C(w, v) = E Ω c (wv + κ∇w • ∇v) dx , (4)
where κ is a constant essentially introduced for dimensionality purposes [START_REF] Dhia | The Arlequin method as a flexible engineering design tool[END_REF], and where the linear forms

f m : W m → R and f M : W M → R are defined, respec- tively, by f m (v) = Ω m α m f E[v]dx + Γ m D α m gE[v]dx and f M (v) = Ω M α M f E[v]dx + Γ M D α M gE[v]dx.
The weights are chosen such that:

             α m , α M ≥ 0 in Ω m ∪ Ω M , α m + α M = 1 in Ω m ∪ Ω M , α m , α M constant in (Ω m ∪ Ω M ) \ Ω c . (5)
The functional spaces are

W M = L 2 (Θ M , H 1 M ) with H 1 M = {w ∈ H 1 (Ω M ), w |Γ M D = 0}
, for the meso-scale space, and

W m = L 2 (Θ, H 1 m ), with H 1 m = {w ∈ H 1 (Ω m ), w |Γ m D = 0}
, for the micro-scale space. Note that the equalities on the boundary conditions should be understood as P M -almost surely and almost-everywhere in Ω M , and P-almost surely and almost-everywhere in Ω m , respectively. The so-called mediator space W c is defined in the Section 3.2.

One can consider that the system (3) consists of three equations: (i) one governing the behavior of the microscale model, weighted by α m (x) and with a loading arising in the coupling volume Ω c embodied in the operator C, (ii) one governing the behavior of the meso-scale model, weighted by α M (x) and with a loading opposite to the previous in the coupling volume Ω c , and (iii) one enforcing the weak compatibility between the two solutions w m (x) and w M (x).

Mediator space

There are two natural choices for the mediator space: (i) the restriction of the functional space for the micro-scale model over the coupling volume

W (i) c = L 2 (Θ M , H 1 (Ω c ))
, and (ii) the restriction of the functional space for the meso-scale model over the coupling volume W (ii) c = L 2 (Θ, H 1 (Ω c )). However, the former choice (i) leads to an unstable mixed problem when no Dirichlet boundary condition is enforced on the micro-scale model (Γ m D = ∅) for example. This is an important case, that includes in particular zooms Ω m ⊂ Ω M with ∂Ω m ∩ ∂Ω = ∅.

The latter choice of mediator space (ii) does lead to a stable mixed formulation. The coupling operator C induces in that case solutions that are such that w m = w M almost everywhere in Ω c and P-almost surely. Based on observations on Figure 4 for example, we consider that this is too demanding on the solutions. Figure 4 presents an example of micro-scale mono-model and meso-scale mono-model solutions obtained by solving Eq. ( 1) and ( 2), respectively, for a realization of k m and a corresponding upscaled version of it k M for a classical 1D traction problem. The parameter fields correspond one to other, although k m presents a higher frequency content than k M , and we observe that the solutions behave similarly. Choosing W (ii) c as mediator space would essentially impose that the two solutions overlap in the coupling volume, among other things limiting the frequency content of the micro-scale solution w m to that of the coarser meso-scale solution. An appropriate choice of mediator space should rather try to enforce a weaker equality between u m and u M in the coupling volume, allowing the two solutions w m and w M of the coupled problem (3) to oscillate at different frequencies. The mediator space is therefore chosen as the sum of a stochastic field related to the meso-scale variation, which imposes a strong equality constraint on the lowfrequency parts of u m and u M , and a stochastic variable related to the high-frequency fluctuations, impos-ing only a weak constraint on that part of the spectrum of u m and u M :

W c = L 2 (Θ M , H 1 (Ω c )) * ⊕ L 2 (Θ m , R), (6)
where the Bochner space

L 2 (Θ M , H 1 (Ω c )) * = L 2 (Θ M , H 1 (Ω c ))\{v | E M [ Ω c
vdx] = 0} explicitly excludes function with a deterministic spatial average. This exclusion is necessary in order that there be no intersection between the two spaces above. This also explains the introduction of matrices S c for the numerical approximation of system (3) (see Section 4 for more details, and in particular Equation ( 11)).

After some algebraic manipulations, this choice of the coupling space translates into the following weak equality constraint in the last line of Equation ( 3 [START_REF] Ghanem | Numerical solution of spectral stochastic finite element systems[END_REF] On the one hand, this equation imposes that the solutions w m and w M be equal almost everywhere in Ω c and almost surely with respect to the low-frequency fluctuations of the medium (P M -almost surely). On the other hand, it imposes that the average in space (over Ω c ) of the high frequency fluctuations of w m vanishes P malmost surely.

): ∀Ψ = ψ(x, θ M ) + θ m ∈ W c , with ψ(x, θ M ) ∈ L 2 (Θ M , H 1 (Ω c )) * , θ m ∈ L 2 (Θ m , R), 0 = C (E m [Ψ], w M -E m [w m ]) -E m θ m Ω c E M [w m -E m [w m ]]dx

Arlequin coupled solution

Each of the solutions w m and w M is spatially supported only over a subdomain of Ω. We discuss here how to transform these localized solutions into global solutions. The micro-scale information is available only within Ω f m = Ω m \Ω c , where the micro-scale parameter k m is actually defined, and where the weight function actually puts emphasis on it. The micro-scale coupled solution is therefore only defined within this subdomain

u arl m ∈ L 2 (Θ, H 1 0 (Ω f m )) as u arl m = α m w m + α M w M , (8)
assuming extension by 0 of the functions wherever they are not naturally defined. On the other hand, the meso-scale information is available everywhere, either through k M , or through k m with also contains that mesoscale information. We can therefore define a global meso-scale coupled solution

u arl M ∈ L 2 (Θ M , H 1 brok ) as u arl M = α m E m [w m ] + α M w M , (9)
where H 1 brok = {w ∈ H 1 brok (Ω), w |Γ D = 0}, and discontinuities in the solution u arl M may appear when the weight functions α m (x) and α M (x) are not continuous.

Numerical implementation

To approximate the solution (w m , w M , Φ) of Equation (3), a mesh T m of n m elements E m is constructed over Ω m , a mesh T M of n M elements E M is constructed over Ω M , and a mesh T c of n c elements E c is constructed over Ω c . The following functional spaces are introduced:

V H m = {v ∈ P 1 (E m ), v |Γ m D = 0}, V H M = {v ∈ P 1 (E M ), v |Γ M D = 0} and V H c = {v ∈ P 1 (E c
)}, composed of linear polynomials on the elements of the different meshes (higher-order polynomials could obviously be used). We associate to these functional spaces the bases: and{v c i } 1≤i≤n c . The mixed system (3) is transformed, after space discretization, into the random matrix system:

{v m i } 1≤i≤n m , {v M i } 1≤i≤n M ,
E [A(θ m , θ M )U(θ m , θ M )] = F, ( 10
)
where θ m and θ M indicate dependency on Θ m and Θ M , respectively, and where

A(θ m , θ M ) =                    A m (θ m , θ M ) 0 C m S m 0 0 A M (θ M ) -C M -S M 0 C T m -C T M 0 0 S T c S T m -S T M 0 0 0 0 0 S c 0 0                    , (11)
with, for n = {m, M},

A n,i j (θ) = Ω n α n (x)k n (x, θ)∇v n i (x) • ∇v n j (x)dx , ( 12
)
C n,i j = Ω c v c i v n j + κ∇v c i • ∇v n j dx , (13)
and, for n = {m, M, c},

S n, j = Ω c v n j . (14)
The load vector is

F = [F m F M 0 0 0] T , where, for n = {m, M}, F n, j = Ω n α n f v n j dx + Γ n D α n gv n j dx. The unknown vector is defined by U(θ m , θ M ) = [U m (θ m , θ M ) U M (θ M ) Ψ M (θ M ) Ψ m (θ m ) Λ] T , where the unknown fields are such that w m (x, θ m , θ M ) = U m, j (θ m , θ M )v m j (x), w M (x, θ M ) = U M, j (θ M )v M j (x), Φ(x, θ m , θ M ) = Ψ M, j (θ M )v c j (x) + Ψ m (θ m )
, and the additional scalar Λ is a Lagrange multiplier ensuring that Ω c Ψ M, j v c j dx = 0. This last equation is necessary to restrict the elements of L 2 (Θ M , H 1 (Ω c )) * to non-constant functions, as described in Section 3.2.

This matrix system can be solved in two different ways. This first consists in solving it using a looped Monte Carlo approach. For each realization of k M , a number of Monte Carlo realizations of k m are generated corresponding to that k M and the statistics (with respect to Θ m ) of the solution are estimated for a fixed θ M using a condensation approach similar to what is done in [START_REF] Cottereau | A stochastic-deterministic coupling method for continuum mechanics[END_REF]. Solving for several realizations of k M then provides the statistics with respect to both Θ m × Θ M . In that approach, the matrices that are inverted for each of the Monte Carlo samples are small, since the micro-scale domain is expected to be much smaller than the mesoscale domain. However, these matrices are Schur complements and in general are not sparse, so that iterative inversion algorithms are usually less efficient.

Another approach consists in considering that each entry of the solution vector U is a random variable in L 2 (Θ, R), solving the full system for each Monte Carlo sample, and obtain the solutions by taking the appropriate averages for each element of U. The inconvenient of that approach is that the matrices to invert are larger, but on the other hand, they are sparse. From first tests that we have performed, it seems that the second approach is more interesting in terms of CPU time. However, these tests were performed for rather large Ω m (with respect to Ω M ) so that they may not be fully objective. More tests are required to clarify this issue.

Numerical examples

We present here two applications. The first one consists in a simple 1D traction problem. The 1D setting allows to compare the micro-scale mono-model, mesoscale mono-model and coupled solutions. A second application presents a 2D problem, more relevant for real problems, where localization of stresses calls for a micro-scale model localized around a geometrical feature.

1D bar in traction

The first application is mainly introduced to compare the solutions obtained by solving the micro-scale mono-model boundary value problem (1), the mesoscale mono-model boundary value problem (2) and the Arlequin mixed problem (3). In particular, the 1D setting allows to observe in detail the behavior of the different solutions in the coupling volume. We consider a bar defined over the domain Ω = [0 1] and submitted to a constant unit bulk load f = 1 and to a differential of Dirichlet boundary conditions. All quantities are non-dimensionalized for notational simplicity.

Micro-scale mono-model problem. We consider a micro-scale mechanical parameter k m (x) statistically homogeneous, with (trimmed, see Section 2.3) lognormal first order marginal law, expectation k m = 1, variance σ 2 m = 2.5 × 10 -2 , triangular PSD and correlation length m = 1 × 10 -2 . The micro-scale mono-model solution u m verifies, P-almost surely, and almost everywhere in Ω:

d dx k m du m dx + f = 0 , (15)
and the boundary conditions are u m (0) = 0 and u m (1) = 1 almost surely. The analytical solution for the microscale mono-model solution is:

u m (x) = x 0 k * -f x k m (x ) dx , (16)
where k * verifies 1 = 1 0 (k *f x )/k(x )dx almost surely, and can be approximated when m << |Ω| by the homogenized parameter, which is equal to the harmonic average in 1D:

k * = E[k -1 m ] -1 = k m /(1+σ 2 m /k 2 m ) ≈ 0.
976. Finally, we have the following equation for the gradient: Using a linear finite element method with 1000 homogeneous interval elements, and 10, 000 Monte Carlo samples, one can estimate the statistics of the microscale mono-model solution. Figure 5 displays the average, the 90%-confidence interval, and one realization for that solution and its gradient. A pc-confidence interval for a random variable X is defined as:

du m dx = k * -x k m (x) . (17)
IC X = E[X] ± E[(X -E[X]) 2 ] 1 -pc . ( 18
)
On Figure 5, one can observe the rapid fluctuations of the micro-scale mono-model solution. As the observations are much clearer for the gradient than for the solution itself, we will only observe gradients from now on.

Meso-scale mono-model problem. We now consider a meso-scale model for the same 1D traction bar problem. We consider a meso-scale parameter field k M (x), obtained by upscaling, as described in Section 2, of the micro-scale parameter field k m (x) described above.

The meso-scale parameter field follows a (trimmed) lognormal first-order marginal law with a trimmed triangle PSD with correlation length set to M = 2 × 10 -1 , variance σ 2 M = 1.2 × 10 -3 and average value k M = 0.976 (in order to have the same homogenized coefficient k * for both models). The meso-scale mono-model boundary value problem is the same as in Equation ( 15), using the meso-scale parameter field k M (x), and with a solution u M . The statistics of that meso-scale mono-model solution are estimated using the same numerical method and parameters as for the micro-scale mono-model. Figure 6 displays the average, the 90%-confidence interval, and one realization for the gradient of that solution. As expected, the fluctuations of the gradient of the mesoscale mono-model solution u M are smoother than for the micro-scale mono-model solution u m , and the confidence interval is smaller. . For simplicity, the two meshes are embedded in Ω c , although this is not a restriction of the method. The mesh corresponding to the spatial discretization of the mediator space W c follows the mesh of Ω M (therefore with 5 elements). The number of Monte Carlo samples is set to 100, 000, using 1000 samples of k M (x) and, for each sample of k M (x), 100 samples of k m (x), as described in Section 4. The parameter κ is set to 10 -3 .

The weight functions are set as in Equation ( 5), and linear between 10 -3 and (1 -10 -3 ) in the coupling zone. The total computation time was 200 s on a four 3.3 GHz processor computer. The raw output of the Arlequin coupled system are the two solutions w m (x) and w M (x). Their gradients are plotted in Figure 7 without any post-treatment. For one realization, the expected behavior that w M be a lowfrequency filtered version of w m is observed. Also, as induced by Equation ( 7), the ensemble average (with respect to P) of the two solutions matches in the coupling domain Ω c . Comparisons and comments. The micro-scale and meso-scale coupled solution can be constructed according to Equations ( 8) and ( 9), respectively. As stated in its definition, the micro-scale coupled solution u arl m is only defined in Ω m \Ω c , where the micro-scale information is actually available and put emphasis on (through the weight functions). On the contrary, the meso-scale coupled solution u arl M is reconstructed over the entire domain. The two coupled solutions u arl m and u arl m are plotted in Figures 8 and9, respectively. On the same figures, each of them is compared to its mono-model counterpart, in terms of average and confidence interval. A remarkable match is observed. It is interesting to note in Figure 9 that there is a smooth transition in the coupling area Ω c between a gradient defined over large elements (with gradient constant over length 0.2) and gradient defined over small elements (with gradient constant over length 0.001). This smooth transition is due to the introduction of the weight functions that balance the energy between two meshes with different sizes.

2D indented plate

Let now consider the example of a plate in traction. The plate occupies a domain Ω that is inside the box [-1, 1]×[0, 1]. This plate is indented by a notch of 1 unit length width and 0.5 unit length depth. There is no bulk modulus. The boundary conditions are u(x = -1, y) = 0 and u(x = 1, y) = 2, and ∇u • n = 0 on the other edges (n being the normal to the edge). The aim is to localize the full stochastic model in the bottom of the notch, the meso-scale model expresses itself on the remaining of the structure. The mechanical parameter is chosen to follow a lognormal law of unit mean and variance σ 2 m = 0.025 with a triangular correlation model of correlation length m = 0.05. The micro-scale area is localized to Ω m =..., nombre d elment The meso-scale model expresses itself on the remaining of the domain. Its correlation length is M = 0.1 and its variance σ 2 M = 0.01 . The domain is meshed with triangular element with maximum edge size equal to 0.1. The mesh is refined near the micro-scale zone in order to have an efficient coupling. The quantities of interest will be the displacement and its gradient (proportional to the stress). Results will be displayed on the entire structure and along the line y = 0.5. Figure 9: Comparison of the gradient of the meso-scale mono-model solution u M (x) and of the meso-scale coupled solution u arl M : average (dashed and solid lines, respectively), 90%-confidence interval (dashed line and grey shade, respectively) and one realization of the Arlequin solution (solid line).

Conclusion

We presented here a novel coupling approach for the approximation solution of complex multiscale stochastic problems. This coupling method is based on the Arlequin framework, with a particular choice of mediator space that can be seen as an extension of a stochasticdeterministic coupling space introduced in [START_REF] Cottereau | Localized modeling of uncertainty in the Arlequin framework[END_REF][START_REF] Cottereau | A stochastic-deterministic coupling method for continuum mechanics[END_REF]. The chosen coupling strategy ensures that the solutions of the micro-and meso-scale models are equal almost everywhere in the coupling volume and almost surely with respect to the low-frequency fluctuations of the medium, and that the average in space of the high frequency fluctuations of the micro-scale solution vanishes. The numerical implementation is performed using two Monte-Carlo loops intertwined.

As this coupling approach stands in the Arlequin framework, it can be adapted to the case when the two models are different, in the sense that they are governed by different sets of partial differential equations. For example, it may be interesting to consider the coupling of a micro-scale 3D elastic model with a meso-scale beam model. Likewise, atomistic-to-continuum coupling approach may be approached. Although the mediator spaces should be modified accordingly, this type of coupled problems, with appropriate levels of randomness on each subdomain, would be extremely interesting in terms of applications.

  lines) and 2 (upper figure), respectively in 1D and 2D, along with an example of a triangular PSD in Figure 1 (upper figure).

Figure 1 :

 1 Figure 1: PSD (upper figure) and trajectories (lower figure) of two 1D centered normal fields at different scales. The dashed lines correspond to a micro-scale field k m (x) with triangular spectrum, unit correlation length m = 1 m, and unit variance σ 2 m = 1 (N/m) 2 . The solid lines correspond to the corresponding meso-scale random field k M (x) with trimmed triangular spectrum, correlation length M = 3 m, and variance σ 2 M = 1/3 (N/m) 2 .

Figure 2 :

 2 Figure 2: Trajectories of two 2D centered normal fields at different scales. The upper figure corresponds to a micro-scale field k m (x) with triangular spectrum, unit correlation length m = 1 m, and unit variance σ 2 m = 1 (N/m) 2 . The lower figure corresponds to the corresponding meso-scale random field k M (x) with trimmed triangular spectrum, correlation length M = 2 m, and variance σ 2 M = 1/2 (N/m) 2 .

Figure 3 :

 3 Figure 3: General setting of the Arlequin model: overlapping microscale domain Ω m and meso-scale domain Ω M and coupling volume Ω c .

Figure 4 :

 4 Figure 4: Gradients of the micro-scale mono-model solution u m (x) (solid line) and meso-scale mono-model solution u M (x) (dashed line) for a bar under traction and unit bulk load. The micro-scale mechanical parameter follows a (trimmed) log-normal first-order marginal density, triangular PSD with m = 3 × 10 -3 m and σ 2 m = 2.5 × 10 -2 (N/m) 2 . The meso-scale mechanical parameter is obtained as described in Section 2 for M = 4.5 × 10 -3 m and σ 2 M = 1.65 × 10 -2 (N/m) 2 .

  Note that we have E[u m ] = x(3x)/(2k * ) and E[du m /dx] = (3 -2x)/(2k * ).

Figure 5 :

 5 Figure 5: Micro-scale mono-model solution (upper figure) and its gradient (lower figure): average (dashed lines), 90%-confidence interval (grey shades) and one realization (solid lines).

Figure 6 :

 6 Figure 6: Gradient of the meso-scale mono-model solution u M (x): average (dashed line), 90%-confidence interval (grey shade) and one realization (solid line).

Figure 7 :

 7 Figure7: Gradients of the micro-scale w m (x) and meso-scale w M (x) solutions of the Arlequin coupled problem: average (dashed lines), 90%-confidence interval (grey shades) and one realization (solid lines).

Figure 8 :

 8 Figure 8: Comparison of the gradient of the micro-scale mono-model solution u m (x) and of the micro-scale coupled solution u arlm : average (dashed and solid lines, respectively), 90%-confidence interval (dashed line and grey shade, respectively) and one realization of the Arlequin solution (solid line).

Figure 10 :

 10 Figure 10: Coarse (red points) : ∆x = ∆y = 0.1, 1 = 0.5, γ = 0.2, fine mesh (blue lines): ∆x = ∆y = 0.01, 2 = 0.05, coupling area between green and red lines.

Acknowledgements

This work was partially supported by the ANR project TYCHE (Advanced methods using stochastic modeling in high dimension for uncertainty modeling, quantification and propagation in computational mechanics of solids and fluids), with project number ANR-2010-BLAN-0904.


R Ohayon
email: roger.ohayon@cnam.fr

C Soize

R Sampaio
email: rsampaio@puc-rio.br

Variational-Based Reduced-Order Model in Dynamic Substructuring of Coupled Structures Through a Dissipative Physical Interface: Recent Advances

Variational-based reduced-order model in dynamic

substructuring of coupled structures through a dissipative physical interface: Recent advances

Introduction

The mathematical aspects related to the variational formulation, existence and uniqueness, finite element discretization of boundary value problems for elastodynamics can be found in [12,23,[START_REF] Hughes | The finite element method: Linear static and dynamic finite element analysis[END_REF][START_REF] Oden | An introduction to the mathematical theory of finite elements[END_REF][START_REF] Ohayon | Structural dynamics in Encyclopedia of Applied and Computational Mathematics[END_REF][START_REF] Sanchez-Hubert | Vibration and coupling of continuous systems -Asymptotic methods[END_REF]. General mechanical formulations in computational structural dynamics, vibration and substructuring techniques can be found in [START_REF] Argyris | Dynamics of structures[END_REF][START_REF] Bathe | Numerical methods in finite element analysis[END_REF][START_REF] Bathe | Finite element procedures[END_REF][START_REF] Craig | Fundamentals of structural dynamics[END_REF][START_REF] Geradin | Mechanical vibrations: Theory and applications to structural dynamics[END_REF][START_REF] Meirovitch L ; Meirovitch | Computational methods in structural dynamics. Sijthoff and Noordhoff 53[END_REF]. In structural dynamics and coupled problems such as fluid-structure interaction, general computational methods can also be found in [START_REF] Bazilevs | Computational fluid-structure interaction[END_REF][START_REF] Clough | Dynamics of structures[END_REF][START_REF] Felippa | Partioned analysis of coupled mechanical systems[END_REF][START_REF] Har | Advances in computational dynamics of particles, materials and structures[END_REF][START_REF] Leung | Dynamic stiffness and substructures[END_REF][START_REF] Morand | Fluid structure interaction[END_REF][START_REF] Zienkiewicz | The finite element method for solid and structural mechanics[END_REF] and algorithms for solving large eigenvalue problems in [START_REF] Chatelin | Eigenvalues of matrices -Revised edition[END_REF][START_REF] Sameh | Eigenvalue and singular value problems 608-615 Encyclopedia of parallel computing[END_REF][START_REF] Saad | Numerical methods for large eigenvalue problems[END_REF]. For computational structural dynamics in the low-and medium-frequency ranges and extensions to structural acoustics, we refer the reader to [START_REF] Ohayon | Structural acoustics and vibration[END_REF][START_REF] Ohayon | Advanced computational vibroacoustics[END_REF] and for uncertainty quantification (UQ) in computational structural dynamics to [START_REF] Soize | Stochastic models of uncertainties in computational mechanics[END_REF].

The problem considered here is the construction of a reduced-order model for linear vibration of a dissipative structure subjected to prescribed forces and composed of two main linear dissipative substructures connected through a physical flexible viscoelastic interface (linking substructure).

Let us recall that the concept of substructures was first introduced by Argyris and Kelsey in 1959 [START_REF] Argyris | The analysis of fuselages of arbitrary cross-section and taper: A DSIR sponsored reserach program on the development and application of the matrix force method and the digital computer[END_REF] and by Przemieniecki in 1963 [START_REF] Przemieniecki | Matrix structural analysis of substructures[END_REF] and was extended by Guyan and Irons in 1965 [START_REF] Guyan | Reduction of stiffness and mass matrices[END_REF][START_REF] Irons | Structural eigenvalue problems -elimination of unwanted variables[END_REF]. In 1960 and 1965, Hurty [START_REF] Hurty | Vibrations of structural systems by component mode synthesis[END_REF][START_REF] Hurty | Dynamic analysis of structural systems using component modes[END_REF] considered the case of two substructures coupled through a geometrical interface, for which the first substructure is represented using its elastic modes with fixed geometrical interface and the second substructure is represented using its elastic modes with free geometrical interface completed by static boundary functions of the first substructure. Finally, Craig and Bampton in 1968 [START_REF] Craig | Bampton MCC (1968) Coupling of substructures for dynamic analyses[END_REF] adapted the Hurty method in order to represent each substructure of the same manner consisting in using the elastic modes of the substructure with fixed geometrical interface and the static boundary functions on its geometrical interface. Improvements have been proposed with many variants [START_REF] Agrawal | Mode synthesis technique for dynamic analysis of structures[END_REF][START_REF] Balmes | Optimal Ritz vectors for component mode synthesis using the singular value decomposition[END_REF][START_REF] Bathe | On non-linear dynamic analysis using substructuring and mode superposition[END_REF][START_REF] Bourquin | Numerical study of an intrinsic component mode synthesis method[END_REF][START_REF] Castanier | Characteristic constraint modes for component mode synthesis[END_REF][START_REF] Farhat | On a component mode method and its application to incompatible substructures[END_REF][START_REF] Hale | A procedure for improving discrete substructure representation in dynamic synthesis[END_REF][START_REF] Hintz | Analytical methods in component modal synthesis[END_REF][START_REF] Lindberg | Component mode synthesis using undeformed interface coupling modes to connect soft and stiff substructures[END_REF]53,[START_REF] Meirovitch | Rayleigh-Ritz based substructure synthesis for flexible multibody systems[END_REF][START_REF] Suarez | Improved fixed interface method for modal synthesis[END_REF][START_REF] Voormeeren | Generalized methodology for assembly and reduction of component models for dynamic substructuring[END_REF], in particular for complex dynamical systems with many appendages considered as substructures (such as disk with blades) Benfield and Hruda in 1971 [START_REF] Benfield | Vibration analysis of structures by component mode substitution[END_REF] proposed a component mode substitution using the Craig and Bampton method for each appendage. Another type of methods has been introduced in order to use the structural modes with free geometrical interface for two coupled substructures instead of the structural modes with fixed geometrical interface (elastic modes) as used in the Craig and Bampton method. In this context, MacNeal in 1971 [START_REF] Macneal | A hybrid method of component mode synthesis[END_REF] introduced the concept of residual flexibility which has then been used by Rubin in 1975 [START_REF] Rubin | Improved component-mode representation for structural dynamic analysis[END_REF]. The Lagrange multipliers have also been used to write the coupling on the geometrical interface [START_REF] Markovic | Reduction of substructural interface degrees of freedom in flexibility-based component mode synthesis[END_REF][START_REF] Ohayon | Dynamic substructuring of damped structures using singular value decomposition[END_REF][START_REF] Park | Partitioned component mode synthesis via a flexibility approach[END_REF][START_REF] Rixen | A dual Craig-Bampton method for dynamic substructuring[END_REF]. Dynamic substructuring in the medium-frequency range has been analyzed [START_REF] Kassem | Structural partitioning of complex structures in the medium-frequency range -An application to an automotive vehicle[END_REF][START_REF] Herran | Optimal component mode synthesis for medium frequency problem[END_REF][START_REF] Sarkar | A substructure approach for the midfrequency vibration of stochastic systems[END_REF][START_REF] Soize | Dynamic substructuring in the medium-frequency range[END_REF]. On the other hand, uncertainty quantification is nowadays recognized as playing an important role in order to improve the robustness of the models for the low-frequency range and especially, in the medium-frequency range in the context of substructuring techniques [START_REF] Brown | Probabilistic component mode synthesis of nondeterministic substructures[END_REF][START_REF] Hinke | Component mode synthesis as a framework for uncertainty analysis[END_REF][START_REF] Hong | Parametric reduced-order models for predicting the vibration response of complex structures with component damage and uncertainties[END_REF][START_REF] Mignolet | Nonparametric stochastic modeling of structures with uncertain boundary conditions / coupling between substructures[END_REF][START_REF] Ohayon | Advanced computational dissipative structural acoustics and fluid-structure interaction in low-and medium-frequency domains -Reduced-order models and uncertainty quantification[END_REF][START_REF] Soize | Random uncertainties model in dynamic substructuring using a nonparametric probabilistic model[END_REF][START_REF] Soize | Stochastic reduced-order model in low-frequency dynamics in presence of numerous local elastic modes[END_REF]. Reviews have also been performed [START_REF] Craig | A review of time domain and frequency domain component mode synthesis method in Combined experimental-analytical modeling of dynamic structural systems[END_REF][START_REF] De Klerk | General framework for dynamic substructuring: History, review, and classification of techniques[END_REF][START_REF] Leung | Dynamic stiffness and substructures[END_REF]. It should be noted that all these above dynamic substructuring methodologies, which have been developed for the discrete case (computational model), have also been reanalyzed in the framework of the continuous case (continuum mechanics) by Morand and Ohayon in 1979 [START_REF] Morand | Substructure variational analysis for the vibrations of coupled[END_REF] for which details can be found in [START_REF] Morand | Fluid structure interaction[END_REF] for conservative systems and by Ohayon and Soize in 1998 [START_REF] Ohayon | Structural acoustics and vibration[END_REF] for the dissipative systems.

A physical flexible interface (the linking substructure) between two coupled substructures, modeled by an elastic medium, has been considered by Kuhar and Stahle [44] considering a static behavior of the junction. In this paper, a generalization of this work is presented in a more general framework for a viscoelastic dynamic behavior of the junction using existing component mode synthesis methods (see [START_REF] Ohayon | Clarification about component mode synthesis methods for substructures with physical flexible interfaces[END_REF] for computational details). We consider a structure composed of two dissipative main substructures coupled through a physical flexible viscoelastic linking substructure by two geometrical interfaces. The boundary value problem is written in the frequency domain. The linking substructure is modeled in the context of the general linear viscoelasticity theory [START_REF] Bland | The theory of linear viscoelasticity[END_REF][START_REF] Truesdell | Mechanics of solids, Vol III, Theory of viscoelasticity, plasticity, elastic waves and elastic stability[END_REF], yielding damping and stiffness sesquilinear forms depending on the frequency, while the two main dissipative substructures are modeled in the context of linear elasticity with an additional classical viscous damping modeling which is assumed to be independent of the frequency.

We present a variational-based reduced-order model in dynamic substructuring, adapted to computational dynamics, for linear elastodynamic of a dissipative structure composed of two main dissipative substructures coupled through a physical flexible viscoelastic interface constituting the linking substructure. A reduced-order model is constructed using the structural modes of the two main substructures with free geometrical interfaces and, for the linking substructure, using an adapted frequency-dependent vector basis with fixed geometrical interfaces and an appropriate static lifting operator with respect to the geometrical interfaces. It should be noted that the linking substructures models which are used, generally correspond to a rough modeling of the real linking systems and consequently, uncertainties induced by modeling errors can be introduced [START_REF] Mignolet | Nonparametric stochastic modeling of structures with uncertain boundary conditions / coupling between substructures[END_REF]. Another interest of using free structural modes of the two main substructures is also to allow a direct dynamical identification of the main substructures using experimental modal analysis [START_REF] Mayes | Experimental modal substructuring to couple and uncouple substructures with flexible fixtures and multi-point connections[END_REF][START_REF] Gindlin | Experimental modal substructuring to estimate fixed-base modes from tests on a flexible fixture[END_REF][START_REF] Ewins | Modal testing: theory, practice and applications[END_REF][START_REF] Jezequel | A hybrid method of modal synthesis using vibration tests[END_REF][START_REF] Liu | Substructure synthesis via elastic media Part I: Joint identification[END_REF][58][START_REF] Morgan | Calculation of component made synthesis matrices from measured frequency response functions, part 2: Application[END_REF][START_REF] Nobari | A new approach to modal-based structural dynamic-model updating and joint identification[END_REF][START_REF] Reynders | System identification methods for (operational) modal analysis: Review and comparaison[END_REF][START_REF] Urgueira | Dynamic analysis of coupled structures using experimental data[END_REF]. Such a reduced-order model is very useful for sensitivity analysis, design optimization and controller design for vibration active control [START_REF] El-Khoury | Recent advances on vibration control of structures and their dynamic loading[END_REF][START_REF] Inman | Vibration with control[END_REF][START_REF] Lim | Adaptive control strategy for dynamic substructuring tests[END_REF][START_REF] Perdahcioglu | Dynamic substructuring and reanalysis methods in a surrogate-based design optimization environment[END_REF][START_REF] Tu | Dynamics, control and real-time issues related to substructuring techniques: Application to the testing of isolated structure systems[END_REF].

2 Displacement Variational Formulation for Two Substructures Connected with a Linking Substructure

Description of the mechanical system and hypotheses

This paper deals with the linear vibration of a free structure, around a static equilibrium configuration which is taken as a natural state (for the sake of brevity, prestresses are not considered but could be added without changing the presentation), submitted to prescribed external forces which are assumed to be in equilibrium at each instant. The displacement field of the structure is then defined up to an additive rigid body displacement field. In this paper, we are only concerned in the part of the displacement field due to the structural deformation. 3 with sufficiently smooth boundaries. The structure Ω is composed of two substructures Ω 1 and Ω 2 perfectly connected through interfaces Γ 1L and Γ 2L by a linking substructure Ω L (see Fig. 1). The boundaries are such that

1 L Γ 1L Γ L 2 Ω 2 Ω 1 L n L n n n 2 Ω Fig. 1 Two substructures Ω 1 and Ω 2 connected with a linking substructure Ω L Let Ω 1 , Ω 2 and Ω L be three open bounded domains in R
∂Ω 1 = Γ 1L ∪ Γ 1 , ∂Ω 2 = Γ 2L ∪ Γ 2 , ∂Ω L = Γ 1L ∪ Γ L ∪ Γ 2L . We then have Ω = Ω 1 ∪ Γ 1L ∪ Ω L ∪ Γ 2L ∪ Ω 2 and ∂Ω = Γ 1 ∪ Γ L ∪ Γ 2 .
The physical space is referred to a cartesian reference system and the generic point is denoted as x = (x 1 , x 2 , x 3 ). A frequency domain formulation is used, the convention for the Fourier transform being v(ω) = R e -iωt v(t) dt where ω denotes the real circular frequency, v(ω) is in C and where v(ω) denotes its conjugate. The convention of summation over repeated indices is used and v ,j denotes the partial derivative of v with respect to x j . In u • v, the dot denotes the usual Euclidean inner product on R 3 extended to C 3 .

For r in {1, L, 2}, the space of admissible displacement fields defined on Ωr with values in C 3 (resp. in R 3 ) is denoted by C Ωr (resp. R Ωr ). For substructure Ωr, the test function (weighted function) associated with u r is denoted by δu r ∈ C Ωr (or in R Ωr ). The space R Ωr is the real Sobolev space (H 1 (Ωr)) 3 and the space C Ωr is defined as the complexified Hilbert space of R Ωr . Let R r rig be the subspace (of dimension 6) of C Ωr spanned by all the R 3 -valued rigid body displacement fields which are written as u r (x) = t + θ × x for all x in the closure Ωr of Ωr, in which t and θ are two arbitrary constant vectors in R 3 . Let R Ω be the space of admissible displacement fields defined on Ω with values in R 3 . Let R rig be the subspace (of dimension 6) of R Ω spanned by all the R 3 -valued rigid body displacement fields which are written as u(x) = t + θ × x for all x.

Each substructure is a three-dimensional dissipative elastic medium. The linking substructure Ω L is modeled in the context of the general linear viscoelasticity theory while the two main dissipative substructures Ω 1 and Ω 2 are modeled in the context of linear elasticity with an additional classical viscous damping modeling which is assumed to be independent of the frequency. For r in {1, L, 2}, for all fixed ω and for each point x, the displacement field is denoted by u r (x, ω) = (u r 1 (x, ω), u r 2 (x, ω), u r 3 (x, ω)), the external given body and surface force density fields applied to Ωr and Γr are denoted by g Ωr

(x, ω) = (g Ωr 1 (x, ω), g Ωr 2 (x, ω), g Ωr 3 (x, ω))
and g Γr (x, ω) = (g Γr 1 (x, ω), g Γr 2 (x, ω), g Γr 3 (x, ω)) respectively. Let be

g Ω (x, ω) = r=1,L,2 1 Ωr (x) g Ωr (x, ω) ,
(1)

g ∂Ω (x, ω) = r=1,L,2
1 Γr (x) g Γr (x, ω) .

(

) 2
in which 1 B is the indicator function of set B. Since we are only concerned in the part of the displacement field due to the structural deformation, it will be assumed that the given external forces are such that, for all u in R rig ,

Ω g Ω (x, ω) • u(x) dx + ∂Ω g ∂Ω (x, ω) • u(x) ds(x) = 0 . ( 3
)

Constitutive equations

For r in {1, L, 2}, the linearized strain tensor is defined by

ε ij (u r ) = 1 2 (u r i,j + u r j,i ) . ( 4
)
(i) For r in {1, 2}, the constitutive equation for substructure Ωr, which is assumed to be made up of an elastic material with linear viscous term, is written as

σ r tot = σ r + iω s r , ( 5
)
where σ r is the elastic stress tensor defined by σ r ij (u r ) = a r ijkh (x) ε kh (u r ) and where iω s r is the viscous part of the total stress tensor such that s r ij (u r ) = b r ijkh (x) ε kh (u r ). The mechanical coefficients a r ijkh (x) and b r ijkh (x) depend on x but are independent of frequency ω and verify the usual properties of symmetry, positiveness and boundedness (lower and upper).

(ii) For r = L, the constitutive equation for linking substructure Ω L , which is assumed to be made up of a linear viscoelastic material, is written as

σ L tot = σ L + iω s L , ( 6
)
where σ L is the elastic part of the total stress tensor defined by

σ L ij (u L ) = a L ijkh (x, ω) ε kh (u L
) and where iω s L is the viscous part of the total stress tensor such that

s L ij (u L ) = b L ijkh (x, ω) ε kh (u L ).
The mechanical coefficients a L ijkh (x, ω) and b L ijkh (x, ω) depend on x and on frequency ω and, for all fixed ω, verify the usual properties of symmetry, positiveness and boundedness (lower and upper). At zero frequency, a L ijkh (x, 0) is the equilibrium modulus tensor (which differs from the initial elasticity tensor). For more details concerning the properties of tensors For all fixed ω and for r in {1, L, 2}, the equilibrium equation for substructure Ωr is written as

-ω 2 ρ r u r i -{σ r tot } ij,j = g Ωr i in Ωr for i = 1, 2, 3 , (7)
in which ρ r is the mass density depending on x (which is assumed to be strictly positive and bounded), with the boundary condition,

{σ r tot } ij n r j = g Γr i on Γr for i = 1, 2, 3 , ( 8
)
in which the vector n r = (n r

1 , n r 2 , n r
3 ) is the unit normal to ∂Ωr, external to Ωr.

Coupling conditions

The coupling conditions of the linking substructure Ω L with substructures Ω 1 and Ω 2 on Γ 1L and Γ 2L are written as

u 1 = u L on Γ 1L , u 2 = u L on Γ 2L . ( 9
)
σ 1 tot n 1 = -σ L tot n L on Γ 1L , σ 2 tot n 2 = -σ L tot n L on Γ 2L . ( 10
)

Boundary value problem in displacement

For all fixed ω, the boundary value problem consists in finding the displacement fields u 1 , u L and u 2 , verifying, for each r equal to 1, L or 2, the equilibrium equation Eq. [START_REF] Balmes | Optimal Ritz vectors for component mode synthesis using the singular value decomposition[END_REF] with the constitutive equation defined by Eq. ( 5) (for r = 1, 2) or by Eq. ( 6) (for r = L), with the Neumann boundary condition defined by Eq. ( 8) and with the coupling conditions defined by Eqs. ( 9) and (10).

Variational formulation of the boundary value problem

Definition of the sesquilinear forms of the problem

For r in {1, L, 2}, a sesquilinear form on C Ωr × C Ωr is defined by

m r (u r , δu r ) = Ωr ρ r u r • δu r dx . ( 11
)
The sesquilinear form m r is continuous positive definite Hermitian on C Ωr × C Ωr .

(i) For r in {1, 2}, two sesquilinear forms on C Ωr ×C Ωr , independent of frequency ω are defined by

k r (u r , δu r ) = Ωr a r ijkh (x) ε kh (u r ) ε ij ( δu r ) dx , ( 12
) d r (u r , δu r ) = Ωr b r ijkh (x) ε kh (u r ) ε ij ( δu r ) dx . ( 13
)
(ii) For r = L, two sesquilinear forms on C Ωr × C Ωr , depending on frequency ω, are defined by

k L (u L , δu L ; ω) = ΩL a L ijkh (x, ω) ε kh (u L ) ε ij ( δu L ) dx , ( 14
)
d L (u L , δu L ; ω) = ΩL b L ijkh (x, ω) ε kh (u L ) ε ij ( δu L ) dx . ( 15
)
Taking into account the usual assumptions related to the constitutive equations, for r in {1, L, 2}, the sesquilinear forms k r and d r are continuous semi-definite positive Hermitian on C Ωr × C Ωr , the semi-definite positiveness being due to the presence of rigid body displacement fields.

(i) For r in {1, 2}, for all δu r in C Ωr , k r (u r , δu r ) and d r (u r , δu r ) are equal to zero for any u r in R r rig . (ii) For r = L, for all fixed frequency ω and for all δu L in C ΩL , k L (u L , δu L ; ω) and d L (u L , δu L ; ω) are equal to zero for any u L in R L rig .

Definition of the antilinear forms of the problem

For r in {1, L, 2} and for all fixed frequency ω, it is assumed that g Ωr and g Γr are such that the antilinear form δu r → f r (δu r ; ω) on C Ωr , defined by

f r (δu r ; ω) = Ωr g Ωr (x, ω) • δu r (x) dx + Γr g Γr (x, ω) • δu r (x) ds(x) , ( 16
)
is continuous.

Definition of the complex bilinear forms of the problem

(i) For r in {1, 2} and for all fixed ω, the complex bilinear form z r on C Ωr × C Ωr is defined by

z r (u r , δu r ; ω) = -ω 2 m r (u r , δu r ) +iω d r (u r , δu r ) +k r (u r , δu r ) . ( 17
)
(ii) For r = L and for all fixed frequency ω, the complex bilinear form z L on C ΩL × C ΩL is defined by

z L (u L , δu L ; ω) = -ω 2 m L (u L , δu L ) +iω d r (u L , δu L ; ω) + k L (u L , δu L ; ω) . ( 18
)

Variational formulation of the boundary value problem

The variational formulation of the boundary value problem in u 1 , u L and u 2 is defined as follows. For all fixed real ω, find (u

1 (ω), u L (ω), u 2 (ω)) in C Ω1 × C ΩL × C Ω2
verifying the linear constraints u 1 (ω) = u L (ω) on Γ 1L and u 2 (ω) = u L (ω) on Γ 2L , such that, for all (δu 1 , δu L , δu 2 ) in C Ω1 × C ΩL × C Ω2 verifying the linear constraints δu 1 = δu L on Γ 1L and δu 2 = δu L on Γ 2L , we have

z 1 (u 1 (ω), δu 1 ; ω) + z L (u L (ω), δu L ; ω) +z 2 (u 2 (ω), δu 2 ; ω) = f 1 (δu 1 ; ω) +f L (δu L ; ω) + f 2 (δu 2 ; ω) . ( 19
)
Under the given hypotheses relative to the constitutive equations and the hypothesis on the external given forces defined by Eq. ( 3), the existence and uniqueness of a solution can be proven for all real ω. This solution corresponds only to deformation of the structure (i.e. without rigid body displacements).

Reduced-Order Model

The method is based on the use of the variational formulation defined by Eq. ( 19).

The reduced-order model is carried out using (i) for r = 1, 2, the eigenmodes of substructure Ωr with free interface Γ rL , (ii) a frequency-dependent family of eigenfunctions for linking substructure Ω L with fixed interface Γ 1L ∪ Γ 2L and (iii) the elastostatic lifting operator of Ω L with respect to interface Γ 1L ∪ Γ 2L at zero frequency.

3.1 Eigenmodes of substructure Ωr with free interface Γ rL for r = 1, 2

For r = 1, 2, a free-interface mode of substructure Ωr is defined as an eigenmode of the conservative problem associated with free substructure Ωr, subject to zero forces on ∂Ωr. The real eigenvalues λ r ≥ 0 and the corresponding eigenmodes u r in R Ωr are then the solutions of the following spectral problem: find λ r ≥ 0, u r ∈ R Ωr (u r = 0) such that for all δu r ∈ R Ωr , one has

k r (u r , δu r ) = λ r m r (u r , δu r ) . ( 20
)
It can be shown that there exist six zero eigenvalues 0 = λ r -5 = . . . = λ r 0 (associated with the rigid body displacement fields) and that the strictly positive eigenvalues (associated with the displacement field due to structural deformation) constitute the increasing sequence 0 < λ r 1 ≤ λ r 2 , . . .. The six eigenvectors {u r -5 , . . . , u r 0 } associated with zero eigenvalues span R rig (space of the rigid body displacement fields). The family {u r -5 , . . . , u r 0 ; u r 1 , . . .} of all the eigenmodes forms a complete family in R Ωr . For α and β in {-5, . . . , 0; 1, . . .}, we have the orthogonality conditions

m r (u r α , u r β ) = δ αβ , ( 21
) k r (u r α , u r β ) = δ αβ λ r α , ( 22
)
in which each eigenmode u r α is normalized to 1 with respect to m r and where

ω r α = √ λ r α is the eigenfrequency of mode α.
3.2 Frequency-dependent family of eigenfunctions for linking substructure Ω L with fixed interface Γ 1L ∪ Γ 2L

Let us introduce the following admissible space,

R 0 ΩL = u L ∈ R ΩL u L = 0 on Γ 1L ∪ Γ 2L . ( 23
)
At given frequency ω, a fixed-interface vector basis of linking substructure Ω L is defined as an eigenfunction of the conservative problem associated with Ω L with fixed interface Γ 1L ∪Γ 2L , subject to zero forces on ∂Ω L . A real eigenvalue λ L (ω) > 0 and the corresponding eigenfunction u L (ω) in R ΩL are then the solution of the following spectral problem: find λ L (ω) > 0 and nonzero u L (ω) in R 0 ΩL such that for all δu L ∈ R 0 ΩL , we have

k L (u L (ω), δu L ; ω) = λ L (ω) m L (u L (ω), δu L ) . ( 24
)
It can be shown that there exist a strictly positive increasing sequence of eigenvalues, 0 < λ L 1 (ω) ≤ λ L 2 (ω), . . ., and a corresponding family {u L 1 (ω), u L 2 (ω), . . .} of eigenfunctions which constitutes a complete family in R 0 ΩL . For α and β in {1, 2, . . .}, we have the orthogonality conditions

m L (u L α (ω), u L β (ω)) = δ αβ , ( 25
)
k L (u L α (ω), u L β (ω); ω) = δ αβ λ L α (ω) , ( 26
)
in which each eigenfunction u L α (ω) is normalized to 1 with respect to m L .

Remarks. The reduced-order model will be constructed in using a finite family

U L (ω) = {u L 1 (ω), . . . , u L NL (ω)} of the sequence {u L 1 (ω), u L 2 (ω), . . .} of the eigenfunc- tions associated with the first N L eigenvalues 0 < λ L 1 (ω) ≤ . . . ≤ λ L NL (ω).
In addition, such a reduced-order model will be constructed for analyzing the response of the structure for frequency ω belonging to a given frequency band of analysis, B = [ω min , ωmax], with 0 ≤ ω min < ωmax. In practice, the response is calculated for the frequencies belonging to the finite subset B = {ω 1 , ω 2 , . . . , ωμ} of μ sampling frequencies of band B, for which μ can be of several hundreds or of the order of one thousand.

(i) In the above formulation, the generalized eigenvalue problem defined by Eq. ( 24) must be solved for all ω in B. If the number μ of frequencies is not too high (that is generally the case for the linking substructure), such a computation remains feasible. It should be noted that the use of massively parallel computers facilitates the analysis of such numerical problem depending on continuous parameter ω.

(ii) For solving the generalized eigenvalue problem defined by Eq. ( 24) for all ω in B, the numerical cost can be reduced using the following procedure. First, Eq. ( 24) is solved for ω belonging to the subset B = {ω 1 , . . . , ω μ } of chosen frequency points in B with μ μ, called master frequency points. Then, an approximation U L (ω) of U L (ω) is calculated for all ω in B\B by using an interpolation procedure based on the values U L (ω 1 ), . . . , U L (ω μ ) at the μ frequency points. Finally, for each ω in B\B , the generalized eigenvalue problem defined by Eq. ( 24) is projected on the subspace spanned by the finite family U L (ω) which yields a reduced-order generalized eigenvalue problem of dimension N L and which allows an approximation λ L 1 (ω), . . . , λ L NL (ω) of λ L 1 (ω), . . . , λ L NL (ω) to be calculated. Such a procedure can be found in [START_REF] Amsallem | An online method for interpolationg linear parametric reduced-order models[END_REF].

(iii) Another way consists in replacing the above interpolation procedure by the following construction of a frequency-independent basis adapted to band B. It consists in extracting the larger family of linearly independent functions from the family U L (ω 1 ), . . . , U L (ω μ ) at the μ master frequency points.

(iv) If k L (•, • ; ω) slowly varies for ω in band B, a well adapted frequencyindependent basis for all ω in B, consists in choosing U L (ω) for an arbitrary ω in B but in such a case convergence with respect to N L must be carefully checked.

(v) More generally, any subset of N L functions extracted from a Hilbertian basis of the admissible space R 0 ΩL can be used.

Elastostatic lifting operator of Ω L with respect to interface Γ 1L ∪ Γ 2L at zero frequency

We consider the solution u L S of the elastostatic problem at zero frequency for linking substructure Ω L subjected to prescribed displacement fields u Γ1L on Γ 1L and u Γ2L on Γ 2L , and zero force on Γ L . We introduce the following sets of functions,

R Γ1L,Γ2L = (H 1/2 (Γ 1L )) 3 × (H 1/2 (Γ 2L )) 3 , ( 27
)
R uΓ 1L ,uΓ 2L ΩL = { u L ∈ R ΩL u L = u Γ1L on Γ 1L ; u L = u Γ2L on Γ 2L with (u Γ1L , u Γ2L ) ∈ R Γ1L,Γ2L } . ( 28
)
Displacement field u L S satisfies the following variational formulation,

k L (u L S , δu L S ; 0) = 0 , u L S ∈ R uΓ 1L ,uΓ 2L ΩL , ∀ δu L S ∈ R 0 ΩL , (29)
corresponding to the following boundary value problem,

{a L ijkh (x, 0) ε kh (u L )} ,j = 0 in Ω L for i = 1, 2, 3 , (30)
with the Neumann and Dirichlet boundary conditions,

a L ijkh (x, 0) ε kh (u L ) n L j = 0 on Γ L for i = 1, 2, 3 , (31)
u L = u Γ1L on Γ 1L ; u L = u Γ2L on Γ 2L . ( 32
)
The problem defined by Eq. ( 29) has a unique solution u L S which defines the linear continuous operator S L from R Γ1L,Γ2L into R uΓ 1L ,uΓ 2L ΩL (called the elastostatic lifting operator at zero frequency),

(u Γ1L , u Γ2L ) → u L S = S L (u Γ1L , u Γ2L ) . ( 33
)
Let R stat ΩL be the subspace of R uΓ 1L ,uΓ 2L ΩL constituted of all the solutions of Eq. (29) (i.e. the range of operator S L ). It can then be proven that

R ΩL = R stat ΩL ⊕ R 0 ΩL . Finally, introducing the complexified spaces C stat ΩL and C 0 ΩL of R stat ΩL and R 0 ΩL , it can be proven that C ΩL = C stat ΩL ⊕ C 0 ΩL . ( 34
)

Construction of a reduced-order model

The following reduced-order model can then be constructed using the elastostatic lifting operator and performing a Ritz-Galerkin projection with the free-interface modes of substructures Ω 1 and Ω 2 , and the fixed interface modes of linking substructure Ω L . Let N 1 , N L and N 2 be finite integers. For all fixed ω, the following finite projections u 1,N1 (ω), u L,NL (ω) and u 2,N2 (ω) of u 1 (ω), u L (ω) and u 2 (ω) are introduced as follows,

u 1,N1 (ω) = N1 α=-5 q 1 α (ω) u 1 α , ( 35
) u L,NL (ω) = S L (u N1 Γ1L (ω), u N2 Γ2L (ω)) + NL α=1 q L α (ω) u L α (ω) , ( 36
) u 2,N2 (ω) = N2 α=-5 q 2 α (ω) u 2 α , (37)
in which u N1 Γ1L (ω) and u N2 Γ2L (ω) are such that

u N1 Γ1L (ω) = N1 α=-5 q 1 α (ω) u 1 α|Γ1L , ( 38
) u N2 Γ2L (ω) = N2 α=-5 q 2 α (ω) u 2 α|Γ2L . (39)
Note that Eq. ( 36) is due to the property defined by Eq. ( 34). The corresponding test functions are then written as, (

δu 1,N1 = N1 α=-5 δq 1 α u 1 α , ( 40
)
) 44
Substituting Eqs. [START_REF] Hintz | Analytical methods in component modal synthesis[END_REF] to [START_REF] Kuhar | Dynamic transformation method for modal synthesis[END_REF] into Eq. ( 19) yields the following variational reducedorder model of order N = N 1 + 6+N L + N 2 + 6. For all fixed Ω, find (q 1 (ω), q L (ω), q 2 (ω)) in C N1+6 × C NL × C N2+6 such that, for all (δq 1 , δq L , δq 2 ) in C N1+6 × C NL × C N2+6 , we have z red N (q 1 (ω), q L (ω), q 2 (ω), δq 1 , δq L , δq 2 ; ω) = f N (δq 1 , δq L , δq 2 ; ω) , [START_REF] Leung | Dynamic stiffness and substructures[END_REF] in which q 1 = (q 1 -5 , . . . , q 1 0 , q 1 1 . . . q 1 N1 ), q L = (q L 1 , . . . , q L NL ), q 2 = (q 2 -5 , . . . , q 2 0 , q 2 1 . . . q 2 N2 ), δq 1 = (δq 1 -5 , . . . , δq 1 0 , δq ).

Concluding remarks and research perspectives

A continuum-based substructuring techniques has been presented for the linear dynamic analysis of two substructures connected with a physical flexible viscoelastic interface, for which each substructure is reduced using its free-interface elastic modes. Concerning the research perspectives, for such a dynamical system, the physical flexible viscoelastic interface generally presents model uncertainties induced by imperfect coupling boundary conditions with the substructures, the viscoelastic model used for of the material and the geometrical parameters. The complexity of such physical interface model can require advanced dynamic multiscale methods in micro-macro mechanics for materials [START_REF] Liu | Nanomechanics and materials -Theory, multiscale methods and applications[END_REF] and requires to take into account model uncertainties induced by modeling errors. Such implementation using the nonparametric probabilistic approach of model uncertainties, coupling recent advanced research concerning uncertainty quantification for viscoelastic structures [START_REF] Ohayon | Advanced computational vibroacoustics[END_REF][START_REF] Soize | Time-domain formulation in computational dynamics for linear viscoelastic media with model uncertainties and stochastic excitation[END_REF] and uncertain coupling interface methodologies [START_REF] Mignolet | Nonparametric stochastic modeling of structures with uncertain boundary conditions / coupling between substructures[END_REF], is in progress. In addition, the introduction of smart materials in the physical interface would be of prime interest for micromechanical systems.

Γ1L

  δu L,NL (ω) = S L (δu N1

Acknowledgment

This research was partially supported by Brazil-France project CAPES-COFECUB Ph672/10.