Keywords

1 Introduction

Thin and elongated structures are widely encountered in the industry. Beams in the construction industry and shells and plates in the automotive and aeronautical industries are typical examples. Their modeling with the Finite Element (FE) Method is expensive when using a classical formulation, based on the straightforward volume momentum equation and 3D displacement fields. Indeed, the constraint on the aspect ratio of the finite elements implies that the small thickness of the physical domain controls the minimum size of the elements overall. The cost of solving the corresponding FE system may then become prohibitive. This difficulty can be circumvented by starting from formulations specifically designed for thin elements. Based on appropriate kinematical hypotheses on the displacement fields, such as assuming rigid sections for a beam, lower-dimensional formulations can be obtained. Hence a 3D elasticity problem in an elongated domain becomes a 1D beam problem over the mid-fiber. Likewise, a 3D elasticity problem in a flat domain becomes a 2D shell problem over the mid-surface. In these theories, the structural parameters (e.g. mass per unit length, moment of inertia) are analytically derived from the solid parameters (e.g. Young’s modulus, Lamé parameters) and the geometry. In the sense that the full-scale 3D elastic model is transformed into a lower-dimensionality structural model, this transformation can be seen as an upscaling process.

When the solid parameters are heterogeneous, the derivation of a homogenized structural model is not so obvious. Analytical solutions exist for specific structures, such as periodic (Caillerie and Nedelec 1984; Kohn and Vogelius 1984; Buannic and Cartraud 2001; Cecchi and Sab 2002; Cartraud and Messager 2006; Grédé et al. 2006; Mistler et al. 2007; Mercatoris et al. 2009) or laminated (Hohe and Becker 2001; Rabczuk et al. 2004; Liu et al. 2006). However, these techniques work well for specific structures and in a particular range of application (for instance, when the mechanical functions of the core and faces of a composite structure are well differentiated). When the fluctuations of the parameters do not present any such simple structure (as in concrete for example), computational homogenization can still be used (Coenen et al. 2010). It is a direct extension of the classical numerical techniques for approximating homogenized coefficients in solid mechanics. The main difference lies in the geometry of the samples, which imposes that the typical sample spans the entire structure along the small dimension(s) and the test boundary conditions (Dirichet, Neumann or periodic) are only applied along the large dimension(s). Finally, when the heterogeneous parameters are modeled as random fields, to the best of our knowledge, there has been no proposal in the literature as to how to treat homogenization of structural models. This case would be the equivalent for solid-to-beam homogenization of the classical solid-to-solid random homogenization problem (Papanicolaou and Varadhan 1981; Huet 1990; Sab 1992; Bourgeat and Piatnitski 2004; Tartar 2009).

This papers aims at proposing a numerical homogenization technique that yields the homogenized parameters of a structural model from the given stochastic fields of parameters of an underlying solid model. It should be pointed out that the method proposed works for solid-to-beam homogenization in all the cases discussed above (homogeneous and heterogeneous, deterministic and stochastic). The main idea is to start from a chosen (a priori erroneous) set of homogenized parameters for the parameters of a structural model, and couple this model to a solid model with the input (stochastic) set of parameters, in a simple geometrical and loading numerical setup. If the coupled system yields the same solution as a mono-model structural parameter with the same set of (chosen) parameters, it is assumed to mean that the chosen model does correspond to the homogenized model. Else, a new set of structural parameters is chosen and the same experiment is repeated until convergence. This method is an application, for structural models, of a numerical homogenization technique introduced recently (Cottereau 2013a,b) for more classical random homogenization of solid models.

The core of this homogenization technique is a coupling method for structural and solid models. Many of those have been developed in the past: enforcing directly the structural hypothesis of rigid sections at the interface between the two models through so-called transition elements (Surana 1980; Bathe and Bolourchi 1980; Cofer and Will 1991; Gmür and Schorderet 1993; Dávila 1994), possibly adding some elasticity to the interface (Osawa et al. 2007; Xue et al. 2009; Song and Hodges 2010), enforcing continuity of the mechanical work at the interface (McCune et al. 2000; Shim et al. 2002), or using the Arlequin method in which the coupling is localized in a volume rather than over a surface (Ben Dhia 1998; Ben Dhia and Rateau 2001; Rateau 2003; Ben Dhia and Rateau 2005; Ben Dhia 2008; Barthel and Gabbert 2010; Rousseau et al. 2010; Qiao et al. 2011; Ghanem et al. 2013). The “weakness” of the coupling (in the sense of the strength of the kinematical constraint imposed by the homogeneous structural model onto the heterogeneous solid model) is essential for the success of the homogenization experiment, so we will consider here the Arlequin coupling.

The next section (Sect. 2) describes the two models that will be used in our method: (i) the stochastic heterogeneous solid model that we are trying to homogenize over an elongated domain, and (ii) the deterministic homogeneous beam model that is the target model. We will limit ourselves in this paper to a beam model, but extension to shell and plate models is expected to be straightforward. The following section (Sect. 3) describes the Arlequin coupling method for these two models. It is somehow a union of previous papers on the Arlequin coupling of deterministic beam and solid models (Rateau 2003; Ben Dhia and Rateau 2005; Barthel and Gabbert 2010; Rousseau et al. 2010; Qiao et al. 2011; Ghanem et al. 2013) on the one hand, and deterministic and stochastic solid models (Cottereau et al. 2010, 2011; Zaccardi et al. 2013; Le Guennec et al. 2013; Cottereau 2013b) on the other hand. Although this coupling method is not the main objective of this paper, and because no similar method can be found in the literature, we believe it is interesting to describe it to some level of detail. Finally, Sect. 4 describes the core of the paper, which is the homogenization method.

2 Description of the Mono-models

In this section, we describe the two models that will be considered: a stochastic continuum mechanics (solid) model and a deterministic Timoshenko beam model. We also highlight the kinematical hypothesis that allows to go from the solid model (when it is assumed deterministic and homogeneous) to the beam model. Throughout, we will indicate quantities related to the solid model with an ‘s’ index, and the quantities related to the beam model with a ‘b’ index.

2.1 Stochastic Solid Model

Let Ω s be a domain of \(\mathbb{R}^{3}\) with a smooth boundary ∂ Ω s , separated into a partition \(\partial \varOmega _{s} =\varGamma _{D} \cup \varGamma _{N}\). The domain Ω s is filled with an elastic and isotropic solid, loaded in the bulk by \(\boldsymbol{f}_{s}\) and on the surface Γ N by \(\boldsymbol{g}_{s}\) (both assumed deterministic), and kinematically constrained along Γ D . The Lamé parameters λ and μ of the solid are modeled as positive, second-order, mean-square continuous stochastic fields indexed on \(\mathbb{R}^{3}\), and defined on probability spaces \((\varXi,\mathcal{A},P)\), where Ξ is a set of events, \(\mathcal{A}\) is a σ-algebra of elements of Ξ and P is a probability measure over \(\mathcal{A}\). Under a small perturbations hypothesis, the weak formulation of the stochastic boundary value problem reads: find \(\boldsymbol{u}_{s} \in \mathcal{V}_{s}\) such that:

$$\displaystyle{ \mathbb{E}\left [\int _{\varOmega _{s}}\boldsymbol{\sigma }[\boldsymbol{u}_{s}]:\boldsymbol{\varepsilon } [\boldsymbol{v}_{s}]d\boldsymbol{x}\right ] =\int _{\varOmega _{s}}\boldsymbol{f}_{s} \cdot \mathbb{E}[\boldsymbol{v}_{s}]d\boldsymbol{x} +\int _{\varGamma _{N}}\boldsymbol{g}_{s} \cdot \mathbb{E}[\boldsymbol{v}_{s}]d\boldsymbol{x},\quad \forall \boldsymbol{v}_{s} \in \mathcal{V}_{s}, }$$
(1)

where \(\boldsymbol{\varepsilon }[\boldsymbol{u}] = 1/2(\nabla \boldsymbol{u} + \nabla ^{T}\boldsymbol{u})\) is the infinitesimal strain tensor, the superscriptT denotes the transpose operator, \(\boldsymbol{\sigma }[\boldsymbol{u}] =\lambda \, \mathrm{Tr}\,\boldsymbol{\varepsilon }[\boldsymbol{u}]\boldsymbol{I} + 2\mu \boldsymbol{\varepsilon }[\boldsymbol{u}]\) is the Cauchy stress tensor, \(\boldsymbol{I}\) is the identity tensor, and Tr denotes the trace operator. Equivalently, one can use the Young’s modulus E and the Poisson’s ratio ν instead of the Lamé constants. These pairs of parameters are linked through:

$$\displaystyle{ \lambda = \dfrac{E\nu } {(1+\nu )(1 - 2\nu )}\,,\quad \mu = \dfrac{E} {2(1+\nu )}\,. }$$
(2)

The functional space is \(\mathcal{V}_{s} = \mathcal{L}^{2}(\varXi,\mathcal{H}_{0}^{1})\), with \(\mathcal{H}_{0}^{1}\,=\,\{\boldsymbol{v} \in (\mathcal{H}^{1}(\varOmega _{s}))^{3},\boldsymbol{v}_{\vert \varGamma _{D}}\,=\,\boldsymbol{0}\}\). Endowed with the appropriate inner product and norm, \(\mathcal{V}_{s}\) is a Hilbert space. Using Lax-Milgram theorem, it can be proved that the problem (1) has a unique solution \(\boldsymbol{u}_{s}\) (see for instance Babuška et al. 2004). An approximation of that solution can then be obtained, for example, by using a Stochastic FE method (Ghanem and Spanos 1991; Stefanou 2009) or a Monte Carlo approach (Robert and Casella 2004).

2.2 Deterministic Beam Model

A beam is a structure whose axial extension is much larger than any dimension orthogonal to it. The cross-sections are defined by intersecting the beam with planes orthogonal to its axis. We define the neutral fiber of this beam as the line joining the centers of mass of all the sections. For simplicity, we consider here a beam whose neutral fiber \(\mathcal{F}_{b}\) is straight and with constant sections, that we denote \(\mathcal{S}\). The beam occupies the domain \(\varOmega _{b} = \mathcal{F}_{b} \times \mathcal{S}\in \mathbb{R}^{3}\). Under the hypotheses of homogeneous symmetric cross-sections, the geometrical centers are coincident to the centers of mass and inertia (see Fig. 1a for an illustrative example). We assume throughout that the neutral fiber is aligned along \(\boldsymbol{e}_{1}\) at rest. A Cartesian reference system is adopted, with base vectors \(\boldsymbol{e}_{i}\ 1 \leq i \leq 3\), and corresponding coordinates \(x \in \mathcal{F}_{b}\) and \((y,z) \in \mathcal{S}\). Any vector \(\boldsymbol{v}\) can be developed into its axial and section parts as \(\boldsymbol{v} = v_{1}\boldsymbol{e}_{1} +\boldsymbol{ v}_{\perp }\). We limit ourselves all along to small transformations around the initial position.

Fig. 1
figure 1

An example of straight beam with a rectangular cross-section and of a coupled solid-beam model in 2D. (a) Beam model. (b) Beam-solid model

Full size image

The classical Timoshenko beam theory (Timoshenko 1922; Oñate 2013) assumes that the cross-sections behave as rigid bodies, although they do not necessarily remain perpendicular to the neutral fiber. This kinematical hypothesis leads to parameterize the 3D displacement field \(\boldsymbol{u}_{b}(x,y,z)\) of the beam as a function of two 1D functions: the displacement of the neutral fiber \(\boldsymbol{u}_{0}(x)\) and the rotation vector of the cross-sections \(\boldsymbol{\theta }(x)\):

$$\displaystyle{ \boldsymbol{u}_{b}(x,y,z) =\boldsymbol{ u}_{0}(x) +\boldsymbol{\theta } (x) \times \boldsymbol{ x}_{\perp }, }$$
(3)

where \(\boldsymbol{x}_{\perp } = [0\;y\;z]^{T}\) gives the location of a point in a cross-section.

The beam is subjected to a linear force \(\boldsymbol{f}_{l}\) and moment \(\boldsymbol{c}_{l}\) (that would be equal respectively to \(\int _{\mathcal{S}}\boldsymbol{f}_{s}d\boldsymbol{x} + \int _{\partial \mathcal{S}}\boldsymbol{g}_{s}d\boldsymbol{x}\) and \(\int _{\mathcal{S}}\boldsymbol{x}_{\perp }\times \boldsymbol{ f}_{s}d\boldsymbol{x} + \int _{\partial \mathcal{S}}\boldsymbol{x}_{\perp }\times \boldsymbol{ g}_{s}d\boldsymbol{x}\) if the beam were modeled as a solid). On the Neumann extremities of the mean fiber \(\mathcal{F}_{b}^{N}\), force \(\boldsymbol{F}_{b}\) and moment \(\boldsymbol{C}_{b}\) loads are also enforced (that would correspond to \(\int _{\mathcal{S}}\boldsymbol{g}_{s\vert \varGamma _{N}}d\boldsymbol{x}\) and \(\int _{\mathcal{S}}\boldsymbol{g}_{s\vert \varGamma _{N}} \times \boldsymbol{ x}_{\perp }d\boldsymbol{x}\), respectively). Assuming an isotropic and elastic behavior, the balance of momentum for each section leads to the following weak formulation: Find \(\boldsymbol{u}_{0} = u_{1}\boldsymbol{e}_{1} +\boldsymbol{ u}_{\perp }\in \mathcal{V}_{b}\) and \(\boldsymbol{\theta }=\theta _{1}\boldsymbol{e}_{1} +\boldsymbol{\theta } _{\perp }\in \mathcal{V}_{b}\) such that for all \(\boldsymbol{v}_{0} = v_{1}\boldsymbol{e}_{1} +\boldsymbol{ v}_{\perp }\in \mathcal{V}_{b}\) and \(\boldsymbol{\gamma }=\gamma _{1}\boldsymbol{e}_{1} +\boldsymbol{\gamma } _{\perp }\in \mathcal{V}_{b}\):

  • Axial momentum equation

    $$\displaystyle{ \int _{\mathcal{F}_{b}}E_{b}Su_{1}^\prime v_{1}^\prime dx =\int _{\mathcal{F}_{b}}f_{1}v_{1}dx + \left (F_{b1}v_{1}\right )_{\partial \mathcal{F}_{b}}; }$$
    (4)

  • Torsion momentum equation

    $$\displaystyle{ \int _{\mathcal{F}_{b}}\mu _{b}J_{1}\theta _{1}^\prime\gamma _{1}^\prime dx =\int _{\mathcal{F}_{b}}c_{1}\gamma _{1}dx + \left (C_{b1}\gamma _{1}\right )_{\partial \mathcal{F}_{b}}; }$$
    (5)

  • Bending momentum equation

    $$\displaystyle\begin{array}{rcl} & & \int _{\mathcal{F}_{b}}E_{b}\boldsymbol{J}\boldsymbol{\theta }^\prime_{\perp }\cdot \boldsymbol{\gamma }^\prime_{\perp } + G_{b}S(\boldsymbol{u}^\prime_{\perp } +\boldsymbol{ e}_{1} \times \boldsymbol{\theta }_{\perp }) \cdot (\boldsymbol{v}^\prime_{\perp } +\boldsymbol{ e}_{1} \times \boldsymbol{\gamma }_{\perp })dx \\ & & \quad =\int _{\mathcal{F}_{b}}\boldsymbol{f}_{l} \cdot \boldsymbol{ v}_{\perp }^{c} +\boldsymbol{ c}_{ l} \cdot \boldsymbol{\gamma }_{\perp }dx + \left (\boldsymbol{F}_{b\perp }\cdot \boldsymbol{ v}_{\perp } +\boldsymbol{ C}_{b\perp }\cdot \boldsymbol{\gamma }_{\perp }\right )_{\partial \mathcal{F}_{b}}, {}\end{array}$$
    (6)

where all the integrals are one-dimensional and the notation a′ denotes the derivative of a quantity a with respect to the variable x. The geometrical parameters are

$$\displaystyle{ S = \int _{\mathcal{S}}d\boldsymbol{x},\quad \boldsymbol{J} =\int _{\mathcal{S}}(\vert \vert \boldsymbol{x}_{\perp }\vert \vert ^{2}\boldsymbol{I} -\boldsymbol{ x}_{ \perp }\otimes \boldsymbol{ x}_{\perp })d\boldsymbol{x}, }$$
(7)

and \(J_{1} =\boldsymbol{ J}: (\boldsymbol{e}_{1} \otimes \boldsymbol{ e}_{1}) =\int _{\mathcal{S}}\vert \vert \boldsymbol{x}_{\perp }\vert \vert ^{2}d\boldsymbol{x}\), where \(\boldsymbol{I}\) is the identity tensor in \(\mathbb{R}^{3}\). If the beam formulation were derived from a solid model with homogeneous mechanical parameters, the beam mechanical parameters would be E b  = E, μ b  = μ and G b  = τ μ, where τ is a shear reduction parameter accounting for the non-uniformity of the shear stress along the cross-section (Oñate 2013). For a non-homogeneous solid model, the derivation of the parameters of the corresponding beam model is not obvious. Assuming (for notational simplicity) homogeneous Dirichlet boundary conditions for both the displacement and rotation fields, the functional space is \(\mathcal{V}_{b} =\{\boldsymbol{ v} \in (\mathcal{H}^{1}(\mathcal{F}_{b}))^{3},\boldsymbol{v}_{\partial \mathcal{F}_{b}} =\boldsymbol{ 0}\}\). We also define \(\mathcal{W}_{b} =\{\boldsymbol{ w} =\boldsymbol{ u} +\boldsymbol{\theta } \times \boldsymbol{x}_{\perp },\,\boldsymbol{u} \in \mathcal{V}_{b},\,\boldsymbol{\theta }\in \mathcal{V}_{b}\}\). Endowed with the inner product of \(\mathcal{H}^{1}(\varOmega _{b})\), \((\boldsymbol{w}_{1},\boldsymbol{w}_{2})_{b} = \int _{\varOmega _{b}}\boldsymbol{w}_{1} \cdot \boldsymbol{ w}_{2} + \nabla \boldsymbol{w}_{1}: \nabla \boldsymbol{w}_{2}d\boldsymbol{x}\), and the corresponding norm, \(\mathcal{W}_{b}\) is a Hilbert space. Using Lax-Milgram theorem (Ern and Guermond 2004), the problem (4)–(6) can be shown to have a unique solution \((\boldsymbol{u}_{0},\boldsymbol{\theta }) \in \mathcal{W}_{b}\). This unique solution can be approximated by the Finite Element method (Hughes 1987; Zienkiewicz and Taylor 2005).

3 Coupling Method in a Stochastic Framework

In this section, we consider a mechanical problem posed over a domain \(\varOmega \in \mathbb{R}^{3}\), and a quantity of interest that can be estimated using the stochastic heterogeneous solid model described in Sect. 2.1. Further, we assume that the complexity of this model is only required over a limited region in order for the quantity of interest to be well evaluated. Hence, we propose to use a coupled model: (i) fine-scale stochastic heterogeneous model over part of the domain, and (ii) coarser deterministic homogeneous beam model over the rest of the domain. The coupled model is developed in the Arlequin framework. 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.

3.1 Arlequin Formulation

The domain Ω is divided into two overlapping subdomains Ω s and \(\varOmega _{b} = \mathcal{F}_{b} \times \mathcal{S}\) such that \(\varOmega _{s} \cup \varOmega _{b} =\varOmega\) (see Fig. 1b). We select a coupling volume \(\varOmega _{c} \subset (\varOmega _{s} \cap \varOmega _{b}) \in \mathbb{R}^{3}\), over which the two models are assumed to exchange information, and introduce its mean fiber \(\mathcal{F}_{c}\) such that \(\varOmega _{c} = \mathcal{F}_{c} \times \mathcal{S}\). For notational simplicity, we assume Dirichet boundary conditions only on the beam model \(\varGamma _{D} \subset \partial \mathcal{F}_{b}\). Forces are imposed in the bulk \(\boldsymbol{f}_{s}\) and on the boundary \(\boldsymbol{g}_{s}\) (on Γ N  ⊂ ∂ Ω s ) for the solid model and along \(\mathcal{F}_{b}\) for the beam model (\(\boldsymbol{f}_{l}\) and \(\boldsymbol{c}_{l}\)). The mixed Arlequin problem reads: find \((\boldsymbol{u}_{s},\boldsymbol{u}_{b},\boldsymbol{\varPhi }) \in \mathcal{V}_{s} \times \mathcal{W}_{b} \times \mathcal{W}_{c}\) such that

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} a_{s}(\boldsymbol{u}_{s},\boldsymbol{v}) + C(\boldsymbol{\varPhi },\boldsymbol{\varPi }(\boldsymbol{v})) =\ell _{s}(\boldsymbol{v}), \quad &\forall \boldsymbol{v} \in \mathcal{V}_{s} \\ a_{b}(\boldsymbol{u}_{b},\boldsymbol{v}_{b}) - C(\boldsymbol{\varPhi },\boldsymbol{v}_{b}) =\ell _{b}(\boldsymbol{v}_{b}),\quad &\forall \boldsymbol{v}_{b} \in \mathcal{W}_{b} \\ C(\boldsymbol{\varPsi },\boldsymbol{\varPi }(\boldsymbol{u}_{s}) -\boldsymbol{ u}_{b}) = 0, \quad &\forall \boldsymbol{\varPsi }\in \mathcal{W}_{c} \end{array} \right., }$$
(8)

where the forms \(a_{s}: \mathcal{V}_{s} \times \mathcal{V}_{s} \rightarrow \mathbb{R}\), \(a_{b}: \mathcal{W}_{b} \times \mathcal{W}_{b} \rightarrow \mathbb{R}\), \(C: \mathcal{W}_{c} \times \mathcal{W}_{c} \rightarrow \mathbb{R}\) are defined by:

$$\displaystyle{ a_{s}(\boldsymbol{u},\boldsymbol{v}) = \mathbb{E}\left [\int _{\varOmega _{s}}\alpha _{s}\;\boldsymbol{\sigma }[\boldsymbol{u}]:\boldsymbol{\varepsilon } [\boldsymbol{v}]d\boldsymbol{x}\right ], }$$
(9)
$$\displaystyle\begin{array}{rcl} a_{b}(\boldsymbol{u}_{b},\boldsymbol{v}_{b})& =& \int _{\mathcal{F}_{b}}\alpha _{b}\left \{E_{b}Su_{1}^\prime v_{1}^\prime +\mu _{b}J_{1}\theta _{1}^\prime\gamma _{1}^\prime + E_{b}\boldsymbol{J}\boldsymbol{\theta }^\prime_{\perp }\cdot \boldsymbol{\gamma }^\prime_{\perp }\right. \\ & & \left.+G_{b}S(\boldsymbol{u}^\prime_{\perp } +\boldsymbol{ e}_{1} \times \boldsymbol{\theta }_{\perp }) \cdot (\boldsymbol{v}^\prime_{\perp } +\boldsymbol{ e}_{1} \times \boldsymbol{\gamma }_{\perp })\right \}dx{}\end{array}$$
(10)

with \(\boldsymbol{u}_{b} = (u_{1}\boldsymbol{e}_{1} +\boldsymbol{ u}_{\perp }) + (\theta _{1}\boldsymbol{e}_{1} +\boldsymbol{\theta } _{\perp }) \times \boldsymbol{ x}_{\perp }\) and \(\boldsymbol{v}_{b} = (v_{1}\boldsymbol{e}_{1} +\boldsymbol{ v}_{\perp }) + (\gamma _{1}\boldsymbol{e}_{1} +\boldsymbol{\gamma } _{\perp }) \times \boldsymbol{ x}_{\perp }\), and

$$\displaystyle{ C(\boldsymbol{u}_{b},\boldsymbol{v}_{b}) = \mathbb{E}\left [\int _{\varOmega _{c}}\left (\boldsymbol{u}_{b} \cdot \boldsymbol{ v}_{b} +\kappa \boldsymbol{\varepsilon } [\boldsymbol{u}_{b}]:\boldsymbol{\varepsilon } [\boldsymbol{v}_{b}]\right )dx\right ] }$$
(11)

where κ is a constant essentially introduced for dimensionality purposes (Ben Dhia and Rateau 2005). Note that for functions \(\boldsymbol{u}_{b}\) and \(\boldsymbol{v}_{b}\) of \(\mathcal{W}_{c}\), decomposed as above, we have

$$\displaystyle\begin{array}{rcl} C(\boldsymbol{u}_{b},\boldsymbol{v}_{b})& =& \int _{\mathcal{F}_{b}}\mathbb{E}\left [Su_{1}v_{1}+S\boldsymbol{u}_{\perp }\cdot \boldsymbol{ v}_{\perp }+J_{1}\theta _{1}\gamma _{1}+\boldsymbol{J}\boldsymbol{\theta }_{\perp }\cdot \boldsymbol{\gamma }_{\perp }+\kappa \left (Su_{1}^\prime v_{1}^\prime + J_{1}\theta _{1}^\prime\gamma _{1}^\prime\right.\right. \\ & & \left.\left.+\boldsymbol{J}\boldsymbol{\theta }^\prime_{\perp }\cdot \boldsymbol{\gamma }^\prime_{\perp } + S(\boldsymbol{u}^\prime_{\perp } +\boldsymbol{ e}_{1} \times \boldsymbol{\theta }_{\perp }) \cdot (\boldsymbol{v}^\prime_{\perp } +\boldsymbol{ e}_{1} \times \boldsymbol{\gamma }_{\perp })\right )\right ]dx. {}\end{array}$$
(12)

The projector \(\boldsymbol{\varPi }: \mathcal{V}_{s} \rightarrow \mathcal{W}_{c}\) is defined by

$$\displaystyle{ \boldsymbol{\varPi }(\boldsymbol{v}) =\langle \boldsymbol{ v}\rangle +\langle \nabla \times \boldsymbol{ v} + \frac{1} {2}(\nabla \cdot (\boldsymbol{e}_{1} \times \boldsymbol{ w}))\boldsymbol{e}_{1}\rangle \times \boldsymbol{ x}_{\perp } }$$
(13)

where \(\langle v\rangle =\int _{\mathcal{S}}v(\boldsymbol{x})dx/S\) for any scalar, vector, or tensor v. Note that any rigid body displacement field in the form \(\boldsymbol{u}_{0}(x) +\boldsymbol{\theta } (x) \times \boldsymbol{ x}_{\perp }\) is conserved by the projection. The linear forms \(\ell_{s}: \mathcal{V}_{s} \rightarrow \mathbb{R}\) and \(\ell_{b}: (\mathcal{V}_{b})^{2} \rightarrow \mathbb{R}\) are defined, respectively, by

$$\displaystyle{ \ell_{s}(\boldsymbol{v}) =\int _{\varOmega _{s}}\boldsymbol{f}_{s} \cdot \mathbb{E}[\boldsymbol{v}_{s}]d\boldsymbol{x} +\int _{\varGamma _{N}}\boldsymbol{g}_{s} \cdot \mathbb{E}[\boldsymbol{v}_{s}]d\boldsymbol{x}\,, }$$
(14)

and (see Sect. 2.2 for the definition of the linear forces and moments)

$$\displaystyle{ \ell_{b}(\boldsymbol{v},\boldsymbol{\theta }) =\int _{\mathcal{F}_{b}}\left \{f_{l,1}v_{1} + c_{l,1}\gamma _{1} +\boldsymbol{ f}_{l,\perp }\cdot \boldsymbol{ v}_{\perp }^{c} +\boldsymbol{ c}_{ l,\perp }\cdot \boldsymbol{\gamma }_{\perp }\right \}dx }$$
(15)

The weight functions α s (x, y, z) and α b (x) in Eqs. (9) and (10) are chosen such that:

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} \alpha _{s} = 1 \quad &\mathrm{in}\,\varOmega _{s}\setminus \varOmega _{b} \\ \alpha _{s}(x,y,z) =\tilde{\alpha } _{s}(x)\quad &\mathrm{in}\,\varOmega _{s} \cap \varOmega _{b}, \\ \tilde{\alpha }_{s}\,,\alpha _{b} > 0 \quad &\mathrm{in}\,\varOmega _{s} \cap \varOmega _{b}, \\ \tilde{\alpha }_{s} +\alpha _{b} = 1 \quad &\mathrm{in}\,\varOmega _{s} \cap \varOmega _{b}.\\ \quad \end{array} \right. }$$
(16)

The functional spaces are \(\mathcal{V}_{s} = \mathcal{L}^{2}(\varXi,(\mathcal{H}^{1}(\varOmega _{s}))^{3})\) and \(\mathcal{V}_{b} = \{\boldsymbol{v},\boldsymbol{\theta }\in (\mathcal{H}^{1}(\mathcal{F}_{b}))^{3}\), \(\boldsymbol{v}_{\vert \varGamma _{D}} =\boldsymbol{\theta } _{\vert \varGamma _{D}} = \mathbf{0}\}\), and the so-called mediator space \(\mathcal{V}_{c}\) is defined as:

$$\displaystyle\begin{array}{rcl} \mathcal{V}_{c}& =& \left \{(\boldsymbol{v}(x) +\boldsymbol{\xi } _{T}) + (\boldsymbol{\gamma }(x) +\boldsymbol{\xi } _{R}) \times \boldsymbol{ x}_{\perp }\;\vert \;\boldsymbol{v},\boldsymbol{\gamma }\in (\mathcal{H}^{1}(\mathcal{F}_{ c}))^{3},\right. \\ & & \ \left.\boldsymbol{\xi }_{T},\boldsymbol{\xi }_{R} \in (\mathcal{L}^{2}(\varXi, \mathbb{R}))^{3}\right \}. {}\end{array}$$
(17)

This choice of mediator space ensures that the resulting mixed formulation (8) is well-posed. The restriction of \(\mathcal{V}_{s}\) to the coupling zone would be another possible option. The resulting mixed formulation would equally be well-posed, but the condition \(C(\boldsymbol{\varPsi },\boldsymbol{u}_{s} - (\boldsymbol{u}_{0} +\boldsymbol{\theta } \times \boldsymbol{x}_{\perp })) = 0\) would be imposed in a much stronger manner, since the dimensionality of the mediator space when discretizing would be much larger. In particular, this would force the average of the solid solution to follow the kinematics of the beam model, which is not desirable because the mechanical parameters of the solid model are heterogeneous, so that it is not reasonable to assume that the sections remains rigid.

One can consider that the system (8) consists of three equations: (i) one governing the behavior of the stochastic solid model, weighted by α s (x) and with a loading arising in the coupling volume Ω c embodied in the operator C; (ii) one governing the behavior of the beam model, weighted by α b (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 \(\boldsymbol{u}_{s}(\boldsymbol{x})\) and \(\boldsymbol{u}_{b} =\boldsymbol{ u}_{0}(x) +\boldsymbol{\theta } \times \boldsymbol{x}_{\perp }\).

3.2 Finite Element Discretization

Based on the previous continuous weak formulation (8), we now consider the discretization and the resulting matrix system. The domain Ω s is split into elements \(\mathcal{E}_{s}\) giving a mesh \(\mathcal{T}_{s}\) with n s degrees of freedom (DOFs). The beam fiber \(\mathcal{F}_{b}\) is split into elements \(\mathcal{E}_{b}\) to form a mesh \(\mathcal{T}_{b}\) with n b DOFs. Finally, \(\mathcal{F}_{c}\) is split into elements to form a mesh \(\mathcal{T}_{c}\) with n c DOFs. All the fields in System (8) are approximated by fields that are globally continuous and polynomials by parts over the elements of the relevant meshes. In particular, we consider the following finite-dimensional functional spaces: \(\mathcal{H}^{1,H}(\varOmega _{s}) =\{ v \in \mathcal{C}^{0}(\varOmega _{s}),v \in \mathbb{P}^{1}(\mathcal{E}_{s})\}\), \(\mathcal{H}^{1,H} = (\mathcal{H}^{1,H}(\varOmega _{s}))^{3}\), \(\mathcal{V}_{s}^{H} = \mathcal{L}^{2}(\varXi,\mathcal{H}^{1,H})\), \(\mathcal{H}^{1,H}(\mathcal{F}_{b}) =\{ v \in \mathcal{C}^{0}(\mathcal{F}_{b}),v \in \mathbb{P}^{2}(\mathcal{E}_{b})\}\), \(\mathcal{V}_{b}^{H} =\{\boldsymbol{ v},\boldsymbol{\theta }\in (\mathcal{H}^{1,H}(\mathcal{F}_{b}))^{3},\boldsymbol{v}_{\vert \varGamma _{D}} =\boldsymbol{\theta } _{\vert \varGamma _{D}} = \mathbf{0}\}\), \(\mathcal{H}^{1,H}(\mathcal{F}_{c}) =\{ v \in \mathcal{C}^{0}(\mathcal{F}_{c}),v \in \mathbb{P}^{2}(\mathcal{E}_{c})\}\), and \(\mathcal{V}_{c}^{H} =\{ (\boldsymbol{v} +\boldsymbol{\xi } _{T}) + (\boldsymbol{\gamma }+\boldsymbol{\xi }_{R}) \times \boldsymbol{ x}_{\perp },\;\boldsymbol{v},\boldsymbol{\gamma }\in (\mathcal{H}^{1,H}(\mathcal{F}_{c}))^{3},\boldsymbol{\xi }_{T},\boldsymbol{\xi }_{R} \in (\mathcal{L}^{2}(\varXi, \mathbb{R}))^{3}\}\), where \(\mathbb{P}^{1}(A)\) and \(\mathbb{P}^{2}(A)\) represent, respectively, the sets of linear and quadratic polynomials over domain A. The consideration of quadratic polynomials over the beam elements simplifies the discretization of the projection operator \(\boldsymbol{\varPi }\). Note that we discretize here only along the space dimension, because we will use the Monte Carlo method (Robert and Casella 2004) for the random dimension. The (scalar) bases associated respectively with \(\mathcal{H}^{1,H}(\varOmega _{s})\), \(\mathcal{H}^{1,H}(\mathcal{F}_{b})\) and \(\mathcal{H}^{1,H}(\mathcal{F}_{c})\), are denoted by: \(\{v_{i}^{s}(\boldsymbol{x})\}_{1\leq i\leq n_{s}}\), \(\{v_{i}^{b}(x)\}_{1\leq i\leq n_{b}}\), and \(\{v_{i}^{c}(x)\}_{1\leq i\leq n_{c}}\). After space discretization, the mixed system (8) may be written:

$$\displaystyle{ \mathbb{E}[\mathrm{\boldsymbol{A}}(\xi )\boldsymbol{U}(\xi )] =\mathrm{ \boldsymbol{F}} }$$
(18)

where ξ indicates dependency on Ξ, and where

$$\displaystyle{ \mathrm{\boldsymbol{A}} = \left [\begin{array}{*{10}c} \boldsymbol{A}_{s}(\xi ) & \boldsymbol{0} &\boldsymbol{P}\boldsymbol{C}&\boldsymbol{P}\boldsymbol{C}_{\xi }& \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{A}_{b} &-\boldsymbol{C}&-\boldsymbol{C}_{\xi }& \boldsymbol{0} \\ (\boldsymbol{P}\boldsymbol{C})^{T} & -\boldsymbol{C}^{T} & \boldsymbol{0} & \boldsymbol{0} &\boldsymbol{S}_{c}^{T} \\ (\boldsymbol{P}\boldsymbol{C}_{\xi })^{T}&-\boldsymbol{C}_{\xi }^{T}& \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{S}_{c} & \boldsymbol{0} & \boldsymbol{0} \end{array} \right ]. }$$
(19)

In that matrix, we have, for the stiffness matrix of the solid model, for 1 ≤ i, k ≤ n s and 1 ≤ j, ℓ ≤ 3:

$$\displaystyle{ \boldsymbol{A}_{s,(ij,k\ell)}(\xi ) =\int _{\varOmega _{s}}\alpha _{s}\boldsymbol{\sigma }[v_{i}^{s}\boldsymbol{e}_{ j}]:\boldsymbol{\varepsilon } [v_{k}^{s}\boldsymbol{e}_{\ell}]d\boldsymbol{x}\,, }$$
(20)

and, the stiffness matrix of the beam model:

$$\displaystyle{ \boldsymbol{A}_{b} = \left [\begin{array}{*{10}c} E_{b}S\boldsymbol{A}_{b}^{1s}& \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & G_{b}S\boldsymbol{A}_{b}^{1} & \boldsymbol{0} & G_{b}S\boldsymbol{A}_{b}^{1v} \\ \boldsymbol{0} & \boldsymbol{0} &\mu _{b}J_{1}\boldsymbol{A}_{b}^{1s}& \boldsymbol{0} \\ \boldsymbol{0} &G_{b}S\boldsymbol{A}_{b}^{1vT}& \boldsymbol{0} &G_{b}S\boldsymbol{A}_{b}^{0} + E_{b}\boldsymbol{(JA)}_{b}^{1} \end{array} \right ], }$$
(21)

with, for 1 ≤ i, k ≤ n b , and 2 ≤ j, ℓ ≤ 3, \(\boldsymbol{A}_{b,ik}^{1s} =\int _{\mathcal{F}_{b}}\alpha _{b}(v_{i}^{b})^\prime(v_{k}^{b})^\prime dx\), \(\boldsymbol{A}_{b,(ij,k\ell)}^{0} = (\boldsymbol{e}_{\ell} \cdot \boldsymbol{ e}_{j})\int _{\mathcal{F}_{b}}\alpha _{b}v_{i}^{b}v_{k}^{b}dx\), \(\boldsymbol{A}_{b,(ij,k\ell)}^{1} = (\boldsymbol{e}_{\ell} \cdot \boldsymbol{ e}_{j})\boldsymbol{A}_{b,ik}^{1s}\), \(\boldsymbol{A}_{b,(ij,k\ell)}^{1v} = (\boldsymbol{e}_{\ell} \times \boldsymbol{ e}_{j}) \cdot \boldsymbol{ e}_{1}\int _{\mathcal{F}_{b}}\alpha _{b}(v_{i}^{b})^\prime v_{k}^{b}dx\) and \(\boldsymbol{(JA)}_{b,(ij,k\ell)}^{1} =\boldsymbol{ J}: (\boldsymbol{e}_{\ell} \otimes \boldsymbol{ e}_{j})\;\boldsymbol{A}_{b,ik}^{1s}\). Note that the transpose sign in Eq. (21) is defined in the sense that \(\boldsymbol{A}_{b,(ij,k\ell)}^{1vT} =\boldsymbol{ A}_{b,(k\ell,ij)}^{1v}\).

The coupling matrix for the beam model is given by

$$\displaystyle{ \boldsymbol{C} = \left [\begin{array}{*{10}c} S\boldsymbol{C}_{1}^{s}& \boldsymbol{0} & \boldsymbol{0} & \boldsymbol{0} \\ \boldsymbol{0} & S\boldsymbol{C}_{1} & \boldsymbol{0} & S\boldsymbol{C}_{1}^{v} \\ \boldsymbol{0} & \boldsymbol{0} &J_{1}\boldsymbol{C}_{1}^{s}& \boldsymbol{0} \\ \boldsymbol{0} &S\boldsymbol{C}_{1}^{vT}& \boldsymbol{0} &S\boldsymbol{C}_{0} +\boldsymbol{ (JC)}_{1} \end{array} \right ] }$$
(22)

with, for 1 ≤ i, k ≤ n c , and 2 ≤ j, ℓ ≤ 3, \(\boldsymbol{C}_{1,ik}^{s} =\int _{\mathcal{F}_{b}}v_{i}^{b}v_{k}^{b} +\kappa (v_{i}^{b})^\prime(v_{k}^{b})^\prime dx\), \(\boldsymbol{C}_{1,(ij,k\ell)} = (\boldsymbol{e}_{\ell} \cdot \boldsymbol{ e}_{j})\boldsymbol{C}_{1,ik}^{s}\), \(\boldsymbol{C}_{0,(ij,k\ell)} = (\boldsymbol{e}_{\ell} \cdot \boldsymbol{ e}_{j})\int _{\mathcal{F}_{b}}v_{i}^{b}v_{k}^{b}dx\), \(\boldsymbol{C}_{1,(ij,k\ell)}^{v} = (\boldsymbol{e}_{\ell} \times \boldsymbol{ e}_{j}) \cdot \boldsymbol{ e}_{1}\int _{\mathcal{F}_{b}}(v_{i}^{b})^\prime v_{k}^{b}dx\) and \(\boldsymbol{(JC)}_{1,(ij,k\ell)} =\boldsymbol{ J}: (\boldsymbol{e}_{\ell} \otimes \boldsymbol{ e}_{j})\boldsymbol{C}_{1,ik}^{s}\). Observe that the structure of the coupling matrix is very close to that of the beam stiffness matrix, with unit material parameters and an additional block diagonal contribution. We additionally introduce the projection matrix

$$\displaystyle{ \boldsymbol{P} = [\boldsymbol{P}^{u_{1} }\;\boldsymbol{P}^{u_{\perp } }\;\boldsymbol{P}^{\theta _{1}}\;\boldsymbol{P}^{\theta _{\perp }}] }$$
(23)

where \(\boldsymbol{P}_{(ij,k)}^{u_{1}} = (\boldsymbol{e}_{j} \cdot \boldsymbol{ e}_{1})\langle v_{i}^{s}(x_{k},\boldsymbol{x}_{\perp })\rangle\), \(\boldsymbol{P}_{(ij,k\ell)}^{u_{\perp }} = (\boldsymbol{e}_{ j} \cdot \boldsymbol{ e}_{\ell})\langle v_{i}^{s}(x_{ k},\boldsymbol{x}_{\perp })\rangle\), \(\boldsymbol{P}_{(ij,k)}^{\theta _{1}} =\langle \nabla \cdot v_{i}^{s}(x_{k},\boldsymbol{x}_{\perp })(\boldsymbol{e}_{1} \times \boldsymbol{ e}_{j})\rangle /2\), and \(\boldsymbol{P}_{(ij,k\ell)}^{\theta _{\perp }} =\langle \nabla \cdot v_{ i}^{s}(x_{ k},\boldsymbol{x}_{\perp })(\boldsymbol{e}_{j} \times \boldsymbol{ e}_{\ell})\rangle\). We also get \(\boldsymbol{C}_{\xi }\) by summing the coordinates of \(\boldsymbol{C}\) over all the DOFs in the columns (because ξ T and ξ R are constant functions (in space) over the coupling domain).

The vector \(\boldsymbol{S}_{c}\) in Eq. (19) is used to remove the over-parameterization of the functional space \(\mathcal{V}_{c}\). Indeed we note that the subspace of constant (in space) deterministic functions of \(\mathcal{V}_{c}\) can be described either only with \(\boldsymbol{v}\) or \(\boldsymbol{\xi }_{T}\). We therefore impose a condition that \(\int _{\mathcal{F}_{c}}\boldsymbol{v}dx =\boldsymbol{ 0}\). It is also possible to impose rather \(\mathbb{E}[\boldsymbol{\xi }_{T}] = 0\) but this would be less trivial in a Monte-Carlo-based simulation. Likewise, there is redundancy between the elements \(\boldsymbol{\theta }\times \boldsymbol{x}_{\perp }\) and \(\boldsymbol{\xi }_{R} \times \boldsymbol{ x}_{\perp }\), and we impose \(\int _{\mathcal{F}_{c}}\boldsymbol{\theta }dx =\boldsymbol{ 0}\). We therefore obtain \(\boldsymbol{S}_{c}\) by summing \(\boldsymbol{C}\) along all the DOFs of the lines.

The load vector is \(\boldsymbol{F}^{T} = [\boldsymbol{F}_{s}\;\boldsymbol{F}_{b}\;\boldsymbol{0}\;\boldsymbol{0}\;\boldsymbol{0}]\), where

$$\displaystyle{ \boldsymbol{F}_{s,(ij)} =\int _{\varOmega _{s}}\alpha _{s}v_{i}^{s}\boldsymbol{f}_{ s} \cdot \boldsymbol{ e}_{j}dx +\int _{\varGamma _{N}}\alpha _{s}v_{i}^{s}\boldsymbol{g}_{ s} \cdot \boldsymbol{ e}_{j}dx, }$$
(24)

and

$$\displaystyle{ \boldsymbol{F}_{b}^{T} = [\boldsymbol{F}_{ b}^{u_{1} }\;\boldsymbol{F}_{b}^{u_{\perp } }\;\boldsymbol{F}_{b}^{\theta _{1} }\;\boldsymbol{F}_{b}^{\theta _{\perp } }], }$$
(25)

with \(\boldsymbol{F}_{b,i}^{u_{1}} = \int _{\mathcal{F}_{ b}}\alpha _{b}v_{i}^{b}f_{ 1}dx\), \(\boldsymbol{F}_{b,(ij)}^{u_{\perp }} =\int _{ \mathcal{F}_{b}}\alpha _{b}v_{i}^{b}\boldsymbol{f}_{ l} \cdot \boldsymbol{ e}_{j}dx\), \(\boldsymbol{F}_{b,i}^{\theta _{1}} = \int _{\mathcal{F}_{ b}}\alpha _{b}v_{i}^{b}c_{ 1}dx\) and \(\boldsymbol{F}_{b,(ij)}^{\theta _{\perp }} =\int _{ \mathcal{F}_{b}}\alpha _{b}v_{i}^{b}\boldsymbol{c}_{ l} \cdot \boldsymbol{ e}_{j}dx\) for 1 ≤ i ≤ n b and 2 ≤ j ≤ 3. Finally the unknown vector is decomposed as:

$$\displaystyle{ \boldsymbol{U}(\xi )^{T} = [\boldsymbol{U}_{ s}(\xi )\;\boldsymbol{U}_{b}\;\boldsymbol{U}_{c}\;\boldsymbol{U}_{\xi }(\xi )\;\boldsymbol{\varLambda }] }$$
(26)

where \(\boldsymbol{U}_{b}^{T} = [\boldsymbol{U}_{b}^{u_{1}}\;\boldsymbol{U}_{b}^{\theta _{1}}\;\boldsymbol{U}_{b}^{u_{\perp }}\;\boldsymbol{U}_{b}^{\theta _{\perp }}]\), \(\boldsymbol{U}_{c}^{T} = [\boldsymbol{U}_{c}^{u_{1}}\;\boldsymbol{U}_{c}^{\theta _{1}}\;\boldsymbol{U}_{c}^{u_{\perp }}\;\boldsymbol{U}_{c}^{\theta _{\perp }}]\) and \(\boldsymbol{U}_{\xi }^{T} = [\boldsymbol{U}_{\xi _{T}}\;\boldsymbol{U}_{\xi _{R}}]\).

The System (18) can then be solved by the Monte Carlo approach, or through a condensation technique of the deterministic part of \(\boldsymbol{A}\) onto its random part. More details on these two approaches can be found in Cottereau et al. (2011) and Le Guennec et al. (2013). The approximate solution of the solid model is then:

$$\displaystyle{ \boldsymbol{u}_{s}(\boldsymbol{x},\xi ) =\sum _{ i=1}^{n_{s} }\sum _{j=1}^{3}\boldsymbol{U}_{ s,ij}(\xi )v_{i}^{s}(\boldsymbol{x})\boldsymbol{e}_{ j}, }$$
(27)

and the approximate solution of the beam model \(\boldsymbol{u}_{b}(\boldsymbol{x}) =\boldsymbol{ u}_{0}(x) +\boldsymbol{\theta } (x) \times \boldsymbol{ x}_{\perp }\) is:

$$\displaystyle{ \boldsymbol{u}_{b}(\boldsymbol{x}) =\sum _{ i=1}^{n_{s} }v_{i}^{s}(x)\Bigg\{\boldsymbol{U}_{ b,i}^{u_{1} }\boldsymbol{e}_{1}+\boldsymbol{U}_{b,i}^{\theta _{1} }\boldsymbol{e}_{1}\times \boldsymbol{x}_{\perp }+\sum _{j=2}^{3}\left (\boldsymbol{U}_{ b,(ij)}^{u_{\perp } }\boldsymbol{e}_{j} +\boldsymbol{ U}_{b,(ij)}^{\theta _{\perp } }\boldsymbol{e}_{j} \times \boldsymbol{ x}_{\perp }\right )\Bigg\}. }$$
(28)

4 Homogenization of a Stochastic Solid Model into a Beam Model

In the previous section, we have described a way of coupling a solid model with stochastic fluctuating mechanical parameters to a beam model with deterministic mechanical parameters. Supposedly, the parameters of the beam model are an upscaled version of the solid parameters. However, as discussed in the introduction, it is not clear how to define the parameters of the deterministic beam given the parameters fields of the solid model. This section aims at proposing a method to do so.

The criterion that we will consider for selecting the upscaled model (the beam model) is based on the idea that its behavior would be the same if considered alone or coupled to its micro-scale version (the solid model). In some sense, if the beam model is well chosen, its mechanical behavior at the macro-scale (for quantities of interest relative to the beam model) should not feel any difference if coupled or not to the solid model it upscales. Note that we do not pretend that this criterion ensures uniqueness of the upscaled model, although it does seem reasonable. The technique we propose here is an extension of the technique proposed in Cottereau (2013a,b).

In terms of implementation, we propose a very simple iterative approach. Starting from an initial guess of parameter vector \(\boldsymbol{p}_{b} = [E_{b}\;\mu _{b}\;G_{b}]^{T}\), we choose a set of boundary conditions, and we compute for each boundary conditions:

  1. 1.

    The solution of a coupled beam-solid model numerically using the Arlequin technique presented in the previous section (Fig. 2); and

  2. 2.

    The solution of a beam model alone, analytically.

The two solutions are then compared in terms of energies and the values of the parameter vector \(\boldsymbol{p}_{b}\) are updated in order to decrease that difference, for instance using the Nelder-Mead technique (Lagarias et al. 1998).

Fig. 2
figure 2

Coupling configuration for the homogenization problem: the beam model is present everywhere \(\mathcal{F}_{b} \times \mathcal{S} =\varOmega\), and the sample microstructure Ω s is placed in the middle

Full size image

As in the classical numerical homogenization technique, the set of boundary conditions that is chosen influences the homogenized vector that is obtained after convergence. At least two approaches can be proposed, using Dirichlet or Neumann boundary conditions, generalizing classical results of homogenization in elastic media. Finally, the choice of initial condition for the parameter vector might also influence the convergence value, or at least the rate of convergence. Two reasonable initial choices would be

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} \boldsymbol{p}_{b} =\quad &[\mathbb{E}[E]\;\mathbb{E}[\mu ]\;\tau \mathbb{E}[\mu ]] \\ \boldsymbol{p}_{b} =\quad &[\mathbb{E}[E^{-1}]^{-1}\;\mathbb{E}[\mu ^{-1}]^{-1}\;\tau \mathbb{E}[\mu ^{-1}]^{-1}]\end{array} \right. }$$
(29)

generalizing the classical Hashin and Shtrikman bounds in linear elasticity (Huet 1990). The general pattern of the numerical homogenization scheme is summarized in Algorithm 1, considering Dirichlet boundary conditions and arithmetic averages for the mechanical parameters.

Algorithm 1 : Algorithmic description of the proposed iterative technique for the numerical homogenization of a random solid model into a beam model

5 Conclusion

We have proposed in this paper a new coupling technique and a new homogenization method. The coupling technique deals with a stochastic solid model and a deterministic beam model, while the homogenization method allows to upscale a stochastic solid beam into a deterministic beam model. Numerical simulations of the proposed homogenization technique will have to be performed in order to discuss the influence of the various numerical parameters involved (choice of initial parameters, boundary conditions). Extension to nonlinear beam models will also be considered, in the context of seismic engineering. Indeed, numerical studies at the micro-scale have shown that strong apparent damping appears in free vibrations of heterogeneous concrete beams (Jehel and Cottereau 2012). Understanding the upscaling of such a micro-scale nonlinear solid model into a beam nonlinear model would allow to reduce significantly the numerical costs associated with the simulation of full-scale buildings.