Minh Vuong Le

Julien Yvonnet
email: julien.yvonnet@univ-paris-est.fr

Nicolas Feld

Fabrice Detrez

The Coarse Mesh Condensation Multiscale Method for parallel computation of heterogeneous linear structures without scale separation

Keywords: Multiscale methods, composites, parallel calculations, CMCM

HAL is

Introduction

Composite materials constitute a mature technology in many industrial fields including the aircraft and automotive industries. With the constant increase in computer performance, numerical methods play a crucial role in the study of heterogeneous structures, including those made of composites. They allow the analysis of these materials without performing expensive laboratory experiments. In some situations, it may be useful to solve the structural problem with a full description of all heterogeneities. For example when the characteristic size of the heterogeneities is not much smaller than the dimensions of the sample, classical homogenization methods fail to describe the local fields and up to a certain precision even the global response. However, solving even small samples with all heterogeneities in complex materials, like woven composites, is a challenge as the corresponding finite element problem can involve billions of degrees of freedom.

One possible strategy for tackling this issue is parallel computing based on domain decomposition methods. In the past years, many techniques have been proposed, including Finite Element Tearing and Interconnecting (FETI) [START_REF] Farhat | A method of finite element tearing and interconnecting and its parallel solution algorithm[END_REF] and Balanced Domain Decomposition (BDD) [START_REF] Tallec | Domain decomposition methods for large linearly elliptic three-dimensional problems[END_REF], where the BDD method is referred to as a primal approach and FETI as a dual approach. The above methods are based on the decomposition of the structure into nonoverlapping subdomains. FETI and BDD were initially developed for homogeneous structures, and later extended to heterogeneous structures using preconditioners [START_REF] Rixen | A simple and efficient extension of a class of substructure based preconditioners to heterogeneous structural mechanics problems[END_REF] and initialization [START_REF] Gosselet | On the initial estimate of interface forces in FETI methods[END_REF]. However when dealing with structures containing large heterogeneities and high contrast of phase properties, these methods often exhibit poor performance. More specifically, when the subdomain interface intersects the heterogeneities, the interface problem lacks an efficient preconditioner. Several techniques have been proposed to alleviate these issues such as FETI-Geneo (FETI-generalized eigenvalues in the overlaps) [START_REF] Spillane | Solving generalized eigenvalue problems on the interfaces to build a robust two level FETI method[END_REF], multipreconditioned FETI method (MPFETI) [START_REF] Gosselet | Simultaneous FETI and block FETI: Robust domain decomposition with multiple search directions[END_REF] or MultiPreconditioned algorithm (AMPFETI) [START_REF] Spillane | An adaptive multipreconditioned conjugate gradient algorithm[END_REF][START_REF] Bovet | Adaptive multipreconditioned FETI: scalability results and robustness assessment[END_REF]. Another technique, the Large Time Increment (LATIN) method [START_REF] Ladevèze | A micro-macro and parallel computational strategy for highly heterogeneous structures[END_REF][START_REF] Ladevèze | The latin multiscale computational method and the proper generalized decomposition[END_REF], is a non-incremental iterative computational strategy where local (nonlinear) problems solutions in subdomains are updated through a global linear problem through interfaces conditions and an appropriate search direction to ensure the global convergence.

Other parallel strategies have been proposed, such as the algebraic multigrid method (AMG) [START_REF] Stüben | Algebraic multigrid (AMG): experiences and comparisons[END_REF][START_REF] Ruge | Efficient solution of finite difference and finite element equations by algebraic multigrid AMG[END_REF][START_REF] Ruge | Algebraic multigrid[END_REF][START_REF] Stüben | A review of algebraic multigrid[END_REF], developed to solve efficiently large linear system of equations. Such approach is a purely algebraic matrix-based approach used to deal with large sparse linear systems and can be considered as a solver to various finite element discretization problems without any geometrical or physical background. The main principle of AMG lies in the coarsening of a given linear system of equations using a coarse grid in order to reduce the problem size.

Multiscale methods constitute another class of approaches to solve the fully detailed heterogeneous structures. For example, in the method proposed by by Zohdi et al. (see e.g. [START_REF] Zohdi | A domain decomposition method for bodies with heterogeneous microstructure basedon material regularization[END_REF], [START_REF] Zohdi | A method of substructuring large-scale computational micromechanical problems[END_REF]), the concepts of homogenization are used in an iterative way. Each iteration consists in solving the homogeneous macro structure, using the effective elasticity tensor, which is computed from substructure problems. The macro strain solution is then applied to the substructure interfaces as Dirichlet boundary conditions and an interface corrector is used to minimize the interface error. Other multiscale methods or subdomain methods with appropriate reduction have been proposed, such as e.g. in [START_REF] Hautefeuille | A multi-scale approach to model localized failure with softening[END_REF][START_REF] Huerta | Proper generalized decomposition solutions within a domain decomposition strategy[END_REF][START_REF] Weinan | Heterogeneous multiscale methods: a review[END_REF][START_REF] Abdulle | The heterogeneous multiscale method[END_REF]. Another approach which has similarities with the method proposed in this work is the Multiscale Finite Element method [START_REF] Hou | A multiscale finite element method for elliptic problems in composite materials and porous media[END_REF][START_REF] Efendiev | Multiscale finite element methods: theory and applications[END_REF], where shape functions obtained from fine scale calculations in each coarse mesh elements substitute the polynomial shape functions. However, the treatments to reduce discontinuities across interfaces induce discontinuities in the displacement field in the reconstructed solution at the fine scale.

Another approach is the direct use of homogenization methods to include the effects of strain gradient in the case where scales are not separated. This situation occurs when the wavelength associated with the strain and stress fields at the micro scale is of the same order of magnitude as the wavelength of the prescribed loads or the characteristic dimensions of the structure. In such situations, first order computational homogenization (see e.g. [START_REF] Zohdi | An Introduction to Computational Micromechanics: Corrected Second Printing, 1st Edition[END_REF]) fails to capture boundary effects or localized fields near defects. For these purposes, strain gradient computational homogenization approaches have been proposed, including Cosserat-type generalized continuum media [START_REF] Feyel | A multilevel finite element method (FE 2 ) to describe the response of highly non-linear structures using generalized continua[END_REF][START_REF] Forest | Cosserat overall modelling of heterogeneous materials[END_REF], second-order computational homogenization [START_REF] Kouznetsova | Multi-scale constitutive modeling of heterogeneous materials with gradient enhanced computational homogenization scheme[END_REF][START_REF] Kouznetsova | Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy[END_REF][START_REF] Yuan | A micromechanical approach of nonlocal modeling for media with periodic microstructures[END_REF][START_REF] Bouyge | A micromechanically based couple-stress model of an elastic two-phase composite[END_REF], asymptotic expansion-based approaches [START_REF] Tran | A micromechanics-based approach for the derivation of constitutive elastic coefficients of strain-gradient media[END_REF], extended computational homogenization with more refined strain gradient fields [START_REF] Tognevi | A multi-scale modeling method for heterogeneous structures without scale separation using filter-based homogenization scheme[END_REF] or nonlocal homogenization methods [START_REF] Hui | A nonlocal homogenization model for wave dispersion in dissipative composite materials[END_REF].

In this paper, we introduce a new multiscale method called Coarse Mesh Condensation Multiscale Method (CMCM), which shares analogies with the FE 2 method but with aim to solve directly a heterogeneous structure. The FE 2 method was originally proposed in [START_REF] Feyel | Multiscale FE 2 elastoviscoplastic analysis of composite structure[END_REF][START_REF] Feyel | FE 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials[END_REF][START_REF] Feyel | A multilevel finite element method (FE 2 ) to describe the response of highly non-linear structures using generalized continua[END_REF]. In this method, the main idea is to solve a heterogeneous nonlinear structure problem by means of a macroscopic mesh, whose strain state defines boundary conditions on representative volume elements (RVE) associated to a lower scale, which after solving the local problem provides the macro stress through averaging. One appealing feature of FE 2 is that the local (fine scale) problems can be solved in parallel, as being independent of each others. FE 2 has been widely applied and extended to many problems including secondorder homogenization, transient conduction problems, model reduction, damage localization or topology optimization [START_REF] Kouznetsova | Multi-scale second-order computational homogenization of multi-phase materials: a nested finite element solution strategy[END_REF][START_REF] Yvonnet | The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains[END_REF][START_REF] Ozdemir | Computational homogenization for heat conduction in heterogeneous solids[END_REF][START_REF] Geers | Multi-scale computational homogenization: Trends and challenges[END_REF][START_REF] Coenen | A multi-scale approach to bridge microscale damage and macroscale failure: a nested computational homogenization-localization framework[END_REF][START_REF] Xia | Concurrent topology optimization design of material and structure within fe2 nonlinear multiscale analysis framework[END_REF].

In this work, we use analogies with this idea to solve heterogeneous structures without scale separation trough an efficient approximation of the fine scale solution by means of parallel computations on subdomains. Even though possibly applied to nonlinear problems, we restrict in this work to linear problems and show that in this case the technique leads to an efficient one-iteration scheme where local solution (in subdomains decomposing the structure) can be solved in parallel through off-line calculations. Here, the solution of fine scale subdomain problems are related to the degrees of freedom of a coarse mesh which has much fewer degrees of freedom than the full problem (by several order of magnitudes). The summary of the paper is as follows. In section 2, we first introduce notations used in this paper. In Section 3, we describe CMCM and provide practical application details in 2D and 3D. Numerical examples involving both academic validation tests as well as large industrial examples are presented and discussed in section 4.

Preliminary: notations

Vectors and second order tensors, as well as matrices, are denoted by bold letters A. Third order tensors are denoted by calligraphic uppercase letters G, fourth-order, fifth-order and sixth-order tensors are denoted by double case letters A. Double contraction of indices for second order tensors A and B is denoted by A : B = A i j B i j , dot product for two vectors a and b by a • b = a i b i , and simple contraction of indices for a second order tensor A and a vector b is denoted by (Ab) i = A i j b j . The gradient operator is denoted by ∇(.) and the divergence operator by ∇ • (.). Let u be defined as the displacement vector and x as a material coordinate, we define the linearized strain as:

ε i j = 1 2 ∂u i ∂x j + ∂u j ∂x i , (1)
the second gradient displacement tensor by

G i jk = ∂ 2 u i ∂x j ∂x k , (2)
and the strain gradient tensor as:

∇ε i jk = 1 2 ∂ 2 u i ∂x j ∂x k + ∂ 2 u j ∂x i ∂x k . (3)
We denote the analogous macro quantities ε, Ḡ and ∇ε as the macro strain, macro second gradient of displacements and macro strain gradient tensors, respectively. The key idea of CMCM approximation is to construct an approximation of the strain and stress solution fields at the finest scale in subdomains whose solutions can be computed in parallel through off-line calculations. First, the heterogeneous structure is decomposed into subdomains Ω α , α = 1, 2, ..., N sub where N sub denotes the number of subdomains, and such that Ω = α Ω α (see Fig. 1 (a)). Then, each subdomain is discretized with a mesh of finite elements associated with the fine scale. In addition, a coarse mesh is defined to discretize the structure, independently of the heterogeneities (see Fig. 1 (b)). Dirichlet boundary conditions are then defined over each subdomain, and associated with a small number of parameters, gathered in a vector ḡα (see section 3.2). We define these parameters such that they minimize the error in the least square senses between the strain field in the subdomain and the strain field in the elements of the coarse mesh covering the subdomain. In the linear case, a linear relationship can be established between displacements of the coarse mesh and the strain field in the subdomains. Note that this approximation depends on the location of the point x in the element of the coarse mesh and is then richer than the simple linear combination of subdomain elementary solutions (see details in section 3.2.1). Then, the solution in the subdomains can be condensed at the nodes of the coarse mesh and the problem can be solved on this coarse mesh only to define the fine scale solution by post-treatment. The main steps are summarized below and described in more details in the following.

Coarse Mesh Condensation Multiscale Method (CMCM)

General description

(i) Solve localization problems in each subdomain Ω α (parallel computations).

(ii) Determine the relationship between the dofs of the coarse mesh and the reduced vector ḡα in each subdomain Ω α (parallel computations).

(iii) Solve the global problem on the coarse mesh.

(iv) Re-localization of fine scale fields from global problem solution.

Step (i): Subdomain problem 3.2.1. Localization problem on subdomains

We first define the problem to be solved on each individual subdomain Ω α whose boundary is denoted by ∂Ω α (see Fig. 1) (a). A general expression of Dirichlet boundary conditions is introduced as:

u i (x) = k ḡα k d α ik (x) on ∂Ω α , ( 4
)
where ḡα is the reduced parameters vector. The size of ḡα is assumed to be small as compared to the total number of dofs on the boundary ∂Ω α in order to perform a reduced condensation of the internal dofs.

In this work, we adopt the particular form for relation (4) as:

u i (x) = εi j x j + 1 2 Ḡi jk x j x k on ∂Ω α , (5)
where εi j is a macroscopic strain tensor and Ḡi jk is a macroscopic third-order second gradient displacement tensor, which can also be re-expressed as a function of an effective (homogeneous) first gradient of the strain tensor ∇ εi jk through (see e.g. [START_REF] Mindlin | On first strain-gradient theories in linear elasticity[END_REF][START_REF] Auffray | A complete description of bi-dimensional anisotropic strain-gradient elasticity[END_REF]):

Ḡi jk = ∇ εi jk + ∇ εik j -∇ ε jki . (6)
In addition, we restrict the approximation (5) to the x j x k , ( j k) terms to reduce the number of parameters, i.e. terms products of x 2 i are set to zero. The corresponding obtained solutions will be referred to as Second-order CMCM in the example (section 4). For comparison, another approximation of (5) keeping only the linear terms (i.e. setting all Ḡ term products to zero) will be considered and referred to as CMCM in the examples.

Special attention must be paid to the case of a homogeneous subdomain, i.e. containing only one material phase. In that situation, we must ensure that the boundary conditions [START_REF] Spillane | Solving generalized eigenvalue problems on the interfaces to build a robust two level FETI method[END_REF] do not induce spurious fluctuations, i.e. that the solution in Ω α derives from (5) in the form:

ε(x) = ε + ∇ε • x in Ω α . (7)
Considering that one subdomain contains a single material phase with elastic properties C 0 , applying the Hooke's law and taking the divergence of ( 7) one obtains:

∇ • C 0 : ε(x) = ∇ • C 0 : [ε + ∇ε • x] = f, (8)
which is verified if

f i = C 0 i jkl ∇ εkl j . (9)
Then, body forces must be prescribed in addition to the boundary conditions to ensure null fluctuations in the case of a homogeneous subdomain. One possible choice for C 0 is (see [START_REF] Yvonnet | Computational second-order homogenization of materials with effective anisotropic strain gradient behavior[END_REF]) C 0 i jkl = C, where C is the homogenized elastic modulus defined by

C = 1 V Ω α C(x) : A(x)dΩ, (10)
and where A(x) is the classical localization tensor. Then, the localization problem is defined as follows: Given ḡα = {ε, ∇ε}, find ε(x) in Ω α such that:

∇ • (C(x) : ε(x)) = f, (11)
f i = C i jkl ∇ εkl j , (12)
u(x) = ε • x + 1 2 Ḡ : x ⊗ x on ∂Ω α . ( 13
)
We can see that with this definition, if the subdomain is homogeneous, then the solution is self-equilibrated for the above quadratic boundary conditions. Problem (11)-( 13) is then classically solved by finite elements as follows.

The weak form of the problem to be solved on the subdomain Ω α is given by: search u ∈ H 1 (Ω α ) satisfying the boundary conditions [START_REF] Ruge | Algebraic multigrid[END_REF] and such that:

Ω α ε(u) : C(x) : ε(δu)dΩ = Ω α f(x) • δudΩ ∀δu ∈ H 1 0 (Ω α ) ( 14
)
where H 1 and H 1 0 are the usual Sobolev vector spaces. Introducing classical FEM discretization in [START_REF] Stüben | A review of algebraic multigrid[END_REF], we obtain a linear system in the form:

K α u = F α (15)
with

K α = Ω α B mT (x)C(x)B m (x)dΩ, F α = Ω α N mT (x)fdΩ, ( 16
)
where N * and B m are matrices of shape functions and of shape functions derivatives on the fine mesh in the subdomain, respectively (see Fig. 49), and C(x) is the matrix form associated with C(x). It is worth noting that for one subdomain Ω α , the linear systems to be solved for each component of ḡα involve the same matrix K α , which thus has to be assembled only once.

In case the when subdomain boundaries cross the interfaces, large errors can be generated by the boundary conditions [START_REF] Ruge | Algebraic multigrid[END_REF]. In that case, an extension of the method consist in prescribing the boundary conditions on the external boundary ∂ Ωα of an extended subdomain Ωα (see Fig. 2). However, the localization tensor is only evaluated in the initial subdomain Ω α ). We define a parameter β = h ext /L 0 , where L 0 is the initial length of the subdomain Ω α (considered as square subdomain in this work for the sake of simplicity). Note that this process is similar to the oversampling technique in MSFEM [START_REF] Hou | A multiscale finite element method for elliptic problems in composite materials and porous media[END_REF].

hext L 0

Reduced parameters g α in the 2D case

In 2D, after neglecting terms in x 2 i , (13) leads to:

u 1 (x) = ε11 x 1 + ε12 x 2 + 1 2 Ḡ112 + Ḡ121 x 1 x 2 , (17)
u 2 (x) = ε12 x 1 + ε22 x 2 + 1 2 Ḡ212 + Ḡ221 x 1 x 2 . (18)
After simplifications, we obtain:

u 1 (x) = ε11 x 1 + ε12 x 2 + ∇ε 112 x 1 x 2 , (19)
u 2 (x) = ε12 x 1 + ε22 x 2 + ∇ε 221 x 1 x 2 . ( 20
)
In the Second-order CMCM case, the boundary conditions of each Ω α subdomain are parameterized by 5 independent scalar values gathered in the ḡα vector:

ḡα = [ ε11 , ε22 , ε12 , ∇ ε112 , ∇ ε221 ] T . ( 21
)
As the problem to be solved over Ω α is linear, applying the superposition principle allows the strain field to be expressed in Ω α as:

[ε(x)] = A α (x)ḡ α ∀x ∈ Ω α . ( 22
)
where [ε(x)] is the column-vector notation associated with the second-order strain tensor ε = 1 2 ∇u + ∇ T u and A α (x) is a localization matrix (see section 3.2.2).

The matrix A α (x) is therefore a 3 × 5 matrix whose columns [ε i (x)] are the strain vector solution of ( 11)-( 13) for ḡα i = 1 and ḡα j = 0, i j:

A α (x) = [ε 1 (x)], [ε 2 (x)], [ε 3 (x)], [ε 4 (x)], [ε 5 (x)] . (23)
In the basic CMCM case, only the first three terms of ḡα and consequently the first three columns of A α are computed.

Reduced parameters g α in the 3D case

In 3D, using ( 13) and ( 6), we obtain, after simplifications and neglecting terms in x 2 i :

u 1 (x) = ε11 x 1 + ε12 x 2 + ε13 x 3 + ∇ ε112 x 1 x 2 + ∇ ε113 x 1 x 3 + G 123 x 2 x 3 , (24)
u 2 (x) = ε12 x 1 + ε22 x 2 + ε23 x 3 + ∇ ε221 x 1 x 2 + ∇ ε223 x 2 x 3 + G 213 x 1 x 3 , (25)
u 3 (x) = ε13 x 1 + ε23 x 2 + ε33 x 3 + ∇ ε331 x 1 x 3 + ∇ ε332 x 2 x 3 + G 312 x 1 x 2 . (26)
In the Second-order CMCM case, ḡα contains 15 independent parameters (and in the basic CMCM case, only the first 6 are needed):

ḡα = ε11 , ε22 , ε33 , ε12 , ε13 , ε23 , ∇ ε112 , ∇ ε113 , ∇ ε221 , ∇ ε223 , ∇ ε331 , ∇ ε332 , G 123 , G 213 , G 312 T .
Thus 15 (resp. 6) linear problems need to be solved on each subdomain. Moreover all these problems can be solved in parallel. The matrix A α (x) is in that case a 6 × 15 (resp. 6 × 6) matrix.

It is worth noting that in the case where the strain gradient parameters are applied, the subdomain needs to be centered in order to obtain the correct deformation mode.

Step (ii): Relation between coarse mesh-displacements and subdomain boundary conditions

To relate the solutions in the subdomains to the coarse mesh, a relationship between the vector of boundary conditions ḡα of subdomain Ω α and the dofs in one element Ω e of the coarse mesh, denoted by ūe , must be established. For this purpose, we minimize the distance between the strain approximation provided by [START_REF] Efendiev | Multiscale finite element methods: theory and applications[END_REF] and the strain in each element of the coarse mesh individually (see Fig. 1), which is given by:

[ε(x)] = Be (x) ūe , ( 27
)
where Be is a matrix of shape functions derivatives of one element Ω e of the coarse mesh. Then the problem to be solved is given by ḡα = Arg min J

with

J = Ω e A α (x)ḡ α -Be ūe 2 dΩ, (29)
where Ω e is the domain associated with the element e in the coarse mesh, and (a) 2 = a•a, a being a vector. Minimizing J with respect to ḡα gives (writing ḡα i ≡ g i , A α i j ≡ A i j , Be i j ≡ B i j and ūE inc ≡ u i for alleviating the notations):

∂J ∂g m = 0, m = 1, 2, ..., N g , (30) or
;

Ω e 2A i j ∂g j ∂g m A i j g j -B i j u j dΩ = 0. (31)
Then:

Ω e A im (x)A i j (x)dΩg j = Ω e A im (x)B i j (x)u j dΩ m = 1, 2, ..., N g . ( 32
)
We obtain a linear system of equations in the form:

G α ḡα = H αe ūe ( 33
)
and then:

ḡα = (G α ) -1 H αe ūe , (34)
with:

G α = Ω α (A α (x)) T A α (x)dΩ, (35)
and

H αe = Ω e (A α (x)) T Be (x)dΩ. (36)
Finally, we obtain the approximation of the fine scale strain in each element Ω e of the coarse mesh covering a subdomain Ω α , using [START_REF] Efendiev | Multiscale finite element methods: theory and applications[END_REF] and [START_REF] Feyel | Multiscale FE 2 elastoviscoplastic analysis of composite structure[END_REF], as:

[ε(x)] = A α (x) (G α ) -1 H αe ūe = M α (x)H αe ūe , (37) or
[ε(x)] = M α (x)H αe ūe (38) with M α (x) = A α (x) (G α ) -1 . (39)

Step (iii): Global problem

We now consider the problem to be solved on the coarse mesh over the entire structure (see Fig. 1 (b)). Assuming zero body forces, the governing equations of the problem in the structure are given by:

                   ∇ • σ(x) = 0 ∀x ∈ Ω, σ(x) = C(x) : ε(x) ∀x ∈ Ω, u(x) = ū * ∀x ∈ ∂Ω u , σ • n = F * ∀x ∈ ∂Ω F , (40) (41) (42)
where ū * and F * are prescribed displacements and tractions on the corresponding Dirichlet and Neumann boundaries ∂Ω u and ∂Ω F . Using vector forms of strain tensors, the weak form of the system of equations ( 40) can be expressed as follows:

Ω [ε(u)] T C(x)[ε(δu)]dΩ = ∂Ω F F * • δudΓ. ( 44
)
Considering the same approximation for test functions δ ūe as in [START_REF] Coenen | A multi-scale approach to bridge microscale damage and macroscale failure: a nested computational homogenization-localization framework[END_REF]:

[ε(δu)] = M α (x)H αe δ ūe , (45)
and substituting [START_REF] Coenen | A multi-scale approach to bridge microscale damage and macroscale failure: a nested computational homogenization-localization framework[END_REF] and ( 45) in ( 44) we obtain:

δ ūeT α Ω α (H αe ) T (M α (x)) T C(x)M α (x)H αe dΩ ūe = δ ūeT ∂Ω F N T (x)F * dΓ. ( 46
)
Owing to the arbitrariness of δ ūeT , we then obtain a linear system of equations in the form:

K ū = F ( 47
)
where:

K = α Ω α (H αe ) T (M α (x)) T C(x)M α (x)H αe dΩ. ( 48
)
In practice, K is assembled from elementary matrices computed in each micro element m covered by an element e defined in its associated domain Ω e of the coarse mesh (see Fig. 3). Given the subdomain Ω α containing the fine mesh element e m , the matrix H αe is constant in Ω α and then the elementary matrix K can be expressed as:

Km = (H αe ) T Ω m (M α (x)) T C(x)M α (x)dΩH αe (49)
where Ω m is the domain associated with one fine mesh element m and

F = ∂Ω F NT (x) F * dΓ, ( 50
)
where N is the shape function matrix of elements in the coarse mesh. The Dirichlet and Neumann boundary conditions are classically prescribed on the coarse mesh. Once ū is known, the strain field on the fine mesh in each subdomain can be reconstructed by [START_REF] Feyel | FE 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials[END_REF] and the stress using:

[σ(x)] = C(x)M α (x)H αe ūe . (51)
In this work, we did not consider the case when one micro element m cut the boundaries of the domain Ω e occupied by a coarse mesh element. This point is reported to future studies. The algorithm for solving the coarse mesh problem is summarized in Algorithm 2. -Solve ( 11)-( 13) to obtain A α (x) and store 3:

-Compute G α using [START_REF] Yvonnet | The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains[END_REF] and store 4:

-Compute M α (x) using (39) and store

5:

LOOP over all macro elements e M ∈ Ω α 6:

-Compute H αe using (36) and store -Find micro elements m ∈ Ω e 3:

-Find the subdomain α contaning m 4:

-Given H αe , M α (x):

5:

LOOP over all micro elements m ∈ Ω e 6:

-

Compute Km = (H αe ) T Ω m M α (x) T C(x)M α (x)dΩ H αe 7:
Assemble Km in K 8:

END LOOP 9: END LOOP In this subsection, we show analogies between the present method and the FE 2 method, originally proposed in [START_REF] Feyel | A multilevel finite element method (FE 2 ) to describe the response of highly non-linear structures using generalized continua[END_REF] to solve nonlinear multiscale problems with separated scales. In FE 2 (see Fig. 49 (a)), a relationship between macro strain ε and the macro stress σ is obtained numerically at each Gauss point of the coarse mesh by: (i) prescribing Dirichlet boundary conditions depending on ε on the boundary of an RVE associated with the element; (ii) solving the local problem on the RVE; (iii) averaging the stress in the RVE to obtain σ. In CMCM (see Fig. 49 (b)), a group of elements is covering a subdomain. Solving a minimization problem (see section 3.3), Dirichlet boundary conditions are prescribed on the boundary of the subdomain. The solution is then solved on the subdomain to obtain the stress field. The main differences are summarized as follows: (a) in FE 2 , the local calculations are computed on an RVE associated with the microstructure. In CMCM, the local calculations are performed in subdomains whose union is the fully detailed heterogeneous structure; (b) In FE 2 , the parameters defining the boundary conditions on the local problem (RVE) are associated with a single integration point in the mesh at the maccroscale. In CMCM, the parameters associated with boundary conditions to be applied on the local problem (subdomain) depend on the strain field of all elements of the coarse mesh covering the subdomain; (c) in FE 2 , the local problem is used to define the macro strain -macro stress relationship. In CMCM, the local problem is used to relate the strain in the coarse mesh to the full stress field in all elements in the subdomain (fine scale) covered by the coarse mesh elements and the notion of macro stress is no more necessary. Both methods share the following features: (a) all local computations can be performed in parallel; (b) the local solution (fine scale) can be reconstructed given the macro (coarse) mesh solution; (c) the macro (coarse) mesh problem involves much fewer degrees of freedom as compared to solving the full problem.

However it is necessary to make it clear that CMCM is not a special case of FE 2 . Even though the method can be potentially applied to nonlinear problems, we only treat linear problems in this paper. The treatment of nonlinear problems would involve the same idea but applied to the linearized problem in a Newton solving procedure. In such, the appealing parallel solving feature of the method would be maintained. However, the developments related to nonlinear problems are out of the scope of the present work and are reported to future studies.

As another remark, note that in CMCM the full microstructure has to be known everywhere in the structure, in contrast with FE 2 . However, we recall that this method applies to structures where scale cannot be separated, and where the size of heterogeneities becomes not negligible as compared to the size of the structure, as for example in woven composites. In industrial application, it is nowadays common to obtain microtomography images of the full structure up to the level of the yarns.

Numerical examples

In this section, we analyze and demonstrate the potential of the method through both academic and simplified industrial examples of realistic scale. In the present work we only consider periodic heterogeneous structures, even though the technique is not restricted to this class of problems. The application to non-repeatable structures will be discussed in a separate study. In the context of periodic subdomains, the local problem ( 11)- [START_REF] Ruge | Algebraic multigrid[END_REF] has to be solved only once (see section 3.2). In the following, three solutions are considered for comparison:

• A reference solution obtained by a direct finite element calculation, where the structure is fully meshed, covering all the details of the heterogeneities. The reference solution of the largest 3D problem is obtained with the iterative AMPFETI method [START_REF] Spillane | An adaptive multipreconditioned conjugate gradient algorithm[END_REF][START_REF] Bovet | Adaptive multipreconditioned FETI: scalability results and robustness assessment[END_REF].

• A solution using the present method without using strain gradient (denoted by CMCM).

• A solution using the present method with strain gradient (denoted by CMCM).

An energy global error is defined as:

Err E = Ω ε re f (x) -ε CMCM (x) : C(x) : ε re f (x) -ε CMCM (x) dΩ Ω ε re f (x) : C(x) : ε re f (x)dΩ (52)
where ε re f denotes the strain field obtained by the reference solution, and ε CMCM denotes the corresponding strain field obtained from CMCM. Alternatively, a global L 2 -error is defined as:

Err L2 = Ω u re f (x) -u CMCM (x) . u re f (x) -u CMCM (x) dΩ Ω u re f (x) . u re f (x) dΩ (53)
where u re f denotes the displacement field obtained by the reference solution, and u CMCM denotes the corresponding displacement field obtained from CMCM.

2D periodic composite structure with cylindrical fibers

We consider the academic structure depicted in Fig. 5 (a), containing 4 periodic cylindrical fibers. Here, the structure is subdivided into 4 subdomains (see Fig. 9 (b)). The size of the square structure is L = 180 mm, and the diameter of the fibers is D = 54 mm. The matrix is assumed to be isotropic elastic with coefficients E mat = 1 Mpa and ν mat = 0.25. The inclusions are also isotropic, with Poisson's ration ν inc = 0.25. We analyze several values of the inclusion Young modulus: E inc = 5 MPa, which is roughly the ratio of phase properties found in typical ceramics/ceramics composites, E inc = 10 3 MPa (large ratio) and an extreme case E inc = 10 6 MPa to analyze the capabilities of the method in that case, and especially in the case when the subdomains cross the interfaces, as such case is known to be a difficult one for iterative FETI domain decomposition methods (see [START_REF] Spillane | Solving generalized eigenvalue problems on the interfaces to build a robust two level FETI method[END_REF]). The structure is subjected to the following Dirichlet boundary conditions:

u(x) =        u x = x 2 y α u y = -x 3 3α on ∂Ω, (54)
where α = 10 6 mm 2 and ∂Ω is the whole external boundary of the structure. We first solve the problem with the present method without the strain gradient enhancement (CMCM). The reference finite element mesh describing the entire structure including all heterogeneities is composed of 73130 linear triangular elements, corresponding to 73852 degrees of freedom. In order to investigate the convergence of the proposed method, 2 cases will be considered in this example:

• For a fixed number of subdomains, we vary the number of coarse mesh elements.

• For a fixed number of coarse mesh elements, we vary the number of subdomains.

Convergence with respect to the number of coarse mesh elements

In this first case, we fix the number of subdomains to 4 and increase the number of coarse mesh elements to analyze the convergence of the error. The case E inc = 10 6 MPa is first chosen (most defavorable case). In this work, we use 4-node bilinear elements and 8-node trilinear elements in 2D and 3D respectively for the coarse mesh. Five coarse meshes are used, whose numbers of elements and dofs are provided in Table 2.

Fig. 6 shows a comparison of the strain field ε 11 (x) solution obtained with CMCM with the 5 meshes and the reference solution. We can note that the local solution, reconstructed from [START_REF] Feyel | FE 2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials[END_REF], is significantly improved when the coarse mesh is refined.

In Figs. 7 and8, the global energy and L 2 errors are plotted as a function of the number of coarse mesh dofs, and for several values of β, the parameter which defines the size of the oversampling (see section 3.2.1). We can see from Figs. 7 and 8 that the convergence is observed with both respect to the number of coarse mesh dofs and to β.

Case 2: fixing number of coarse mesh dofs, changing the number of subdomains

In this case, the number of coarse mesh elements is fixed and equal to 502, corresponding to 1305 dofs. The number of subdomains is then increased as described in Fig. 9. In the case of 16 subdomains, we can note that the interfaces cross the inclusions.

In this test, we investigate the influence of β on the relocalized solution for the case with 16 subdomains (Fig. 9 (c)). Several values of β (β = 0.16, 0.33, 0.47, 0.82 and 1) are used. Figs. 11 and12 show the stress fields with respect to β in the cases E inc = 10 6 MPa and E inc = 5 MPa. The evolution of global energy error and L 2 error versus β is given in Fig. 10. We can see that the obtained global errors are very high when the interfaces cross the inclusions (i.e. β = 0.16, 0.33 and 0.47). However, when the extended subdomains cover the whole inclusions (i.e. β = 0.82 and 1), the global errors are significantly reduced. Error also decrease with respect to the phase properties contrast. More interestingly, for contrasts of the order of 5 (as e.g. in ceramics/ceramics composites), the error is low even for β = 0.

We now use the converged solution in the case of 16 subdomains (i.e. β = 1) and compare to the cases of 1 and 4 subdomains. Fig. 13 (a),(b),(c) and (d) show the relocalized stress field of the refence solution and of the CMCM solution, for 1, 4 and 16 subdomains, respectively in the case E inc = 10 6 MPa and in Figs. 14 in the case E inc = 5 MPa. Fig. 15 shows the evolution of global strain energy error in function of number of subdomains. We can see that when the number of subdomains increases, the corresponding global error increases. This is due to the fact that some errors are localized at the interfaces between subdomains, thus more subdomains we have, more errors are likely to be committed. However, it is worth noting that in the third case where interface crosses inclusions, the global error is 0.032 which is still reasonable considering a high contrast between inclusion and matrix properties (10 6 ).

In conclusion, from the numerical results, we can see that refining the coarse mesh indeed reduces the global error while increasing the number of subdomain increases the global error, due to the fact that the chances that the subdomains cross the interfaces are higher. When inclusions cut the interfaces we have seen that the use of the larger subdomains (β > 0) (see section 3.2.1) might drastically reduce the errors. If this case can be avoided, e.g. by means of appropriate choices of subdomains not crossing the interfaces, the errors can also be reduced. These errors also reduce with the contrast of phase properties between the inclusions and the matrix. it is also worth noting that defining an error indicator in this framework and an associate mesh adaptation would be useful and might deserve investigations in future works.

Comparison with a simple method using fine integration and submodelling

In this section, we compare the present method with a simple other approach using a coarse mesh and relocalization. In this method, the same fine mesh is defined over the whole to describe the microscale. The same coarse mesh as in Fig. 1 (a) is used. Here, the problem is first solved on the coarse mesh, but an exact integration is performed on the fine mesh. Then, a "submodelling" step is performed, by applying the obtained coarse mesh strain field as first order boundary conditions u(x) = ε • x (see Eq. 27 for the expression of ε) on a domain of fine mesh elements contained in the corresponding coarse mesh element, in order to obtain the displacement as well as strain and stress fields at the micro scale. It is worth noting that this simple approach does not require preliminary calculations and thus takes much less computational times. To perform fair comparisons, an extended subdomain method, as described in Fig. 2 is employed. Results are presented in Figs. [START_REF] Zohdi | A method of substructuring large-scale computational micromechanical problems[END_REF], 17 and 18. The same example as in the previous section is chosen with a coarse mesh containing 256 elements (see Table 2). The contrast between inclusion and matrix properties is taken as E inc /E mat = 10 6 . Fig. 16 shows the evolution of the global error in the energy norm and L 2 error as a function of β, which defines the size of the extended subdomain. It can be shown that the error can be reduced from CMCM as comparison whith such simple approach. Figs. 17 and 18 show re-localized strain and stress fields obtained by both approaches for β = 1, showing that the simple approach based on fine integration on the fine mesh is not able to capture correctly local fields while the present CMCM provides acceptable re-localized fine scale fields. In this example, we consider a composite beam under bending (see Fig. 19a). The objective of this academic example is to appreciate the gain obtained by introducing the strain gradient effects. The dimensions of the beam are L = 21 mm and H = 1 mm. Each fiber has a diameter of 0.4 mm and is positioned at the center of a square subdomain of size H, as depicted in Fig. 19b. The mechanical properties of fibers and matrix are the same as in the previous example. The applied loading is a pressure field mimicking the reaction force of contact with cylinders. The corresponding fields are applied in the vicinity of three points with coordinates (x c i , y c i ) from left to right as follows: (0.5, 0), (10.5, 1) and (20.5, 0) and are provided as:

p i (x) = p 0 i        1 - x -x c i x p 2        , (55)
where p 0 2 = 20 MPa, p 0 1 = p 0 3 = 10 MPa and x p = 1 mm. To remove rigid body motions, the nodal y-displacements at nodes (0, 0) and (L, 0), and the nodal x-displacement at node (0, H) are blocked. Three different coarse meshes are considered (Fig. 20). The number of elements and the corresponding number of dofs are listed in Table 3 Here, the parameter β (see section 3.2.1) was chosen to be zero. Different components of the local strain field ε(x) are compared with the reference solution, the CMCM and the Second-order CMCM in Figs. 21, 22 and 23. To better quantify the accuracy of each solution, the strain and stress (1, 1) components are plotted on the line (x = L/2) in Fig. 24. In these figures, only the most converged solutions, on coarse mesh 3, are displayed. In Fig. 25, the logarithmic error between the reference solution on a fine mesh and the reconstructed solution using GFE 2 is shown. We can note that in this case, the Second-order CMCM gives a good convergence to the reference solution when refining the coarse mesh, while the CMCM does not converge.

In this example, the method with strain gradient leads to a more accurate solution at the same computational costs than the first-order method, as we recall that no additional degrees of freedom are introduced for that purpose.

In the next example, we consider a three dimensional problem to evaluate the capability of the method to handle a structure that is closer to practical applications.

3D three point bending of a woven composite structure

In this example, we show the capability of the method to handle industrial problems with a fine description of local heterogeneities within the structure. We consider the 3-point bending of a plain woven composite, as depicted in Fig. 26. The dimensions of the structure are L = 50 mm, b = 10 mm and h = 1.1 mm. As in the previous example, the loading is applied as pressure fields distributed along three lines (x = 3, z = 0), (x = 25, z = 1.1) and (x = 47, z = 0). The field distribution is described in Eq. 55. In order to remove rigid body motions, the nodal z-displacements on line (x = 0, z = 0), (x = L, z = 0), the nodal y-displacements at nodes (0, 0, 0), (L, 0, 0) and the nodal x-displacements at node (0, 0, 0) are blocked.

In woven composites, there are three main scales of interest: the micro scale related to an explicit description of the fibers which compose the yarns, the meso scale related to the yarn's weaving, and the macro scale related to the whole structure. In this example, we consider a mesoscale description of the structure. Such simulations are usually avoided as not computationally tractable. However, we show that using CMCM, we are now able to solve the full problem including all the details of the macro and meso scales. In order to provide a reference solution, the problem is both solved using CMCM and the high performance parallel domain decomposition method AMPFETI (Adaptive Multi-Preconditioned Finite Element Tearing and Interconnect i.e. AMPFETI [START_REF] Spillane | An adaptive multipreconditioned conjugate gradient algorithm[END_REF] which has been implemented in the finite element software Z-set at Safran Tech [START_REF] Bovet | Adaptive multipreconditioned FETI: scalability results and robustness assessment[END_REF]. The solution obtained with the AMPFETI method is performed on a mesh involving 63 × 10 6 linear tetrahedral elements, corresponding to 37 × 10 6 dofs, while the CMCM computation involves a total of 37.1 × 10 6 dofs, distributed over 652 subdomains including 59.2 × 10 3 dofs (and 16.6 × 10 3 trilinear hexahedral elements). The discretizations of both methods are of comparable sizes. In CMCM, the global problem is computed on a coarse mesh involving only 16.6 × 10 3 trilinear hexahedral elements corresponding to 60.6 × 10 3 dofs. The mesh of a periodic subdomain at the meso scale is generated with the TexGen software [START_REF] Lin | Modelling and simulating textile structures using texgen[END_REF], on a domain with dimensions (2 × 2 × 0.22 mm 3 ), using linear tetrahedral elements. For the sake of simplicity, the parameter β (see section 3.2.1) was chosen as zero. The parameters used to generate this representative unit cell are described in Table 4. For the comparison of the local fields, we re-localize one block column at the center of the structure (see Fig. 28) and compare the strain and stress fields within the red section. The behavior of the matrix is considered as elastic and isotropic (see Table 5). The yarns are considered as linear elastic orthotropic, and their properties are provided in Table 6.

Young's modulus E (MPa) Poisson's ratio ν 4000 0.3 The constitutive relation of the stress and strain in the yarns is given in matrix form as:

                         σ 11 σ 22 σ 33 σ 12 σ 13 σ 23                          =                                  1/E 11 -ν 12 /E 22 -ν 13 /E 33 0 0 0 -ν 12 /E 11 1/E 22 -ν 23 /E 33 0 0 0 -ν 13 /E 11 -ν 23 /E 22 1/E 33 0 0 0 0 0 0 1/G 12 0 0 0 0 0 0 1/G 13 0 0 0 0 0 0 1/G 23                                  -1                          ε 11 ε 22 ε 33 2ε 12 2ε 13 2ε 23                          . ( 56
)
These orthotropic properties are defined in a frame attached to the material orientation of the yarns. In each main (warp and weft) yarn direction, the above tensor is further slightly rotated according to their normal orientation, which varies due to weaving form. However, in this example we simplify the problem by assuming that each yarn only has one constant direction. To be specific, for the warp yarn elements (along x), the above tensor is unchanged, but for the weft yarn elements (along y), it is rotated 90 • . Fig. 34 shows the computational time of the local and global problem for different number of processors. It is worth noting that all local problems are independent, and thus can all be performed in parallel. We can then observe that the computational times are decreased linearly with the number of available cores We can clearly see on the different results that the agreement between AMPFETI and CMCM is satisfying, and that the use of the gradient enhancement improves the agreement with the reference solution. For comparison, the AMPFETI calculation took 10 minutes on 744 cores. The CMCM took 73 minutes in total (including parallel calculations and solving the problem on the coarse mesh), but on a standard 24 cores workstation.

Large scale simulation involving 1.3 Billion of dof

In this example, we demonstrate the capabilities of the method to handle fully detailed structures with large dimensions, and a larger scale ratio between structure dimensions and microstructural details, as found in some industrial applications. The structure has the same characteristics as the one presented in the previous example, but involves 49 × 10 × 10 subdomains (see Fig. 35) as defined in Fig 27 (b). The material properties and the boundary conditions are the same as in the previous section. The complete model involves roughly 1.3 billion dofs. It is obvious than in such situation, most available FEM solvers are not able to handle such a large simulation even with high performance computing. The coarse mesh involves 96,800 trilinear hexahedral elements corresponding to 321,489 dofs (Figure 37), indicating that there is still room for possible refinement of this mesh to improve accuracy. Here again, the parameter β (see section 3.2.1) was chosen as zero. To provide a more accurate solution in the vicinity of the load, the coarse mesh is refined in some regions near the pressure fields, as depicted in Fig. 37(a). For illustration, we relocalize the local strain and stress fields in two block columns of the structure (see Fig. 36). As we have already shown that, for such bending-dominated cases, the Second-order CMCM method improves the accuracy without additional costs, we have here only used the Second-order CMCM method. The total simulation time was 20 minutes for each subdomain off-line calculation (performed in parallel) and the coarse mesh calculation took roughly 5 days, on 32 cores. Then, the present method has a very high potential to study fully detailed composite structures without any assumptions on scale separation and periodicity.

Conclusions

In this work, we have introduced a new method for parallel computations of large heterogeneous structures, called Coarse Mesh Condensation Method (CMCM). In this technique, the key idea is to construct an approximation of the full field solution in the structure at the scale of heterogeneities by combining solutions computed on a fine mesh on subdomains decomposing the structure and a solution computed on a coarse mesh. By analogy with the FE 2 method [START_REF] Feyel | A multilevel finite element method (FE 2 ) to describe the response of highly non-linear structures using generalized continua[END_REF], elements of the coarse mesh provide information for applying Dirichlet on the boundaries of subdomains covering the structure. As a result, the full-field solution is approximated over the whole structure at the cost of a finite number of calculations on subdomains which can be solved in parallel and a coarse mesh problem with a low number of dofs. In this paper, we have only treated the linear case. In this situation, matrix relationships can be obtain to relate the dofs of the coarse mesh and the dofs of the fine mesh subdomains solutions, resulting in a one-iteration scheme only. An extension of the approach to strain gradient has been proposed to enhance the solution in case of global bending of the structures. The accuracy of the method has been tested through benchmark problems involving subdomains crossing the interfaces in the case of high contrast between the phase, which is known to constitute a difficult case for iterative domain decomposition methods. Finally, we have presented an industrial-sized example of a composite beam involving 1.3 billion dofs which has been computed on a standard 32-core standard workstation to show the potential of the method.

x

  : Position vector at the fine scale u(x) : Displacement vector at the fine scale ε : Linearized strain tensor σ : Cauchy stress tensor G : Tensor of second gradient of displacements ∇ε : Strain gradient tensor C(x) : Elastic tensor C 0 (x) : Auxiliary elastic tensor ε : Macroscopic linearized strain tensor Ḡ : Macroscopic second gradient of displacements tensor ∇ε : Macroscopic strain gradient tensor ḡα : Vector of parameters defining Dirichlet boundary conditions on the boundary of Ω α Ω α : Subdomain α A α (x) : Localization tensor in Ω α [ε(x)] : Vector for associated with ε [σ(x)] : Vector form associated with σ C(x) : Matrix form associated with C(x) Ω m : Domain associated with an element m in the fine mesh Ω e : Domain associated with an element e in the fine mesh u m : Vector of nodal displacements in one element m of the fine mesh ūe : Vector of nodal displacements in one element e of the coarse mesh

Figure 1 :

 1 Figure 1: Schematic description of the CMCM method; (a) decomposition of the structure into subdomains; (b) discretization of the structure with a coarse mesh; (c) elementary (off-line) problems to be solved over the subdomains.

Figure 2 :

 2 Figure 2: Illustration of an extended subdomain.

Figure 3 :

 3 Figure 3: Domain Ω α , coarse mesh element e associated with a domain Ω e and fine mesh element m associated with a domain Ω m .

1 :

 1 LOOP over all subdomains Ω α (parallel) 2:

1 :

 1 LOOP over all coarse mesh elements e (parallel) 2:

Algorithm 2 :

 2 Online calculations.

3. 5 .Figure 4 :

 54 Figure 4: (a) Schematic description of FE 2 method [33] and of CMCM.

Figure 5 :

 5 Figure 5: (a) Composite structure: geometry; (b) subdomain used for off-line calculations; (c) subdomain finite element mesh.

Coarse mesh 1

 1 Coarse mesh 2 Coarse mesh 3 Coarse mesh 4 Coarse mesh 5

Figure 6 : 13 Figure 8 :

 6138 Figure 6: ε 11 (x) of reference and different coarse meshes.

Figure 7 :

 7 Figure 7: Evolution of energy error with respect to the number of coarse mesh dofs.

Figure 9 :

 9 Figure 9: Structure divided into (a) 1 subdomain, (b) 4 subdomains and (c) 16 subdomains.

Figure 10 :

 10 Figure 10: Evolution of energy error (left) and L 2 error (right) in function of different values of β in the case of 16 subdomains with contrast E inc /E mat = 10 6 , 10 3 and 5.

1 Figure 11 : 1 Figure 12 :Figure 13 :Figure 14 :

 1111121314 Figure 11: σ 11 (x) using different values of β in the case of 16 subdomains with E inc /E mat = 10 6 .

Figure 15 :

 15 Figure 15: Evolution of energy error in function of number of subdomains.

Figure 16 : 20 (Figure 17 :Figure 18 : 21 4. 2 .Figure 19 :

 1620171821219 Figure 16: Evolution of energy error (left) and L 2 error (right) in function of different values of β with contrast E inc /E mat = 10 6 using CMCM and submodeling.

22

 22

(a) Coarse mesh 1 (b) Coarse mesh 2 (c) Coarse mesh 3 Figure 20 :

 12320 Figure 20: Coarse meshes for global calculation of the beam: (a) coarse mesh 1, (b) coarse mesh 2, and (c) coarse mesh 3.

Figure 21 :

 21 Figure 21: Relocalized strain solution ε 11 (x).

Figure 22 :

 22 Figure 22: Relocalized strain solution ε 22 (x).

Figure 23 :component σ 11 Figure 24 :

 231124 Figure 23: Relocalized strain solution ε 12 (x).

Figure 25 :

 25 Figure 25: Minimum displacement (left) and maximum stress (right) using different coarse meshes.

Table 4 :

 4 Yarn spacing (mm) Yarn width (mm) Fabric thickness (mm) Gap size 1 Parameters used in TexGen for the generation of the texture form.

Figure 26 :

 26 Figure 26: 3D woven composite beam structure problem: geometry and boundary conditions.

Figure 27 :

 27 Figure 27: Finite element mesh of: (a) the macro structure and (b) one subdomain at the meso scale.

Figure 28 :

 28 Figure 28: Illustration of the section used for result comparison.

Figure 29 :

 29 Figure 29: Macro strain solution ε33 (x) on the deformed coarse mesh using the Second-order CMCM.

Figure 30 :

 30 Figure 30: Relocalized strain solution ε 22 (x): (a) reference solution (AMPFETI); (b) basic-CMCM; (c) Second-order CMCM.

Figure 31 :

 31 Figure 31: Relocalized strain solution ε 33 (x): (a) reference solution (AMPFETI); (b) basic-CMCM; (c) Second-order CMCM.

Figure 32 :

 32 Figure 32: Relocalized stress solution σ 11 (x)(MPa): (a) reference solution (AMPFETI); (b) basic-CMCM; (c) Second-order CMCM.

Figure 33 :

 33 Figure 33: Relocalized stress solution σ 22 (x)(MPa): (a) reference solution (AMPFETI); (b) basic-CMCM; (c) Second-order CMCM.

Figure 34 :

 34 Figure 34: Computational time in function of number of cores.

Figure 35 :

 35 Figure 35: Structure decomposed into 49 × 10 × 10 subdomains and involvin 1.3 × 10 9 dofs.

Figure 36 :

 36 Figure 36: Subdomains chosen for relocalization of strain and stress fields.

Figure 37 :

 37 Figure 37: Coarse mesh: (a) global view and (b) zoom-in at the center of the structure.

Figure 38 :

 38 Figure 38: Macro strain solution ε33 (x) on the deformed coarse mesh.

Figure 39 :

 39 Figure 39: Relocalized solution: ε 11 (x) (left) and ε 22 (x) (right).

Figure 40 :

 40 Figure 40: Relocalized solution: ε 33 (x) (left) and σ 11 (x)(MPa) (right).

Figure 41 :

 41 Figure 41: Relocalized solution: σ 22 (x)(MPa) (left) and σ 33 (x)(MPa) (right).

Table 1 :

 1 Notations

Table 2 :

 2 Number of elements and dofs for each coarse mesh.

Table 3 :

 3 ). Number of elements and of dofs for each mesh used in calculation (T3: linear triangular element; Q4: bilinear quadrilateral element).

		subdomain Coarse mesh 1 Coarse mesh 2 Coarse mesh 3
	Element type	T3	Q4	Q4	Q4
	Number of elements	5000	21	84	336
	Number of dofs	5202	88	258	850

Table 5 :

 5 Mechanical properties of the matrix.

	E 11 (MPa) E 22 (MPa) E 33 (MPa) ν 12 ν 23	ν 31	G 12 (MPa) G 23 (MPa) G 31 (MPa)
	194400	8200	8200	0.3 0.3 0.0126	7000	3100	7000

Table 6 :

 6 Mechanical properties of the yarns

Acknowledgements

The authors would like to thank Professor Tarek Zohdi at University of California, Berkeley, for his comments and suggestions concerning an early preprint of this manuscript. They are also grateful to F. Feyel, C. Rey and T. Rose for their help in proofreading and improving this article.


MD Norma Beatriz Romero
email: nb.romero@institut-myologie.org

PhD Marc Bitoun
email: m.bitoun@institut-myologie.org

 Centronuclear myopathies

Introduction

Congenital myopathies with central nuclei or centronuclear myopathies (CNM) are genetically inherited disorders classically identified by the presence of an abnormally high number of muscle fibres with nuclei placed in rows in the central part of the fibre. The clinical presentations of patients are extremely heterogeneous ranging from severe hypotonia in newborns to a relatively late onset of muscle weakness with extraocular muscles involvement in adolescent and young adults.

The early classification of centronuclear myopathies was mainly based on the age of onset and the severity of clinical symptoms. The first group identified during the sixties was the early-adult onset with moderate and mild forms, and then the group composed of severe cases detected in the neonatal period, on male newborns, with a recessive X-linked inheritance [see review in [START_REF] Fardeau | Congenital myopathies[END_REF] ]. This recessive X-linked myotubular myopathy (XLMTM) is caused by mutations in the MTM1 gene [START_REF] Laporte | A gene mutated in X-linked myotubular myopathy defines a new putative tyrosine phosphatase family conserved in yeast[END_REF] encoding the myotubularin 1 (MTM1). On the other hand, numerous sporadic patients and some families with pedigrees showing an obvious autosomal inheritance have been recognized [START_REF] Jeannet | Clinical and histologic findings in autosomal centronuclear myopathy[END_REF] . Thereafter, two causative genes of autosomal CNM were identified, i.e. the DNM2 gene [START_REF] Bitoun | Mutations in dynamin 2 cause dominant Centronuclear Myopathy[END_REF] encoding the dynamin 2 (DNM2) in the autosomal dominant CNM (AD-CNM) and the BIN1 gene 5 coding for the amphiphysin 2 (BIN1) in the autosomal recessive CNM (AR-CNM). However, a lot of sporadic or familial CNM cases

are not yet genetically resolved, implying the existence of additional causative genes.

Noteworthy, a heterozygous RYR1-mutation has been found in a single patient with the initial diagnosis of CNM and core-like lesions in the muscle biopsy [START_REF] Jungbluth | Centronuclear myopathy due to a de novo dominant mutation in the skeletal muscle ryanodine receptor (RYR1) gene[END_REF] , and recessive RYR1-mutations has been reported in some patients with congenital myopathy harbouring a peculiar morphological pattern with prominent nuclear internalisation and large areas of myofibrillar disorganisation (personal communication in the 164th International ENMC Workshop/January 2009) [START_REF] Jungbluth | On behalf of the Centronuclear (myotubular) myopathy consortium[END_REF] . Afterwards, it has been established that patients in whom an initial diagnosis of centronuclear myopathy was established because the high number of internalised nuclei observed in the muscle biopsy performed in early age, indeed, correspond to a form of AR-RYR1-related congenital myopathy [START_REF] Wilmshurst | RYR1 mutations are a common cause of congenital myopathies with central nuclei[END_REF][START_REF] Bevilacqua | Recessive RYR1 mutations cause unusual congenital myopathy with prominent nuclear internalisation and large areas of myofibrillar disorganisation[END_REF] .

Here, we report an overview of the clinical and morphological features of the MTM1-DNM2and BIN1-related CNM and discuss the pathophysiological hypotheses emerging from recent studies using tissue from patients and animal models for these three main CNM genes.

Clinical and morphological features in CNM

Recessive X-linked Myotubular myopathy

The term myotubular myopathy was initially used by Spiro et al, in 1966 10 ; and had defined a distinct congenital myopathy because the morphologic alterations of the muscle fibre closely resembling the myotubular stage of development. Even though at the beginning this term was overall used to the group of congenital myopathies with central nuclei, the term myotubular myopathy was secondarily restricted to the X-linked recessive form (XLMTM), affecting mainly male patients [see review in [START_REF] Fardeau | Congenital myopathies[END_REF] ]. The affected newborn boys present mostly with severe hypotonia, wasting muscle and generalized weakness at birth, almost always associated with respiratory insufficiency requiring immediate ventilator support and swallowing difficulties needing a feeding tube to prevent early death due to aspiration (Figure 1). Antenatal signs as reduced foetal movements, polyhydramnios and thin ribs are typically found. Ocular muscle involvement, moderate ptosis and external ophthalmoplegia, are often observed after the neonatal period. The affected infants are usually macrosomic associating asymmetric muscle bulk and may have diverse malformations comprising pyloric stenosis, inguinal hernia and undescended testes. The neonatal form of XLMTM carries a poor prognosis, the large majority of patients die within the first months of life, but a relatively small proportion of boys may be less affected in the neonatal period and survive into childhood and even adulthood. Interestingly, some affected boys acquire the ability to walk with a quite autonomous life [START_REF] Biancalana | Characterisation of mutations in 77 patients with X-linked myotubular myopathy, including a family with a very mild phenotype[END_REF][START_REF] Chanzy | Unusual good prognosis for Xlinked myotubular myopathy][END_REF][START_REF] Tosch | Novel molecular diagnostic approaches for X-linked centronuclear (myotubular) myopathy reveal intronic mutations[END_REF] . However, many patients surviving through the infancy usually remain severely impaired and require intensive physiotherapy and adapted rehabilitation (permanent ventilation, brace with head support, even feeding tube).

The morphologic analysis of muscle biopsies shows a characteristic high number of small muscle fibres with centralised nuclei and a predominance of type 1 fibres [START_REF] Romero | Centronuclear myopathies: a widening concept[END_REF] . Frequently there are also some large type 1 fibres which correspond to remaining Wohlfart B Fibers. On transverse muscle sections crossing the fibre between two adjacent nuclei, the central area of the fibres displays reduced myofibrillar ATPase reaction and increased oxidative enzymes activity and glycogen staining.

Noticeable, in sections reacted for oxidative enzymes activity, numerous muscle fibres show a pale halo at the periphery of the fibre, and the few large type 1 Wohlfart B Fibers fibres show a strong oxidative staining (Figure 1). On longitudinal sections, chains of nuclei are observed in the central part of the fibre. The ultrastructural analyses show that for the most part the myofibrils are restrained in the peripheral zones of the fibres. By contrast, the central areas of myofibres are mainly occupied by aggregates of mitochondria and glycogen particles whereas the myofilaments are lacking (Figure 1). By immunohistochemistry we have recognized a positive labelling mainly in the central areas of the muscle fibres with antibodies for DHPR1s (the dihydropyridine receptor-1subunit) a Ca 2+ channel of the T-tubule, and RYR1 (the ryanodine receptor type 1) a Ca 2+ channel of the sarcoplasmic reticulum.

The wholly cytoplasm was positive for desmin with increased staining in the central areas of the fibres; and the caveolin3 the immunolabelling was normally observed in the sarcolemma (Figure 5).

Recently, the presence of necklace fibres in muscular biopsies was observed in sporadic lateonset MTM1-related centronuclear myopathy [START_REF] Bevilacqua | Necklace" fibers, a new histological marker of late-onset MTM1-related centronuclear myopathy[END_REF] . The clinical symptoms are mild during childhood, and worsened after the first or second decades of the live. Necklace fibres are characterized by a basophilic ring deposit following the contour of the cell, in which myonuclei are aligned (Figure 2). Under electron microscopy, the necklaces show myofibrils with oblique orientation and they have smaller diameter than the myofibrils placed in the normally structured surrounding areas [START_REF] Romero | Centronuclear myopathies: a widening concept[END_REF][START_REF] Bevilacqua | Necklace" fibers, a new histological marker of late-onset MTM1-related centronuclear myopathy[END_REF] .

Autosomal dominant DNM2-related centronuclear myopathy

Several clinical phenotypes of autosomal dominant DNM2-related CNM have been reported, they range from mild forms in adult patients to severe forms in infants. The classical phenotype of centronuclear myopathy corresponds clinically to mild or late-childhood and adult onset forms [START_REF] Bitoun | Mutations in dynamin 2 cause dominant Centronuclear Myopathy[END_REF] , but then, an intermediate form in adults with early onset of the disease has been also reported [START_REF] Bitoun | A new centronuclear myopathy phenotype due to a novel dynamin 2 mutation[END_REF] . In paediatric population initially we described sporadic cases corresponding with the severe and earlyonset clinical form [START_REF] Bitoun | Dynamin 2 mutations cause sporadic centronuclear myopathy with neonatal onset[END_REF] ; afterwards we demonstrated a marked variability in clinical presentation severity of CNM associated with mutations in DNM2 [START_REF] Susman | Expanding the clinical, pathological and MRI phenotype of DNM2-related centronuclear myopathy[END_REF] . In general, adolescent and young adult patients with moderate phenotype had normal pregnancy and delivery but motor milestones are typically delayed, especially walking and then climbing stairs and running. Moderate skeletal muscle weakness associated with ptosis and ophthalmoplegia are constant. On the other hand, paediatric patients usually have generalised weakness, hypotonia, moderate degree of facial weakness with open mouth, ptosis and ophthalmoplegia (Figure 3). Achilles tendon contractures, scoliosis, reduced jaw opening, mild pes cavus and pronounced atrophy of calves and thenar eminences often complete the clinical phenotype [START_REF] Bitoun | Dynamin 2 mutations cause sporadic centronuclear myopathy with neonatal onset[END_REF][START_REF] Susman | Expanding the clinical, pathological and MRI phenotype of DNM2-related centronuclear myopathy[END_REF] .

The analysis of muscle biopsies of DNM2-related CNM shows a characteristic triad of morphological features: a) a radial arrangement of sarcoplasmic strands with oxidative enzyme reactions conferring a spokelike appearance, b) significant nuclear centralisation and internalisation (Figure 3), c) type 1 muscle fibre predominance and hypotrophy. Interesting, this complete histopathological triad of features has been exclusively found in DNM2-associated CNM. However, in young DNM2-related children the characteristic triad of features could be absent or quantitatively less important [START_REF] Bitoun | A new centronuclear myopathy phenotype due to a novel dynamin 2 mutation[END_REF] . Increase in fibro-adipose endomysial tissue may be present in DNM2-related CNM; this makes sometimes the initial pathological diagnosis quite difficult. At electron microscopy, the central nuclei appear without morphological abnormality and the radial distribution of the intermyofibrillar sarcoplasmic strand is easily recognized (Figure 3). Typically, the diameter of the myofibrils became progressively smaller from the periphery to the central zone of the fibre, giving rise to radiating appearance [START_REF] Fardeau | Congenital myopathies[END_REF][START_REF] Romero | Centronuclear myopathies: a widening concept[END_REF] ; in addition, the central internuclear spaces are occupied by mitochondria, rough endoplasmic reticulum, Golgi complex and glycogen particles. By immunohistochemistry we have documented a diffuse positive labelling in the wholly cytoplasm of the muscle fibres with antibodies for DHPR1s, RYR1 and desmin. In a few fibres, the central areas have a desmin immunostaining moderately increased; and for caveolin3, the immunolabelling was normally observed in the sarcolemma (Figure 5).

We reported the clinical and muscle imaging findings of ten patients with late-onset CNM harbouring DNM2 mutations [START_REF] Fischer | Characterization of the muscle involvement in dynamin 2-related centronuclear myopathy[END_REF] . In all ten patients, muscle computer tomography assessment showed a characteristic temporal course with early and predominant distal muscle involvement and later affection of the posterior thigh compartment and gluteus minimus muscles. The recognition of this specific imaging patterndistinct to the reported patterns in other congenital myopathiesmay enable a better selection for direct genetic testing.

Autosomal recessive BIN1-related centronuclear myopathy

Although a few cases were indentified with autosomal recessive BIN1-related centronuclear myopathy so far, these patients present with a large clinical variability from the severe to moderate phenotypes [START_REF] Nicot | Mutations in amphiphysin 2 (BIN1) disrupt interaction with dynamin 2 and cause autosomal recessive centronuclear myopathy[END_REF][START_REF] Claeys | Phenotype of a patient with recessive centronuclear myopathy and a novel bin1 mutation[END_REF][START_REF] Mejaddam | Severe phenotype of a patient with autosomal recessive centronuclear myopathy due to a BIN1 mutation[END_REF][START_REF] Bohm | Case report of intrafamilial variability in autosomal recessive centronuclear myopathy associated to a novel BIN1 stop mutation[END_REF] . Pregnancy and birth can occur without major events, however delayed motor milestones, difficulties to run, walking and climb stairs since childhood are common antecedents.

Muscular atrophy, diffuse weakness, facial diplegia, ptosis and varying degrees of ophtalmoplegia are usually present.

The muscle biopsies analysis from BIN1-related CNM shows numerous rounded small type 1 fibres, with central nuclei in the large majority of them [START_REF] Romero | Centronuclear myopathies: a widening concept[END_REF] . Interesting, clusters of centrally placed nuclei and rows of central nuclei have been observed in some muscle fibres [START_REF] Romero | Centronuclear myopathies: a widening concept[END_REF] (Figure 4). The electron microscopy findings show central placed nuclei and an internuclear spaces occupied by an amorphous material containing mitochondria, sarcotubular structures and glycogen particles (Figure 4). By immunohistochemistry, we demonstrated a strictly positive labelling in the central areas of the muscle fibres with antibodies for DHPR1s, RYR1 and desmin, giving a small colourless area with dark dotted underlining, mainly around the central nuclei. The cytoplasm is also labelled with these three antibodies. Noticeable, the caveolin3 immunolabelling was observed normally in the sarcolemma but strangely associated with increased staining in the central areas of the fibres (Figure 5).

CNM genes and pathophysiological hypotheses

The recessive XLMTM with severe neonatal phenotype and the late-onset form with necklace fibres is caused by mutations in the MTM1 gene, located on Xq28 chromosome and composed of 15 exons coding for a 603 amino-acids protein [START_REF] Laporte | A gene mutated in X-linked myotubular myopathy defines a new putative tyrosine phosphatase family conserved in yeast[END_REF] . MTM1 mutations lead to the decrease or absence of the gene product in most of the cases. Although XLMTM affects above all newborn and children males; rare female infants have been reported, usually associated with large DNA rearrangements. In addition, mainly symptomatic female carriers have been related due to a probable skewed Xinactivation. The MTM1 gene encodes the myotubularin 1 (MTM1) which is a 3-phosphoinositides phosphatase for the phosphatidylinositol (3,5) bisphosphate (PI(3,5)P2) and the phosphatidylinositol (3) monophosphate (PI3P) leading to the production of PI5P and PI, respectively. The PI3P and PI3,5P2 are second messengers located at the membrane surface of the endosomal system and have been involved in membrane remodelling and trafficking [START_REF] Nicot | Endosomal Phosphoinositides and Human Diseases[END_REF] .

The autosomal dominant CNM is due to heterozygous mutations in the DNM2 gene at the 19p13 locus composed of 22 exons. Mutations in this gene have been also identified in rare intermediate and axonal forms of the Charcot-Marie-Tooth (CMT) peripheral neuropathy [START_REF] Zuchner | Mutations in the pleckstrin homology domain of dynamin 2 cause dominant intermediate Charcot-Marie-Tooth disease[END_REF] . DNM2 is a large GTPase mainly involved in membrane trafficking through its function in the formation and release of nascent vesicles from biological membranes. DNM2 participates in clathrin-dependent and clathrin-independent endocytosis and intracellular membrane trafficking (from endosomes and Golgi apparatus). In addition, DNM2 tightly interacts with, and is involved in the regulation of actin and microtubule networks [START_REF] Durieux | Dynamin 2 and Human diseases[END_REF] .

The autosomal recessive CNM is due to mutations in the BIN1 gene 5 located to the locus 2q14 and composed of 20 exons. Like MTM1 and DNM2, BIN1 is ubiquitously expressed but specific BIN1 muscular isoform exists. BIN1 belongs to the endocytosis machinery in which the protein is involved in membrane curvature and tubulation via its BAR (Bin/Amphiphysin/Rsv167) domain and recruitment of endocytosis-associated proteins via its SH3 (SRC homology-3) domain such as DNM2 [START_REF] Dong | Expression and purification of dynamin II domains and initial studies on structure and function[END_REF][START_REF] Kojima | Regulation of Bin1 SH3 domain binding by phosphoinositides[END_REF] .

It is interesting to note that despite the differences in the clinical presentation and their evolution, and the pathological features in muscle biopsy, all known gene products causing centronuclear myopathy are involved in membrane remodelling, endocytosis and intracellular membrane trafficking. Indeed, impairment of endocytosis and membrane trafficking was reported in cultured cells expressing MTM1 and DNM2 mutants [START_REF] Zuchner | Mutations in the pleckstrin homology domain of dynamin 2 cause dominant intermediate Charcot-Marie-Tooth disease[END_REF][START_REF] Chaussade | Expression of myotubularin by an adenoviral vector demonstrates its function as a phosphatidylinositol 3-phosphate [PtdIns(3)P] phosphatase in muscle cell lines: involvement of PtdIns(3)P in insulinstimulated glucose transport[END_REF][START_REF] Bitoun | Dynamin 2 mutations associated with human diseases impair clathrin-mediated receptor endocytosis[END_REF][START_REF] Tanabe | Dynamic instability of microtubules requires dynamin 2 and is impaired in a Charcot-Marie-Tooth mutant[END_REF][START_REF] Durieux | A centronuclear myopathy-dynamin 2 mutation impairs skeletal muscle structure and function in mice[END_REF] . The question which remains to be explored is how trafficking impairment due to mutation in ubiquitous proteins can induce a tissue-specific phenotype. On the other hand, it was reported that the muscle specific isoform of amphiphysin 2 is concentrated at T-tubules in mouse and drosophila and was involved in the organization of this plasma membrane invagination acting in excitation-contraction coupling [START_REF] Lee | Amphiphysin 2 (Bin1) and T-tubule biogenesis in muscle[END_REF][START_REF] Razzaq | Amphiphysin is necessary for organization of the excitation-contraction coupling machinery of muscles, but not for synaptic vesicle endocytosis in Drosophila[END_REF] . Interestingly, MTM1 was also located to the T-tubules in mouse [START_REF] Buj-Bello | AAVmediated intramuscular delivery of myotubularin corrects the myotubular myopathy phenotype in targeted murine muscle and suggests a function in plasma membrane homeostasis[END_REF] and zebrafish [START_REF] Dowling | Loss of myotubularin function results in T-tubule disorganization in zebrafish and human myotubular myopathy[END_REF] and knock-down of myotubularin in these species leads to disorganisation of the T-tubule system and defect in excitation-contraction coupling [START_REF] Dowling | Loss of myotubularin function results in T-tubule disorganization in zebrafish and human myotubular myopathy[END_REF][START_REF] Al-Qusairi | T-tubule disorganization and defective excitation-contraction coupling in muscle fibers lacking myotubularin lipid phosphatase[END_REF] . More recently, abnormalities in T-tubule morphology was also reported in mouse muscle overexpressing a CNM-related DNM2 mutant [START_REF] Cowling | Increased expression of wild-type or a centronuclear myopathy mutant of dynamin 2 in skeletal muscle of adult mice leads to structural defects and muscle weakness[END_REF] . In addition, defects in T-tubule and triad structures have been reported in muscle biopsies from MTM1-, BIN1-and DNM2-CNM patients [START_REF] Nicot | Mutations in amphiphysin 2 (BIN1) disrupt interaction with dynamin 2 and cause autosomal recessive centronuclear myopathy[END_REF][START_REF] Dowling | Loss of myotubularin function results in T-tubule disorganization in zebrafish and human myotubular myopathy[END_REF][START_REF] Al-Qusairi | T-tubule disorganization and defective excitation-contraction coupling in muscle fibers lacking myotubularin lipid phosphatase[END_REF][START_REF] Cowling | Increased expression of wild-type or a centronuclear myopathy mutant of dynamin 2 in skeletal muscle of adult mice leads to structural defects and muscle weakness[END_REF][START_REF] Toussaint | Defects in amphiphysin 2 (BIN1) and triads in several forms of centronuclear myopathies[END_REF] . Altogether, these data argue for a common pathomechanism in CNMs in which T-tubule dysfunction leads to an excitation-contraction coupling impairment. Further studies will be necessary to determine if the Ttubule defects are associated with an impairment of intracellular trafficking processes in which MTM1, DNM2 and BIN1 have been involved or if these defects could be related to a muscle-specific function of the three CNM proteins which remains to be determined.

A novel pathophysiological mechanism was recently reported in the MTM1-CNM [START_REF] Hnia | Myotubularin controls desmin intermediate filament architecture and mitochondrial dynamics in human and mouse skeletal muscle[END_REF] . Hnia et al.

showed that MTM1 is a desmin-binding protein and that MTM1 mutants alter the desmin network architecture and mitochondrial homeostasis. In agreement with this hypothesis we show here that desmin network is disorganized in the three forms of CNM (Figure 5). Indeed, the immunocytochemical staining for desmin in muscle cross-sections from CNM patients has revealed, besides a diffuse cytoplasm localisation, a particular distribution of the protein which appears increased in the central areas of the fibres, mainly in BIN1-and relatively less in MTM1-related CNM.

The presence of high immunolabelling in the central areas of muscle is also observed for DHPR and RYR1.

RYR1-linked congenital myopathy with multiple internal nuclei and myofibrillar disorganisation

Whereas the histopathological alterations in skeletal muscle increase with age, muscle biopsies performed in young patients, sometimes could not show all structural abnormalities that characterize the morphological phenotypes of congenital myopathies. In this context, a group of patient sharing high number of internalised nuclei and large and poorly defined areas of myofibrillar disorganisation in the muscle biopsy performed in early age was originally classified in the group of CNM not linked to the DNM2 gene [START_REF] Romero | Centronuclear myopathies: a widening concept[END_REF] . In these patients, clinical symptoms usually onset during the neonatal period with a predominant axial and proximal weakness of diverse severity, associated with facial weakness, bilateral ptosis and ophthalmoparesis, occasional restrictive respiratory syndrome and contractures. The muscle biopsies analysis revealed unusual pattern characterised by significant myonuclei internalisation and muscle fibres showed large areas devoid of ATPase activity and irregular oxidative enzymatic reaction; some fibres presenting with a fuchsinophilic granular precipitate as purple dusty fibres. At electron microscopy, the large zones of myofibrillar disorganisation frequently occupied the whole myofibre cross-section and extended to a few sarcomeres in length, Zdisk streaming and accumulated Z-disk proteins and depleted of mitochondria are frequently observed.

In this group of patients, a heterozygous RYR1-mutation has been found in a single patient with the initial diagnosis of CNM and core-like lesions in the muscle biopsy [START_REF] Jungbluth | Centronuclear myopathy due to a de novo dominant mutation in the skeletal muscle ryanodine receptor (RYR1) gene[END_REF] , and recessive RYR1mutations has been recently reported in the other patients [START_REF] Wilmshurst | RYR1 mutations are a common cause of congenital myopathies with central nuclei[END_REF][START_REF] Bevilacqua | Recessive RYR1 mutations cause unusual congenital myopathy with prominent nuclear internalisation and large areas of myofibrillar disorganisation[END_REF] suggesting that this phenotype is a form of AR-RYR1-related congenital myopathy.

Figures Figure 1

Newborn boy with severe X-linked myotubular myopathy: Note severe hypotonia, ptosis and respiratory distress requiring assisted ventilation and gastric feeding (A). Cryostat transverse muscle sections (B, C) show many fibres with central nuclei resembling myotubes (B, HE); and the oxidative staining reveals the fibres with a dark central region which are regularly surrounded by a paler peripheral halo (C, NADH-TR). Under the electron microscope, the transverse sections show the nuclei in the centre of the fibre often bordered by mitochondria, glycogen and some tubular structures (D). In sections crossing the muscle fibre between two adjacent nuclei, the central area of the fibres displays strikingly a reduction of myofilaments (E).

Figure 2

 2 Figure 2Seven-year-old girl affected by DNM2-related CNM: Note Mild facial weakness, bilateral ptosis and open mouth (A). Cryostat transverse muscle section shows a high percentage of small rounded

Figure 3 Late

 3 Figure 3 Late-onset MTM1-related CNM: Transverse cryostat muscle sections showing typical necklace fibres with a ring deposit following the contour of the cell in the subsarcolemmal area in which myonucleus are aligned (A, HE; B, GT). Ultrastructural photographs of transverse section of a necklace fibre showing the nucleus aligned with the necklace which corresponds with myofibrils of smaller diameter and oblique orientation, increased number of mitochondria, tubules and glycogen (C).

Figure 4 BIN1

 4 Figure 4BIN1-related CNM: Note the cluster of centrally placed nuclei in a muscle fibre (A, HE) and rows

Figure 5

 5 Figure 5 Muscle biopsies sections of patients with severe X-linked myotubular myopathy (A,B,C,D), and DNM2-(E,F,G,H) and BIN1-related CNM patients (I,J,K,L) showing Immunohistochemical findings of DHPR (A,E,I), RYR1 (B,F,J), desmin (C,G,K) and caveolin3 (D,H,L). Muscle sections in A, B and C (MTM1-) demonstrate muscle fibres with positive expression of either of DHPR, RYR1 and desmin with a labelling increased in the central areas of the fibres. Muscle sections in E, F and G (DNM2-) reveal in general a relatively homogeneous and diffuse labelling of the cytoplasm. However, muscle sections in I, J and K (BIN1-) show a labelling strictly increased in the central areas of the fibres with an intense coloration in the cytoplasm. Noticeable, the immunolabelling for caveolin3 was observed normally in the sarcolemma in MTM1-(D) and DNM2-(H) related CNM, but it was peculiarly associated with increased staining in the central areas of the fibres in BIN1related CNM (L).

Acknowledgements

We sincerely thank Professor Michel Fardeau for advice, and the patients and the parents of the patients. We wish to thank the laboratory engineers and technicians for the excellent technical work: L. Manéré, M.T. Viou, E. Lacène, M. Beuvin, G. Brochier, D. Chauveau, B. Prudhon.