N Moës

M Cloirec

P Cartraud

J.-F Remacle

A computational approach to handle complex microstructure geometries

Keywords: Homogenization, surface of discontinuity, X-FEM, periodicity

In multiscale analysis of components, there is usually a need to solve microstructures with complex geometries. In this paper, we use the extended finite element method (X-FEM) to solve scales involving complex geometries. The X-FEM allows one to use meshes not necessarily matching the physical surface of the problem while retaining the accuracy of the classical finite element approach. For material interfaces, this is achieved by introducing a new enrichment strategy. Although the mesh does not need to conform to the physical surfaces, it needs to be fine enough to capture the geometry of these surfaces. A simple algorithm is described to adaptively refine the mesh to meet this geometrical requirement. Numerical experiments on the periodic homogenization of two-phase complex cells demonstrate the accuracy and simplicity of the X-FEM.

INTRODUCTION

The eXtended Finite Element Method simplifies greatly the analysis of structures with complex geometry since the mesh is not required to match this geometry. This property is particularly useful when homogenizing the behavior of micro-structures, complex by nature. We focus here our attention to the application of X-FEM to micro-scale problems involved in the homogenization theory. The homogenization theory is a powerful mean for analyzing and modeling heterogeneous structures. Moreover, in the case of periodic linear media, this theory can be rigorously justified, through convergence results, see e.g. [START_REF] Bensoussan | Papanicolaou Asymptotic Analysis for Periodic Structures[END_REF], [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF]. Thus, as the ratio of the period length to the structure length DRAFT goes to zero, it can be shown that the actual solution of the heterogeneous structure tends to the solution of a problem written for a structure with an homogenized behavior. Furthermore, it turns out that the effective properties are obtained from the solution of a boundary value problem to be solved on a period of the structure, which will be called the basic cell problem in the sequel.

Since the basic cell problem is a boundary value problem, classical numerical methods can be used for computing their solution, see [START_REF] Bőhm | A short introduction to basic aspects of continuum micromechanics[END_REF] for a state of the art in this matter.

The majority of studies have been based on the Finite Element Method (FEM), see [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF], [START_REF] Michel | Effective properties of composite materials with periodic microstruture: a computational approach[END_REF] and references herein, and only few papers present a Boundary Element Method implementation, see e.g. [START_REF] Kamiński | Boundary element method homogenization of the periodic linear elastic fiber composites[END_REF], [START_REF] Okada | Homogenization method for heterogeneous material based on boundary element method[END_REF]. Let us also mention the Fast Fourier Transforms based numerical method [START_REF] Moulinec | A numerical method for computing the overall response of nonlinear composites with complex microstructure[END_REF]. This method is well suited for periodic homogenization, and is very efficient compared to FEM, especially in the linear range [START_REF] Michel | Effective properties of composite materials with periodic microstruture: a computational approach[END_REF], provided that no voids or rigid inclusions are present in the basic cell.

The advantages of the FEM in micromechanical analysis are indeed the same as in standard engineering problems: its flexibility, and its applicability to nonlinear problems, anisotropic materials, and arbitrary geometries. Following [START_REF] Bőhm | A short introduction to basic aspects of continuum micromechanics[END_REF], one may distinguish, in FEM based periodic homogenization studies, four main groups.

In most of the studies, the mesh models the heterogeneities boundaries in the unit cell, i.e. only one phase is present within each finite element. The main drawback of such an approach is the explicit microstructure modeling, which leads to problems for generating the mesh for complex geometries, and requires the use of sophisticated tools, see e.g. [START_REF] Wentorf | Automated modeling for complex woven microstructures[END_REF].

One way to overcome these difficulties is to use a digital image based FEM technique, as initially proposed in [START_REF] Hollister | Homogenization theory and digital imaging: a basis for studying the mechanics and design principles of bone tissue[END_REF]. It consists in using a uniform mesh that has the same resolution as the digital image, and then to identify each pixel or voxel as a finite element. Such an approach, however, leads to models which are computationally expensive.

A third approach also uses regular meshes, but the interfaces between constituents may be represented independently of element boundaries. For an element cut by an interface, the heterogeneity is treated at the integration point level [START_REF] Zohdi | A description of macroscopic damage through microstructural relaxation[END_REF]. As we shall see in the numerical experiments, this strategy yields a reasonable rate of convergence for the homogenized parameters of the basic cell but suffers a slow rate of convergence for the quality of the overall stress distribution over the basic cell (energy norm in the stresses).

Another technique is the Voronoi cell finite element model [START_REF] Ghosh | Multiple scale analysis of heterogeneous elastic structures using homogenization theory and Voronoi cell finite element method[END_REF], in which an element incorporates the effect of an embedded inclusion in the matrix. Both stress and displacement fields need to be discretized with this technique.

The eXtended Finite Element Method provides yet another way to solve the basic cell problem. The advantage of this approach is to retain all the advantages of the finite element approach (applicability to nonlinear and anisotropic constitutive laws, wide range of codes already written, robustness, ...) while considerably easing the meshing step. Indeed, the physical surfaces of the problem do not need to be meshed. They are taken into account by enriching the finite element approximation space through the partition of unity technique [START_REF] Melenk | The Partition of Unity Finite Element Method: Basic theory and applications[END_REF]. The adequate choice of the enrichment function for cracks was discussed in [START_REF] Belytschko | Elastic Crack Growth in Finite Elements with Minimal Remeshing[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF], for intersecting cracks and voids in [START_REF] Daux | Arbitrary branched and intersecting cracks with the eXtended Finite Element Method[END_REF]. Concerning material interfaces, an enrichment was introduced in [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF]. This latter paper was also the first one to use a level set representation of surfaces in the X-FEM context. This representation not only reduces the surface storage to a usual finite element field but also provides a natural way to express the enrichment. Note that in order to build the level set representation of the material interfaces, the CAD geometries of these surfaces are not necessarily needed. All that is needed is a function returning for one (x,y,z) point the distance to the closest material interface. In other words, the X-FEM approach coupled to the level set representation of the surfaces not only simplifies the mesh step but also the geometrical data pre-processing. In this paper, we improve the enrichment functions for material interfaces and obtain a convergence rate very close to the one obtained with regular finite elements (i.e. conforming meshes).

The remainder of this paper is organized as follows. The basic cell problem is described in Section 2. The X-FEM approach to solve the basic cell problem is detailed in Section 3 and a new enrichment function for material interface is introduced and compared to previous approaches. The fourth section uses the X-FEM to homogenize three different basic cells: a cell with orthogonal non-woven fibers, a cell with woven fiber and, finally, a cell with randomly placed spherical inclusions.

THE BASIC CELL PROBLEM

The formulation of the basic cell problem for composites with periodic microstructure can be derived in a systematic way using the two-scale asymptotic expansion method [START_REF] Bensoussan | Papanicolaou Asymptotic Analysis for Periodic Structures[END_REF], [START_REF] Sanchez-Palencia | Non homogeneous media and vibration theory[END_REF], or following a process which is also valid for random media [START_REF] Suquet | Elements of homogenization for inelastic solid mechanics[END_REF], [START_REF] Michel | Effective properties of composite materials with periodic microstruture: a computational approach[END_REF]. The latter approach is used in this paper.

The microstructure description is achieved through the definition of a representative volume element, containing all the geometrical and material data. In the case of a periodic medium, the representative volume element is simply a basic cell Ω, which forms the composite by spatial repetition.

At the macroscopic scale, stress and strain fields are denoted by Σ and E, which are the average of the corresponding microscopic field, σ and e, on the basic cell (the macroscopic and microscopic scales are denoted by x and y, respectively):

   Σ(x) =< σ(x, y) >= 1 |Ω| Ω σ(x, y)dΩ E(x) =< e(x, y) > (1)
The homogenized behavior relating Σ to E is built using a localization process which consists in submitting the composite structure to a given macroscopic stress or strain state. In the case of a periodic microstructure, such a problem can be studied on a single basic cell, thanks to periodic boundary conditions. Moreover, the local periodicity of the strain e(x, y) with respect to the microscopic scale y, and the relation E(x) =< e(u(x, y)) > leads to the following decomposition of the displacement field:

u(x, y) = E(x).y + u m (y), u m periodic on ∂Ω (2)
Therefore, with the data E, the basic cell (or localization) problem is defined by: Find σ, e, u m such that:

             divσ(y) = 0 σ(y) = a(y) : e(y) = a(y) : E + e(u m (y)) u m periodic σ.n anti-periodic (3)
where "periodic" ("anti-periodic") means that values on opposite sides of the boundary ∂Ω are equal (opposite). The problem (3) admits a unique solution, up to a rigid body translation. Once problem (3) is solved, the strain localization tensor D is obtained

e(y) = D(y) : E (4)
and the effective stiffness a hom may be computed as:

a hom =< a(y) : D(y) > (5)
For a linear elastic medium, the complete determination of a hom requires the solution of problem (3) for 6 independent data E. One way to proceed is to consider successively the elementary macroscopic strain states corresponding to one component of E equal to unity, the others being zero. Thus, introducing the second order tensor I ij defined by:

I ij kh = 1 2 (δ ik δ jh + δ ih δ jk ) (6)
and denoting u m ij the solution of problem (3) with the data I ij , one has:

D khij = e(u m ij (y)) kh (7)
The problem (3) corresponds to the strain approach, the data being the macroscopic strain E. One can also use a stress approach, with the data Σ, thus leading to effective compliance A hom , which is equal to the inverse of a hom .

It is important to mention that, in the framework of periodic homogenization, non periodic materials (i.e. with some randomness in the geometrical shape of the heterogeneities and their spatial distribution) may be studied. In that case, however, one has to use a sufficiently large basic cell, in order to obtain a statistically representative response and accurate effective properties. In this paper, such examples of basic cells will be treated.

The variational formulation of the basic cell problems (3), for a data E = I ij is given by:

   Find u m ij periodic such that a(u m ij , v) = -a(I ij , v) ∀v periodic (8)
where we use the bilinear form:

a(u, v) = 1 |Ω| Ω e(u) : a : e(v)dΩ (9)
The effective properties are determined from:

a hom ijkl = a(I kl + u m kl , I ij + u m ij ) = a(I kl , I ij ) -a(u m kl , u m ij ) (10)
Since in this paper a displacement approach is used, the right-hand side of ( 8) is known. Therefore, the only specific feature of problem ( 8) is the periodicity condition [START_REF] Michel | Effective properties of composite materials with periodic microstruture: a computational approach[END_REF].

X-FEM DISCRETIZATION

In this paper, we consider the homogenization of two-phase basic cells. The material interface separating the two phases is described by a level set function.

The level set representation is then used to express a jump in the strain field within the elements crossed by the material interface.

Level set representation

Consider a two-phase basic cell. The two phases are separated by an interface which we may locate either explicitely (set of CAD entities) or implicitely by assigning to each node I of the mesh, the distance φ I to the interface (with a positive sign if node I is located in one phase and a negative sign if node I is located in the other phase). Next, we may interpolate these nodal informations using the finite element shape functions, N I (x), yielding the level set expression:

φ(x) = I φ I N I (x) (11)
where the sum over I indicates a sum over the nodes of the mesh (more precisely only the nodes belonging to the element containing point x). The isozero of the level set function φ approximates the true location of the interface. A strategy for constructing the level set function in the case where the interface is known only by a set of points is described in [START_REF] Belytschko | Structured extended finite element methods of solids defined by implicit surfaces[END_REF].

With the eXtended Finite Element Method, the mesh does not need to conform to the interface but the mesh has to be fine enough to locate precisely the interface and to accurately resolve the displacement field i.e. two kinds of error appear: geometrical and numerical errors. Since a level set interpolated on the finite element mesh is used to locate the interface, one can notice that both geometrical and numerical errors are reduced as the mesh gets finer (as with the FEM). As an example, Figure ?? shows the geometry of the basic cell for a landing helicopter grid located on the deck of an aircraft carrier.

Using the coarse mesh, Figure 2 (left), the iso-zero level set obtained is shown Figure 2 (right). It can be seen that the geometrical representation is poor. A finer mesh, Figure 3 (left), gives a much better representation of the surface location, Figure 3 (right).

The finer mesh was obtained using a simple adaptive mesh refinement strategy. The idea is to reduce the size of the elements located close to the physical surface if the local radius of curvature of the surface is smaller than the characteristic element size. The radius of curvature ρ on an element is computed by the second gradient of the level set interpolation over the element [START_REF] Sethian | Level Set Methods & Fast Marching Methods:Evolving Interfaces in Computational Geometry[END_REF][START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets.Part I: Mechanical model[END_REF] and the characteristic element size l c is computed by l c = (V * d!)

)
where φ is the level set value at the centroid of the element and l tot the side length of the basic cell. The dimensionless parameters α and β govern the extent and level of refinement. Based on the elements tagged for refinement, the mesh can be refined where needed. In order to meet the condition l c > αρ starting from one initial coarse mesh, several intermediary meshes are usually needed. As meshes become finer, the evaluation of the curvature ρ becomes more accurate. The process ends when no element is tagged.

The mesh shown Figure 3 was obtained with α = 0.1 and β = 0.03. The gmsh meshing tool [START_REF] Remacle | Gmsh finite element grid generator[END_REF] and the aomd mesh database library [START_REF] Remacle | Algorithm Oriented Mesh Database[END_REF][START_REF] Remacle | AOMD: Algorithm Oriented Mesh Database[END_REF] were used to refine the mesh based on the elements tagged.

Enrichment strategy

Once the level set function is defined, we can express the X-FEM approximation. The classical finite approximation over the basic cell is given by

u FEM = I u I N I (x) (13)
Z Y X Fig. 2. A coarse mesh and the corresponding iso-zero level set.

Z Y X Fig. 3. A refined mesh and the corresponding iso-zero level set.

If the finite element mesh conforms to the material interface, the (linear) approximation above yields an O(h) convergence rate in the energy norm (provided the solution is smooth) [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF]. On the contrary, as will be shown in the numerical experiments, if the finite element does not conform to the interface, a poor rate of convergence is obtained. For instance, a two-phase 1D problem with a mesh non-conforming to the interfaces gives an asymptotic rate of convergence as poor as O( √ h) [START_REF] Moës | Investigation of the interactions between the numerical and the modeling errors in the Homogenized Dirichlet Projection Method[END_REF]. In order to avoid poor rates of convergence, the X-FEM adds an enrichment to the classical finite element approximation through a partition of unity technique [START_REF] Melenk | The Partition of Unity Finite Element Method: Basic theory and applications[END_REF]:

u X-FEM = I u I N I (x) + u enr , u enr = J a J N J (x)F(x) (14)
The additional degrees of freedom a J are added at the nodes for which the support is split by the interface. The function F is called the enrichment function. It is discontinuous in its derivative across the interface. In [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF], the enrichment is defined as the absolute value of the level set function which indeed has a discontinuous first derivative on the interface:

F 1 (x) = | I φ I N I (x)| (15)
A similar choice [START_REF] Belytschko | Structured extended finite element methods of solids defined by implicit surfaces[END_REF], is to define u enr as

u enr = J a J N J (x)(F 1 (x) -F 1 (x J )) (16)
A closer look at the two enrichments above indicates that they will yield the same approximation space. However, the conditioning number of the two resulting matrices will be different in general. It was shown in [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF] and [START_REF] Belytschko | Structured extended finite element methods of solids defined by implicit surfaces[END_REF] that a smooting of F 1 away from the element layer containing the interface somewhat improved the convergence. The effect of this smoothing in 1D is illustrated Figure 4. For 2D problems, the smoothing strategy is detailed in [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF].

In this paper, we suggest another choice for the enrichment function:

F 2 (x) = I |φ I |N I (x) -| I φ I N I (x)| ( 17
)
This enrichment is shown Figure 4 in the 1D case. For two-and three-dimensional problems, the enrichment function F 2 is a ridge centered on the interface and has zero value on the elements which are not crossed by the interface.

Remark: If the mesh conforms to the material interface, no nodes are enriched since no element is cut by the material interface. The X-FEM then does behave as the FEM. On an element cut by the interface, the integration needs to be performed carefully. First, we divide the element into subdomains matching the material interface as described in [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]. Then, the number of integration points over each subdomain is chosen so that the integration is "exact"(note that F (x)N J (x) is polynomial). On the elements, not cut by the interface but neighbouring an element cut (for instance element A, Figure 4) , no subdivision needs to be performed but the order of the integration needs to be chosen with care since the enrichment acts on the element.

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Convergence tests

In order compare the convergence rate of the classical and extended finite element method, we consider a 2D circular and a 3D spherical inclusion under tension in an infinite domain. These benchmark problems were already considered in [START_REF] Sukumar | Modeling Holes and Inclusions by Level Sets in the Extended Finite Element Method[END_REF] and [START_REF] Belytschko | Structured extended finite element methods of solids defined by implicit surfaces[END_REF] in 2D and 3D, respectively. In order to compare the numerical and exact solution, we use the energy norm. Let u h be an approximate solution and u ex the exact solution, the "exact" error is defined by:

ε = Ω e(u h -u ex ) : a : e(u h -u ex )dΩ
Ω e(u ex ) : a : e(u ex )dΩ [START_REF] Okada | Homogenization method for heterogeneous material based on boundary element method[END_REF] Concerning the material properties, the Poisson ratio is set to 0.25 for the inclusion and to 0.3 for the matrix. The Young modulus ratio between the inclusion and matrix is set to 10. Figures 5 and6 show the decrease in the energy error with respect to the exact solution in four different cases:

• A finite element mesh conforming to the material interface (denoted FEM);

• A finite element computation with a non conforming mesh (denoted FEMnon-conforming). On the elements containing the material interface, the integration of the bilinear form is performed using the appropriate constitutive law at each integration point. • The extended finite element method using the enrichment F 1 improved by a smoothing (denoted X-FEM-1 + smoothing); • The extended finite element method using the new enrichment, F 2 , given equation ( 17) (denoted X-FEM-2).

It can be seen that the new enrichment X-FEM-2 improves somewhat the previous enrichment strategy X-FEM-1. The X-FEM convergence rate is now very close to the "optimal" finite element convergence. We also note, as expected, the poor convergence rate of the non-conforming FEM approach.

Although, the error norm used is a global norm, most of the error contribution comes from the element cut by the interface as can be seen in Figure 7 showing the contribution to the error of the disc (or sphere), D(r), of radius r.

ε(r) = D(r) e(u h -u ex ) : a : e(u h -u ex )dΩ Ω e(u ex ) : a : e(u ex )dΩ (19)
The error norm thus depends directly on the quality of stresses along the -

FEM non conforming (alpha =0.58) XFEM 1 (+ smoothing) (alpha = 0.7) XFEM 2 (alpha = 1) FEM (alpha = 1) Fig. 5. Convergence rates for a 2D circular inclusion problem. The alpha value indicates the rate of convergence with respect to the average element size h.

-

FEM non conforming (alpha = 0.42) XFEM 1 (+ smoothing) (alpha = 0.65) XFEM 2 (alpha = 0.91) FEM (alpha = 1) Fig. 6. Convergence rates for a 3D spherical inclusion problem. The alpha value indicates the rate of convergence with respect to the average element size h. material interface.

Periodicity conditions

To take into account the periodicity condition, we use meshes which are periodic. The displacement constraints on the faces of the unit cell are condensed out of the stiffness matrix at the assembly stage. We mentioned earlier that a node is enriched if its support is crossed by the interface. The support must take into account the periodicity of the mesh. For instance, Figure 8, the support of node K is based on the four shaded elements. Thus node K will be enriched since its support is crossed by the material interface. Nodes K and L have the same number of degrees of freedom with the same values.

NUMERICAL EXPERIMENTS

We consider three different cell geometries: an non-woven composite cell with orthogonal fibers, a woven composite cell and a randomly filled cell with spherical inclusions. In each case, we compare the X-FEM results to existing simulations based on the classical finite element method.

cell

This example is taken from [START_REF] Chung | Asymptotic expansion homogenization for heterogeneous media: computational issues and applications[END_REF]. The cell geometry is shown given Table 1 for the X-FEM and FEM analysis. Figure 10 compares the shear xy mode obtained with the two approaches. It is clear from these plots that the displacement field gradient is indeed discontinuous at the material interface. The X-FEM mesh is a uniform 4-node tetrahedron mesh shown Figure 11 whereas the FEM mesh used for comparison is shown Figure 13. Note that the X-FEM mesh used is composed of many more elements than the FEM mesh. The X-FEM mesh used is uniform as shown Figure 11 whereas the FEM mesh was optimized. Note that the strategy designed in section Section 3.1 could be used to optimize the X-FEM mesh. Figure 9 shows the iso-zero of the level set field.

Woven cell

The second example is based on the numerical experiment carried out in [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF].

The fibers geometry is defined by a disc following a sinus path. The geometry of the unit cell is shown Figure 12 for "two plies". The finite element mesh designed in [START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF] is shown Figure 13 (for one ply). For this example, the X-FEM mesh used is the same as for the non-woven cell (Figure 11). The boron

              (20)
Table 1 Homogenized behavior for the non-woven composite: X-FEM/FEM results.

        (21)
Table 2 Homogenized behavior for the woven composite: X-FEM/FEM results.

fibers and aluminum matrix have the following properties:

E f = 400GP a, ν f = 0.3, E m = 72GP a, ν m = 1/3.
The homogenized behavior obtained by the X-FEM and FEM computations are given in Table 2 and the shear xy mode obtained with the X-FEM is shown Figure 14.

Random cell

As a last example, we consider a random media filled with spherical inclusions. This example was solved by [START_REF] Michel | Homogénéisation en mécanique des matériaux 1[END_REF] using a Fast Fourier Transform approach. In order to take into account the randomness of the spatial distribution of the inclusions, many basic cells have to be considered for the computation of the homogenized properties. Therefore, the X-FEM approach is particularly effective since the same mesh can be used for all the simulations (whereas the level set field must be regenerated for each draw). Moreover, it is also applicable to voids (in which case the enrichment is not used and the voids are taken into account using the strategy designed in [START_REF] Daux | Arbitrary branched and intersecting cracks with the eXtended Finite Element Method[END_REF][START_REF] Belytschko | Structured extended finite element methods of solids defined by implicit surfaces[END_REF]).

The Figures 15 and16 show draws of 8 and 32 particles, respectively. The draws were forced to produce a periodic cell structure. The location of the sphere centers are randomly generated. If a sphere cannot be introduced in the cell because it intersects other spheres already present, a new center location is generated. If a new sphere intersects the cell boundary, its periodic "images" are also introduced.

The particules and matrix properties are E p = 70GP a, ν p = 0.2 and E m = 3GP a, ν m = 0.35, respectively and the particule volume ratio is set to 26.78%. The X-FEM mesh used is still the uniform mesh shown Figure 11 and the shear xy mode obtained with the X-FEM is shown Figure 17. The homogenized stiffness obtained along the x axis (a hom 1111 ) is 7.611 for 8 particles and 7.711 for 32 particles. These results are very close to the ones reported in [START_REF] Michel | Homogénéisation en mécanique des matériaux 1[END_REF]. The latter where computed with many more draws (and thus more reliable): a 1111 = 7.675 for 8 particles, 7.664 for 64 particles and 7.667 for 420 particles.

CONCLUSIONS

The paper has demonstrated the applicability of the X-FEM to the homogenization of periodic basic cells. Since the mesh does not need to conform to the material interfaces, complex and random cell geometries can easily be computed with very limited meshing effort. The results obtained are very close to direct FEM and Fast Fourier Transform approaches. Moreover the X-FEM is also applicable to cells containing voids and phases with nonlinear constitutive laws. However, in order to reasonably locate the interface, the mesh must not be too coarse. A simple geometrical adaptivity scheme has been proposed. Finally, a new enrichment to introduce discontinuous strains inside an element has been developped. The initial results are promising since this enrichment provides a very close convergence rate to the "optimal" FEM convergence rate.

Fig. 1 .

 1 Fig. 1. Unit cell geometry for an helicopter landing grid.

Fig. 4 .

 4 Fig. 4. Several choices for the enrichment function.

Fig. 7 .

 7 Fig.7. Error contribution ε(r) with respect to r/R for the 2D circular inclusion problem (left) and the 3D spherical inclusion problem (right). In the both cases, R is the radius of the inclusion.

Fig. 8 .

 8 Fig. 8. The support of a node located on the side of a 2D cell.

Figure 9 .

 9 Although several fibers are involved only one level set function is needed. The cell side length is 2 and the fibers radius is 0.45. The material properties are representative of E-glass fibers and epoxy resin matrix (in GPa): E f = 72.0, G f = 27.7, ν f = 0.30, E m = 3.5, G m = 1.3, ν m = 0.35. The homogenized stiffness is

Fig. 9 .

 9 Fig. 9. The unit cell of non-woven composite.

Fig. 10 .Fig. 11 .

 1011 Fig. 10. Shear xy deformation modes obtained by the X-FEM (left) and FEM (right) computations.

Fig. 12 .

 12 Fig. 12. Woven cell geometry.

Fig. 13 .

 13 Fig.[START_REF] Michel | Homogénéisation en mécanique des matériaux 1[END_REF]. Mesh designed in the paper by Kikuchi et al.[START_REF] Guedes | Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods[END_REF] for the FEM analysis of the woven unit cell.

Fig. 14 .

 14 Fig. 14. Shear xy deformation modes obtained by the X-FEM the woven microstructure.

Fig. 15 .

 15 Fig. 15. Four different draws with 8 particles.

Fig. 16 .

 16 Fig. 16. Two different draws with 32 particles.

Fig. 17 .

 17 Fig. 17. Shear xy deformation modes obtained by the X-FEM for two cells with 8 spheres (top) and 32 spheres (bottom).

  1/d where V is the element volume and d the problem dimension. An element is tagged for refinement if l c > αρ and |φ| < βl tot (12


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From local to global. New paths for publishing in Africa 1

How can one's voice be heard on a global scale? Historically, the writing, selling, and reading of books has been a central means through which stories have been shared across distance. Despite the wide-ranging nature of print, however, in Africa, books and publishers still struggle to get their share of attention. Not only is the volume of African book production dramatically underrepresented in the world 2 but the African publishing industry has also had to contend with the ongoing legacies of former colonial monopolies. At the same time, if printing books and selling them across national borders has always been one of the main stumbling blocks to the African publishing trade, then it is reasonable to think that the new technologies and media that have emerged at the turn of the millenium might enable the written word to travel more easily across the continent and beyond, expanding the reach and circulation of African publishing, and African knowledge, today.

In this paper, I would like to look at these issues with reference to Benedict Anderson's seminal work, Imagined communities 3 . In doing so, my focus will not be on nationalism, as is so often the case in studies of Anderson, but rather the other major and interconnected notions which appear in his work, "capitalism"; "print-languages" and "imagined communities", as terms important in the publishing field where one publisher has to sell books to a community of readers that constitutes itself through one of the print-languages available in the country. In Africa, these are often European languages like English, French and Portuguese though other languages such as Arabic or Kiswahili are also important. Following Anderson, my argument questions the constitution of communities that identify themselves through shared discourses and values. Do such communities exist? If texts published in Africa constitute a part of the fabrication of these communities, how do they reach people inside and beyond national boundaries? In answering these questions, issues of uneven trading relationships, identity and new technologies will be taken into account, as will the relationship between local and global: is it possible to "relocate" African literature on the continent, and to what extent? What can we learn from the popular success of a book serie like Adoras in Ivoiry-Coast or of the use of new media by the Tanzanian author Eric Shigongo? In a world where dematerialization is ever-developing, does the distribution of African cultural production become easier?

Communities of readers, print-languages and states policies

If the written word has historically held less weight than the spoken word in African cultures, it is nonetheless true that some rulers and politicians saw its possibilities for empowerment early on. On return from London, in 1902, for instance, Ham Mukasa underlined how impressed the Katikiro, the Prime minister of Buganda, had been by the books he had seen at the University library 4 . The idea that the printed word was powerful as well as useful had gained more and more recognition since the first missionaries had begun to turn books into more familiar objects. After the wave of independences in the 1960s in Africa, a significant number of politicians aimed to introduce cultural policies where books had a role to play. They were helped by the UNESCO observatory on cultural policies which was also active in supporting the publishing industry in Africa (especially in the 1970-1980). One notable example of this trend was the establishment, by Leopold Sedar Senghor, president of Senegal and man of letters in his own right, of the Nouvelles Editions Africaines in 1972. It is important to note that, from the beginning, this publishing house was linked to France as five French publishers were financially part of it. On the other hand, founded in the era of Pan-Africanism, the publishing house was also based on a partnership between Senegal, Ivory Coast and Togo. Elsewhere, Tanzania's Julius Nyerere, himself a teacher and a translator, gave in his Arusha Declaration a paramount place of importance to culture and developed both Kiswahili as a national language and the printing industry as a means of education and in order to spread his message of Ujamaa. In Algeria, too, the state was very much involved in the publishing industry, having nationalized the Hachette group in Algeria. This act had led, in 1966, to the birth of the Société Nationale d'Edition et de Diffusion (SNED) that became ENAL (Entreprise National du Livre Algérien) in 1983. This state entreprise controlled a great part of what has been published until 1998 (when it disappeared and let more space to independent publishers).

Given this context, it is reasonable to wonder to what extent the strong national policies developed around the education of citizens in the early years of independence had an impact or contributed significantly to the development of books on the continent. As state policies have been extremely diverse in Africa, it is of course impossible to give a precise answer; at the same time, one can point out that globally the spread of education in Africa resulted in an increased number of readers on the continent. The ultimate effect of this effort to improve literacy varied across countries, however; some countries encouraged education in African languages (Tanzania is one of the best examples, where Kiswahili is taught to all children from primary school), while others relied only on European ones. The importance of this distinction is not because, as some would say, only African languages can sustain a community of African readers 5 but rather because publishing in African languages is somewhat less affected by the risk of having its economy directly challenged by the publishing houses of the former colonies: if publishers provide books in non-European languages that people can read, they have fewer opportunities to compete with Western publishers. This supposes, of course, that some African languages have developed into print-languages, though in reality only a small subset have, including Kiswahili, Arabic, Afrikaans, Amhara and to a lesser extent Yoruba, Wolof, Bambara and Fula 6 .

The question of an economic competition with publishers from the Western world is important because even if independent African states were willing to build up a national publishing industry, they were not always able to secure this activity. If Pan-African cooperation was the rule in the beginning, this sort of cooperation has since largely eroded, especially during the 1980s and 1990s. For example Nouvelles Editions Africaines ceased to trade in 1988, becoming instead the more nationally-bounded Nouvelles Editions Africaines du Sénégal. With the economic crises in West Africa and the economic liberalisation of countries in East Africa, most African states cancelled entire sections of cultural affairs, leaving new independent publishers to cope with the work, which was not without difficulties, notably the cost of raw materials such as paper. Despite the Florence Agreement7 , the cost of paper remains prohibitive to this day, preventing African publishers from being able to issue products that could be sold at a cheap price. Indeed, even with the growth of the middle classes in various countries in Africa, the cost of books remains beyond the average household's means. Thus, it seems that the emergence of a community of readers is perhaps not the main issue facing African publishers. Rather, it is the lack of financial support from the state and at the local scale which leaves African publishers in a relatively non-competitive position at the global level.

"Under pressure": challenging ongoing difficulties by offering new alternatives

During the 1990s, a period in which a severe depression hit Francophone Africa, numerous states tended to limit their involvement in the cultural field. At the same time, structural adjustment programs established by the World Bank contributed to take over the school curricula, at the expenses of local publishers. Far from being intimidated by the rather gloomy prospects, however, independent publishers have continued to emerge and new publishing houses were founded: Jamana (1988, Mali) ; Edilis (1992, Ivory Coast) ; Sankofa & Gurli (1995, Burkina-Faso) ; Donniya (1996, Mali). Others were to follow in the 2000' as Barzakh (2000, Algeria); Eburnie (2001, Ivory Cost) ; Proximité (2002, Cameroun)… These publishing houses, all of which are still currently active, have worked hard to develop their lists and establish a steady readership. If never joining any kind of supra-national union, nearly all of them nevertheless became members of the French-language network of the Alliance Internationale des éditeurs indépendants8 and gained much from the meetings among peers from all over the world and the advocacy actions led by the structure. Despite their efforts, however, difficulties remain, including the relatively small proportion of potential book buyers on the African continent and the problems associated with the distribution of books. In some countries only a few existing bookshops are available to sell locally-produced material, a situation which is made worse when publishers attempt to find markets outside of their home countries. Indeed, the inadequacy of the book distribution system is one of the big issues for publishers operating on the African continent. In order to develop readerships, it is essential that books are well-publicised so that potential readers are made aware of their existence and that a physical infrastructure exists to allow the purchase of books. Neither of these two points is easy in the current African context, and both are made more difficult in a transnational context in which it is often challenging simply to get information regarding what is published, let alone find an outlet through which to purchase specific books, even if African Books Collective or Afrilivres do help a lot in increasing the visibility of the African book production, as we shall see later on.

These difficulties are all the more important since they tend to reinforce the dynamics of extraversion rooted in the fact that both printing activity and the publishing field in Africa have emerged through the missionary presence and then through the structures of colonization. To the present day, ties between African and western institutions remain close. We mentioned above that French publishers were financial partners of the Nouvelles Editions Africaines since its birth, as for today, Edicef (a subsidiary of Hachettewhich means Lagardère) is one of the major investors in Nouvelles Éditions Ivoiriennes. Relationships are often uneven as larger and better-resourced northern publishers share the same playing-field with their counterparts on the continent. Politics of "developing" books and reading in Africa thus sometimes only sustain foreign domination of the African market at the expense of independent local publishers. In the context of francophone Africa, for example, some publishing houses like Gallimard (through its collection "Continents noirs") or Présence Africaine are able, through various policies, to sell their books in Africa at reduced prices. 9 While advantageous for local readers, this policy doesn't help local publishers gain visibility in their own countries. The same logic is at work with a structure like the Centrale de l'Edition via its Programme Plus, supported by the French Ministry of Culture and Communications, which makes it possible to buy books published in France at 65% cheaper that the original price. As for booksellers from Francophone Africa, they also benefit from the guaranties offered by the Coface, a French export-credit insurer. Raphaël Thierry, in his well-documented essay on the book market in Africa, writes : "Si l'on s'en réfère aux chiffres du rapport 2008 de la Centrale de l'Edition, l'Afrique représente, pour le marché français 38 980 000 euros d'exportation d'ouvrages contre 1 030 000 euros d'importation, soit un rapport d'environ un ouvrage importé contre trente-neuf exportés. » 10 Such figures speak for themselves. And private initiatives do not make things easier for African publishers. Consider the case of L'Harmattan, the Parisian publishing house which recently started a network of bookshops in countries like Senegal, Mali, Burkina Faso and many others, mostly selling their own Paris-printed publications (including unsold books from France). 11 As a consequence, in Africa, and especially in its francophone countries, it is often easier to access books published in Europe over those published on the continent. The publications are more visible (famous African writers are still published in Paris, even if it is true that some of them -Patrice Nganang, Henri Lopes, Boubacar Boris Diop, Ken Bugul to name but a feware also published in Africa), it's easier to buy them, and, as we have just seen, sometimes they are even cheaper. The schoolbook market is a particularly blatent example of the European grip on the book trade in francophone Africa. A highly lucrative field, recent studies suggest that 85% percent of the textbook market share is monopolized by Hachette International in the region, 12 a figure made all the more significant when we recall the importance of education in the development and shaping of reader-communities. The fact is all the more harmful as this schoolbook market is nearly the only profitable sector where publishing is concerned. In comparison, in Anglophone countries like Kenya or Tanzania, books and textbooks eligible for school curricula are most of the time issued locally in English as well as in Kiswahili. 13 Working hand in hand to achieve success?

This rather gloomy picture could be daunting for many if it weren't for the stubbornness of publishers who are bringing together their energy and trying to determine solutions. If publishing in major European languages may be a drawback (as we've seen, it can means a more direct competition with Western publishers), it can also be taken as an advantage, as shared languages also induce transnational networks that can enable wider cooperation and make it possible to work and think collectively. The African Book Collective, founded in 1989, is a good example of how an Oxford-based structure can help African publishers to gain a wider visibility on the continent and abroad. From its 17 original founding members, the association, which is owned and governed by African publishers 14 , has grown up to include now 149 members, mostly from Anglophone Africa but hosting more and more publishers from Francophone countries. English then serves as an efficient tool to highlight the African production, mostly in English, but not only. This successful initiative led to the creation, in 2002, of the French-speaking network Afrilivres 15 which is also committed to increasing the visibility of African books. Despite any difficulties which it has faced, the association's platform provides its 33 publishers with a space where their books can be seen, even if they can't be ordered directly through the website (unlike ABC website that enables readers to buy online). It is to note that the issue of e-commerce is an important one, but that banking agreements are still hindering the African publishers' activity, as most payment sites (like Paypal for example) do not acknowledge African banks. It thus stifles the development of e-commerce and in fact, most of the time, when African publishers want to sell abroad (or even in their own countries), they often have no other option but to load with copies whatever car or luggage they have and to carry themselves their production. Institutions like L'Oiseau indigo diffusion, launched by Isabelle Gremillet who used to work for Actes Sud, is dedicated to enable the diffusion of African books, mostly in French, but also in Arabic or Wolof 16 . Pollen Diffusion, a structure specialized in the diffusion/distribution of independent publishers, also assume this role for a few publishing Houses from Maghreb (Elyzad, Tarik editions, Yomad). Another way to circumvent the difficulties associated with book publishing in Africa is the emergence of initiatives such as "Terres solidaires" developed by the Alliance des Editeurs Indépendants and which allows African publishers to co-edite a text previously published in France. Initially, it was a discussion with Véronique Tadjo, who wrote L'ombre d'Imana, a text dealing with the genocide in Rwanda, that started the adventure. The author was concerned that her book, published in France, had nearly no chance to be found and read in Africa. The will to "give back" to the South texts written by African authors and also for an African audience led to the co-publishing of L'Ombre d'Imana, and later to the collection "Terres solidaires". Beside the "relocation of culture", the initiative offers answers to various problems 13 Ivory Coast is the only francophone country in Africa to organize national tenders. On African publishers have to face through a variety of methods. Amongst these, Terres solidaires offers African readers the possibility to read books by high-profile authors first published in France at a competitive prices, thereby bridging the gap between African writers publishing in the west and their audiences on the African continent. 17 Because the Alliance des Editeurs Indépendants purchases the rights of publication for Africa from the original French publishers, where willing 18 , the prices of the republished book can be adjusted to suit the needs of the African market. For example, titles like Mandela et moi or Loin de mon père are sold at 2 500 CFA or 4,3 DM (roughly 3 euros 19 ). Secondly, books published through the Alliance are available in each of its member countries, offering de facto an intraafrican system of distribution and making the collective's work more visible. For example, Jazz et vin de palme, written by Emmanuel Dongala, was distributed in the bookstore of the Algiers airport, while the latest title issued, La Saison de l'ombre, by Leonora Miano, will be co-published by eight African publishers and distributed in 18 countries on the continent.

Initiatives of this sort are important not just for the work that they do, but also for the way in which each successful enterprise might serve as a point of inspiration for the creation of further enterprises. For example, François Nkeme, who is the co-founder of the Cameroonian publishing house Ifrikiya, has worked with the Alliance on several co-editions of the collection "Terres solidaires". In 2008, he worked in collaboration with Editions Tropiques to co-publish Je suis né en prison 20 , a collection of short stories written in tribute to Nelson Mandela. Working in cooperation in this context serves as a form of apprenticeship and training in the marketing and distribution of books, as well as a means of cost-sharing through a form of wholesale printing in which each publisher may choose how many copies to pay for from a larger consignment, with the sole constraint being that at least 2000 copies are printed in total. For example, for the co-publishing of De l'autre côté du regard, a book written by Ken Bugul and initially published in France by Le Serpent à Plumes, Editions Eburnie (Abidjan, Ivory Coast) purchased 700 copies and Khoudia (Dakar, Senegal) 200. Initiatives like this demonstrate the ways in which, in a global economy, African publishers think and operate in ways which move beyond the local, often using the resources at hand to offset the asymmetrical relationship between Africa and the West in the realm of book production. New means of communication help, mostly for the debate between co-editors, but alsoof coursein the transmission of computer files.

Publishing books with profit 17 The information flows are clearly unequal: while books printed in Africa rarely attract the attention from the Western world, on the other hand, the African literary scene whose books are published in Europe enjoys a wide visibility on the continent: media from the Western world (RFI being the most important for francophone Africa) as well as local journalists are voicing the successes obtained in France or elsewhere, so that most of the people will most likely know about Alain Mabanckou and Leonora Miano than Nafissatou Dia Diouf or Badiadji Horrétowdo. As a consequence, most readers would like to be able to read these famous writers. Members of the Alliance des éditeurs indépendants, if they can propose local writers to be published in co-edition, usually agree easily upon a "big name" published in France but already identified in West-africa. It is more difficult to find a consensus upon a "local" name that won't be able to unite readers in the same way in various African countries. 18 If rather small independent publishing houses are often willing to take part in such an operation, a major publishing house like Le Seuil for example, is steadily and categorically rejecting this option. 19 For further details : http://www.alliance-editeurs.org/+-terres-solidaires-+?lang=en (accessed January 2016) 20 Edmont VII Mballa Elanga (dir.), Je suis né en prison. Yaoundé : Editions Ifrikiya/Tropiques, 2008, 205 p. Both publishers are based in Cameroon.

If the development of readerships and markets remains challenging in the African context, it is nonetheless untrue to imagine the African continent merely as a place of 'book famines' and wasted potential and it would be wrong to think that the virtuous relationship induced by collective action only goes from Western world to African publishers. Kamel Daoud's Meursault, contre-enquête originally published in Algeria by Sofiane and Selma Hadjadj, the co-founders of Barzakh, 21 is one example of the complex and multi-faceted relationship between publishers. Later published in France, the book was eventually translated into English and released in the United States, where it ranked in the top ten of Publishers Weekly's Best Book of 2015. While the American publication of the book was certainly a result of its French release, the case of Meursault, contre-enquête nonetheless demonstrates the ways which African publications can and do garner interest from Western readers and publishers. Indeed, the international success of the book led not only to a series of debates around it in Algeria, its original country of publication, but also generated a far greater interest in the book, reflected in record sales figures for fiction of 10,000 copies.

It is important to keep in mind such figures, even if they are somewhat exceptional, because as with every business, the publishers' trade must be economically viable. In fact, in some countries, popular literature remains an important and successful force. In Côte d'Ivoire, for example, the Nouvelles Éditions Ivoiriennes 22 launched a very popular collection of romance novels, Adoras, in 1998. Often referred to as "the African Harlequin" (in reference to the Canadian company Harlequin Enterprises that publish romance paperbacks), the books are nonetheless rooted in West African contexts and attract a large readership, mostly composed of women [START_REF] Moudileno | The Troubling Popularity of West African Romance Novels[END_REF] . One of the most successful writer of the collection is Isaïe Biton Koulibaly whose novel, La parenthèse délicieuse, sold 3255 copies, while one of his short stories, Ah ! Les femmes, 5000 copies in just one month after its release (in 2012). This success is also explained by the fact that Nouvelles Éditions Ivoiriennes is partly owned by Lagardère (Hachette), giving it a firm institutional and structural basis. This statement is particularly true as commercialisation is concerned: all titles are distributed by NEI, Edipress and Edicef (Hachette international), a structure that helps to explain its strong economic position. It also explains why some titles of the Adoras collection are among the very few to be proposed by Amazon (through the Kindle format). At the same time, the local networks of Isaïe Biton Koulibaly are also of importance. In addition to being a journalist in both local and international papers and working at the public radio and television, the author has a strong business acumen, as well as a keen sense of marketing. In the case of Adoras, success thus appears to be the result of both social criteria (stories attuned to the taste of readers and rooted in an African context), economical status and distribution strength of the publisher (closely linked to the ex-colony) and a keen sense of marketing and self-promotion of the author. But as far as popular literature is concerned, other factors can come into play, like the reading habits, that is often boosted by the facts that in some countries, books are also written in national languages. It is the case, for example, in Tanzania and Nigeria 24 .

In Tanzania, where a truly popular literature, brewed in crimes and passion novels, has developed and attracts a lot of readers since long, a new turn has been taken with an author like Eric Shigongo, one of the latest and most prolific writers at that time. Just as Isaïe Biton Koulibaly, Shigongo has strong connections with journalism, as he is the owner and chief executive officer of Global Publishers, a company he co-founded in 1998 with his partner Abdallah Mrisho Salawi. Since the 1990s, Tanzania has experimented the privatization of media, giving rise to a multiplication of newspapers and also to new ways of publishing: writing in newspaper is, in fact, one of the most popular form of writing nowadays, allowing writers to reach a greater number of readers and playing a major roleas before in Europe or elsewherein the constitution of communities of readers. As reported by Uta Reuster-Jahn, in Tanzania, newspapers print-runs reach between 45 000 and 60 000 copies for an average of 250 000 readers each 25 . Being a powerful medium, newspaper is often a good option for writers who publish in "feuilleton" before issuing books. In Shigongo's case, this strategy proved incredibly successful. Writing serials published in the newspapers of his own media company, his texts are widely read, as these are distributed in the whole country. As the author enjoys a page for his books on the website of Global Publishers, he can exchange extensively with readers and gain an informed opinion about their tastes and the reception of his texts. 26 Afterwards, he can choose which serials can be turned into books. Thus far, this has been the case for three of them: The Last days of my Life (Siku za mwisho wa Uhai wangu); The President Loves my Wife (Rais anampenda Mke wangu) and Blood and Tears (Damu na Machozi), 27 all of which, despite their titles, are written predominantly in Kiswahili and feature a strong combination of romance, suspense, crime and passion. In a conversation with Uta Reuster-Jahn, Shigongo himself puts the figure of 120 000 copies sold for the The Last Days of My Life and The President Loves my Wife. This huge figure, even if somewhat inflated, is an indicator of how new digital strategies can achieve success. In Shigongo's case, the reasons are many:

-a well-established strategy: visually, books of a high quality whose size and style strongly remind the reader of international best-sellers; -a nationwide distribution as he uses his sale representatives for his book as well as his newpapers; -at the same time, books are also clearly advertised in the newspapers and on the Global Publishers'website; -printed in India, the book are affordable for Tanzanian as the price is about 4000 ShT (namely around 2,50 euros).

These endeavors raise questions around what 'independent' means in a context in which support from metropolitan publishing houses, advertisement and patronage remains essential. Nonetheless, they give hope that related marketing and distribution structures may be developed by and for small and more demanding publishing houses such as Ankara Press/Cassava Republic Press. At the same time, these endeavours also indicate both the extent to which a real market for reading continues to exist in Africa and the importance of new media in capturing this market. Based on these brief examples, a series of conclusions can be drawn. First, popular literature is deeply shaped by writers very much aware of global trends, as well as marketing. They adapt to an African market where social issues are important and distinct from those of the Western world28 . Second, they use diverse media to reach a greater audience, mastering the new possibilities that technology offers and taking advantage of a strategy where the written word travels from newspaper to blog and cinema. Most of these strategies may be used by publishers, even if they are not working on the same kind of publications.

African publishers : New tools for the future ?

Major upheavals in history have been generated by the emergence of a new medium: after the invention of the press (and Victor Hugo writing that book was to replace the cathedral), is digital publishing ready to take over paper edition in Africa? It is easy to become overly optimistic when considering the possibilities produced by new technology in Africa, where they remain unevenly distributed. Some cities like Nairobi, Cape Town and Dar es Salaam are highly connected, as are many capitals in francophone Africa. Yet, when one gets out of the town and into rural areas, the situation changes; in some countries like DRC and South Sudan, internet access can be rare, even in major cities 29 . At the same time, mobile phones remain prevalent across the continent. Despite concerns around uneven access, digital technologies have become an increasingly important tool for African publishers, one which more and more are experimenting with. 30 It is easy to see why: digital publishing offers solutions to two of the major problems which face print publishing, namely cost and distribution.

New reading habits are also developing in Africa, especially amongst children, driven by the provision by some NGOs of Amazon Kindles and other tablets. Worldreader, an American NGO, claimed through its website to have distributed 7395 tablets (and a total of 1.2 million digital books). 31 The NGO has also developed a Worldreader Mobile application that will provide access to digital library through an ordinary mobile phone. In Europe, Foundation Orange is also developing projects for Africa32 . If this news is rather good for readers, it also underlines how important it is for African publishers to think about the new possibilities offered by technology, especially given the risk, should such projects be left to Western companies and NGOs, of ongoing extraversion. African publishers are nowadays quite aware of the importance of mastering these new media. For the moment, their main interest is to use the great visibility allowed by Internet. Publishers from Anglophone Africa are already quite used to having their own websites, and structures like Kwani? or Writivism, among others, have succeeded to give visibility to literary creation thanks to their presence on the web 33 . This is not yet always the case of publishing houses from francophone Africa. To improve the situation, the Alliance Lab organized a workshop in Dakar in June 2015 to train publishers of francophone Africa on web technologies to help with the marketing of their publications. This is a first step, and a few editors (Le Fennec éditions ; Editions Amalion ; …) now have websites allowing people to order or buy online.

The most significant action in francophone Africa is undoubtedly the creation, in 2006, of the Nouvelles éditions numériques africaines (NENA) followed, in 2013, by the Librairie numérique africaine. 34 . Most recently, the emergence of Kusoma 35 , a Senegalese-based platform which works as an online publisher, bookshop and library underlines how people from the editing world in Africa have grasped the capital importance of "turning digital". NENA, founded by Marc-André Ledoux, works with different African publishers, mostly, but not exclusively, from Senegal, as a Dakar-based institution, and propose a catalogue of more than 400 books in Epub format. You can order digital books ranging from those by well-known authors (Henri Lopes, Francis Bebey) to those by young author like Mutt-Lon, noted for his novel Ceux qui sortent la nuit [START_REF] Mutt-Lon | The novel was awarded the Kourouma Prize in 2014[END_REF] or the above mentioned Isaïe Biton Koulibaly, directly, using a userfriendly interface and benefiting from affordable prices (around 2500 CFA for a book in Epub format, i.e, slightly less than four euros). Marc-André Leroux says that more and more authors and publishing houses are willing to work in partnership with NENA [START_REF] Ledoux | interview with[END_REF] . He also states that digital publishing is in no way an opponent to paper publishing. On the contrary, and as we have already seen in Eric Shigongo's case, one format can find its own public and lead to the other. In fact, it may be important to take into account that texts are gaining strength when adapted to various formats including newspaper, books, movies, digital and also audiobooks.

While paperbooks have never truly found a place in the daily life of most Africans, new opportunities are emerging as texts are gradually getting more and more separated from printed materials. In Europe, more and more teenagers read online [START_REF]As for the French case, see the CNL report "les Français et la lecture[END_REF] , and one may think that in Africa, where printed books are less

Publishing in Africa… in African languages?

But of course technology does not stand for everything. Equally important is the task of making voicesand different voices -heard. As Ngugi wa Thiong'o wrote in Decolonizing the mind, it is also important to propose one's ideas through African languages. Not only because it strengthen the language in itself (that can thus gain also more readers) but because every language also encapsulate a specific worldview that has a right to take part in the intellectual debates. It has become more commonplace in the twenty-first century to view languages like English, French, Portuguese and Spanish as African languages. These languages have been part of national culture for many decades now, often hold national status and are often the language of instruction in educational institutions. At the same time, these languages have not been passively accepted by African populations; rather, they have been subject to a process of adaptation, where specifically African forms of each language have developed across the continent. But African is also rich with an incredible number of indigenous languages, sharing very different situations in terms of number of speakers and status. If it would be sheer utopia to think that all these languages will someday become standard languages for publishing, it is nonetheless important to consider the current situation, given the opportunities provided by new technologies. Historically, indigenous languages have been hampered by the lack of standardization in orthography, leading to a situation where languages easily spoken were not so easily written or read. These facts are still very true today but slowly new projects are emerging, both with respect to well-established African languages such as Kiswahili, with its long history as a written language, and in smaller ones, like fe'efe'e. Significantly, computer technologies make it possible to write in every language: Unicode, for example, with its 110 000 characters may be used to transcribe more than one hundred languages. 39 If you work with a Mac, you can set up more than eighty languages on your computer, and with an Iphone, these figures go up to more than two hundred. Of course, the potential to write in a language is not the same thing as publishing in a language, but it is still important to keep in mind that more than once, changes have come only after the development by tools. Indeed, a current example today comes from Aline-Léonie Chouapi, who is translating Patrice Nganang's book, Mont Plaisant, into fe'efe'e, a bamileke language, using Clafrica software, a literal manifestation of Ngugi's plea that, for African languages to develop, they must be put into use and enriched through the engagement of a writer or translator who decides to work in African languages.

Publishing in African languages would not be the only solution to the problems plaguing African publishers and, indeed, a series of serious challenges remain in the project of indigenous publishing. At the same time, work is being done in this regard and it is often writers who take a central role in this process. Along with Ngugi wa Thiong'o, Senegalese writer Boubacar Boris Diop is another public intellectual who stands as an example and whose experience of publishing Doomi Golo in Wolof has led him to think about the relative potentials and pitfalls of writing in his mother tongue 40 technologies diversify the options available to writers and publishers, and a growing interest has emerged from the African publishers themselves. The first edition of the SAELLA (Salon de l'Ecrit et du Livre en Langues Africaines) that took place in Bamako (Mali) in 2016 41 is a prime example. For three days, professionals, academics, institutions and NGO exchanged ideas and debated on publishing in African languages in a clear sign that the question remains vital. Two panels dedicated to "African languages and communication technologies" also took a specific interest in digital issues. Finally, the birth of "Ceytu", a collection edited by Boubacar Boris Diop which aims to publish major works of francophone literature in Wolof translation 42 is another example of the ways in which indigenous languages are operating in the landscape of publishing. Significantly, Ceytu's co-publishers, Zulma and Mémoire d'encrier, are Canada and France-based, that "imagined communities" in our contemporary world are perhaps less and less related to nation-state. It enhances how a "local" language can go "global" thanks to networks and the fact that, as Bernard Mouralis puts it: "Désormais, ce qui compte, en matière d'écriture et d'édition, c'est la connexion bien plus que la localisation, qui peut s'avérer bien difficile à repérer." 43 Equally, translation plays a role in the possibilities of dialogue and the development of larger communities, as numerous examples from the African continent make plain.

It is interesting to note that François Nkeme, from the late Ifrikiya, has sold the rights of Walaandé to three coeditors (Med Ali, Tunisia; Dar-al Farabi and Al Intishar in Lebanon). The book has been translated in Arabic and will gain more readers in countries where polygamy is also an important issue to debate. A proof that African publishers themselves are expanding their horizon and take part, through their publications, to a more global debate even if, from a Western point of view, it can go indetected.

The publishing industry has always had a major role to play in the defence of diversity at local and global scales. In Africa, publishers still have to fight against serious competition from their Western counterparts, where the textbook battle remains the most importantand challengingpart of the job, not only economically but also symbolically as education plays a key role in shaping dominant views and values. If literary publishers in Africa have proven that it is possible to publish successfully on the continent, this sometimes occurred through the use of strategies and patterns for marketing and production developed elsewhere. New technologies seem to offer a real opportunity for publishing in Africa, but will that lead to more diversity or, on the contrary, a reign of global standardization? The question is still open. Nathalie Carré 41 SAELLA, Salon de l'Ecrit et du Livre en Langues Africaines. Bamako : January 20-23, 2016. 42 Three major writers -Césaire, Le Clézio and Mariama Bâhave thus already been translated and published. 43

  this question, see Lauwerier, Thibault L'influence de la Banque mondiale sur les politiques d'éducation de base en Afrique de l'Ouest francophone: les cas du Mali et du Sénégal de 1980 à 2010. Thesis defended at the Université of Geneva, 2013.

14

See http://www.africanbookscollective.com (accessed January 2016).

15

See http://www.afrilivres.net (accessed June 2016). 16 See for example Papyrus edition, a house that published in French, Pulaar, and Wolof (http://www.loiseauindigo.fr/category/liste-editeurs/editionspapyrus accessed January 2016)

  24 In Tanzania, most books are published in Kiswahili, and even if the reading of Swahili classics remains for most Tanzanians rather confined to the school time, detective and crime stories are numerous and quite successful. In Nigeria, even if most of the production is written in English, Yoruba literature is quite dynamic, with such authors as David O. Fagunwa, I.B. Akinyele, Akinwumi Isola, Akinlade Kola and, more recently Akinlabi Banjo. There is also a tradition of translation between English and Yoruba, Soyinka's works being translated in Yoruba by Akinwumi Isola and Soyinka himself translating into English Fagunwa's works (see The Forest of a Thousand Daemons and In the Forest of Olodumare). To have a glimpse on Yoruba literature, see http://www.sunshinenigeria.com/yoruba (accessed June 2016). I am indebted to George Alao for this online reference.25 Uta Reuster-Jahn, The newspaper serials in Tanzania : the case of Eric Shigongo. See tab "hadithi" (stories in Kiswahili) of the Global Publishers website (http://globalpublishers.co.tz accessed January 2016). The website itself, full with adverts (and widely advertising Shigongo's novels) is an excellent example of new marketing strategies.27 The Last days of my Life is out of print, The President Loves my Wife has been published in 2003 and Damu na Machozi in 2005.

	Swahili Forum 15 (2008), p.
	25-50. The paper can be read online :
	http://www.qucosa.de/fileadmin/data/qucosa/documents/9104/15_04_Reuster-Jahn.pdf (accessed
	January 2016).

26

  . New 39 Conference presentation by Huguette Rigot, International Workshop "Enseigner les littératures en langues "minorées" : Politique, outils et pratique", Paris: INALCO, 23-24 September, 2015. 40 Diop has said and written much on the subject. For a brief bibliography and further information, see my contribution "Between mother tongue and 'ceremonial tongue': Boubacar Boris Diop and the selftranslation of Doomi Golo" in International Journal of Francophone Studies, vol. 18, n°1, 2015, pp. 101-114.

  Bernard Mouralis, "Ecrire et publier en Afrique : (re)localisation, postcolonial, mondialisation" in Awezaye, Imorou, et de Meyer (eds) Ecrire et publier en Afrique francophone. Enjeux et perspectives, French Studies in Southern Africa, n° 44-2-2014, p. 24..

The Florence Agreement(1950) and its Protocol (known as "Protocol of Nairobi", 1976), sponsored by Unesco, aim at making it easier to "import educational, scientific or cultural materials. They reduce tariff tax, currency and trade obstacles to the international circulation of these materials permitting organization and individuals to obtain them from abroad with less difficulty and at less cost.) Guide for the application of the Florence Agreement and its Protocol, portal.unesco.org/culture/ accessed January 2016.

See: http://www.alliance-editeurs.org/-les-editeurs-#rl23 (accessed June 2016).

Gallimard has decided to sell its books in Africa to the single tariff of 9,50 euros (roughly half of the average price in France), which is still quite a significant amount for francophone Africa. Books published by Présence Africaine sometimes benefit from subsides to lower their prices in Africa.

Raphaël Thierry, Le marché du livre africain et ses dynamiques littéraires. Le cas du Cameroun. Pessac : Presses Universitaires de Bordeaux, coll. « Littératures des Afriques », 2015, p. 19. « If we refer ourselves to the report published by the Centrale de l'Edition in 2008, the figures are such as follow : French publishers sell for 38 980 000 euros in Africa, whereas, in the same time, importation of books published in Africa amounts to 1 030 000 euros. The ratio is thus 39 books exported per 1 book imported."

For further information, see www.harmattan.fr/groupeharmattan/harmattan-etranger.html (accessed January 2016).

Brigitte Perruca, « La France règne en maître sur le marché des manuels scolaires en Afrique francophone » in Le Monde, 10th of June, 2010, quoted in Raphaël Thierry, op. cit., p. 40.

The role of women, gender questions are much debated, through various point of view. For example, the most successful book by Ifrikiya is Walaandé, written by a fula woman from Cameroon on the issue of polygamy. 5000 copies of the book has been sold in Cameroon in five years. Djaïli Amadou Amal, Walaandé, l'art de partager un mari. Yaoundé : Ifrikiya, 2010.

In 2012, Radio Okapi reported that less than 1% of the population in DRC had access to Internet (see: http://www.radiookapi.net/economie/2012/11/09/rdc-moins-de-1-de-la-population-seulement-accesinternet. Accessed June 2016).

 30 For example, in a recent interview, François Nkeme who founded Proximité (Cameroon) mentioned that smartphones, tablets, and laptops were less expensive now, so it was important to take account of these new media. (unpublished personal exchange with François Nkeme, 24/03/

2016). 31 http://www.worldreader.org (accessed January 2016).

For further informations : Matthieu Joulin et Octavio Kulesz, « Le don de livres numériques : quelles logiques et quels enjeux ? » in Takam Tikou, 2015 available online : http://takamtikou.bnf.fr/dossiers/dossier-