Robin Alais

Petr Dokládal

Etienne Decencière

Bruno Figliuzzi

Function Decomposition in Main and Lesser Peaks

Keywords: Mathematical morphology, Extinction values, Tree representation, Dynamics, Gray-scale thinnings

This article shows how the dynamics extinction value can be used to compute the decomposition of a function as a sum of simpler components. We show that this decomposition induces a hierarchical segmentation of the domain of definition, and a new partial ordering on nonnegative functions. Removing some of the components according to different criteria leads to new morphological operators. Their properties are discussed and illustrated in the last section. In particular, we see that thresholding on the supports' areas simplifies textured zones, while retaining perceptually salient elements of the image.

Introduction

Since their introduction by Vachier and Meyer [START_REF] Vachier | Extinction value: a new measurement of persistence[END_REF], extinction values have been used as a measure of importance for the extrema of a signal [START_REF] Angulo | Automatic analysis of DNA microarray images using mathematical morphology[END_REF][START_REF] Meyer | On the regularization of the watershed transform[END_REF]. When introduced, they were defined with respect to a granulometry; the extinction value of a maximum is the parameter of the smallest opening to completely erase it. The concept was then generalized [START_REF] Vachier | Extraction de caractéristiques, segmentation d'image et morphologie mathématique[END_REF]; for instance, volumic extinction values can be defined, although there is no corresponding granulometry.

While extinction values are typically used to select relevant markers, prior to watershed segmentation, this article introduces a new function decomposition associated to extinction values. In this article, we focus on a specific one, dynamics [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF], but the idea can be easily extended to other ones.

The input signal can be written as a sum of simpler components that we will denote as main and secondary peaks, following a mountain climbing analogy. Moreover, the inclusions of the peaks' supports define a forest structure. This representation is different from state-of-the-art techniques such as the max-tree [START_REF] Salembier | Antiextensive connected operators for image and sequence processing[END_REF] or the tree of shapes [START_REF] Ballester | The tree of shapes of an image[END_REF].

Peaks are then attributed various measures of importance; removing or altering the least important peaks (in the sense of those criteria) leads to new operators, some of which we detail in this work. These operators differ from classic attribute openings or thinnings [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF]. In particular, they do not obey the threshold decomposition principle [START_REF] Henk | Theoretical aspects of gray-level morphology[END_REF]: we extend binary connected operators to gray-level operators by considering the supports of our peaks, instead of considering threshold sets.

All of the methodology presented in this article is applicable on signals defined on any finite simple graph; however, in the context of image processing, signals are usually defined on a regular grid in a two-or three-dimensional space. The decomposition that we introduce and all related operators have the additional attractive property of being invariant by isometries (rotations, translations, symmetries and combinations thereof).

Notations

In the following, E denotes a finite, connected, simple graph. In most image processing applications, typically E would be a subset of Z 2 with a rectangular or hexagonal grid. For 3-D images, it could as well be a subset of Z 3 with any arbitrary connectivity.

For a node x ∈ E, N(x) denotes the neighborhood of x, i.e. all nodes y ∈ E that share an edge with x.

The support of a function f is the set of vertices on which it takes non-zero values:

supp( f ) = {x ∈ E| f (x) = 0}.

Decomposition in Main and Lesser Peaks

In this section, we introduce a new function decomposition, that we call the decomposition in main and lesser peaks, by two different, equivalent ways of computing it. The first approach is based on successive geodesic reconstructions [START_REF] Beucher | Segmentation d'Images et Morphologie Mathématique[END_REF] from the global maxima of a signal -as this transformation in a way 'ignores' maxima of lesser values, which is in our case the desired behavior. The second approach is much more closely related to the theory of extinction values [START_REF] Vachier | Extinction value: a new measurement of persistence[END_REF][START_REF] Vachier | Extraction de caractéristiques, segmentation d'image et morphologie mathématique[END_REF], and although we focus here solely on dynamics [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF], it can easily be extended to other extinction values. We define Γ ( f ) as the geodesic reconstruction of f with markers X M .

Decomposition via Successive Geodesic Reconstructions

Let C 1 , C 2 , ... C K be the K connected components of supp(Γ ( f )), and for

1 k K, let f k M be the restriction of Γ ( f ) on C k : f k M = Γ ( f ) × 1 C k In particular, Γ ( f ) = ∑ K k=1 f k M .
If f is identically zero, we take Γ (0) = 0 and the above summation is empty. If f is not identically zero, we call the f k M s the main peaks of function f .

Note that by construction, Γ is an anti-extensive operator: Γ ( f ) f , so the residue f -Γ ( f ) is a nonnegative function, to which we can apply operator Γ again. By iterating the process until the residue is zero, we obtain a first way of computing our decomposition (see algorithm 1).

Algorithm 1 Reconstruction-based algorithm Given a function f ∈ Fun(E, R + ): initialize with i = 0, R 0 = f . repeat Let d i = max(R i )
Compute the main peaks of R i :

Γ (R i ) = K i ∑ k=1 f k d i R i+1 ← R i -Γ (R i ) i ← i + 1 until R i = 0
Let I be the greatest integer such that R I = 0; the previous algorithm yields our decomposition:

f = I ∑ i=0 K i ∑ k=1 f k d i
where, by construction, d 0 = M = max f . Definition 2. We call the f k d 0 the main peaks of f , while for i 1, we call the f k d i lesser peaks of f . It is easy to show that the previous algorithm converges in a finite number of steps: keeping the same notations as above, we have:

∀x ∈ X M , f (x) = Γ ( f )(x) = M (1)
If f = 0, X M = / 0 and the support of f -Γ ( f ) is strictly included in the support of f . Since we chose E to be a finite graph, the number of steps required for the algorithm to converge is upper-bounded by the cardinal of E.

Equation 1 also implies that the sequence

(d i ) 1 i I is strictly decreasing: d 0 > d 1 > . . . > d I .

Decomposition via Razings

The second way of computing the decomposition is quite similar to the watershed algorithm by flooding [START_REF] Beucher | The morphological approach to segmentation: the watershed transformation[END_REF], except we raze the relief instead of flooding it. The idea is that we start from the regional maxima and gradually expand labels by following descending edges. When two or more regions meet, we stop propagating the labels that started from the lowest maxima, subtract the razed peak from the original function and propagate the remaining label. If two or more regions meet that started at the samehighest -altitude, we merge the regions and replace their labels by a single one. This way of handling equality cases can lead to 'twin peaks', like the green component in figure 1, and it is debatable; many image processing algorithms would arbitrarily break such ties, considering for instance the lexicographical order on the coordinates when working on a rectangular subset of Z 2 , or the area extinction value, as was proposed by Grimaud [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF]. The former approach is not invariant by rotation (swapping the coordinates will break ties in a different way) and is not applicable to the general case when the domain of definition is a finite simple graph; the latter is more attractive, but does not handle the case where the area extinction values are equal, too. Our way of handling ties ensures invariance by rotation when working on a regular grid, is well-defined, and as we shall see, yields interesting properties, in particular regarding the supports of the peaks.

Let us consider again a function f ∈ Fun(E, R + ). If f = 0 its decomposition in peaks is just an empty summation; in the following, let us assume f is not identically zero and that its support is connected (if it is not, we simply apply the following procedure on each connected component).

Let us consider the set X RM of all regional maxima of f . A regional maximum of f is a connected set X ⊂ E such that f takes the same value on each point of X and stricly lower values on neighboring points of X.

Let X 1 RM , . . . X K RM be the K connected components of X RM , and let L = {l 1 , ..., l L } = f (E) \ {0} the set of values ('levels') taken by f on its support, sorted in increasing order: 0 < l 1 < l 2 < . . . < l L .

The algorithm then goes as follows: we start by defining K functions f 1 , . . . , f K which are simply the restrictions of f on the connected components X 1 RM , . . . , X K RM :

∀k ∈ {1, . . . , K}, f k = f × 1 X k RM
We will then consider the connected components of the upper level sets of f , and progressively assign new points to one of the f k s until we have f = ∑ K k=1 f k . Let us denote by F l = {x ∈ E| f (x) l} the upper set of function f at level l; for each i from L to 1, for each connected component C of F l i , let us consider all k's such that supp( f k ) ⊂ C (there is at least one such k).

There are two cases to consider: if only one of the f k has a maximum strictly greater than all others (max( f k ) > max k =k max( f k )), we consider it a potential main peak and assign to the others, which are necessarily secondary peaks, their final values:

∀k = k : f k ← f k -l i Finalize f k f ← f -∑ k =k f k f k ← f × 1 C
If there is only one f k whose support is a subset of C, obviously we would just assign it the value f × 1 C . Else, if there is a tie between two or more peaks (max(

f k 1 ) = max( f k 2 ) = . . . = max( f k m ) = max k max( f k ))
, we first merge all these higher peaks together:

f k 1 ← f k 1 ∨ f k 2 ∨ . . . ∨ f k m f k 2 ← 0 f k 3 ← 0 . . . f k m ← 0 then as before: ∀k / ∈ {k 1 , k 2 , . . . , k m } : f k ← f k -l i Finalize f k f ← f -∑ k =k 1 f k f k 1 ← f × 1 C
In the previous instructions, 'finalize' f k means that when a peak f k is overtaken by a higher peak f k at level l i , we subtract l i from it at this iteration, but then we will not change its value at any subsequent iteration.

After the last iteration, we have f = ∑ K k=1 f k ; this does not mean there are necessarily K peaks, as some terms of the summation can be zero. After removing those, regrouping the terms with the same maximum value d i and reindexing, we can rewrite the summation as

f = I ∑ i=0 K i ∑ k=1 f k d i
as we had before. When we are not concerned with the values d i , we may skip the double summation and write f = J ∑ j=1 f j for ease of reading.

Note that although this algorithm is mathematically valid for any finite simple graph E and for any nonnegative real-valued function f , it can be more easily rewritten in the more restrictive -but more often encountered -case where E is a regular grid and f takes values in a finite set, typically {0, . . . , 255} for 8-bit images. Efficient algorithms exist for simulating floodings or razings, based on hierarchical queues [START_REF] Beucher | Segmentation d'Images et Morphologie Mathématique[END_REF]. Although this paper does not aim at providing a thorough algorithmic study, this remark means that computing the decomposition using this algorithm is roughly as time-expensive as computing the watershed transformation of an image.

Proposition 1 Nature of the peaks

Every peak p obtained by our decomposition (p = f k d i for some i and k) satisfies the properties that:

1. supp(p) is connected 2. Γ (p) = p
Conversely, each function q satisfying these properties is by itself its own peak decomposition.

Proof. The first point is obvious by construction of the second algorithm. The fact that Γ (p) = p comes from the fact that a function f k obtained by the second algorithm cannot have local maxima that are not global maxima, else they would have been assigned another component f k .

The reciprocal sense is basically the definition given in the first algorithm.

We denote by P the set of functions from E to IR + satisfying those properties. In particular, peaks have no local maxima that are not global maxima.

A function c : P → {0, 1} will be referred to as an intrinsic criterion. In particular, any binary criterion [START_REF] Breen | Attribute openings, thinnings, and granulometries[END_REF] T (defined not on the space of functions but on the space of subsets of E) can be associated a criterion c T on P by considering the support of a peak p: c T (p) = T (supp(p)).

Proposition 2 If x m is the index of a local maximum of f with dynamics d, then x m is the index of a global maximum of f k d for some k, and f k d (x m ) = d.

The proof can be found in the original article defining the notion of dynamics [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF], as the original proposed algorithm is similar to the second algorithm we propose; the originality of ours lies in the decomposition it computes along with the dynamics.

A noticeable difference is how equality cases are handled: in [START_REF] Grimaud | New measure of contrast: the dynamics[END_REF], the author considers that two close maxima of the same value being assigned the same dynamics is undesirable behavior, since dynamics are often used as a way to select markers prior to segmentation, typically using the watershed algorithm. In this context, selecting many similar close extrema (usually minima, actually) is indeed undesirable behavior.

Since we chose not to break ties, our definition of the dynamics of a maximum is the one usually found in the literature: it is the difference in altitude between this maximum and the maximal lowest altitude of a point on a path joining it to another, strictly higher, maximum.

The dynamics of the global maximum (or global maxima) of a function f is illdefined; the most common convention is to assign it the value max( f )min( f ); other authors assign it an infinite value; if dynamics are to be used for ranking extrema by order of importance, the essential point is just to assign it a higher value than all local ones. In order to be consistent with the definitions of our algorithms, we will adopt the convention that the dynamics of the global maximum is simply its value. This is consistent with the fact that the value 0 plays a specific role in our decomposition. For instance, if the support of f is not connected, f and f +λ , where λ > 0 may not have the same main peaks ( f may have several, while f + λ necessarily has only one). Following our mountain-climbing analogy, the value 0 can be thought of as the sea level; even if the support of f is equal to the entire domain E, meaning that for all x in E, f (x) > 0, our reference level is not the minimum of f , but 0. Proposition 3 For any two components f k d i and f k d j , supp( f k d i ) and supp( f k d j ) are either nested or disjoint.

In particular, if supp( f ) is connected, then the number of main peaks K 0 = 1, Γ ( f ) = f d 0 and:

∀i > 1, ∀k ∈ {1, . . . , K i } supp( f k d i ) ⊂ supp( f d 0 )
This can be proved by induction; consider the razing-based algorithm: at the beginning, all supports are disjoint. Supports can only increase with iterations, and when peaks meet at level l i , they are either merged together (if there is a tie) or some are finalized and the greater is assigned the value l i > 0 on the supports of the other.

Proposition 4 For any two components f d i and f d j :

supp( f d i ) ⊂ supp( f d j ) ⇒ d i < d j

Proposition 5 Stability by truncation

Let us consider a function f ∈ Fun(E, IR + ) and its decomposition in main and lesser peaks f

= ∑ I i=0 ∑ K i k=1 f k d i . Let us now consider a function g = ∑ I i=0 ∑ K i k=1 δ k i f k d i , where δ k i ∈ {0, 1}, that is, g is a truncated summation of the components of f .
The peaks of g, as defined in the first section, are simply the peaks of f that appear in the summation; the decomposition of g can be written:

g = I ∑ i=0 ∑ 1 k K i δ k i =1 f k d i
In section 4, we present several operators based on the idea of keeping only certain peaks, depending on certain criteria. The previous proposition asserts that, provided the criteria only depend on intrinsic properties (that is, the choice of δ k i only depends on f k d i ), these operators are idempotent.

Induced Tree Structure

Definition 3. If supp( f ) is connected, then supp( f 1 ) = supp( f ), so proposition 3 naturally defines a tree structure on the components. Let f d j be a child node of f d i (we drop the superscripts for ease of reading) if:

-supp( f d j ) ⊂ supp( f d i ) -There is no k such that supp( f d j ) ⊂ supp( f d k ) ⊂ supp( f d i )
If supp( f ) is not connected, the previous relationship yields a forest structure, with a tree for each connected component.

Proposition 6 Stability by anamorphosis

The forest decomposition is stable by any nonnegative anamorphosis that goes through the origin; if u is an increasing, invertible function from IR + to IR + with u(0) = 0, then f and u( f ) yield the same forest structure.

Ordering Induced by the Decomposition

Let us consider f and g two functions from E into IR + , and their decompositions in main and lesser peaks, with the peaks labeled respectively from 1 to J f and from 1 to J g :

f = J f ∑ j=1 f j , g = J g ∑ j=1 g j
We write f g if there exists α : {1, . . . , J f } → {1, . . . , J g } such that:

∀ j ∈ {1, . . . , J g }, ∑ i|α(i)= j f i g j (2)
It can be easily seen that f g ⇒ f g, but the reverse is not true in general.

Unfortunately, the couple (Fun(E, IR + ), ) is not a lattice, nor even a semi-lattice: neither the infimum nor supremum of two functions are guaranteed to exist.

4 Morphological Operators on (Fun(E, IR + ), )

In this section, we present some operators based on intrinsic criteria. All these operators are anti-extensive and idempotent; however, they lack the property of being increasing when considering the usual ordering on functions, so they are not morphological filters on the lattice (Fun(E, IR + ), ) (they are only thinnings).

They are increasing, though, if considering the order introduced in the previous section. It must be kept in mind that since (Fun(E, IR + ), ) is not a lattice, they stand outside the usual framework of mathematical morphology, but they can be thought of as morphological filters on a poset.

Definition 4. General form for criterion-defined operators

Given a function f , its peak decomposition

f = I ∑ i=0 K i ∑ k=1 f k d i
and an intrinsic criterion c, we define:

γ c : Fun(E, IR + ) → Fun(E, IR + ) f → ∑ I i=0 ∑ K i k=1 c( f k d i ) f k d i
It is straightforward that any such operator is anti-extensive and idempotent; it is not, in general, increasing, be it for the usual order or the peak-induced order ; we must add the constraint that the intrinsic criterion c is itself increasing (for either order, as they are the same on P). Fortunately, this requirement is easily met.

Definition 5. Peak dynamics thresholding

This tranformation has a parameter δ , which is the minimum dynamics of peaks to be kept; the intrinsic criterion is a threshold above this value. Formally:

c δ : P → {0, 1} p → 1 if max(p) δ 0 else Definition 6.
Peak area thresholding This tranformation has a parameter A, which is the minimum area (cardinal of the support) of peaks to be kept; the intrinsic criterion is simply a threshold above this area. Formally:

c A : P → {0, 1} p → 1 if # supp(p) A 0 else Definition 7.
Peak volume thresholding This time, the parameter V is the minimal volume (sum of the values) of peaks to be kept; the intrinsic criterion is again a simple threshold above this parameter. Formally:

c A : P → {0, 1} p → 1 if ∑ x∈E p(x) V 0 else
These operators are illustrated in Fig. 4, in one dimension. Figure 5 illustrates peak area thresholding in two dimensions, and compares it to area opening, as defined by Vincent [START_REF] Vincent | Morphological area openings and closings for grey-scale images[END_REF]. Gray-scale area opening razes all maxima until they become large enough flat zones, reducing the overall contrast of the image. Peak area thresholding, on the other hand, does not affect high or isolated maxima, thus preserving the most salient bright structures.

Conclusions and perspectives

In this article, we presented the methodology and mathematical foundations for obtaining a function decomposition based on a particular extinction value: dynamics. Instead of assigning a scalar value to each maximum, we assign it a function, that we call a peak, associated to this maximum. Those peaks can in turn be assigned different measures: maximum value, area, volume; binary criteria can also be computed on their supports. We exhibit some properties of our decomposition, and present a new order on functions induced by it.

Although we focused here on a dynamics-driven decomposition, similar decompositions can as easily be obtained by considering other extinction values in the definition of section 3.2. Area or volume extinction values will lead to different decompositions, and may be more sensible than dynamics, depending on the application and on the expected type of noise. Comparison between classical area opening and peak area thresholding; the original image is shown on the top left, the second column shows the result of a gray-scale area opening of size 350 and its residue; the rightmost column shows the result of a peak area thresholding of size 350 and its residue (the contrast of the residues has been enhanced for better visualization). We can see that peak area thresholding flattens textured zones, such as the hair, forehead and right cheek, but does not alter the most salient ones, such as the right eye or the reflection on the hat.

Fig. 1 .and f 2 9 .

 19 Fig. 1. Example of the elementary operation: the maximum of f is M = 9, reached at X M = {5, 24, 28}. There are two connected components in the support of the reconstruction Γ ( f ), so we obtain two functions f 1 9 and f 2 9 . One is shown in blue, the other in green.

Figure 1

 1 Figure 1 illustrates operator Γ and the notion of main peaks.It is easy to show that the previous algorithm converges in a finite number of steps: keeping the same notations as above, we have:

Fig. 2 .

 2 Fig. 2. Example of the decomposition

1 Fig. 3 . 1 2 and f 2 2

 1312 Fig. 3. Example of the decomposition and its associated tree: we have f 7 in blue, f 4 in yellow, f 1 2

Fig. 4 .

 4 Fig. 4. From top to bottom: original signal, peak dynamics thresholding with δ = 3, peak area thresholding with A = 3, peak volume thresholding with V = 15.

Fig. 5 .

 5 Fig.5. Comparison between classical area opening and peak area thresholding; the original image is shown on the top left, the second column shows the result of a gray-scale area opening of size 350 and its residue; the rightmost column shows the result of a peak area thresholding of size 350 and its residue (the contrast of the residues has been enhanced for better visualization). We can see that peak area thresholding flattens textured zones, such as the hair, forehead and right cheek, but does not alter the most salient ones, such as the right eye or the reflection on the hat.

In order to introduce the concepts, we considered nonnegative functions and the value zero as our reference level; however, it is easily generalized to real-valued functions by considering the positive and negative part and applying the decomposition to each, yielding self-dual operators.


Benoît Lé

Grégory Legrain

Nicolas Moës

Finite Element Analysis and Design Mixed dimensional modeling of reinforced structures

Keywords: Reinforced structures, Reinforcements, Extended Finite Element Method, Bar elements, Mixed dimensional modeling

Using thin reinforcements is a common way to strengthen structures, as in reinforced concrete for example. From a numerical point of view, dealing with these reinforcements is tedious, because of their diameter which is usually small compared to the characteristic dimensions of the structures, therefore requiring very fine meshes to represent them accurately. In this paper, a new approach allowing to mix a volumic and a lineic modeling of the reinforcements is proposed. Fine meshes with a volumic representation of the reinforcements are used in the zones of interest of the structure, whereas coarser meshes associated to lineic elements are used in the rest of the structure in order to decrease the computing times. A methodology to ensure the transition between both modeling is proposed, so that the results in the zones of interest are similar to the results that would be obtained with a full volumic representation of the reinforcements. The efficiency of the method is illustrated on several examples, involving linear elasticity and plasticity.

Introduction

Including "almost-1D" reinforcements, that is to say elements with one dimension much greater than the two others, is a common way to strengthen structures. A well-known example is reinforced concrete structures, where steel reinforcements are used in the zones where tensile stresses prevail, to make up for the low tensile strength of concrete. Numerical simulation of such structures is quite challenging from a geometrical point of view. Indeed, the diameter of the reinforcements is usually small compared to the characteristic dimensions of the structures (at least one order of magnitude smaller in the example of reinforced concrete structures), requiring very fine finite element meshes to represent them accurately. As a result, the necessary CPU resources and computation times can become very high.

In order to take into account these reinforcements with reasonable computing costs, different approaches were developed in the literature. The smeared model [START_REF] Suidan | Finite Element Analysis of Reinforced Concrete[END_REF] consists in adding the stiffness of the reinforcements to the volumic elements containing them, introducing orthotropy in the reinforcements direction. This approach is well suited for structures were the reinforcements are perfectly bonded and are arranged in a regular pattern. More recently, David [START_REF] David | Homogenized interface model describing inhomogeneities located on a surface[END_REF] developed a membrane model for regularly spaced reinforcements, based on asymptotic expansions, which is more accurate and allows to take into account loss of bond. In the discrete model, 1D bar elements are added along the edges of the volumic elements. It is more flexible than the smeared approach, since the layout of the reinforcements does not need to be regular anymore, but their paths still need to follow the nodes of the volumic mesh. Finally, the embedded approach [START_REF] Hartl | The 3D Computational Modeling of Reinforced and Prestressed Concrete Structures[END_REF][START_REF] Gomes | Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the finite element method[END_REF] allows any reinforcements layout, independently of the volumic mesh.

These approaches give good global results, but are not designed to get accurate local results around the reinforcements, as shown in [START_REF] Jason | Truss vs solid modeling of tendons in prestressed concrete structures: Consequences on mechanical capacity of a Representative Structural Volume[END_REF] for instance. To sum up, there are two possibilities to model the reinforcements: using a 3D volumic mesh, which would give the most accurate results but is incompatible with industrial studies because of the too large computing costs, or using a smeared or a 1D representation of the reinforcements, decreasing the computing costs but leading to inaccurate local results. The approach proposed in this paper rests upon the idea that, when performing the finite element analysis of any structure, it is often possible to identify zones of interest, that is to say, parts of the structures where accurate results are wanted (because of stress concentrations, or because these are critical parts for which one wants to know how it will deteriorate, etc). Usually, fine meshes are used in such zones, whereas coarser meshes are used in the rest of the structure to decrease computing costs. Now, applying this idea to reinforced structures, we propose in this paper a method that allows to use a volumic representation of the reinforcements in the zone of interest, and a 1D representation in the rest of the structure. The volumic representation in the zone of interest will allow to get results as accurate as possible, especially close to the reinforcements, whereas the 1D representation will allow to use coarser meshes away from the zone of interest, therefore reducing computing times. The transition between both representations will ensure that the results in the fine zone will be close to the results that would have been obtained with a full volumic representation of the reinforcements in the whole structure.

Combining 1D/2D and 3D models in the same computation has already been done, as in [START_REF] Monaghan | Coupling 1D Beams to 3D Bodies[END_REF][START_REF] Mccune | Mixed-dimensional coupling in finite element models[END_REF] where coupling 1D beams or 2D plates to 3D bodies was achieved by equating the work at the interface between both representation, resulting in constraint equations between 1D/2D and 3D degrees of freedom. In [START_REF] Nguyen | Nitsche's method method for mixed dimensional analysis: conforming and non-conforming continuum-beam and continuumplate coupling[END_REF], Nitsche's method [START_REF] Hansbo | Nitsche's method for interface problems in computational mechanics[END_REF] is applied to the beam/solid and plate/solid coupling. However, the main difficulty with the existing methods is the need to ensure the compatibility between the beam/plate particular kinematic and the volumic one. In this paper, we will consider structures where the reinforcements have a small enough diameter so that we can assume that they behave like bar element, working in tension/compression. However the proposed approach would still be useful if the bending energy was taken into account [START_REF] Sadek | A three dimensional embedded beam element for reinforced geomaterials[END_REF].

This paper is organized as follow: sections 2 and 3 introduce the numerical tools involved in the proposed method, as well as their limitations. In section 4, the solution to combine these tools in order to answer the issues raised above is explained. Some results to illustrate this method are shown in section 5. Finally conclusions are drawn in section 6.

Volumic model

The most direct approach to model reinforcements would be to mesh them using volumic finite elements. This process may be complicated, considering their small diameter, their number, and their path which may be complicated (intersecting or tangent reinforcements, curved paths, etc). However, the eXtended Finite Element Method (X-FEM) can be used, as done in the work of [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF]. This method was first introduced by Moës et al. [START_REF] Moës | A finite element method for crack growth without remeshing[END_REF] to deal with crack propagation, with meshes independent of the crack path. Based on the concept of Partition of Unity [START_REF] Babuška | The partition of unity method[END_REF], it relies on the use of enrichment functions to introduce discontinuities into the classical finite elements, as well as level-set functions to locate these discontinuities. The enrichment strategy proposed in [START_REF] Moës | A computational approach to handle complex microstructure geometries[END_REF], dedicated to the analysis of inclusions, is used in this paper for the volumic part of the reinforcements, and will be reminded below.

The eXtended Finite Element Method (X-FEM)

Consider the bidimensional problem depicted in figure 1 a., a circular inclusion made of a material A, in a rectangular plate made of a material B. The plate is meshed with elements whose nodes do not coincide with the material interface. The set I denotes the nodes of the mesh, whereas J is the set of the nodes which need to be enriched. J is defined as the set of nodes belonging to the elements crossed by the material interface (cf. figure 1 a). The X-FEM approximation consists in finding a displacement solution of the form: a. b.

Figure 1: Representation of a circular inclusion in a square plate, using the X-FEM method. a. Definition of the enriched nodes. b. Localization of the interface using a level-set function

u h (x) = i∈I N i (x)u i + j∈J N j (x)F (x)a j u i , a j ∈ R 2 (1)
Where (N k ) k=i,j are the classical finite element approximation functions of node k, u i are the classical finite element degrees of freedom, a i are the enriched degrees of freedom and F is the enrichment function. In this paper we consider that the two materials are perfectly bonded: therefore, u h must be continuous over the domain. However, because of the change of material properties, the strain will be discontinuous at the interface. The enrichment function F is thus chosen continuous, but with a discontinuous derivative.

To define F , the position of the interface must be determined. A level-set function φ is introduced to locate the interface Γ between material A and B so that:

Γ = {x ∈ R 2 : φ(x) = 0} (2)
φ(x) is chosen to be positive if x is outside Γ, negative if x is inside Γ and equals zero if x is on Γ. An example of level-set defining the inclusion of figure 1 a. is given in figure 1 b. The main example of level-set function is the signed distance function to the interface:

φ(x) = ± min x Γ ∈Γ x -x Γ (3)
In the particular case of a cylindrical reinforcement, we use the following level-set function: where r is the radius of the reinforcement and Λ is its center-line (the 1D curve defining the reinforcement path through its center). From a numerical point of view, φ is discretized using the linear finite element shape functions N i :

φ(x) = min x Λ ∈Λ x -x Λ -r (4)
φ(x) = i∈I N i (x)φ i (5)
where the φ i are the nodal values of the level-set function. Now that the position of the interface is known, the enrichment function can be defined. We consider the ridge function from [START_REF] Moës | A computational approach to handle complex microstructure geometries[END_REF]:

F (x) = i∈I N i (x)|φ i | - i∈I N i (x)φ i (6)
F is shown in 1D on figure 2. Some authors also proposed to treat material interfaces at the integration point level [START_REF] Zohdi | A description of macroscopic damage through microstructural relaxation[END_REF]. However, as illustrated in appendix 7, this approach has a lower convergence rate than the X-FEM. On the contrary, for a given mesh density, the X-FEM and conforming FEM lead to similar results without meshing burden, which motivates the use of the X-FEM here.

Numerical aspects

A first remark about the use of level-sets to implicitly represent interfaces is the fact that they are linearly approximated (cf. figure 3) ; the value of φ i is known at each node i of the mesh. If node i is inside Γ, φ i < 0, if node i is outside Γ, φ i > 0. The intersection of Γ with the edges of mesh can be easily computed using the linear interpolation on the triangular element (cf. equation ( 5)). This implies that the quality of the geometries discretized using level-set functions will depend on their curvature (the less the curvature, the better the approximation).

The second remark is about numerical integration. Since the derivative of u h is now discontinuous inside the elements crossed by Γ, classical Gauss integration rules are not accurate anymore. The elements which are crossed by Γ are divided into conforming subcells (cf. figure 3), then Gauss integration is performed using the Gauss integration Material A Material B a.

b.

Figure 3: Representation of an interface over a mesh using level-set functions. Black thick lines: mesh edges. a. Blue dashed line: real interface b. Green triangle nodes: φ(x) < 0. Red square nodes: φ(x) > 0. Blue dots: φ(x) = 0. Green dashed line: approximate geometry represented with the level-set. Each triangle appearing in b. will be used for the numerical integration points of each subcell. Note that subcells do not bring any new degrees of freedom. The only purpose is to reach a proper level of accuracy for the integration with a very limited number of integration points.

Limitations of the 3D approach

Now we consider the 2D case of a rectangular plate, with a unique reinforcement (cf. figure 4). Considering that it is perfectly linked to the plate material, it can be assimilated to an inclusion and modeled using the X-FEM method, as presented above. The level-set function is the one defined by equation [START_REF] Gomes | Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the finite element method[END_REF]. Following the idea presented in the introduction, two meshes are considered: a fine mesh, and a second one where the right part was derefined (assuming that it is out of the zone of interest). The approximated geometry obtained by the level-set is given for both meshes in figure 4. With the fine mesh, the reinforcement is correctly represented (in 2D it is a flatten rectangle with width 2r). On the other hand, the reinforcement approximated on the second mesh shows strong geometrical errors, almost disappearing where the mesh is really coarse. The corresponding finite element computation will obviously give wrong results, both locally and globally. These matter islands are due to the nodes whose support (set of elements connected to a node) is cut twice by Γ, as illustrated in figure 5. The support of A is cut twice by Γ. Let us have a closer look at the intersection between Γ and edges AB and AC of the mesh. Because of equation ( 4), the value of φ A is equal to the distance from A to the lower part of Γ. The intersection between Γ and AB is correctly computed, but because the value of φ A underestimates the distance from A to the upper part of Γ, the intersection between Γ and AC is closer to A than it should be. It is important to emphasize that this definition ensure that if there are some errors in the approximation of Γ, it will always result in local underestimation of the volume of the reinforcement (some "matter" can be removed, but not added). Finally, the elements where the reinforcement seems to disappear are due do the fact that there is no node inside Γ, therefore the sign of φ does not change, making it unable to locate the inner part of the reinforcement.

Linking truss and 3D model

Another way to model reinforcements without using a costly 3D representation is to use bar elements. The reinforcements are meshed with lineic element, whose nodes are not necessarily matching the nodes of the volumic mesh, cf. figure 6. A reinforcement node a is inside a cubic volumic element whose nodes are called (b i ) i=1,...,8 . The position of a inside the volumic element is x a . Considering perfect bond between both materials, the reinforcement degrees of freedom are linked to the volumic degrees of freedom according to equation [START_REF] Mccune | Mixed-dimensional coupling in finite element models[END_REF]:

u a = 8 i=1 N i (x a )u bi (7)
In section 4.1 we will see that bar elements will be used with X-FEM in the same computation. Therefore, in the case where a bar node is located inside an enriched volumic element, (7) must be modified in order to take into account the X-FEM enrichment (1):

u a = 8 i=1 (N i (x a )u bi + N i (x a )F (x a )a bi ) (8)
The bar degrees of freedom are eliminated from the calculation through equation ( 7) (or equation ( 8) if bar elements are used with the X-FEM enrichment), which is somehow equivalent to considering that the stiffness of the bar elements is added to the stiffness of the surrounding volumic elements (green elements on figure 7 b.). Besides allowing to use coarser meshes around the reinforcements, using bar elements simplifies the meshing procedure since the volumic and lineic meshes can be independent. This method allows to get good global results (force-displacement curve for instance), but is obviously not as accurate as a volumic representation of the reinforcements as shown by [START_REF] Jason | Truss vs solid modeling of tendons in prestressed concrete structures: Consequences on mechanical capacity of a Representative Structural Volume[END_REF]. Indeed, two drawbacks can be underlined:

• The overall stiffness of the structure is overestimated. Let us consider the example of figure 7 a. A reinforcement, denoted Ω inclusion , is embedded in a matrix called Ω matrix . The structure is meshed using coarse 2D elements, not taking into account the interface between Ω inclusion and Ω matrix (cf. figure 7 b.). These 2D elements have the material properties of the matrix material, and the reinforcement is introduced with 1D bar elements having the material properties of the inclusion material. The discretized stiffness of the geometrical domain Ω inclusion is the sum of two terms: obviously, the stiffness of the reinforcements coming from the 1D elements, and a second term which corresponds to the part of the 2D finite elements inside Ω inclusion , but have the material properties of the matrix (gray part in figure 7 b.). Therefore the stiffness of Ω inclusion is overestimated. However, the reinforcements are usually stiffer than the matrix and their diameter is small, so the influence of this extra stiffness should remain small in most of the cases.

• One can see from figure 7 b. that if the size of the 2D elements around the bar elements becomes small, the stiffness of the reinforcements is distributed over a band of vanishing thickness, which may lead to non-physical stress concentrations (see section 5.2.2).

It is also possible to consider that the link between bar and volumic elements is not perfect, by allowing a slip between the bar nodes and the volumic ones [START_REF] Casanova | Bond slip model for the simulation of reinforced concrete structures[END_REF], but in this paper we will consider only perfect bond. 4 Mixing lineic and volumic models

Correction scheme

It was shown in section 2 that X-FEM can be used to obtain a volumic representation of the reinforcements with meshes which do not need to coincide with their boundaries. However, if the mesh is too coarse, some geometrical errors can occur. The solution proposed in this section is to keep the geometry of the reinforcements approximated by X-FEM, which may be accurate in the zone of interest and potentially inaccurate where the mesh is coarse, and to use bar elements as presented in section 3 at the same time. Let us consider the example depicted in figure 8. A reinforcement is modeled using both X-FEM and bar elements, placed along the reinforcement path. In some part of the structure, the mesh is coarse enough so that at least one node has its support cut twice by Γ. The volumic part approximated with X-FEM and the lineic part are both assembled, but the stiffness of each bar element is weighted by a coefficient α e . Considering that each bar element delimitates a segment of the reinforcement (red dashed line in figure 8) ; the real volume of this segment can be computed as:

Ω real e = Sl e ( 9
)
where S is the section of the reinforcement an l e the length of the bar element. Then α e is computed by:

α e = 1 - Ω num e Ω real e (10)
where Ω num e is the volume of the segment of the reinforcement that could be approximated using X-FEM (darken color part inside the red dashed line rectangle in figure 8). Ω num e is computed as follow:

1. The volumic element around each bar element are divided into subelements, using the level set φ, according to section 2.2 To sum up, the bar elements can be seen as a correction to the X-FEM approximation, through the use of coefficient α e computed by (10):

• α e = 0 if Ω num e = Ω real e : the segment of reinforcement is perfectly represented by X-FEM, the bar element is not taken into account, no correction is needed

• α e = 1 if Ω num e = 0: the mesh is very coarse, no node is inside the reinforcement, therefore it "disappears": a classical bar element is assembled

• α e ∈]0; 1[ if 0 < Ω num e < Ω real
e : the reinforcement is only partially represented by the level-set, the bar element brings the missing stiffness as a correction Note that using these modified bar elements does not solve the local issues of X-FEM emphasized in section 2.3: in the coarse zone, the results around the reinforcements will still be inaccurate. However, from a global point of view, the stiffness of the structure in the coarse zone will be sufficiently correct so that the local results in the fine zone are almost identical to the results that would be obtained with a full volumic representation of the reinforcements in the whole structure. The extra stiffness around bar element e mentioned in section 3 is now equal to α e k matrix e , where k matrix e is the stiffness of Ω real e assuming that it is made only of the matrix material. Since α e is smaller than 1, the global stiffness of the structure should be better with the approach proposed in this section than with only 1D bar elements. This will be shown in numerical examples in section 5.

Flowchart

Here is the general procedure to perform the finite element computation of a reinforced structure in linear elasticity with perfect bond, using the method presented in section 4.1:

1. Make a volumic mesh of the structure 2. Make a 1D mesh of the reinforcements 3. Compute the value of the level-set φ using equation [START_REF] Gomes | Some aspects on three-dimensional numerical modelling of reinforced concrete structures using the finite element method[END_REF]. Note that the center-line of the reinforcement Λ can be replaced by a 1D mesh. In this case the value of the level-set at each node of the volumic mesh is computed by a direct distance computation from this node to the 1D mesh, and by subtracting the radius of the reinforcement. Provided that enough bar nodes are used, the discretization error remains low, without influencing the computation times 4. Assemble the stiffness corresponding to the volumic part, according to the procedure given in section 2. -100 6 -Table 1: Material properties

Numerical examples

The approach proposed in section 4 is applied to several examples. To illustrate the interest of the method, we consider structures with reinforcements stiffer than the surrounding matrix. Perfect bond is considered between both materials. In section 5.1 the matrix and the reinforcements are made of linear elastic materials. For the examples of section 5.2, non-linear test cases are considered. Softening models like damage mechanics would be a rather logical possibility as one of the main applications of the proposed method is reinforced concrete. However, these models need some regularization in order to avoid spurious mesh dependency [START_REF] Bažant | Instability, Ductility and Size-effect in Strain-Softening Concrete[END_REF], which is usually done by introducing non-locality in the problem. Taking this non-locality into account close to the reinforcements would require some developments which would be out of the scope of this paper. Therefore, an elastoplastic material with a simple isotropic hardening (J 2 -flow theory [START_REF] Simo | Computational Inelasticity[END_REF]) is considered for the matrix. This will not be representative of any particular kind of physical phenomenon, but will allow to get qualitative non-linear results. The hardening modulus is chosen negative (softening plasticity), to get a behaviour qualitatively similar to damage mechanics. The material properties of the matrix and the reinforcements are summarized in table 1,and the behaviour at Gauss points for the matrix material is shown is figure 9.

For each example, a zone of interest is identified. A mesh with elements fine enough to correctly represent the reinforcements is used in this zone. This mesh will be called the "computation mesh". The method presented in section 4 will be called the "mixed" method in what follows (mix between a volumic representation of the reinforcements with X-FEM and a 1D representation with bar elements). Unless otherwise specified, the obtained results are compared with two other methods:

• A computation performed with the same mesh, but using only truss elements in the whole structure. This method will be called the "bar" method.

• A computation performed with a fine mesh in the whole structure. The element size is the same as the the element size in the zone of interest of the computation mesh. This mesh will be called the "reference mesh". The reinforcements are represented using X-FEM. As the mesh is fine everywhere, the reinforcements are perfectly represented in 3D, therefore these results will be taken as references. This Figure 9: Behaviour at Gauss points of the matrix material method will be called the "volumic" method.

All the meshes used in this section are made of linear elements. For the computation mesh, the element size is 0.005 m in the zone of interest, and 0.05 m outside (cf. figure 11). For the reference mesh, the element size is 0.005 m in the whole Figure 11: Mesh used for the three points bending test in linear elasticity. In the zone of interest, the elements are geometrically superimposed for the computation mesh and the reference mesh.

Elastic test cases

beam. The width of the zone of interest is D = 0.06 m. Note that the element size does not seem really fine in the zone of interest compared to the reinforcements radius. Since in 2D, straight reinforcements are represented with flat rectangles, the mesh does not need to be really fine, which is not the case in 3D as illustrated in section 5.2.2. It just needs to ensure that no node has its support cut twice by the iso-zero of the level-set.

The mesh in the zone of interest is made of squares cut into triangles (see figure 11). This is to ensure that the elements in this zone are exactly superimposed for the computation and the reference meshes, something which is necessary for the comparison between the different computations (cf. equation ( 14)). For this computation the mesh was also refined around the boundary conditions, to account for the singularities resulting from the imposition of a displacement on points.

The vertical displacement v and the normal stress σ along the upper reinforcement are plotted in figure 14, as well as the relative error between the mixed/bar method and the volumic method. For a 1D bar element, the computation of the normal stress is reminded:

σ ij = E l e (u j -u i ).v (11)
where u i and u j are the displacement of the bar nodes. v the unitary vector orienting the bar element, going from node i to node j. The same expression is used for the volumic and mixed methods (in the case of the volumic method, bar elements are used in the computation but their stiffness is null, they are used only to compute [START_REF] Sukumar | Modeling holes and inclusions by level sets in the extended finite-element method[END_REF]). The relative error for the bar and the mixed method compared to the volumic method, are also computed :

η bar/mix v = |v vol i -v bar/mix i | |v vol i | (12)
for the displacement error, computed at each node i, where the subscripts bar/mix/vol refer to the bar, mixed and volumic method respectively, and Figure 12: Approximated geometry of the reinforcements for the three points bending test (zoom on the center of the beam).

Figure 13: Coefficient α e along the reinforcements for the three points bending test.

η bar/mix σ = |σ vol ij -σ bar/mix ij | |σ vol ij | (13)
for the stress error, computed at each bar element, defined by nodes i and j.

The three methods give similar results. The effect of the extra-stiffness mentioned in section 3 can be observed, with a vertical displacement with the bar method smaller than the one obtained with the mixed method, which illustrates the fact that with the mixed method, this extra-stiffness is lower than with a pure bar method.. The error compared to the volumic method are lower with the mixed method than with the bar method, even if they are acceptable in both cases (less than 10%). It can be noticed that in the fine zones around the boundary conditions, the bar method fails to capture the displacement and normal stress, which is not the case for the mixed method. The same conclusions can be drawn for the lower reinforcements.

To get local information about the mechanical field around the reinforcements in the zone of interest, the error for each element e per unit volume is also computed:

η bar/mix e = | Ωe (u bar/mix -u vol ) : E : (u bar/mix -u vol )| | Ωe (u vol ) : E : (u vol )| (14)
η bar/mix e is computed on the reference mesh, by projecting the stress and strain fields obtained with the bar and mixed methods. It must be noticed that this quantity does not really make sense to compare results computed on different meshes, since the stress and strain field are discontinuous at the boundaries between the finite elements. However for this test the meshes are exactly the same in the zone of interest (cf. figure 11), ensuring the relevance of ( 14) in this zone. The obtained errors are plotted in figure 15.

The most interesting observation is that the error in the zone of interest for the mixed method remains low (under 1.0%). For the bar method, it is obviously high around the truss elements, but it can also reach about 5.0% even near the center of the beam. Outside of the zone of interest, the error is lower in the center of the beam for the mixed method even if conclusions must be considered with care since the meshes are not the same; in particular, some numerical artefacts can be observed. In both cases, the errors are high around the reinforcements, which was to be expected since ( 14) is not computed with the same Hooke tensor, depending on how the reinforcement is represented. Promising conclusions can be drawn from this example: in the zone of interest, the results obtained with the mixed method and the volumic method are very close to each other. The importance of using a volumic representation of the reinforcement (at least in the zone of interest) compared to a 1D representation was also emphasized. As expected, in the rest of the structure, the results are not as accurate as results that would be obtained with a fine mesh in the whole structure, however it seems that some improvement can be observed compared to the bar method.

Bending of a 3D post

This example illustrates how the mixed method can handle many reinforcements with complex geometries. The mesh of the post, the 1D mesh and the approximated geometry of the reinforcements are shown in figure 16. It is reminded that the only necessary information for the representation of the reinforcements is a 1D mesh and their radius, which is particularly interesting in this test case with many reinforcements and complex paths. Indeed, representing several 3D reinforcements which may be tangent to each other as shown in figure 16 d. with a conforming mesh is very laborious. The lower surface at the base of the post is clamped, whereas the top of the post is subjected to a shearing load. The zone of interest is supposed to be at the base of the beam, where the stress is expected to be the highest. To simplify the computations, only the reinforcements crossing the zone of interest are modeled using the mixed method (that is to say, the vertical reinforcements and two horizontal frames). The other reinforcements are modeled by 1D truss elements. The results of this computation are still referred as the "mixed" method (even if it is not true for all the reinforcements). The computational cost for a volumic representation of the reinforcements would be too high, therefore there is no reference computation for this example. Von Mises stresses in the zone of interest, in several cross-sections of the post, are shown in figure 17. Far away from the reinforcements, no significant differences can be observed between the bar and the mixed method. However, with the mixed method, a slight decrease of the Von Mises stress can be observed around the reinforcements boundaries (green dashed frames in figure 17).

Plastic test cases

In this section the mixed method is illustrated on non-linear examples. The matrix is made of an elastoplastic material with isotropic hardening. To avoid spurious appearance of plasticity around the part of the reinforcements which are only partially represented by the level-set, the plasticity is restrained to the zone of interest whereas the rest of the structure is assumed linear elastic. Usually the reinforcements are stronger than the surrounding matrix and yield for much higher loading, in some applications they do not yield at all; therefore they will be considered linear elastic in what follows. Figure 18: Top: Mesh used for the three points bending test. Bottom: geometry of the reinforcements approximated with level-set functions.

Three points bending 2D plastic

The test case of section 5.1.1 is considered with an elastoplastic matrix. The width of the zone of interest is D = 0.3 m. The element size in the zone of interest is 0.005 m. Corresponding computation meshes are shown in figure 18. The geometry of the reinforcements approximated with level-set functions on both meshes is represented as well. The plastic strain at the end of each computation for the different meshes and methods are given in figure 19.

In each computation, the plastic strain starts to develop under the loading zone, then at the bottom of the beam. However, with the bar method, it tends to reach the middle of the beam, whereas for the volumic method, the plasticity is confined around the reinforcements. The volumic method and the volumic part of the reinforcements in the mixed method introduce some heterogeneities, and therefore a stress field different from the bar method, leading to these differences in the results. With the mixed method, the plastic strain in the zone of interest is similar to the one obtained with the volumic method.

3D Four points bending test

The next example is a 3D beam subjected to 4 points bending, with the following dimensions: length L = 5 m, height h = 1 m, thickness t = 0.5 m. It is reinforced with four reinforcements (cf. figure 20) of radius r = 2 9 t, placed at a distance c v = 0.25 m from the upper and lower faces of the beam, and c h = 0.125 m from the lateral faces. The distance between the middles of the two loading zones is b = 0.75 m and the width of the zone of interest is D = 1 m .

Using symmetry conditions, only a quarter of the beam is modeled; the displacement along the x-axis of the horizontal face located at x = 0 is blocked, as well as the displacement along the z-axis of the horizontal face located at z = L/2. The element Figure 19: Plastic strain at the end of the computation. Top: Bar method. Middle: Mixed method. Bottom: Volumic method.

Figure 20: Geometry of the four points bending test and boundary conditions size is 0.1 m, except in the zone of interest where it was reduced to 0.05 m to better capture the non-linear effects. Note that this size is too large to represent correctly the reinforcements, but in order to decrease the computing times, the mesh was refined only around the reinforcements with a size of 0.005 m (cf. figure 21). For the computation mesh, this local refinement is concentrated in the zone of interest. The geometry approximated with the level-set function is represented in figure 22. The plastic strains in the zone of interest in different cross-sections of the beam and in the reinforcements plane, are given in figure 23. One can observe that when using the bar method, the bar elements tend to concentrate the plastic strain, near the nodes of the 1D mesh. This phenomenon was not observed in the example of the section 5.2.1. Indeed, in two dimensions finite elements with a size with the same order of magnitude than the reinforcements radius were enough to correctly represent them. In three dimensions, the curvature due to the cylindrical geometry of the reinforcements requires smaller elements (in this example the ratio between the elements size around the reinforcements and their radius is around 1 20 ). This leads to stress concentrations around the 1D bars. Here again, the absence of heterogeneity leads to a plastic strain which is higher with the bar method, even far from the reinforcements. However, the results with the mixed method are almost the same as the ones obtained with the volumic method, which were the desired results.

Finally, to quantify the gain in CPU times, a sequential elastic computation was The additional cost of the mixed method is quite low compared to the bar method, which is interesting considering that in the zone of interest, the obtained results are the same than the one obtained with the volumic method.

Bar Mixed Volumic

Figure 23: Plastic strain at the end of the computation in several cross-section views.

Punching of a 3D slab

The last example is a 3D slab. It is reinforced with two layers of perpendicularly and regularly spaced reinforcements of radius 0.015 m. Each layer is made of 21 reinforcements in the x-axis direction and 21 reinforcements in the y-axis direction (for a total of 84 reinforcements). The upper layer of reinforcement is located 0.032 m from the upper boundary of the slab, and the lower layer is located 0.04 m from the lower boundary of the slab (distances measured from the center-line of the reinforcements). The dimensions of the slab are 3 × 3 × 0.25 m (cf. figure 24). The zone of interest is a parallelepiped with dimensions 0.3 × 0.3 × 0.25 m located at the middle of the slab, under the loading zone. The element size is 0.01 m in the zone of interest, and 0.15 m in the rest of the structure.

To decrease the computing time due to the treatment of the level-set and enrichment, the reinforcements crossing the zone of interest are modeled using the mixed method as described in section 4 (that is to say, 3 reinforcements by layer, in each direction), whereas the others are represented using only 1D bar elements. The geometry of these reinforcements approximated by X-FEM is shown in figure 25. For this test case, the number of degrees of freedom would be too high for a volumic representation, so there will be no reference computation. The lateral sides of the slab are clamped, and a vertical loading is applied at the center, on a square surface of dimensions 0.15 × 0.15 m.

The plastic strain in different planes (two horizontal planes crossing the upper and lower layer of reinforcements, and one vertical plane with a normal oriented along the x-axis, crossing the slab at x = 1.5) is shown in figure 26 for the bar and mixed methods.

In this case, the element size and the reinforcements radius are close to each other, which explains why plastic strain concentration is not observed around the reinforcements for the bar method. However, differences can still be observed between the results obtained with the mixed and bar methods, showing the importance between a volumic and a 1D representation of the reinforcements.

Toward a coupling with mesh adaptation methods

The potential of the proposed approach is finally illustrated on a last example. Indeed, the coupling of this mixed dimensional model with mesh adaptation strategies is very appealing, as the scale of representation of the reinforcements changes dynamically with element size. Consider the three point bending test presented in section 5.1 in which Figure 24: Geometry of the slab test damaged zones are introduced thanks to the Thick Level Set (TLS) method [START_REF] Moës | A level set based model for damage growth : the thick level set approach[END_REF]. The TLS approach is a method to regularize damage models, which allows a natural transition to discontinuous failure, relying on several key elements:

• A representation of the damage variable d through an auxiliary field φ, which is a level-set function. d is written as a known function (parameter of the model) d(φ), usually called the damage profile

• A characteristic length c , corresponding to the regularization length as in most of the non-local damage models [START_REF] Bažant | Instability, Ductility and Size-effect in Strain-Softening Concrete[END_REF] • An enrichment with the ramped Heaviside function of the isoc of φ which corresponds to the position of the cracks, to introduce discontinuities of the displacement field [START_REF] Bernard | Damage growth modeling using the Thick Level Set (TLS) approach: Efficient discretization for quasi-static loadings[END_REF] Note that no damage evolution is considered here: the damage zone is fixed (and frozen) at a given state of φ and the mechanical fields arising from this state are computed. φ is chosen in order to avoid fully damaged zone (corresponding to d = 1), which would involve the treatment of the coupling between the enrichment representing the material interface (see section 2.1) and the ramped Heaviside enrichment (which will be the topic of further investigations). The quadratic damage profile proposed by [START_REF] Parrilla Gómez | Comparison between Thick Level Set (TLS) and cohesive zone models[END_REF] is used. Adapted meshes (see figure 27) are considered in order to track the damaged zone during its evolution and show the influence of this refinement on the representation of the mechanical fields. The resulting axial strain fields xx along the axis of the beam are given, both for a standard 1D representation of the reinforcements and for the proposed mixed-dimensional method. Comparing those fields for different damaged states highlights the benefits of the proposed approach over classical bar models: mechanical fields are improved in the damaged area (i.e. where the accuracy of the model is of prime importance), especially around the reinforcements. The switch between bar and volumic Figure 25: Geometry of the reinforcements approximated by level-set functions models is handled automatically during the evolution of the damaged zone. Note also that, like in the previous examples, the length of the bar elements has to be carefully chosen as it can lead to stress concentration spots when this size is too large with respect to the elements.

Conclusion

A new approach to study reinforced structures is proposed, based on the idea that when performing a finite element analysis, it is most of the time possible to identify zones of interest. Fine meshes are used in these zones to get accurate results, whereas coarser elements are used in the rest of the structure to decrease computation times. In this framework, X-FEM is used to model the reinforcements, simplifying the meshing process. X-FEM alone would give inaccurate results, because the geometry of the reinforcements would not be correctly represented in the coarse zone. The proposed solution consists in assembling simultaneously 1D truss elements, multiplying their elementary stiffness matrix by a coefficient depending on the local accuracy of the representation of the reinforcements by the level-set functions. This approach was validated through several examples, where the same results were obtained in the zone of interest, whether this approach or a full volumic representation of the reinforcements on a fine mesh is used.

The proposed method allows to combine a volumic and a lineic representation of the reinforcements in the same computation. In the zone of interest, where the meshes are fine, a 3D representation of the reinforcements guarantees accurate results. In the rest of the structures, the use of coarser meshes associated with lineic elements decreases the computing times. The transition between the 3D and the 1D representation is such that the results in the zone of interest are almost identical to the results that would be obtained with a full volumic representation. Besides, several other advantages can be underlined:

• It is easy to implement, provided that one has a basic X-FEM library (enrichment functions and level-set, as well a element cutting for numerical integration). It can also be noticed that compared to most of the approach coupling 1D/2D model to 3D model, it is not necessary to know exactly the interface between the 1D and the 3D zones; here both 1D and 3D models coexists, the transition between both is insured by the coefficients α e .

• The computational costs remain low: the computation of the coefficients α e needs to be done only once, at the beginning of the analysis, therefore the computing costs related to this step are low, even in the case of a non linear analysis (unless some remeshing is involved).

• It is simple to use. One just needs to know the zones of interest, which is a common issue when performing a finite element analysis, and the 1D path of the reinforcements. The 3D part of the reinforcements is represented implicitly by X-FEM, simplifying the meshing process, whereas the 1D mesh is independent from the volumic mesh.

It must be emphasized that the efficiency of the proposed approach is strongly dependent on the common sense of the user. For instance, one should should ensure that the mesh is fine enough in the zones of interest, and should not have a transition between 1D and 3D representation in these zones (as a reminder, the coefficients α e corrects the global stiffness in the areas where X-FEM fails to represent the reinforcements, but does not provide accurate local results in these zones). As the geometry approximated by level-sets is piecewise linear 2.2, the density of the mesh in the zones of interest should be the same as the one that should be used to represent the reinforcements with a classical 3D mesh (that is to say, with nodes coinciding with the material interface). At least one order of magnitude between the reinforcement radius and the size of the elements in the fine zones seems to be a minimum in 3D.

This approach was developed considering perfect bond between the reinforcements and the surrounding matrix, however it is often not the case, as in reinforced concrete for instance where the steel-concrete interface is not perfect. If the reinforcements are represented using X-FEM, an approach similar to what is done in [START_REF] Moës | Extended finite element method for cohesive crack growth[END_REF] for cohesive crack growth could be considered. The Ridge enrichment function could be replaced by a Heaviside enrichment function, to get a discontinuous displacement at the interface but with "cohesive" forces representing the bond between the reinforcements and the matrix. For reinforcements represented using bar elements, it is possible to consider size, the non-conforming FEM is not as accurate as the conforming FEM, which is not the case of the X-FEM.

Figure 2 :

 2 Figure 2: Ridge enrichment in 1D. The enriched nodes are the diamond nodes.

Figure 4 :Figure 5 :

 45 Figure 4: Using level-set to represent reinforcements. Left: with a fine mesh in the whole structure. Right: with a coarse mesh in part of the structure. Top: overview. Bottom: zoom in the transition between the fine and the coarse mesh.

Figure 6 :

 6 Figure 6: Reinforcement node inside a volumic element

Figure 7 :

 7 Figure 7: Redistribution of the reinforcements stiffness in the volumic elements. a. Domains definition. b. Mesh of the domains, using 1D bar elements.

2 .

 2 The subelements close to the extremities of the segment are cut once again by two planes normal to the bar element at the bar nodes 3. Ω num e is then the sum of the volumes of all the subelements resulting from step 1 and 2 which are inside Ω real e . These elements can be found easily because the value of φ is smaller or equal to zero at each of their nodes.

Figure 8 :

 8 Figure 8: Computation of α e

2 5.

 2 For each bar element, compute coefficient α e 6. Assemble the stiffness corresponding to the 1D part of the reinforcements, multiplying the stiffness of each bar element by α e and taking into account the relation (

5. 1 . 1

 11 Three points bending 2D elasticThe first example is a three points bending test in 2D, under plane strain assumption. The geometrical characteristics of the test are given in figure10: length L = 5.1 m, height h = 0.8 m, thickness t = 1 m. The beam is strengthened with a reinforcement of radius 0.016 m in the upper part, and a reinforcement of radius 0.02 m in the lower part. The zone of interest is supposed to be at the center of the beam, and has a width D.

Figure 10 :

 10 Figure 10: Geometry of the three points bending test and boundary conditions

Figure 14 :Figure 15 :

 1415 Figure 14: Displacements, normal stresses and errors in the upper reinforcement. The black dashed lines locate the limits of the zone of interest

Figure 16 :Figure 17 :

 1617 Figure 16: a. 3D mesh of the post. b. Overview of the 1D mesh and of the approximated geometry of the reinforcements. The red lines correspond to the 1D mesh of the reinforcements. c. Zoom around the zone of interest. d. Zoom around the zone of interest, close to the 3D part of the reinforcements.

Figure 21 :

 21 Figure 21: Meshes used for the four points bending test. Left: mesh used to test the bar and mixed methods. Right: mesh used to test the volumic method.

Figure 22 :

 22 Figure 22: Approximation of the reinforcements in the four points bending test, using the mixed method

Figure 26 :

 26 Figure 26: Plastic strain at the end of the computation in several cross-section views.

Figure 27 :

 27 Figure 27: Axial strain xx for different "remeshing" steps. From a. to d. : mixed method. e. bar method at the step corresponding to d. 32

Figure 28 :

 28 Figure 28: Convergence rates for the 2D inclusion problem.

Table 2 :

 2 Number of degrees of freedom and computing times.

		Number of de-	Pre-	Assembling Solving
		grees of freedom	treatment
	Bar	117909	2.68 s	14.63 s	11.06 s
	Mixed	155190	36.96 s	141.90 s	36.96 s
	Volumic 482677	142.01 s	262.57 s	50.97 s

Acknowledgment

The support of the ERC Grant XLS n0 291102 is gratefully acknowledged.

a slip between the bar nodes and the surrounding volumic elements, and to introduce bond forces as in [START_REF] Casanova | Bond slip model for the simulation of reinforced concrete structures[END_REF]. Future work will focus on how to deal with bond slip in the zones where both X-FEM and bar representation are superimposed (0 < α e < 1).

Appendix

In order to illustrate the interest of using the eXtended Finite Element Method, a convergence test is performed on the bidimensional problem of a circular inclusion in a rectangular plate similar to the one depicted on figure 1. The error is defined by

where u ex is the exact solution and u h an approximate solution. u h is calculated with three different methods:

• The classical finite element method (FEM) with a mesh conforming to the material interface.

• A finite element computation with an arbitrary mesh, not conforming to the material interface. The numerical integration is performed using the procedure described in section 2.2, i.e. with subcells but no enrichment.

• A computation with the same non-conforming mesh, using the X-FEM.

The error is plotted for the three methods on figure 28. It can be observed that the X-FEM allows to recover a convergence rate which is close to the conforming FEM, whereas the non-conforming FEM gives a poor convergence rate. For a given element size, the non-conforming FEM is always the less accurate method.

In order to get information about the local accuracy of each method, the local error for each element of the mesh e is defined:

The local error for each method, for a ratio h r = 1 4 where h is the element size and r the radius of the inclusion, is shown on figure 29. For the three methods, the local error is concentrated around the material interface. However, the error obtained with the X-FEM is of the same order of magnitude than the error obtained with the conforming FEM, which is of course lower than the error obtained with the non-conforming FEM. Note that the chosen ratio h 4 = 1 r is quite representative of a typical element size required to study a circular inclusion (good compromise between the approximation of the curvature of the geometry and a reasonable mesh refinement). One could have imagined that for a fine enough mesh, the error committed with the non-conforming FEM would be small, which is not the case here. This example shows that, for a practical element


Théry Béord
email: thery.beord@laposte.net&thery@unikl.edu

AN ANALYZIS OF THE FILIPINO-ENGLISH DIGLOSSIA THROUGH SOCIAL REPRESENTATIONS OF LANGUAGES

Keywords: Linguistic representation, diglossia, historical forgetting, neo-colonialism, English in the Philippines

We propose in this article to analyze the linguistic representations associated with the two official languages of the Philippines: Filipino and English -both at the heart of the educational system. Referring to the systems of values and affects attached to languages, the concept of linguistic representationor social representations of languageshas been adopted by sociolinguists and educational specialists in so far as it plays a key role in the process of identity construction as well as in the transmission of language. A collective interview organized in one of the most prominent university in Manila has allowed us to bring forward the social representations of Filipino and English among students before measuring their respective importance through the administration of a questionnaire. The study of these representations among socially privileged Filipino will bring us to question the language configuration and more specifically the place of English in the social structure of this former US colony marked by the permanency of extreme social inequalities. We will see that the medium of instruction and its representations are a tool to analyze neocolonial issues.

INTRODUCTION

Since its introduction as the sole language of education by the US President McKinley in 1900, English has remained pre-eminent in the Philippines. At the core of the assimilation, the implementation of a school system patterned on the American model was also the occasion to rally the local elite. Reversely, the latter found an opportunity to back up its privileged position after the changeover of power which saw the Spanish Crown leave the archipelago more than three centuries after Magellan first landed [START_REF] Tupas | Bourdieu, historical forgetting, and the problem of English in the Philippines[END_REF]. Thereafter, the language of instruction has represented a major political issue especially since the independence gained in 1946. It has consistently crystallized the opposition between nationalists and advocates of neoliberalism. If one were to summarize the evolution of the language policy in education throughout the 20 th century, one could argue that the Philippines have moved on from a monolingual system under the American colonial occupation to a bilingual system that associates the Filipino language to English, and recently to a multilingual system that provides space for regional languages at the beginning of education. Along with the promotion of Filipino to unify the archipelago, English has shift from the position of first to second language. Whether it will now shift from a second to a foreign language remains to be seen (Gonzales, 1994). Indeed, the nation-building is challenged since the 1970's by a major evolution: the implementation of a labour export program by the State to tackle unemployment [START_REF] Hau | Rethinking History and "Nation-Building" in the Philippines[END_REF]. The expanding of economic globalization at the turn of the century has enshrined the transnational labour flows as well as the dependence to the remittances of Oversea Filipino Workers (OFW). The latter have been described by the former President Corry Aquino as "the new heroes". Together with the inflation of the Philippine diaspora that reaches currently 10 million Filipinos, the surge of the Business Process Outsourcing (BPO) has made it necessary to maintain and develop the mastery of English. The coexistence of Filipino and English in the educational system for different agendas raises questions. Such bilingualism can be analysed in terms of diglossia which means that they may compete or conflict with each other. Values assigned to them are most likely to vary in nature and in importance or even to establish a hierarchy. The task we have set ourselves consists in measuring how these languages are perceived by students and determining the functions, affects or values attributed to each. In order to bring forward such linguistic representations we have organized a collective interview with students coming from one of the university that trains the elite of the country. The results discussed in this paper concern therefore the upper part of the Philippine society which belongs to the Tagalog ethnic coming from the capital and its region.

Literature review

The notion of linguistic representation reached linguistics in the late 1990's when its role in language appropriation and transmission was acknowledged [START_REF] Matthey | Les représentations de l'apprentissage des langues et du bilinguisme dans l'institution éducative[END_REF][START_REF] Moore | Les représentations des langues et de leur apprentissage[END_REF]. Since then it has often been utilized in the Francophone area to analyse situations of diglossia, namely cases of conflicting coexistence of languages, either in a postcolonial context (see Maurer, 2013, for a recent example) either in the context of regionalism in Europe (see Lafont, 2000, for the Occitan-Catalan school). Diglossia differs from a theoretic neutral bilingualism in the sense that it describes the linguistic configurations in dynamic terms, entailing an unequal distribution between a dominating language and a dominated onewhere the former is in the process of absorbing the latter. The functions, prestige or images associated to the languages in presence structure a system of positive or negative values and thus determine a hierarchy which in turn induce the transmission or substitution of these languages (ibid). Thus, the paradigm of linguistic representations constitutes a mean to identify the process of diglossia, its dynamics and, more acutely, its structuring ideology (ibid).

METHODOLOGY

The modus operandi of our work can be described as follow: a collective interview with some twenty students recorded and transcribed, the preparation of a questionnaire based on the interview, the submitting of a questionnaire to another group of thirty-four students in the same university and the analysis of the results. The survey took place in one of the most prestigious universities in the country with students practicing both Filipino and English. As a first step, the group of students has been invited to express its views on these languages through two simple and neutral questions: "what do you think of English? What do you think of Filipino language?" The interview has been transcribed in its entirety. From this discourse the representations of languages has been identified, sorted and condensed with full respect for the student's formulations. Based on this, questionnaires have been prepared for each language, consisting of twenty propositions synthetizing the respondent's representations. They have been presented under the form of a twenty items-table, with a blank cell at the end of each line. Three days after the collective interview the questionnaires have been submitted to evaluation to another group. After inviting students to read and ensuring that they understand the items, we have asked them:

-to grade +2 the four propositions best suited to evoke the language -then to grade -2 the four propositions which evoke the least the language -to grade +1 the four propositions which evoke rather well the language -to grade -1 the four propositions which evoke poorly the language -to grade 0 the four remaining propositions which have been neither rejected nor selected From this evaluation one can measure the importance of the selected representations among the group in terms of adherence or rejection of the item. Indeed a calculation the results' average reveals the adherence to the item, with a maximum of +2 (case where all respondents graded +2 an item) and a minimum of -2 (case where all respondents graded -2 an item).

FINDINGS

The results of the survey can be presented under the form of two tables where the representations are ranked in descending order.

Table 1. Linguistic representations of the Filipino language Table 2. Linguistic representations of the English language

Representation of the language adherence score (a) English is a bridge to communicate with other nationalities 1.74 (b) English is the teaching language used in school and in university 1.21 (c) In the Philippines, English is mostly spoken in the urban developed cities 1.18 (d) In the Philippines, if you are fluent in English, people look at you differently 1.15 (e) We are surrounded by English language everywhere we go 0.79 (f) It is easier read and write in English 0.79 (g) English is a language of convenience 0.68 (h) To be a good English speaker shows that you have a good academic background 0.47 (i) In the Philippines, to be a good English speaker shows that you belong to the higher social class 0.29 (j) In the Philippines, when you are fluent in English, people believe that you have a superiority complex -0.03 (k) English threatens the maintenance of the Filipino identity -0.12 (l) We are forced to place English on a higher pedestal -0.47 (m) English doesn't really give you a root -0.62 (n) English is a common means of communicating within the linguistic groups of the archipelago -0.76 (o) English is a language we are forced to talk starting high school -0.91 (p) To be a good English speaker shows that you are rich -0.94 (q) To be forced to learn English makes it harder to like it -0.97 (r) English fits English the best, Filipino fits Filipinos the best -1.06 (s) English is the ultimate language -1.09 (t) English is difficult to learn -1.26

As one can observe in the figures above Filipino and English are hierarchically organized and dominated by a single representation supported by three closer elements. With the acceptance of Filipino as the Representation of the language adherence score (1) Filipino language contributes to the prospect of the national identity or national consciousness 1.64 (2) Filipino is a really rich language 0.94 (3) Filipino language is connecting us with the history of the Philippines 0.82 (4) Filipino language is a connection to the Filipinos 0.79 (5) Filipino language is poetic 0.45 (6) Filipino is the language of Filipinos 0.42 (7) Filipino is a mixture of many of the languages of the Philippines 0.3 (8) Filipino language fits the best to Filipino culture 0.24 (9) Filipino is heavily based in Tagalog 0.21 (10) In the Philippines, Filipino language is useful when you travel in the provinces 0.18 (11) It's easier to speak in Filipino with the maid or the driver -0.18 (12) New generations don't really know Filipino language -0.24 (13) Filipinos tend to forget their national language -0.42 ( 14) You cannot find all words to express yourself in Filipino -0.48 (15) Filipino language makes you feel special or unique -0.52 (16) If you can convey your thoughts in Filipino it shows how educated you are -0.52 (17) Filipino language is hard -0.61 (18) It easier to speak Filipino with the family -0.76 (19) It irritates me that Filipino is a forbidden language at school -1 (20) Many words in Filipino are outdated -1.27

uncontested symbol of nationhood (1), it is the integrative function of the language that is highlighted. Being a relatively recent nation-state confronted with a multidimensional fragmentation, this dimension is a significant indicator of the existence of a national awareness. It means that the ethno-linguistic divisions of the country (about 200 languages and dialects) as well as its geographic dispersal (7107 islands) are overcome by the national language which plays well its role of interethnic communication (4). On the other hand it endorses a mythic role by connecting the people with the past of the archipelago's history (3). This is marked by over three centuries of Spanish colonization followed by half a century of American one. One can note however that putting into words the national consciousness is done very prudently. The formulation proposed in the questionnaire, taken from the interview, indicates indeed the ongoing process of the national identity construction described as being in a phase of "prospect". The language is appreciated for its richness (2) and its beauty (5) at the same time as its putative image of outdatedness is unequivocally rejected (20). Filipino language is the subject of an emotional investment reinforced by its proximity with Tagalog language (9) which is widely spoken in the capital and its region.

By contrast, English appears clearly as an instrumental language as it is at once the language of communication with abroad (a) and the medium of education (b). But whereas English finds its roots in the American colonial period, the issue of its colonial features doesn't emerge during the interview. No reference is made to the imported or foreign nature of this language in the country: the American linguistic bequest, it seems, remains unspoken, nested in the limbs of the collective sub-conscious. Yet, should we analyse this oversight as a mark of acclimatisation or appropriation? The clear reject of the cognition stating that "English doesn't really give you a root" tends to prove it as well as the image of a language "easy to read and write" (f). But if we are to believe the French sociologist [START_REF] Bourdieu | Reproduction in Education, Society and Culture[END_REF] the forgetting of colonial configuration is in fact a political imperative structuring the society and revealing its social inequities. Theorizing what he calls "symbolic violence", Bourdieu describes how historical forgetting, as misrecognition, represents a practice of symbolic violence that imposes legitimate significations covering up underpinned power relationships (1970). Eventually it allows the institutionalisation of a power that remains unknown. Bourdieu states that "forgetting prevents us from drawing connections between past and present conditions. It also prevents us from historicizing the present" [START_REF] Bourdieu | Reproduction in Education, Society and Culture[END_REF]. As such the forgetting could appear as a strategy to maintain social structures favourable to the dominant classes with the implicit consent of the dominated classes. If the place of English in Philippines history is not debated by our group of students it is by far a major issue among Philippine scholars most specifically in the post-colonial context. Among them, [START_REF] Tupas | Bourdieu, historical forgetting, and the problem of English in the Philippines[END_REF] has identified the politics of forgetting as a "national problem" in the archipelago. In this way historical misrecognition would be at the basis of the social inequalities and English the "last bastion of soporific colonialism" (ibid). The role of social divider of English arises indeed strongly in the interview to such an extent that we have identified five cognitions which refers to varying degrees to the social status conferred by English (d, h, I, j, p). Two of them are formulated so that they are criticizing the role of English in the society. In addition to the afore-mentioned refusal to reject English as "a root", students also disapproved the cognition according which they "are forced to place English on a higher pedestal" (l). Given that we have highlighted the great number of cognitions putting forward the feeling of superiority attached to English, the question might be asked whether this pressure does not exist or whether our respondents are not exposed to it because of their social origin. The fact that they recognize leaving in an Anglophone environment where they are "surrounded by English language everywhere [they] go" seems to be indeed an indicator of their position in society. Be that as it may, the student's rejection of these criticisms towards English shows their acceptation of the language and even more their refusal to challenge its preeminent position.

CONCLUSIONS

Although it has long been marginal Filipino appears as the undisputed language of the nation in the group of students. It is considered as an element facilitating the unification of the country despite a multidimensional fragmentation. In addition to performing mythical and symbolic functions Filipino is able to ensure the intertribal communication. It would be relevant to extend this kind of survey to other provinces of the archipelago to confront the adherence to the national language in the capital to more local forms of identity anchored in an ethnical frame and in a regional language. On the other hand the linguistic representations of English give us a glimpse of the social gap that undermines the foundation of the Philippines. Much more importantly, it shows that English has a class dimension in a society where social inequalities are extreme. For these students enrolled in the university of the political and economical elite, English is the language of daily environment. The survey shows their uncritical acceptance of English and more precisely the absence of aware link with the American imperialism. English appears neither as a foreign language neither as a sign of the continuous hegemony of the USA in the country until our time. The historical forgetting seems to legitimize the maintenance of English and through it the neo-colonial social structures. Thus, the Anglophone environment in which the elite live perpetuates an economical advantage by endowing the language with a social prestige. Here again it would be valuable to compare the representation of English of this group with representations of lower part of the society leaving in a non-Anglophone context.