A Martin

S Brisard

Homogenization of fiber materials with equivalent inclusion method

Keywords: Homogenization, Conductivity, Equivalent inclusion method, Lippmann-Schwinger equation

 est une alternative intéressante aux méthodes traditionnelles de calcul en champ complet, pour des microstructures de type matrice/inclusion. Brisard et al. [2] ont introduit une formulation variationnelle de cette méthode, bien adaptée à l'homogénéisation. Dans cet article, cette méthode variationnelle est mise en oeuvre dans le cas des fibres, dont l'élancement permet de décrire le champ de polarisation avec peu de degrés de liberté, en assimilant les fibres à des éléments linéiques. Les résultats sont présentés ici sur quelques fibres, pour des problèmes de conductivité. Ils peuvent être étendus à des problèmes d'élasticité.

Introduction

Recently, Spie batignolles and EuroMC have developped an electromagnetic shielding process for confidential facilities, based on a concrete reinforced by conducting fibers [START_REF] Mardiguian | The intelligent concrete : A new, economical technique for architectural shielding of buildings[END_REF]. Within the framework of homogenization, we study the effective electric conductivity of such heterogeneous materials.

Our goal is to propose an estimation of the homogenized conductivity, that improves on 'mean field' models, while being less costly computationally than 'full field' methods.

In order to get around this difficulty, we have chosen an intermediate technique, the equivalent inclusion method, based on Eshelby's work [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] and extended to several inclusions by Moschovidis et al. [START_REF] Moschovidis | Two-ellipsoidal inhomogeneities by the equivalent inclusion method[END_REF]. We use a variational form of the equivalent inclusion method, introduced by Brisard et al. [START_REF] Brisard | A variational form of the equivalent inclusion method for numerical homogenization[END_REF], whose benefits are the following : (i) the resulting linear system is well-posed, (ii) the numerical solution converges to the exact solution, (iii) the method can provide rigorous bounds on the apparent properties of the statistical volume element. This method is called EIM in the following. The application of such method can be found in electrostatics (homogenized conductivity) but also in mechanics (homogenized stiffness), thermal science... In this paper, we present the implementation of the EIM in the case of fibres in 3D, which is a case which would require too much computational ressources for a full field simulation. We take advantage of fibre aspect ratio to reduce the number of DOF. We assess the accuracy of the method through 3D conductivity problems involving a limited number of cylinders.

The paper is organized as follows. Section 2 is a presentation of the variational form of the equivalent inclusion method [START_REF] Brisard | A variational form of the equivalent inclusion method for numerical homogenization[END_REF] and of its implementation in the case of fibres. In Section 3, we present a first application on one fibre and another application with two fibres.

Equivalent inclusion method for fibres

In this section, we implement the variational form of the equivalent inclusion method introduced by Brisard et al. [START_REF] Brisard | A variational form of the equivalent inclusion method for numerical homogenization[END_REF] in the case of linear conductivity, for cylindrical inclusions.

The modified Lippmann-Schwinger equation is first introduced in Sec. 2.1 and its Galerkin discretization (for cylinders) is presented in Sec. 2.2.

Following Ostoja-Starzewski [START_REF] Ostoja-Starzewski | Material Spatial Randomness : From Statistical to Representative Volume Element[END_REF], we use the terminology statistical volume element to denote a large (but finite) realization of the random heterogeneous material under consideration.

The modified Lippmann-Schwinger equation

Within the framework of electric conductivity, we consider a statistical volume element Ω of a random heterogeneous material. The conductivity at x ∈ Ω is σ(x) (symmetric, positive definite, second-order tensor) ; besides, E(x), φ(x) and j(x) denote the electric field, electric potential and volumic current, respectively, at point x.

The apparent conductivity of the SVE Ω, σ app , is found from the solution to the following boundaryvalue problem

Ω : div j = 0, (1)
Ω : j = σ • E, (2)
Ω : E = grad φ, (3)
with boundary conditions that must be specified. The boundary conditions chosen here, in order for the equivalent inclusion method to be applicable, are mixed boundary conditions, previously discussed within the framework of linear elasticity [START_REF] Brisard | New boundary conditions for the computation of the apparent stiffness of statistical volume elements[END_REF][START_REF] Brisard | A variational form of the equivalent inclusion method for numerical homogenization[END_REF].

In order to apply these boundary conditions, the SVE Ω must be embedded in a homogeneous, infinite matrix with conductivity σ 0 ; χ denotes the indicator function of Ω (χ(x) ∈ {0, 1} and χ(x) = 1 iff x ∈ Ω). It was shown in Brisard et al. [START_REF] Brisard | New boundary conditions for the computation of the apparent stiffness of statistical volume elements[END_REF] that the introduced mixed boundary conditions (not stated in the present paper for the sake of conciseness) alow to reformulate problem ( 1)-( 3) as a unique integral equation of the Lippmann-Schwinger type [START_REF] Willis | Bounds and Self-Consistent Estimates for the Overall Properties of Anisotropic Composites[END_REF][START_REF] Zeller | Elastic Constants of Polycrystals[END_REF] :

σ -σ 0 -1 • τ + Γ ∞ 0 (τ -χ τ ) = E, (4)
where the macroscopic electric field E is prescribed at infinity :

x → +∞ : E(x) → E, (5)
and the main unknown is the polarization

τ τ = (σ -σ 0 ) • E. (6)
In the above modified Lippmann-Schwinger equation, Γ ∞ 0 denotes the Green operator, associated with the conductivity σ 0 . It maps the field τ onto the field Γ ∞ 0 (τ ). In the remainder of this paper, it will always be assumed that the reference material is isotropic : σ = σ 0 1. Then, we have the following expression of the Green operator [4,[START_REF] Torquato | Effective Electrical Conductivity of Two-phase Disordered Composite Media[END_REF]

Γ ∞ 0 (τ )(x) = P 0 • τ (x) + PV x y∈Ω G 0 (y -x) • τ (y) d 3 y, (7)
where

P 0 = 1 3σ 0 and G 0 (r) = 1 -3n ⊗ n 4πσ 0 r 3 (8)
(r = r and n = r/r). In the above equation, "PV x " refers to the principal value at x for spherical excluded regions

PV x y∈Ω f (y) d 3 y = lim δ→0 y∈Ω y-x ≥δ f (y) d 3 y (9)
From the volume average τ of the solution τ to the modified Lippmann-Schwinger equation ( 4), the apparent conductivity is readily retrieved. Indeed, using Eq. ( 6)

σ app • E = σ 0 • E + τ = σ 0 • E + τ (10)
To close this section, it is emphasized that the equivalence between, on the one hand problem (1)-( 3) with mixed boundary conditions and, on the other hand, the modified Lippmann-Schwinger equation ( 4) holds for SVEs Ω of ellipsoidal shape ; of course, the limit for large SVEs of the apparent conductivity does not depend on the radii of the ellipsoid. In the remainder of this paper, the microstructures under consideration are statistically isotropic. It is therefore natural to use spherical SVEs.

Discretization of the Lippmann-Schwinger equation

The Lippmann-Schwinger equation ( 4) is first reformulated as a variational problem as follows Find τ ∈ V such that, for all ∈ V :

• (σ -σ 0 ) -1 • τ + • Γ ∞ 0 (τ -χ τ ) = E • (11)
Following a Galerkin approach, we then introduce the finite-dimensional discretization subspace V p of V, (p ∈ N is a discretization parameter to be defined), and search for a solution of (11) in V p , denoted τ p .

In the following, we consider a spherical SVE Ω ⊂ R 3 that hosts N cylindrical inhomogeneities Ω 1 , . . . , Ω N embedded in a homogeneous matrix Ω 0 . We introduce the common radius R and common total length 2L of the fibres. Then, Ω α is a circular cylinder centered at x α ∈ Ω, oriented by the unit-vector n α (see Fig. 2). The aspect ratio e of the fibers is defined as : e = L/R. The inhomogeneities do not overlap. The corresponding indicator functions χ α (α = 0, . . . , N ) are

χ α (x) =    1 if x ∈ Ω α , 0 otherwise. ( 12
)
It will be convenient to introduce volume averages over inhomogeneity Ω α

• α = 1 V α Ωα •(x) d 3 x, (13)
where V α denotes the volume of Ω α . Finally, f α = V α /V denotes the volume fraction occupied by inhomogeneity α within the SVE.

Within the framework of the EIM, the reference material must coincide with the matrix. In other words, the SVE Ω is embedded in an infinite, homogeneous material with same conductivity σ 0 as the matrix.

Similarly, each inhomogeneity α = 1, . . . , N has homogeneous conductivity σ α .

For the applications considered in this paper, it is assumed that the matrix is isotropic : σ 0 = σ 0 1. The discretization space V p is generated by a finite number of linearly independent functions supported on the inhomogeneities. More precisely, we seek the following decomposition for the trial function τ p

τ p (x) = N α=1 Kp-1 k=0 Ψ k α (x) τ k α , (14)
where K p is the number of scalar shape functions Ψ k α that are supported in Ω α and τ k α are constant, unknown vectors.

Typically, for slender cylindrical inhomogeneities, the longitudinal coordinate z α defined as follows

z α = x -x α • n α ( 15
)
plays a specific role. This suggests to consider polarizations that are polynomials of high-order p of the longitudinal coordinate, and polynomials of low-order q of the two other, transverse, coordinates.

Here, only the simplest case q = 0 was investigated. In other words, the discretization space V p is now the space of tensor fields that are, over each inhomogeneity Ω α , polynomial of the local longitudinal coordinate z α . We then have K p = p and

Ψ k α (x) = χ α (x) z k α . (16)
Plugging this decomposition into the discrete variational problem, and testing with test functions p ∈

Ω φ = E • x j = σ 0 • E j = σ α • E F -Problem on a RVE
V p decomposed similarly, the following linear system is derived

l M k+l α σ α -σ 0 -1 +S kl α -f α M k α M l α P 0 •τ l α + β =α,l T kl αβ -f β M k α M l β P 0 •τ l β = M k α E, (17)
where

M k α = 1 + (-1) k 2 k + 1 L k , ( 18
)
while T kl αβ denotes the linear mapping over the space of 3d vectors defined below

T kl αβ • τ = Ψ k α Γ ∞ 0 Ψ l β τ α . (19)
Eq. ( 19) shows that T kl αβ • τ is the Ψ k α -weighted average of the (opposite of the) electric field induced over inclusion α by the non-uniform polarization Ψ l β τ supported in inclusion β. As such, T kl αβ will be called influence tensors.

For α = β, these tensors measure the electric field induced on inclusion α by a polarization applied to the same inclusion α. The T kl αα will be called self-influence tensors, and denoted S kl α .

The evaluation of these tensors T kl αβ and S kl α will be discussed in the next Section.

Finally, Eq. ( 10) is applied to the solution to the above linear system to derive the EIM estimate of the apparent conductivity, σ EIM

σ EIM • E = σ 0 • E + α,l f α M k α τ k α . (20)
x α

x β r αβ r α z α n α

F -Computation of the influence tensors

where

w 0 = r 12 + Lζ 2 n 2 -Lζ 1 n 1 . (29)
Upon integration with respect to

0 ≤ ρ 1 , ρ 2 ≤ 1 and 0 ≤ θ 1 , θ 2 ≤ 2π U kl 12 = L k+l+3 8 R 2 L 2 1 -1 1 -1 ζ k 1 ζ l 2 w 0 3 dζ 1 dζ 2 , (30)
V kl 12 = 3L k+l+3 8 R 2 L 2 1 -1 1 -1 ζ k 1 ζ l 2 w 0 ⊗ w 0 w 0 5 dζ 1 dζ 2 . (31)
In the above expressions, the first integration (with respect to ζ 1 ) can be performed analytically, the second integral (with respect to ζ 2 ) is evaluated numerically.

Self-influence tensors

Using Eqs ( 7) and ( 19), the self-influence tensor is given by

S kl α = M k+l α P 0 + 1 V α x∈Ωα Ψ k α (x) PV x y∈Ω β Ψ l β (y) G 0 (y -x) d 3 y d 3 x. (32)
Eq. (32) shows that the singularity cannot be removed in the integral expression of the self-influence tensors, which makes their analytical evaluation difficult. Rather than attempting a numerical evaluation of these integrals, and observing that this tensor can be precomputed off-line prior to the full EIM calculation, we used a numerical approach based on a finite element analysis.

The resulting problem is formulated over the axisymmetric domain Ω 1 , although the loading is not axisymmetric. Still, it is possible to reduce the 3d problem to a finite set of 2d problems [START_REF] Wilson | Structural Analysis of Axisymmetric Solids[END_REF][START_REF] Percy | Application of Matrix Displacement Method to Linear Elastic Analysis of Shells of Revolution[END_REF]. These 2d variational problems were implemented within the FEniCS framework [START_REF] Logg | Automated Solution of Differential Equations by the Finite Element Method, The FEniCS Book[END_REF][START_REF] Alnaes | The FEniCS Project Version 1.5[END_REF].

- 3 Applications with a few fibres

Problem with one fibre

The equivalent inclusion method introduced above is first applied to Eshelby's inhomogeneity problem [START_REF] Eshelby | The determination of the elastic field of an ellipsoidal inclusion, and related problems[END_REF] for a circular, cylindrical inhomogeneity. More precisely, we consider a single inhomogeneity Ω 1 centered at the origin and embedded in a homogeneous, infinite matrix. The system is subjected to a uniform electric field E = Ee z at infinity, where E is a scalar constant and e z denotes the axis of revolution of the inhomogeneity. Both matrix and inhomogeneity have isotropic conductivities σ 0 and σ 1 , respectively. The contrast of conductivities is σ 1 /σ 0 = 10 6 , while the aspect ratio is L/R = 50.

We compared the results of EIM with finite element calculation. To present this finite element results, we compute the average field on the cylinder cross section, along the fibre longitudinal coordinate z.

The EIM field, which is uniform in fibre section, is shown for every z. We can see on figure 3 a good agreement between finite element calculation and EIM method, for the longitudinal component of the electric field. Moreover, we can see that even for quadratic polynomials, the EIM estimation is close to the finite element solution.

Two parallel cylinders

Here we consider two parallel cylinders submitted to an external field E = e z at infinity, where e z corresponds to the orientation of the cylinders. The distance between cylinders is 2R. We work with an aspect ratio of e = 50 and a contrast of 10 6 . We can see on figure 4 that even for cylinders near from each other, their interaction is well capted and there is no need to search for high polynomial orders.

Conclusion

The method presented in this paper is an adaptation of the variational form of the equivalent inclusion method [START_REF] Moschovidis | Two-ellipsoidal inhomogeneities by the equivalent inclusion method[END_REF][START_REF] Brisard | A variational form of the equivalent inclusion method for numerical homogenization[END_REF], in the case of fibres. It is based on the same principles, but it takes advantage of the fibre aspect ratio. The tensors involved in the linear system to be assembled are pre-calculated via a combination of 2D finite element calculations, analytical calculations and numerical integrations. The examples presented in this paper with few cylinders show a good agreement between this approximate method and more accurate finite element calculations, even for a small number of degrees of freedom.

This work will now be extended to large assemblies of conducting fibres, where full field approaches become too expensive.

  cylinders with aspect ratio 50 and contrast 10 6

Integral expression of the influence and self-influence tensors 2.3.1 Influence tensors

Plugging Eq. ( 16) in Eq. ( 19), and using Eqs. [START_REF] Mardiguian | The intelligent concrete : A new, economical technique for architectural shielding of buildings[END_REF] and [START_REF] Moschovidis | Two-ellipsoidal inhomogeneities by the equivalent inclusion method[END_REF] (which give the expression of the Green operator for isotropic reference material), we obtain :

with

and

with

We then introduce the local cylindrical coordinates (r α , θ α , z α ) (α = 1, 2, see Fig. 2)

where e r,α is the unit radial vector, while θ α is the polar angle with respect to a fixed (unspecified) direction. We also introduce the following dimensionless variables

Then

We were not able to derive a closed-form expression of these integrals for two cylinders. However, assuming that the radius R is small compared to the smallest distance between the two cylinders, a multipole expansion can be produced. For low volume fractions of, this assumption is certainly verified for most pairs of fibers. It is then assumed that R ρ 2 e r,2ρ 1 e r,1 Lζ 2 n 2 -Lζ 1 n 1 + x 2x 1 in the above expression. Then U kl 12 and V kl 12 read


Alain Queffelec

Pierrick Fouéré

Ludovic Bellot-Gurlet

Benoît Bérard

Jean-Baptiste Caverne

Christian Stouvenot

Loup Bernard

Queffelec Alain

Fouéré Pierrick

Bellot-Gurlet Ludovic

Bérard Benoît

Caverne Jean- Baptiste

Stouvenot Christian

Lapidary craftsmanship in the Antilles during the Ceramic Period, a renewed and updated analysis

ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

If the study of ceramic production is at the foundation of Caribbean archaeology and the production of shell tools and ornaments exceeds in quantity the production of these same objects made of stone during the Ceramic period, it must be noted that the study of lithic raw materials has also been included since the 1990s as one of the approaches that allows us to understand questions of territory and interactions between groups [START_REF] Knippenberg | Stone artefact production and exchange among the Lesser Antilles[END_REF][START_REF] Rodriguez | Puerto Rico and the caribbean pre-saladoid "crosshatch connection[END_REF][START_REF] Rodriguez Ramos | What is the Caribbean? An archaeological perspective[END_REF][START_REF] Watters | Maritime trade in the prehistoric eastern Caribbean, in: The Indigenous People of the Caribbean[END_REF].

Within these stone material productions, ornamental objects such as beads and pendants have been the subject of a relatively large number of work [START_REF] Boomert | Gifts of the Amazons: Green stone pendants and beads as items of ceremonial exchange in Amazonia and the Caribbean[END_REF][START_REF] Cody | Distribution of exotic stone artifacts through the Lesser Antilles : their implications for prehistoric interaction and exchange[END_REF][START_REF] Crock | Archaeology of Trants, Montserrat. Part 4. Flaked stone and stone bead industries[END_REF][START_REF] Harrington | A West Indian gem center[END_REF][START_REF] Murphy | Pre-Columbian gems and ornamental materials from Antigua, West Indies[END_REF][START_REF] Narganes Storde | La lapidaria de Sorce, Vieques y Tecla, Guayanilla, Puerto Rico[END_REF][START_REF] Narganes Storde | La lapidaria de la Hueca, Vieques, Puerto Rico[END_REF][START_REF] Rodriguez | Arqueologia de Punta Candelero, Puerto Rico[END_REF][START_REF] Roobol | Petrography and source of some Arawak artifacts from Jamaica[END_REF]. However, the Caribbean islands form a fragmented, varied environment, both in terms of geology and geography, but also in terms of our knowledge of Amerindian lapidary production. In order to consolidate our knowledge about this part of the archaeological record, and to integrate into the archaeological corpus the lapidary productions found in the sites of the French Caribbean islands, recent data have been produced for the lapidary artifacts coming from several archaeological sites such as Gare Maritime [START_REF] Queffelec | Local production and long-distance procurement of beads and pendants with high mineralogical diversity in an early Saladoid settlement of Guadeloupe (French West Indies)[END_REF], Vivé, Morel and Anse à la Gourde (Queffelec et al., 2020a). These data incorporate mineralogical characterizations and comprehensive iconographic documentation, and supplement to a pioneering work on Antigua [START_REF] Murphy | Pre-Columbian gems and ornamental materials from Antigua, West Indies[END_REF] and recent work published for Pearls and several Dominican Republic sites (Falci et al., 2020b(Falci et al., , 2020a)). Combining these robust data and extensive bibliographic research, it has been possible to create an integrated database for Ceramic age lapidary ornaments in the Antilles that will allow for a better understanding of the geographical, temporal and cultural dynamics associated with these particular material productions in the future.

Results

Recent work in the French islands of the Antilles has made it possible to inventory, document and characterize the entire production of stone ornaments from the Ceramic period. This includes the Early Ceramic sites of Gare Maritime, Vivé, Morel and Hope Estate, as well as the Late Ceramic site of Anse à la Gourde (Queffelec et al., 2020a(Queffelec et al., , 2020b[START_REF] Queffelec | Local production and long-distance procurement of beads and pendants with high mineralogical diversity in an early Saladoid settlement of Guadeloupe (French West Indies)[END_REF]. In addition to these sites, which have yielded several dozen objects and have been published elsewhere, many sites that yielded smaller numbers of artifacts have also been studied (Figures 12345). This further highlighted the diversity of materials and types of artefacts produced, particularly in Early Ceramics: beads, pendants, bead-pendants, made of carnelian, diorite, amethyst, serpentine, calcite and other gems. All these objects were subjected to mineralogical characterization, mainly by Raman spectroscopy, but also by other methods when necessary. These analytical approaches have been particularly important in clarifying the mineralogical natures of the rocks used by the Amerindians. This is illustrated, for example, by the demonstration of the use of sudoite, a mineral of the chlorite family, for the production of several objects found in several archaeological sites. This material, unknown in the world as a gemstone, was thus the subject of a particular study in order to confirm this gemological attribution (Queffelec et al., 2021a). All these first-hand data and data from an intensive bibliographic search were integrated into a database. The structure of this database is based on 4 tables (islands, sites, beads, sources) and has been published in a dedicated article (Queffelec et al., 2021b) following recommendation by Peer Community In Archaeology [START_REF] Marwick | Open data on beads, pendants, blanks from the Ceramic Age Caribbean[END_REF]. The coding system in the database (also used in Figures 1 to 5) is constructed as follows. It starts with two letters corresponding to the island on which it was found, followed by a two-digit number corresponding to the number of the site on that island, and finally a three-or four-digit number representing the number of the object in that site. Thus, object GD-02-003 corresponds to the third object (003) studied from Morel (02), an archaeological site located in Guadeloupe (GD).

In order to disseminate this information to all interested researchers, online tools were used. This database is available online in English and French as a spreadsheet (https://osf.io/qbnkm and https://osf.io/zpwt8 respectively), and in the form of a Filemaker online database (Figure 6) accessible at this address: https://fm02.db.huma-num.fr/fmi/webd/PACEA_PAAF (clic on « se connecter en tant qu'invité » below the blue button and use the quick search bar for example). The geographical distribution of these lapidary artifacts can also be viewed on the online geographic information system (GIS) platform ArkeoGIS (Figure 7) (https://arkeogis.org/en/), and it can be interrogated on OPENARCHAEO (http://openarchaeo.huma-num.fr/explorateur/home). Users must register first to access this geographical application, since it is controlled in order to prevent archaeological looting. We hope to disseminate this work, share it with the scientific community and, if possible, bring together colleagues with other primary data at their disposal. This work has so far identified 8103 artifacts from 95 sites (Table 1), spread over the entire Caribbean arc (Figure 8). Concerning the sites located on the French islands, some of which unpublished, it has been possible to identify 21 of them in the Guadeloupe -Marie-Galante -La Désirade archipelago, 7 in Martinique, and 5 in Saint-Martin (Figure 9). Despite some difficulties in attributing a specific archaeological period of the Ceramic age for some artifacts or even some sites, the distribution of the number of sites and objects for each period shows an extremely marked preference for the Early Ceramic and, to a lesser extent, the Middle Ceramic period (Figure 10). It is interesting to compare this pattern with the number of known archaeological sites in the Caribbean Arc, which increases between the Early and Late Ceramic periods [START_REF] Curet | Caribbean paleodemography: population, culture history, and sociopolitical processes in ancient Puerto Rico[END_REF][START_REF] Hofman | The Post-Saladoid in the Lesser Antilles[END_REF][START_REF] Keegan | The Caribbean before Columbus[END_REF], what reinforce the fact that there is a high concentration of site and artifacts in the earlier period. We also note that the distribution of objects per siteis not at all homogeneous and that only a few sites form the majority of the corpus inventoried, when the vast majority of the sites have yielded less than 10 artifacts (Figure 11). Only 5 sites really stand out as having provided more than 500 objects each: Sorcé -La Hueca, Pearls, Sorcé, Trants and Punta Candelero, three of which are located in Puerto Rico/Vieques. Beads dominate the inventory to a large extent (Figure 12), being twice as numerous as pendants, followed to a lesser extent by fragments of raw material (which correspond to any fragment of rock most probably intended for lapidary production but not yet having reached the preform stage). This relative scarcity of initial elements and waste from the chaîne opératoire, although it may be a real archaeological fact, could also be linked to the lack of recording/mentioning of these objects at the various stages of archaeological research, from excavation (absence of sieving) to publication (no mention of raw materials). Only five sites account for 88% of the objects classified as raw material, but it should be noted that the absence of a count of raw material shards in the very rich sites of Puerto Rico and Vieques, although indicated as being present "in large quantities" (Narganes Storde, 1995), obliterates a certainly not insignificant part of the archaeological record because they cannot be included in our database. The more rarely identified object types include unperforated plates, bead-pendants, which are mostly longitudinally and transversely perforated cylinders, a few unidentified objects (often because they are still preforms), and, finally, labrets, which are rather thin and long objects, but whose attribution as a labret remains speculative. Among the different types of beads, the most numerous are the discoid kind, followed by cylindrical ones (Figure 13). Plano-convex beads were found in similar numbers to barrelshaped beads, the other types being much rarer. In view of the collections studied in detail on the French Caribbean islands, this third position of plano-convex beads is surprising. However, it should be noted that their number is mainly driven by a few sites such as Sorcé -La Hueca which accounts for 72% of their numbers, while barrel-shaped beads are more evenly distributed although the Pearls site accounts for 56% of the total. The lapidary production of pendants in the Ceramic period is largely dominated by zoomorphic forms (91% of the corpus) among which frogs are widely prevalent (Figure 14). Geometric pendants of various shapes are the second most common type. Other forms are very rare and not very diverse. It must also be noted that, although this database certainly lists a very large proportion of the archaeological sites of the Ceramic period that have yielded stone ornaments, there is still considerable room for improvement in the data it contains. Another current limitation to be highlighted I that the distribution of the raw materials used clearly indicates the predominance of the indeterminate in the database: they represent almost half of the corpus! This lack of knowledge further underlines the need for additional studies including precise raw material identification to increase the strength of our interpretations of diffusion networks in the future, especially in the richer sites that have not been subject to such work. When focusing on the different raw materials identified so far, the corpus is largely dominated by diorite and serpentine, followed quite far behind by rock crystal and then by a group formed by calcite, carnelian, jadeite, turquoise and amethyst (Figure 15). As this database is largely based on an inventory drawn from the literature, these proportions of material differ from those that can be established solely from the set of sites located in the French islands where the raw materials have been systematically analyzed (Queffelec et al., 2020a(Queffelec et al., , 2020b[START_REF] Queffelec | Local production and long-distance procurement of beads and pendants with high mineralogical diversity in an early Saladoid settlement of Guadeloupe (French West Indies)[END_REF]. In particular, the work carried out on these sites did not identify a single jadeite bead or pendant, and it is very likely that several of the green materials that we were able to identify through Raman spectroscopy (paragonite, sudoite, clinochlore for example) could be found in the "jadeite category" of several authors who did not carry out these mineralogical analyses.

Discussion et conclusion

By combining the analytical characterization and comprehensive documentation of Ceramic period lapidary production in the French islands of the Antilles with an intensive review of the scientific literature on the subject for the other islands, the establishment of a database of lapidary production at the scale of the islands of the Antilles allows for the robust validation of ideas that have already been established for a long time by archaeologists of the Caribbean area. It is now possible to attest and quantify that the Early Ceramic and Early/Middle Ceramic periods provided far more lapidary artifacts than the more recent periods, as already observed by several authors [START_REF] Bérard | The Saladoid[END_REF][START_REF] Hofman | Networks Set in Stone: Archaic-Ceramic interaction in the early pre-Colonial northeastern Caribbean[END_REF][START_REF] Hofman | Island Rhythms: The Web of Social Relationships and Interaction Networks in the Lesser Antillean Archipelago between 400 B.C. and A.D[END_REF][START_REF] Knippenberg | Stone artefact production and exchange among the Lesser Antilles[END_REF][START_REF] Rodríguez Ramos | The humanization of the insular Caribbean[END_REF]. Beyond this first chronological approach, such an inventory also makes it possible to highlight other information in terms of typology and raw materials used: beads are twice as numerous as pendants, discoid beads largely dominate over cylindrical beads, pendants are more than 90% zoomorphic and 85% of them represent frogs. Finally, this work has also highlighted the very important work that remains to be done on these objects, both in terms of mineralogical characterization (almost half of the beads are not determined at this level) and detailed typological documentation. Further studies of these data are planned in order to compare more precisely the distribution of types, raw materials, and combinations of these two parameters, in time and space. For this, diversity analyses, ordering by means of seriations, comparisons thanks to correspondence analysis comparisons, and network analyses will be performed. Such documentation could also allow stylistic studies on pendants, by making a large number of photographs accessible to the greatest number of people, although on this point too, much work remains to be done.

Our proposal to use open online tools will hopefully encourage Caribbean archaeologists to participate in making the database even more exhaustive. It is indeed always possible to add new data and this database will be updated regularly, in particular during future studies of archaeological series from sites located outside the French islands of the archipelago.

List of figures

List of tables

Figure 1 .

 1 Figure 1. Photos and technical drawings of lapidary artifacts from diverse archaeological sites in Guadeloupe (Anse Vinaigri, Anse Ste Marguerite and Butel)

Figure 6 .

 6 Figure 6. Screenshot of one of the entries in the online database (artifact MA-02-037).

Figure 7 .

 7 Figure 7. Screenshot of the geographical distribution of lapidary artifacts on the ArkeoGIS platform.

Figure 8 .

 8 Figure 8. Location of the sites registered currently in the database, with the spot size relative to the number of artifacts inventoried.

Figure 9 .

 9 Figure 9. Distribution of the archaeological sites documented in the database for the archipelago of Guadeloupe (A), Saint-Martin (B) and Martinique (C)

Figure 11 .

 11 Figure 11. Distribution of the number of artifacts per site with an insert, in Log scale, of the sites having delivered at least 10 artifacts.

Figure 12 .

 12 Figure 12. Distribution of the number of artifacts inventoried by major types.

Figure 13 .

 13 Figure 13. Distribution of the number of beads for each of the subtype.

Figure 14 .

 14 Figure 14. Distribution of the number of pendants for each subtype. The contribution of frogs to zoomorphic pendants is shown as a superimposed pattern (black dots).

Figure 15 .

 15 Figure 15. Distribution of the number of artifacts by raw material with an insert, in Log scale, of the raw materials used to produce at least 10 artifacts.

Figure 1 .Figure 4 .

 14 Figure 1. Photos and technical drawings of lapidary artifacts from diverse archaeological sites in Guadeloupe (Anse Vinaigri, Anse Ste Marguerite and Butel) Figure 2. Photos and technical drawings of lapidary artifacts from diverse archaeological sites in Guadeloupe (Anse à la Barque and Anse Bertrandl) Figure 3. Photos and technical drawings of lapidary artifacts from diverse archaeological sites in Guadeloupe (Ilet Gosier, Grand Carbet, Les Mineurs, Cathédrale Basse-Terre, Allée Dumanoir) Figure 4. Photos and technical drawings of lapidary artifacts from diverse archaeological sites in Marie-Galante (Cocoyer, Stade José Bade, Cadet 2) and La Désirade (Morne Cybèle, Petite Rivière) Figure 5. Photos and technical drawings of lapidary artifacts from diverse archaeological sites in Martinique (Anse Trabaud, Rue Pory-Papy, Macabou, Le Diamant, Rue Perrinon-Doume) Figure 6. Screenshot of one of the entries in the online database (artifact MA-02-037).

Figure 7 .

 7 Figure 7. Screenshot of the geographical distribution of lapidary artifacts on the ArkeoGIS platform.

Figure 8 .

 8 Figure 8. Location of the sites registered currently in the database, with the spot size relative to the number of artifacts inventoried.

Figure 9 .

 9 Figure 9. Distribution of the archaeological sites documented in the database for the archipelago of Guadeloupe (A), Saint-Martin (B) and Martinique (C) Figure 10. Distribution of the number of sites (A) and artifacts (B) in the regional database by archaeological period.

Figure 11 .

 11 Figure 11. Distribution of the number of artifacts per site with an insert, in Log scale, of the sites having delivered at least 10 artifacts.

Figure 12 .

 12 Figure 12. Distribution of the number of artifacts inventoried by major types.

Figure 13 .

 13 Figure 13. Distribution of the number of beads for each of the subtype.

Figure 14 .

 14 Figure 14. Distribution of the number of pendants for each subtype. The contribution of frogs to zoomorphic pendants is shown as a superimposed pattern (black dots).

Figure 15 .

 15 Figure 15. Distribution of the number of artifacts by raw material with an insert, in Log scale, of the raw materials used to produce at least 10 artifacts.

Table 1 .

 1 List of archaeological sites currently included in the database.

					Number
	Island	Site	Period	Culture	of	Reference
					artifacts
	Anguilla	Whitehead's Bluff	Archaic	Casimiroïde / Ortoiroïde	2	Crock_2019
	Antigua	Elliot's (PH-03)	(Middle?) Ceramic	Middle Cedrosan Saladoid	64	Murphy-etal_2000
	Antigua	Mill Reef (PH-01)	Late Ceramic	Mamoran Troumassoid (Mill Reef)	2	Hoffman_1970
	Antigua	Royall's (JO-11)	(Middle?) Ceramic	Middle Cedrosan Saladoid	199	Murphy-etal_2000
	Antigua	Doig's (PA-15)	Early/Middle Ceramic	Early/Middle Cedrosan Saladoid	43	Gent&deMille_2003
	Antigua	Winthorpe Bay	(Middle?) Ceramic	Middle/Late Cedrosan Saladoid	1	deMille-etal_1999
	Aruba	Tanki Flip	Late Ceramic	Dabajuroid	2	Rostain_1995
	Barbuda	Seaview	Early/Middle Ceramic	Early/Middle Cedrosan Saladoid	18	Kendall-etal_2011
	Bahamas	Minnis-Ward	Late/Final Ceramic	Lucayan	1	Blick-etal_2010
	Bonaire	Wanapa	Ceramic		3	Haviser_1990
	Carriacou	Grand Bay	(Middle?) Ceramic	Late Cedrosan Saladoid	17	Sutty_1990
	Curacao	De Savaan	Ceramic		4	Haviser_1990
	Dominique	Soufrière	Early Ceramic	Early Cedrosan Saladoid	1	Bérard_2009
	Guadeloupe	Gare maritime	Early Ceramic	Huecan Saladoid	59	Romon-etal_2013
	Guadeloupe	Morel	Early/Middle Ceramic	Cedrosan Saladoid/huecan	61	Delpuech-etal_1996
	Guadeloupe	Anse à la Gourde	Late Ceramic	Troumassoid	28	Delpuech-etal_1997
	Guadeloupe	24 rue Schoelcher	Early Ceramic	Early Cedrosan Saladoid	1	Etrich_2003a
	Guadeloupe	Allée Dumanoir	Early/Middle Ceramic	Early/Middle Cedrosan Saladoid	2	Etrich_2003b
	Guadeloupe	Anse à la Barque	Undetermined	Undetermined	1	Turpin_2015
	Guadeloupe	Anse Bertrand	Undetermined	Undetermined	2	Turpin_2015
	Guadeloupe	Anse Ste Marguerite	Undetermined	Undetermined	4	Delpuech_2007
	Guadeloupe	Cathédrale	Early/Middle Ceramic	Saladoid	6	Bonnissent&Romon_2004
	Guadeloupe	Butel	Undetermined	Undetermined	1
	Guadeloupe	Grand Carbet	Early Ceramic	Early Cedrosan Saladoid	1	ToledoIMur_2003
	Guadeloupe	Ilet Gosier	Late/Final Ceramic	Troumassoid	1	Romon-etal_2003
	Guadeloupe	Les Mineurs	Undetermined	Undetermined	3
	Guadeloupe	Plage de Roseau	Contact	Cayo	2	LeLay_2013
	Guadeloupe	Anse Vinaigri	Undetermined	Undetermined	3
	Guadeloupe	La Ramée	Early/Middle Ceramic	Saladoid	3	Casagrande_2013
	Grenada	Pearls	Early/Middle Ceramic	Saladoid	1412 Murphy-etal_2000
	Grenada	Grand Anse Beach	Early/Middle Ceramic	Saladoid	3	Cody_1993
	Grenada	Caliviny Island	Early/Middle Ceramic	Undetermined	3	Bullen&Bullen_1968

Table 1 .

 1 List of archaeological sites currently included in the database.

Acknowledgments

The authors of this article wish to thank all the archaeologists who conducted the fieldwork and therefore permitted this type of study to exist. We also thank all the people who made it possible for us to access the archaeological artifacts in museums and curation centers: Susana Guimaraes, Katarina Jacobson, and Jean-Luc Bernos (Musée Edgar Clerc), Thierry Dorival (DAC Martinique), Gwenola Robert (DAC Guadeloupe), François Petit (owner of Hope Estate), Peggy Deroze and Claudia Daribo (Musée d'Archéologie et de Préhistoire de Martinique). The study of the archaeological artifacts would not have been possible without the help of Christian Stouvenot et Gwenola Robert (DAC Guadeloupe), Susana Guimaraes, Katarina Jacobson (Musée Edgar Clerc), Thierry Dorival (DAC Martinique), et François Petit (owner of Hope Estate). We also would like to thank Philippe Jullien and Jean-Philippe Droux for their help with the integration of the database in ArkeoGIS. This work was carried out within the framework of the Programme Collectif de Recherche Parures Amérindiennes dans les Antilles Françaises, funded by the Ministry of Culture (DAC Guadeloupe and DAC Martinique) and the Region of Guadeloupe. This work benefited from the environment of the GPR Human Past.


Yacoub Dali

Marc Tupin

Philippe Bossis

Michèle Pijolat

Françoise Valdivieso

Application of a magnetic suspension balance to the oxidation study of the zirconium based alloys under high pressurewater vapour

Keywords: high pressure, kinetics, magnetic suspension, oxidation, steam, thermogravimetry, Zircaloy-4

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Zirconium based alloys are widely used as cladding material for fuel rods of water-cooled nuclear reactors because of their low neutron capture cross-section, good corrosion resistance in operating conditions and satisfying mechanical properties. The first family of zirconium based alloys is the Zircaloys, starting with Zircaloy-1, a sponge zirconium containing 1.5 mass% Sn. Tin was added to eliminate the negative effect of nitrogen. This element increases the strength but reduces the corrosion resistance [START_REF] Parfenov | Corrosion of zirconium and zirconium alloys in Myers[END_REF]. A hazardous pollution of a Zircaloy-1 with an austenitic stainless steel (18% Cr and 8% Ni) led to the alloy called Zircaloy-2 which contains 1.2-1.7 mass% of Sn, 0.07-0.2 mass% Fe, 0.05-0.15 mass% Cr, 0.09-0.16 mass% O and 0.03-0.08 mass% Ni. Chromium and iron improve the corrosion resistance in water and steam, and the nickel improves the corrosion resistance in steam [START_REF] Cox | [END_REF]. The last alloy developed is the Zircaloy-4 which does not contain any nickel because of its detrimental role in the hydrogen absorption [START_REF] Hillner | Hydrogen absorption in Zircaloy during aqueous corrosion, Effect of Environment[END_REF].

Numerous studies in the last decades were performed to develop these alloys and to elucidate their particular corrosion kinetic behaviours [START_REF]Waterside Corrosion of Zirconium in Nuclear Power Plants[END_REF]. For the first kinetic regime called the pre-transition, it is generally assumed that the growth of the oxide during this stage is controlled by the diffusion of the oxygen through the oxide. The mass gain curve in this case corresponds to a sub-parabolic law [START_REF] Kofstad | High temperature corrosion in[END_REF]. After this period, an acceleration of the mass gain results in the change in the oxide morphology. This phenomenon is called kinetic transition. Finally the mass gain law tends to become linear.

The oxidation mechanism of Zircaloy-4 is not yet well understood. For a long time, studying the corrosion process of zirconium alloy in high temperature and high pressure conditions has been done by discontinuous measurements of the mass gain. To improve the understanding of this corrosion process it is of great interest to perform at high pressure steam in situ measurements. Recently, we designed and built a new experimental device, based on a Rubotherm magnetic suspension balance, that allows to access continuously to the mass gain during the oxidation of zirconium based alloys in pure water vapour under high pressure. In this paper, experimental installation will be presented in first place and its principle will be detailed. Secondly, the qualifications tests will be described and analyzed.

Previously, some classical thermogravimetric microbalance tests on zirconium alloys were carried out but only at low partial pressure [START_REF] Tupin | [END_REF][START_REF] Pétigny-Putigny | Comparaison de l'oxydation de deux alliages à base de zirconium par diffraction des rayons X in-situ et ex-situ: texture, phase, contrainte[END_REF]. For high pressure, the corrosion experiments performed in autoclaves consist of discontinuous mass gain measurements at different oxidation time. The only way to follow in situ the mass gain in these extreme conditions is to separate the balance and the measurement cell, from the corrosive medium and to couple magnetically the balance to the sample as proposed by Gast in 1959 [START_REF] Th | [END_REF]9].

Experimental

As described in Figure 1 the high pressure water vapour device is made up of different parts: S St te ea am m g ge en ne er ra at to or r The steam generator consists of a one-liter autoclave connected to different atmospheres, inert gas (helium or argon), mixed helium-oxygen at 20% and diluted hydrogen at 3% in helium. The temperature, the hydrogen partial pressure and the whole pressure are controlled by thermocouples and pressure sensors. With this autoclave 150 bars of pure water vapour can be generated (maximal pressure of the Magnetic Suspension Balance (MSB)).

H He ea at ti in ng g s sy ys st te em m The oxidation experiments are done around 400-450°C under high pressure steam. A three-zone furnace guarantees an isothermal zone of about 12 cm, covering the entire surface of tested samples (2•10 cm). Heating collars surrounding the different connecting tubes (reaching 400°C) allow to avoid condensation. They are regulated to be always over the water liquid-vapour equilibrium temperature. M Ma ag gn ne et ti ic c s su us sp pe en ns si io on n b ba al la an nc ce e Developed by RUBOTHERM and based on a magnetic coupling system, the magnetic suspension balance is composed of:

• the microbalance (accuracy of 10 -5 g);

• the electromagnet;

• the magnetic suspension composed of a permanent magnet, a sensor core and a coupling cone;

• the coupling house at which the sample is hung via a platinum wire.

The electromagnet is suspended to the microbalance, and maintains in levitation the magnetic suspension. As shown on Figure 2, three positions are used to perform the mass gain measurements:

• OFF position: the magnetic suspension is not attracted by the electromagnet and so lays down on the coupling house;

• Zero position (ZP): there is no coupling between the magnetic suspension and the sample, this position corresponds to the tare;

• Measure position (MP): the sample is coupled to the magnetic suspension; the mass of the sample is deduced from the current flowing in the electromagnet (and measured in this position by the microbalance).

The succession of the two positions (ZP-MP) leads to in situ acquirement of the mass gain. T Th he er rm mo os st ta at ti in ng g c ci ir rc cu ui it t As described in the Figure 1, two thermostating circuits are used in this experiment. In the first one, a special high temperature oil flows inside the double wall chamber that allows to maintain the temperature up to 315°C in this part of the device and thus to avoid the water condensation. The second circuit is a cooling one; liquid water at 15°C flows inside a cooler diaphragm placed underneath the microbalance limiting the temperature in the microbalance. The circulation of the cold water is performed by a cryothermostat. P Pr re es ss su ur re e a an nd d t te em mp pe er ra at tu ur re e s se en ns so or rs s A two points temperature sensor is localised underneath the sample. These thermocouples allow the acquisition of two temperatures near the sample (the points are 5 cm apart). The temperature of the double wall chamber is controlled by an external thermocouple Two pressure sensors -connected to the autoclave cell -allow measuring the variation of the hydrogen pressure and of the total pressure (water vapour, inert gas and hydrogen). C Co on nn ne ec ct ti in ng g t tu ub be es s a an nd d v va al lv ve es s All the connections and valves are in a stainless steel tube adapted to high temperature and high pressure conditions, the introduction of gases in the tube-reactor is regulated by a high temperature and high pressure long valve (AT231). D Da at ta a a ac cq qu ui is si it ti io on n The Mess pro software allows to operate manually and also to perform measurement by programming different time segments (the maximum acquisition time is 2 months). All the electronic components are connected to the microbalance at ambient conditions.

Qualification tests

Specific tests have been performed to validate the experimental measurements, in particular signal accuracy, and to estimate the temperature stability, the buoyancy effect, the signal reproducibility and the effect of natural convection. The qualification tests were conducted on a Zircaloy-4 experimental alloy (sheet of 0.5 mm thickness) provided by AREVA NP CEZUS. Its composition is given in Table 1.

S Si ig gn na al l a ac cc cu ur ra ac cy y To validate the mass gain values measured by the thermobalance (magnetic suspension balance), three corrosion experiments at 415°C and different pressures have been performed on Zy4. These values of mass gains have been compared to mass variations of the samples measured after and before the corrosion with a classical balance (Sartorius) working at room temperature. The mass gain values obtained with the two techniques and their discrepancy are given in It should be noticed that about 2 g difference in mass between the ordinary balance and the thermobalance comes from the fact that the mass measured by the microbalance takes into account the mass of the sample but in addition the mass of the hook and the coupling house. From Table 2, the mass gain determined by the ordinary balance is in good agreement with the mass variation measured with the thermobalance. In consequence, the signal accuracy of our experimental device is validated up to 50 bars. S St ta ab bi il li it ty y o of f t te em mp pe er ra at tu ur re e Our experiments are generally done under isothermal conditions over a long time period. That is the reason why different tests were done to verify the thermal stability inside the tube-reactor with a thermocouple located 1 cm underneath the sample. The temperature variations during 10 days of experiment for different temperature plateaus are shown in Figure 3. The temperature fluctuations do not exceed ± 2 °C during the isothermal plateau, and temperature stability is consequently relatively satisfying. B Bu uo oy ya an nc cy y e ef ff fe ec ct t Doing experiments under high pressures needs a quantification of the buoyancy effect. Indeed, by increasing the pressure, the density of the reactive gas changes leading to an apparent decrease in the mass gain as observed in Figure 4 for five pressure plateaus at respectively 19, 30, 50, 80 and 100 bars, at 415°C. The apparent mass loss resulting from the buoyancy effect can be theoretically calculated and then compared to the experimental results.

Pressure increase/bar

Cal. Mass loss due to the buoyancy/g Mass loss given by the experimental results/g 19-30

4.2 10 -3 5.5 10 -3 30-50 8.9 10 -3 9.0 10 -3 50-80 14.8 10 -3 20 10 -3 80-100 11.3 10 -3 11 10 -3

T Ta ab bl le e 3 3: : V Va al lu ue es s o of f l lo os ss s m ma as ss s, , t th he eo or re et ti ic ca al ll ly y c ca al lc cu ul la at te ed d a an nd d e ex xp pe er ri im me en nt ta al ll ly y m me ea as su ur re ed d

Table 3 and Figure 5 sum up the experimental mass loss and the calculated one considering the Archimedes' principle. The apparent mass change during a pressure change from P1 to P2 due to buoyancy was calculated by the following formula (1):

2 1 ( , ) ( , ) m V P T P T ρ ρ Δ = × - ( 1
)
where Δm is the mass variation during the pressure drop (mg), V the volume of the object (cm 3 ), and ρ(P 1 ,T) and ρ(P 2 ,T), the densities of the steam, respectively at the pressures P 1 and P 2 and the temperature T (mg cm -3 ). For instance, for the pressure change from 30 to 50 bars, we have to take into account the buoyancy effect on the masses of the sensor coil, the coupling house and the sample. But all these components are not submitted to the same temperature, the sensor coil and the coupling house are actually at 320°C while the sample is corroded at 415°C. Consequently, the mass variations per volume unit are not the same because the steam densities are different at these temperatures. For this pressure drop, the mass loss can be detailed for each component:

• sensor coil:

⏐Δm⏐=V×⏐ρ(50b,320°C)-ρ(30b,320°C)=0.82×(20.8-10.8)=0.82mg with V=0.82cm The superheated water densities come from [START_REF] Meyer | ASME Steam Tables in American Society of Mechanical Engineers[END_REF]. The masses measured during the ZP measurement step and the MP one describe respectively the mass loss of the sensor coil and those of the three preceding elements. The mass variations indicated on the Figure 5 correspond to the difference between the ZP measurement and the MP one. Thus the mass loss due to the buoyancy observed during the pressure drop results only from those calculated on the coupling house and the sample. It is equal to 8.9 mg and is so very close to the experimental value (Table 5).

As shown on Figure 5, except for the surrounded point with approximately 20 mg, theoretical calculations are in relatively good agreement with the experimental results. It is so possible to correct the mass signal from the buoyancy effect during any experiment that includes temperature or pressure modification. 6, the measurements for two experiments performed in the same conditions (415°C and 2 bars) are clearly reproducible. The oxide thickness is deduced from the mass gain by the following formula (2):

2 2 2 0 1 ZrO ZrO O M m X S P M Δ = × (2)
where ρ ZRO2 is the mass density of zirconia (5.6 g cm -3 for the monoclinic phase), M ZrO2 and M O2 are respectively the molar masses of zirconia and oxygen (g mol -1 ), S 0 is the sample reacting surface (cm 2 ). N Na at tu ur ra al l c co on nv ve ec ct ti io on n Natural convection phenomena are often frequent and observed during thermogravimetric measurements because of gas density variations inside the experimental system, related to cold points and more generally to temperature differences between two parts of the device. The Rayleigh adimensional number allows to estimate qualitatively the convection phenomena in a given experimental system. It is defined as follows:

2 3 p a g C b T R K β ρ η Δ =
with g: acceleration of gravity (m s -2 ), β: thermal expansion coefficient (K -1 ), C p : heat-storage capacity (J kg -1 K - 1 ), ρ: density (kg m -3 ), b: length of the nonisothermal zone (m), ΔT: temperature difference (K), η: dynamic viscosity of gas (Pa s), K: thermal conductivity of gas (W m -1 K -1 ).

According to the literature [START_REF] Meyer | ASME Steam Tables in American Society of Mechanical Engineers[END_REF], the threshold value of the Rayleigh number, beyond which the natural convection must be taken into account in this kind of reactors, is about 40000. Rayleigh number values were calculated for three different temperatures, 300, 400 and 500°C, and three different pressures 10, 50 and 100 bars applied to the sample, considering that the temperature difference (DT) between the magnetic suspension and the sample is in general around 100°C. The results of these calculations are presented in Table 4.

The Rayleigh number increases with increasing pressure and the obtained values are always above the threshold which means theoretically that convection should take place in our tube-reactor. Whatever, when the pressure is above 30 bars, in practice, the convection currents disturb significantly the mass signal. For instance, as shown on Figure 7, the thickness variation vs. time for an isobaric experiment at 50 bars is relatively noisy compared to the one at 2 bars. For this pressure range, it is thus important to limit these convection currents as well as possible. In order to reduce the natural convection phenomena, the aim is to break the currents of convection flowing inside the reacting chamber, with a protection cage around the sample, by creating obstacles such as an alumina tube.

Conclusions

A high pressure thermogravimetric installation was developed allowing to study in situ the oxidation kinetics of the zirconium based alloys under pure water vapour until 50 bars of pressure at 415°C. Accuracy of measurements is about 5•10 -5 g under 2 bars, and 10 -4 g under 50 bars. The reproducibility of measurements was verified and the deviation regarding post test weighing measurements at room temperature is about 5•10 -5 g.

  Figures

Table 2 .

 2

			Sn/	Cr/	Fe/		O/	Zr/
	Alloy				Mass%
	Zircaloy-4	1.46	0.107	0.22		0.126	Bal
	T Ta ab bl le e 1 1: : C Ch he em mi ic ca al l c co om mp po os si it ti io on n o of f t th he e s st tu ud di ie ed d a al ll lo oy y
		'Ordinary' balance	Thermobalance		Mass difference
	P/bar	initial mass/g	final mass/g	Δm1/g	initial mass/g	final mass/g	Δm2/g	Δm2-Δm1/g
	2	8.10108	8.11145	0.01037	10.21394	10.22435	0.01041	0.00004
	10	8.13409	8.14889	0.0148	10.23718	10.25195	0.01477	-0.00003
	50	7.87105	7.88219	0.0114	9.95504	9.96155	0.01115	0.00001
	T Ta ab bl le e 2 2: : M Ma as ss s g ga ai in ns s o ob bt ta ai in ne ed d w wi it th h b ba al la an nc ce e a at t r ro oo om m t te em mp pe er ra at tu ur re e ( (' 'o or rd di in na ar ry y' ' b ba al la an nc ce e) ) a an nd d w wi it th h t th he er rm mo ob ba al la an nc ce e f fo or r
	e ex xp pe er ri im me en nt ts s a at t d di if ff fe er re en nt t p pr re es ss su ur re es s

Acknowledgements

The authors acknowledge the technical support from Paul Finet of SPECTRA SERIA and from Michael Holm of Rubotherm.


Ophélie Renaud

Dylan Gageot
email: dgageot@yubik.io

Karol Desnos

Jean-François Nezan

SCAPE: HW-Aware Clustering of Dataflow Actors for Tunable Scheduling Complexity ⋆

Keywords: Dataflow model, Hierarchy, Granularity, Clustering

This paper introduces a fast method to generate high performance parallelized code from a dataflow specification of an application. Dataflow Models of Computation (MoCs) are efficient programming paradigms for expressing the parallelism of an application. Traditionally, mapping and scheduling methods for dataflow MoCs rely on complex graph transformations to explicit their parallelism which can result in complex graph for embarrassingly parallel applications. For such applications, state-of-the-art mapping and scheduling techniques are prohibitively complex, while the exposed parallelism often exceeds the parallel processing capabilities of the target architecture. We propose SCAPE, an automated method to control the complexity of the pre-scheduling graph transformation by using information from the architecture and application models. By decreasing the complexity of the graph, the mapping scheduling task is accelerated at the potential expense of the produced schedule. Our method offers a limited and controlled decrease of the schedule quality while enabling mapping and scheduling execution time between 1 and 2 orders of magnitude faster than state-of-the-art techniques.

Introduction

Digital signal processing technology emerged in the 1960s and has grown rapidly, becoming more complex over the years, particularly with the arrival of machine learning applications a decade ago. To meet the ever-increasing need for computing power and speed of execution of these applications, developers first sought to increase the frequency of individual Processing Elements (PEs) and then turned to heterogeneous multicore embedded systems.

Exploiting in an optimized way the maximum parallelism of such multicore target architectures is very challenging. The development of parallel code is tedious and is not adapted to manage hardware and software upgrades during the exploitation phase of the project. Tools such as Simulink [START_REF] Klikpo | Modeling multi-periodic simulink systems by synchronous dataflow graphs[END_REF] and Xilinx AI Engine Technology [START_REF] Alok | Architecture apocalypse dream architecture for deep learning inference and compute-versal ai core[END_REF] are then investigated to automate the rapid deployment of new algorithms on the computing system. Both tools are based on the dataflow approach consisting in the modeling of the algorithms by a graph in which nodes, called actors, represent the calculations and directed arcs, called First In First Out queue (FIFO) buffers, represent the data, called tokens, exchanges between nodes.

The automated generation of parallel code from such dataflow models requires solving several NP-Complete problems, especially for resource allocation. Calculations are distributed on the PEs of the target architecture and will read and write, during the execution of an application, on FIFO buffers assigned to a range of memory addresses. The resource allocation choices can be made at compile time or at runtime. The allocation at runtime leads to performance overhead and unpredictable application behavior. For these reasons, this paper investigates methods that allocate resources at compile time, during software synthesis. The software synthesis process is responsible for translating a dataflow model into a prototype.

Classic resource allocation methods involve three phases: The mapping consists in distributing actors on the PEs of the target. The Scheduling consists in ordering the execution of actors on the PEs. The Timing associates to each actor a start time and an end time, useful to calculate the long time average throughput, also called latency, of the application. The time required for the mapping and scheduling process grows exponentially with the number of PEs and the number of nodes and edges of the dataflow graph [START_REF] Lee | Scheduling strategies for multiprocessor real-time dsp[END_REF].

The Scaling up of Clusters of Actors on Processing Element (SCAPE) method is introduced in this paper which is a hierarchy-based clustering method that transforms an application to match its degree of parallelism to the parallel computation capabilities of the targeted architecture. The method offers as many clustering configurations as there are hierarchy levels in the Synchronous Dataflow (SDF) input graph which gives the user the possibility to choose the required granularity for a reduced software synthesis time.

Section 2 presents dataflow MoCs, the traditional mapping and scheduling method and the state-of-the-art clustering heuristics. Section 3 describes the proposed SCAPE method and the backbone of the resulting code. Section 4 outlines experimental results on several granularity clustering configurations showing a tradeoff between design space exploration time and produced schedule latency. Finally, Section 5 concludes this paper.

Context and related work

SDF based dataflow MoCs

The most studied dataflow MoC is the SDF [START_REF] Lee | Static scheduling of synchronous data flow programs for digital signal processing[END_REF] illustrated in the fig. 1, in which the integer numbers on the input and output ports of actors are the rate of tokens respectively consumed and produced by actors at each execution of the actors.

Actor Delay

Buffer FIFO

Rate of tokens produced

Rate of tokens consumed

C 1 1 1 A 2 B 2 D 1 E 1 2 2

Fig. 1. SDF MoC semantics

An SDF graph is usually transformed into an equivalent Single rate Directed Acyclic Graphs (SrDAG) [START_REF] Sih | Scheduling to account for interprocessor communication within interconnection-constrained processor networks[END_REF] to map and compute a periodic deadlock-free schedule iterated infinitely. A repetition of a periodic schedule is called a graph iteration. The single rate transformation consists in duplicating the actors by the number of firings specified by the schedule, and in adding special actors to distribute or gather tokens, so that the production and the consumption rates on each FIFO are equals. Then, DAG transformation consists in breaking cyclic data-paths. To be consistent, all cyclic data paths must contain at least one initial data token also called delays. Therefore, breaking a cycles means replacing FIFOs with delays by a pair of special actors which backup delayed tokens and read the backed up tokens.

An extension of the SDF model is the Parameterized and Interfaced SDF (PiSDF) model [START_REF] Desnos | PiSDF: Parameterized & Interfaced Synchronous Dataflow for MPSoCs Runtime Reconfiguration[END_REF]. In this paper, the feature of interest of the PiSDF model is its support for hierarchy. The hierarchy feature allows the internal behavior of actors to be specified by a subgraph instead of C code. The PiSDF MoC defines interfaces of a subgraph as input and output data ports of the parent hierarchical actor. Interfaces allow the transmission of tokens between hierarchical levels. The hierarchy is used to represent different levels of granularity of the computations that compose an application, the lower levels of hierarchy being the finer granularity.

These SDF based models have two main advantages justifying their interest. The first one is to express the three types of parallelism [START_REF] Zhou | Scheduling of parallelized synchronous dataflow actors[END_REF], two of which are used in this paper: task and data parallelism (pipeline being the third).

-Task parallelism is expressed by two actors belonging to parallel data-paths like actors A and B in the fig. 1. As there is no data-path between these actors, they can be fired at the same time. -Data parallelism is expressed when several firings of a single actor are independent from each other. If enough data tokens are present in the input FIFOs, then several firings can be executed concurrently. An example is the actor C in the fig. 1 which can be executed 2 times when A and B are executed 1 time.

The second advantage is that the model is independent of the target architecture. An application is represented once and executed on all types of architecture (single cores, multicores with shared or distributed memories, FPGA, etc.).

Classic flattening method

This paper focuses on tools that have chosen to allocate resources at compile time, also called static allocation. Software synthesis can be modeled by a workflow. The typical static scheduling workflow is composed of four main tasks: flattening, SrDAG transformation, mapping and scheduling, then code generation.

The flattening task of the workflow consists in putting all the actors of a graph at the same level which means all hierarchical actors are replaced by their subgraph. To keep the functionality of the application in fig. 2, token rates consumed and produced in the initial subgraph are scaled up on upper levels. Tools usually flatten the whole graph to execute the rest of the process which brings the finer level of granularity to the top-level graph.

The SrDAG transformation task is used to reveal parallelism on flattened graph fig. 2. It highlights the minimal number of firings of each actor to return the graph back to its original state given by the calculation of the Repetition Vector (RV) q. Here, actor B is fired 4 times per graph iteration, so q(B) = 4. It also emphasizes the interdependencies between the actors, which is useful to calculate the execution order of the actors, allowing them to be iterated infinitely without generating a deadlock. In the figure, the named nodes are the actor instances and the unnamed nodes are the special actors responsible for distributing or gathering the data tokens. The excessive complexity of the SrDAG increases the mapping opportunities which results in a better distribution of the computations on the different PEs and reduces the latency of the application on the target. Since the mapping opportunities are limited to the number of PEs in the architecture, exposing more parallelism than the number of PEs is unnecessary and time consuming. [START_REF] Pino | A hierarchical multiprocessor scheduling framework for synchronous dataflow graphs[END_REF]. This is why reducing the exposed parallelism to the number of PE will most likely be sufficient to fully exploit the architecture parallelism, while being much simpler to map and schedule.

Example 1. Considering a machine learning application: Squeezenet neural network whose SDF model is composed of 70 actors, its SrDAG transformation results in 5452 actors. Mapping this application on an architecture composed of 8 PEs with a greedy algorithms requires considering and evaluating 8 mapping choices. Here the degree of parallelism is up to 1000 which is an needlessly fine granularity.

Cluster of SDF actors

A way to reduce the complexity of mapping and scheduling algorithms is to reduce the number of actors to map in the srDAG, without altering behavior of the application. This reduction can be achieved using clustering techniques, wich transform the input graph by grouping actors with a particular behavior. Since grouping two or more actors into a single equivalent hierarchical actor may change the behavior of the application, or even create deadlocks, clustering rules have been introduced in [START_REF] Pino | A hierarchical multiprocessor scheduling framework for synchronous dataflow graphs[END_REF]. These rules are illustrated in Figure 3 where SDF graphs are represented with rectangular actors and the corresponding precedence graphs with round actors. A cluster must respect the execution order of the actors defined by the precedence rules (a), the initial tokens must be considered (b) and there must be no simple path from a node of the precedence graph to another one that contains more that one arc(c). A simple path is the one which does not visit any node along the path more than once. A method to cluster SDF actors is presented in [START_REF] Bhattacharyya | Apgan and rpmc: Complementary heuristics for translating dsp block diagrams into efficient software implementations[END_REF] which introduced the Pairwise Grouping of Adjacent Nodes for Acyclic graph (APGAN) algorithm. Considering an acyclic SDF graph G = ⟨A, F ⟩ where A is a set of stateless actors and F is a set of FIFO, the algorithm can be summarized as follows: A cluster hierarchy is constructed by clustering exactly two adjacent vertices at each step. At each clusterization step, the chosen pair of adjacent actors have the maximum repetition count value ρ see Definition definition 1, associated to their inter-connected edge.

Definition 1. If Z is a subset of actors in a connected, consistent SDF graph: ρ(Z) ≡ gcd({q(A) | A ∈ Z})
APGAN candidates should respect the clustering rules that can be verified by applying a reachability matrix [START_REF] Bhattacharyya | Scheduling synchronous dataflow graphs for efficient looping[END_REF]. Then repeat the process until the end of the opportunities. Execution of hierarchical actor resulting from a clustering operation is assumed to be atomic and is thus mapped and scheduled as a whole on a single core. In order to execute such cluster actor, it is thus necessary to compute a sequential schedule of all actor firings that belong to the cluster. APGAN algorithm also provides special clustering schedules that are nested looped schedules whose specificity is to make sequential the behavior of a cluster. APGAN clustering technique focuses on single-core optimization. Four clustering techniques are presented in [START_REF] Pino | A hierarchical multiprocessor scheduling system for dsp applications[END_REF]. The first one empowers the user to select improper groups of actors, manual methods are tedious and may introduce deadlocks. The second one consists in clustering SDF subgraphs as long as possible. The third one is the Unique Repetition Count (URC) clustering technique developed in section 3.1. The last one is an adaptation of Sarkar's multiprocessor DAG scheduling heuristic [START_REF] Sarkar | Partitioning and scheduling parallel programs for execution on multiprocessors[END_REF] based on macro dataflow model in which the program is partitioned into tasks at compile time and the tasks are scheduled on processors at run time. All of these methods focus on the efficient reduction of the complexity of a graph without considering parallelism. The method introduced in this paper involves automatically generated architectureadaptive parallel cluster instances.

SCAPE method

The objective of the SCAPE method is to apply graph transformation to the SDF graph of an application, prior to its mapping and scheduling. To do so, clustering of actors within the input hierarchical SDF graph aims at reducing the complexity of the derived SrDAG used during mapping and scheduling. The SCAPE method aims at preserving the parallelism of the application so that it matches the parallel computing capabilities of the target architecture.

Design space exploration optimisation

The SCAPE method is composed of three steps: configuration of the granularity, identification of particular patterns that will be the subject of clustering, and scaling up of the last clusters on the target architecture.

Configuration of the granularity

The SCAPE method takes as input the PiSDF graph of n hierarchy levels that models the application and an integer value corresponding to the number n c of hierarchy levels that the user wants to group coarsely. The output of the new method is a transformed graph with the RV q associated with the actors located in the subgraphs on n -n c level reduced to the number of PE that compose the architecture. A graph on n levels will have n+2 possible configurations of n -n c levels.

-Level 0 configuration: it is the state-of-the-art configuration where the entire graph is flattened before producing the SrDAG for scheduling. level n+1 clustering configuration: it is grouping the entire graph into a single actor, thus resulting in a mono-core schedule. level 1 clustering configuration: it corresponds to generate groups on the bottom levels and reduce the RV q associated with the actors located in the subgraphs on this level to the number of PE.

level l | l ∈ [2, n+1[ clustering configuration: it corresponds to coarsely grouping bottom levels, generate groups on the just upper levels and reduce the RV q of actors on this level to the number of PE fig. 4.

Level number

Fig. 4. Configuration based on clustering levels

Identification of particular patterns that will be the subject of clustering The SCAPE method considers two patterns:

URC pattern: It's a sequence of at least two sequential actors with the same repetition count value ρ see Definition definition 1 without internal state. Such a pattern can be the object of a cluster if the consistency of the graph is preserved that can be verify by applying a reachability matrix. The method transforms the graph G by replacing this identified group with a hierarchical actor whose behavior is specified by a subgraph containing the identified group. This way the newly created hierarchical actor executes once per iteration and the elements it contains keep their initial execution number. Thus the SrDAG transformation of this piece of graph which would have resulted in 3 × 8 = 24 actors is presently 1 actor.

Single Repetition Vector (SRV) pattern: It's a single actor that does not belong to an URC candidate, with a RV q greater than or equal to the number of PEs of the target architecture.

Example 3. We consider the graph G shown in fig. 5 which contains an actor E with a RV q equal to 16 and a target architecture which contains 4 cores. The method transforms the graph G by replacing this identified actor by a hierarchical actor whose behavior is specified by a subgraph containing the identified actor. This way the hierarchical actor executes once per iteration and the element it contains keeps its native execution number. Thus the SrDAG transformation of this piece of graph which would have resulted in 16 actors is presently 1 actor. These two identified patterns will drastically reduce the size of the SrDAG and consequently make the application intrinsically sequential.

Scaling up of cluster

The final step, called the scaling, consists of creating clusters of actors with a RV q matching the target architecture. The scaling is done on the hierarchical actors located on the subgraphs at the level n c input integer value. Level 0 and n+2 clustering are not subject to scaling. According to [START_REF] Lee | Static scheduling of synchronous data flow programs for digital signal processing[END_REF] to preserve the consistency of a graph G, on each FIFO f the rates of consumed and produced tokens cons and prod and the RV q of the source and sink actors src and snk are linked by the equation:

q(src(f )) × prod(f ) = q(snk(f )) × cons(f ) (1)
To calculate the scaling, the RV of the hierarchical actor q(h a ) shall be equal to the greatest common divisor of the RVs of the actors of the subgraph C flattened just above the number of PE n P E .

q(h a ) = gcd(q(a ∈ C) | q(h a ) ≥ n P E (2)
In case the hierarchical actor contains a FIFO with a number of delay D, special care must be taken when scaling the actor. In particular, if the hierarchical actor is directly connected to a delayed FIFO or indirectly via a special actor or an interface connected to a delayed FIFO. If one condition holds true then the calculation of the scaling is indexed on the delay value such as the rates of consumed tokens on the delayed FIFO cons(f ha d ) has to be less than or equal to the delay value D.

q(h a ) = gcd(q(a ∈ C)) | cons(f ha d ) ≤ D (3)
In order to keep the consistency of the graph, the rates of tokens consumed and produced on the input and output ports by the hierarchical actor in(h a ) and out(h a ), f for final and i for the initial value, are scaled as follow:

in(h a ) f = in(h a ) i × q(h a ) i /q(h a ) f out(h a ) f = out(h a ) i × q(h a ) i /q(h a ) f (4)
Thus the actors from the subgraph are executed q(a ∈ C)/q(h a ) times.

Example 4. We consider the graph G shown in fig. 5 and a target architecture with 4 PEs. As the RV q of the URC cluster is q(a ∈ U RC) = 8, then the scaling will be gcd(8, 4) = 4. Thus, the hierarchical actor URC executes 4 times and the subgraph elements twice per iteration. Respectively as the RV q of the SRV cluster is q(a ∈ SRV ) = 16, then the scaling will be gcd(16, 4) = 4. Thus, the hierarchical actor SRV executes 4 times and the subgraph elements 4 times per iteration. From the input graph, the classic "flattening" approach obtains a size of the SrDAG of 50 actors, the "SCAPE" approach obtains a size of the SrDAG of 10 actors. The method reduces both the number of actors related to calculations and the number of special actors related to data transfers on the different instances. The complexity of the graph has been divided by 5, which considerably reduces the mapping and scheduling time of the tool without compromising the parallelism.

Code generation

The code generated by the classic flattening approach in our tool [START_REF] Heulot | An experimental toolchain based on high-level dataflow models of computation for heterogeneous mpsoc[END_REF] takes the form of a specific C file for each target PE. Every file contains first of all a part dedicated to the initialization of the application which includes the definition of the allocated buffers, actors and FIFOs initialization functions such as delay initialization. The second part of these files is a loop representing the thread containing the scheduled firing of actors. It is a function call implementing the behavior of the actor. Up until now, the tool did not support code generation for optimized actor groups. A plugin has been created to answer this new constraint. Thanks to this plugin, a cluster of actors is translated by nested function calls depending on whether the group contains other groups and the firing instances of the group elements are translated by "for" loops fig. 6.

Experiments

The purpose of this section is to show that the proposed method offers a tradeoff between reducing mapping and scheduling time, also called analysis time, while preserving the latency of applications in comparison to the classic flattening method. The proposed method has been implemented in open source projects into Parallel and Real-time Embedded Executives Scheduling Method (PREESM) rapid prototyping framework. Compared to other mapping and scheduling frameworks, the absolute analysis time may seem high for DAGs with a few thousand actors. This time is due to the language used, Java, and heavyweight implementation choices made by PREESM. Nevertheless, the comparison of the evolution of analysis times, which relates to their complexity, would remain valid with faster implementations of both the state-of-the-art corresponding to "Level 0" and the proposed technique. The experiments are performed on a desktop computer with an 8-core Intel i7-8665U processor and 31,2 GB of RAM.

Experimental setup

Figure.7 presents the "analysis" and latency measured for the stereo application. The application has 2 levels of hierarchy, so there are 4 possible clustering configurations as explained in the section section 3.1. Three image processing application use-cases such as Stereo, Stabilization and Squeezenet were used to conduct the experiments on architectures with a 1, 2, 4, 8 or 16 of homogeneous cores summarized in the table Table.1. These models were chosen because they do not contain too many delays, which impends the scaling opportunities of the SCAPE method. These applications have between 2 and 3 levels of hierarchy. For each number of cores, only the result giving the best latency was kept, among all levels of the SCAPE method.

Analysis time evaluation

The experimental results depicted in Fig. 7 compare in red the state-of-the-art configuration and the different shades of blue for the different levels of clustering configuration up to 0. The analysis time curves are shown on the left side of Fig. 7. The curves representing the clustering configuration on different levels are between two extremes. The highest curve named "Level 0" represents the more complex graph with a time that increases with the number of cores due to the time that the mapping and scheduling algorithm takes to map, schedule and allocate memory to each of the elements of the SrDAG. The lowest curve: the fully cluster configuration remains constant and fast whatever the number of cores but no parallelism.

Latency evaluation

The latency curves are shown on the right side of figure Fig. 7. There are still two extreme curves: the "level 0" curve whose complexity allows to distribute the actors on the different cores. That's why the latency decreases with the number of cores. The level 3 clustering configuration, because it is sequential, has the longest latency and is architecture-independent. Thus the different clustering Results depicted in table 1 correspond to the ratio between the times obtained on the state-of-the-art configuration, "level 0", and those obtained on the clustering configuration that offers the best compromise between analysis time and latency. A value greater than 1 is a speedup. The values obtained on the Squeezenet application on level 0 configuration are estimated with an exponential function. The process exceeded the RAM memory capacity of the used machine, due to the large number of actors in the SrDAG on the state-of-theart configuration, and was unable to complete after 48h. Hence the relevance of the method that allows to provide analysis and executable code even on very complex applications. 1. Comparison of analysis time and latency between the classic flattening approach and best configurations of SCAPE method on three use-cases

Conclusion

This paper presents a new method to reduce mapping and scheduling time while preserving the parallelism of SDF graphs. It consists in reducing the size of the graph by clustering actors reproducing particular patterns and then reducing the firing instances of these clusters on the target architecture. The method allows the user to choose the potential expense of the produced schedule and reduce the analysis time accordingly. Experimental results show that for a significantly improved analysis time we obtain a slightly deteriorated latency of the generated code. In addition, the methods enable mapping and scheduling massively parallel applications which were too complex for state-of-the-art approaches. Potential directions for future work include identifying and clustering more complex patterns and automating the search for the optimal level of clustering, without needing to try all configurations.

Fig. 2 .

 2 Fig. 2. Classic flattening process: 3 SDF actors turn into 10 SrDAG actors

Fig. 3 .

 3 Fig. 3. (a) illustrate the violation of the first precedence shift condition, (b) illustrate the violation of the hidden delay condition, and (c) illustrate the violation of the cycle introduction condition

Example 2 .

 2 Considering the graph G shown in fig. 5 which contains a sequence of actors B, C, D, and each FIFO connecting these actors presents the same repetition count value ρ, ρ(A, B) = gcd(1, 4) = 1 and ρ(Ω, C) = gcd(1, 2) = 1.

Fig. 6 .

 6 Fig. 6. Considering the graph from fig. 4 and a two-core architecture (a) illustrates the code generation on clustering configuration with level 0, and (b) illustrates the code generation on clustering configuration on 2 levels

Fig. 7 .

 7 Fig. 7. Comparison of analysis time and latency between the classical flattening approach and three configurations of SCAPE method on Stereo application on several architectures

⋆ This work was supported by DARK-ERA (ANR-20-CE46-0001-01).