El Bartali

J.-F Witz

Buffiere J Y ✯2

Keywords: synchrotron x-ray tomography, in-situ, fatigue, digital volume correlation

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.

Characterisation of 3D damage mechanisms of a cast Al alloy during in-situ high temperature low cycle fatigue tests using Synchrotron X-ray tomography and digital volume correlation

INTRODUCTION

In the automotive industry, the Lost Foam Casting (LFC) process is gradually replacing the traditional die casting process for producing engine parts in Al alloys. The advantages of LFC are that it allows to optimize the geometry of cast parts as well as to reduce the production cost. One drawback, however, is that LFC generates a coarser microstructure and more porosity than, for example, die casting because of slower cooling rates. In this work the low cycle fatigue mechanisms of a LFC Aluminium-Silicon alloy (A319) used for making automotive cylinder heads have been studied. In situ fatigue tests monitored by synchrotron tomography have been carried out at (moderately) high temperatures representative of service conditions.

EXPERIMENTAL METHOD

Low Cycle Fatigue tests monitored by synchrotron X-ray tomography have been carried out at ID19 beamline (ESRF) using an in-situ test rig developed at MATEIS laboratory [START_REF] Dézécot | In situ 3D characterization of high temperature fatigue damage mechanisms in a cast aluminum alloy using synchrotron X-ray tomography[END_REF] to study crack initiation and growth at 250 ➦C in dog bone specimens extracted from automotive cylinder heads. The scan duration was 45 s with a 2.75 µm voxel size. Uniaxial fatigue tests at constant stress amplitude (R=0.1) were performed with a maximum stress of the order of 150% of the yield stress at the corresponding temperature. 3D displacement fields within the samples were measured by Digital Volume Correlation (DVC) all along the fatigue life. In the present study, a 3D image correlation platform developed at LML laboratory, YaDICs, has been used [START_REF] Dahdah | Damage Investigation in A319 Aluminium Alloy by X-ray Tomography and Digital Volume Correlation during In Situ High-Temperature Fatigue Tests[END_REF].

RESULTS

The reconstructed 3D images allowed visualizing the various elements of a highly three dimensional microstructure, i.e. eutectic AlAl2Cu, iron based intermetallics phases and eutectic Si. The latter cannot be imaged with laboratory tomography (figure 1a); all these constituents formed a natural speckle for DVC. Through the measured displacement fields the development and localization of strain could be monitored during low cycle fatigue. It is shown that the complex 3D microstructure of the alloy produces heterogeneous local strain fields which can be correlated with damage initiation and propagation (figure 1b andc). The analysis reveals, for the first time, local strain levels responsible for damage initiation within the various components of the microstructure.

  Dahdah, N Limodin, Ahmed El Bartali, Jean-Francois Witz, Eric Charkaluk, Jean-Yves Buffiere To cite this version: Nora Dahdah, N Limodin, Ahmed El Bartali, Jean-Francois Witz, Eric Charkaluk, et al.. Characterisation of 3D damage mechanisms of a cast Al alloy during in-situ high temperature low cycle fatigue tests using Synchrotron X-ray tomography and digital volume correlation. 3rd International Conference on Tomography of Materials and Structures (ICTMS 2017), Jun 2017, Lund, Sweden. hal-01669924

Figure 1 :

 1 Figure 1: In situ low cycle fatigue testing of a cast Al alloy at T=250 ➦C; the uniaxial cyclic tensile load is vertical (a) 3D reconstruction of the complex microstructure of the alloy showing pores (in black) Si and intermetallic particles; the red arrow indicates a crack that has formed after one fatigue cycle on a Si particle (b) vertical displacement field in voxels (c) map of Von Mises strain showing strain localisation in the pore vicinity (scale bar in real values).


Anthony Gravouil

Alain Combescure

Thomas Elguedj

Emilie Ferrié

Jean-Yves Buffière

Wolfgang Ludwig

T E L G U E D J

Application of X-FEM to 3D Real Cracks and Elastic-Plastic Fatigue Crack Growth

Keywords: X-FEM, level sets, X-ray microtomography, fatigue crack growth, plasticity, contact

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

Introduction

It is now well established that the eXtended Finite Element Method is of great interest for evolving discontinuities, in particular for industrial applications [START_REF] Wyart | Substructuring FE-XFE approaches applied to three dimensional crack propagation[END_REF][START_REF] Moës | A finite element method for crack growth without remeshing[END_REF]. Indeed, no initial mesh and remeshing techniques are needed during the evolution of the interfaces [START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level sets, Part II : Level set update[END_REF][START_REF] Elguedj | A mixed Augmented Lagrangian eXtended Finite Element Method for modelling elastic-plastic fatigue crack growth with frictional contact[END_REF][START_REF] Sukumar | Extended Finite Element Method for Three-Dimensional Crack Modeling[END_REF]. Furthermore, in a more general point of view, X-FEM plus level set representation of the interfaces reveals to be of great interest in the aim to couple experimental data with numerical simulations [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF][START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level sets, Part II : Level set update[END_REF][START_REF] Sukumar | Extended Finite Element Method for Three-Dimensional Crack Modeling[END_REF]. This can be highly illustrated in the case of 3D fatigue crack growth simulations where the initial 3D "real crack" is extracted from tomographic images with a spatial resolution of the order of 1 µm [START_REF] Ferri | Threedimensional study of fatigue crack propagation of a semi-elliptical crack: 3D crack visualization using X-ray microtomography and numerical simulation of crack propagation using the Extended Finite Element method[END_REF]. In such a case, X-FEM is coupled with a level set representation of the crack and a specific algorithm is developed in the aim to define the initial level sets from tomographic images. One can also notice that it is not necessary to impose the compatibility between the meshes of the structure and the support of the level sets. Furthermore, a robust and accurate technique is proposed to compute the stress intensity factors [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF]. Based on experimental observations and quantitative analysis of crack propagation in the bulk, a "3D" crack growth law can be established to predict the crack front shape evolution in the bulk of the material. From a more general point of view, this study shows that coupling X-ray microtomography to X-FEM provides a promising tool to assess the 3D behaviour of arbitrary shaped cracks and to perform direct comparisons of "experimental" and "simulated" crack shapes during propagation.

In a second example, it is shown that even with elastic-plastic behaviour coupled with contact and friction, non-remeshing property can be preserved for instance for 2D mixed-mode plastic fatigue crack growth [START_REF] Ferri | Threedimensional study of fatigue crack propagation of a semi-elliptical crack: 3D crack visualization using X-ray microtomography and numerical simulation of crack propagation using the Extended Finite Element method[END_REF][START_REF] Elguedj | Appropriate extended functions for X-FEM simulation of plastic fracture mechanics[END_REF]. For that purpose, an augmented Lagrangian formulation in the X-FEM framework for the treatment of contact and friction with elastic-plastic behaviour is proposed. On the one hand, the numerical integration is adapted in order to properly integrate the high order terms in the enrichment basis, and to have a fine knowledge of the stress state around the tip to precisely model plasticity. On the other hand, the strategy of enrichment used for linear elastic X-FEM fatigue simulation has to be modified [START_REF] Elguedj | Appropriate extended functions for X-FEM simulation of plastic fracture mechanics[END_REF]. In this respect, the integration strategy is designed in order to avoid the projection of stresses and internal variables as the crack evolves to ensure the reliability of the method. Numerical issues are addressed. In particular, one shows the ability of the method to model the phenomenon of crack closure under cyclic tension [6,[START_REF] Sander | Finite element analysis of fatigue crack growth with interspersed mode I and mixed mode overloads[END_REF]. Indeed, this can have a great influence on the propagation of the crack with fatigue loading.

2 Application of X-FEM to 3D Real Cracks

Experimental Observations

The propagation of a semi-elliptical crack in the bulk of an ultrafine-grained Al-Li alloy has been investigated using synchrotron radiation X-ray microtomography. In this material, the studied crack, despite its small dimension, can be considered as a "microstructurally long" and described in the frame of the linear elastic fracture mechanics. The main advantage of using this alloy for the present study is that it exhibits an exceptionally linear crack path compared to ingot metallurgy Al alloys. The ultrafine grains promote homogeneous deformation and prevent crystallographic cracking so that the crack shape is not disturbed by microstuctural features, at least not at the level of the spatial resolution employed in this study. In situ fatigue tests monitored by X-ray microtomography were carried out on beamline ID 191 of the European synchrotron radiation facility (ESRF) in Grenoble, France. This experimental station is dedicated to high-resolution X-ray imaging and features a highly coherent X-ray beam, a precision mechanics sample stage and a high-resolution detector system. During a 3D tomographic scan, the sample is rotated over 180 in order to acquire 1500 two-dimensional (2D) projection radiographs. From this set of projections, a quantitative 3D map of the attenuation coefficient distribution within the sample is produced by means of a standard filtered backprojection tomographic reconstruction algorithm. The spatial resolution obtained in the reconstructed images is of the order of 1 µm, a value comparable to the resolution of an optical microscope. One can also notice that the 3D scans are done with the maximal load: it ensures that the crack is open. However, even with a locally closed crack, diffraction effect allows to measure a crack opening until 100 nm. The sample was imaged in a dedicated fatigue machine designed at INSA Lyon to perform in situ fatigue test at the ESRF. The mechanical design of the cyclic tension loading mechanism enables operation at cycling frequencies of up to 80 Hz, minimising thereby the cycling time required for a fatigue test.

The geometry of the sample used is represented in Figure 1. A thin (2 µm) rectangular notch, 100 µmw i d ean d20µm deep, was produced in the sample using focused ion beam machining. This notch is located at the centre of one of the specimen faces and acts as a crack initiation site.

The in situ fatigue test was performed in air, at constant stress amplitude with σ max = 220 MPa, a stress ratio R =0 .1, a frequency of 40 Hz and at room temperature. In the fine-grained alloy studied here, the crack size (which is in the range 100-500 µm) is at least 100 times the grain size, and can hence be considered as microstructurally long.

Implicit Representation of the Crack Using the Level Set Method

From the previous experiments, 3D images are obtained and exploited in order to proceed to a simulation of the crack propagation. In this respect, one represents the crack by the use of two level sets.

Indeed, an implicit representation of the crack using the level sets method is very well suited to the eXtended Finite Element Method [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF][START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level sets, Part II : Level set update[END_REF].

The crack front is defined as the intersection between a surface C crack that defines the crack surface and a surface C front that defines the crack front. These two surfaces, defined for a semi-elliptical crack, are displayed in Figure 4.I n the general case, these surfaces can have arbitrary 3D shape, and are defined by means of level sets functions. In order to couple the level set representation of the crack with the 3D images which come from X-ray microtomography, a specific strategy is proposed and is summarized in Figures 2 and3.

The first step consists in a segmentation of the 3D image in order to extract the crack. The result is a set of voxels which defines a volume linked to the crack. The second step consists in the definition of a Boolean field calculated from the segmentation: a voxel outside the crack is initialized to 0 and a voxel inside the crack is initialized to 1. The main difficulty is now to be able to define a surface from this crack volume. In this respect, for the third step, it is needed first to "model" the crack as a surface with a predefined length scale. As a consequence, a length parameter is defined, and an intermediate triangulation step is done according to this length scale (in practice greater than the thickness of the crack).

Indeed, it seems essential in practice to model the crack as an explicit non-planar surface before to define the level sets. From the set of triangles, the 1D elements of the front are extracted with the local basis along the crack front. The fourth and the fifth steps consist then in defining the crack and the front level sets from the set of triangles and the set of 1D elements with their respective local basis.

In this respect, reinitialization equations (to the sign distance function) and orthogonalization equations proposed in [START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level sets, Part II : Level set update[END_REF] are used according to Figure 3 in order to compute the two initial 3D scalar fields linked to the crack and the front (see Figure 4). In particular, this approach can be done in a narrow band close to the crack.

Strategy of Enrichment and SIFS Calculation

In FE methods, the presence of cracks in a structure must be taken into account in mesh generation: the mesh must adequately define the crack geometry. Special elements and considerable mesh refinement near the crack tip are necessary to capture accurately the asymptotic displacement fields. The X-FEM alleviates the shortcomings associated with meshing the crack surface by representing the crack geometry using additional functions called enrichment functions. The enrichment method, as described in [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF], can be summarised as follows. The displacement field calculated in the structure "without crack" is locally modified by adding, at specific nodes, the nodal values of the enrichment functions. These functions define the crack geometry by "modelling" the discontinuity introduced by the crack in the displacement field. As the discontinuities at the crack front and along the crack surface are different, two enrichment functions are necessary to model the entire crack [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF].

H(x)= 1f o r φ>0, -1f o r φ<0, (1)
{F J (x)}≡ √ r sin θ 2 , cos θ 2 , sin θ 2 sin θ, cos θ 2 sin θ . (2)
The function H, defined as a generalised Heaviside function, models the displacement field along the crack surface. It is used to enrich the nodes of the element that are cut by the crack surface.

The enrichment F J is used to enrich the nodes of the elements that contain the crack front. F J consists of a span of functions which incorporate the radial and angular behaviour of the asymptotic crack-tip displacement fields, where the local coordinate system can be extracted from the level sets [START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level sets, Part II : Level set update[END_REF]. Furthermore, the nodal values of the crack level set and the front level set give the precise location of the crack and control whether a node has to be enriched or not (see Figure 5).

Concerning the calculation of the local stress intensity factor values (K I , K II and K III ) along the crack front, the interaction integral is used on a Jdomain defined in the local basis of the crack front [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF][START_REF] Gravouil | Non-planar 3D crack growth by the extended finite element and level sets, Part II : Level set update[END_REF]. The J-domain is completely independent of the mesh of the structure and its size depends on the curvature of the crack front and the size of the finite elements cut by the front. On the boundary, due to plane stress effect, the interaction integral is not used and stress intensity factors in the bulk are extrapolated from the bulk to the boundary [START_REF] Ferri | Threedimensional study of fatigue crack propagation of a semi-elliptical crack: 3D crack visualization using X-ray microtomography and numerical simulation of crack propagation using the Extended Finite Element method[END_REF]. In Figure 6, we illustrate the non-deformed mesh and the deformed mesh of the specimen submitted to a fatigue load after 14000 cycles.

Development of a 3D Local Paris Law

After 14,000 fatigue cycles, a crack initiated at the notch was detected on a 2D radiograph. The in situ fatigue test was performed at constant stress amplitude with σ max = 220 MPa, a stress ratio R =0.1, and a frequency of 40 Hz. A tomographic scan of the part of the specimen containing the crack was acquired and corresponds to the "first scan". During crack propagation, nine scans were recorded, reconstructed and segmented to obtain 3D renditions of the crack at different stages of its evolution. Projections in the (X-Y )planeof seven 3D renditions (chosen among the nine) are displayed in Figure 7.T h e dotted lines on the 3D crack renditions represent the location of the crack front at the previous step. The position along the crack front is defined by the angle x as shown in Figure 7(a). Qualitative and quantitative information on the evolution of the 3D crack geometry can be obtained.

The crack size at the surface (2c), for ω =0 • , and in the bulk (a), for ω =9 0 • , are measured on the 3D renditions of the crack. The fatigue crack growth rates at the surface, dc/dN , and in the bulk, da/dN , as a function of the stress intensity factor range ∆K =(K max -K min )aresh o w ni nFigure 9. The values of K max and K min are calculated for ω =0 • and for ω =90 • using domain integral techniques [START_REF] Moës | Non-planar 3D crack growth by the extended finite element and level sets, Part I : Mechanical model[END_REF].

Thus for the same ∆K, the crack propagates faster in the bulk than at the surface implying that, at least for the specimen geometry used here, the crack growth behaviour is anisotropic. Thus, a fatigue crack growth law determined at the surface does not account for the crack behaviour in the bulk. Furthermore, if used for fatigue life calculation, the surface experimental law would lead to a non-conservative prediction.

One possible reason for the observed crack growth anisotropy between the surface and the bulk is variation of the closure stress along the crack front: the crack will then propagate faster in the bulk because the closure stress in this region of the crack front is smaller and hence the effective driving force is higher than at the surface.

Elber [6] proposed a modified Paris equation to account for the effect of closure on the crack growth rate: da/dN = C(∆K eff ) m where ∆K eff = As a first approach, the variation of K op (ω) is taken to be linear with the angle ω and is therefore given by K op (ω)=aω + b. The value of the constants a and b are determined from the values of K op at the surface K op (0 • )a n da t the deepest point K op (90 • ). Data from the literature are used to determine K op (0 • ): the closure response of an Al-Li powder metallurgy alloy, very similar to the alloy studied here, was investigated in [7]. The value of K op measured at the surface of CT specimens, for R = K max /K min =0 .1, is found to be equal to 0.4K max . We assume here that the ratio K op (ω =0 • )/K max remains constant during the fatigue test and that there is no closure at the deepest point for ω =9 0 • ,w h i c hg i v e sK op (90 • )=0 .1K max (90 • ). The evolution of the closure stress along the crack front is given by the linear relation:

K op = - 0.3K max 90 • ω +0.4K max . (3)
The 3D crack growth law taking into account the variation of the closure stress along the crack front is given by

da dN (θ)=C 0.3 90 • ω +0.6 K max m (4)
with C =1 0 -9.2 mm.cycle -1 .(MPa.m 0.5 ) -m and m =3 .51 for the previous aluminium alloy. In Equations ( 3) and ( 4), the angle ω, expressed in degrees, is defined for 0 • <ω<90 • and K max (ω) is obtained from a polynomial interpolation of the values calculated by the X-FEM as described in the previous section. This means that, for the sample geometry investigated here, a single Paris law can be used to predict the observed crack growth anisotropy provided the variation of the closure stress along the crack front is taken into account.

From a more general point of view, this study has shown that coupling X-ray microtomography to X-FEM provides a promising tool to assess the 3D behaviour of arbitrary shaped cracks and to perform direct comparisons of "experimental" and "simulated" crack shapes during propagation. Such data, to the best of the authors' knowledge, are currently lacking in the literature.

In this study, closure effect is taken into account empirically in the Paris law. An extension of this work consists in using explicitly a elastic-plastic behaviour with contact and friction in the simulation. This is the aim of the next section.

Extension of X-FEM to Elastic-Plastic Crack Growth

Strategy of Enrichment

The main purpose of this paragraph is to treat the case of bulk and interface non-linearities in the framework of the eXtended Finite Element Method. The presented method will focus on the case of plasticity combined with frictional contact and is applied to fatigue crack growth analysis. The aim of the method is to use the X-FEM to model the well known fatigue crack closure under cyclic tension phenomenon first observed by Elber [8].

In the case of material non-linearities, several issues have to be addressed. In a previous paper [START_REF] Elguedj | Appropriate extended functions for X-FEM simulation of plastic fracture mechanics[END_REF] an appropriate elastic-plastic enrichment basis was developed to allow the X-FEM to deal with plasticity. In the present paper this approach is coupled with the treatment of frictional contact, and the problems associated with plastic crack propagation are explored. The aim of the study is to develop a general strategy for a wide family of elastic-plastic behaviour (kinematics and/or isotropic hardening, etc.). In this approach, neither remeshing nor thermodynamic field interpolation during propagation of the crack are needed. Furthermore, the crack front is enriched with specific basis in order to use sufficiently coarse mesh compared to FEM. In fact, the aim is to be able to take into account closure effect and confined plasticity with a good accuracy during propagation of the crack: indeed, this can have a great influence on the fatigue crack growth. In order to accurately model the confined plasticity, a geometrical subdividing zone is defined with an estimation of the plastic zone which can be smaller than the spatial discretization assumed to be fixed (see Figure 11). In the same way, during the propagation of the crack, one defines new sub-elements between the two configurations according to the estimation of the new plastic zone (see Figure 12): only the Gauss points of the new sub-elements are needed (which allows a good accuracy of the plastic behaviour during the crack growth with no remeshing and no interpolation field) [START_REF] Elguedj | A mixed Augmented Lagrangian eXtended Finite Element Method for modelling elastic-plastic fatigue crack growth with frictional contact[END_REF]. Concerning the strategy of enrichment of the element containing the crack tip, an appropriate enrichment basis for fatigue crack growth simulation with confined plasticity is used [START_REF] Elguedj | Appropriate extended functions for X-FEM simulation of plastic fracture mechanics[END_REF]:

where n is linked to the hardening of the elastic-plastic model. Indeed, the plastic asymptotic HRR fields can be well represented by the following basis (non-linear elasticity). In this respect, such enrichment can be well suited for elasto-plasticity when the load is increasing. Furthermore, numerical simulations show that the following basis gives accurate results for specimens Fig. 12. Evolution of the sub-elements between the two configurations submitted to loading and unloading even with coarse mesh for a wide range of elastic-plastic hardening [START_REF] Elguedj | Appropriate extended functions for X-FEM simulation of plastic fracture mechanics[END_REF].

In order to take into account contact and friction between the crack faces, interface elements are implemented along the crack with their own displacem e n ta n dt r a c t i o nfi e l d s( t and w). In this respect, a three fields Augmented Lagrangian formulation is introduced which allows to couple the interface elements with the displacement and the Cauchy stress fields (u and σ) [START_REF] Elguedj | A mixed Augmented Lagrangian eXtended Finite Element Method for modelling elastic-plastic fatigue crack growth with frictional contact[END_REF]:

Ω σ (i) n : ε(u * )dΩ = ∂ 2 Ω F d .u * dS + Γ Λ (i) n .u * | Γ dS + Γ Λ * .(u (i) n | Γ -w (i) n )dS + Γ (t (i-1) n + αw (i-1) n ).w * dS - Γ (Λ (i) n + αw (i) n ).w * dS ∀(u * ,w * ) ∈ U 0 , ∀Λ * ∈ L 0 , (6)
where σ and ε are respectively the Cauchy stress and the strain fields; t and w respectively the load and displacement fields along the crack faces; F d the prescribed load; u * , w * and Λ * respectively the virtual displacement field in the bulk, the virtual displacement field on the crack faces and the virtual Lagrange multiplier field; α the penalty term (for the normal and the tangential problem) and Λ the Lagrange multiplier of the Augmented Lagrangian formulation. Finally, concerning the update of the local enrichment during propagation the following strategy is retained: in the aim to preserve the history of the thermodynamics variables, all enrichments are retained during crack growth [START_REF] Réthoré | An energy-conserving scheme for dynamic crack growth using the extended finite element method[END_REF]. As a consequence, the change of discretization is performed by imposing the new crack segment to be closed (new enrichments are initialized to zero; the new t and w must be compatible with u and σ). Furthermore, the new stress and internal variables are initialized with elastic conditions thanks to the moving subdividing zone (see Figure 13). One considers as a first example a Compact Tension Shear (CTS) specimen, submitted to mode I fatigue crack growth with interspersed overload. A numerical and experimental investigation of that specimen is fully presented in [START_REF] Sander | Finite element analysis of fatigue crack growth with interspersed mode I and mixed mode overloads[END_REF]. In this example, one wants to compare the results obtained by a classical FE calculation presented in [START_REF] Sander | Finite element analysis of fatigue crack growth with interspersed mode I and mixed mode overloads[END_REF] with the proposed method. In Figure 14,t h e geometry and boundary conditions are presented, and one also illustrates the mesh of the specimen with a zoom around the crack tip for the FE simulations and the X-FEM simulations.

The material is chosen similar to the one in [START_REF] Sander | Finite element analysis of fatigue crack growth with interspersed mode I and mixed mode overloads[END_REF]: 2.1011 Pa for the Young's modulus, 0.3 for the Poisson's ratio, 200 MPa for the yield strength, 534 MPa for the hardening coefficient, 0.27 for the hardening exponent n,and 90 MPa.m 1/2 for the critical mode I SIF. In this comparison, the X-FEM mesh as been chosen 10 times coarser than the FEM mesh. Furthermore, a standard von Mises plasticity with isotropic hardening (Newton+radial return scheme) has been considered here [START_REF] Elguedj | A mixed Augmented Lagrangian eXtended Finite Element Method for modelling elastic-plastic fatigue crack growth with frictional contact[END_REF]. The specimen is submitted to a total crack growth of ∆a =0 .5m m( 2 0s t e p si n FEM, 2 steps in X-FEM), then submitted to an overload with a ratio of 2.5 and then a total growth of delta = 0.5 mm. One can conclude on this example that, even with a 10 time coarser element length and a 10 time coarser time discretization, very good agreements can be obtained between FEM and X-FEM on the vertical displacement close to the crack tip before and after the propagation of the crack. One can also notice the closure of the crack due to the overload and the confined plasticity (after the propagation).

As a second example, one considers a mode I Compact Tension (CT) specimen with a loading ratio of 0.1. The geometry and dimensions of the specimen are given in Figure 17. The material is chosen to be similar to the previous example. The specimen is submitted to a tension cyclic loading in order to have a stabilized stress state around the tip, then a growth of ∆a =0.05 mm is imposed at maximum load, and so on. In Figure 18, one illustrates the ability of the method to simulate closure effect close to the crack tip. In Figure 19, one illustrates stress-strain curves obtained on a Gauss point close to the crack tip during the propagation and the cycles of stabilization. One can notice that, due to the fact that no remeshing is done during the propagation of the crack, it is very easy to follow the stress-strain evolution of the material in a fixed point, which can not be possible with the same accuracy if you proceed to remeshing and interpolations.

Concerning the time dependence of the problem, this case can be considered as a two scale approach: a fixed "coarse" scale linked to the geometry (structure and crack) and a "fine" scale linked to the non-linear properties of the material (confined plasticity, contact and friction between the crack faces) which follows the crack tip during the propagation. In this respect, no remeshing is needed; however, new integration points are needed (it consists to initialize the new internal variables to zero according to the linear elastic behaviour outside the crack tip plastic zone), and new interface elements are needed.

Numerical simulations show that the proposed basis of enrichment can give accurate results for specimens submitted to loading and unloading even with coarse meshes for a wide range of hardening. However, because of conditioning issues, the crack extension can not be small compared to the element size (discrete successive plastic zone). For cyclic loadings with elastic-plastic crack growth, the question of the optimal enrichment basis is still open. One can also notice that the effects of non-singular terms of the asymptotic fields like T-stress are not taken into account. Furthermore, numerical issues suck as locking, plastic incompressibility and convergence rate need specific studies.

Conclusions

The propagation of a semi-elliptical crack in the bulk of an ultrafine-grained Al-Li alloy has been investigated using synchrotron radiation X-ray microtomography. In this material, the studied crack, despite its small dimension, can be considered as a "microstructurally long" and described in the frame of the linear elastic fracture mechanics. The extended finite element method was used to calculate the stress intensity factors along the crack front taking into account the three-dimensional geometry extracted from the tomographic images. For the same nominal value of the stress intensity factor range, crack propagation was faster in the bulk than at the surface. The observed anisotropy is attributed to the variation of the closure stress along the crack front between surface and bulk. The experimentally observed fatigue crack propagation is compared to numerical simulations. Good agreement is found when a linear variation of closure stress along the crack front is taken into account in the "3D crack propagation law" used for the simulation. In a second example, it is shown that even with elastic-plastic behaviour coupled with contact and friction, non-remeshing property can be preserved for instance for 2D mixedmode plastic fatigue crack growth. The main idea consists in using a specific nonlinear enrichment basis which allows to take into account the asymptotic behaviour around the crack tip. Because of the treatment of multiple nonlinearities (plasticity, contact), the numerical integration is adapted in order to properly integrate the enrichment basis and to have a fine knowledge of the stress state around the crack tip. Second, the propagation strategy consists in keeping the history of the variables such as stresses and plastic strains. Finally, an original strategy was designed in order to avoid projection of stresses and internal variables as the crack evolves to ensure the reliability of the method. In the presented examples, the method shows its ability to model the physical phenomena that are present in fatigue crack growth.

Fig. 1 .

 1 Fig. 1. Geometry of the sample used for the in situ fatigue test monitored by X-ray microtomography

Fig. 2 .

 2 Fig. 2. From X-ray microtomography to X-FEM

Fig. 3 .

 3 Fig. 3. Initialization of crack and front level sets

Fig. 4 .

 4 Fig. 4. Representation of the 3D scalar fields linked to the crack and front level sets

Fig. 5 .

 5 Fig. 5. Set of elements completely cut by the crack; set of elements cut by the front; 2D representation of the crack extracted from the crack level set

Fig. 6 .

 6 Fig. 6. Representation of the non-deformed and deformed mesh of the specimen submitted to fatigue loading

Fig. 7 .Fig. 8 .

 78 Fig. 7. 3D segmentation operation of the crack from the 3D tomographic image during propagation

Fig. 9 .

 9 Fig. 9. Comparisons between experimental, numerical and analytical solutions from 20000 to 22000 cycles

Fig. 10 .

 10 Fig.10. Three-dimensional effect (difference between surface and bulk)

Fig. 11 .

 11 Fig. 11. Geometrical subdividing zone with an estimation of the plastic zone

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Local enrichment and initialization strategy

Fig. 15 .

 15 Fig. 15. Mesh of the CTS specimen with zoom around the crack tip

Fig. 17 .

 17 Fig. 17. Geometry and X-FEM mesh for the mode I CT specimen

Fig. 19 .

 19 Fig. 19. Stress-strain evolution during crack growth


Ikram Senoussaoui

Mohammed Kamel Benhaoua

Houssam-Eddine Zahaf

Toward memory-centric scheduling for PREM task on multicore platforms, when processor assignments are specified

Keywords: Real-time, Scheduling, PREM, multicore

HAL is

I. INTRODUCTION

Contention on memory resources was the subject of many research works [START_REF] Alhammad | Time-predictable execution of multithreaded applications on multicore systems[END_REF], [START_REF] Maia | Schedulability analysis for global fixed-priority scheduling of the 3-phase task model[END_REF], [START_REF] Rosen | Bus access optimization for predictable implementation of real-time applications on multiprocessor systems-on-chip[END_REF], [START_REF] Yao | Memory-centric scheduling for multicore hard real-time systems[END_REF], [START_REF] Tabish | A real-time scratchpad-centric os with predictable inter/intra-core communication for multi-core embedded systems[END_REF]. Estimating precise bounds on timing delays due to contention is difficult. In general, it is hard to define what are the tasks that might interfere with each other, which likely leads to include scenario's that might never occur, therefore over-estimating the worst-case interference. A second approach tends to prevent interference by enforcing space and time isolation (e.g. time partitioning schemes).

The Acquisition-Execution-Restitution model (AER) [START_REF] Durrieu | Predictable flight management system implementation on a multicore processor[END_REF], and the PRedictable Execution Model (PREM) [START_REF] Pellizzoni | A predictable execution model for cots-based embedded systems[END_REF] represent a good alternative. In the PREM model, the task is modeled in two phases: (i) a memory phase where all data are transferred from main memory to the local memory and vice-versa, (ii) and a pure computation phase where the loaded data are processed. Memory-transactions are scheduled so that the memory bus is accessed without interference (time isolation), while pure computations are executed on the platform cores. In this paper, we focus on the PREM model.

Several difficult challenges are encountered when scheduling real-time application modeled by the PREM model on multicore architectures. Finding a proper way to schedule memory phases is the most important difficulty since the bus controllers available on commercial platforms are not dedicated to real-time scheduling, they support very simple policies for example: First-In-First-Out (FIFO) scheduling policy. A second issue is the complexity of the scheduling problem: we need to consider at the same time scheduling of bus memory access and scheduling on the different cores.

In this paper, we propose novel method for assigning artificial deadlines to memory phases of a set of periodic PREM tasks, so to avoid memory contention, in particular, contention between the communication bus and the main memory. We do not address task to core assignment in this paper, therefore we use classical bin-packing heuristics to compute the task to core allocation. Finally, we provide a large set of synthetic experiments to demonstrate the efficiency of the proposed heuristic in terms of schedulability, as well as the required analysis time.

This paper is organized as follows: in the following section we review the state of the art. We present the hardware and task models in Section III. Further, we detail in Section IV how the artificial deadlines are assigned to memory phases. Within Section V, we give a quick overview how tasks can be allocated to cores. Results and simulations are described in Section VI. We draw conclusions in Section VII.

II. RELATED WORK

Modern COTS-based embedded systems include multiple active components (CPU cores and I/O peripherals) that can independently initiate access to shared resources (like main memory and inter-connection buses) which cause contention leading to timing degradation [START_REF] Pellizzoni | Predictable execution model: concept and implementation[END_REF]. These architectures feature generally, a single port main memory shared among all CPU cores and peripherals, therefore, it is hard to guarantee the absence of memory conflicts during execution. In [START_REF] Pellizzoni | Worst case delay analysis for memory interference in multicore systems[END_REF], authors have shown that the worst case execution time (WCET) of a task can increase linearly with the number of suffered cache misses, due to contention for access to main memory. To solve this problem, new execution models making use of pre-fetching techniques have been proposed in the literature. These techniques have been investigated first by Rosen et al. [START_REF] Rosen | Bus access optimization for predictable implementation of real-time applications on multiprocessor systems-on-chip[END_REF] in the context of worst-case execution time computation and bus access optimization. The authors show in [START_REF] Adzer Hoogeveen | Preemptive scheduling in a two-stage multiprocessor flow shop is nphard[END_REF] that pre-fetching techniques improve the cache/scratchpad locality and reduce average execution times.

The PRedictable Execution Model (PREM) from Pellizzoni et al [START_REF] Pellizzoni | A predictable execution model for cots-based embedded systems[END_REF] was first proposed to co-schedule both memory requests from CPU and I/O with computations on uniprocessor platforms with multi-level caches. It splits tasks in memory/computation phases. This execution model drastically reduces the variability of memory-contention latency by explicitly controlling memory accesses during memory phases. PREM has been extended later in [START_REF] Yao | Memory-centric scheduling for multicore hard real-time systems[END_REF] to partitioned multicore/processor platforms where the core isolation is provided through a coarse-grained TDM memory schedule.

Different memory-aware scheduling policies for PREM tasks are evaluated in [START_REF] Bak | Memory-aware scheduling of multicore task sets for real-time systems[END_REF] by simulating synthetic task systems on a multicore platform. The authors of this work used different combinations of TDMA slots (different slot sizes) and different priority-based schedulers for the memory and computation phases on a partitioned multicore platform.

The Acquisition-Execution-Restitution model (AER) [START_REF] Durrieu | Predictable flight management system implementation on a multicore processor[END_REF] is a generalization of the PREM model where each task is divided into three distinct parts: two memory phases (reading and writing) and a computation phase.

We believe that the memory hierarchies of modern multicores can further boost the effectiveness of the PREM model, and, in general, improve WCET predictability. The reason is that, platform designers replace traditional data caches with explicitly managed memories such as scratchpads (SPMs). The key point is that the behavior of explicitlymanaged scratchpads is also much more predictable than that of caches, because access latency is independent of the access pattern [START_REF] Lickly | Predictable programming on a precision timed architecture[END_REF], [START_REF] Puaut | Scratchpad memories vs locked caches in hard real-time systems: a quantitative comparison[END_REF], [START_REF] Wasly | A dynamic scratchpad memory unit for predictable real-time embedded systems[END_REF].

The schedulability problem for globally scheduled PREM tasks using the memory phase prioritizing concept is addressed in [START_REF] Yao | Global real-time memory-centric scheduling for multicore systems[END_REF]. The work in [START_REF] Alhammad | Memory efficient global scheduling of real-time tasks[END_REF] proposes a global fixed-priority scheduling algorithm for a set of sporadic PREM tasks, it considers co-scheduling a separate DMA component to perform transfers from main memory to scratchpad and viceversa. Melani et al. [START_REF] Melani | Exact response time analysis for fixed priority memory-processor co-scheduling[END_REF] proposed exact response time analysis for fixed-priority scheduling on a single core with a fully preemptive DMA engine.

Much works have been done on the communication and task scheduling problem on single/multi-processor architectures. A good overview of the common real-time scheduling methods is given in [START_REF] Korst | Scheduling periodic tasks[END_REF]. The relevant scheduling approaches can be classified on: event-based scheduling, time-triggered scheduling and hybrid approaches.

One of the most effective techniques to schedule dependent tasks on multicore platforms is to assign artificial deadlines and offsets in order to enforce precedence constraints [START_REF] Houssam-Eddine | The hpc-dag task model for heterogeneous real-time systems[END_REF]. The most popular heuristic algorithms are fair distribution and proportional distribution. We propose in this paper a technique for assigning artificial delays to memory phases to avoid memory contention.

III. SYSTEM MODEL

A. Architecture model

We consider in this work multicore architectures composed of m cores. Each core has access to a private scratchpad memory. Cores are connected through a single shared bus to a global shared memory. Before starting the computation, the different tasks trigger memory copies between main and scratchpad memories, explicitly by the system designer.

We assume that all memory is directly accessible to all cores via different address spaces. An example of such architecture is the Infineon Aurix TC397 [START_REF] It | Aurix 32-bit microcontrollers for automotive and industrial applications[END_REF].

Core1 Core2 Core3 Core4 SPM1 SPM2 SPM3 SPM4
Main memory C-Bus Fig. 1: Multicore target platform.

In Figure 1, we report an architecture compound of 4 cores. Each core is connected to its own scratchpad memory. Different scratchpads are connected using a single bus (C-bus) to the main memory.

B. Task model

We consider a set T of n independent periodic tasks, i.e., T = {τ 1 , τ 2 , • • • , τ n }. According to the semantics defined in PREM, each task τ i is divided in two phases namely: memory phase and computation phase. The later, can not start before the completion of the memory phase. A task τ i is characterized by tuple τ i = (M i , C i , D i , T i ): M i is the task worst case dataprefetch time. It represents the required time that the task τ i needs to load data from memory and/or I/O devices; C i is the task worst case computation time. This phase can be preempted contrary to the memory phase; D i is the task's relative deadline. Each instance of task τ i must finish its execution no later than D i time units after its activation; T i is the task period, it represents the exact time interval between two consecutive activations of τ i .

The computation phase of tasks does not access the shared bus, thus contentions when accessing the shared bus do not exist between computation phases and memory phases.

We denote by H the taskset hyperperiod, i.e. the system's period. It is defined as the least common multiple between all periods of tasks

H = LCM (T 1 , T 2 , • • • , T n ).
The task utilization is given as:

U i = u m i + u c i
where, u m i = Mi Ti (resp. u c i = Ci Ti ) denotes the memory phase (resp. computation phase) utilization.

The total utilization of taskset T is computed as follow:

U T = n i U i IV. DEADLINE-BASED MEMORY-PROCESSOR CO-SCHEDULING
We address in this paper the bus contention problem by avoiding memory requests overlap entirely. In this section, we will show how to determine artificial deadlines to memory phases so that they do not overlap during runtime. We consider all the memory requests as non-preemptive tasks having a period and an artificial deadline δ i and scheduled on the communication bus using Earliest Deadline First (EDF) scheduler. The schedulablity on the different cores will be tested using EDF algorithm. The analysis is based on the processor demand function [START_REF] Sanjoy K Baruah | Algorithms and complexity concerning the preemptive scheduling of periodic, realtime tasks on one processor[END_REF], [START_REF] Pellizzoni | Feasibility analysis of real-time periodic tasks with offsets[END_REF], and considers only the computation phases. The computed artificial deadlines will be used as offsets of the computation phases.

A. Artificial deadline assignment heuristic (ADA)

One strategy to solve the artificial deadline assignment problem could be to search exhaustively among all possible artificial deadline combinations in a Pareto-Front fashion. In a such strategy, the objective would be both to ensure a nonpreemptive scheduling of memory phases to avoid the overlap on the communication bus and to minimize their artificial deadlines in order to improve the schedulability on cores. Although the Pareto-Front based approaches are exact and have the advantage to find all the realizable solutions for the considered problem, they suffer from large computational complexity, as the search space may potentially be very large. In fact, any realizable solution can be accepted from real-time perspective. We present in this section a weightbased artificial deadline assignment heuristic for computing deadlines of memory phases. The search is done for each task τ i in the interval

I i = D i -C i -M i as shown in figure 2. Ii M i C i a i Di Ti δ i
Fig. 2: The task artificial deadline search interval.

We will assume that each task τ i is characterized by a weight w i such that w i < 1 (the weight is the same for all the tasks). The idea is to search for the solution by deceasing the memory phase artificial deadline for each task using its weight and to stop the search after the first realizable solution.

Algorithm 1 implements our method for a set of PREM tasks. Lower bound lb i and upper bounds ub i are computed for every task. S is the current solution, it contains the task set with assigned artificial deadlines, it is initialized to empty Algorithm 1 artificial deadlines assignment(T , S,stop)

curr d = curr d * w head Sol 16: until (curr d < lb head S ∧ stop = true )
before the algorithm starts, and S remained is its remainder, it contains the taskset before the assignment of the artificial deadlines and it is full at the beginning of the algorithm.

The algorithm computes the artificial deadlines for all the tasks in S remained . The artificial deadline of every task τ i is equal to ub i in the beginning. At each iteration, the algorithm selects the task in the head and moves it to S (line [START_REF] Jeffay | On non-preemptive scheduling of periodic and sporadic tasks[END_REF][START_REF] Korst | Scheduling periodic tasks[END_REF][START_REF] Lickly | Predictable programming on a precision timed architecture[END_REF]. Once, S remained for the current solution is empty, the algorithm performs the non-preemptive EDF schedulability test of [START_REF] Jeffay | On non-preemptive scheduling of periodic and sporadic tasks[END_REF] to assess the feasibility on the bus. If the solution is feasible on the bus, then the algorithm saves the solution into S realizable , sets the booleen variable stop to true and exit successfully (line 2-8). The schedulability is tested, further, on the different cores using EDF algorithm (IV-B).

If the schedulablility test fails on the bus for the current solution S, the algorithm ignores the solution, and continues to search in the solution space by decreasing the artificial deadline for the task in the head τ i using its weight while fixing the artifitial deadline of the other tasks, until reaching the lower bound lb i .

The main difference between algorithm 1 and a pure Pareto-Front based algorithm when solving our problem is at lines (5 and 15): (i) algorithm 1 uses the task weight w i , to decrements the task artificial deadline rather than reduce it by 1, and (ii) if a realizable solution is found, then, algorithm 1 stops the research immediately and returns the solution rather than compare it with the Pareto-set using the Pareto dominance concept and adding it to the Pareto-set if not dominated.

In our prototype, we chose np-EDF for scheduling the memory phases on the communication bus and EDF for computation phases on the cores. However, our strategy is general enough that other schedulers can be easily plugged in our framework. As long as offsets are assigned to act as precedence constraints between memory phases and computation phases, the choice of the scheduler is arbitrary.

B. Schedulability analysis on cores

We use in this paper, the Earliest Deadline First algorithm (EDF) to schedule computation phases on cores. Our analysis is based on the processor demand criterion [START_REF] Pellizzoni | Feasibility analysis of real-time periodic tasks with offsets[END_REF] and considers only computation phases with offsets. The processor demand function for computation phases with offsets is defined as follows:

df (t 1 , t 2 ) = n i=1 ∆ i (t 1 , t 2 ) • C i (1)
Where:

∆ i (t 1 , t 2 ) = t 2 -φ(C i ) -D i T i - t 1 -φ(C i ) T i + 1 (2)
It is the amount of time demanded by the tasks (computation phases) in interval [t 1 , t 2 ) that the core must execute to ensure that no task misses its deadline. Considering a fully preemptive single core scheduler, a necessary and sufficient condition for a set of tasks (computation phases) to be schedulable by EDF consists in checking that the demand never exceeds the length of the interval. lemma 1: (Baruah et al. [START_REF] Sanjoy K Baruah | Algorithms and complexity concerning the preemptive scheduling of periodic, realtime tasks on one processor[END_REF]). The taskset T is feasible on a single core (U T ≤ 1) if and only if:

∀0 ≤ t 1 < t 2 ≤ H, df (t 1 , t 2 ) ≤ t 2 -t 1 (3)
V. ALLOCATION HEURISTICS

The tasks are assumed to be already assigned to the different cores of the platform. In this paper, we use the bin-packing heuristics: Best-Fit (BF) and Worst-Fit (WF) to perform taskcore allocation (Algorithm 2) which is likely similar to the one used in [START_REF] Senoussaoui | Allocation of real-time tasks onto identical core platforms under deferred fixed preemptionpoint model[END_REF]. BF and WF sort the cores by capacity. For BF the cores are sorted in a decreasing order of their utilizations, whereas in the case of WF, they are sorted in increasing order of utilization.

First, the algorithm sorts PREM tasks according to their deadlines or their periods, then, for each task, it sorts cores according to parameter alloc (Best-Fit or Worst-Fit order) (line 4). Then, it tests the possibility of allocating the selected task on each core in turn. If the allocation is successful, the next PREM task is tested, otherwise the algorithm tries the next core (line [START_REF] Sanjoy K Baruah | Algorithms and complexity concerning the preemptive scheduling of periodic, realtime tasks on one processor[END_REF][START_REF] Durrieu | Predictable flight management system implementation on a multicore processor[END_REF][START_REF] Emberson | Techniques for the synthesis of multiprocessor tasksets[END_REF][START_REF] Adzer Hoogeveen | Preemptive scheduling in a two-stage multiprocessor flow shop is nphard[END_REF][START_REF] Houssam-Eddine | The hpc-dag task model for heterogeneous real-time systems[END_REF][START_REF] It | Aurix 32-bit microcontrollers for automotive and industrial applications[END_REF][START_REF] Jeffay | On non-preemptive scheduling of periodic and sporadic tasks[END_REF]. If the studied PREM task cannot be allocated on any core, then, the algorithm fails (line [START_REF] Korst | Scheduling periodic tasks[END_REF][START_REF] Lickly | Predictable programming on a precision timed architecture[END_REF][START_REF] Maia | Schedulability analysis for global fixed-priority scheduling of the 3-phase task model[END_REF]. If all PREM tasks have been allocated, the algorithm returns the tasks allocation.

VI. RESULTS AND DISCUSSIONS

In this section, we present the performances of the proposed approach. We conducted experiments with randomly generated workloads to evaluate the proposed approach using different task partitioning heuristic. Consistently with previous results in the literature, WF outperforms BF in all simulated scenario, therefore, we only report the results for WF for clarity of presentation. We study the impact of the workload size on schedulability and the require time to complete the analysis.

Algorithm 2 Heuristics(T , P, alloc[BF, W F ], order)

A. Task set generation

The task set generation process takes as input n the number of tasks and the target total utilization U T . It starts by generating the utilizations of the n tasks by using the UUniFast-Discard [START_REF] Emberson | Techniques for the synthesis of multiprocessor tasksets[END_REF] algorithm. We varied the baseline utilization from 0.2 to P (number of available cores) with a step of 0.4. For every utilization u i , the algorithm generates the memory phase utilization u m i using a random value stall = ( Mi Ci+Mi ) that represents the utilization of the memory phase: for example, a task with a memory stall of 0.1 spends 10% of its WCET in the memory phase. The range of random stall in our experience is: lev = [0.10, 0.20]. We fixed the task weight to 0.5. We highlight that the total workload comprises also the memory phases, hence, the computation workload on the cores is smaller than the total utilization U T .

For each utilization, we use 50 tasksets per utilization. We generate 10 tasks per taskset. The task period is selected randomly from a predefined list of periods: {80,100,150,200,240,300, 400,500, 600,800}, so to establish an upper bound to the hyperperiod. The task deadline is set to 70% of the task's period.

B. Results of synthetic task set experiments

In this section, we evaluate the performance of our proposed approach against a Pareto-Front based approach which uses the task weight to search all the realizable solutions. Tasks are allocated on 4 cores by Worst-fit (WF) heuristic.

Figure 3 shows the schedulability rate and the required analysis time of schedulable tasksets as a function of the utilization. We simulated the memory stall [0.1 -0.2) since it is the closest to reality. The same performance behavior is noticed for our proposed approach (WF-ADA) and Pareto-Front based approach (WF-PF) since they search for solutions exactly the same way. WF-ADA as well as WF-PF (in this experience) are not exact because they check only a subset of the design space, however, their performances results in term of schedulablity are acceptable compared to an exact approach (example: a Pure Pareto-Front based approach or an ILP-based approach [START_REF] Senoussaoui | Contention-free scheduling of prem tasks on partitioned multicore platforms[END_REF]).

As expected, WF-PF based approach require more time than our proposed approach WF-ADA, even for small tasksets of 10 tasks and using a task weight equal to 0.5, Unfortunately, it increases exponentially with the number of tasks, and this is why we limited our analysis to 10 tasks per taskset. It is, therefore, unfeasible to run WF-PF (and also the exact approaches) on large tasksets during design time when the parameters of the tasksets may change frequently. The runtime of WF-ADA is still acceptable (a few seconds in average).

VII. CONCLUSION AND FUTURE WORK

Multicore systems using ScratchPad Memories (SPMs) are attractive architectures for executing time-critical embedded applications, because they provide both predictability and performance. However, they are likely subject to contention and require a particular attention when supporting hard realtime constraints. In this paper, we proposed technique based on artificial deadline assignment for contention avoidance. Our experiments show a significant improvement in the system performances compared to state-of-the-art (example: FIFO based buses). As future work, we plan to take into account allocation strategies in the analysis phase for AER task model.

Fig. 3 :

 3 Fig. 3: Assignment artificial deadlines algorithms performances

1 :

 1 Require: S realizable 2: if S remained ≡ ∅ then

	3:	if dbf analysis np(S) then
	4:	S realizable ← S
	5:	stop ← true
	6:	end if
	7:	return true;
	8: end if
	9: curr d ← (d head S = 0)?(ub head S ) : d head S
	10: repeat
	11:	Sol.copy(S)
	12:	d head Sol ← curr d
	13:	remove(d head Sol , Sol remained )
	14:	artificial deadlines assignment(T , Sol,stop)
	15:

1 :

 1 sort tasks(T , order) 2: for (∀τ i ∈ T ) do

	3:	allocated← false
	4:	sort cores(P, alloc)
	5:	for (∀p ∈ P) do
	6:	if (schedulable(τ i , p)) then
	7:	allocate task into core p
	8:	allocated ← true
	9:	break;
	10:	end if
	11:	end for
	12:	if (allocated= false) then
	13:	No task allocation is found
	14:	end if
	15: end for
	16: return tasks allocation