Nathanaël Savalle
email: n.savalle@civil.uminho.pt

Christine Monchal
email: christine.monchal@geolithe.com

Eric Vincens
email: eric.vincens@ec-lyon.fr

Sten Forcioli
email: sten.forcioli@geolithe.com

Paulo B Lourenço

Static and seismic design of Dry Stone Retaining Walls (DSRWs) following Eurocode standards

Keywords:

Dry Stone Retaining Walls are structures made of rubble stones assembled without mortar and have been present worldwide for centuries. Today, they still constitute an attractive alternative to building techniques involving higher embodied energy, such as reinforced concrete walls. This study uses a pseudo-static approach to give design recommendations to maintain this built heritage and allow its modern construction. Both non-seismic (Eurocode 7) and seismic (Eurocode 8) cases are addressed. The present work confirms that a seismic design is not critical and is therefore not required for zones with a design acceleration below 0.05g. In addition, this work highlights the significant positive effect of the stone bed inclination and the internal wall face batter. Finally, depending on the wall site conditions and the seismic zone associated with the project, general design recommendations are given to optimise the volume of stones used, which are illustrated in the case of France. These recommendations based on pseudo-static analyses are already usable in practice for low to moderate seismic areas as the required retaining wall dimensions can be easily implemented on-site. In addition, it is also shown that the actual French recommendations for these walls fully comply with Eurocode 7.

Introduction

Dry stone structures have been built in most regions of the world, sometimes shaping typical and valuable landscapes. These vernacular structures are made of rubble stones carefully assembled by hand and without mortar. Dry Stone Retaining Walls (DSRWs) are likely to constitute the most representative part of this built heritage, allowing agricultural activities on terraces and traffic on rural roads in mountainous or sloped areas. Therefore, DSRWs play an essential economic role in these regions that benefit less from globalisation and major investments. In addition, they also hold a high cultural value, sometimes labelled by UNESCO (e.g., Douro's Valley in Portugal or the Lavaux's Terraces in Switzerland). In fact, the art of dry stone walling, knowledge and techniques were designated as Intangible Cultural Heritage of Humanity by UNESCO in 2018. However, these structures have often faced a lack of maintenance in recent decades and require urgent repair.

Given the need to preserve and repair old DSRWs, several research studies have been conducted mainly in Europe. Experimental works [START_REF] Villemus | Experimental assessment of dry stone retaining wall stability on a rigid foundation[END_REF][START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Colas | Assessing the two-dimensional behaviour of drystone retaining walls by full-scale experiments and yield design simulation[END_REF], analytical [START_REF] Mundell | Limit-equilibrium assessment of drystone retaining structures[END_REF][START_REF] Alejano | Stability of granite drystone masonry retaining walls: I. Analytical design[END_REF][START_REF] Colas | Yield design modelling of dry joint retaining structures[END_REF][START_REF] Terrade | Upper bound limit analysis of masonry retaining walls using PIV velocity fields[END_REF] and numerical studies [START_REF] Harkness | Numerical modelling of full-scale tests on drystone masonry retaining walls[END_REF][START_REF] Powrie | Deformation and failure modes of drystone retaining walls[END_REF][START_REF] Claxton | Rigid block distinct-element modelling of dry-stone retaining walls in plane strain[END_REF][START_REF] Walker | Plane strain numerical model for drystone retaining walls[END_REF][START_REF] Alejano | Stability of granite drystone masonry retaining walls: II. Relevant parameters and analytical and numerical studies of real walls[END_REF][START_REF] Oetomo | Modeling the 2D behavior of drystone retaining walls by a fully discrete element method[END_REF][START_REF] Li | Upper-Bound Analysis for Stone Retaining Wall Slope Based on Mixed Numerical Discretization[END_REF][START_REF] Pulatsu | Advanced analysis of masonry retaining walls using mixed discrete-continuum approach[END_REF] focused on the static mechanical behaviour of 2D sloped DSRWs, while other studies investigated the 3D mechanical behaviour of these walls in case of a concentrated traffic load [START_REF] Mundell | Behaviour of drystone retaining structures[END_REF][START_REF] Le | 3D homogenised strength criterion for masonry: Application to drystone retaining walls[END_REF][START_REF] Le | Assessing the Three-Dimensional Behaviour of Dry Stone Retaining Walls by Full-Scale Experiments[END_REF][START_REF] Quezada | 3D failure of a scale-down dry stone retaining wall: a DEM modelling[END_REF]. In France, these researches led to two practical handbooks that include design rules for DSRWs retaining slopes. These are valid for any country with similar building techniques, which can be found worldwide [START_REF] Capeb | Pierres sèches: guide de bonnes pratiques de construction de murs de soutènement[END_REF][START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF]. However, even though the recommendations are used in practice and recognised by the drystone masonry and civil engineering communities, they do not consider seismic action. Only a few study cases have been investigated according to the past French seismic recommendations [START_REF] Afps | Recommandations AFPS 90[END_REF][START_REF] Le | Stabilité des murs de soutènement routiers en pierre sèche : Modélisation 3D par le calcul à la rupture et expérimentation échelle 1[END_REF]. Moreover, the validation of the recommendations according to Eurocode 7 [START_REF] Afnor | Geotechnical design -Part 1: General rules[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Ouvrages de soutènement -Murs[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Fondations superficielles[END_REF] has not been investigated exhaustively, even if partly considered in the latest DSRWs French handbook [START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF].

To address the seismic design of DSRWs in slopes, the authors developed a pseudo-static analytical tool based on Coulomb's wedge theory, which was validated by pseudo-static scaled-down laboratory experiments [START_REF] Savalle | Pseudo-static scaled-down experiments on dry stone retaining walls: Preliminary implications for the seismic design[END_REF]. The first section of this paper revises the analytical method, while the second section provides a comparison with the current standards on geotechnical engineering and seismic engineering, Eurocode 7 & 8, respectively [START_REF] Afnor | Geotechnical design -Part 1: General rules[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Ouvrages de soutènement -Murs[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Fondations superficielles[END_REF][START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 5: Foundations, retaining structures and geotechnical aspects[END_REF][START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings[END_REF]. Finally, following the Eurocodes, recommendations are given for designing DSRWs in areas ranging from very low to high seismicity.

Analytical method

The analytical method relies on the limit-equilibrium theory under plane strain conditions (Fig. 1). The DRSW is characterised by a height H, a base width B, an external slope to the vertical of λ v and an internal slope to the vertical of λ m . The bed inclination of the wall is referred to as α and the backfill slope as β. Contrary to the static limit-equilibrium approach of Villemus [START_REF] Villemus | Etude des Murs de Soutènement en Maçonnerie de Pierre Sèches[END_REF], the analytical method includes seismic forces modelled as equivalent horizontal pseudo-static actions. Briefly, the pseudo-static equilibrium of a Coulomb's wedge of soil is first computed to obtain the pseudo-static active earth pressure [START_REF] Coulomb | Essai sur une application des regles des maximis et minimis a quelques problemes de statique relatifs a l'architecture[END_REF][START_REF] Okabe | General theory on earth pressure and seismic stability of retaining wall and dam[END_REF][START_REF] Mononobe | On Determination of Earth Pressures during Earthquakes[END_REF]. The wall's equilibrium is then computed, stating the possible types of failure: an internal sliding or toppling mode [START_REF] Savalle | Pseudo-static analytical model for the static and seismic stability of dry stone retaining walls[END_REF].

Fig. 2 describes the backfill-wall system, with all geometrical and mechanical parameters clarified. The wall is characterised by a homogeneous medium (Fig. 1) with its homogeneous unit weight γ rw (accounting for voids between stones) and joint frictional angle φ rw . Similarly, the backfill is modelled as a homogeneous medium described by its unit weight γ f , cohesion C f and friction angle φ f . Compared to other retaining structures, the particularity of DSRWs lies in a failure line developing through the dry joints, which is modelled as an equivalent straight failure line with an inclination ω (Fig. 1). In practice, the inclination of this failure line from the bed joints is limited by a maximum value of approximately 20 • , see details in [START_REF] Savalle | Pseudo-static scaled-down experiments on dry stone retaining walls: Preliminary implications for the seismic design[END_REF]. This inclination is also different from the homogenised inclination θ of the failure line crossing the backfill, which mainly depends on material (soil friction and cohesion), geometrical (slope of the backfill) and seismic (pseudo-static accelerations) parameters, as explained below.

The mechanical system has three unknowns (θ, ω and h g ) that should be determined to compute the earth pressure F δ . According to the Coulomb soil's wedge theory, for each combination of these parameters, the limit equilibrium of the soil's wedge D 1 D 2 D 3 can be calculated (Fig. 2b-c) to evaluate the active earth pressure F δ . In particular, the weight P f of the soil is proportional to the wedge area (triangle D 1 D 2 D 3 ) and is applied at the gravity centre of the triangle D 1 D 2 D 3 . Similarly, the pseudo-static action (inertial force due to the seismic motion) F f is also proportional to the wedge area and applied at its gravity centre. The backfill frictional reaction R φ application point and intensity are unknown, but its orientation is given by the backfill friction angle φ f since the limit-equilibrium is assumed. Similarly, the orientation, yet not the application point, of the backfill cohesive reaction R C is known. Regarding the backfill-wall interface, the interface cohesive reaction R Cint has an unknown point of application and a known orientation. The intensities of the cohesive forces (R C and R Cint ) are proportional to the cohesive strength and the length of the interface (D 1 D 3 and D 1 D 2 ). Hereafter, the interface cohesive strength (R Cint ) always equals zero, as the drain directly behind a DSRW is usually made of dry cohesiveless gravel.

Finally, the earth pressure F δ has a known orientation δ (internal face of the wall) and application point but unknown intensity. The intensity is deduced from the mechanical equilibrium of the soil's wedge (Fig. 2c). In the absence of pseudo-static action F f and cohesive resistance, the application point of the earth pressure is located at one-third of the height of the retaining structure. Then, adding cohesive effects decrease the application point height while adding a pseudo-static action increases its height. The reader can refer to the literature for deeper insights into the location of the application point of earth pressure in this case [START_REF] Savalle | Pseudo-static analytical model for the static and seismic stability of dry stone retaining walls[END_REF][START_REF] Ishibashi | Dynamic earth pressures with different wall movement modes[END_REF][START_REF] Ichihara | Earth pressure during earthquake[END_REF].

The analytical method also accounts for the tensile cracks that classically occur at the top of cohesive backfill and reduce the cohesive forces (R C and R Cint ). In addition, in the present work, the presence of dead loads on top of the backfill and saturated retained backfill [START_REF] Savalle | Pseudo-static analytical model for the static and seismic stability of dry stone retaining walls[END_REF] can be accounted for, yet not described here for brevity.

In a second stage of the calculation, the equilibrium of the wall itself is computed (Fig. 3) including its own weight and the pseudo-static horizontal action, applied at the centre of gravity of the studied portion of wall A 2 A 3 D 1 E, corresponding to the part above the failure line D 1 E. The interface actions F δ and R Cint (as well as their line of actions) are derived from the previous stage of calculation. Subsequently, the wall equilibrium is computed in the X and Y directions to evaluate the base reaction R b . The equilibrium in terms of momentum gives the application point of R b . Finally, the stability of wall portion A 2 A 3 D 1 E is computed considering a toppling mode of failure (e.g. application point inside the wall, i.e. l b > 0, or any other criteria defined in the following sections) or a sliding mode of failure. For this last point, the base reaction R b is projected on the plane defined by the orientation of the bed joints (axes X s and Y s in Fig. 3), and then Mohr-Coulomb criterion is checked. Note that the bed joints' orientation is updated because of the possible internal rotation of stones inside DSRWs (see [START_REF] Savalle | Pseudo-static analytical model for the static and seismic stability of dry stone retaining walls[END_REF] for more details). Finally, the unknowns of the system (θ, ω and h g ) are optimised for each failure mode to find the most critical situation for the criterion checked. Several automated iterations involving the base width B of the Fig. 1. DSRW with its geometric parameterisation a): actual backfill-wall system; b): modelled backfill-wall system. The present sign convention makes the internal batter (λ m ) negative.

wall allows to identify the minimum B value that barely satisfies the stability criteria.

The method has been validated on scaled pseudo-static experiments on a tilting table, using dry joint brick retaining walls retaining a sandy backfill [START_REF] Savalle | Pseudo-static scaled-down experiments on dry stone retaining walls: Preliminary implications for the seismic design[END_REF]. Fig. 4 gives the results, showing that the developed pseudo-static approach is as precise as more sophisticated Discrete Element Modelling (DEM) simulations [START_REF] Savalle | Experimental and numerical studies on scaled-down dry-joint retaining walls: Pseudo-static approach to quantify the resistance of a dryjoint brick retaining wall[END_REF].

As additional validation, the analytical method was used to model two sets of experimental campaigns carried out on full-scale DSRWs with 1) a hydrostatic load [START_REF] Villemus | Experimental assessment of dry stone retaining wall stability on a rigid foundation[END_REF]; 2) a dry backfill load, as displayed in Fig. 5 [START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Colas | Assessing the two-dimensional behaviour of drystone retaining walls by full-scale experiments and yield design simulation[END_REF]. Table 1 describes the geometric and mechanical parameters of the experiments, along with the analytical results, which are in excellent agreement for both campaigns. Moreover, the developed analytical approach provides a similar level of accuracy to the results of Villemus et al. [START_REF] Villemus | Experimental assessment of dry stone retaining wall stability on a rigid foundation[END_REF] and Colas et al. [START_REF] Colas | Yield design modelling of dry joint retaining structures[END_REF][START_REF] Colas | Yield design of dry-stone masonry retaining structures -Comparisons with analytical, numerical, and experimental data[END_REF].

Design of DSRWs following Eurocodes

The present section aims at designing DSRWs using the analytical method and the partial safety factors (actions, material properties and resistance) from Eurocode 7 and 8 [START_REF] Afnor | Geotechnical design -Part 1: General rules[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Ouvrages de soutènement -Murs[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Fondations superficielles[END_REF][START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 5: Foundations, retaining structures and geotechnical aspects[END_REF]. First, it is emphasised that in a seismic context, a pseudo-static method (like the one presented above) will never accurately predict the true time evolution of the dynamic response or resistance of a real DSRW during an earthquake. Therefore, it is only used as a simplified design method proposed by Eurocode 8 to give fast seismic assessment of retaining walls. No partial safety factor related to the method is considered since no constant bias Fig. 2. Parametrisation of the mechanical system and geometrical equilibrium of the soil's wedge in order to compute the earth pressure F δ. [START_REF] Savalle | Pseudo-static analytical model for the static and seismic stability of dry stone retaining walls[END_REF]. has been found in the validation processes between theoretical and experimental results. Moreover, only the wall's internal sliding and toppling failures are considered: the bearing capacity of the foundation soil at ultimate or serviceability limit states are assumed not to be reached. Similarly, the passive soil is considered infinitely rigid, which is reasonable according to Alejano et al. [START_REF] Alejano | Stability of granite drystone masonry retaining walls: I. Analytical design[END_REF]. Finally, liquefaction and the failure of the entire soil slope are disregarded. Table 2 presents the safety factors used for the computations as well as those from the French professional rules, which are similar [START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF].

Regarding non-seismic verifications, the Ultimate Limit State Equilibrium (ULS EQU), the Structural/Geotechnical Ultimate Limit State with the second approach (ULS STR/GEO), see [START_REF] Wang | Evaluating foundation design concepts of Eurocode 7 & 8[END_REF], and the Serviceability Limit State (SLS) are examined. The Ultimate Limit State (ULS SEISM) is applied for the seismic verification.

For the toppling verification of the SLS and the ULS STR/GEO, the design criterion to satisfy corresponds to the maximum eccentricity (noted e) of the transmitted load through the wall, as stated in Eurocode 7 [START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Ouvrages de soutènement -Murs[END_REF][START_REF] Afnor | Justification des ouvrages géotechniques -Normes d'application nationale de l'Eurocode 7 -Fondations superficielles[END_REF]. In the context of the ULS EQU and SEISM verifications, the partial safety factors γ m (γ φ' and γ C' ) for the materials are only applied to the backfill soil properties and not to the friction between blocks (as mentioned in [START_REF] Wang | Evaluating foundation design concepts of Eurocode 7 & 8[END_REF][START_REF] Pecker | Earthquake resistant design of foundations: new construction[END_REF][START_REF] Koseki | Seismic stability of reinforced soil walls[END_REF]). The friction between blocks is linked to the wall resistance, thus to the resistance safety factor γ R . As a consequence, applying a safety factor for the block material (block-block friction) would penalise twice the same parameters, which is not in agreement with the framework of Eurocodes. Moreover, in the seismic verification (ULS SEISM), the safety factors for materials are linked to the degradation of the shear strength of soils at high strain and/or in the presence of pore pressures. These are unlikely to occur for dry block-block joints [START_REF] Pecker | Earthquake resistant design of foundations: new construction[END_REF][START_REF] Koseki | Seismic stability of reinforced soil walls[END_REF][START_REF] Fardis | Seismic design, assessment and retrofitting of concrete buildings: based on EN-Eurocode[END_REF].

Non-seismic case

Computations discarding the seismic action are carried out on five walls, whose main characteristics are presented in Table 3. The walls are built of stones with geological natures representative of European and, in particular French, geology (molasse or marl sandstone, schist, limestone and granite). They include typical variations of essential DSRW properties (geometrical shape and soil resistance). Note that their height is typical of relatively high DSRWs. The backfill-wall interface friction angle δ is equal to the soil friction angle, given the dry gravel drain placed behind each DSRW [START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Colas | Assessing the two-dimensional behaviour of drystone retaining walls by full-scale experiments and yield design simulation[END_REF]. Both the internal batter λ m and the stone bed inclination α are initially equal to zero. The minimum required (bottom) widths for the walls given by the Eurocodes are then computed and compared to the recommended widths from ENTPE (Eds) et al. [START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF].

The recommendations from ENTPE (Eds) et al. [START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF] are very close to the maximum value recommended by the Eurocode standards with * V refers to Villemus [START_REF] Villemus | Experimental assessment of dry stone retaining wall stability on a rigid foundation[END_REF] with hydrostatic loading and C to Colas [START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Colas | Assessing the two-dimensional behaviour of drystone retaining walls by full-scale experiments and yield design simulation[END_REF] with backfill loading, while l refers to limestone blocks, s to schist blocks and g to granite blocks. † This experiment failed. However, the wall resisted at least a loading corresponding to a backfill height of 2.17m. ‡ This information is based on experimental results, using a default value. Details can be found in [START_REF] Villemus | Experimental assessment of dry stone retaining wall stability on a rigid foundation[END_REF][START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Colas | Assessing the two-dimensional behaviour of drystone retaining walls by full-scale experiments and yield design simulation[END_REF] for the experiments and in [START_REF] Savalle | Pseudo-static analytical model for the static and seismic stability of dry stone retaining walls[END_REF][START_REF] Savalle | Comportement sismique des murs de soutènement de talus en pierre sèche[END_REF] for the analytical method. § A combined sliding-overturning failure has been defined if the critical theoretical heights of the two failure modes were within a range of ± 5%.

Table 2

Partial safety factors for non-seismic and seismic safety verifications; Annex A of the Eurocode 7 [START_REF] Afnor | Geotechnical design -Part 1: General rules[END_REF] and ENTPE (Eds.) et al. [START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF].

Eurocode 7 [START_REF] Afnor | Geotechnical design -Part 1: General rules[END_REF] French professional rules [ slightly larger and conservative values: on average + 5% and maximum + 10 % difference. The most critical limit state and the corresponding failure mode differ according to the studied walls, justifying the consideration of all possible limit states. Finally, even if only five case studies are shown for brevity, the authors carried out a more comprehensive set of typical walls with identical conclusions.

Seismic case

Global analysis

The same walls are assessed according to Eurocode 8 [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings[END_REF][START_REF] Fardis | Seismic design, assessment and retrofitting of concrete buildings: based on EN-Eurocode[END_REF], assuming their design is provided by the most non-seismic critical case mentioned above. Keeping all other geometrical parameters constant, the extra-width required to withstand the seismic loading is computed (see Table 2 for the associated partial safety factors). Here, the authors recall that this section does not aim to describe precisely the dynamic behaviour of DSRWs against earthquakes but aims to provide a safe design width for DSRWs built in a seismic context. For this reason, the simplified pseudo-static modelling strategy is considered. The following equations give the pseudo-static design accelerations (horizontal a h and vertical a v ) [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings[END_REF]:

a h = a gR *S*S T *γ I r , a v = 0.5*a h (1)
where a gR is the reference acceleration for the considered seismic zone; S is the soil amplification factor (the maximum value being S = 1.8); S T is the topographic amplification coefficient; γ I is the importance coefficient of the structure; and r is the seismic behaviour factor [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings[END_REF]. The latter parameter considers the wall's ability to move during the seismic motion and dissipate seismic energy before collapsing [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 5: Foundations, retaining structures and geotechnical aspects[END_REF][START_REF] Fardis | Seismic design, assessment and retrofitting of concrete buildings: based on EN-Eurocode[END_REF]. Scaled-down experiments on a shaking table suggested a conservative value of r = 1.5 [START_REF] Savalle | Dynamic behaviour of drystone retaining walls: shaking table scaled-down tests[END_REF], corresponding to walls able to handle small movements before collapsing according to Eurocode 8 [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 5: Foundations, retaining structures and geotechnical aspects[END_REF]. In addition, numerical simulations using DEM approach have concluded that r = 1.5 was indeed conservative enough for the French seismicity [START_REF] Savalle | Dynamic Numerical Simulations of Dry-Stone Retaining Walls: Identification of the Seismic Behaviour Factor[END_REF]. In fact, after a numerical validation step using scaled-down experiments with dynamic time-history analyses, the numerical model was applied to full-scale structures with various dynamic time-history analyses (signal and backfill-wall parameters). Then, comparing the obtained numerical resistance and the estimated (analytical) pseudo-static resistance, the seismic behaviour factor r for each configuration was determined. Combining all the results, a mean value of r = 2.0 and a minimum value of r = 1.8 have been found. Therefore, a conservative recommended value of r = 1.5 was proposed since Eurocode only advices three values for r namely 1, 1.5 and 2. Even though this study has focused on the French case with moderate seismicity, the obtained outcome is considered acceptable elsewhere in Europe, especially given the safety margin taken at each step in [START_REF] Savalle | Dynamic Numerical Simulations of Dry-Stone Retaining Walls: Identification of the Seismic Behaviour Factor[END_REF]. In all analyses, the studied DSRWs are assumed to belong to the normal importance class. Particular conditions that rarely occur on-site are excluded (e.g., walls near very high buildings, highways, hospitals or energy facilities). It means that the importance coefficient γ I is lower than (or equal to) 1, which also implies no topographic amplification (S T = 1). Depending on the reference acceleration a gR , the most critical design acceleration reads:

a h,max = a gR *1.8 1.5 = a gR *1.2, a v,max = 0.5*a h,max = a gR *0.6 (2)
Fig. 6 depicts the extra-width required to fulfil a seismic design compared to the previously non-seismic design for each of the five walls (Table 3). Eurocode 8 states that seismic design is not required below a threshold of a h = 0.05g [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings[END_REF]: actually, the current calculations show that even if accounted for, the seismic case is not critical for the wall stability (Fig. 6). For higher design accelerations, walls sensitive to a sliding failure (case of schist wall, Fig. 6a) tend to have larger required extrawidths than those sensitive to a toppling failure (Fig. 6b-c). Finally, for high seismic hazard regions in Europe and France, see Fig. 7 (a h = 0.25 -0.4g), extra-widths reach very high values (+200%), reflecting the inadequacy of the pseudo-static method for high reference accelerations, which may provide unpractical recommendations for on-site works. Here, it is noted that wall disintegration is not considered, meaning that in high seismicity areas, additional prevention actions may be required.

A larger set of walls was analysed to obtain more general results. Walls geometries are still given in Table 3, but each geometry uses three different types of stone (schist, limestone, and molasse). Additionally, the analysis considers three typical bed inclinations α (0 • , 10%=5.7 • and 20%=11.3 • ) for each wall to improve the sliding resistance of DSRWs [START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Villemus | Etude des Murs de Soutènement en Maçonnerie de Pierre Sèches[END_REF][START_REF] Brady | Analysis of the stability of masonry-faced earth retaining walls[END_REF][START_REF] Colas | 2D modelling of a dry joint masonry wall retaining a pulverulent backfill[END_REF]. Fig. 8 gives the maximum (and mean) required extrawidths for DSRWs built with different kinds of stone depending on the stone bed inclination α. Here, the local peaks are related to vertical asymptotes for a peculiar response: when the design acceleration a h equals a h,critical (Eq. ( 3)), the whole retained soil slope loses its static equilibrium leading to infinite forces impossible to be sustained by the wall. Hence, the pseudo-static design requires huge wall widths close to this critical acceleration.

a h,critical = sin(φ r -β) + Cr cosφ γ r *Hcosβ cos(φ r -β) + 0.5*sin(φ r -β) (3)
For clarity, the maximum and mean values displayed in Fig. 8 ignore walls around and after their critical acceleration a h,critical , yet still resulting in small visible local peaks. For example, the first peaks for a design acceleration of about 0.22g observed in Fig. 8 correspond to Wall 5, the only case with a non-zero backfill slope β. Therefore, in practice, one may prefer to build a higher wall with a flat retained backfill than a smaller wall with an inclined backfill, especially in high seismic hazard regions. Finally, Eq. ( 3) highlights the inherent limitation of the design approach proposed by Eurocodes, which does not apply for high design acceleration a h , basically larger than 0.25-0.4g depending on the parameters of the backfill.

Analysing Fig. 8, it is noted that increasing the stone bed inclination is very efficient in reducing the extra-widths according to a pseudo-static design, although this parameter was not integrated into the non-seismic

Table 3

Main characteristics and required widths of five typical DSRWs according to Eurocode 7 (non-seismic case) criteria; maximum values are given in bold. Comparisons with the recommendations of ENTPE (Eds) et al. [START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF] are also given. Each column heading displays the critical failure mode (T: toppling; S: sliding). Granite DSRWs have similar properties as schist walls, and the values shown for schist can be used. * The slope inclination (backfill slope) refers to the backfill directly retained by the wall and not the global slope of the site.

French recommendations [START_REF] Capeb | Pierres sèches: guide de bonnes pratiques de construction de murs de soutènement[END_REF][START_REF] Entpe | Artisans Bâtisseurs en Pierre Sèche (ABPS), Ecole des Ponts ParisTech, IFSTTAR, and Fédération Française du Bâtiment (FFB), Technique de construction des murs en pierre sèche : Règles professionnelles. ENTPE, Artisans Bâtisseurs en Pierre Sèche[END_REF]. Table 4 details the maximum required extra-widths for each bed inclination according to different reference accelerations (a gR ) representative of Europe. A non-exhaustive description of the concerned regions is also given for each reference acceleration. For most western, northern, and eastern Europe, where the reference acceleration is smaller than a gR = 0.11g (low to medium seismicity), the maximum required extra-width reaches about 50%. If the wall is more resistant to sliding (not built with schist or with a nonzero bed inclination), the maximum extra-width decreases to about 35%. Finally, for reference accelerations larger than 0.25g, due to the inherent approximations of the pseudo-static approach, the maximum extra-widths exceed 100% of the non-seismic design which seems non reasonable in practice. For these cases, the pseudo-static modelling approach may also be inadequate, given the weak consideration of dynamics and, additionally, the fact that wall disintegration is ignored.

In conclusion, the systematic use of a stone bed inclination α is recommended. Whenever possible, an external batter equal to the stone bed inclination to facilitate the construction process is also suggested. For low to medium seismic regions (a gR < 0.11g), a stone bed inclination α of 10% is suitable, whereas, for larger seismic hazards (a gR ≅ 0.16-0.2g), a value of 20% can help to reduce the required extra-widths significantly.

French case study

This section provides a more detailed case study for France, chosen as an illustrating example of low to medium seismicity European countries. The section helps to understand the trends and practical [START_REF] Giardini | Mapping Europe's seismic hazard[END_REF]. The threshold a gR = 0.04 g is also highlighted in the legend. National seismic zonation maps in Eurocode 8 currently differ from this map. seismic design for DSRWs, depending on three different but typical cases for the backfill-wall condition. The first case (MAX) corresponds to the previous European study, i.e., large amplification for the soil (S = 1.8) and a standard construction (importance factor γ I = 1.0). The second (ROCK) represents the most critical case when the DSRW is directly founded on the bedrock (S = 1.0), considered a reference case for foundation conditions. The last one (RURAL) corresponds to walls of less importance (γ I = 0.8), built far from any road or building. In practice, according to the French regulations, these specific cases (RURAL) are not subjected to seismic regulations. However, this work gives reference values, which are helpful for DSRWs stakeholders. These three configurations are analysed according to the four seismic zones of metropolitan France (a gR = 0.04, 0.07, 0.11 and 0.16g), leading to a total of 12 case studies (Table 5).

Table 6 sums up the maximum (and mean) extra-widths (among the different walls and stones type) obtained for each seismic situation depending on the stone bed inclinations. In seismic zones S1 to S3, one can again note that bed inclination α dramatically impacts the results; however, a bed inclination of 20% does not provide a significant increase in resistance compared with an inclination of 10% (see also Fig. 8b-c). Therefore, as usual in the South of France, an inclination of the stone bed of 10% is recommended: seismic design requires no more than 40% extra-width (compared to a non-seismic design). In seismic zone S4, walls built with a stone bed inclination of 10% require an extrawidth of 90%, which induces substantial extra costs for the wall construction. In this case, one should either use a steeper bed inclination or conduct a specific analytical computation to optimise the geometry of the DSRW. However, if the wall is built far away from roads and buildings (RURAL) and with an inclination bed of 10%, the maximum expected extra-width only reaches 50%. On the contrary, if the wall is directly founded on the bedrock (ROCK), the seismic required extrawidth drops to a maximum of 30%. If both conditions are fulfilled (RURAL & ROCK), the required extra-widths do not exceed 20% (case not addressed in Table 6). Finally, the general recommendations of Table 6 (maximum values) can be readily used for practical non-seismic and seismic design of DSRWs without requiring more detailed computations.

As already noted, specific analytical computations should be carried out for the situation MAX-S4 or more critical seismic implantations instead of using the general approach with the maximum values

Table 4

Maximum extra-widths for DSRWs required by a pseudo-static seismic design compared to a non-seismic design depending on the reference acceleration (a gR ). The analysis only covers DSRWs belonging to the normal class of importance (γ I ≤ 1.0). According to the seismic hazard map of European countries, the second column gives the corresponding regions of each reference acceleration. The reader is referred to Figs. 7 and8. displayed in Table 6. More specifically, one should pay attention to specific parameters of the DSRW in the design that plays a critical role in the seismic computation (see full details in [START_REF] Savalle | Comportement sismique des murs de soutènement de talus en pierre sèche[END_REF]). Apart from the positive influence of the stone bed inclination and the negative effect of the retained slope angle already highlighted, a positive internal wall batter is recommended (this is the case of a self-stable wall). To support this recommendation, Walls 1 to 5 (with various stone types and bed inclinations) have been designed to withstand a specific seismic acceleration (0.05g, 0.10g … and 0.35g). In a second step, each wall section geometry has been modified, adding an internal batter (λ m = 5%, 10% and 15%) but keeping the same surface area as before. It means that the total volume of stones used in that case is the same but that the geometry of the wall section is different (i.e., with a larger width at the base). Finally, the maximum acceleration withstood by the walls with an internal batter is compared to the maximum acceleration found for those without the batter (Fig. 9). The curves correspond to the average values found throughout the different walls and stones. Only positive values have been found, meaning a systematic improvement of the seismic resistance when adding an internal batter. This improvement is particularly significant for walls with non-zero stone bed inclination and low (S2) to moderate (S3) seismic hazard regions.

To conclude, in zone S2, no DSRW extra-width is required to fulfil a seismic design if both an inclination bed of 10% and an internal batter of at least 10% are used. This means that an adequate choice for the wall geometry compensates for the extra resistance required to satisfy a seismic design in case of low seismicity.

Table 5

Horizontal seismic design accelerations a h for different critical cases (as a proportion of g = 9.81m.s -2 ). The vertical acceleration a v is systematically taken equal to 0.5*a h .

Conclusions

The present study addresses non-seismic and seismic designs of Dry Stone Retaining Walls (DSRWs) according to the European standards (Eurocodes) used for conventional retaining walls while proposing an adapted methodology (e.g. including internal failure of DSRW). It has been shown that the current French recommendations for the nonseismic design of DSRWs comply with Eurocode 7 (geotechnical engineering) standards being slightly more conservative than the latter. Moreover, in very low seismic zones (a h < 0.05g), non-seismic limit states are the most critical states for the design of DSRWs, which confirms that a seismic design is not required in these regions [START_REF] Afnor | Eurocode 8): Design of structures for earthquake resistance -Part 1: General rules, seismic actions and rules for buildings[END_REF]. This is generally not the case in zones of higher seismicity where the seismic design according to Eurocode 8 (seismic engineering) standard is almost always the most critical.

The study revealed different geometrical optimisation options for DSRWs. It is highly recommended to use systematically: i) a bed inclination of at least 10%; ii) a flat retained backfill; iii) and a positive internal batter of at least 10%. These three geometric parameters have a significant impact on seismic design. If these recommendations are followed in low seismic zones (up to a h = 0.08g), there is also no increase in dimensions to fulfil a seismic design.

The pseudo-static approach generally gives practical global recommendations for low to moderate seismic hazard zones (up to a h = 0.2g). In addition, a specific geometrically optimised (as stated above) pseudostatic design can still produce affordable recommendations in more critical cases (up to a h = 0.3g). However, for higher design acceleration or particularly critical cases, the pseudo-static approach for the seismic design leads to values higher than 50% for the extra-widths. In this case, dynamic time history computations are recommended to obtain more accurate results that account for wall disintegration failure. Indeed, this failure mode may be critical for high seismicity areas, particularly if combined with poor execution conditions. Finally, as an example illustrating European countries, France is used as a case study of seismic design applied to DSRWs in low to moderate seismicity areas. In metropolitan France, where many DSRWs can be found, the expected extra-width provided by seismic design does not exceed 40% for walls located in low seismic zones presenting a stone bed inclination of 10%, in case the foundation is not on the bedrock. In the same conditions, walls built in moderate seismic hazard zones need either a stone bed inclination of 20% or a specific analytical computation to optimise the section. Finally, walls directly founded on the bedrock, in the case of low and moderate seismicity areas, require a maximum of 30% extra width to fulfil a seismic design.

Fig. 3 .

 3 Fig. 3. Equilibrium of the dry stone retaining wall [34].

Fig. 4 .Fig. 5 .

 45 Fig. 4. Comparison of DEM simulation, analytical simulation and experimental tilting tests [37]. Fig. 5. Experimental toppling failure obtained by Colas [39].

Fig. 6 .

 6 Fig. 6. Extra-width required by a seismic design compared to a non-seismic design for the studied walls. Walls are presented according to the type of stone used; a) schist; b) limestone; c) molasse (marl sandstone).

Fig. 7 .

 7 Fig. 7. European map of reference acceleration a gR (return period of 475 years) according to Giardini et al. [49]. The threshold a gR = 0.04 g is also highlighted in the legend. National seismic zonation maps in Eurocode 8 currently differ from this map.

Fig. 8 .

 8 Fig. 8. Maximum (continuous line) and mean (dashed line) required extra-widths for the studied DSRWs depending on the type of stone and the stone beds inclination α: a) α = 0%; b) α = 10%; c) α = 20%.

Fig. 9 .

 9 Fig. 9. Effect of the internal batter λ m (keeping the same area for the wall section) on the seismic resistance of a DSRW for different bed inclinations a) α = 0%; b) α = 10%; c) α = 20%.

Table 1

 1 Parameters of the full-scale DSRWs experimental campaigns from[START_REF] Villemus | Experimental assessment of dry stone retaining wall stability on a rigid foundation[END_REF][START_REF] Colas | Full-scale field trials to assess dry-stone retaining wall stability[END_REF][START_REF] Colas | Assessing the two-dimensional behaviour of drystone retaining walls by full-scale experiments and yield design simulation[END_REF]. Experimental and analytical results are also given.

	Name*	V1l	V2l	V3l	V4l	V5s	C1g †	C2s	C3s	C4l
	Geometrical parameters
	Height H (m)	2	1.95	4	2	4.25	2.5	2.5	2.5	2.5
	Base width B (m)	0.9	0.91	1.8	0.9	1.8	0.6	0.6	0.7	0.65
	External batter λ v (%)	15	0	15	12	15	6	6	6	6
	Stone bed inclination α ( • )	0	0	0	4	8.5	3.4	3.4	9.1	9.1
	Backfill slope β ( • )	NA	NA	NA	NA	NA	26.4	31.7	32.6	34.9
	Mechanical parameters Wall unit weight γ rw (kN/m 3 )	15.4	14.9	15.7	15.7	18.0	21.0	20.0	20.0	21.8
	Blocks friction ϕ rw ( • )	36	36	36	36	28.5	27	25	25	35
	Internal rotation ( • ) ‡ Backfill unit weight γ f (kN/m 3 )	5 NA	5 NA	5 NA	5 NA	5 NA	5 14.9	5 14.9	5 14.9	5 14.9
	Backfill friction ϕ f ( • )	NA	NA	NA	NA	NA	37.7	37.7	37.7	37.7
	Experimental results
	Critical height (m)	1.74	1.9	3.37	1.94	3.62	>2.17	2.41	2.96	2.95
	Failure mode (S/T)	S	S/T	S	S/T	S	NA	S/T	T	T
	Analytical results
	Critical height (m)	1.73	1.84	3.48	1.86	3.68	2.86	2.69	3.02	2.82
	Failure mode (S/T) §	S	S/T	S	S/T	S	T	S/T	T	T
	Difference to exp.	-1%	-3%	þ3%	-4%	þ2%	NA	þ11%	þ2%	-4%

Table 6

 6 Influence of the stone bed inclination α on the extra width required for seismic design. Maximum values, together with average values in parentheses, are given.

N.Savalle et al.

Funding

This work was partly financed by FCT / MCTES through national funds (PIDDAC) under the R&D Unit Institute for Sustainability and Innovation in Structural Engineering (ISISE), under reference UIDB / 04029/2020. This study has also been partly funded by the STAND4-HERITAGE project (New Standards for Seismic Assessment of Built Cultural Heritage) that has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (Grant agreement No. 833123), as an Advanced Grant. In addition, the authors want to acknowledge the French Ministry of Education and Research for their financial support through a PhD grant attributed to the first author. The opinions and conclusions presented in this paper are those of the authors and do not necessarily reflect the views of the sponsoring organisations.

Data availability

The data arising from the analytical simulations presented in the paper is not available publicly.

Data availability

Data will be made available on request.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Raphaël Préa
email: raphael.prea@cea.fr

Philippe Fillion
email: philippe.fillion@cea.fr

Laura Matteo
email: laura.matteo@cea.fr

Gédéon Mauger
email: gedeon.mauger@cea.fr

Anouar Mekkas
email: anouar.mekkas@cea.fr

Cathare-3 V2

CATHARE-3 V2.1: The new industrial version of the CATHARE code

Keywords: System code, CATHARE, Multi-field Computation, Axial Pump, 3D Module

INTRODUCTION

In 1979, CEA, EDF and FRAMATOME launched the CATHARE project. The goal to the CATHARE project is to develop a best-estimate thermal-hydraulic code for safety analysis of nuclear reactors. In 1987, CATHARE-1 V1.3L was the first CATHARE version validated for small break Loss of Coolant Accident (LOCA). In 1996, CATHARE-2 V1.3L became the first CATHARE version qualified for safety studies (including large break LOCA). In December 2003, CATHARE-2 V2.5 was implemented in real time plant simulators. CATHARE-3 is the new version of the CATHARE code. Its development has begun in 2006 as part of the NEPTUNE project launched by the CEA, EDF, FRAMATOME and IRSN in 2001 [START_REF] Guelfi | NEPTUNE: A New Software Platform for Advanced Nuclear Thermal Hydraulics[END_REF]. An initial roadmap for CATHARE-3 was presented in 2009 [START_REF] Emonot | CATHARE-3: A new system code for thermalhydraulics in the context of the NEPTUNE project[END_REF]; an evaluation version V1.0 was delivered in 2012 and a validated version V2.0 was delivered in 2016. The first industrial version of the code validated for the Pressurized Water Reactors (PWRs), and partially validated for sodium applications, V2.1, was delivered in December 2019, in the continuity of the reference multi-concept CATHARE-2 [START_REF] Geffraye | CATHARE 2 V2.5_2: A single version for various applications[END_REF]. This article first shows the current status of the CATHARE-3 code and the main new features of this industrial version. Then, the planned developments are presented.

CURRENT STATUS AND MAIN NEW FEATURES

In this part, the current status of the code is first presented, then, the main new features of the code are described, related to new needs for the PWR safety analysis and for reactor designs:

-The 3 fields modelling capabilities, -The predictive pump model, -The Multi-fluid and multi-reactor modelling tools, -The 3D modelling capabilities

Current status: Continuity with CATHARE-2 Code

CATHARE-2 is the current industrial version of the CATHARE code; it is mainly use to safety analysis [START_REF] Geffraye | CATHARE 2 V2.5_2: A single version for various applications[END_REF] and into real-time simulator of nuclear power plant [START_REF] Pialla | Integration of CATHARE, a 'state-of-the-art' code, in real-time simulators at EDF/SEPTEN[END_REF]. The main goal of CATHARE-3 code is to be the successor of CATHARE-2 and to be a modular tool that can evolve. To perform this objective, a new language have been chosen for CATHARE-3 with a rigorous plan to ensure non-regression between CATHARE-2 and CATHARE-3.

A new coding language

CATHARE-2 benefits from an object-oriented conception, but its implementation is in FORTRAN so that many features of modern object-oriented languages (memory management, polymorphism, and encapsulation) are simulated. The code is therefore not as extensible, readable or maintainable as it could be with a true object-oriented language. Another important limitation of CATHARE-2 architecture relies in the fact that it is composed of two executables:

-A reader that reads the data file, do some preprocessing and writes a "memory image input file" and a FORTRAN file describing the transient to be simulated. -A calculation executable obtained by linking the previously generated FORTRAN subroutine with CATHARE's library. This executable reads the "memory image input file" and runs the simulation. It has been decided to develop switch language with CATHARE-3 and to have a C++ code. This C++ architecture have been obtained with the following process:

-From 2006 to April 2019, the development of CATHARE-3 was made in FORTRAN, starting with CATHARE-2 code and adding new features. In 2012, the merge of the reader and the calculation code was achieved and a 'python' translator of the input deck (needed as CATHARE-3 input decks are slightly different from CATHARE-2 due to the suppression of the reader) was written. -From 2012 to April 2019, a FORTRAN to C++ translator was developed, including optimizations of the code, ensuring a binary non-regression between the FORTRAN and C++ results at each version of CATHARE-3. -Since April 2019, new developments of CATHARE-3 are written in C++. The main advantage of this progress is that it make possible to perform daily non-regression tests between the current development version of CATHARE-3 and the current industrial version of the CATHARE code: CATHARE-2 V2.5.

Non-Regression between CATHARE-2 and CATHARE-3

As explain in the previous part, non-regression tests are daily perform between CATHARE-3 and CATHARE-2. For the V2.1 version of CATHARE-3, more than 2000 tests are perform, including more than one hundred Integral Effect Tests (IET) and Separate Effect Tests (SET) from the CATHARE-2 validation matrix. Industrial tests are also included in non-regression tests. They confirm that the two-fluid 6-equation model of CATHARE-3 is reliable, as most of the results superimpose with those of the CATHARE-2. These tests are included in Non-Regression reports delivered with the code. CATHARE-3 is already used for safety studies in the framework of the ASTRID project [START_REF] Anderhuber | Validation of CATHARE-3 system code for sodium twophase flow application: R&D path and simulation of key tests from the SIENA program[END_REF] [39] .

Three-Field Model

For flow conditions where both droplets and liquid film are present, the two-fluid model of CATHARE uses a correlation for the entrainment fraction, i.e. the ratio of the droplet flowrate to the total liquid flowrate, on the Steen-Wallis model. This model has some limitations, for example in annular-mist flow, because it cannot take into account the different behavior of the droplets and the liquid film on the wall, which may have very different velocities including counter-current velocities. This is also the case for stratified-mist flows encountered in hot legs, or for the estimation of the de-entrainment phenomenon in the upper plenum due to the drops impinging the structures. In such cases, a 3three-field model in which the liquid phase is split into a continuous field (film) and a dispersed field (droplets), each having its own volume fraction, velocity and temperature seems more likely to simulate the behavior of the flows.

A three-field model is being developed in CATHARE-3, based on an extension of the two-fluid model. Three sets of mass, momentum and energy conservation equations are solved for the three fields: the gas or vapor, the continuous liquid, and the dispersed liquid flowing in the form of entrained droplets. This model has been first developed and assessed for vertical two-phase flows especially for dryout prediction. Specific correlations for the entrainment, deposition and for the estimation of the droplet size have been developed or introduced during the 2000' and validated against several experiments in tubes (e.g Wurtz, Bennett) [START_REF] Jayanti | Calculation of dry out and post-dry out heat transfer in rod bundles using a three field model[END_REF] and rod-bundles (e.g. THTF, BFBT) [START_REF] Valette | Analysis of boiling two phase flow in rod bundle for NUPEC BFBT benchmark with 3field NEPTUNE system code[END_REF]. More recently, models of entrainment and deposition due to Okawa have been developed in the code, and models of the initial entrained fraction, i.e. the entrainment at the onset of annular two-phase flow, which has a significant impact on the prediction of the dryout, have been assessed against experiments, such as pre-dryout tests performed at KTH, completing the validation matrix for this application of the three-field model [START_REF] Spirzewski | Prediction of annular flows in vertical pipes with new correlations for the CATHARE-3 three-field model[END_REF].

The architecture of the code allows to select the three-field model in some elements of the circuit, and to keep the calculation of the flow with the 6-equation model where the droplet are not present. This is the case for example for LBLOCA application, where the calculation of the BETHSY 6.7c integral test, simulating the refilling and reflooding phases of this type of accident, has been performed using both the 3-field and the 6-equation models [START_REF] Valette | Revisiting large break LOCA with the CATHARE-3 three-field model[END_REF]. The three-field model is used to simulate the production of small droplets in the core at the quench front during the reflooding phase of such transients, and the liquid carryover by a strong steam flow towards the upper plenum, the hot legs and the steam generators. These phenomena, including the vaporization of the droplets in the steam generators creates the "steam binding" effect, influencing the reflooding rate. Specific models of droplet entrainment and deposition in the core, adapted to the reflooding phase conditions (namely inverted annular flow downstream the quench front, low pressure) are been developed in CATHARE-3, and validated against PERICLES and RBHT reflooding tests. ROSCO oscillating reflooding tests are also part of the validation matrix. For these tests, the threefield model better predicts, with respect to the 6-equation model, the top-down quenching where falling films and upward drop flow coexist in the upper part of the core. For the de-entrainment phenomenon in the upper plenum, SEROPS 2 experiments, devoted to study the droplets behavior in this part of the vessel, have been used to develop simple correlations in the CATHARE-3 3D module, taking into account the impact of droplets onto the upper plenum structures and the deposition of the droplets due the gravity. In the last years, one-dimensional models have been introduced in CATHARE-3 for the entrainment and deposition of the droplets in the hot legs, including both stratified and annular flow regimes. These models have been developed and validated using air-water experiments performed in large diameter horizontal pipes, including REGARD and Williams databases [START_REF] Fillion | Development of the CATHARE 3 three-field model for simulations in large diameter horizontal pipes[END_REF]. The deposition model takes into account the two mechanisms acting on the droplet deposition in such geometries: turbulent diffusion and gravity. For the latter mechanism, the deposition constant is related to the terminal falling velocity of the droplet, and accounts the droplet diameter, estimated using a specific correlation. MHYRESA scaled experiments, including the elbow and the inclined part of the hot leg, will be used to complete these 3-field models.

A Predictive 1D-Model for Rotodynamic Pumps

The primary circuit of nuclear reactors is composed of huge components such as primary pumps, which play a great role in reactor transients dynamics. For several years, the CEA is studying Generation IV reactors such as the Sodium cooled pool-type fast reactor concept. In this frame, hypothetic accidental transients like 'primary pump seizure' are analyzed using the CATHARE-3 code [START_REF] Chenaud | Computational thermal hydraulic schemes for SFR transient studies[END_REF]. Throughout this type of transients, cavitation may occur in non-affected primary pumps and the affected (locked rotor) pump may operate in reverse conditions in terms of flow rate and rotational speed. However, neither the primary pump 4-quadrant performances nor the cavitation behavior of the pump are known at the reactor design step. Experimental constraints due to liquid sodium working fluid enhance even more these considerations concerning the measured pump performances. For this purpose, CEA and interested industrial partners (Framatome, ArianeGroup, CETIM and EDF) decided at the end of year 2015 to implement a predictive transient, two-phase flow rotodynamic pump model in the CATHARE-3 code and to validate results at component scale (pump) and system scale (loop, reactor) with respect to available experimental data (Matteo et al. [START_REF] Matteo | Development and validation of a onedimensional transient rotodynamic pump model at component scale[END_REF][13][14] [START_REF] Matteo | Qualification of the CATHARE-3 one-dimensional transient rotodynamic pump model on derap two-phase cavitating tests[END_REF][16] [17][18][19]). An attempt of a similar modelling was done earlier in the CATHARE-2V1.3L version of the code with the work of De Crecy [START_REF] Crecy | 2 phase -2 pressure model application to an axial pump model[END_REF] [21], Van den Hove and Geffraye [START_REF] Van Den Hove | The CATHARE code one-dimension pump model[END_REF], Coste [START_REF] Coste | Use of the CATHARE 1D pump model to predict the two-phase characteristic of a pump for petroleum fluids[END_REF] and Geffraye and Bestion [START_REF] Geffraye | Assessment of the CATHARE 1D pump model[END_REF]. Focus was mainly made on two-phase flow modelling and less on rotating machinery physics. This led to difficulties for the model to be predictive for various pump geometries (radial to mixed pumps). This model has not been maintained and is no longer available. By reconsidering the approach and developing a new rotodynamic pump model into the last version of the CATHARE code (CATHARE-3), it is intended to obtain a predictive and generic model. The originality of the present project is to build a real 1D model -the mean streamline in the different parts of the pumps is meshed-able to predict the performance of mixed and radial flow pumps in purely liquid or gas/liquid, and four quadrant operations. The 6-equation system representing classical two-phase flows in pipes [START_REF] Faydide | Two-phase flow modeling with thermal and mechanical non equilibrium[END_REF] is used as a basis to model each part of the centrifugal pump: suction, impeller, diffuser, volute and discharge pipe. The system is discretized according to one curvilinear direction, what allows following a mean flow path. The example of the impeller inter-blade channel is given in Fig. 1. For the suction and discharge pipes, there is no need to implement other source terms in the six-equation system. On the contrary, new source terms have to be added to correctly represent the impeller, diffuser and volute parts in order to take into account the pressure rise due to centrifugal acceleration in the moving part, the losses specific to pump geometries and the flow redress in the fixed parts.

Fig. 1 : The mean flow path in the impeller inter-blade channel [14]

As the impeller is a moving part, is it solved according to the rotating frame and the associated calculated velocities are the relative ones. On the contrary, the calculated velocities are the absolute ones in the fixed parts of the pump. Consequently, a frame change has been implemented between fixed parts and the rotating one. This is operated using the velocity triangles at frame interfaces (suction-impeller and impellerdiffuser). See reference [START_REF] Matteo | Modelling of a centrifugal pump using the CATHARE-3 onedimensional transient rotodynamic pump model[END_REF] for more details on the modelling (expression of governing equations and source terms in particular). When modelling a rotodynamic pump using the new CATHARE-3 1D-pump model, the representation shown on Fig. 2 is obtained using meridian sections [START_REF] Matteo | Modelling of a centrifugal pump using the CATHARE-3 onedimensional transient rotodynamic pump model[END_REF]. Each part of the pump (suction, impeller, diffuser, volute and discharge pipe) is meshed. To be able to represent pumps without diffuser or multi-stage turbomachines using the same modelling structure, the diffuser and volute parts are defined in the same CATHARE-3 one-dimensional element.

Fig. 2 : 1D-pump structure: four 1D elements (diffuser and volute belong to the same element)

A non-dimensional head curve is represented Fig. 3 with the pressure coefficient versus the flow rate coefficient for computations [START_REF] Matteo | Modelling of a centrifugal pump using the CATHARE-3 onedimensional transient rotodynamic pump model[END_REF] and experimental data [START_REF] Duplaa | Experimental study of a cavitating centrifugal pump during fast startups[END_REF]. Two main conclusions can be drawn regarding these results : first, the similarity laws [START_REF] Stepanoff | Pompes centrifuges et pompes hélices[END_REF] are well respected by the CATHARE-3 1D pump model for the three considered rotational speeds (1000, 2000 and 3000 rpm) as the maximal relative error is 12% when considering all cases; second, off-design flow situations can be predicted (very low or very high flow rates).

A Multi-Fluid and Multi-Reactor Concept Modeling Tool

The same multi-fluid and multi-reactor concept strategy as for the CATHARE-2 code [START_REF] Geffraye | CATHARE 2 V2.5_2: A single version for various applications[END_REF] is being adopted for CATHARE-3 developments. This is in order to model advanced concepts such as gas-cooled fast reactors [START_REF] Bertrand | Synthesis of the safety studies carried out on the GFR2400[END_REF][29] and sodium-cooled fast reactors [5][30]. The modular structure allows the coolant circuits of any experimental facility or nuclear plant to be represented by assembling 0-D, 1-D and 3-D hydraulic modules. Thermal and hydraulic sub-modules such as thermal walls, heat exchangers, pumps, compressors, turbines, valves, fluid source or sink can be added to these main modules. Similarly, the same quality and validation procedures [START_REF] Bentivoglio | Validation of the CATHARE 2 code against experimental from Brayton cycle plants[END_REF][32], as well as numerical resolution scheme and pre/post-processing tools (GUITHARE) are used for any application. All these legacy features have been reported in the new version of the code, but new ones have been developed in addition. These new developments mainly aimed to model real gas circuits used in such applications as Brayton cycle power conversion systems [START_REF] Mauger | On the dynamic modeling of Brayton cycle power conversion systems with the CATHARE-3 code[END_REF]. For highpressure nitrogen Brayton cycles (180 bar), the CATHARE-2 calculations led to significant uncertainties due to the use of the ideal gas equation of state and models. Indeed, the way to model nitrogen in CATHARE-2 is based on the standard two-phase fluid model (residual fluid named HIGHXNC) adapted to deal with very high fraction of ideal non-condensable gas. In the operating range of the studied Brayton cycle, the maximal errors were about 10% for density, 24% for heat capacity, 28% for viscosity and 37% for thermal conductivity.

In the framework of the NEPTUNE project, the CEA built the NEPTUNE platform on a component architecture [START_REF] Guelfi | NEPTUNE: A New Software Platform for Advanced Nuclear Thermal Hydraulics[END_REF]. The same component can be used by different NEPTUNE thermal-hydraulics codes such as CATHARE-3 or NEPTUNE-CFD. Especially, the EOS component has been developed to compute the thermodynamic and transport properties of the fluids from several in-house code libraries such as CATHARE-2, THETIS, FLICA-4, etc. This allowed to implement REFPROP [START_REF] Lemmon | NIST Standard Reference Database 23: Reference Fluid Thermodynamic and Transport Properties-REFPROP, Version 9.1[END_REF] as a new library in the EOS component in order to use the nitrogen real gas equation of state in CATHARE-3. REFPROP core property routines compute thermodynamic and transport properties of many fluids using a Helmholtz energy approach mainly intended for scientific use. Moreover, new methods have been developed in the EOS component to access properties only available in the REFPROP libraries such as speed of sound and entropy.

The CATHARE-3 architecture was made in order to perform three-phase flow calculations, i.e. adding dispersed phases into the two original continuous phases (especially droplets in the continuous gas phase).

It was therefore possible to choose between two-phase flow (standard CATHARE-2 model) and three-phase flow (advance model). By using this more flexible architecture, a single-phase flow model has been developed for monophasic calculations. This new feature is available for both liquid and gas single-phase flow calculations. This single-phase flow functionality mainly contributed to the reduction of computational time and the resolution of some numerical issues due to the presence of useless residual phases. Indeed, in this singlephase case, the liquid phase properties are no longer calculated as well as some other properties such as surface tension, which is particularly expensive in terms of computing time with REFPROP. In addition, new real gas models have been implemented for the turbomachinery [START_REF] Mauger | Description of an Improved Turbomachinery Model to be Developed in the CATHARE 3 code for ASTRID Power Conversion System[END_REF] and the sonic flow calculation [START_REF] Mauger | On the dynamic modeling of Brayton cycle power conversion systems with the CATHARE-3 code[END_REF] in order to be consistent with this new equation of state. With regard to the turbomachinery model, performance characteristics are calculated with the real speed of sound and the gas entropy that are available in REFPROP. Moreover, considering the high Reynolds number in such gas piping, specific singular [START_REF] Idel'cik | Handbook of Hydraulic Resistance[END_REF] and regular [START_REF] Zigrang | Explicit approximations to the solution of Colebrook's friction factor equation[END_REF] pressure drop correlations have been implemented in the CATHARE-3 code taking into account pipe roughness and abrupt flow area changes. The impact of this new CATHARE-3 models has been investigated on several case studies such as the experimental DIADEMO facility [START_REF] Mauger | Modelling of the DIADEMO experiment with the CATHARE code for ASTRID gas power conversion system studies[END_REF], a high-pressure nitrogen Brayton cycle [START_REF] Mauger | On the dynamic modeling of Brayton cycle power conversion systems with the CATHARE-3 code[END_REF] and a complete sodium-cooled fast reactor transient on a loss of power supply case [START_REF] Droin | CATHARE3 transient analysis of an innovative Power Conversion System based on the Brayton cycle modelled with real gas Equations Of State[END_REF]. New application and validation cases are already being considered such as the modeling of an organic Rankine cycle with Novec 649 used as a working fluid, a supercritical CO2 Brayton cycle, a twophase flow ammonia cooling system or a natural gas supply system.

3D Modelling Capabilities

This section presents the evolution of the 3D capabilities with CATHARE-3, first for the 3D modelling of the whole PWR vessel, then, for the sub-channel analysis.

From CATHARE-2 to CATHARE-3 for a Modular Modelling of the Vessel

The 3D module of CATHARE-2 has been validated for both whole reactor pressure vessel [START_REF] Dor | Assessment of the CATHARE 3D module for LBLOCA simulation[END_REF] and some subcomponents of the pressure vessel such as the core [START_REF] Morel | Validation of the CATHARE code against PERICLES 2D BOIL-UP Tests[END_REF] or the downcomer [START_REF] Dufrêche | Assessment of the CATHARE 3D module with UPTF downcomer refill tests[END_REF] regions from the small to the large break LOCAs. A global overview of the CATHARE-2 3D module is available in [START_REF] Dor | CATHARE 3D Module from CATHARE-2 V2.5_3 to CATHARE-3 V1.0[END_REF]. One of the main goal of CATHARE-3 is to improve 3D modeling of the vessel using thin and non-conformal structured meshes [START_REF] Emonot | CATHARE-3: A new system code for thermalhydraulics in the context of the NEPTUNE project[END_REF].

Fig. 5 : Reactor vessel modelled by CATHARE-2 (middle) and CATHARE-3 (right)

In CATHARE-2, only one cylindrical 3D grid is used for the simulation of the whole pressure vessel (see Fig. 5 center) and it has been proved to be sufficient for simulating any kind of breaks LOCA transients. However, for a better flexibility of the modeling, for the simulation of some transients with significant transverse flow in the core, for neutronic-coupled simulations, or any other 3D flow investigations, a Cartesian 3D meshing of the core and an adapted meshing of the plena would significantly improve the simulation capabilities. In CATHARE-3, it is possible to couple different kinds of 3D modules (one for each component: core, downcomer, plena...). A typical set of grids is (see Fig. 5 right):  Cylindrical 3D grids for all the vessel components except the core,  Cartesian 3D grids (one cell per assembly in a horizontal section and 40 axial meshes) for the core. As explained in [START_REF] Ruby | CATHARE-3: First computations of a 3-inch break Loss-Of-Coolant Accident using a modular multi-3D modeling of a PWR vessel[END_REF], new optimized resolution methods have been developed in CATHARE-3 to couple 3D modules, and the input data-deck syntax has also been adapted to these new functionalities [START_REF] Dor | CATHARE 3D Module from CATHARE-2 V2.5_3 to CATHARE-3 V1.0[END_REF]. Creating such type of modeling can be complicated for users, so the CATHARE team has improved the GUITHARE tool (based on the SALOME platform [45]) and the layout data of the 3D modules. GUITHARE stands for Graphical User Interface for caTHARE, it is a common tool for CATHARE-2 and CATHARE-3, including a syntax translator from CATHARE-2 to CATHARE-3 and now adapted to these new 3D capabilities. Most users have a CATHARE-2 modeling of the pressure vessel, in which the 3D vessel is made up of a single 3D module. Using GUITHARE, it is possible to directly translate then split the CATHARE-3 Mono-3D dataset into multiple 3D elements. GUITHARE allows the pre-and post-processing of multiple 3D modules in the same study. Several 3D modules can be grouped together to form a macro-module that is displayed as if it was a single 3D module (see Fig. 5right). GUITHARE also helps creating Cartesian-cylindrical junctions; the standard way to define the junction between two 3D modules is to let the user define the geometry and let the code compute the connectivity tables between meshes. This is well adapted when the connection is simple and does not require particular treatments. However, the case of a junction between a Cartesian-meshed core and a cylindrical-meshed plenum is much more complex and we have decided that GUITHARE would be able to calculate the characteristics of the meshing intersection between the two grids and give them to CATHARE-3 for the calculation. Thanks to the new syntax abilities of CATHARE-3, it is possible to write a compact description of the geometry and the power distribution. Indeed, CATHARE-3 allows a multiple creation of some objects made into a loop (with the REPEAT directive). For example, it can be used for the fuel rod creation in input data deck with a parametric distribution of the radial power. A LOCA computation using these new 3D functionalities of CATHARE-3 is available in [START_REF] Préa | CATHARE-3: a first computation of a 3-inch break loss-of-coolant accident using both Cartesian and cylindrical 3D meshes modelling of a PWR vessel[END_REF].

Sub-channel Analysis with CATHARE-3

For core applications, the CATHARE-3 3D module includes also extended capabilities to improve the accuracy of simulations of light water reactor accidents. Both 3D modules of CATHARE 2 and CATHARE-3 are based on a porous version of the two-fluid 6-equation model, obtained from a double average process (first a time-average filtering, then a space average accounting fluid-solid interactions) of the local instantaneous two-phase balance equations. But, contrary to CATHARE-2, the 3D module in CATHARE-3 features models for turbulent dispersion and diffusion terms appearing in the 3D porous equations, which can affect the temperature map in a rod bundle, and the void dispersion phenomena is taking into account by a mixing term in the momentum balance equations [START_REF] Chandesris | Rod bundle thermalhydraulics mixing phenomena: 3D analysis with CATHARE 3 of various experiments[END_REF]. These models can be activated optionally.

The 3-D validation matrix for the diffusion and dispersion terms includes two OECD/NRC benchmarks: PSBT [START_REF] Valette | Analysis of subchannel and rod bundle PSBT experiments with CATHARE 3[END_REF] and BFBT [START_REF] Valette | Analysis of boiling two phase flow in rod bundle for NUPEC BFBT benchmark with 3field NEPTUNE system code[END_REF] with sub-channel calculations. The latter tests are simulated using the 3-field model of the code (see paragraph 2.2). A semi-empirical relation for the turbulent diffusion-dispersion terms of the energy equation, using an algebraic model for the turbulent kinetic energy, has been established for rod bundle with PWR type geometries and calibrated from PSBT tests with mixing vane grids [START_REF] Valette | Analysis of subchannel and rod bundle PSBT experiments with CATHARE 3[END_REF] [START_REF] Fu | Analysis of flow in rod bundles with velocity and temperature radial dissymmetry using CATHARE-3[END_REF]. The assessment of the energy diffusion-dispersion model has also been performed against the PNNL 2×6 rod bundle single-phase tests, at low velocity [START_REF] Fu | Analysis of flow in rod bundles with velocity and temperature radial dissymmetry using CATHARE-3[END_REF]. Weiss experiments, consisting of two 14x14 square lattice rod bundles with unbalanced velocities at the inlet of the assemblies [START_REF] Fu | Analysis of flow in rod bundles with velocity and temperature radial dissymmetry using CATHARE-3[END_REF], and AGATE tests, performed in a 5x5 PWR-type rod bundle with two truncated partial grids placed at two different elevations [START_REF] Chandesris | Rod bundle thermalhydraulics mixing phenomena: 3D analysis with CATHARE 3 of various experiments[END_REF], both featuring crossflows between the subchannels, are also used to assess the friction terms and/or the momentum turbulent diffusion and dispersion terms in single-phase conditions. The void dispersion effects, related to the double averaging process of the interfacial momentum term, are taken into account using a semi-empirical correlation and validated against PSBT two-phase tests series. The validation matrix at subchannel scale is completed by some rod bundle experiments, in adiabatic conditions in GRAZIELLA loop and with heated rods in OMEGA test section [START_REF] Chandesris | Rod bundle thermalhydraulics mixing phenomena: 3D analysis with CATHARE 3 of various experiments[END_REF].

Sub-channel simulations require meshing with several thousands of cells, and the historical CATHARE linear solvers developed for 1D simulations or 3D modelling using a coarse mesh become inefficient in these cases. To find a more efficient resolution method, requiring less memory space, being less time consuming, and getting more flexibility in the resolution method, we have chosen to interface CATHARE-3 to the PETSC library, providing interfaces to linear solvers such as MUMPS [50]. Thanks to these numerical methods, it is now possible to perform simulations with a large number of 3D cells, up to 20,000. This capability has been used to perform sub-channel calculations with CATHARE-3, modeling some experimental facilities as PERICLES-2D [START_REF] Alku | Modelling of turbulent effects in LOCA conditions with CATHARE-3[END_REF] or LSTF core [START_REF] Préa | Rod bundle thermalhydraulics mixing phenomena: 3D analysis with CATHARE-3 of ROSA-2/LSTF experiment[END_REF].

FUTURE DEVELOPMENTS

Physical modelling

For PWR applications, CATHARE-3 modelling is mostly based on the revision 6.1 physical models of CATHARE-2. This revision (i.e. a consistent package of closure laws validated against SETs and IETs) gives satisfactory results. However, some improvements are identify to meet the state of art, to extent the validation or to reduce the uncertainties of the models. Three main directions have been drawn to meet these requirements:

 Improvement of the 6-equation model. Some closure laws could be revised, aiming at reducing the uncertainties or some weaknesses of the code. Some of them have been already included into the revision 6.2, already available in option for the version 2.1 of CATHARE-3. This revision includes improvements of the flashing model at low pressure, the 2006 Loop-up table for the estimation of the critical heat flux, and minor modifications for the formulation of the interfacial friction in the 3D module. For the next revisions of the models, one of the objectives is to better predict the critical flowrate at the break, especially in low-subcooled conditions, by the development and the assessment of mechanistic models for the flashing [START_REF] Vernassiere | A mechanistic flashing model for critical flowrate prediction using CATHARE 3 system code[END_REF]. The formulation of the interfacial friction in the core at low pressure, typical to loss of residual heat removal conditions, is being revisited [START_REF] Lopez | Validation of a drift-flux model used in the CATHARE code for rod bundle geometry at low pressure and low liquid flow conditions[END_REF]. Current developments also concern the improvement of the flow map, especially for transition from and to stratified flows. Long-term researches could be the use of additional equation for the estimation of the turbulent kinetic energy for the transition and/or the use of an equation for the transport of interfacial area as suggested in [START_REF] Bestion | On the modelling of two-phase flow in horizontal legs of a PWR[END_REF].  Improvement of the 3D model. One of the main direction concerning the physical research is relative to simulation of 3D flows, namely in the downcomer and in the core. Dedicated experiments, such as METERO-V, will provide data for the validation and development of the friction terms and turbulent-dispersion terms involved in possible crossflows in the core, occurring during the reflooding phase or the uncovering phase of LOCAs [START_REF] Bestion | Improved PWR LOCA simulation through refined core 3D simulation -An advanced 3D modelling and the associated METERO program[END_REF].  Improvement of the 3-field model. The current 3-field model will be improved and extended to other parts of the circuit (e.g. near the injection system of the cold leg). The correlations developed for the 3-field could be also derived for the 6-equation model, namely for the entrainment fraction. As far as new concepts of reactors will be study using CATHARE-3, development or extension of physical models is needed. This is case for example the passive systems for the PWRs, or the Generation IV concepts including SFRs.

Informatics Developments

In recent years, multi-scale numerical simulations are increasingly needed to provide high fidelity analysis of PWRs. These simulations often require fine multidimensional modeling to better represent physical phenomena. While it is not practical to compute the whole PWR system with a multi-dimensional resolution, a multi-scale numerical simulation of a PWR, where CATHARE-3 modules (0D, 1D and 3D) having different length scales could be used simultaneous, is useful for an accurate thermal hydraulic simulation with reasonable computing time. In order to do this, it is important to extend the numerical capabilities of CATHARE-3 3D module to perform component-scale simulations, in conjunction with physical modelling improvements (see paragraph 3.1). With this aim in mind, several point will be addressed:

 A new architecture for 3D simulations: the CATHARE-3 3D module would be based on the CEA's open-source software development platform -TRUST [57]. This object-oriented platform provides mesh management for parallel computation, and numerical integration methods such as PolyMAC which could be used for general meshes [START_REF] Gerschenfeld | Development of a polyhedral staggered grid mesh scheme application to subchannel and CFD SFR thermal-hydraulics[END_REF]. By using the TRUST development environment, CATHARE-3 3D code will be developed by following the same software design paradigms used for other TRUST development efforts.  Application Programming Interface for software coupling: multi-scale and multi-physical calculations are one of the main targets of the NEPTUNE project. CATHARE-3 has been coupled successfully to various software by means of a common coupling interface, called ICoCo. This interface makes it possible to use each software as a callable library through C function calls: these calls allow a small C++ or Python program, known as the "supervisor" to perform and control a calculation. In the future, this interface will be extended for use of CATHARE-3 in the full scope and engineering simulation platforms for Operator training.  Improvement of the Graphical User Interface: As presented in part 2.5, GUITHARE is the Graphical User Interface for the CATHARE code. For the component-scale approach, this tool will be complemented by the development of a preprocessor for CATHARE software, which is integrated in GUITHARE interface. This preprocessor (see Fig. 6) will be based on SALOME [45] platform capable of generating meshes on reactor core geometries defined by a set of component scale technological objects. It already includes a 3-field model under development and partially validated for dry-out prediction and for reflooding conditions during a LBLOCA, A predictive pump model is included in the code, a well as new capabilities for multi-fluid and multi-reactor modelling. One of the major improvement concerns the 3D modelling capabilities for the simulations of the whole PWR vessel, and for the sub-channel analysis with a dedicated validation against rod bundle experiments. Future developments aim at continuing to improve or to extend the capabilities of the code. These developments will address the physical modelling (both 6equation and multi-field models, 3D modelling), the architecture and the interface of the code for the 3D simulations and coupled simulations.

Fig. 3 :Fig. 4 :

 34 Fig. 3 : Pressure coefficient versus flow rate coefficient (experimental data from Duplaa [26])

Fig. 6 :

 6 Fig. 6 : GUITHARE Pre-Processing prototype

ACKNOWLEDGMENTS

CATHARE code is developed in the framework of the NEPTUNE project, supported by CEA, EDF, FRAMATOME and IRSN.


Abdelaziz Amara Korba

Abdelwahab Boualouache

Bouziane Brik

Rabah Rahal

Yacine Ghamri-Doudane

Sidi Mohammed Senouci

Apprentissage fédéré pour la détection des attaques Zero-Day dans les réseaux V2X 5G et 5GB

Keywords: 5G, V2X, IoV, Sécurité, Attaques, Détection, Apprentissage Fédéré

Le déploiement de véhicules connectés et automatisés (CAV) utilisant les réseaux 5G et 5G-B les rend vulnérables à de nouveaux vecteurs d'attaques. Afin de détecter et de mitiger ces attaques, plusieurs systèmes de détection d'intrusions (IDS) basés sur l'apprentissage automatique ont été proposés, dont la grande majorité utilise l'apprentissage profond supervisé. Cependant, la principale limite de ce type de solution est son incapacité à détecter des attaques différentes de celles vues lors de l'apprentissage de l'IDS, ou de nouvelles attaques, également appelées attaques Zero-day. De plus, l'apprentissage du modèle de détection nécessite une collecte et un étiquetage importants des données, ce qui augmente le coût de communication et soulève des problèmes de confidentialité. Afin de pallier ces problèmes, nous proposons dans cet article un nouveau IDS basé sur l'apprentissage fédéré non supervisé préservant la confidentialité, et dont l'apprentissage du système ne nécessite que le trafic réseau légitime. L'expérimentation de la solution proposée sur un jeu de données récent montre que le système proposé présente un taux de détection élevé tout en minimisant le taux de faux positifs et le délai de détection.

Introduction

Les réseaux de cinquième génération (5G) et au-delà de la 5G (5GB) promettent de révolutionner les systèmes de transport intelligents (STI) avec des communications à latence ultra-faible et à bande passante élevée. Cependant, les véhicules connectés et automatisés (CAV) à différents niveaux d'automatisation seront confrontés à de nouveaux vecteurs d'attaques provenant des technologies 5G/5GB, conduisant ainsi à des situations dangereuses pour les usagers de la route. L'intelligence artificielle, et l'apprentissage automatique en particulier apparaît comme un allié clé de la cybersécurité [START_REF] Boualouache | A Survey on Machine Learning-based Misbehavior Detection Systems for 5G and Beyond Vehicular Networks[END_REF]. Divers systèmes de détection d'intrusion (IDS) basés sur l'apprentissage profond ont été proposés pour protéger les STIs des attaques. La plupart de ces IDSs s'appuient sur un apprentissage supervisé et centralisé [RAKGZ + 22]. L'apprentissage centralisé nécessite une collecte et un étiquetage importants des données, ce qui augmente le coût de communication, et soulèvent des problèmes de confidentialité. Pour pallier ces limitations, des recherches récentes [BE22b, LZZ + 21] ont exploité le potentiel de l'apprentissage fédéré (FL), qui a montré des résultats prometteurs dans divers domaines. Par contre, tous les IDS à base de FL proposés pour les ITS reposent sur des techniques d'apprentissage supervisé. Une limitation importante de l'utilisation de ces techniques est leur incapacité à détecter des attaques différentes de celles vues lors de la phase d'apprentissage et des attaques de type ≪ zero-day ≫. D'autant plus, la plupart des approches existantes supposent que les IDS déployés sur les CAVs (clients FL) détiennent des données préalablement étiquetées, ce qui n'est pas toujours vrai.

Puisque un CAV exécute un ensemble d'applications bien connues (sécurité, commodité, commerciales, etc.), sont comportement réseau doit présenter un degré élevé de régularité tant qu'il n'est pas attaqué ou défectueux. De même, une attaque doit altérer son modèle de communication. Dans cet article, nous proposons un IDS à base d'apprentissage fédéré non-supervisé pour modéliser le comportement réseau légitime d'un CAV (ou attendu), et ainsi détecter les attaques comme des anomalies. L'apprentissage fédéré de l'IDS est orchestré par le serveur Multi-access Edge Computing (MEC) pour améliorer l'efficacité de l'apprentissage et minimiser la latence. L'expérimentation de la solution proposée sur un jeu de données récent montre que le système proposé présente un taux de détection élevé tout en minimisant le taux de faux positifs et le délai de détection.

Solution proposée

Nous avons développé un système de détection d'intrusion (IDS) basé sur un auto-encodeur (AE) entraîné de manière fédérée, exclusivement sur des flux réseau légitimes. Pour cela, nous avons utilisé un serveur MEC qui a permis d'orchestrer l'apprentissage en distribuant les modèles et en agrégeant les gradients. Le modèle initial de l'AE a été distribué par le serveur MEC à chaque CAV, qui l'a ensuite entraîné sur ses propres données locales. Ces données ont été constituées d'un ensemble de flux extraits à partir des paquets capturés localement par le CAV. Pour chaque flux, un ensemble d'attributs a été calculé. Les cycles d'apprentissage (round) ont été orchestrés par le serveur MEC et exécutés par l'AE sur l'ensemble des flux locaux.

Extraction des flux et calcul des attributs

Tout d'abord, pour identifier un flux réseau, nous utilisons une combinaison de cinq informations de l'entête du paquet, à savoir : l'adresse IP source, l'adresse IP de destination, le numéro de port source, le numéro de port de destination et le type de protocole de couche transport. Pour chaque flux extrait, un ensemble d'attributs est calculé selon une fenêtre temporelle donnée (ex. 100 secondes). Les attributs sont constitués principalement de statistiques (moyenne, médiane, écart-type, etc.) calculées à partir des informations des en-têtes des couches réseau et transport de la pile TCP/IP, tels que le nombre d'octets envoyés/rec ¸us par seconde, l'intervalle moyen entre deux paquets successifs, la taille maximale des paquets envoyés/rec ¸us, etc.

Modélisation du trafic légitime et détection d'anomalies

Le modèle Auto-Encoder (AE) [START_REF] Hinton | Autoencoders, minimum description length and helmholtz free energy[END_REF] est un modèle d'apprentissage profond non supervisé qui comprime les vecteurs d'entrée sous forme de vecteurs de code à l'aide d'un ensemble de poids de reconnaissance, puis les reconvertit en m (m < d) nombre de neurones reconstruits en utilisant un ensemble de poids génératifs. Il y a deux parties principales dans l'architecture de l'AE : l'encodeur et le décodeur. L'encodeur réduit la dimension des vecteurs d'entrée (x i ∈ R d ) au nombre de neurones qui forment la couche cachée. L'activation du neurone i dans la couche cachée est donnée par :

h i = f θ (x) = s( n ∑ j=1 W input i j x j + b input i ) (1)
Ou x est le vecteur d'entrée, θ les paramètres W input , b input , W est une matrice de poids du codeur de dimension m × d, ou b est un vecteur de biais de dimension m. Ainsi, le vecteur d'entrée est codé en un vecteur de faible dimension. Le décodeur fait le mappage de la représentation cachée de faible dimension h i à l'espace d'entrée initial R d par la même transformation que le codeur.

L'erreur de reconstruction de l'AE est utilisée comme score d'anomalie. Les flux réseau avec des erreurs de reconstruction importantes sont considérés comme des flux malveillant (anomalies). Après l'apprentissage, le modèle reconstruira exceptionnellement bien les flux légitimes, mais pas les flux malveillant qu'il n'a jamais vus. Algorithm 1 décrit le processus de détection d'anomalies en utilisant les erreurs de reconstruction de l'auto-encodeur. L'écart absolu médian (MAD) a été utilisé pour le calcul du seuil α, parce qu'il est moins susceptible d'être faussé par les valeurs aberrantes.

Formatting a submission for AlgoTel

Algorithm 1: Détection du trafic malveillant 1 BEGIN 2 INPUT : dataset légitime X, dataset malveillant x (i) i ∈ {1, ..., N}, Seuil : α = MSE benign val + 5 × MAD(MSE benign val ) 3 OUTPUT : RE (Erreur de reconstruction) ∥x -x∥ 4 φ , θ ← entraîner l'AE avec le dataset légitime X 5 for i ∈ {1, ..., N} do 6

RE(i) = x (i) -g θ ( f φ (x (i) )) 7 if RE(i) > α then 8 x (i) est un flux malveillant

Évaluation des performances

Nous avons testé notre solution sur le jeu de données [START_REF] Rahal | Towards the development of realistic dos dataset for intelligent transportation systems[END_REF], soixante-quatre attributs ont été calculés pour les flux légitimes et malveillant extraits à partir du trafic brut (fichiers PCAP). Tenant compte du fait qu'une fenêtre temporelle d'échantillonage plus petite permet une détection plus rapide, nous avons fixé la fenêtre temporelle à 1 seconde. Nous avons implémenté un auto-encodeur profond avec trois couches cachées. La fonction de loss d'erreur quadratique moyenne (MSE) binaire a été utilisée. Pour évaluer les performances de détection de notre solution, en plus de l'accuracy, nous avons aussi considéré le F1-Score et le taux de faux positifs (FPR).

Globalement, notre solution présente un taux de détection élevé avec un faible taux de faux positifs. Le tableau 1 montre comment la l'accuracy moyenne, le F1-score et le taux de faux positifs varient en changeant le nombre de CAVs participants à l'apprentissage fédéré. L'apprentissage avec trois CAV a permis d'obtenir à peu près les mêmes performances tout en réduisant le temps d'apprentissage d'environ 30%. D'après la figure 1a, notre solution détecte légèrement mieux l'attaque SynFlood. Cela peut s'expliquer par le nombre relativement important d'attributs liées au protocole TCP (comparé au protocole UDP). Bien que 99,99 % de F1-Score et 0,01 % de FPR aient été obtenus en utilisant l'apprentissage supervisé dans [START_REF] Rahal | Towards the development of realistic dos dataset for intelligent transportation systems[END_REF], notre solution présente des performances comparables en utilisant uniquement le trafic légitime. Nous avons aussi comparé notre modèle fédéré avec le même modèle entraîné de fac ¸on centralisée. La comparaison des résultats illustrée dans la figure 1b montre que le modèle fédéré a donné des performances de détection remarquablement similaires à celles du modèle centralisé.

Conclusion

Dans cet article, nous avons proposé un nouveau IDS basé sur un modèle d'auto-encodeur profond, qui exploite la prédictibilité du trafic réseau légitime des CAVs pour détecter les attaques. En utilisant un apprentissage fédéré orchestré par le serveur MEC, l'IDS que nous avons proposé ne nécessite ni étiquetage ni partage de données entre les CAV. Nous avons démontré que notre IDS présente des performances de

  Processus d'apprentissage fédéré assisté par MEC Le problème d'apprentissage fédéré est formulé comme un problème d'optimisation fédérée et résolu à l'aide de l'algorithme FedAvg [MMR + 17]. En effet, en utilisant ses données locales, chaque CAV calcule le gradient moyen pour un cycle d'apprentissage. D'autre part, le serveur MEC agrège les mises à jour locales rec ¸us et retransmet le modèle global aux CAVs participants. Ce processus est répété pendant un certain nombre de rounds, défini initialement par le serveur MEC.

  FIGURE 1 -Comparaison des performances prédictives

TABLE 1 -

 1 Évaluation des performances de détection

	Nb. Clients Accuracy F1-Score	FPR	Temps (mn)
	10	87.94%	91.21%	6.95%	14.80
	8	87.94%	91.22%	6.98%	11.53
	6	87.95%	91.23%	7.06%	9.32
	3	87.94%	91.21%	6.92%	4.43

† 5G-INSIGHT (ANR-20-CE25-0015-16)