Kamel Mekhnacha
email: kamel.mekhnacha@probayes.com

David Raulo
email: david.raulo@probayes.com

Robust multi-target sensing/tracking in the Bayesian Occupancy Filter framework

Keywords: Bayesian inference, multi-sensor data fusion, multi-target tracking, occupancy grid. MOTS-CLÉS : Inférence Bayésienne, fusion multi-capteurs, suivi de cibles multiples, grille d'occupation

We present the "Bayesian Occupancy Filter" (BOF) and the "Fast Clustering-Tracking" algorithms as a framework for robust sensing and multi-target tracking using multiple sensors. Perceiving of the surrounding physical environment reliably is a major demanding in smart systems requiring a high level of safety such as car driving assistant, autonomous robots, and surveillance. The dynamic environment need to be perceived and modeled according to the sensor measurements which could be noisy. To fit such a requirement, we propose a hierarchical approach in which two filtering layers are used:

(i) Robust grid-level sensor fusion using the "Bayesian Occupancy Filter" algorithm in order to construct an occupancy/velocity grid representation of the environment.

(ii) Robust object-level tracking using the "Fast Clustering-Tracking".

RÉSUMÉ. Nous présentons les algorithmes "Bayesian Occupancy Filter" (BOF) et "Fast Clustering-Tracking" comme un cadre pour la perception robuste et le suivi de cibles multiples en utilisant plusieurs capteurs. Percevoir l'environnement physique d'une manière fiable est une exigence clef dans les systèmes intelligents nécessitant un grand degré de sécurité tels que l'assistante à la conduite automobile, la robotique autonome et la surveillance. L'environnement dynamique doit être perçu et modélisé en utilisant les mesures capteurs généralement bruitées. Pour répondre à cette exigence, nous proposons une approche hiérarchique dans laquelle deux niveaux de filtrage sont utilisés: (i) Fusion robuste de capteurs en utilisant l'algorithme "Bayesian Occupancy Filter" dans lequel l'environnement dynamique est représenté par une grille d'occupation/vitesse.

(ii) Suivi robuste d'objets en utilisant l'algorithme "Fast Clustering-Tracking".

Introduction

It has been shown that the dynamic environment can be efficiently and robustly represented by the Bayesian Occupancy Filter (BOF) [START_REF] Tay | An Efficient Formulation of the Bayesian Occupation Filter for Target Tracking in Dynamic Environments[END_REF]. In the BOF framework, this environment is decomposed into a grid-based representation [START_REF] Moravec | Sensor fusion in certainty grids for mobile robots[END_REF] in which both the occupancy and the velocity distributions are estimated for each grid cell. In such a representation, concepts such as objects or tracks do not exist and the estimation is achieved at the cell level. However, the objectlevel representation is mandatory for applications needing high-level representations of obstacles and their motion.

To achieve this, a natural approach is to perform clustering on the BOF output grid in order to extract objects. We present in this paper a novel clustering-tracking algorithm. Its main ideas are:

1) using the prediction result of the tracking module as a form of feedback to the clustering module in order to avoid searching in the complete grid, 2) using both occupancy and velocity estimates in order to better separate the extracted clusters. This algorithm reduces drastically the complexity of the data association. Compared with the traditional data association approaches such as the joint probabilistic data association (JPDA) algorithm [START_REF] Bar-Shalom | Tracking and Data Association[END_REF], our algorithm demands less computational cost, so as to be suitable for environments with large amount of dynamic objects and/or a large amount of sensor observations. The proposed sensing/tracking framework is based on a hierarchical approach in which two filtering layers are used (Fig. 1). In this architecture, the output grid of the BOF filter (i.e, the probability distributions over the occupancy and over the velocity of the cell) is used as input for the object tracker by extracting object hypothesis from the grid.

The paper is organised as follows. In the next section, the "Bayesian Occupancy Filter" (BOF) algorithm is briefly described. The "Fast Clustering-Tracking" algorithm is presented in section III. Finally, conclusions are drawn in section IV.

Bayesian Occupancy Filter (BOF)

The Bayesian Occupancy Filter (BOF) is represented as a two-dimensional gridbased decomposition of the environment. Each cell of the grid contains two probability distributions. The probability distribution over the occupancy of the cell, and the probability distribution over its velocity. Given a set of input sensor readings, the BOF algorithm allows to update the occupancy/velocity estimates of each grid cell.

BOF is a special implementation of the Bayesian filter approach [START_REF] Jazwinski | Stochastic Processes and Filtering Theory[END_REF]) [START_REF] Thrun | Probabilistic Robotics[END_REF]. This approach addresses the general problem of recursively estimating the posterior probability distribution P (X k | Z k ) of the state X of a system conditioned on its observation Z. This posterior distribution is obtained in two stages: prediction and estimation. The prediction stage computes an a priori prediction of the target's current state known as the prior distribution. The estimation stage then computes the posterior distribution by using the prediction with the current measurement of the sensor.

In the case of the BOF, using this prediction/estimation scheme allows filtering out false alarms, miss-detections, and localization errors in sensors data readings. Figure 2 shows an example of BOF output using a computer vision car detector. The input of the BOF in this case is, for each time step, a set of bounding boxes corresponding to the detected vehicles (red boxes). The output represents a grid of occupancy probability (blue-to-red mapped color) and mean velocity (red arrows) estimates.

The Bayesian model presented in the following text is a reformulation of the one we presented in [START_REF] Tay | An Efficient Formulation of the Bayesian Occupation Filter for Target Tracking in Dynamic Environments[END_REF]. The aim of this reformulation is to make clearer the strong link between the discretization of the space and the discretization of the velocity, which reduces the number of the used random variables and makes the model easier to explain. The key idea of the model is to represent the 2D space using a regular grid. Given this space discretization and assuming that objects do not overlap, the velocity of a given c cell at a time t is directly linked to the identity of its antecedent cell A c from which the content of cell c moved between t -1 and t. In other words, we can define the velocity of a given cell by providing the index of its antecedent. Therefore, estimating the velocity of a given cell is equivalent to estimating a probability table over all its possible antecedents. Possible antecedents of a cell are defined by providing a neighbourhood from which the cell is reachable in a time step. This model applies also to velocities needing more than one time step for a neighbour cell to reach c. However, for simplicity we will assume only one-step velocities (neighbours reaching c in one time step).

The BOF model

Variables

For a given cell having c ∈ Y as index in the grid, let:

-A t c ∈ A c ⊂ Y represents each possible antecedent of cell c over all the cells in the grid domain Y. The set of antecedent cells of cell c is denoted by A c and is defined as a neighbourhood of the cell c.

-A t-1 c ∈ A c ⊂ Y the same as A t c but for the previous time step. -O t c ∈ O ≡ {0, 1} is a boolean variable representing the state of the cell in terms of occupancy at time t, either [O c = 1] if occupied, [O c = 0] if empty.
Given the independency hypothesis, the occupancy of each cell at time t is considered apart from the occupancy of its neighbouring cells at time t.

-Z t i ∈ Z, 1 ≤ i ≤ S ∈ N
, is a generic notation for measurements yielded by each sensor i, considering a total of S sensors yielding a measurement at the considered time instant.

Joint distribution factors

The following expression gives the decomposition of the joint distribution of the relevant variables according to Bayes' rule and dependency assumptions:

P (A t-1 c A t c O t c Z t 1 • • • Z t S ) = P (A t-1 c )P (A t c | A t-1 c )P (O t c | A t-1 c ) S i=1 P (Z t i | A t c O t c ). [1]
The parametric form and semantics of each component of the joint decomposition are as follows:

-P (A t-1 c ) is the probability for a given neighbouring cell A c to be the antecedent of c at time t -1. In order to represent the fact that cell c is a priori equally reachable from all possible antecedent cells in the considered neighbourhood, this probability table is initialised as uniform and is updated in each time step.

-

P (A t c | A t-1 c
) is the distribution over antecedents at time t given the antecedent of cell c at t -1. It represents the prediction (dynamic) model over velocity. If we assume a perfect constant velocity hypothesis between the two time frames t -1 and t, this distribution is simply:

P (A t c | A t-1 c ) = P (A t-1 A t-1 c
).

In other words, the predicted probability is simply the probability at the preceding time instant for the antecedent at t -1.

Considering imperfect constant velocity hypothesis may be done by introducing:

-E ∈ {0, 1} ≡ "There was a prediction error", -P (E) = ǫ the probability of violating the constant velocity hypothesis (a parameter of the model).

If we define:

c E) = (1 -ǫ)P (O t-1 A t-1 c ) + ǫ U(O t c ) = (1 -ǫ)P (O t-1 A t-1 c ) + ǫ/2. -P (Z t i | A t c O t c
) is the direct model for sensor i. It yields the probability of a measurement given the occupancy O t c and the antecedent (velocity) A t c of cell c. Measurements for all sensors are assumed to have been taken independently from each other.

For sensors providing measurements depending exclusively of occupancy, this distribution can be written as P (Z t i | O t c ). In the same manner, for sensors providing measurements depending exclusively of the velocity, this distribution can be written as P (Z t i | A t c ).

Occupancy and velocity estimation using the BOF model

At each time step, the estimation of the occupancy and velocity of a cell is answered through Bayesian inference on the model given in Equation [1]. This inference leads to a Bayesian filtering process. In this context, the prediction step propagates cell occupancy and antecedent (velocity) distributions of each cell in the grid to get the prediction P (O t c A t c ). In the estimation step, P (O t c A t c ) is updated by taking into account the observations yielded by the sensors

S i=1 P (Z t i | A t c O t c ) to obtain the a posteriori state estimate P (O t c A t c | [Z t 1 • • • Z t S ]
). This allows by marginalization to compute

P (O t c | [Z t 1 • • • Z t S ]) and P (A t c | [Z t 1 • • • Z t S ]
) that will be used for prediction in the next iteration. It's important to notice that the distribution P (A t c ) over velocity is updated even when no velocity sensors are available. Indeed, suppose we have only one occupancy sensor described by the model

P (Z t OCC | O t c
). The a posteriori distribution

P (A t c | [Z t OCC ]
) leads to the formula:

P (A t c | [Z t OCC ]) ∝ A t-1 c ∈Ac P (A t-1 c )P (A t c | A t-1 c ) O t c ∈{0,1} P (O t c | A t-1 c )P ([Z t OCC ] | O t c ).
This allows to update the velocity distribution even when no velocity sensors are available. In this case, the update is based exclusively on the occupancy observations.

When an additional velocity sensor

P (Z t VEL | A t c
) is available, it should be used to update the estimate [2] as follows:

P (A t c | [Z t OCC Z t VEL ]) ∝ P (A t c | [Z t OCC ])P ([Z t VEL ] | A t c ).

The "Fast Clustering-Tracking" algorithm

In many applications, the object level representation is demanded. We propose to use a layered architecture as shown in Fig. 1 to obtain this representation. In our former work [START_REF] Tay | An Efficient Formulation of the Bayesian Occupation Filter for Target Tracking in Dynamic Environments[END_REF], the data association is achieved using a classical JPDA algorithm. However, in a cluttered environment with a large number of moving objects, the JPDA [START_REF] Bar-Shalom | Tracking and Data Association[END_REF] suffers from the combinational explosion of hypothesises. To overcome this problem, we propose a novel object detecting and tracking algorithm. This algorithm could be roughly divided into a clustering module, an ambiguous association handling module and a tracking and track management module.

Clustering

The clustering module takes the occupancy/velocity grid of the BOF as the input and extracts object level reports from it. A natural algorithm to achieve this is to connect the eight-neighbor cells according to an occupancy threshold occ_threshold. In addition to the occupancy values, a threshold of the Mahalanobis distance between the velocity distributions vel_threshold is also employed to distinguish the objects that are close to each other but with different moving velocities.

In order to avoid searching for clusters in the whole grid, we use the predicted targets' states as a form of feedback. For a given target with ID id, the predicted state is used to define a region of interest (ROI) in which the clustering process starts. After a starting point with an occupancy probability value greater than the occ_threshold is found in the ROI, the id is propagated in the ID grid using the connectivity criterion among the non-associated cells (cells with ID = 0).

A report for the tracker is a 4-dimentional observation corresponding to the position and the velocity of an extracted cluster. The 2D position component of this vector is computed as the mass center of the region corresponding to the cluster pixels (cells) set. We also compute the corresponding covariance matrix representing the uncertainty of the observed position. The 2D velocity component is simply the weighted mean of the estimated velocities of all cells of the cluster. It comes also with a covariance matrix representing the uncertainty of the observation velocity.

Re-clustering and tracks merging

During the clustering process, three possible situations need to be considered.

-Case 1: no cell with P (occ) ≥ occ_threshold is found. The target has not been observed and no association is needed.

-Case 2: a cluster C of non-associated cells having ∀c(i, j) ∈ C, P (occ(i, j)) ≥ occ_threshold is extracted. These cells are then associated to the target id: ∀c(i, j) ∈ C, ID(c i , c j ) = id. This situation occurs when there is no ambiguity in the association. This is an advantageous situation allowing a fast clustering-association procedure. Fortunately, this case is the most frequent one when applying the algorithm to the real data.

-Case 3: cells having P (occ(i, j)) ≥ occ_threshold exist. However, they have already been assigned to other IDs. In this conflicted case, an observation (cluster) could be possibly generated by two (or more) different targets.

The first two cases are normal cases, however, the third case is refereed to as an ambiguous association case which need to be dealt with in a special manner. The ambiguous association could occur in the following two situations:

1) Different targets are being too close to each other and the observed cluster is in fact the union of more than one observations generated by different targets.

2) The different tracked targets are corresponding to a single object and should be merged into one.

We take a re-clustering strategy to deal with the first situation and a cluster merging strategy to deal with the second one.

Suppose when an ambiguous association occurs, a set of tracks T 1 , T 2 , • • • , T m are the potential candidates to be associated to the observed cluster. We have to cut up the extracted cluster and generate a sub-cluster (possibly empty) for each candidate. This re-clustering is achieved by a k-means [START_REF] Bishop | Pattern recognition and machine learning[END_REF] algorithm using a simple Cartesian distance. The considered distance is taken between the center of the subcluster and a given cell. In this way, the first cause of the ambiguous association is handled.

To deal with the second cause of the ambiguous association, we introduce a concept of "alias" which is in the form of a two-tuples to represent the duplicated tracks. When an ambiguous association between two tracks T i and T j is detected, an alias ALIAS(T i , T j ) is initialized and added to a potential alias list.

At each frame, the tracker updates this list by confirming or disproving the existence of each alias hypothesis ALIAS(T i , T j ) according to the observation of the ambiguous association. At a given time step t, if the ambiguous association occurs between T i and T j , and the alias ALIAS(T i , T j ) is found in the potential alias list, the probability P t (S(T i , T j )) is increased by a confirming step using a Bayesian filtering approach as follows:

P t (S | F ) = P t-1 (S) × P (F | S) P t-1 (S) × P (F | S) + [1 -P t-1 (S)] × P (F | ¬S)
, where:

-S ≡ "the T i and T j tracks are alias for the same object".

-F ≡ "an ambiguous association between the tracks T i and T j is observed".

The probability values P (F | S) and P (F | ¬S) are constant parameters of the tracker. The former denotes the probability of observing an ambiguous association when the two concerned tracks are alias of the same object and is set to a constant value 0.8. The second denotes the probability of falsely observing an ambiguous association and is set to 0.1.

When ALIAS(T i , T j ) is found in the potential alias list but is not observed as an ambiguous association, its probability is decreased in a similar manner:

P t (S |¬F ) = P t-1 (S) × P (¬F | S) P t-1 (S) × P (¬F | S) + [1 -P t-1 (S)] × P (¬F | ¬S)
.

According to the probability P t (S(T i , T j )), the decision of merging of tracks T i and T j could be made.

New tracks creation

For new targets creation, we introduce a concept "cluster seed" to define a cell in the BOF grid where we will try to find, for each step, a new (non-associated) cluster. Indeed, the searching for potential new targets is after all the existing tracks are processed. Thus, only non-associated cells will be processed to extract clusters as the observations for the potential new targets. The "cluster seed" concept is general and can be implemented via various strategies. The simplest strategy is to insert a possible seed in each cell of the grid. However, more sophisticated strategies could be more efficient. For example, cluster seeds could be inserted only in entrance regions of the monitored area.

Tracks updating and deleting

The prediction and estimation of the targets are accomplished by attaching a Kalman filter [START_REF] Kalman | A New Approach to Linear Filtering and Prediction Problems[END_REF] with each track. Once associated to a given track, a report (Gaussian distributions for both position and velocity) corresponding to an extracted cluster is used as an observation to re-estimate the position and velocity of the track in a prediction-update step. For non-observed tracks, only a prediction step is taken by applying the dynamic model to the estimation result of the precedent time step.

The deleting of tracks is also achieved in a Bayesian manner. If an existing track T is associated with a given report (cluster), its existence probability is increased using the following formula:

P t (E | O) = P t-1 (E) × P (O | E) P t-1 (E) × P (O | E) + [1 -P t-1 (E)] × P (O | ¬E)
, where:

-E ≡ "the target T exists".

-O ≡ "the target T has been observed (associated)".

The parameters P (¬O | E) and P (O | ¬E) are the tracker miss-detections and false alarms probabilities respectively.

If an existing target is not associated with any report (cluster), its existence probability is decreased in the similar way: P t (E | ¬O) = P t-1 (E) × P (¬O | E) P t-1 (E) × P (¬O | E) + [1 -P t-1 (E)] × P (¬O | ¬E) .

According to the existence probability, the track deleting operation is achieved by applying a deleting threshold on it.

Conclusion

We presented a novel sensing/tracking algorithm for the BOF framework. This algorithm takes the occupancy/velocity grid of the BOF as input and extracts the objects from the grid with a clustering module which takes the prediction of the tracking module as a feedback to reduce the computational cost. A re-clustering and merging module is proposed to deal with the ambiguous data associations. The extracted objects are then tracked and managed in a probabilistic way. The experiment results show that the presented algorithm is robust as well as computationally efficient so as to be suitable for cluttered environment.

Our approach has been applied on real data and achieved satisfied results in several conditions. The proposed algorithms have been used in several driving assistance projects in both highway and cluttered urban environments. The used sensor modalities include: (i) multi-layer lidars, (ii) computer vision detection algorithms (Fig. 2), and (iii) stereovision-based 3D sensors.

According to confidentiality agreements of the on-going projects, we could not provide these works in the publications. However, we are now preparing new experiments using our own data sets in order to be able to present quantitative experimental results in future publications.

Figure 1 .

 1 Figure 1. Robust Sensing/Tracking system architecture.

Figure 2 .

 2 Figure 2. Example of BOF output using a computer vision car detector as input (red boxes). The images are provided by a camera mounted on the moving ego-vehicle. The BOF output is projected back on the image. It represents a grid of occupancy probability (blue-to-red mapped color) and the mean velocity (red arrows) estimates.


 Chapter 1 School divisions and inequalities in funding, access to schools and staffing 1.1

Introduction

Socio-spatial inequalities are defined as "disparities experienced as injustices" [START_REF] Brunet | Les Mots de la géographie, dictionnaire critique[END_REF]. They can cover such a wide range of unequal situations that the use of the notion of division is possible. These inequalities or divisions break up the French space into several types of areas (Le Bras 2013), with radically heterogeneous living Chapter written by Patrice CARO. conditions and demo-socio-economic characteristics (opposition between French overseas territory and Metropolitan France for example). The scope of initial training and schooling does not escape such divisions, as shown by the "Géographie de l'école" published regularly by the French Ministry of National Education (MEN 2017) and reports published at the end of 2018 by the Conseil national d'évaluation du système scolaire (French national council for the evaluation of the school system) or Cnesco [START_REF] Schneider | État des lieux des inégalités scolaires d'origine territoriale en France[END_REF]). An atlas of school divisions was presented in 2010 based on these scores from metropolitan France and overseas French territories (Caro and Rouault). An update of this atlas 1 has been available online for a few months, with the authors reaffirming that: "Academic divisions are the result of processes that lead to a breakdown in the equality of position or educational pathways between students. They can result from spatial or social processes, or their interactions, they manifest themselves for students through delay, academic difficulty, failure, drop-out."

If we select certain dimensions of the functioning of the school system, what divisions appear when we spatialize recent statistical data in a multiscalar logic? After examining disparities in funding by local authorities, it will be necessary to focus on inequalities in access to schools, then on those relating to teacher profiles according to communes and regional education authorities to confirm the existence of school divisions or to qualify this statement.

Disparities in funding from communities

Departments and regions are a source of educational inequality through their investments in school transport, construction and renovation of schools. Departments that contain a metropolis benefit from its public transport network. Those which are more rural must spend more on this area, especially in mountain areas. In 2015, departmental operating expenses per school child were highest in Morbihan, Cantal and Charente. The departmental councils in charge of middle schools spent on average 5.4% of their operating expenditure budgets on education, while social action absorbed 63% [START_REF] Simon | Les budgets primitifs des départements[END_REF]. The situation was different with regard to capital expenditure on education, with an average of 15.2% of budgets devoted to this item.

57 million was the figure spent on transport in 2015 in Oise and Pas-de-Calais, compared with less than 1 million in Gers, Hauts-de-Seine, Haute-Vienne and Seine-1. https://fracturesscolaires.fr/ (accessed June 2019).

Saint-Denis (see Figure 1.1). It was only 6.6 million in Lozère, 13.2 million in Aveyron and 15 million in Nièvre. This is a long way from the very high spending rate seen in Seine-Maritime. On a per capita basis, these expenses distinguish mountain departments such as Jura (96 €/pc, for an average of 30 €/pc), Corse-du-Sud (90 €/pc), Lozère (81 €/pc 2 ) or Ardèche (76 €/pc). Guyana and Mayotte are part of this group. On the other hand, the expenditure per capita ratio is lowest in departments with a metropolitan area with numerous public transport networks (Val-d'Oise: 3 €/pc; Rhône: 4 €/pc; Var: 8 €/pc; Val-de-Marne: 10 €/pc; Bouches-du-Rhône: 10 €/pc; Nord: 14 €/pc). With the total expenditure of the regional councils in 2015 for public and private high schools, the scale is changing. Metropolitan regions are isolated with more than 4 million inhabitants such as Île-de-France (880 million), Nord-Pas-de-Calais (almost 3. Source: Insee. 400 million), PACA (377 million) or Rhône-Alpes (309 million), from small rural regions with less than 1.5 million inhabitants such as Auvergne (1.32 million), Basse-Normandie (1.37 million), Corsica (0.32 million) or Limousin (0.78 million).

Expressed in euros per high school student, the highest operating expenditure is found in Limousin, Auvergne, Normandy, Corsica, Champagne-Ardenne, Burgundy and Picardy (see Figure 1.2). All these regions without metropolitan areas (no urban areas with at least 700,000 inhabitants) spend more than the average. The presence of one or more metropolitan areas (PACAs) explains lower expenses since a large number of high school students benefit from the metropolitan public transport networks.

Expressed in euros per capita, these expenses highlight the overseas regions with ratios from 124 euros (Réunion Island) to 144 euros (French Guiana), for an average of 81 euros. This financial effort is linked to the demographic growth at work in these territories, which is stronger than in metropolitan France. The average annual rate of population change between 2007 and 2012 was +2.4% in French Guiana and +1% for Réunion Island, compared to +0.5% in metropolitan France 4 .

Whether it is for departmental (middle schools) or regional (high schools) expenditure, French Guiana has the highest per capita expenditure ratios, with Guadeloupe. On the other hand, regions such as Franche-Comté and all its departments, except Haute-Saône, have low ratios, below average. This is partly due to the age structure of the population. The youthfulness of the population of some overseas departments contrasts with the aging population of rural regions without metropolitan areas such as Burgundy or Franche-Comté (34% of 0-14 year-olds in French Guiana in 2013, 24% on Réunion Island and 18% in Jura or Martinique).

Apart from this effect of the age structure, departmental expenditure is a source of educational inequality since funding for the construction of schools, their renovation, equipment and school transport varies from 1 to 4 euros for middle schools (per capita), the key period for enrolment and from 1 to 2.6 euros for high schools. This range varies from 1 to 4 euros for operating expenses for high schools (per high school student). These differences reflect political and budgetary choices made by the local authorities in charge of schools. The current expenditure on education per local authority does not prejudge the financial efforts made by these public actors in the previous period (before 2015) or those that will be committed for the coming years.

The analysis of the cumulative investment expenditure of regions and departments from 2004 to 2014 reveals differences between departments that were greater than between regions (5 for 3.5see Figures 1.1 and 1.2). The departments and regions of 4. https://www.insee.fr/fr/statistiques/zones/1405599. the far northwest, but also those of Burgundy and Lorraine, stand out for investment expenditure below or around the average (less than 5,200 euros/middle school student in Ille-et-Vilaine, less than 11,600 euros/high school student in Brittany). On the other hand, some rural departments such as Haute-Marne or Landes, but also much more urban departments such as Nord, Hauts-de-Seine or Isère, spend more than the average for their middle school students (more than 10,000 euros/school in Bouchesdu-Rhône and more than 15,000 euros/high school student in Guyana or Midi-Pyrénées).

At a more local level, the role of communities of communes (EPCI) and communes in terms of PEDTs, PELs, PESLs, PREs 5 and the implementation of extracurricular activity times (TAPs) is also a source of inequality (David 2008). According to the Ministry of National Education, during extracurricular activity times (TAPs), "students will be able to access sports, cultural and artistic activities that could develop their intellectual curiosity and enhance the pleasure of learning and being at school" (MEN 2013) 6 . PELs and PREs have the same objectives, namely to compensate for the lack of access to sports, cultural or artistic activities for some pupils from disadvantaged social backgrounds. It is not possible to map France, including overseas France, in terms of the municipalities and communities of municipalities currently involved in these student support schemes.

However, it can be argued that a complete reorganization of the systems, developed by some municipalities and communities of municipalities since 2013 (implementation of the new school rhythms, Decree No. 2013-77 of 24 January 2013), is under way with the current movement to merge communities of municipalities tending to become increasingly expanded and populated. In addition, the recent relaxation of the decree has had an impact. This first approach to inequalities in funding and endowment by department and regional education authority shows that there are gaps between rural and urban areas in metropolitan France, and between overseas areas and those in metropolitan France. While their effects on school careers can be discussed (direct or indirect effects), the effects of inequalities in access time to schools are probably clearer and less debatable. 6. La réforme des rythmes scolaires à l'école primaire, press kit. Available at: http://www.education.gouv.fr/.

Spending more time on school transport is probably a disadvantage for some isolated rural high school students. Living more than an hour from the nearest vocational school can affect individual and family choices in terms of orientation.

Inequalities in access to schools

Not all pupils have a middle or high school in their commune, canton or community. A very large number therefore commute, stay close to the school for week periods at a time, between boarding school and the parents' commune of residence. Some geographers refer to spatial capital as the accumulation by certain individuals, or even families, of knowledge and know-how about mobility [START_REF] Ripoll | Sur la dimension spatiale des inégalités : contribution aux débats sur la « mobilité » et le « capital spatial[END_REF].

The following maps show the distances between municipalities that are not equipped and schools without taking into account the time distances to be covered by road to access the service. In rural areas without mountainous constraints, these mapped distances are a fairly accurate reflection of road travel times, while in mountain environments, the situation is different.

The purpose of the following maps is to provide a first look at the relative distance of the population from schools in France (metropolitan France, Mayotte, Guadeloupe, Martinique, Réunion Island, French Guiana).

The distances used in this section are calculated according to the actual geographical coordinates of the middle schools and high schools.

The principle used is that of calculating the Euclidean distance (straight distance between two points, or "as the crow flies" distance) between a place and the nearest identified school. Thus, for each spatial unit, we have a calculated distance that is then represented according to 5 classes in a range of colors of increasing intensity: from 0 to 5 km; from 5 to 10 km; from 10 to 15 km; from 15 to 20 km; more than 20 km. Distances are calculated for three types of schools: middle schools, high schools and vocational schools, whose location is obtained through their geographic 7 coordinates. 7. Source: ministère de l'Éducation nationale, de l'Enseignement supérieur et de la Recherche, May 2016.

The reference spatial unit is chosen according to the scale of analysis. For metropolitan France, the choice was made for the municipality, thus providing sufficient precision to assess the fine variations in distance observed over the territory. These maps therefore represent the distance from each municipality to the nearest school. For the overseas departments, the choice was made on an infra-communal scale with a regular subdivision of the area into a grid of 500 m on each side. For this scale, these maps therefore represent the distance from each location (500 m cell on each side) to the nearest school.

These initial results effectively provide a relevant and global overview of the relative differences in the distance between the population and schools. Nevertheless, this information must be qualified and we therefore remain cautious in the interpretation of the results.

Indeed, the principle of Euclidean distance, which is quick to implement and generally informative, does not allow in some cases to account for the distances "experienced" by the travelling population. This is particularly evident in certain territories that are difficult to access (mountainous areas, among others), where "distance" and "distance-time" have very different meanings. A desirable extension of this work in the future could therefore be the more refined calculation of a network distance (travel distance on traffic networks), from which time distances (travel times on networks) could be derived.

Similarly, this initial assessment of distances should be refined according to the overall traffic conditions in the territory where population travel is multifaceted: integrating public transport networks, the different modes of travel (car, motorcycle, pedestrians, etc.), the traffic and speeds used, which determine travel times, etc.

These maps therefore constitute a first exploration of a problem of access to educational places, with multiple dimensions that will then have to be analyzed globally.

Box 1.1 Comments on the methodology: Euclidean distance maps to schools (source: Mohand Medjkane and Thierry Saint-Gerand -UMR IDEES CNRS -Caen-Normandy University)

Along with the regional education authorities of French Guiana and Réunion Island, those of Reims, Dijon, Montpellier, Grenoble, Toulouse and Corsica are the only ones to have areas where pupils from municipalities that are not equipped are located more than 20 km as the crow flies from the training offer (see Figure 1.3). The diagonal of the empty space, oriented northeast/southwest, reappears logically here. The distances as the crow flies to the nearest general and technological high school show the same areas of low density as in the case for middle schools, but in these areas the number of municipalities concerned is higher. The number of regional education authorities concerned by this phenomenon is also higher. To the "rural" regional education authorities of Reims, Dijon, Montpellier, Grenoble, Toulouse and Corsica, all the others should be added, with the exception of the more "urban" ones of Strasbourg, Rouen, Créteil, Versailles, Paris and Martinique. The location of the equipped prefectures, central in their department, highlights many communes located on the departmental borders (from 16 to more than 20 km from the high school). This applies to the margins of the departments of Meuse and Marne, Haute-Marne and Haute-Saône, Puy-de-Dôme and Creuse, Ardèche and Lozère, etc.

It is more relevant to map inequalities of access to the same educational establishments, but taking into account road access times under normal traffic conditions (INSEE METRIC method). This approach highlights the same areas in metropolitan France that are disadvantaged in terms of road accessibility. The latter underestimates the transport times that some pupils, particularly schoolchildren, have to endure, as school transport follows non-linear collection routes. The most disadvantaged areas in terms of access times to middle school are mainly rural mountain areas, often located at the margins of certain regions (see Figure 1.4). These areas also include municipalities classified by INSEE as having very low density. Road access times from a municipality that is not equipped to the nearest one equipped with a public or private middle school thus vary from a few minutes to more than 30 minutes in 2015, or even more than one hour, for some Corsican municipalities in particular.

The center and north of PACA 8 (hinterland of the Alpes-Maritimes, Alpes-de-Haute-Provence, Hautes-Alpes), as well as the whole south-east of Rhône-Alpes (Savoie, Isère, Drôme) are in the same situation. To the south-east of the Massif Central, access times are also among the longest in Ardèche, Lozère, Aveyron or Haute-Loire, at the margins of the Auvergne, Languedoc-Roussillon and Rhône-Alpes regions. The same applies near the Spanish border in the Pyrénées-Orientales, Aude and Ariège. Further north, access times are unfavorable, but to a lesser extent (20 to 30 minutes), along the entire diagonal of the empty space, i.e. from the Meuse and Ardennes to the Dordogne and Lot, via the Haute-Marne, the Côte-d'Or, the Allier, the Corrèze or the Cantal. Here again, regional borders coincide with difficulties of 8. Provence-Alpes-Côte d'Azur region. road access. The diagonal of the empty space coincides with the contact between Champagne-Ardenne and Burgundy, Burgundy and Auvergne, Auvergne and Limousin, Limousin and Aquitaine.

In total, the centre of gravity of this group of disadvantaged areas with regard to road access times is located in the south-eastern quarter of metropolitan France, somewhere between Languedoc-Roussillon (Occitania), PACA and Auvergne-Rhône-Alpes. The low population density and mountainous terrain in these areas generate a looser network of middle schools than elsewhere, resulting in longer access times. The regions of the Grand Est and Bourgogne-Franche-Comté are also affected by this phenomenon, but for a smaller number of municipalities.

These same areas in the same regions constitute the heart of rural France far from general and/or technological high schools and vocational high schools (see figures 1.5 and 1.6), but the access time between equipped and unequipped communes is increasing. The notion of a school divide thus makes sense because the same areas under low urban influence combine unfavorable accessibility for all types of schools. In addition to accessibility to schools, an equally important dimension for schooling lies in the human resources available in these schools.

Teachers with different profiles depending on the municipalities and regional education authorities

In 2015, young teachers (under 30 years of age) and unqualified staff were concentrated in the regional education authorities of Créteil, Versailles, Amiens and Guiana, where the majority of the risks of school failure and drop-out is partly located (Boudesseul et al. 2016).

The profile of teachers working in middle and high schools partly reflects the attractiveness of the regional education authorities in the south, west and capital [START_REF] Veschambre | Les professeurs du secondaire public : essai de géographie d'un groupe social[END_REF]. Rennes, Bordeaux, Toulouse or Montpellier are attractive centers in the analysis of the inter-academic movement of high school teachers [START_REF] Hilary | Enseignants de collège et lycée publics en 2013 : panorama d'un métier exercé par 380 000 personnes[END_REF]. On the other hand, the Créteil, Versailles and Amiens trio is the cluster seen as having the most deterring features.

Like the migratory flows observed from census to census, many teachers want to access a brighter regional education authority where student success is above average and where a metropolis has developed, such as Rennes, Nantes, Bordeaux, Toulouse or Grenoble. It is striking to note that the Paris-Créteil regional education authorities held, in 2012 as in 1990, the record for the highest number of teachers over 50 years of age (24.6% in Paris in 1990) and the lowest (11.1% in Créteil in 1990(MEN-DEPP 1993, p. 114)).

The local composition of the teaching staff can be understood via the proportion of young people under 30 years of age and that of those over 50 years of age at the level of all high schools in the same commune. Academically, there is a radical opposition between Paris and its peripheries (Versailles and Créteil). The proportion of teachers over 50 years of age varies by a factor of two between Paris and Créteil (see Figure 1.7).

In the first case, the regional education authority has the highest proportion of teachers over 50 years old due to an ageing teaching staff population with low turnover, while in the second case, it is a high turnover regional education authority with many vacancies. The youngest begin working more often than in Paris. The regional education authorities of French Guiana, Lille and Amiens share the same profile as the Parisian suburbs, for the same reasons. Some newly qualified staff assigned to French Guiana leave their posts, based on rivers, for example, which generates a high turnover (commune of Saint-Laurent du Maroni in north-western French Guiana in a border position). COMMENT ON FIGURE 1.7 -In 2015, the Créteil regional education authority had 37.1% of teachers under 35 years of age, the proportion of those over 50 years of age was 22%. On the other hand, the Paris regional education authority had a 41.7% ratio of teachers over 50 years old, almost double that of Créteil.

For the regional education authorities of Montpellier, Guadeloupe, Martinique, Limoges and Corsica, the ratio increased considerably over the period. The age structure was therefore more unbalanced than it was with, for example, 12 times more teachers over 50 than teachers under 30 in Montpellier in 2014 compared to about 5 times more in 2005. The other regional education authorities retain their profile, the relative youth of the teachers in Amiens, French Guiana, Versailles and Créteil contrast with the maturity of the latter in Paris, Bordeaux and Rennes. Among the factors that explain this age structure, spousal employment opportunities in the different regions and attachment to region of birth could play a role. COMMENT ON FIGURE 1.8 -In the Montpellier regional education authority, in 2005, the ratio was 4.7, so there were 4.7 times more teachers aged 50 and over than teachers aged under 30. In 2014, this ratio stoodat 12, the proportion of teachers aged 50 and over in the regional education authority was 12 times higher than that of those under 30.

This first observation must be refined at the communal and cantonal level by summing up all middle and high school teachers, public or private, according to their age (see Figure 1.9). In middle school, the proportion of teachers under 30 years of age is highest in Dugny in Seine-Saint-Denis (56% in 2014), which is part of the Créteil regional education authority, while it does not reach 1% in Vannes in the Rennes regional education authority. The municipality of Dugny has 10,200 inhabitants 9 , more than 38% of the inhabitants over 15 years of age are without a diploma and 48.5% of households are taxed on their income. Vannes (53,032 inhabitants) 10 is much more socially privileged (27.1% of people have no diploma and 57.8% are taxed households). Does the youth of teachers systematically match up with social housing and the impoverishment of communes? This is true in the case of Parisian regional education authorities (Botton and Miletto 2018).

The average share of teachers under 30 years of age per canton in public middle schools was around 12% in 2014. This is the case, for example, in a canton such as Bollène (regional education authority of Aix-Marseille). The proportion of young teachers is highest in the 1st and 2 nd Parisian suburbs (more than 40% in Grigny or Evry), especially in the eastern and northern parts, the poorest in Île-de-France (more than 40% in Le Bourget, La Courneuve, Villepinte, Stains or Pierrefitte-sur-Seine). The margins of the Paris urban area are equally affected (radius of 70 to 80 km from Paris), towards Yonne (Dijon regional education authority), Eure (Rouen regional education authority), Loiret (Orléans-Tours regional education authority), or Marne (Reims regional education authority). The same is true in Mayotte and in the coal basin of the Lille regional education authority, which is still undergoing industrial conversion. The demographic growth and the mass of young people to be enrolled in school in these areas leads to many vacancies. On the other hand, all municipalities in large urban areas have situations close to the average, such as Fort-de-France, Reims, Nantes or Besançon. The attractive regional education authorities, with regard to the wishes for the assignment and transfer of teachers over the long term , concentrate municipalities where the proportion of young people is lower than the average or equivalent (Atlantic and Mediterranean coasts, regional education authorities of Rennes, Nantes, Bordeaux, Toulouse, Aix-Marseille, etc.). These last three regional 9. https://www.insee.fr/fr/statistiques/1405599?geo=COM-93030. 10. https://www.insee.fr/fr/statistiques/1895081. education authorities (Bordeaux, Toulouse, Aix-Marseille), as well as the one in Nice, already stood out in 1990-1991 with ratios of teachers over 50 years of age of more than 18% (MEN-DEPP 1993, p. 29).

Figure 1.9. Percentage of middle school, general high school (LEGT) and public professional teachers under 30 years of age per equipped canton in 2014

The proportion of teachers working in public middle schools under 30 years of age is highest in the canton of Le Bourget (49.5%) in the Seine-Saint-Denis department (Académie de Créteil). The proportion of teachers working in public general and technological high schools under 30 years of age is highest in the canton of Mitry-Mory (34.1%). The proportion of teachers working in public vocational high schools under 30 years of age reaches its maximum in the canton of Suresnes (25%) in the department of Hauts-de-Seine (Versailles regional education authority), while less than 1% work in Bastia, La Rochelle or Aix-en-Provence.

The observation is identical to the examination of data relating to public general and technological high schools (LEGT), always at canton level. The record for young teachers is held by Mitry-Mory (34.1%), with less than 2% in Biarritz, Dinard, Gap, Royan or Anglet, for example. Conversely, the ratio of teachers under 30 years of age for this category of schools is at its lowest in Gap (less than 1%). This small tourist resort in the Southern Alps is at odds with Le Bourget in terms of social fabric (29.9% of people aged 15 and over without a diploma in Gap compared with 42.4% in Le Bourget in 2013).

The range of differences narrows when focusing on the share of young teachers in public vocational high schools. The canton of Suresnes (Versailles regional education authority) is in an extreme situation with 25% of teachers under 30 years of age. In the heart of this canton, the municipality of Suresnes is home to a disadvantaged, impoverished and vulnerable population (risk of unemployment), since the proportion of inhabitants over 15 years of age without a diploma (20.5% in 2013) 11 and the household income tax rate at the same date (75.9%) are well above the average. At national level, the opposition between Parisian peri-urban and urban areas in the "province" is confirmed. A major difference appears with this indicator, since many small and medium-sized urban areas (from 30,000 to 150,000 inhabitants in 2012, such as Alençon 68,000 inhabitants, or Colmar 130,000 inhabitants), with the exception of Brittany, are distinguished from the most populated, metropolitan or not, where the ratios are lower (Nantes, Reims, Bourges, Bourges, Besançon, Valence, etc.). Brittany has little turnover its teaching staff due to the stability of teachers in their positions. It is difficult to enter, which is one of the reasons why the ratios of young people are high at the gates of the Rennes regional education authority (Orne, Sarthe, etc.). We are as close as possible to the day when we will be able to access this regional education authority (sufficient number of points in the allocation mechanism) 12 . With the regional education authorities of Versailles and Créteil, those of Lille, Amiens, Orléans-Tours, Nancy-Metz and Strasbourg are the youngest. They all have an economic and social profile marked during the 19 th and 20 th centuries by intense industrialization, a source of development of vocational education.

After the inequalities linked to teachers' ages, those linked to their status are analyzed at cantonal level (proportion of non-accredited teachers), but this time the 11. https://www.insee.fr/fr/statistiques/1405599?geo=COM-92073.

12. An inter-academic scale defined nationally and academic scales established by the rectors and the vice-rector allow the classification of requests as well as the elaboration of movement projects. However, they are only indicative. The scales first of all reflect the legal and regulatory priorities for processing the requests of certain staff: bringing together spouses, disabled officials and staff working in urban areas where particularly difficult social and security problems arise, school mapping measures, etc. They give priority to elements related to legal priorities and, to a lesser extent, take into account other elements related to: personal situation; career situation (length of service and post); the individual situation of the staff member. numbers are aggregated for middle, LEGT and vocational, public and private high schools (see Figure 1.10). In 2014, more than 86% of teachers working in the commune of Grand-Santi in French Guiana did not have a qualification. The entire overseas departments of France are in a somewhat comparable situation, except for Réunion Island (see Table 1.1). Saint-Laurent-du-Maroni has 225 teachers, more than three-quarters of whom do not have a qualification. In metropolitan France, the regional education authorities of Versailles and Créteil have the same problem as those overseas, namely a difficulty in attracting and retaining their teachers. Cantons such as Le Palais, Annot and Ile d'Yeu have a record number of unqualified teachers (50 to 60%). To compensate for the lack of candidates to fill their posts, the education officers of these regional education authorities use individual temporary workers and other unqualified staff. In 1990, the proportion of unqualified staff among high school teachers in the Créteil education authority was already a record 13.6% compared with 12.4% in Versailles (then average 9.7% (MEN-DEPP 1993, p. 114)).

Between 2004 and 2014, the number of unqualified teachers, all types of establishments combinedmiddle schools, high schools, public and privateincreased slightly (+3% on average), especially in Seine-Saint-Denis and Île-de-France in general. It also increased in overseas France, from 8.7% to more than 15%. In metropolitan France, as a general rule, growth is all the stronger as the cantons and communes are located at the margins of their departments, i.e. in the rural periphery under low urban influence. These areas, often poorly served by trains or highways and therefore poorly connected to large urban areas, are not places of work or residence sought by qualified teachers. Spousal employment is probably more difficult to find than in urban areas at the top of the urban hierarchy [START_REF] Veltz | Mondialisation, villes et territoires[END_REF]. This is the case, for example, of the margins of Haute-Marne (border with Vosges), a department that itself constitutes a margin at the borders of the Paris basin and the regional urban system of the Grand Est, which are particularly affected by the growth in the rate of unqualified staff. The same applies to the borders between Haute-Garonne and Ariège, as well as to the borders of Morbihan and Ille-et-Vilaine, or between south-east French Guiana and Brazil. These peripheries, which are poorly urbanized and therefore poorly equipped (private services, transport infrastructure), do not host a varied and diversified labor market, all of which works against the establishment of working households with higher education qualifications, as is the case for teachers. On the other hand, a decline in the rate of unqualified staff benefits areas well connected to transport networks such as the southern part of the department of Manche. This explanatory hypothesis does not exhaust the subject.

Conclusion

How much of France is included when we analyze inequalities in the distribution of financial and human resources and inequalities in access to schools? After examining the inequalities in terms of financial allocation by local authorities to middle and high schools, the inequalities in road access to these establishments, and the disparities in the location of human resources in these establishments such as teachers, the use of the notion of the school divide does not seem excessive given the differences observed between cantons, departments and regional education authorities.

If we were to retain only one dimension of these divisions, it would undoubtedly be the one relating to the rate of unqualified teachers. The share of this type of teacher increased, as we have seen, on average in metropolitan France between 2004 and 2014, from 4.2% to 6.9%, but it multiplied by 1.7 in the overseas departments and regions over the same period, and went from 25.8% to 39.3% in French Guiana. The average metropolitan rate in this area is 2.2 times lower than that of the overseas departments and regions.

By broadening the scope, a school divide isolates the French overseas communities from metropolitan France, and within these overseas communities, French Guiana and Mayotte are clearly different from the other French overseas departments and territories. In metropolitan France, the regional education authorities of Amiens, Créteil and Versailles are opposed by their characteristics to those of Paris, Rennes, Montpellier or Bordeaux (age of teachers for example). On a finer scale, in metropolitan France, a divide separates mountainous rural areas from those on the crossing of the empty space, to the urban network and its peri-urban fringes, when we focus on inequalities in road access to educational establishments.

But this multiscalar observation needs to be confirmed by taking into account statistical data of the same nature over a decade to measure the possible stability of these divisions, a reduction in them or an aggravation. In addition, indicators such as those on road accessibility to schools would benefit from being refined by type of high school, for example (e.g. by providing industrial or tertiary training or by providing rare training). [START_REF] Schneider | État des lieux des inégalités scolaires d'origine territoriale en France[END_REF]). Quartiers, égalité, scolarité. Cnesco, Paris. Boudesseul, G., Caro, P., Grelet, Y., Minassian, L., Monso, O., Vivent, C. (2016).

Bibliographical references

Botton

Atlas des risques sociaux d'échec scolaire : l'exemple du décrochage, France métropolitaine et DOM. Céreq, Marseille.

Figure 1 . 1 .

 11 Figure 1.1. Total expenditure of departmental councils in 2015 for middle schools by department (in millions of euros) (source: DGCL 2015)

Figure 1 . 2 .

 12 Figure 1.2. Total expenditure of the regional councils in 2015 for public and private high schools per student and per region (in euros per student) (source: DGCL 2015)

  5. PEDT: projet éducatif territorial, territorial educational project; PEL: projet éducatif local, local educational project; PESL: projet éducatif social local, local social educational project; PRE: programme de réussite éducative, educational success program.

Figure 1 . 3 .

 13 Figure 1.3. Euclidean distance to the nearest middle school and the nearest high school (LEGT), public or private, in metropolitan France, in 2016, for unequipped municipalities

Figure 1 . 4 .Figure 1 . 5 .Figure 1 . 6 .

 141516 Figure 1.4. Road access time from an unequipped commune to the commune equipped with the nearest middle school in 2016

Figure 1 . 7 .

 17 Figure 1.7. High school teachers: proportion of over-50s and under-35s, at the beginning of the 2015 academic year, according to the regional education authorities (source: MENESR DEPP, based on "Salary slips", December 2015)

Figure 1 . 8 .

 18 Figure 1.8. High school teachers: change in the ratio of over-50s to under-30s from 2005 to 2014 (sources: Geography of schools in 2011 and social balance sheet of the French Ministry of National Education, Higher Education and Research 2013-2014; data: MENJVA-MESR DEPP, 2010 pay file and MENESR-DEPP, 2014 pay file).

Figure 1 . 10 .

 110 Figure 1.10. Ratio of teachers in public and private middle schools and high schools with unqualified staff among all teachers in 2004 and 2014 studied by canton


Georges Gras

Jean-François Jaulent

Note on 2-rational fields

d'un corps de nombres 2-rationnel.

Introduction and history

The notions of ℓ-rational field and ℓ-regular field (for a prime number ℓ and a number field K), independently introduced by A. Movahhedi and T. Nguyen Quang Do in [MN], and by G. Gras and J-F. Jaulent in [GJ], coincide as soon as K contains the maximal real subfield of the field of ℓth roots of unity, thus especially for ℓ = 2.

• The ℓ-regularity expresses the triviality of the regular ℓ-kernel of K (i.e. the kernel, in the ℓ-part of the universal group K 2 (K), of Hilbert symbols attached to the non-complex places not dividing ℓ).

• The ℓ-rationality traduces the pro-ℓ-freeness of the Galois group

G K := Gal(M K /K) of the maximal pro-ℓ-extension ℓ-ramified ∞-split M K of K (i.
e. unramified at the finite places1 not dividing ℓ and totally split at the infinite places).

More precisely, let c K be the number of complex places of K; let µ K (resp. µ K l ) be the ℓ-group of roots of unity in K (resp. in the localization K l ); and let

V K := {x ∈ K × | x ∈ K ×ℓ l ∀l | ℓ & v p (x)
≡ 0 mod ℓ ∀p ∤ ℓ∞} be the group of ℓ-hyperprimary elements in K × . Then, with these notations, from [START_REF] Jaulent | Corps p-rationnels, corps p-réguliers et ramification restreinte[END_REF]Th.1.2] or [G 3 , IV.3.5, III.4.2.3], the ℓ-rationality of K may be expressed as follows:

Theorem and definition 0. The following conditions are equivalent:

(i) The Galois group G K is a free pro-ℓ-group on 1 + c K generators. (ii) The abelianization G ab K of G K is a free Z ℓ -module of dimension 1 + c K . (iii)
The field K satisfies the Leopoldt conjecture (for the prime ℓ) and the torsion submodule T K of G ab K is trivial. (iv) One has the equalities:

V K = K ×ℓ & rk ℓ (µ K ) = l|ℓ rk ℓ (µ K l ).
When any of these conditions is realized, the number field K is said to be ℓ-rational.

Remark. In case ℓ = 2, it follows from the condition (iv) above that a 2rational field has a single place above 2.

The premises of the notion of ℓ-regularity go back to the works of G. Gras, mainly to his note on the K 2 of number fields [G 2 , II, § 2; III, § § 1, 2], whereas the notion of ℓ-rationality appears (in a hidden form) in the work of H. Miki [Mi] concerning the study of a sufficient condition for the Leopoldt conjecture, as well as those of K. Wingberg [W 1 , W 2 ], concerning the same condition.

Movahhedi's thesis and the above papers [GJ, MN] characterised the going up for ℓ-rationality in any ℓ-extension in terms of ℓ-primitivity of the ramification (a definition given in [G 2 , III, § 1] from the use of the Log function defined in [G 1 ]), a property which was unknown in the preceding approaches.

For instance, this gives immediately that if K is a ℓ-extension of Q, a N.S.C. for K to be ℓ-rational is that K/Q be ℓ-ramified, or that K/Q be {p, ℓ}-ramified, where p = ℓ is a prime ≡ 1 mod (ℓ) such that p ≡ ±1 mod (8) if ℓ = 2 and p ≡ 1 mod (ℓ 2 ) if ℓ = 2 (cf. [G 3 , IV.3.5.1] giving Jossey's examples [Jo]).

A synthesis of these results is given in [JN] and a systematic exposition is developped in the book of G. Gras ([G 3 , III, § 4, (b); IV, § 3, (b); App., § 2]).

Various generalizations of these notions have been studied by O. Sauzet and J-F. Jaulent (cf. [JS 1 , JS 2 ]), especially in the case ℓ = 2 which is, as usual, the most triky; in particular, they introduce the notion of 2-birational fields.

Very recently, J. Jossey [Jo] introduced a notion of ℓ-rationality which is incompatible with the classical one (for ℓ = 2, as soon as K contains real embeddings) and is unlucky since it does not apply to the field of rationals Q.

For these reasons, to avoid any confusion, we propose to speak, in his context, of 2-superrational fields. More precisely: Definition 1. Let K be a number field with r K real places and c K complex places, M ′ K the maximal 2-ramified pro-2-extension of K, and M K the maximal subextension of M ′ K totally split at the infinite places. We say that K is: (i) 2-superrational, if G ′ K := Gal(M ′ K /K) is pro-2-free; (ii) 2-rational, if its quotient G K := Gal(M K /K) is pro-2-free.

The purpose of the next section is to determine the structure of the Galois group G ′ K when the number field K is 2-rational. Our proof relies on the functorial properties of ℓ-ramification theory.

According to the conventions of the ℓ-adic class field theory (cf. [G 3 , Ja]), we never speak of ramification at infinity but of complexification of real places.

This argument is equivalent to the use of the formulas of Šafarevič (cf.[Sa] or[START_REF] Neukirch | Cohomology of number fields[END_REF] Th.8.7.3])

2 Main Theorem: description of G ′ K = Gal(M ′ K /K)

Our very simple result has the following statement:

Theorem 2. Let K be a 2-rational number field having r K real places and c K complex places. The Galois group

of (1 + c K ) copies of the procyclic group Z 2 and of r K copies of Z/2Z.

Corollary 3. The 2-rational number fields which are 2-superrational are the totally imaginary ones.

Proof. Consider the quadratic extension L = K[i] generated by the 4th roots of unity. It is 2-ramified over K, thus thanks to the going up theorem of [GJ, MN] (cf. e.g. [START_REF] Jaulent | Corps p-rationnels, corps p-réguliers et ramification restreinte[END_REF]Th. 3.5] or [G 3 , IV.3.4.3, (iii)]), it is 2-rational, then 2superrational since it is totally imaginary. In other words, the Galois group

As in [Jo], the results of W. Herfort and P. Zalesskii (cf. [START_REF] Herfort | Finitely generated pro-2-groups with a free subgroup of index 2[END_REF]Th.0.2]) give the existence of a finite familly (F i ) i=0,...,k of free pro-2-groups on respectively d 0 , . . . , d k generators (where k is the number of conjugacy classes of subgroups of order 2 in G ′ K ), such that:

In particular, the abelianisation G ′ ab K of G ′ K admits the direct decomposition:

Since the 2-rational field K satisfies the Leopoldt conjecture, we get

, so that T ′ K is generated by the decomposition groups of the real places of K which are deployed, a key argument of class field theory (cf. [Ja] or [G 3 , III.4.1.5]) giving k = r K . Now the pro-2-decomposition of G ′ K clearly shows that the minimal number of generators d(G ′ K ) and of relations r(G ′ K ), defining G ′ K as a pro-2-group, are:

It is well-known (cf. e.g. [G 3 , App., Th.2.2, (i)]) that one has 2 :

Thus we obtain d 0 = 1 + c K , giving d i = 0 for 1 ≤ i ≤ k, then the expected result.


domain: spi phys

text:
Virgile Charton

Pierre Trontin

Gilles Aouizerate

Philippe Villedieu

Semi-empirical modelling of erosion phenomena for ice crystal icing numerical simulation

The aim of this work is to develop a semi-empirical model for erosion phenomena under ice crystal condition, which is one of the major phenomena for ice crystal accretion. Such a model would be able to calculate the erosion rate caused by impinging ice crystals on accreted ice layer. This model is based on Finnie [1] and Bitter [2] [3] solid/solid collision theory which assumes that metal erosion due to sand impingement is driven by two phenomena: cutting wear and deformation wear. These two phenomena are strongly dependent on the particle density, velocity and shape, as well as on the surface physical properties such as Young modulus, Poisson ratio, surface yield strength and hardness. Moreover, cutting wear is mostly driven by tangential velocity and is more effective for ductile eroded body, whereas deformation wear is driven by normal velocity and is more effective for brittle eroded body. Several researchers based their erosion modelling on these two phenomena such as Hutchings et al.

 and Arabnejad et al.

[6] for cutting and deformation erosion.

The main work of this paper is to develop an erosion model for ice crystal impingement based on these two phenomena, and to show its capability to predict accretion shape by simulating experimental cases from the National Research Council of Canada (NRC). NRC's Currie et al. ice crystal experiments [7] [8] realized in warm aerodynamic conditions, such as the one encountered in high icing severity areas of a turbofan engine, show accretion severity for a large range of liquid water content to total water content. In order to validate the erosion model based on solid/solid collision, this paper presents the simulation of the lower melting rate experiment. Results show fair agreement with experimental data and allow us to propose pertinent further work.

Introduction

Aircraft icing is studied since the early 20 th century, as it is a major weather hazard for aircraft operation. During the 90s, ice crystal icing has been identified [START_REF] Strapp | The Ice Particle Threat to Engines in Flight[END_REF], and the regulation accordingly modified. The mechanisms involved in ice crystal icing are not fully understood and several ongoing projects such as MUSIC-Haic [START_REF] Commission | MUSIC-haic[END_REF] are still needed to improve the knowledge on ice crystal icing and to build robust modelling tools able to predict the location and the severity of the accretion.

A major phenomenon that drives ice crystal icing is the erosion caused by impinging crystals on accreted ice deposits. For now, empirical models are used to calculate the erosion rate as the one proposed by Trontin et al. [START_REF] Villedieu | Glaciated and mixedphase ice accretion modeling using ONERA 2D icing suite[END_REF] [START_REF] Baumert | Experimental and numerical investigations on aircraft icing at mixed phase conditions[END_REF] in Onera's 2D icing code based on Currie et al. experimentations [START_REF] Currie | Experimental Studies of Mixed-Phase Sticking Efficiency for Ice Crystal Accretion in Jet Engine[END_REF] [START_REF] Currie | Experimental Results for Ice Crystal Icing on Hemispherical and Double Wedge Geometries at Varying Mach Numbers and Wet Bulb Temperatures[END_REF]. These empirical models were developed with few experimental data, and even if they have good agreement with the data used for their calibration, it can't be assumed that they will be able to predict well ice accretion size and shape for new experimental cases. The objective of this paper is to study if a model backed by solid-solid erosion can be proposed. Such a model should be more reliable for the simulation of new cases.

The first part of this paper briefly introduces the existing erosion model implemented in Onera's 2D icing code IGLOO2D, and highlights the improvements needed. The second part of this paper is dedicated to solid/solid collision theory, which allows us to calculate the volume eroded by the two identified mechanisms: deformation wear and cutting wear. The third part presents simulation results and discussion around them, especially compared to IGLOO2D ones. Finally the last part introduces future work, in particular the particle size and the liquid water dependencies.

Onera's 2D icing code erosion modelling

IGLOO2D icing code takes into account the influence of erosion phenomenon on ice shape by calculating an erosion rate which corrects the accretion rate given by mass and energy balance equations (IGLOO2D extended Messinger's equations are presented in [START_REF] Villedieu | Glaciated and mixedphase ice accretion modeling using ONERA 2D icing suite[END_REF]).

Erosion means here material loss caused by crystals impact on ice layer. During the framework of High Altitude Ice Crystal project (HAIC) [START_REF] Commission | HAIC[END_REF] Trontin et al. [11] [12] developed an empirical model based on Currie et al. [8] [7] observations in order to calculate this material loss. This model gives the erosion rate 𝑚̇𝑒 𝑟 as:

𝑚̇𝑒 𝑟 = 𝜂 𝑒𝑟 × 𝑚̇𝑖 𝑚𝑝 (1)
Where 𝑚̇𝑖 𝑚𝑝 is the impacting mass rate and 𝜂 𝑒𝑟 the erosion ratio. HAIC erosion model gives a value of 𝜂 𝑒𝑟 , between 0 and 1, according to:

𝜂 𝑒𝑟 = 𝐸 ( 𝑉 𝑡 𝑉 0 ) 2 ( 𝛼 𝑙0 𝛼 𝑙0 -min(𝛼 𝑙 , 𝛼 𝑙0 ) ) (1 + (𝑙 0 𝑘) 2 ) (2)
Where 𝑉 𝑡 is the tangential impact velocity, 𝛼 𝑙 the wall liquid ratio, and 𝑘 the wall curvature. 𝐸, 𝑉 0 , 𝛼 𝑙0 and 𝑙 0 are empirical parameters only calibrated on Currie's experiment. One can notice that the velocity dependency is linked to the tangential component only, which means that negligible erosion should be predicted for quasinormal impact. Also the curvature term is a numerical correction which reflects the smoothing effect of erosion phenomenon. The new model presented furtherly is aimed at reducing the empiricism of the original one by giving more physical meaning to the adjustable parameters. In addition, it is expected to have a larger domain of validity.

Semi-empirical erosion modelling

This section presents the semi-empirical erosion model inspired by Finnie [START_REF] Finnie | On the velocity dependence of the erosion of ductile metals by solid particles at low angles of incidence[END_REF] and Bitter [START_REF] Bitter | A study of erosion phenomena -Part I[END_REF] [3] solid-solid collision theory. According to these authors, erosion phenomenon caused by a solid particle impact on a wall is mainly due to two phenomena: the deformation wear and the cutting wear. For the sake of simplification, and because 29/03/2018 experimental data are better exploitable in 2 dimensions, the following sections will use a 2D approach. However, the semiempirical model developed in this paper can be implemented in 3D simulation codes as well as in 2D's ones.

Deformation wear

According to Bitter [START_REF] Bitter | A study of erosion phenomena -Part I[END_REF], deformation wear is the consequence of the particle kinetic energy that is absorbed by the body during the impact.

As long as the maximum corresponding stress, which occurs on the center of the contact area, does not reach the yield strength of the material, only elastic deformation occurs and no material is removed. However, if the yield strength is exceeded during the collision, plastic deformation happens at the place of maximum stress. Hence, a large number of particle impacts will form a plastically deformed surface with an increased hardness that may lead to material failure due to the creation of lateral and radial cracks [START_REF] Commission | HAIC[END_REF] (Figure 1). The eroded volume is calculated resolving energy balance equation using the following assumptions:

1) Impinging particles are spherical 2) No particle fragmentation or plastic deformation occurs within the particle 3) Impact velocity is low enough, so that the energy dissipated as heat is negligible 4) The eroded volume is equivalent to the volume of the formed crater

For elastic-plastic impact, the provided kinetic energy is divided into three energies absorbed in the substrate (Figure 2):

𝑄 = 1 2 𝑚𝑉 𝑛 2 = 𝑄 𝑒 + 𝑄 𝑝𝑒 + 𝑄 𝑝 ( 3
)
Where 𝑄 is the kinetic energy, 𝑄 𝑒 the energy absorbed in the region of elastic deformation, 𝑄 𝑝𝑒 the potential energy of the elastic deformation in the area subjected to a plastic-elastic load, and 𝑄 𝑝 the energy required to the formation of the permanent indentation. In the case of an elastic-plastic impact, the energy 𝑄 𝑒 absorbed in the region of elastic deformation corresponds to the amount of kinetic energy absorbed with a velocity 𝑉 𝑒𝑙 at which the elastic limit is just reached:

𝑄 𝑒 = 1 2 𝑚𝑉 𝑒𝑙 2
Where 𝑚 is the particle mass. The elastic velocity 𝑉 𝑒𝑙 is expressed using Hertz elastic collision theory [START_REF] Bitter | A study of erosion phenomena -Part I[END_REF]:

𝑉 𝑒𝑙 = 𝜋 2 2√10𝜌 𝑅 𝑒𝑙 5 2 ( 1 -𝑞 𝑠𝑢𝑏 2 𝐸 𝑠𝑢𝑏 + 1 -𝑞 𝑝𝑎𝑟𝑡 2 𝐸 𝑝𝑎𝑟𝑡 ) 2 ( 4
)
Where 𝜌 is the particle density, 𝑅 𝑒𝑙 , called yield strength (or elastic limit), is the largest load that can be exerted upon the body without causing plastic deformation, 𝑞 and 𝐸 are respectively Poisson's ratio and Young's modulus of particle (part) and substrate (sub).

𝑄 𝑝𝑒 is the potential energy of the elastic deformation in the area subjected to the elastic-plastic load (green area), whereas 𝑄 𝑝 is the energy of the plastic deformation (red area). Using the definition of the potential energy, one has:

𝑄 𝑝𝑒 = ∫ 𝐹 ⃗ . 𝑑ℎ ⃗⃗⃗⃗⃗ ℎ 𝑒𝑙 0 Where 𝐹 = 1 2
𝑅 𝑒𝑙 𝜋𝑟 𝑝 2 is the average force applied during the impact (at the beginning of the impact, the pressure is null and when the elastic limit is reached, the impact pressure is equal to 𝑅 𝑒𝑙 ). Using the Pythagorean theorem and assuming a small depth of indentation, the following geometrical relation between the elastic-plastic contact area radius 𝑟 𝑝 , the indentation depth 𝐻 and the particle radius 𝑅 can be written:

𝑟 𝑝 2 = 2𝑅𝐻
Consequently, this leads to:

29/03/2018

𝑄 𝑝𝑒 = ∫ 1 2 𝑅 𝑒𝑙 𝜋𝑟 𝑝 2 𝑑ℎ 𝑒𝑙 ℎ 𝑒𝑙 0 = ∫ 𝑅 𝑒𝑙 𝜋𝑅𝐻𝑑ℎ 𝑒𝑙 = 𝑅 𝑒𝑙 𝜋𝑅𝐻ℎ 𝑒𝑙 ℎ 𝑒𝑙 0
Similarly, one can express 𝑄 𝑝 with 𝐹 = 𝑅 𝑒𝑙 𝜋𝑟 𝑝 2 (elastic limit is already reached):

𝑄 𝑝 = ∫ 𝑅 𝑒𝑙 𝜋𝑟 𝑝 2 𝑑𝐻 𝐻 0 = ∫ 𝑅 𝑒𝑙 𝜋2𝑅𝐻𝑑𝐻 = 𝑅 𝑒𝑙 𝜋𝑅𝐻 2 ℎ 𝑒𝑙 0
Using the following relation between 𝑉 𝑒𝑙 and ℎ 𝑒𝑙 from the Hertzian theory [START_REF] Bitter | A study of erosion phenomena -Part I[END_REF]:

𝑅ℎ 𝑒𝑙 2 = 15 8 𝑚 𝜋𝑅 𝑒𝑙 𝑉 𝑒𝑙 2
The previous expression of the energies involved imply:

𝑄 𝑝𝑒 = 𝑅 𝑒𝑙 𝜋𝑅𝐻ℎ 𝑒𝑙 = √𝜋𝑅 𝑒𝑙 𝑅ℎ 𝑒𝑙 2 √𝜋𝑅 𝑒𝑙 𝑅𝐻 2 = √ 15 4 𝑄 𝑒 𝑄 𝑝
Finally, using the energy balance equation ( 3):

𝑄 = 𝑄 𝑒 + √ 15 4 𝑄 𝑒 𝑄 𝑝 + 𝑄 𝑝 = (√𝑄 𝑒 + √𝑄𝑝) 2 -0.061√𝑄 𝑒 𝑄 𝑝
Since 𝑄 𝑒 is generally small in relation to 𝑄 𝑝 , the second term of the right-hand side of this equation may be neglected. It follows that:

𝑄 𝑝 = (√𝑄 -√𝑄 𝑒 ) 2 = 1 2 𝑚(𝑉 𝑛 -𝑉 𝑒𝑙 ) 2
Bitter [START_REF] Bitter | A study of erosion phenomena -Part I[END_REF] defines then the deformation wear factor 𝜀 as the amount of energy needed to remove one unit volume of material using deformation wear. Therefore the volume eroded by deformation wear 𝑉𝑜𝑙 𝐷 is given by:

𝑉𝑜𝑙 𝐷 = 1 2 𝑚(𝑉 𝑛 -𝑉 𝑒𝑙 ) 2 𝜀 (5)
While 𝑉 𝑛 ≤ 𝑉 𝑒𝑙 , the collision is purely elastic and no erosion occurs, and if 𝑉 𝑛 > 𝑉 𝑒𝑙 , the eroded volume is given by 𝑉𝑜𝑙 𝐷 . Equation ( 5) shows a dependency to the mass of the particle. If many particles impact the substrate with the same characteristics, one can take 𝑚 as equal to the total impinging mass.

Cutting wear

According to the description of cutting wear by Finnie [START_REF] Finnie | On the velocity dependence of the erosion of ductile metals by solid particles at low angles of incidence[END_REF], Bitter proposed the following mechanism [START_REF] Bitter | A study of erosion phenomena -Part II[END_REF] : if impact energy allows the formation of a permanent indentation, the tangential component of the velocity will make that indentation larger and will also pile up material around the crater. The piled up material is then eventually removed by continuous particle impacts (Figure 3). The hypotheses used for the cutting eroded volume calculation are the same as the one used previously. Thus, if plastic deformation occurs for an acute angle, one also has a shear stress due to the tangential velocity 𝑉 𝑡 , applied to the surface 𝐴 𝑥 of the particle section which has penetrated into the substrate. Finnie [START_REF] Finnie | On the velocity dependence of the erosion of ductile metals by solid particles at low angles of incidence[END_REF] or Arabnejad et al. In order to obtain an expression of the surface 𝐴 𝑥 , equations of the particle motion originally developed by Finnie et al. [START_REF] Finnie | On the velocity dependence of the erosion of ductile metals by solid particles at low angles of incidence[END_REF] can be resolved:

𝑚 𝑑 2 𝑥 𝑑𝑡 2 + 𝑃. 𝐴 𝑥 = 0 𝑚 𝑑 2 𝑦 𝑑𝑡 2 + 𝑃. 𝐴 𝑦 = 0
Where 𝑃 is the impact pressure which is assumed to be equal to the material hardness [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF]. The area 𝐴 𝑦 can be geometrically calculated from the contact area radius 𝑟 𝑦 , which can be expressed using the indentation depth 𝑦 and the particle radius 𝑅:

𝐴 𝑦 = 𝜋𝑟 𝑦 2 = 2𝜋𝑦𝑅
Then, parameter 𝐾 is introduced as the ratio of the projected surface 𝐴 𝑦 to the projected surface 𝐴 𝑥 (Figure 4). The equations become: 𝑉𝑜𝑙 𝐶 is calculated by:

𝑚 𝑑 2 𝑥 𝑑𝑡 2 + 𝑃2𝜋𝑅 𝐾 𝑦 = 0 𝑚 𝑑 2 𝑦 𝑑𝑡 2 + 𝑃2𝜋𝑅𝑦 = 0
𝑉𝑜𝑙 𝐶 = ∫ 𝐴 𝑥 𝑑𝑥 𝑥 𝑐 0 = ∫ 𝐴 𝑦 𝐾 𝑡 𝑐 0 𝑥 ′ (𝑡)𝑑𝑡 = ∫ 2𝜋𝑅 𝐾 𝑡 𝑐 0 𝑦(𝑡)𝑥 ′ (𝑡)𝑑𝑡
Which can be developed as:

𝑉𝑜𝑙 𝐶 = 2𝜋𝑅 𝐾 𝑉 2 sin(𝜃) cos(𝜃) 𝛽 2 (1 -cos(𝛽𝑡 𝑐 )) + 2𝜋𝑅 𝐾 2 𝑉 2 sin 2 (𝜃) 𝛽 2 (cos(𝛽𝑡 𝑐 ) -1) + 𝜋𝑅 𝐾 2 𝑉 2 sin 2 (𝜃) 2𝛽 2 (1 -cos(2𝛽𝑡 𝑐 ))
Where 𝑡 𝑐 is the time when the cutting action stops. According to Finnie [START_REF] Finnie | On the velocity dependence of the erosion of ductile metals by solid particles at low angles of incidence[END_REF] and Arabnejad et al. [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF], two scenarios exist when 𝑡 = 𝑡 𝑐 :

1) The particle cannot move forward and is trapped into the substrate : 𝑥 ′ (𝑡 𝑐 ) = 0 2) The particle leaves the surface : 𝑦(𝑡 𝑐 ) = 0

Whereas the first scenario corresponds to quasi-normal angle impact, the second one corresponds to low angle impact. The critical angle 𝜃 𝑐 between the two scenarios is given by the boundary condition:

𝑥 ′ (𝑡 𝑐 ) = 0 = 𝑉 cos(𝜃) - 𝑉 sin(𝜃) 𝐾 (1 -cos(𝛽𝑡 𝑐 )) ⇔ cos(𝛽𝑡 𝑐 ) = 1 - 𝐾 tan (𝜃) ⇒ -1 ≤ 1 - 𝐾 tan(𝜃) ≤ 1 ⇒ 𝜃 ≥ arctan ( 𝐾 2 )
While the impinging angle 𝜃 is lower than the critical angle 𝜃 𝑐 = arctan ( 𝐾 2

), the particle leaves the subtract with a non-null tangential velocity, and while 𝜃 is above 𝜃 𝑐 , the particle is stuck in the subtract at the end of the cutting action.

The volume eroded by cutting action 𝑉𝑜𝑙 𝐶 is calculated using the ideal hypothesis that each impact would remove this volume. To take into account the fact that several impacts are usually required to remove a piece of material the cutting erosion efficiency 𝐶 is introduced. Finally, the two conditions on the particle behavior give the following expressions for 𝑉𝑜𝑙 𝑐 :

𝑉𝑜𝑙 𝐶 = { 𝐶 × 𝑚𝑉 2 cos 2 (𝜃) 2𝑃 , 𝑓𝑜𝑟 𝜃 ≥ 𝜃 𝑐 𝐶 × 2𝑚𝑉 2 sin(𝜃) (𝐾𝑐𝑜𝑠(𝜃) -sin(𝜃)) 𝐾 2 𝑃 , 𝑓𝑜𝑟 𝜃 < 𝜃 𝑐 (6)
As for the deformation wear calculation, if many particles impact the substrate, 𝑚 can be taken as equal to the total impinging mass.

Erosion ratio

Trontin et al. [START_REF] Villedieu | Glaciated and mixedphase ice accretion modeling using ONERA 2D icing suite[END_REF] [12], as well as Arabnejad et al. [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF] define the erosion ratio as the subtract eroded mass rate divided by the particle impinging mass rate. Also, one can calculate the erosion rate 𝜂 𝑒𝑟 as follows:

𝜂 𝑒𝑟 = 𝐹 𝑠 𝜌 𝑉𝑜𝑙 𝐶 + 𝑉𝑜𝑙 𝐷 𝑚 (7)
Where 𝐹 𝑠 is the particle sharpness factor, 𝜌 is the material density, and 𝑚 the particle total impinging mass. Table 1 summarizes the different expressions used to calculate the erosion mass rate. Several input variables are needed to be characterized in the ice crystals context:

-Arabnejad et al. [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF] assume that particle sharpness changes the erosion effectiveness through stress concentration. Since the eroded volume calculated in this paper use the hypothesis of spherical particles, this volume has to be corrected in the case of sharp particles using the sharpness factor 𝐹 𝑠 . Based on experimental observations the eroded volume is up to 4 times higher for fully sharp particles. Thus the sharpness factor 𝐹 𝑠 introduced above varies between 1 for fully rounded particles up to 4 for fully sharp particles. Since the NRC RATFac experimentations use ice crystal produced from a grinder one can consider the particle as highly sharped. A value of 4 is taken for the sharpness factor for RATFac experimentations simulations.

-According to Arabnejad et al. [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF], the impact pressure 𝑃 is equal to the material hardness. However, it is complicated to obtain a precise hardness value for ice crystal accretion as few data are available.

Chuzel [START_REF] Parsi | A comprehensive review of solid particle erosion modeling for oil and gas wells and pipelines applications[END_REF] shows that ice hardness double between -10°C and -50°C, and Poirier et al. [START_REF] Chuzel | Caractérisation expérimentale et simulation numérique d'impacts de glace à haute vitesse[END_REF] give experimental measures of ice hardness represented by the following expression: 𝑃(𝑇) = 29/03/2018 (-0.6 ± 0.4)(𝑇 -273.15) + (14.7 ± 2.1). However, according to the author this equation is questionable for a temperature of 0°C. Moreover the ice layer used for these measurements is not produced by accretion process which can affect the mechanical behavior of the ice.

-The ratio of projected area 𝐾 is given by Arabnejad et al. [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF] for sand grain application (sharp particle) as 𝐾 𝑔𝑟𝑎𝑖𝑛 ≈ 0.4. Moreover, according to Arabnejad et al., one has geometrically: 𝐾 𝑠𝑝ℎ𝑒𝑟𝑒 ≈ 2.5 𝐾 𝑔𝑟𝑎𝑖𝑛 . Thus 𝐾 = 𝐾 𝑠𝑝ℎ𝑒𝑟𝑒 = 1 in the presented model, where particles are assumed spherical (𝐾 = 1 even for sharp particles: the sharpness factor will correct the calculated volumes in the case of sharp particles).

-Regarding 𝑉 𝑒𝑙 calculation, ice elastic limit is given by Chuzel [START_REF] Parsi | A comprehensive review of solid particle erosion modeling for oil and gas wells and pipelines applications[END_REF] for atmospheric ice accretion with a value of 𝑅 𝑒𝑙 = 12𝑀𝑃𝑎.

Polycrystalline ice Young modulus and Poisson coefficient are given by Chuzel [START_REF] Parsi | A comprehensive review of solid particle erosion modeling for oil and gas wells and pipelines applications[END_REF] and confirmed by Petrovic [START_REF] Poirier | Ice hardness in winter sports[END_REF]: 𝐸 = 10𝐺𝑃𝑎 and 𝑞 = 0.3. Such values produce a very small 𝑉 𝑒𝑙 and every impact will fulfill the erosion triggering condition 𝑉 𝑛 > 𝑉 𝑒𝑙 .

-Finally the model has two empirical factors which are not characterized in the literature for ice crystal accretion: the deformation wear factor 𝜀 and the cutting efficiency 𝐶. These parameters are calibrated using Currie et al. experiments [7] [8].

However one can use at first approximation the 𝜀 expression given by Arabnejad et al. for cement [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF] which is composed by particle agglomerations as may also be accreted ice: 𝜀 = 𝐹 𝜀 𝑅 𝑒𝑙 2 / 𝐸, where 𝐹 𝜀 is a calibrated constant. The calibration gives : 𝐶 = 0.3 and 𝐹 𝜀 = 1.53 × 10 5 .

Results

This section shows the simulation results of Run 17 (Figure 5) and Run 233 (Figure 7) from Currie et al. experimental series realized at the NRC [START_REF] Currie | Experimental Studies of Mixed-Phase Sticking Efficiency for Ice Crystal Accretion in Jet Engine[END_REF]. These series use an icing tunnel located in an altitude chamber (Research Altitude Test Facility or RATFac). The model presented in this paper does not take into account the amount of liquid water. In order to compare the model with experimental data, it is then necessary to compute a case with low crystal melting ratio such as Run 17 (with 6% of melting ratio) or Run 233 (with 8.6% of melting ratio). The difference between these two Run is the Mach number which is 0.25 for Run 17 and 0.40 for Run 233. Table 2 presents a summary of the experimental conditions used for these two Runs. The result presented Figure 5 shows correctly computed ice shape for Run 17 using the semi-empirical model presented in this paper. The two models (HAIC and the presented model called Sim) give very high accuracy since they are both calibrated with the NRC experimental series of low Mach number to which Run 17 belongs. Figure 6 shows the contribution of deformation erosion and cutting erosion in 𝜂 𝑒𝑟 calculation. One can notice that cutting erosion is higher than deformation erosion which is negligible except near the stagnation point where impacts occur mostly with very small tangential velocity. Run 233 results presented on Figure 7 is also in good agreement with experimental data. Ice height is better predicted using the semiempirical model than HAIC's one. This simulation shows the capability of the presented model to predict ice shapes for higher Mach number.

29/03/2018 The main differences and similarities between the empirical HAIC model and the semi-empirical model presented in this paper are the following: both HAIC and semi-empirical model have a squared velocity dependency, which underline the main kinetic energy dependence of the erosion mechanism. However, whereas HAIC model only consider the tangential component of the particle velocity, the presented model considers also the normal velocity contribution. This allows the simulation to predict the smoothed profile near the stagnation point without using numerical correction (curvature dependent term of the HAIC model). Moreover, the semiempirical model has no liquid water dependency and is not able to predict the erosion rate for high melting ratio cases, while HAIC model is operational for these cases using an empirical function to take into account the amount of liquid water trapped in the ice layer.

Finally, these two models use calibration based on 45µm MVD particles and ice shape are not correctly simulated when the particle size is lowered while keeping the same impingement rate. Thus, one should improve the presented model implementing particle size and liquid water dependency.

Sensitivity study

In order to obtain an idea of the impact of the model parameters on the erosion rate, Figure 8 presents ice shapes obtained for Run 17 using set of parameters represented in Table 3. This sensitivity study is not exhaustive but allows us to know in first approximation which kind of influence have these parameters. When its value increase one can easily notice the increase of eroded material, with a lower influence near the stagnation point where the tangential velocity is negligible. Secondly, increasing the volume eroded by cutting action also decreases the length of the iced surface. (b) The impact pressure 𝑃 has also a direct influence on the volume eroded by cutting action. Ice shape evolution with 𝑃 is as important as the one with 𝐶. (c) The deformation wear factor 𝜀 influences the volume eroded by deformation wear which is not as significant as the volume eroded by cutting wear. (d) The elastic limit 𝑅 𝑒𝑙 takes action in 𝑉 𝑒𝑙 calculation (equation 4), which remains very small compared to 𝑉 𝑛 , and in 𝜀 calculation where it appears squared. Thus ice shape evolution with 𝑅 𝑒𝑙 is similar to the one with 𝜀 but amplified. (e) The ratio of projected area 𝐾 has an influence on the calculation of the limit angle between quasi-normal impact and low angle impact. When 𝐾 varies from 0.5 to 1.5, 𝜃 𝑐 varies from 14° to 37°, which means that the majority of the impacts are occurring with a quasi-normal angle. The value of 𝐾 defines the area where the impact angle is considered as quasi-normal and also the value of the volume eroded in the case of low impact angle (equation 6). For the variation of 𝐾 considered here, there is no influence on ice shape since almost all impacts are 29/03/2018 considered as quasi-normal. In order to characterize 𝐾 influence, simulations with lower impact angle have to be performed.

The majority of these parameters have great influence on ice shape. In addition, their characterization is not an easy work regarding large number of phenomena and coupled dependencies in the accretion process which complicates the feasibility of experimentations as well as the understanding of their observation.

Further work

Liquid water dependency

Currie et al. [START_REF] Currie | Experimental Studies of Mixed-Phase Sticking Efficiency for Ice Crystal Accretion in Jet Engine[END_REF] experimental study shows that above a certain value, icing severity decreases with increasing ice crystal melting ratio (right section of the melting ratio plateau effect shown Figure 9). As a matter of fact the presence of liquid water is very essential for the accretion process in ice crystal condition. Inside jet engines, while none melt ice crystal would rebound to the surfaces, the liquid water of a partially melt ice crystal enable it to stick to the wall and begin the accretion process. However, the ice severity is dropping when the liquid ratio is too high. Too few liquid water on the wall prohibits the sticking of solid particles, while too few ice particles is not enough to cool the surface sufficiently and are then quickly washed away by the liquid water. Trontin et al. erosion empirical model takes this dependency into account [START_REF] Baumert | Experimental and numerical investigations on aircraft icing at mixed phase conditions[END_REF], which is not the case of the presented semi-empirical model. Thus, the model presented in this paper is not able to predict ice accretion with a high liquid ratio.

Particle size dependency

An important weakness of the semi-empirical model presented in this paper lies in the fact that it is not particle size dependent. This means that, the calculated erosion rate is the same if numerous small particle impacts or few large particle impacts as long as impingement mass flux rates are the same. This hypothesis is not realistic and should be reviewed since, based on experimental evidences, several authors agree on the particle size dependency of erosion rate. Regarding sand erosion studies, Oka et al. [START_REF] Petrovic | Mechanical properties of ice and snow[END_REF], Arabnejad et al. [START_REF] Arabnejad | Development of mechanistic erosion equation for solid particles[END_REF], and Parsi et al. [START_REF] Commission | HAIC[END_REF] show that erosion rate usually increase with particle size considering constant impingement rate. Knezevici et al. [START_REF] Oka | Practical estimation of erosion damage caused by solid particle impact -Part 1 : Effects of impact parameters on a predictive equation[END_REF] [19] noticed the same effect for ice crystal accretion. An essential step to improve the erosion model is to implement this particle size effect in the calculation and/or the characterization of the model parameters.

Summary/Conclusions

Erosion rate calculations for ice crystal accretion simulations are usually based on empirical modelling. This study is a first step to provide a more reliable model using the solid/solid collision theory to leverage a semi-empirical model. This theory is mainly built on Finnie et al. [START_REF] Finnie | On the velocity dependence of the erosion of ductile metals by solid particles at low angles of incidence[END_REF] and Bitter [START_REF] Bitter | A study of erosion phenomena -Part I[END_REF] [3] studies, which assume that two main mechanisms are responsible for the erosion caused by solid impacts: the deformation wear and the cutting wear. The model developed is able to predict ice crystal erosion rate with good agreement for NRC's low liquid water to total water content cases. It still has to be assessed using larger databases, especially to take into account the presence of the liquid water, as well as the particle size effect, which have been shown to have significant impact on erosion and accretion phenomena.

Figure 1 :

 1 Figure 1: Expected mechanism of deformation erosion -(a) growth of cone crack and median crack, (b) closure of median cracks and creaction of lateral cracks, (c) eroded crater formed -Adapted from Parsi et al. [13]

Figure 2 :

 2 Figure 2: Deformation erosion energy balance -The kinetic energy is divided into 𝑸 𝒆 (blue area), 𝑸 𝒑𝒆 (green area) and 𝑸 𝒑 (red area)

𝑑𝑥Figure 3 :

 3 Figure 3: Schematic of cutting erosion procedure -(a) before the impact, (b) crater formation and piling material, (c) separation of material from the surface -Adapted from Parsi et al. [13]

Figure 4 :

 4 Figure 4: Schematic of the projected surface 𝑨 𝒙 and 𝑨 𝒚

Figure 5 :

 5 Figure 5: Run 17 ice shape simulation (left) and experimental observation (right) [8] -Comparison between Trontin et al. HAIC empirical model, presented semi-empirical model (Sim), and experimental data ice height measurment (red dash)

Figure 6 :

 6 Figure 6: Erosion rate along the curvilinear abscissa of the studied profile (stagnation point at 𝒔 = 𝟎) -Deformation component (𝛈 𝐃𝐞𝐟 ) and Cutting component (𝛈 𝐂𝐮𝐭 )

Figure 7 :

 7 Figure 7: Run 233 ice shape simulation (left) and experimental observation (right) [8] -Comparison between Trontin et al. HAIC empirical model, presented semi-empirical model (Sim), and experimental data ice height measurement (red dash)

Figure 8 :

 8 Figure 8: Run 17 ice shape using different values of parameter 𝑪 (a), 𝑷 (b), 𝜺 (c), 𝑹 𝒆𝒍 (d), K (e) (a) The cutting efficiency 𝐶 has a high influence on ice shape.When its value increase one can easily notice the increase of eroded material, with a lower influence near the stagnation point where the tangential velocity is negligible. Secondly, increasing the volume eroded by cutting action also decreases the length of the iced surface. (b) The impact pressure 𝑃 has also a direct influence on the volume eroded by cutting action. Ice shape evolution with 𝑃 is as important as the one with 𝐶. (c) The deformation wear factor 𝜀 influences the volume eroded by deformation wear which is not as significant as the volume eroded by cutting wear. (d) The elastic limit 𝑅 𝑒𝑙 takes action in 𝑉 𝑒𝑙 calculation (equation 4), which remains very small compared to 𝑉 𝑛 , and in 𝜀 calculation where it appears squared. Thus ice shape evolution with 𝑅 𝑒𝑙 is similar to the one with 𝜀 but amplified. (e) The ratio of projected area 𝐾 has an influence on the calculation of the limit angle between quasi-normal impact and low angle impact. When 𝐾 varies from 0.5 to 1.5, 𝜃 𝑐 varies from 14° to 37°, which means that the majority of the impacts are occurring with a quasi-normal angle. The value of 𝐾 defines the area where the impact angle is considered as quasi-normal and also the value of the volume eroded in the case of low impact angle (equation 6). For the variation of 𝐾 considered here, there is no influence on ice shape since almost all impacts are

Figure 9 :

 9 Figure 9: Evolution of icing severity versus liquid water to total water content -Adapted from Currie et al. [8]

Table 1 : Expressions used for erosion rate calculation

 1

			Deformation	Cutting
	𝑉 𝑛 < 𝑉 𝑒𝑙		0	0
	𝑉 𝑛 > 𝑉 𝑒𝑙 and 𝜃 ≥ 𝜃 𝑐	1 2	𝑚(𝑉𝑠𝑖𝑛(𝜃) -𝑉 𝑒𝑙 ) 2 𝜀	𝑚𝐶𝑉 2 cos 2 (𝜃) 2𝑃
	𝑉 𝑛 > 𝑉 𝑒𝑙 and 𝜃 < 𝜃 𝑐	1 2	𝑚(𝑉𝑠𝑖𝑛(𝜃) -𝑉 𝑒𝑙 ) 2 𝜀	2𝑚𝐶𝑉 2 sin(𝜃) (𝐾𝑐𝑜𝑠(𝜃) -sin(𝜃)) 𝐾 2 𝑃

Table 2 : NRC experimental conditions for Run 17 and 233 - Mach number M, total water content TWC, total pressure Pt, mean volume diameter MVD, melting ratio Yl

 2

	Run M	TWC	Pt	MVD	Yl
	17	0.25	6 g/m 3	34.5 kPa 45 µm	6 %
	233	0.40	6 g/m 3	34.5 kPa 45 µm	8.6 %

Table 3 : Parameters value for the sensitivity study (wall temperature is 273.15 K)

 3

		50%	75%	100%	125%	150%
				(Ref)
	𝐶 0.15	0.23	0.3	0.38	0.45
	𝑃 7.35 MPa	11.0 MPa	14.7 MPa	18.4 MPa	22.1 MPa
	𝜀	92.3 MJ/m 3 139 MJ/m 3	185 MJ/m 3	231 MJ/m 3	277 MJ/m 3
	𝑅 𝑒𝑙 1.74 MPa	2.61 MPa	3.47 MPa	4.34 MPa	5.21 MPa
	𝐾 0.50	0.75	1.00	1.25	1.50
	Each simulation has the reference value of all parameters but the one
	studied which varies between 50% and 150% of the reference value.
	Values of 𝑅 𝑒𝑙 and 𝑃 are calculated for a temperature of 273.15 K
	which is the temperature of the accreted ice in Run 17 conditions
	(presence of liquid water).

Acknowledgement

This work is conducted in cooperation between ONERA and SAFRAN AIRCRAFT ENGINES (Safran Group). The authors gratefully acknowledge the ANRT (Association Nationale de la Recherche et de la Technologie) which supports part of the PhD.

Contact Information

Virgile Charton Phd student -Safran Aircraft Engines YXTA / Onera DMPE-MH virgile.charton@onera.fr