Sonia Lehman-Frisch

Jean-Yves Authier

Frederic Dufaux

"Draw me your neighbourhood": a gentrified Paris neighbourhood through its children's eyes

Keywords: city, gentrification, social mix, gender, drawings

This article aims to contribute to the international research on the relationship between the child and the city, based on drawings and interviews collected from 4th grade children in a socially mixed gentrified neighbourhood of Paris. Its ambition is to reflect on the methodological contribution of drawings to analyze the children"s representation of the neighbourhood. In doing so, the article also presents the principal facets of a gentrified Parisian district as perceived by the children themselves, by studying to what extent children's gender and diverse social backgrounds impact their neighbourhood representations and experiences.

Introduction

Children studies developed considerably in many northern countries starting in the 1970"s, and particularly since the 1990"s. The question of the relationship between children and their area, and their city especially, has been the subject of considerable research [START_REF] Lynch | Growing Up in Cities[END_REF][START_REF] Aitken | Putting Children in their Place[END_REF], McKendrick 1999[START_REF] Holloway | Children Geographies: Playing, Living, Learning[END_REF], Chawla , Christensen and O"Brien 2003). On the contrary, in French social sciences this question has rarely been touched on [START_REF] Danic | Enfants et jeunes dans les espaces du quotidien[END_REF]. Research regarding the urban experiences of non-adult individuals has focused primarily on the working-class youth from the estates of the banlieues [START_REF] Lepoutre | Coeur de banlieue[END_REF][START_REF] Beaud | 80% au bac… et après ? Les enfants de la démocratisation scolaire[END_REF][START_REF] Kokoreff | La Force des quartiers[END_REF]. And yet in the 1970"s, Marie-José Chombart de Lauwe and her team had carried out a very shrewd study on the relationship between children aged 6 to 14 and their neighbourhoods and towns, in an old district of Paris, in new town neighbourhoods, and in a remote rural village (Chombart de Lauwe 1977). But this research, pioneering as it was, did not inspire others in France. Only a few works, produced by psychologists, at times in conjunction with geographers, have more recently analysed the urban mobility of children, but from a perspective heavily centred on the question of the child"s autonomy and development (Legendre 2006, Enaux and Legendre 2006[START_REF] Depeau | Radioscopie des territoires de la mobilité des enfants en milieu urbain : Comparaison entre Paris intra-muros et banlieue parisienne. Enfances, Familles, générations[END_REF]. This article, based on a Paris case study, therefore aims to develop the contribution of the French social sciences to the international research on the relationship between the child and the city.

More specifically, it presents the first results of a research that aims to compare the urban experience of children (and of their parents) from three gentrified neighbourhoods of Paris, London and San Francisco. Gentrified neighbourhoods, which refer to the formerly working-class central districts transformed by the "return to the city" (Bidou 2003) of the middle and upper echelons of the population (Glass 1963[START_REF] Lees | Gentrification[END_REF], represent relatively rare examples of socially mixed areas within increasingly segregated central cities. As such, they make special platforms for observation, to study the relationshipssymbolic and realwhich children from different social backgrounds, cohabiting in the same area, have with the neighbourhood and the city in which they live.

The few researches which have studied the issue of children in the gentrification process and social diversity have exclusively focused on the point of view of the adult "gentrifiers" [START_REF] Karsten | Family Gentrifyers : Challenging the City as a Place Simultaneously to Build a Career and to Raise Children[END_REF][START_REF] Karsten | Housing as a Way of Life : Towards an Understanding of Middle-Class Families" Preferences for an Urban Residential Location[END_REF][START_REF] Desena | Gentrification and Inequality in Brooklyn : The New Kids on the Block[END_REF]. As the children are social actors in their own right, and as it is established that their perspectives on an area can greatly differ from those of adults [START_REF] Lynch | Growing Up in Cities[END_REF][START_REF] Valentine | Public Space and the Culture of Childhood[END_REF], it is essential to analyze the child"s point of view. How do these children from different social backgrounds perceive the neighbourhood that their parents" residential choices have led them to live in? What use do they make of these neighbourhoods and the city in which they are situated? More specifically, this article seeks to analyze the primary school children's representations of the gentrified Batignolles district in Paris. This is based on material composed primarily of drawings made by the children showing how they perceive their neighbourhood, and subsequent individual interviews with the subjects.

The drawings result from "visual methods". Let us not forget that these were firstand predominantlydeveloped by environmental psychology researchers, in the same vein as the "Place Perception Project" put in place by James Blaut and David Stea at Clark University in the 1970"s. In this perspective, maps, aerial photographs and sketch maps (produced by the children and analyzed according to the principles set out by Kevin Lynch in 1960) became special tools for analyzing the development of spatial awareness and the terms of spatial cognition amongst the children; they helped establish the variations in the environmental knowledge according to the children"s age and sex, while bringing into question the Piagetist hypothesis of the stages of development of the child [START_REF] Blaut | Studies of Geographical Learning[END_REF][START_REF] Hart | Children's Experience of place[END_REF][START_REF] Matthews | Cognitive Maps of Young Children : A comparison of Graphic and Iconic Techniques[END_REF][START_REF] Matthews | Young Children"s Representation of the Environment : A comparison of Techniques[END_REF][START_REF] Matthews | Making Sense of Place : Children's Understandings of Large-Scale Evironmen[END_REF]. These methods were criticized due to the strictness with which the researchers controlled the protocols, which left very little room for the children to express themselves spontaneously, leading to adult-centered interpretations of the spatial representations of the children [START_REF] Young | Adapting visual methods : action research with Kampala street children[END_REF].

With the emergence of the sociological approach, the research has given greater weight to the perspectives of the children of their living space, to the sense and the values they attribute to the places, and to the kinds of bonds they feel; at the same time it has brought to light the diversity of their experiences according to the sociospatial context in which they find themselves (Aitken 1994, Holloway and[START_REF] Holloway | Children Geographies: Playing, Living, Learning[END_REF]. This trend has been accompanied by a diversification in the methods used in research with children, aiming to give the children the most varied and appropriate means to give their own account of their interpretations of an area. Amongst the numerous methods used, the visual methods have pride of place, drawings, maps, photos, etc., often serving as a basis for individual and collective discussions [START_REF] Young | Adapting visual methods : action research with Kampala street children[END_REF][START_REF] Christensen | Children in the city : Home, Neighborhood and community[END_REF][START_REF] Béneker | Picturing the City : Young people"s representations of urban environments[END_REF]. This article"s ambition is twofold. It aims to reflect on the methodological contribution of visual methods, and in particular of the free recall sketch to analyze the children"s representation of the neighbourhood: it emphasizes its assets and its limits through the confrontation with other sources of information (interviews with the children and with their teacher). In doing so, the article presents the principal facets of the gentrified Parisian district of Les Batignolles as the children represent it (type of landscape and uses, the role of the physical perceptions, variations in the scale of the area) by studying to what extent the class and gender of the participating children can explain the similarities and differences in their representations of the neighbourhood.

Field and methods

The Batignolles neighbourhood, in Paris" 17th arrondissement, is situated between the working-class districts to the north and the middle-class districts to the west: it is widely considered an "intermediary" neighbourhood [START_REF] Bernard | Le XVIIème arrondissement de Paris : des cicatrices profondes, une mixité introuvable[END_REF], both in the geographic and the social sense. A working-class neighbourhood strongly anchored by the Cardinet goods station in the 1950"s, the Batignolles subsequently suffered several decades of demographic and socio-economic decline, before a revival, starting in the 1990"s, with growth led by the return to the city of the middle classes. Today it is a densely populated (39,940 inhabitants for 1.44 km2, or 27,740 inhabitants/km2 (INSEE, 2006 census)) and socially heterogeneous neighbourhood: professional and managerial workers make up 39.9% of households, side by side notably with manual workers" households (6.6%), clerical employees (10.6%) and the retired (15.7%), who are for the most part former manual workers or clerical employees themselves. This social diversity is related to the structure of the habitat: in this neighbourhood of modest buildings of five or six storeys in Paris stone, small apartments are still predominant; furnished rentals and social housing also allow accommodation for the less wealthy households. Finally, the structure of incomes gives a good illustration of the social disparity of the neighbourhood"s inhabitants, the incomes of the households in the first decile (approximately €90,300) being ten times greater than those of the last decile (€9,100).

As a gentrified neighbourhood whose inhabitants will happily talk up its "village" character, it has also become (again) a family neighbourhood. After a historic low of 10.7% in 1975, the proportion of children under the age of 15 increased in the 1990"s, in correlation with the gentrification process. Today, they make up for 12.5% of the population of the Batignolles neighbourhood (INSEE, 2006 census). Moreover, shopkeepers have readily recognised the growing importance of children in the neighbourhood. On Saturday or Sunday mornings, the narrow pavements of the rue des Moines or the alleys of the covered Market can be extremely busy, quickly congested with prams and the patter of tiny feet. In fine weather the density of families and children in the neighbourhood"s two parks is extraordinary.

Conversely, during the school holidays, the neighbourhood empties and the weekends seem very dreary. And yet the statistics show that only 22.7% of the households are families with children, a relatively small proportion compared to the 17th arrondissement as a whole and to all of Paris. In other words, although the proportion of families with children is relatively low compared to the rest of the district and the city, they are remarkably visible in the day-to-day life of the neighbourhood.

In this neighbourhood, the children whose graphic representations are analyzed in this article all belong to a CM1 class (Year 5) of a local state primary school of average size (10 classes from CP to CM2 -the equivalent to Year 2 to Year 6 classes in the British school system). Since in France attendance of elementary state schools depends on catchment areas, these schools are, by definition, neighbourhood schools i . Our hypothesis was that, at that age, children (who were aged between 9 or 10 at the time of the study) possess a certain knowledge of their environment, due in particular to relatively independent practices, as well as a basic level of drawing and discursive ability. This age group is also interesting as it precedes the great upheaval of admission to secondary schools. The 27 children in the class, with almost as many girls as boys, are a reflection of the socio-cultural diversity of the rest of the school (table 1) and of the neighbourhood: 11 amongst them are from middle class families (their parents, for example, work as lawyers, sound engineers, journalists, teachers or biologists), whereas 16 other children are from working class families (their parents work in the building trade or the clothing industry or they are caretakers, cleaners, waiters, retired or unemployed). As in the school as a whole (table 2), most of the parents were born in France, but a significant minority come from other European countries, northern Africa, sub-Saharan Africa, and Asia (with several mixed couples). In short, it is well and truly a socio-culturally mixed class, in line with the rest of the school and the neighbourhood. Note also that almost all the children interviewed have been in this same school since the beginning of elementary school, and have lived in the neighbourhood for as long as they can remember (except four children): in other words, they represent a great residential stability.

Table 1 Table 2

Having obtained the agreement of the teacher and the parents in the selected class, we organized a "neighbourhood ii workshop" in May 2009. Having presented to the class our project of "better understanding the neighbourhood and how people (adults and children) live in their neighbourhood today", we distributed sheets of A4 paper with the following instruction: "draw your neighbourhood". This wording, deliberately open, aimed to leave each child the choice of the type and content of his graphic representation. One child asked if it meant making a map or drawing a picture, and we replied to the whole class that it was their choice. Later, we interviewed each child for approximately fifteen minutes, in the school library. We started by asking each child to comment on his/her drawing. The discussion then consisted of displaying a set of seven photos of the neighbourhood (the nearby private school, the park, the local library, the shopping street, the market, the metro station, and an urban square deliberately some distance from the previous places), each one

corresponding to an open questioning on the recognition, the uses, the sociability, the attachment and the mobility issues related to the specific place.

The drawings produced by the children were analyzed according to the method described in a recent article [START_REF] Béneker | Picturing the City : Young people"s representations of urban environments[END_REF]. The first stage consisted of observing the drawings as a whole, to identify the main themes and the most revelatory examples. Then each drawing was analyzed in detail: this consisted, for each theme identified, of studying the precise way in which it was dealt with (or not dealt with) graphically. The individual discussion therefore allowed us to observe the concordance or the discrepancy between the themes of the drawing and those of the interview. Finally, the child"s representations (graphic and discursive) were linked to other information obtained from the teacher or possibly an interviewed parent.

How to represent one's neighbourhood?

On first inspection, the great disparity between the 27 neighbourhood drawings is striking, despite them being drawn by children of a similar age living in the same urban area, and the question arises of the very possibility of using them to analyze the children's neighbourhood, which was one of the motivations of the exercise.

"Detailed" drawings, "sketchy" drawings

In particular, certain graphic offerings, which we can qualify as "sketchy", seem to offer much less material for analysis (careless outline of the drawing, weakness of detail and absence of place names) than others, which, conversely, we can qualify as "detailed". Needless to say, it is not this paper's intention to give a value judgement on the quality of the drawings per se, but we argue that the drawings' contents, the number and the nature of their components, are very interesting to analyse. Thus, during the discussions that followed, several children said that thought their work was incomplete. Constantin iii , for example, showed the (limited) area between the school and his home. The drawing, lacking in detail, simply shows several shapes which act as buildings. Commenting his drawing, he immediately acknowledges its incomplete nature: "So I forgot to put loads of things here because I couldn"t remember". And he proceeds to orally and precisely describes all these places which hadn"t come to his mind at the time of the drawing ("the cafe", "an apartment", "the school", a "motorbike shop", "the park", "the sports field", "the tanning shop"...). The exercise of the free recall mental map, leads to the question of one"s competence to mentally visualise at a given time all the elements one sees as significant, but in their absence.

And yet the individuals have a more or less acute visual memory, not taking into account fatigue, distraction or a simple lack of motivation that might affect their capacity to remember.

Antonia, for her part, produced a rough and careless map featuring a small segment between the school and her home. She also spontaneously admits her dissatisfaction. But for her, if her drawing doesn"t represent what she wished, it is due to a lack of space. Had she had more, she adds, "[she] would have added loads of things…". This brings into play another competence that the children can be more or less able to convey, that of the scalar transposition of the mentally conceived space onto the space of the piece of paper given to them. Another problem is pointed out by Léa: "Actually I didn"t really know what to draw. There should be a restaurant next my house". But the complexity of this object, which she would have liked to represent, made her abandon the idea in her drawing. She preferred instead to represent an imaginary element ("a smaller house"), but which she felt more capable of drawing, and which incidentally is the archetypal house in a child"s drawing.

To graphically represent a space therefore requires calling on several competences (memory and imagination, mastering the scale and the graphic instrument); it is up to each child to decide which trait is more or less important, with varying levels of mastery of these abilities. The drawing produced by the child arises from considerable filtering and does not in itself approach the fullness of the relationship between the children and their neighbourhood. The individual discussions are, therefore, a precious further source: they allow the children both to look back on their graphic productions with a reflexive eye, as well as giving them an opportunity to express their thoughts on the neighbourhood through another medium. They thereby updated the extent of the discrepancies between the graphic and oral representations of the children interviewed about their neighbourhood: certain children produced "sketchy drawings", contrasting with the richness of the discussion, during which they demonstrated fine knowledge, strong practices, and a strong sense of attachment to the neighbourhood, which one would scarcely have suspected from their drawings. Conversely, other children produced "detailed drawings" with specifics about the places, the landscapes, the activities and practices, but were unable to express orally their representation of the neighbourhood, despite the aid of photos and their own drawings.

We know that representations of space, graphic or discursive, are influenced by the medium offered for expressing them. It is also strongly impacted by the experience and the values of the individual as well as his social standing [START_REF] Gould | Mental Maps[END_REF]White 1974, Ramadier and[START_REF] Ramadier | Social Legibility, the cognitive map and urban behaviour[END_REF]. The drawings and discussions cannot easily be interpreted in the absence of data outlining social (parents" profession), family (family status, presence of and relation to siblings) and academic (individual performance) characteristics. For example, Marc"s drawing, which indicates the location of nine of his classmates" homes, might lead us to believe that, for him, the neighbourhood consists of the "spatialisation" of his social relationships developed at school. The discussion with his teacher, revealing that the young boy had "big relational problems" and did not have any friends within the school, leads us to reconsider this interpretation. To finalize, on the condition that the context of the production is taken in account, and that the children are given other means to express themselves, the drawings remain a very rich source for understanding the representations of the neighbourhood, and the diversity of the drawings and their discrepancies with the discussions become full of meaning in themselves.

Map or drawing: a question of intellectual maturity?

The instructions left the children to represent their neighbourhood in the graphic form of their choice. A large majority opted for the drawing, only a couple for the map, and often for a concept bringing together certain elements of a drawing and a map. We can ask ourselves if these forms of representations correspond to a gradient in the knowledge of the neighbourhood, and from there, in the spatial aptitude of the children. This question has been the subject of famous debates: certain psychologists, following Piaget, wanted to demonstrate that the cognitive awareness of a space is exhibited in qualitatively distinct stages according to the age of the child, whereas others have argued that the representation of a space is built in a linear and incremental way as the development of the child progresses [START_REF] Aitken | Putting Children in their Place[END_REF]. At the start of the 1980"s, a geographer seized on this debate and sought to evaluate the impact of a child"s age on its spatial awareness and on the way in which it represented it: he put forward that three types of drawings should be distinguished: "pictorial maps" (or drawings), "pictorial plans" and "plans", which seem to correspond to three different stages of cartographic ability in the child. Indeed, the first type of graphic production demonstrates a certain egoism and self-absorption, the second implicates a certain level of precision despite the persistence of inaccuracies when compared to the reality, whereas the third requires a strong understanding of the relative geographic positions of the elements represented [START_REF] Matthews | Cognitive Maps of Young Children : A comparison of Graphic and Iconic Techniques[END_REF]. Other works have shown in addition that younger children sometimes use more abstract means of representation, while conversely adults tend to represent their environment in a more egocentric way.

Amongst the children of our sample, Marc is one of the only ones to have chosen to represent the neighbourhood in the form of a "plan" (figure 1), and it is undeniably in this representation that the scale, the exactitude and the precision is the most advanced. The space represented covers 11 blocks; it is named (it is for him "the Batignolles neighbourhood"), as is each of the roads within it. The important places in his day-to-day life are precisely located: the two parks, the school (including a plan of the playground, admittedly on an excessive scale compared to the reality), the sports field (also with a plan of the field), as well as the location of his home ("me") and that of his classmates" homes. During the discussion, he justifies his choice of representation: "I thought to myself it"s good to do a map like that, I know it"s to know a bit more of the neighbourhood, so we know, for example we can say there"s the street, like what"s happening and stuff". He therefore sees in the plan the most appropriate means to sum up (and transmit) his knowledge of the neighbourhood. His representation of the neighbourhood points to a very astute understanding of the organization of the space.

Figure 1

At the other end of the scale, Lilian iv (figure 2) chose to represent his neighbourhood in the form of a "pictorial map". This creates an aesthetically pleasing composition: every element (the buildings, a street, a park, people) is meticulously coloured in crayon and harmoniously laid out on the page. However, the graphic design is relatively simplistic, hardly seeking to reflect the complexity of the reality, and, above all, the relative geographic positioning is incoherent. The discussion with Lilian confirms his total lack of sense of space: he struggles to recognize the places presented to him in the photos, and hardly names any of them. What is more, on several occasions he refers to his neighbourhood as Paris.

Figure 2

Prima facie, these two cases seem to confirm a progressive conception of the type of representation adopted by each child to portray its environment: the drawing is characteristic of cognitive abilities still in their infancy, whereas the map points to a more sophisticated conception of space and would be associated with a later phase of intellectual development. And yet several elements suggest that this interpretation be qualified. Firstly, it should not be forgotten that both children are the same age: 10 at the time of the discussion. Secondly, we would expect the academic performance of the children to reflect their spatial awareness: on the contrary, the teacher reveals that Marc is an average student, and Lilian is more successful in school. Whereas Marc"s parents are biologists and chemists, Lilian"s have modest jobs in the building and caretaking trades respectively. Can the children"s social background explain the differences in their spatial awareness? A hypothesis could be presented whereby middle class children are culturally more exposed to cartographic representations than those of the working class. But this is difficult to determine with certainty from our sample, which is quantitatively limited and offers contradictory examples. Let us note also that girls and boys chose one or other of the modes of representation in equal numbers. Finally, the choice of the "plan" over the "pictorial map" is perhaps less a question of cognitive maturity or individual or social competence. The drawing, more than the plan, and beyond the question of spatial organization, enables the landscape dimension of the area to be highlighted.

Representing landscapes... inhabited landscapes

In their entirety, the drawings are in fact a more appropriate form than the maps to appreciate the perception of the urban landscapes.

The urban environment, from the barren to the fertile

Firstly there emerges from the children"s drawings a clear impression of density: often the environment is represented as five or six storey buildings, attached and relatively narrow, with almost inexistent pavements. This is how almost all of the nine children who live close to the avenue de Clichy (or who use it to go to school), as well as a few other children who live strictly within the Batignolles neighbourhood, chose to represent their neighbourhood (six in total). Sophie"s drawing illustrates it well (figure 3): for her, the neighbourhood consists visually of a succession of buildings lined up along the avenue de Clichy. And by drawing a long segment on two levels she accentuates the feeling of density, as the majority of the page is taken up with buildings. Sometimes the height of buildings is exaggerated compared to the reality, reminding us that the small size of the children accentuates for them the effect of verticality. Conversely, it can be that only the ground floors are represented, which may also be linked to the children"s specific field of vision, limited vertically by their height and horizontally by the buildings on one side and the continuous line of parked cars along the pavement on the other.

Figure 3

Beyond the density, the children rarely sought to represent the specificities of the Parisian architecture. They certainly drew apartment buildings rather than detached houses, but the buildings are often represented through the classic symbols used by children to portray houses: a large square for the main part of the building, a triangle for a roof, smaller squares as windows, as well as a rectangle to signify the front door. Olivia (figure 4) is the only one to have paid attention to the architectural details of the façade of her building, notably drawing blue double doors, rounded at the top, as well as several wrought iron balconies. It just so happens that her mother is an architect: we can assume that she raised her daughter"s awareness to the urban landscape, which the majority of the other children do not clearly identify.

Figure 4

If Sophie and Olivia"s drawings give an entirely barren image of the neighbourhood, others leave a place for nature also. The local park, which nine children chose to represent, appear to be the archetypal places of nature within the city. Thus Victoria, who had come from one of France"s overseas territories at the start of the academic year, dedicates no less than a quarter of the page of her pictorial plan to the Martin Luther King park: the rest of the neighbourhood is merely sketched in pencil, whereas the visual importance of the park is accentuated by meticulous colouring with a felt-tip pen, where the green (trees, more plentiful than in reality) and the blue (of the small stream, oversized in the drawing) are predominant. The discussion confirms her sensitivity to nature, which in her case involves a rejection of the neighbourhood, of Paris, which she blames for being "polluted", and a painful nostalgia for her original home: the park seems to be one of the only places where "she feels better". However, sensitivity to nature does not always reflect a rejection of the city. For many children, the park is an important vehicle for their attachment to the neighbourhood: some (mainly girls, as well as a few boys) give it pride of place in their drawings. Others only reflect on it, but at length, during the discussion. They mention the park for its natural qualities (mainly the girls), others for its playful qualities (mainly the boys).

A lively area

The neighbourhood is not for these children a lifeless area, be it barren or fertile. The drawings manifest particular attention to the uses of the area, aside from the park. This is the case in Olivier"s pictorial plan (figure 5), where shops act equally as spatial landmarks within the represented area, identified by their function ("tobacconist"s", "bakery") or by their name (the "Bloc" restaurant). Often the children do not merely settle for showing the location of the shops but also draw their interiors. Olivier notably sketches the clients in the tobacconist"s, leaning on the counter or sitting at a table, as well as the waiter. This sensitivity towards the shopping activities of the neighbourhood, which shows in thirteen drawings, also emerges in a number of the discussions. Whether or not they like shopping, eleven children appreciate the presence of shops within the neighbourhood. Kamel, who is used to doing a certain amount of shopping alone, sees them as practical due to their proximity ("We don"t need to go very far"). Beyond the simply functional aspect of the shops, some children evoke their emotional link to the products sold in these shops, such as Antonia, of Portuguese origin, who speaks with enthusiasm of the market, where she can find Portuguese products.

Figure 5

The school appears on Olivier"s drawing, as it does on that of eleven of his classmates (who have no specific features in terms of gender, social background and the location of their home in the neighbourhood). This is inevitably linked in part to the fact that the drawings were done within its four walls; but it also demonstrates the unquestionable importance of this place in their everyday life. Gaelle thereby draws the school disproportionately, making it occupy the entirety of the block of houses -it only really covers an eighth of itwhereas Mathieu and Marc carefully map the playground. This logically results from the fact that the school takes up a large part of their daily time. It has therefore become the hub of their social relationships. Asked about their friends, the children make a clear distinction between "school friends" (which turn out most of the time to be classmates) and friends from elsewhere, (those from "the building", "the park", "dance class", or "handball"). They clearly categorize them according to the place and activity (and therefore the frequency of their meetings) which lead them to meet, and give their classmates almost systematically the emotional priority. These friendships are maintained essentially during school hours and extracurricular activities, but they sometimes extend to other places: the park (for spontaneous meetings), the household (for personal invitations) or birthday parties. In short, the drawings and discussions highlight the fundamental role played by the school in the children"s neighbourhood.

Figure 6

The landscapes and places represented are often inhabited: fifteen drawings include at least one person, be it a simple representation or a more elaborate one. In his pictorial plan, Salih (figure 6) shows thirteen people going about their varied businesses: a mother pushes the pram of her child, shopkeepers serve their clients, and children walk their dog, play with a ball or skateboard. On the street he draws a driver, a cyclist, and some young people rollerblading. He demonstrates well how different places of the neighbourhood (shops, parks, the school) give rise to intense pedestrian activity, even if he particularly insists on the playful nature of the space.

The discussions confirm that the neighbourhood, for these children, truly is a place of social interactions. Several of them explain that they appreciate it because they "often meet friends", or even family. Most of the time, however, children encounter strangers: these "people of the neighbourhood" are judged as "sometimes nice", and "sometimes not very nice", or "not very kind". One girl tells in particular of a "fight" that happened between children in the park: the episode developed within her a feeling of worry but without affecting her frequenting the area, where she still likes to go with her older sister and her friends. These tensions are also mentioned by Marc: his mother does not allow him to go to the park alone due to the presence of teenagers she describes as "scum": "there"s a football pitch and I play there and sometimes there are bigger boys who kick us out". The Martin Luther King Park, situated on the border of the Batignolles and more working-class Epinettes neighbourhoods, is a space which often reveals tensions linked to age groups and social backgrounds [START_REF] Karsten | Growing Up in Amsterdam : Differentiation and Segregation in Children"s Daily Lives[END_REF]. Despite a few counter-examples, on the whole the children are serene about their cohabitation with the locals and other users of the neighbourhood, be they familiar or strangers.

An embodied perspective

Through their drawings, the children have portrayed a more or less aesthetic image of their neighbourhood. The discussions confirm that the visual sensitivity -which is the basis for their aesthetic judgment in some cases -, and more largely the embodied approach, is an integral part of their relationship with the neighbourhood. Gaelle explains, for example, that she likes her neighbourhood because she finds it "pretty", before adding: "well, there are some things which are pretty, but it"s not that nice".

Certain places attract the attention of the children in particular: the park, the library, the market ("there are loads of lights, and it"s nice, and it makes you want to eat").

Léa admits that she"s very attached to the view she has from her apartment, especially "when the weather"s good". Some children assimilate their aesthetic appreciation of the neighbourhood with the whole city: "Paris is pretty", Antonia considers, whereas Martin calls on his experience of other urban areas (he knows the south of France for having spent his holidays there; he has also been to Eastern Europe for family reasons): "I love this neighbourhood and I love Paris too. (…) It"s the nicest city I"ve seen so far".

Of course, the graphic method emphasizes the visual dimension. But orally, the children also explained the way in which their other senses allow them to comprehend their neighbourhood. Several of them therefore brought up the sound dimension: in general, the neighbourhood is known for its "calm", a term whose connotations are clearly always positive. But Samantha clearly distinguishes between different areas of the neighbourhood according to the noise intensity: she compares the "calm" areas (the library, rue Brochant), which she appreciates, she who dreams of living in the countryside, with the "noisy" places (the park, place Clichy, the market). Olivia, for her part, mentions the noise variations of certain places over time: she appreciates the park when it is calm, but dislikes it when there is "lots of noise" and "lots of people". In other words, the notion of calm is ambiguous: it refers also to a need for tranquility, for separation from the density of the urban bustle. Finally, other children mention their sensitivity to odours. Lilian, trying to describe what he dislikes about the neighbourhood, ends with: "for example, when it smells on the ground". In short, the children have an undeniably sensitive approach to the space: "a large part of their knowledge of the neighbourhood is physical and is thus embedded in their bodies. (…) The children are present in the neighbourhood and they have the neighbourhood "under their skin"" (Rasmussen and Smidt 2003, p.97).

In predominantly choosing the drawing over the map, it may be that the children showed that the neighbourhood is for them a space that is more perceived than conceived, and it would be remiss to interpret the small number of maps as merely pointing to a lack of cartographic skills. In this perspective, the drawing is a more appropriate tool than the map to account for their physical relationship with their neighbourhood.

Territories of different scales

The examination of the graphic productions helps us appreciate the variations in the expanses of the represented spaces. These variations are due in part to the choice of mode of representation: the drawing favours the large scale, and the plan (pictorial or not) the smaller scale. But this is not always the case and other factors account for the varying scope of the territories in the children"s everyday lives.

From the home... The great majority of the children"s graphic productions is linked to the home; it either appears directly in fifteen drawings or a perspective is presented as seen from the home in three more drawings. Unsurprisingly, the home appears as a stronghold of the neighbourhood, qualified by Golledge as a "primary node" or an "anchor point" (1985 cited Aitken 1994 p.61). Three children assimilated the representation of their neighbourhood with that of their building, either by restricting themselves to its façade, or by enlarging its perception from neighbouring buildings.

Figure 7

Thereby Shelihane"s very rough drawing (figure 7) (she uses symbolism which one would expect from a younger child), represents her building, with a tree and two people. It suggests that for her the neighbourhood can be reduced to the exterior façade of her building: it is constituted by what she sees of her "house" when she is "outside". And indeed the child, who clearly struggles to express herself in the presence of the interviewer, leads us to presume a certain weakness in her practices in the neighbourhood. Of North African origin, she lives alone with her mother, who works as a cleaner, in a household which she qualifies as "too small" a studio. She struggles to identify the places presented by the photos, and it emerges that she scarcely frequents places in the neighbourhood other than the school and sometimes the library, accompanied by her mother without fail. Most of the time, due to the cramped nature of her home, she plays in the courtyard of her buildingwhich is the title she gives to her drawingwith her neighbour or her cousins. For her, therefore, the neighbourhood is firmly anchored around the home. Shelihane"s drawing and discussion, despite an apparent lack of quality, reveal a relationship with the neighbourhood in itself very limited and based around the home.

... to the neigbhourhood

The (pictorial) plan form often allowed the other children to represent larger spaces (up to eleven blocks for Marc) than those who chose to do a drawing or pictorial map.

We had expected that the graphic productions would give some indication of the expanse of the children's territory. But it turned out that all of them actually demonstrate a glaring discrepancy with the discussions on this point, the latter leading to presume spatial practices far more spread out than in the drawings. In addition, much more conclusively than the drawings, the interviews illustrated the contrasting scales of the children's territory, by highlighting the differing intensity of uses, the variety of the types of uses, and the diverse level of parental supervision that the children experience in the Batignolles neighbourhood v . The question therefore arises of the factors that might explain theses variations in territory.

The research has put forward several elements to explain this differentiation.

Firstly, it has been established that the girls have less spatial freedom, and, accordingly, less knowledge of the neighbourhood, than the boys: the scope of the "home range" is directly linked to the educational principles of the parents: they require that the girls be more involved in household tasks, which limits their leisure time and space; what is more, considering them to be more vulnerable to sexual abuse, the parents do not allow them to wander as far as the boys [START_REF] Ward | The child in the city[END_REF][START_REF] Hart | Children's Experience of place[END_REF][START_REF] Matthews | Gender, Home Range and Environmental Cognition[END_REF]. More recent studies however tended to show that this trend may be changing in part because the parents" representations are becoming more complex [START_REF] Valentine | Public Space and the Culture of Childhood[END_REF]). While the drawings in our study do not allow to differentiate the expanse of territory of boys and girls, the interviews do confirm that the girls have a territory somewhat more restricted than boys, that their uses of the neighbourhood are less varied and more strictly supervised by adults.

Irrespective of gender, other researchers have shown that working class children benefit from a greater freedom within the neighbourhood, whereas middle class children live instead in an urban environment made up of disconnected places, and are constantly being ferried from one to the other by their parents [START_REF] Larreau | Unequal Childhoods : Class, Race, and Family Life[END_REF][START_REF] Zeiher | Shaping Daily Life in Urban Environments[END_REF]. It is true that, in this class, several children from the more modest families are free to roam the neighbourhood with a greater degree of autonomy, as the interviews reveal it: they go to the park without their parents to meet with friends, are regularly sent to do small shopping errands, and have been getting to school on their own for some time. This independence is relative, however, as, according to the teacher, none of them "hangs out" in the neighbourhood: they only spend a small portion of their out-of-school time there. But this social interpretation of the freedom of children in the public urban space is more subtle: on the one hand, other children of parents from lower income families admit to far more limited practicesin intensity and autonomyand to a very rough knowledge of the neighbourhood; on the other hand, a few middle class children enjoy considerable spatial autonomy.

The greater freedom within the neighbourhood enjoyed by these working-class children appears to go hand in hand with less involvement in out-of-school activities: only 6 out of 16 working class children have such activities. Lilian, son of a caretaker and a worksite foreman, when asked about his activities away from the classroom, answered revealingly: "well I run a lot, I play a lot of football, and I go to the country a lot". He doesn"t, therefore, spontaneously associate the notion of "activities" with those that are organized and supervised by adults. Middle class children, for their part, are undoubtedly involved in more sporting and cultural activities within or outside their neighborhood (9 out of 11 children). But, unlike what has been shown in other contexts, this does not weaken their link to the neighbourhood, and they all demonstrate a clear fondness and a great knowledge of it: they recognize most of the areas shown in the photos, spontaneously naming and positioning them in the space, and giving the street name. On the contrary, we can infer that their multiple activities help to anchor and broaden their territory, as most of these activities happen in the neighbourhood or in close proximity, and because they often go on foot, even if accompanied. Let us note also that the middle classes do not have a monopoly over extra-curricular activities: several children from the more modest families also take part in sporting activities. The "Ateliers Bleus" and leisure centres, sporting and educational activities financed by the City Council and often based at the school, only account for part of these children"s activities. For them as for their middle class classmates, the mobility involved in these activities deepens their relationship with the neighbourhood.

The children in our study have therefore built very contrasting local territories.

Admittedly the size of our sample does not allow us to make clear-cut conclusions or to establish a social categorization of the relationship the children have with their neighbourhood. If the analysis reveals contrasted spaces, which partly result from class and gender differences, the causal link is far from being systematic. In fact, we can put forward a hypothesis whereby the socio-cultural diversity of the neighbourhood, the school, and the class in question, strongly contribute to blurring the dividing lines classically established according to sex, social background or the children"s culture. Alissa provides a perfect example. Her drawing, which represents her building and the neighbouring buildings in the street, may lead one to believe initially that her day-to-day life revolved around her home. Not at all. Alissa, who is 10, lives with her older brother and her mother, a French woman who looks after children for a living, the father having returned to live in his country of origin in North Africa. She enjoys great freedom in the neighbourhood: she goes to school alone and she likes to meet up with her friend in the park or at the library. Alissa appreciates this freedom, organized around sporting activities she really enjoys. This strong sense of independence, undoubtedly encouraged by her father"s absence, upsets the classic girl/boy divide in the usage of the neighbourhood, with external and autonomous practices one might expect from boys. She frequents and knows her close neighbourhood as well as a wider territory, right up to the close suburbs -where her activities take place (figure skating, horse riding) -that she often reaches alone, by public transport. Her activities associate her more with middle class children, but, unlike them, she navigates this vast territory alone for the most part.

Two houses: which neighbourhood?

If the home is the fundamental centre of the child"s neighbourhood, what happens when, for family reasons, the child alternates between two houses? In this class, the majority of the children lives in one household, either because their parents live together (fourteen children), or because they live there with their mother and do not (or rarely) see their father any more (seven children, essentially from working class families). But five children (all from middle class backgrounds) have separated parents who continue to be involved in their everyday lives: they alternate, therefore, at varying rhythms, between the father"s and the mother"s house. This special residential situation, relatively common in today"s society, is not without effect on these children"s relationship with their neighbourhood.

The first question facing these children at the time of the drawing was that of the choice of home and/or neighbourhood to represent. Mathias thereby drew the home of his father, who lives in nearby Epinettes neighbourhood, even though his mother lives closer to the school and within the neighbourhood of the study: "I couldn"t do both, I had to choose. I chose my dad"s". It did not occur to him to represent his neighbourhood on a more reduced scale which would have allowed him to include both homes: his drawing features only his father"s house and its "little garden", more insistent on the exceptional nature of this kind of home in a dense urban environment than its place within the neighbourhood. He does, however, explain that both his houses count for him as part of his neighbourhood. And the teacher explains that she often sees him travelling between the two homes of his parents: he therefore enjoys a certain level of spatial autonomy earlier than some of his classmates. The fact that he has two homes, close but distinct, by adding a "primary node" around which the child"s territory revolves, results in an enlargement of his daily space. This enlargement is even clearer in Olivier"s case. One of his parents lives in the Batignolles, the other in the wider suburbs. When the two parental homes are too far apart, the child must choose one of them for his drawing. The choice of his mother"s home, where he lives during the week, comes naturally to him: "It"s (my neighbourhood) more here because (…) I"m only there every other weekend and I live here and school"s here and (…) all my friends are here". In other words, the neighbourhood is for him the territory encompassing not only the home, but also the school and his friends. Unlike Mathias, he does not demonstrate independence when travelling within the neighbourhood, and shows no great desire to do so: he goes to school alone but prefers it when his sister takes him. Away from his organized activities, he frequents few places in the neighbourhood, his practices seemingly based around his homes. On the other hand, due to his alternation between parental homes, he has a clear conception of the scale of his territory, from the Batignolles neighbourhood, to Paris, to the wider suburbs. For him the scales are associated with different practices (school and extra-curricular activities, leisure outings in Paris and visits to his father), different means of travelling (on foot, by bus, by metro or by car) and different timeframes (the week, the weekend). Children of separated parents thereby shape an enlarged, or even multi-scalar, daily territory.

Conclusion

Putting the drawings of the neighbourhood in perspective with the discussions conducted with the children reveals the richness of the graphic method for analyzing their relationship with their neighbourhood. Unquestionably, these drawings, which are the result of a complex filtering process (as repeated research has shown in recent decades), are heterogeneous, even though they were produced by children of the same age living in the same urban environment. Some are "detailed", that is to say dense and meticulous, whereas others are "sketchy" and offer few elements which enable the relationship with the neighbourhood to be characterized; some are plans, others are pictorial maps, and others even belong to an intermediary category (pictorial plans). This diversity, which can initially seem disconcerting, is worth being interpreted in the light of outside information. Certain "sketchy" drawings thereby appear to represent a true account of the relationship with the neighbourhood for certain children. In addition, the choice of form of representation is not insignificant and must not be seen as a reflection of the level of spatial awareness: on the contrary, it can be that it signifies different projects and contrasting relationships with the neighbourhood. The plan, for example, facilitates a clearer expression of the knowledge and mastery of the organization of the space for a certain child, the neighbourhood for him being conceived above all as a body of places and spatially structured elements. The drawing, in contrast, is more adapted to those children for whom it is the best means of representing the landscape and the atmosphere: for them, the neighbourhood is above all to be lived and felt. This diversity in forms of graphic production is in part a reflection of the diversity in the relationships between the children of the Batignolles CM1 class (Year 5) and the neighbourhood. Where some might insist on the properly urban dimensions of the neighbourhood (density, architectural characteristics), others concentrate on the natural elements (parks, vegetation and animals). Several children associate the notion of neighbourhood not with the landscapes but with the activities and people who liven it up on a daily basis. But the drawings do not demonstrate the variable scope of the neighbourhood according to the children. Only the interviews allow to assess the contrasting expanse of their territory, from the apartment building to an area of several blocks. These differentiations can sometimes be explained along the lines of class, gender, family situation, or the children"s personal history. However, our study reveals that, in this socially diverse gentrified neighbourhood, these traditional lines are sometimes blurred.

It remains to highlight certain common points emanating from the discussions in the experiences that these children have in their neighbourhood, which are not always highlighted in the drawings: on the whole, the neighbourhood is for them a territory based primarily around the home, the school and the park. i It must be pointed out, in addition, that the research studied the families from a local private school in order to gauge the extent of possible avoidance of the state school system by certain families, and the variations this may lead to in their practices and their representations of the neighbourhood.

ii "Quartier" is a term commonly used in France, but it does not carry any single definition. It refers to a concept differentially appropriated by a variety of disciplines within the social sciences; it also refers to a category constructed and used by politics for sociospatial action; it finally refers to the everyday experiences and representations of the people who live in it and use it (Authier, Bacqué, Guérin 2006). In other words, the institutionalized neighbourhood and the experienced neighbourhood do not strictly overlap.

  They are deeply fond of them and their fondness rests upon a functional, social and physical judgement. In other words, the neighbourhood is made up of what they see, of what they do, what they feel, and the people they meet. And it is this combination of different methods, visual and oral, which enables us to comprehend the complexity of the child"s relationship with the urban space.

Table 1 .

 1 A socially diverse school: division of children according to family quotient. Table2. One school, 21 nationalities: division of children according to nationality. Figure 1. Marc's neighbourhood.Figure 2: Lilian's neighbourhood.Figure 3: Sophie's neighbourhood: a dense neighbourhood.Figure 4: Olivia's neighbourhood: landscape and architecture.Figure 5: Olivier's neighbourhood: a shopping district.Figure 6: Salih's neighbourhood: a lively area.Figure 7: Shelihane's neighbourhood: the courtyard of her building.

iii All the names have been changed for anonymity purposes. iv Lilian, with no "e" at the end, is a boy's name in French. v We are in the process of developing these aspects in greater detail in a forthcoming paper.


A few steps towards an emancipation of the spectators of the French banlieues

How can one understand the recurring suburban riots in France over the past three decades and the scale and intensity of the November 2005 riots, in spite of the substantial urban policies adopted by the state since the early 1980s? Mustafa Dikeç, in Badlands of the Republic, provides

some insight helping us to understand this disturbing French paradox. He relates the history of the spatial construction of France"s urban policy between the early 1980s and the middle of the first decade following the year 2000. At the beginning, this policy conceived the banlieues (suburban areas) as "threatened" territories that the state needed to help by implementing temporary policies and promoting local democracy. Gradually the Politique de la ville (urban policies, subsequently PV) changed and was linked with fear of immigration and minorities, considered a threat for the Republic, banlieues turning into "danger zones".

The strength of Mustafa Dikeç"saccount derives from his distinctive theoretical approach which links space and policies, helping to understand France"s urban policies over the past thirty years.

He shows in an empirical and very convincing way, that the banlieues, far from being a given, homogeneous territory, are the result of a construction and the object of political manipulation.

From a rich corpus, he manages to analyze the way some banlieues were established as badlands as descriptive names, spatial indications and statistic categories were specifically created for them, by a whole set of representations which he defines, referring to Jacques Rancière, as the "police".

He shows, in practical terms, how the tough suburbs had received assistance from the State in the early 1980s (by means of a flexible and empirical concept of space) and are now analyzed within a more and more bureaucratic framework. This has resulted in an increased number of neighbourhoods which arestatistically set-up as sensitive areas and whose marginal status is quasi-naturalised. He demonstrates the close links that exist between space and policies, which very few studies related to the PV in France had previously interrogated [START_REF] Genestier | Le sortilège du quartier : quand le lieu est censé faire lien[END_REF][START_REF] Philippe | Instruments et fondements de la géographie prioritaire de la politique de la ville[END_REF][START_REF] Philippe | L'usage des quartiers. Action publique et géographie dans la politique de la ville[END_REF].

The same year as Badlands was published,L'Etatetles quartiers (The State and theNeighborhoods)by the sociologist Sylvie Tissot also came out [START_REF] Sylvie | L'Etat et les quartiers[END_REF]. Both projects were similar in their constructivist approach to France"s PV which highlights the highly political dimension of the territorialising of public initiative. The comparison of the two books allows us to underline the originality of Mustafa Dikeç"s reflection, which, in our opinion, is the consequence of his background as a geographer and as an academic, steeped in the Anglophonetradition. From the outset, their initial point of view appears to be quite different. Sylvie Tissot"s opinion, as a sociologist, is that the "spatializing of social problems" [START_REF] Sylvie | La spatialisation des problèmes sociaux[END_REF] set up by the PV, is highly problematic as it intrinsically downplays social and political relations. Mustafa Dikeç by contrast, as a geographer, assumes that all policies are essentially spatial, as is resistance to such policies (local rallies, riots...). In his reading the problem results not so much from the jurisdiction of urban policies as from their underlying ideologies.

Both authors are highly attentive to the conditions in which the PV is developed. However, whereas the social context and the relationships between the actors involved in the development of the PV are at the heart of Tissot"s analysis, Dikeç demonstrates special sensitivity towards France"s political and cultural specificities both within the national and the international contexts. He shows how these specificitiesaccount for the transformations of the imagined banlieuesand shape the policiestargeted at them on a more or less short term basis,. Heanalyzes more specifically the emergence of what he calls "republican nationalism" in France, with its contradictions. Beginning in the middle of the 1990s, the increasing reference to the merits of equality and social cohesion conceals a confused fear of the immigrant who is now considered a threat and for which the "banlieues" have become the spatial embodiment.

In opposition to Sylvie Tissot, who claims her study to be empirical (although explicitly founded on academic debates on the PV), Mustafa Dikeç roots his empirical analyses in theoretical reflections, most notably the concept of "police" which he borrows from Jacques Rancière. This leads to stimulating philosophical developments, which attests to the skills of the author of Badlandsfor linking theory and empirical research. For French readers, his way of writing in the area of social science is typical of the Anglophone academic tradition. His work is also a good example of the appeal for "French Theory", this "transatlantic hybrid" (Cusset 2004) that originated from a group of French intellectuals and which has had considerably more impact on the social sciences in the United States than in France.

Last, whereas Sylvie Tissot focuses her analysis on the spatial construction of the policies dealing with the "quartiers" (neighbourhoods), Mustafa Dikeç expands his questioning to the practical impact of these policies on the inhabitants of the given spaces, and shows how they suffer from inexorable and degrading confinement through the branding names that have been specially created by the PV. Badlands endeavours to link the voices "at the top" (from the people in charge of the PV) to the voices "at the bottom". The author explains that the voices do not correlate and that the latter have been trying, in vain, to voice their strong will for urban and social integration. We contend that this analysis is in fact the weakest point in the book. Only one chapter out of eight is devoted to these "alternative voices" through a study of an association in Vaulx-en-Velin in the city of Lyon and the unsuccessful attempts of its members to participate in local democracy and have an impact on the future of their neighbourhood.This disjuncture is unfortunate, for it is important to expose how legitimate and vital this participation of the inhabitants of the banlieuesis. In addition, we regret the lack of reference to some studies (Kokoreff, 2003, Pialoux and[START_REF] Maurice | Violences urbaines, violence sociale[END_REF], which explore in depth the alternative voices of the banlieues inhabitants, whose analyses complete and confirm those found in Badlands.

One of our other points of contention, is thatDikeç has hardly engaged with the existing wide range of local interpretations of the PV, though they enable multiple local solidarity regimes [START_REF] Philippe | L'usage des quartiers. Action publique et géographie dans la politique de la ville[END_REF]. According to their means and their political choices, the municipalities play a determining role in this diverse appropriation. Thus we find the possibility of multiple negotiations in the setting up of local projects (that are not simply the embodiment of State policies at a local level) and that may offer overtures to demands for justice. This diverse appropriation of urban policies can help analyze the diverse geographical realities for the 2005 riots: some of the quartiers sensibles remained calm, in spite of their negative social and economic indicators. Thus, we note that due to the proximity of downtown Paris, to the availability of work, to various other sources of income (among which drugs are included), to financial assistance provided by the town council enabling the implementation of social policies and to considerable investment in local democracy -within a strict framework-for the poorest neighbourhoods, a city such as Nanterre is in no way similar to Clichy-sous-Bois.

Working-class banlieues are plural and the constellation of local authorities, everywhere under the constraint of national policies, sometimes manage to deal with programs differently and in collaboration with their community inhabitants. In this regard thegovernment"s focus on particular spaces can be problematic.This approach condemns them to acknowledge failure because "sensitive neighbourhoods" are increasingly the areas of residence of the most precarious, due to the policies themselves. Symmetrically, some of their inhabitants have made use of these places as passageways, using the successive policies to find jobs and apartments and have often chosen to leave the "sensitive areas". The mobility approach changes the perspective for those studying these areas [START_REF] Pierre | Ghetto », « relégation[END_REF].

Since Badlandswas published, the situation has deteriorated even more and PV is facing drastic financial cuts. The main issue the government has been bringing up since the election of Nicolas Sarkozy in 2007 is the demonizing of the suburbs and immigration in France. The PV budgets and human means are constantly getting smaller (the police force included), whereas the president in his speeches points to the banlieues increasingly.This spectacular construction seems to initiate new uses of the banlieues. The cycle of urban renovation is ending and a new story begins, where the banlieues could be used as a regulating means, through fear (to the point of preparing a potential intervention of the army in the banlieues(Belmessous, 2010)). Mustafa Dikeç"s analyses provide welcome tools to help us better understand the impact of the Sarkozy years on France"s banlieues. His account brilliantly initiates, in Rancièreian terms, the emancipation of our sight of these banlieues.

Frédéric Dufaux, Université Paris Ouest-Nanterre Sonia Lehman-Frisch, Université de Cergy-Pontoise


Sundararajan Natarajan
email: sundararajan.natarajan@gmail.com

Stefano Dal Pont

Marc Duflot

Timon Rabczuk

Stéphane Bordas
email: stephane.bordas@alumni.northwestern.edu

Smoothed extended finite element method for continua

Keywords: strain smoothing, extended finite element method, boundary integration, isoparametric mapping

The extended finite element method (XFEM) introduced by Belytschko [1] allows discontinuities to be arbitrarily present in the domain, i.e, discontinuities can be independent of the mesh. However this complicates the numerical integration of the stiffness matrix, because it requires sub-division of elements cut by discontinuity, which in 3D may be computationally costly.

 for finite element method, coined as Smoothed Finite Element Method (SFEM) with the XFEM. The SmXFEM shares the properties of both the SFEM [9] and XFEM. The SmXFEM alleviates the need to compute the derivative of shape functions and numerical integration of singular functions, commonly encountered in linear elastic fracture mechanics. This is achieved by transforming the surface integration to line integration along the boundary of an element. The effectiveness of the method is illustrated by a few benchmark examples taken from linear elastic fracture mechanics. The results obtained with SmXFEM compare well with those of the standard XFEM.

Introduction

The eXtented Finite Element Method (XFEM) is a numerical method to model arbitrary discontinuities in continuous bodies that does not require the mesh to conform to the discontinuities [START_REF] Bordas | Mechanical failure in microstructural heterogeneous materials[END_REF][START_REF] Bordas | Enriched finite element short course : class notes[END_REF]. Recently smoothed finite element method (SFEM) has been proposed by Liu et al. [START_REF] Liu | A smoothed finite element for mechanics problems[END_REF] by means of combining the strain smoothing technique with the conventional FEM technology and further extended by Nguyen et al. [START_REF] Nguyen-Xuan | Smooth finite element methods : Convergence, accuracy and properties[END_REF]. In this method, strain smoothing operation is performed on an entire or part of an element to replace the conventional strains obtained using displacementcompatible equations in FEM. Due to the presence of strain projection process, only shape function is involved in the calculation of field gradients and no isoparametric transformation is required. Hence, the element is allowed to be of arbitrary shape and thus the field domain can be discretised in a more flexible way. Interested readers can refer to work of Nguyen et al. [START_REF] Nguyen-Xuan | Smooth finite element methods : Convergence, accuracy and properties[END_REF] for detailed description about the SFEM. The SFEM is based on strain smoothing stabilization, originally introduced by Chen [START_REF] Chen | A stabilized conforming nodal integration for Galerkin mesh-free methods[END_REF] for meshfree mthods. Refer to [START_REF] Nguyen | Meshfree methods : review and key computer implementation aspects[END_REF] for a recent review on meshfree methods. The SFEM was subsequently extended to plate and shell analyses [START_REF] Nguyen-Xuan | A smoothed finite element method for plate analysis[END_REF][START_REF] Nguyen | A smoothed finite element method for shell analysis[END_REF].

We present a technique to combine the strain smoothing technique to the XFEM, to get the Smoothed eXtended Finite Element Method (SmXFEM) [START_REF] Stephane | Strain smoothing in fem and xfem[END_REF]. Section 2 briefly discusses the basics behind the XFEM and section 3 gives an overview of the proposed method and numerical results are presented in section 4.

Extended Finite Element Method

The main idea in XFEM is to extend the approximation basis by set of enrichment functions, that are chosen based on the local behavior of the problem. For the case of linear elastic fracture mechanics, two such functions are used : Heaviside jump function to capture the jump across the crack face and asymptotic branch functions that span the two-dimensional asymptotic crack tip fields. The enriched approximation for fracture mechanics problems takes the form [START_REF] Belytschko | Elastic crack growth in finite elements with minimal remeshing[END_REF][START_REF] Bordas | Mechanical failure in microstructural heterogeneous materials[END_REF][START_REF] Bordas | Enriched finite element short course : class notes[END_REF] :

u h (x) = ∑ I∈N fem N I (x)q I + ∑ J∈N c N J (x)H(x)a J + ∑ K∈N f N K (x) 4 ∑ α=1 B α (x)b α K (1)
where a J and b K are nodal degrees of freedom corresponding to the Heaviside function H and the near-tip functions, {B α } 1≤α≤4 . Nodes in set N c are such that there support is split by the crack and nodes in set N f belong to the elements that contain a crack tip. These nodes are enriched with the Heaviside and near-tip (branch functions) fields, respectively.

This modification of the displacement field approximation doesn't introduce a new form of the discretised finite element equilibrium equation, but leads to an enlarged problem to solve :

K uu K ua K au K aa u a = f u f a (2)
In the XFEM, the discontinuities are independent of the mesh, i.e., the discontinuities can lie anywhere within or on the element. And the numerical integration of the above equation involves sub-division of elements, over which the integration can be done. Since it is not the scope of the present work, it is suffice to say that the numerical integration in XFEM is cumbersome. For more detailed information, interested readers can refer to the XFEM literature.

Smoothed eXtended Finite Element Method

The SmXFEM is a novel technique that combines the strain smoothing technique proposed by Chen for meshless [START_REF] Chen | A stabilized conforming nodal integration for Galerkin mesh-free methods[END_REF] with the XFEM to model discontinuities without having to remesh locally. SmXFEM uses a similar approximation for the displacement field as the XFEM Equation (1).

The element stiffness matrix writes as the sum of the contributions from each of the nc subcells of the element e

K e = nc ∑ C=1 Z Ω C B T C D B C dΩ ( 3
)
where C ∈ {1, 2, . . . , nc} is the number of the subcell Ω C . All entries in matrix B C in Equation (3) are constants over each subcell Ω C -each of these entries are line integrals calculated along the boundaries of the subcells. From the displacement approximation Equation (1), the strain field is deduced by differentiation of Equation (1)

ε h (x) = ∑ I∈N fem B Ifem (x)q I + ∑ J∈N enr B Jenr (x)a J = [B fem B enr ][q] ε h (x) = [B xfem ][q] (4)
The B xfem matrix in Equation ( 4) includes two terms B fem and B enr corresponding to the standard nodes (FEM) and enriched nodes (ERN). The smoothed enriched stiffness matrix for subcell C, K C xfem is computed by

K C xfem = Z Ω C B T C D B C dΩ = B T C D B C A C ( 5
)
where B C ≡ B xfem , A C is the area of the subcell. And the smoothed enriched element stiffness matrix K e xfem is the sum of the K C xfem , for all subcells, C

K e = nc ∑ C=1 B T C D B C Z Ω C dΩ = nc ∑ C=1 B T C D B C A C ( 6
)
where nc is the number of smoothing subcells of the element.

Numerical Examples

The effectiveness of the proposed method is illustrated by taking two benchmark problems from linear elastic fracture mechanics. We first consider a plate with an edge crack and then examine the case of plate with shear traction.

Edge crack under tension

A plate of dimension 1×2 is loaded by a tension σ = 1psi over the top edge. The displacements along y-axis is fixed at the bottom right corner and plate is clamped at the bottom left corner. The geometry, loading, boundary conditions and domain discretisation are shown in Figure 1. The reference mode I SIF is given by

K I = F( a b )σ √ πa ( 7
)
where a is the crack length, b is the plate width and F( a b ) is an empirical function given as (For ( a b ) ≤ 0.6) 1 shows the normalized stress intensity factor for various discretizations and crack sizes. Note that even for the coarsest meshes, the error is around a few percent and decreases with mesh refinement although the convergence is not absolutely monotonic.

F( a b ) = 1.12 -0.231( a b ) + 10.55( a b ) 2 -21.72( a b ) 3 + 30.39( a b ) 4 (8)

Edge crack under shear stress

As a last example, in order to demonstrate the capabilities of proposed method for a general case with mixed-mode loading, we consider an edge cracked plate subjected to shear load. The plate is clamped at the bottom edge and loaded by a shear traction τ = 1psi over the top edge. The material parameters are Young's modulus, E = 3 × 10 7 and Poisson's ratio ν = 0.25. The geometry, loading and boundary conditions are shown in Figure 2. The reference mixed mode stress intensity factors are :

K I = 34.0 psi √ in (9) K II = 4.55 psi √ in ( 10

Conclusion

In this paper, we presented a method to couple the strain smoothing technique with the XFEM to obtain the SmXFEM and to solve linear elastic fracture mechanics problems. The main advantage of this method is that it avoids the need to integrate the singular functions present in the XFEM stiffness matrix in linear elastic fracture mechanics problems. With the help of a few numerical analysis, we have shown the effectiveness of the proposed method. And since isoparametric mapping is eliminated, the elements can take any arbitrary shape.

Figure 1 -

 1 Figure 1 -Plate with edge crack under tension

Figure 2 -

 2 Figure 2 -Plate with edge crack under shear

Table 2

 2 gives the SIFs for various mesh refinement. It is evident that with mesh refinement, the computed SIFs converge to the reference mixed mode SIFs.

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