Keynote

Vincent Dubois

What do public policies do to their publics?

Introduction

Over the past few months, I have been conducting research on policies against welfare fraud in France, focusing in particular on the recent rise of a reinforced control policy mainly aimed at minimum income and housing benefits recipients and lone mothers. During my fieldwork, I came across a fascinating material in the form of boxes filled to the brim with letters of denunciations reporting the misbehavior of clients to welfare authorities. The authors of these letters are most often neighbors if not relatives of the alleged fraudsters. Intimate knowledge of personal situations is indeed required to document family life, housing conditions or gainful activities. Private conflicts and rivalries are the most common reasons for sending these letters. While a few look like official reports, providing the administration with detailed information and using a formal bureaucratic vocabulary, the vast majority are handwritten, badly misspelled about why Mr. or Mrs. So-and-So should be punished for cheating welfare, often written in capital letters on a torn notebook page. Unlike other countries like the US and the UK, France has no official procedure or incentives for ordinary citizens to report welfare fraud. Such letters have obviously always existed here as elsewhere. But while they are not a new phenomenon, surprisingly enough, their number has recently dramatically increased (twofold between 2007 and 2014).

Why this increase?

A possible partial answer to this question is that it might be an effect of changes in policy discourse over the same period. The last decade, especially during the five years of the Sarkozy administration, has certainly been the most intense period for the stigmatization of "the entitled" (les assistés), described as unwilling to work and suspected of cheating in addition to being lazy, since the construction of contemporary welfare. It would be over-simplistic to posit a single causal relationship between this policy discourse and the attitudes towards welfare expressed in these letters of denunciation. Instead of drawing a direct link between them, it is necessary to reflect on the objective social conditions that provide the foundations for the appropriation of such a discourse, thereby rendering it potentially effective if not "performative" [START_REF] Bourdieu | Language and Symbolic Power[END_REF]. In this case, the precarisation of working-classes individuals and the heightened competition for jobs, housing and welfare benefits among them have been conducive to renewed forms of social and symbolic separation [START_REF] Lamont | The Dignity of Working Men: Morality and the Boundaries of Race, Class, and Immigration[END_REF] between those who work and those who do not, and, among them, those who "deserve" support and behave properly and those who do not. In this socioeconomic context, the policy discourse blaming welfare recipients in general and welfare "fraudsters" in particular can echo the experience, perceptions, meaning-making practices and attitudes of individuals, This discourse can in turn legitimate and encourage denunciation, usually seen as a morally dubious practice in other contexts. In this sense, the study of letters of denunciation can help us reflect on the relationship between policy discourse and what we at this stage may provisionally call the socio-political attitudes of ordinary citizens. This illustrates the point I intend to make in this essay. It is by considering this relationship that I offer some answers to the question: What do public policies do to their publics? Here, emphasis will be placed on the symbolic aspect of the question, with a focus on impacts in terms of principles of vision of the social world. In other words I look into how policy discourse and policy generally speaking affect the way people define themselves, view others, and envision social reality. I explore how and to what extent the categories of policy discourse and the social norms enforced through public policies can impact the representations, dispositions and attitudes of the persons subjected to that discourse, in addition to possible changes in their status. This symbolic aspect is strongly linked with material factors and issues, such as, in my research, welfare provision, living conditions, and the possible penalties against fraudsters. While these factors and issues will be considered, they are not a central concern here.

The most common pitfall of plenary essays consists in listing a set of general, theoretical and abstract propositions, which may fulfill the social function of an introductory ceremony to the academic ritual of a conference or a special issue, but can fall into a scholastic rhetoric disconnected from research practices. The diametrically opposed pitfall is to focus on a specific empirical research providing the audience with information on the topic under scrutiny that might be irrelevant for those with remote research interests. In order to avoid both pitfalls of theoreticism in and empiricism, I have chosen to vary and, as far as possible, combine specific examples from my own research as a basis for some general and conceptual proposals. In turn, I illustrate theoretical statements with concrete cases drawn from my fieldwork. I use my research on social welfare to illustrate four conceptual proposals, from the production of policy discourse to the daily practices of people-processing, and eventually, to the impact these discourses and practices may have on the way people consider the social world.

The social spaces and legitimization of policy-making

My first theoretical proposal consists in positing that the social basis for the making of a policy discourse, and of a policy more generally (i.e. the system of relationships between various categories of actors contributing to shape policy and official discourse), is also the social basis for its legitimization. The best analytical tool to account for this double process is in my view Bourdieu's concept of field.

The concept of field as a tool for critical policy analysis

Policy analysts have invented and used many concepts to analyze the social milieus of policy makers, and more generally the relationships between the individuals involved in policy processes. To mention only a few, these include policy networks, policy community, iron triangle, advocacy coalition, epistemic community and many others. These concepts can of course be very useful, but in my view they show several weaknesses, which may vary from one to the other. First, they remain mainly descriptive, as they lack of conceptual elaboration or are not explicitly part of a broader theoretical model. Second, their use is generally not connected to effective methodological tools. Even the concept of policy network, which certainly is the most strongly related to a sophisticated analytical framework and to quantitative analysis (for a good example, see [START_REF] Laumann | The Organizational State: Social Choice in National Policy Domains[END_REF], is often used metaphorically, with suggestive illustrations rather than systematic empirical evidence.

Lastly, some of these concepts may suffer from a normative bias, sometimes implicit, as when the 'flexibility' of governance networks is celebrated in contrast with the so-called rigidity of state institutions (for further discussion, see the first part of Dubois 2014).

On the other hand, policy analysts almost never use Bourdieu's concept of field. In my view it is also under-used by sociologists who work on public policies (for a rare recent example, see [START_REF] Itçaina | Varietals of Capitalism: A Political Economy of the Changing Wine Industry[END_REF]. And yet I do think that applying this concept to the social and political spaces in which policies are made would be an important contribution towards a critical sociological approach to public policy (for a complete discussion, see my chapter in Dubois 2014).

A field may be summarily defined as a social microcosm: a space of objective positions held by agents with various types and levels of capitals, and a space of competition between these agents. This microcosm is relatively autonomous from the other fields, defined by specific rules, stakes and forms of capital. The academic field, for instance, be it dependent of private or public funding, has its own norms and its agents usually denounce the possible intervention of private companies or politicians in the internal affairs of academia. These agents do not compete for the same trophies as politicians or athletes. They do not use the same assets in their specific competition, and, again, they do not follow the same rules. A field is a system of relations between agents defined by their position within the field, depending on the level and the structure of their capital, this capital being itself composed of all the assets and resources useful for competition within the field (i.e. authority provided by hierarchical status, social capital of relationships with other agents to collaborate with, informational capital of knowledge and expertise, symbolic capital of recognition).

The theory of social fields is not only concerned with describing a social milieu. It is also meant to understand how the outputs of a social milieu (e.g. artworks for the arts field, political statements for the political field) are rooted in its structure. The key concept here is homology. According to Bourdieu's field theory, there is a homology between the objective positions held by social agents within the field and their preferences, choices, stances, discourses. The place where an individual is located in the social space of the field is predictive and explains the options taken by this individual. Or, to put it briefly in a famous phrase known as the Miles' Law: "Where you stand depends on where you sit" [START_REF] Miles | The Origin and Meaning of Miles' Law[END_REF]).

If we move from an individual point of view to a global one, this means that the outputs of a field in a specific period of time (e.g. the variety of artistic genres or possible and thinkable political statements) "reflect", so to speak, the structure of the positions of producers (artists or politicians), namely the social agents of the arts field and of the political field.

Using this analytical tool in the policy world leads to the basic postulate arguing that from a sociological point of view, a public policy can be defined as the product of practices and representations of social agents who contribute to policy making. These practices and representations are determined by the objective position of these agents, their social characteristics, their interests, and by the structure of relations between them, or, in other words, by their position in the field.

This postulate brings us to a general hypothesis. It is quite obvious that the competing options in the definition of a policy relate to the positions and interests of those who defend them, inside the governmental arena (with the competition between various departments), or outside of it (among interest groups and expert agencies, for instance). To go beyond this fairly trivial statement, I propose to account for the correspondence (more precisely: the homology) between the contents of a policy (orientation, style), and the relational structure of the space of agents involved in its production. This means that the compromises and the balance between competing options characterizing a policy match the structure of power relations within the field and the concrete arrangements established on this basis. Following this reasoning, we can propose a sociological and critical explicans (the structure of a policy field) to account for an explicanda (the nature of a given policy).

There are several possible uses of this concept in policy analysis. I list the five main ones below.

1) The concept of field can be applied specifically to the space of bureaucracy, which brings together the official positions defining the state [START_REF] Bourdieu | On the State[END_REF]. This application can be useful on one level to understand how competing dispositions within the system of relationships of a public service determine actual policy orientations and practices (see for instance [START_REF] Woolford | Community Positions, Neoliberal Dispositions: Managing Nonprofit Social Services Within the Bureaucratic Field[END_REF]. Here, the concept of field provides a sociologically informed internal view of the official policy-makers and of their distinctive patterns, problems and competitions.

2) It is also possible to move out of bureaucracy in the strict sense and to use the concept of field to describe the internal structure of social space that is non-bureaucratic but nevertheless significantly contributes to policy making. This was for instance done by Thomas Medvetz in his study of American think tanks [START_REF] Medvetz | Field Theory and Organizational Power: Four Modes of Influence Among Public Policy 'Think Tanks[END_REF].

3) At the intersection of the first two levels, a policy field can be defined as the relational system dedicated to the regulation of a specific policy domain, bringing together agents from the bureaucratic field and agents from other fields, such as representatives of interest groups, CEOs of private companies or independent experts. Bourdieu and Christin did this in their research on housing policy reform in France in the mid-seventies, which remains the main example of a systematic use of the concept of field in the critical analysis of a policy [START_REF] Bourdieu | The Social Structures of the Economy[END_REF], Chapter 2, "The state and the construction of the market").

4) An institutionalized system, including bureaucrats, politicians, experts, interest groups, in which a full range of public policies are made, can also be analyzed through the lenses of field theory. Didier Georgakakis and Jay Rowell gave an excellent example of such an approach in their analysis of the European Union as "the field of Eurocracy" (Georgakakis and Rowell 2013). 5) In many cases, the making of a policy involves a wide range of actors holding heterogeneous positions in various fields, and, therefore, results from the relationsbetween these various fields. In the third possible use of the concept of field I mentioned above, the system of positions and power built in a specific policy domain is considered as a field on its own. At this fifth and last level, I propose to take into account the various social fields involved in the making of a policy and to describe the relations between them in order to analyze the interdependence system they form. To put it in Bourdieu's words, this consists in setting the system of relations (inter-fields) between systems of relations (fields), considering it as the locus for the definition of policy orientations. In this perspective, a policy orientation is viewed as the outcome a) of the internal balance of power within each field involved in the process; b) of the interactions between these fields.

The reinforcement of surveillance and penalties for welfare recipients as an output of interactions between fields

Policy reforms and new programs concerning the surveillance and sanction of welfare clients in France illustrate this role of interdependent various fields. In the following I focus on the four social fields most directly active in the success of this policy orientation.

First, the scientific field of economic expertise contributes to framing policies. It provides intellectual models which can play two roles: they sometimes strongly influence policy-making; they are referred to after the fact by policy-makers in order to confer so-called scientific legitimacy on political orientations defined on different grounds. The field of economic expertise, now dominated by neoclassical orthodox approaches, paved the way for more control in welfare. The success of the concept of inactivity trap, or poverty trap, is a good example of this role. To put it simply: according to this model individuals on welfare calculate their financial interest to decide whether they take a job or stay on welfare. When the level of welfare benefits is "too high", they will prefer to stay on welfare.While this debatable model will not be debated here, I can only observe that it has been widely used as a basis for welfare reform and its legitimization, including the development of control and sanctions as incentives to work. This has been the case in France, as we can see in numerous policy recommendations by economists. To mention only one, Michel Camdessus, former director of the IMF in 2004, in a report on the general economic situation of the country entitled 'The burst: towards a new growth for France', surprisingly devoted numerous pages to urging the government to strengthen control over the unemployed and welfare recipients, directly in line with the inactivity trap model.

Second, in the bureaucratic field, the welfare elite took a decisive managerial turn, beginning in the early 1990s, whose impact has been increasingly visible during the past two decades.

These economic models became all the more influential as a new generation of higher civil servants with a background in management and in economics replaced the previous one, trained in law and attached to the old welfare model. At this level, welfare control was defined as a good management technique. The new welfare officials laid emphasis on financial concerns and imposed neo-managerial references and practices on their organizations and their agents. Within the bureaucratic field, the ministry for finance and budget, the Court of Auditors (Cour des Comptes) and the accounting departments in welfare organizations came to play a decisive role in the management of welfare provision, including a new "risk management" strategy which in practice consists in new tools for monitoring and sanctioning welfare recipients. These orientations were widely reported if not supported in the media field. In my research I found hardly any paper on welfare fraud before the mid-1990s. By contrast, countless amounts were published after this date, especially after the most important reforms of the early 2000s. This chronology shows that the media have not fulfilled an agenda-setting function, urging the government to make reforms. On the contrary, they have generally followed the government on these issues, spurring public support for governmental reforms.

While some left-wing papers were initially critical and used references to George Orwell to denounce a surveillance society, the vast majority of articles denounce welfare fraud as a scourge requiring more control. Not all the press follows this orientation, but crucially, national TV channels and mainstream newspapers do.

Fourth and last, the internal dynamics of the political field appear to play a prominent role.

The right has unsurprisingly promoted the theme of welfare fraud. In 1995, this was to retain the support of the privileged fractions of the conservative electorate and of independent workers. Later, it became an explicit means to gain support from the working classes, among which turnouts are very low. Criticizing the "lazy entitled" became a very common way for right-wing politicians to present themselves as sharing the concerns of workers who can hardly make ends meet, and are supposed and encouraged to be upset with "their neighbor who stays home and makes as much money on welfare". This is in my view a good example of the circular effect of a political discourse, which by repeating the same arguments reinforces if not generates the concerns to which it supposedly responds. This is also an illustration of how right-wing themes spread across the political spectrum. Other issues, such as security and immigration, have become central in the political debate on welfare and beyond. The moderate left which alternates with the right at the government could not avoid addressing them. Its leaders have had to do so in order to appear as credible government officials, tough on crime and fraud, far from the "over-leniency" denounced by their competitors. By doing so, they have progressively included some of their opponents' arguments into their own discourse, and unintentionally contributed to the political success of these themes.

The construction of welfare fraud as a public problem, and the new relevant surveillance policies result from the interaction between these four fields, and not from political will. The importance granted to this issue and the accompanying policy initiatives in the public debate is no way proportional to its objective impact Tax evasion (20 to 25 billion euros per year), which is estimated to approximately amount for ten times the amount of welfare abuse (2 to 3 billion per) year), is far less present in the public debate. It became a prominent issue on the media, political and judicial agendas only after it was revealed in December 2012 that the minister of the Treasury himself (Jérôme Cahuzac) had undeclared bank accounts in Switzerland. While some steps were subsequently taken to fight tax evasion, they were not as significant as the measures against welfare fraud, neither from a political nor from a policy point of view.

Welfare fraud owes its success as an object of public rhetoric to its combination of a wide range of registers, from financial rigor to morals, from expertise to casual conversation and barroom politics. Its targets can vary from the bad mothers who "have children to live on welfare" to the bad immigrants who come to France to abuse the system; it is also part of the delegitimization of welfare in general, even if it, somewhat ironically, the system is depicted as in need of saving because it is supposedly undermined by fraud. This rhetoric peaked under the Sarkozy regime, when welfare fraud and entitlement were contrasted with "the work value" (la valeur travail) promoted as the key element of the governmental program to "redress" French society. This brings us back to my first proposal: the legitimization of a policy orientation rests on the social structure of its definition. If the reinforcement of control over the poor has become an option, it is because it has been promoted simultaneously in various fields, following their own rationales. As a result, it appears as a shared orientation, if not as a self-evident one.

Policy-making as symbolic power

Imposing the legitimate point of view

This legitimization process of a policy orientation is related to the broader process of how in turn policies legitimize a specific view on the social order, and, by doing so, social order itself. My point here is that the legitimization process of a policy pertains to the promotion of a specific principles for approaching problems, situations and people as the legitimate point of view. Welfare reform and its discourse did not invent the opposition between hardworking taxpayers and people staying on welfare or between the deserving and the undeserving poor. However, welfare reform did renew and reinforce these divisions and contributed to have them taken for granted among the general public. This has a concrete impact on the actions of: writing letters of denunciation, for instance, which, in that perspective, has become a more legitimate and easily doable act. This is one of the ways to explore what policies do to their publics: they promote if not impose a specific point of view as the legitimate one. This is the reason why I think symbolic power is at the core of policy, and why reciprocally, policy-making is about exerting symbolic power, meaning that policy plays a key role in the adjustment of mental structures to social structures. This approach does not postulate a mechanical impact of policy discourse on the ways of thinking of the populations subjected to it. There are social conditions for this power to be effective, which vary from one policy domain to another, and according to the resources at the disposal of social agents to resist it. And this power is concretely exerted through concrete practices and dispositifs, to use Foucault's word. In order to illustrate how symbolic power is made concrete through public policy, I will turn now to the processes of classification and category-making at work in policy-making.

Policy as a classification process

The legitimate point of view on social reality is expressed through policy categories, which are both tools for the interpretation of situations and classifications used and enforced in routine bureaucratic practices . Since policy commonly operates by defining the categories through which people and problems are perceived and dealt with, the analysis of classification processes is a needed part of a comprehensive critical policy analysis [START_REF] Yanow | Constructing 'Race' and 'Ethnicity' in America: Category-Making in Public Policy and Administration[END_REF]. In this perspective, I propose a three-step research program: first unveiling the construction of official categories; second, analyzing their mobilization in unequal power relationships by policy agents who process people and handle public problems; and third, identifying the impact of official categories on the people who come to internalize them.

A first step consists in questioning the conventional policy categories, showing their social and political roots and, by doing so, their non-natural, non-necessary character. This follows an anthropological tradition initiated by Durkheim and Mauss, who demonstrated in their seminal work on primitive classifications that categories of perception of the world rest on the structures of the society which uses them [START_REF] Durkheim | Primitive Classification[END_REF]. On this basis, the genetic approach accounts for the historical making of these classifications, and the theory of social fields enables us to establish the relational dynamics which promoted them as legitimate classifications.

In a second step, studying policy categories is about observing how policy-makers and administrations use them and apply them to problems and people. Therein lies a possible convergence between street-level bureaucracy theory and the sociology of symbolic power.

The study of bureaucratic work, and more precisely of what Goffman called "people processing encounters" illustrates how formal rules and official categories are applied to people [START_REF] Goffman | The Interaction Order: American Sociological Association, 1982 Presidential Address[END_REF]. Welfare bureaucrats relate their definition of the situations of the poor they meet to standard criteria and categories, translating the lives of their clients into the language of bureaucratic files. They decide who is to be regarded as a single mother or unemployed, and their classification is imposed on the individuals subjected to it, even if they have a different notion of their own status. This symbolic power of definition is all the more effective as it has concrete and material impacts: receiving benefits or not, being sanctioned or not. It is also all the more effective as the relationship between bureaucrats and welfare clients is unequal. Clients depend on welfare, and they generally have nothing but the weapons of the weak to face bureaucracy: resignation, silence, feigned docility, and sporadic eruptions of verbal or physical violence. This brings me to the social differentiation of these enforcement processes. To give just one example, there are for instance striking differences between bureaucracy's handling of tax evasion [START_REF] Spire | Faibles et puissants face à l'impôt, Raisons d'agir éditions[END_REF]) and its handling of welfare fraud. In the first case, when bureaucrats face individuals from the middle and upper classes, arrangements and negotiation prevail. Tax fraudsters may be charged additional taxes, but are rarely prosecuted, and in the few cases where they are, sentences are generally lenient. In the case of welfare, bureaucrats face individuals from precarious fractions of the lower classes, and no room is left for negotiation: stigmatization and coercion prevail.

Three comments

Accounting for what policies do to their publics requires considering the social patterns of the different "publics". This may seem self-evident, but it is not in policy analysis, even in interpretive and critical policy analysis, in which, for instance, social class is not systematically considered as a factor. Here I would like to advocate for a framework that evidences the entanglement of class, race and gender in the practices of policy-makers and street-level bureaucrats (Watkins-Hayes 2009), in the attitudes of individuals subjected to policy, and in the way they are "processed" by the policy machinery.

Social differentiation is not limited to the question of class, defined as a stable position in the social space. It is also about other aspects defining one's position, such as gender and race, and about social trajectories. Individual relationships to welfare bureaucracy vary according to social mobility trajectories. The clients most prone to displaying aggressive behavior, for instance, are not the most impoverished, long-term welfare clients, but rather those who have recently experienced social downgrading and reject the welfare administration as a symbol of their own decline, and of the social system they view as responsible for this decline. Therefore, individual relationships to welfare bureaucracy also vary according to the stage in the welfare client's "career", from newcomers who lose themselves in red tape and bureaucratic language to regular clients who know the written and unwritten rules and can more easily find their way. These differences impact the intensity and the forms taken by symbolic power; and the ways in which different publics can deal with that power. This leads us to a third remark. Analyzing what public policies do to their publics requires reflecting on how these publics do with the policies they are subjected to. The power relationship between policy agents and ordinary citizens may be unbalanced, but this does not mean that citizens are passive and irreversibly dominated. Depending on their situations, resources, and social dispositions, they can use tactics to escape domination or subvert, at least partially and temporarily. Docility at the desk of a welfare office, for instance, may be nothing but a strategy to obtain compassion and help. This does not mean that there is no bureaucratic domination, but it shows its complexity and its nuances.

Having these nuances and complexity in mind, it is possible to account for the impact policy categories may have on people. This is the third and last stage of a research program on policy categorization processes. One of the most effective ways of addressing this question is to evaluate to what extent individuals categorized by policy in turn refer to these categories to define their situation or identity. In his work on scientific categories Ian Hacking has coined the phrase "looping effect" to designate how "invented" or constructed categories become "real", in the sense that they create new groups -a process he terms "making up people". Similar looping effects occur when policy subjects come to define themselves in reference to the policy categories used to classify them. Here we have a good illustration of symbolic power, with the nuances I mentioned before: the possible internalization of principles of vision and division inculcated through the unequal relationship between policy agents, vested with the power and legitimacy of official policy, and individuals subjected to it. This symbolic power can be violent when the categories enforced contradict subjective views and tend to replace them. I once heard a bureaucrat tell a claimant in a welfare office: "no, you are not regarded as a mother, because you do not meet the criteria of [having] affective ties [with your children]". This may lead some potential claimants to turn down welfare entirely: refusing to be defined and stigmatized as a welfare client is one among the various reasons for the non-take up of welfare benefits. Symbolic power gains effectiveness when official categories match the self-definition of identity or converge with views rooted in personal experience. The official vision then becomes "a well-founded illusion", to use Durkheim's phrase about religion, and is all the more legitimate and unquestioned as it is regarded as natural and self-evident.

Concluding remarks

The diffusion of a policy discourse and practice against fraud, stigmatizing the allegedly antisocial behaviors of people accused of preferring welfare to work, certainly encourages letters of denunciation. Yet there is no direct causal relationship involved. The official symbolic boundaries between workers and the entitled, between deserving and undeserving poor, are aligned with the ordinary symbolic boundaries between "them" and "us", in a context of socioeconomic crisis and tensions between social groupshere, between fractions of the lower classes. Like the gossip studied by Norbert Elias, denunciation letters are a way to assert one's position on the right side of the fence (Elias and Scotson 1994, pp.89-105).

Policy discourse is therefore not the only causal factor. But it does matter, as it legitimates the practice of denunciation, and provides this practice with a general socio-political meaning.

I have argued that a partial answer to the question "What do policies do to their public?" is that citizens internalize and utilize principles of vision and division promoted by public policies. In the process, I have also raised the question of how citizens (make things) do with the policies they are subjected to, which is in my view essential to provide a realistic view of domination processes. This should be complemented with a third question, which could be the topic of another essay: what do the publics do to policies?


Publié dans Proceedings of the Seminar for Arabian studies, Vol. 12 (1982), pp. 39-48.

IRON AGE STONE VESSELS FROM THE OMAN PENINSULA.

A PRELIMINARY NOTE.

Pierre LOMBARD

First discovered in 1964 at Dibba,in the United Arab Emirates (BIBBY 1965 : 353, 354), the Omani Iron Age period was forsaken for years and often eclipsed by the important archaeological results yielded by earlier periods. (1) But if forsaken, it was not entirely forgotten. In 1968, the first sondage carried out in the Al Aïn oasis by Karen FRIFELT and her team revealed a settlement site at Rumeilah, in which archaeological material had clear associations with the few finds collected at Dibba (FRIFELT 1968 : 171). Traces of this already famous « Dibba culture » were even recovered from one of the Hafit cairns which was subsequently re-used (BIBBY 1970 : 315, 316 ;FRIFELT 1970 : 377).

Since this pioneering period, our knowledge of these First Millennium BC cultures has been extended appreciably. New sites were surveyed in 1976 at Shimal and Ghalilah, in the vicinity of Ras-AI-Khaimah (DE CARDI 1976). In the Al Aïn oasis a French survey in 1977 revealed about twelve Iron Age settlements in the Hili-Qattarah area (CLEUZIOU et al. 1977 : 10, 11 and Fig. 4), and since the beginning of 1981, the Rumeilah settlement, threatened by increasing urbanization, is being extensively excavated by the French Archaeological Mission in Abu Dhabi (cf. infra).

Last but not least, two important necropoles have yielded particularly rich material : Qarn Bint Sa'ud, to the north of Hili, and Al Qusais, near Dubaï (cf. infra). One can complete this review by indicating sites discovered by the Harvard Archaeological Survey in Oman (HUMPHRIES 1974 :52),and the interesting fortress of Lizq, presented by G. WEISGERBER (1981) at the present meeting.

But it would be long and tedious to sketch a whole picture of the Oman Peninsula in the first half of the First Millennium BC. The purpose of this paper is rather to consider this Iron Age period in concrete terms through one of its products: carved stone vessels. Obviously this choice is not fortuitous. Stone vessels are present throughout these cultures. They are found either in settlement sites or in graveyards. Their shapes, their decoration and probably their material, too, differ from the well-known productions of previous periods. In fact, these items represent one of the most original features of the Omani Iron Age.

THE SITES

As an illustration of this production, we have chosen specimens from three main sites of the period,already mentioned : Qarn Bint Sa'ud, Al Qusais and Rumeilah. (2)

The Qarn Bint Sa'ud necropolis, located in 1970 by the Danish Expedition, is situated about 15 km north of Hili (FRIFELT 1970 : 378). The site appears as a rocky outcrop, 800m long. Its greatest width is about 200m and it stands 60m above the surrounding sand dunes. From 1972 to 1975, 18 graves were excavated, some by the Danish Expedition and some by a staff of the Department of Antiquities, Al Aïn, led by Walid YASIN. These excavations revealed that the occupation range of the necropolis was quite wide, and in the light of recent works, Qarn Bint Sa'ud appears today as a regional key site where the main phases of the Omani prehistory coexist:

1) The Hafit phase (late IVth-early IIIrd mill. BC), represented here by its distinctive cairns, all located at the foot of the mount ;

2) The period of the IIIrd mill. BC settled culture of the Omani foothills, represented by circular and compartmentalized graves, close to the Hili ones ;

3) The so-called Wadi Suq period, now well illustrated in Oman in the early IInd mill. BC as CLEUZIOU (1979) demonstrates in a recent study ; the bronze material from a rectangular grave seems to be linked with this phase ; 4) Lastly, the Iron Age period, which is in fact represented there only by reoccupation or extension of earlier graves, in spite of what had been previously believed (LOMBARD 1979 : 28).

The Al Qusais necropolis, situated 13 km NE of Dubaï was accidentally discovered and excavated in 1974 by an Iraqi team and more recently ) by a local one (SALMAN 1974 : o-p;Al-Athar 1975 : 52-57;TAHA 1981). The site has yielded about thirty undamaged graves and the burial material is one of the richest collected from this period : pottery, bronze weapons,stone vessels, etc... (TAHA 1981 ;LOMBARD 1981).

The last site, Rumeilah, was apparently the most extensive Iron Age settlement of the Al Aïn oasis, but a large part of it was destroyed last year by bulldozers. Before this unfortunate destruction, this low tell was almost 800m long and 150m wide. In January 1981, the French Archaeological Mission in Abu Dhabi stepped in to protect and excavate the site. Several large mudbrick houses have already been discovered and two different occupation levels in the First Millennium BC have been identified, each characterized by a distinctive architecture and artefacts. This important discovery helps to fill the gap between the Iron Age and the Hellenistic period (BOUCHARLAT and LOMBARD to be published).

TYPOLOGY

We realize the limits of this preliminary study. Until the last few years, Iron Age stone material from the Oman area has been scarcely published. Furthermore,items are now displayed or stocked in various museums (Al Aïn, Dubaï) ,and the relevant documentation is similarly scattered.

As we were unable to work on an exhaustive corpus, and as excavations in progress reveal new specimens every year, one must not consider this « typology » as definitive. However, these carved stone vessels can be classified norphologically into four main representative groups:

1) Flat based cups,with in most cases a horizontal spout. Their height ranges from 4 to 7 cm, and the diameter from 7 to 15cm. (FIG. 34).

2) Larger bowls ,with splayed sides and a fIat base. The rim is generally cut horizontally. Sometimes, it is thickened and displays incised patterns. The height ranges from 4 to 8 cm and the diameter from 10 to 18 cm. (FIG. 34567).

3) The third group is the most distinctive of the period. Vessels are strongly carinated, with convergent sides and a convex base. Several variants can be distinguished by their crosssections : conical, "beehive-shaped", etc... . One specimen, from Al Qusais, is divided into three compartments.

4) The last type is also rather unusual. It appears as a barrel-shaped suspension vessel,with four vertically pierced lugs. The dimensions can be rather large, from 20 to 25cm high. 4 : 13). Curiously, this type is not found at Qarn Bint Sa'ud (at least among the Danish material) but is weIl represented on other sites, notably at Ghalilah, and Al Qusais, where a nice specimen had been found (SALMAN 1974 : Pl. 15 (arabic par); Al-Athar 1975 : Pl. 11).

We must add to these main categories some other shapes, represented here and there by a single specimen (FIG. 4 :14 ,15) and numerous lids , which in some cases greatly outnumbered the collected vases, suggesting an earlier looting of the corresponding vessels. (3)

DECORATION

Almost aIl of the vessels examined present a geometrical incised decoration, probably hand-made with a spike. Except for some specimens, treatment is often coarse. Furthermore it is surprising to notice how such poor workmanship in decoration is often coupled with finely executed shapes.

Incised patterns are usually superficial and, due to the stone wear, less apparent. Illustrated vessels show some of the most distinctive patterns : herring-bones, arches, triangular compositions, and saw-tooth incisions. We have to emphazise this last pattern which seems to be a rather good stylistic and chronological indicator of the period, at least with respect to stone vessels. (4) TECHNOLOGY One of the most interesting problems posed by this production is that of the stone employed for this purpose. The terminology used by archaeologists has often been contradicted by geological reality.

In archaeological publications, soft stone objects are often refered to as « steatite », as opposed to stronger materials like diorite, for example. But a wholesale use of this term made it unreliable : nowadays the term « steatite » is applied to a whole range of minerals of various aspects, textures, colours and hardness. Geologically speaking however, steatite has a very precise definition, being essentially constituted of magnesium silicate. Light in colour, it reveals, when touched, a smoothness from which is derived its usual appellation, « soapstone ».

In his study devoted to the Yahya stone vessels, KOHL (1974), following the report of Xray diffraction analysis, has proposed the term « chlorite » which indicates not a precise mineral but rather a group of metamorphic and complex silicates, characterized by a green colour and a cristalline appearance. In fact, many archaeologists have now agreed upon this term « chlorite », the meaning of which is certainly wider. However, it still does not coincide with aIl the archaeological evidence. KOHL himself admits that 15% of his comparative samples from various Near-Eastern sites were not proved to be chlorite... (1975 : 30).

Iron Age stone vessels found in the Oman Peninsula have never been submitted to any petrographical analysis except for a sample from Rumeilah, precisely proved to be true steatite. Of course, one must beware of any generalization founded on a single rudimentary analysis (5). However,this preliminary indication suggests that the term « chlorite » is no more suitable than the previous one and, on the other hand, strenghtens the hypothesis that the Iron Age items differ slightly from those of the IIIrd millenium BC. In addition to their dark colour and the dense structure of their stone, Bronze Age specimens found at Hili and Umm-An-Nar, for instance are above all different in shape.

We are convinced that the strongly carinated shapes noted in the First Millennium BC at Qarn Bint Sa'ud and elsewhere are explained by a stylistical evolution as weIl by the use of another material, probably softer and more fragile, which dictates the shape of the product. We find it significant that these carinated shapes,quite unusual in the IIIrd millennium BC, appear extensively as from the Wadi Suq period (CLEUZIOU 1979 : fig. l0:3) and, indeed, linked with the use of the same light grey stone discovered at First Millenium BC sites.

Another significant case is that represented by a similar industry still existing today in Mashâd,the pilgrim's city of eastern Iran,where the local use of a soft whitish stone has produced almost similar shapes (WOLFF 1966 : Fig.202). In this connection, KOHL (1975) has suggested already that carving techniques noticed at Mashad seemed to be rather similar to the protohistorical ones. Such an hypothesis appears unsafe for the Iron Age period, especially regarding the use of a similar lathe, hand-operated with a bow (WOLFF 1966 : Fig.205-207;KOHL 1975 : Fig.6). As a fact we did not found such tooling marks on the Omani vessels and we are inclined to share WEISGERBER's opinion (1981) that there is no real evidence for the use of a lathe for carving vessels before the Hellenistic period.

If a slight doubt remains about its precise definition, the source of the stone carved by Iron Age craftsmen is more evident. Chlorite, steatite and derivates abound in the Oman Peninsula. These ultrabasic rocks represent the main geological formation of the Hajjar Mountains, the Ophiolites, locallycalled by geologists "The Semail igneous" (LEES 1928 : 601-604). (Cf. Fig. 2)

These rocks have produced a typical landscape of jagged peaks in the area. After PILGRIM (1908), who first identified this formation at the beginning of this century in the Wadi Semail near Sib, the Oman Ophiolites were more recently studied (ALLEMANN and PETERS 1972 ;WILSON 1973), notably by GLENNlE (et al. 1973) who demonstrated that the main strata of chlorite and steatite were generally situated in the lower part of the massif and more particularly on its western side, therefore not far from our archaeological sites in the foothills.

Carved stone vessels are not new to archaeologists working in the Arabian Peninsula. Our specimens from U.A.E sites appear to be a link in a long regional tradition. In his study devoted to the stone material from the Ryadh Museum, ZARINS (1978 :67) reported the earliest evidences to be those from Janūb Al Matabattihat, a neolithic site in the southwestern Rub-Al-Khali, and those from Al-Khubar, a late Obeid site of Eastern Arabia. We have already mentioned the carved stone vessels at Bronze Age sites as Failaka, Tarūt and in the Oman area. On the other hand, they are found on later preislamic cities (Najrān) or Early Islamic ones (Al-Fau, Saudi Arabia or Murwab, Qatar) (6).

However, this distinctive Iron Age industry poses some questions, which still remain unanswered.

First, the chronological problem. Even though recent discoveries have contributed to an improved understanding of the material culture, they do not allow at present a precise dating. Nevertheless, first comparative studies based mainly on pottery and bronze material (HUMPHRIES 1974 ;LOMBARD 1979) have suggested a reasonable date about 750-650 BC.

On the other hand, what can be said about the social function of these vessels ? We know from excavations of various settlements that they are not only funeral goods. Their abundance alone suggests a rather functional use as opposed to the IIIrd Millenium BC items (LAMBERG-KARLOVSKY 1970 : 66). New excavations in progress on the Rumeilah settlement are likely to shed new light on this topic.

A last problem is that of the diffusion of these Omani products. In Bahrain Island (where rough material is non-existent), similar shaped and decorated objects have been found in the so-called Assyrian graves of the Al-Hajjar necropolis (RICE 1972) as weIl as in some unpublished graves accidentally discovered during the construction of Isa Town.

We have in the future to explain these foreign evidences and possibly to clarify the nature of these contacts.

FIGURE 3 FIGURE 4

 34 FIGURE 3 Stone vessels. N°s 1, 2, 10 : Al-Qusais ; 3 to 9, 11, 12 : Qarn Bint Sa'ud

FIGURE 5 FIGURE 6

 56 FIGURE 5 Stone vessels. N° 21 : Rumeilah ; 22 : Al-Qusais

NOTES (1)

The terminology « Iron Age » is used with reference to the Iranian chronology, particularly Iron Age III which had clear associations with the Omani evidence. In this study « Iron Age » is not applied to the Late First Millennium BC, still considered as « Hellenistic period », in default of a better term.

(2)

I wish to thank here Dr. Karen FRIFELT who has kindly allowed me to study the stone material from Qarn Bint Sa'ud, and Dr. Munir TAHA for discussing the Al-Qusais excavation and material.

(3) I am indebted to Miss Beatrice DE CARDI to have drawn my attention to this interesting detail.

(4)

As a fact, saw-tooth incisions appear to be a very common pattern on later pottery found at hellenistic sites such as Thaj or Ed-Dour.

(5)

Polarizing microscope analysis, by courtesy of Dr. S .FUDRAL, Université de Savoie (Chambéry).

(6)

Personal communication of C. HARDY-GUILBERT, French Archaeological Mission in Qatar.


J Collomb
email: jean.collomb@univ-smb.fr

P Balland

P Francescato

D Leh

Y Gardet

P Saffré

THERMOMECHANICAL OPTIMIZATION OF AN INNOVATIVE LOW INERTIA MOLD WITH RECTANGULAR HEATING CHANNELS

Keywords: Profiled channels, Conformal channels, Rapid molding, Response surface methodology, Evolutionary algorithm

In order to be able to meet industries expectations, especially aeronautical and automotive industries, in terms of production rate, aspect and structural quality for high performance composite injected parts, it is necessary to produce reactive and thermally efficient molds. The objective of this study is to lead a thermomechanical optimization on an innovative low thermal inertia mold with rectangular heating channels. The optimized structure is compared with a reference : a massive conventional mold with circular heating channels. The finite elements model setting up for this study take into account technological constraints related to pressure drop and the use of a pump for fluid circulation. Obtained results highlight the importance of taking into account these technological constraints in the mold design. Thus, according to the choosen heating technology, water or oil heating, the optimized model of the low thermal inertia mold and with rectangular heating channels allows a reduction of the heating time by a factor of 2 for water heating and by a factor of 8.5 for oil heating.

Introduction

Today, aerospace and automotive industries are using more and more structural composite material parts, but current technologies for processing composites, especially with thermoplastic matrix, are not sufficiently competitive in terms of cost, heating rate, and reproducibility. To meet these industries expectations, it is necessary to produce thermally reactive mold capable to absorb high production rates while ensuring structural and appearance quality of molded parts. Conventional plastic injection and composite processes involve injecting a viscous polymer into a hot and massive mold and then ejecting the part. This kind of process makes possible high heating rates by avoiding thermal cycle of the mold, but it is at the origin of defects in the molded parts: internal stresses, poor surface appearance. It is therefore essential to thermally optimize molds in order to meet heating rate expectations, while controlling energy consumption during the cycle. To improve appearance and properties of molded parts, while increasing the mold productivity through increased heating rates, it is possible to optimize two categories of parameters: parameters related to the process: nature of materials (resin and preform), injection pressure, temperature, surface treatments [START_REF] Nian | Warpage control of thin-walled injection molding using local mold temperatures[END_REF][START_REF] Park | Optimization of injection molding process for car fender in consideration of energy efficiency and product quality[END_REF], and parameters related to the mold design (materials, thicknesses, channels). Parameters related to the design of the tools require thorough studies and questioning the design rules of the tools, which until now impose massive structures and straight channels created by drillings [START_REF] Dimla | Design and optimisation of conformal cooling channels in injection moulding tools[END_REF]. Studies have been conducted on the main parameters impacting thermal performance of molds: choice of mold material [START_REF] Asproiu | Influence of mold properties on the quality of molded parts[END_REF][START_REF] Guilong | Analysis of thermal cycling efficiency and optimal design of heating/cooling systems for rapid heat cycle injection molding process[END_REF], addition of insulation to focus the heat of the molded part face, optimization of the network of heating channels [START_REF] Guilong | Analysis of thermal cycling efficiency and optimal design of heating/cooling systems for rapid heat cycle injection molding process[END_REF][START_REF] Xu | Rapid thermal cycling with low thermal inertia tools[END_REF] and the type of heating technology used [START_REF] Wang | Multi-objective optimization design of the heating/cooling channels of the steam-heating rapid thermal response mold using particle swarm optimization[END_REF][START_REF] Agazzi | Optimal cooling design in injection moulding process -A new approach based on morphological surfaces[END_REF][START_REF] Xia | Conjugate heat transfer in fractal tree-like channels network heat sink for high-speed motorized spindle cooling[END_REF]. Current research on thermal optimization of mold parameters is usually carried out maintaining circular section channels with expensive fluidic models or with thermal models that do not take into account technological parameters such as pressure drops. Now, with new technologies of additive manufacturing, it is possible to produce structures with low thermal inertia and rectangular conformable heating channels for better thermal performance. The interest of this concept of mold with rectangular heating channels having been numerically demonstrated [START_REF] Collomb | Thermo-mechanical simulations for the comparison of heating channels geometries for composite material injection tools[END_REF], the objective of this paper is the thermomechanical optimization of a rectangular heating channels mold with a thermal finite element mold taking into account technological parameters: pressure drop , nature of the heat transfer fluid and the regulation of the thermoregulator's power. Methods and models used will be presented first, then the design of experiments and genetic algorithm optimization setting up for this study and finally results and discussions.

Methods and conventional model

The approach put in place for this study consists of:

1-Thermomechanical observations on the massive conventional mold presented in Fig. 1a. 2-Thermomechanical optimization of the rectangular heating channel mold presented in Fig. 1b.

This study is conducted on a representative bidirectional model in order to neglect edge effects and to focus on the heating zone of the mold. This work is based on a conventional plate mold made of stainless steel. To study the influence of the heat transfer fluid on thermal performances of the mold, thermal simulations are carried out initially with water, then in a second time with Therminold D12 oil [11]. Insulating Monolux 500 [13] Composite Carbone-Epoxy [START_REF] Hexcel | HexPly Prepreg Technology[END_REF] λ (W.m -1 .K -1 ) 15 0.18

λ x = 2 λ y = 0.2 Cp (J⋅kg -1 ⋅K -1 ) 450 1130 1000 ρ (kg.m -3 ) 7900 770 1800 E (MPa) 210000 3750 - ν 0.3 0.3 -
Finite element simulations are performed using the Ansys v18.1 software. For thermal simulations, mold, composite, insulation and outer steel plate are modeled using PLANE77 quadratic thermal elements. Volume of the fluid, power of the thermoregulator and regulation are modeled with thermal mass element MASS71. The convective exchange between the thermal mass element and the wall of the mold heating channels is performed using LINK33 elements which are convective links. For mechanical simulations, tooling, insulation and the outer steel plate are modeled with PLANE183 quadratic mechanical elements. The composite is not modeled for mechanical simulations. After testing different mesh sizes, an optimal mesh size of 1 mm was chosen for thermal and mechanical simulations. The mold is considered as symmetrical at the level of the neutral fiber of the composite.

Loading conditions applied for thermal simulations are: an initial temperature of 25°C for all parts, a natural convection on the external face of the outer steel plate: T = 25°C and h = 10 W.m 2 .K -1 [START_REF] Müller-Steinhagen | A-to-Z Guid. to Thermodyn[END_REF], and adiabatic surfaces on symmetry zones (lateral lines and neutral plane of the composite): heat flux = 0 W. A forced convection in mold heating channels is present and temperature is controlled by the heating of the thermal mass element with the thermoregulator power of 12 kW and the convective exchange is carried out by thermal link elements which have a convective coefficient calculated using the method presented in the study of Collomb and al. [START_REF] Collomb | Thermo-mechanical simulations for the comparison of heating channels geometries for composite material injection tools[END_REF] and taking into account the data of the thermoregulator pump, the nature of the heat transfer fluid and the temperature. The temperature set point is 80°C. Phenomena related to the crosslinking of the resin (including exothermic) are not taken into account.

Loading conditions applied for mechanical simulations are: zero vertical displacements of the outer line of the outer steel plate, symmetries on model lateral lines, fluid circulation pressure in the channels of 3.5 bar and injection pressure on the mold cavity surface of 15 bars. The composite is not modeled for these studies.

Thermomechanical responses observed for these simulations are:

t: heating time of the cold point of the cavity mold surface to reach 90% of the set point -T: maximum temperature difference on the molding surface over time u: the maximum vertical displacement of the molding surface.

For the optimization phase, heating time, temperature difference on the cavity molding surface and maximum vertical displacement of the molding surface shall be respectively: minimized, limited to 2°C during the heating phase and limited to 10 μm.

Design of experiment and optimization

The approach implemented to achieve optimization consists of: 1. Finite element simulations based on design of experiments (DoE) 2. Creating thermal and mechanical metamodels using the response surface methodology (RSM) 3. Thermomechanically optimization by genetic algorithm (GA).

The finite element model of the low thermal inertia mold with rectangular heating channel and its optimization variables are presented in Fig. 1b and Table 3. To avoid differential thermal expansion problems, the assumption is made that the thickness of steel above and below heating channels is identical. Construction of the response surfaces is carried out using Ellistat software and Latin Hypercube Sampling (LHD) space filling method in order to increase prediction capabilities of metamodels. The 2D finite element model is inexpensive in terms of computation time, so 100 numerical cases are generated by the LHD method. ANOVA variance analysis is carried out for the construction of metamodels by the response surface methodology.

After creating the thermal and mechanical metamodels, optimization is conducted using a genetic algorithm. The operating principle of evolutionary algorithms is to initialize a population of individuals that, generation after generation, evolve with individuals affected by: selections, crosses and mutations, which tends to improve the quality of individuals regarding to the objective function, leading to the optimal solution. Optimization parameters used for the genetic algorithm are: generation number = 500, individuals number per generation = 150, selection by tournament, individuals number per tournament = 2, tournaments probability = 0.7, crossing probability = 0.7, mutation probability = 0.03.

Results and discussion

Results obtained for the massive reference model for water heating are: heating time of 910 s, temperature difference in molding surface of 0.06°C. For oil heating, results are: heating time of 3659 s, temperature difference in molding surface of 0.03°C. The maximum vertical displacement is 3.99 μm. Change in thermal properties of the fluid between water heating model and oil heating causes a pressure drop increase (due to viscosity evolution), which causes a significant drop of the convective coefficient and therefore a significative thermal performance difference of the structure.

Polynomial metamodels representative of the innovative mold are constructed after having performed thermal and mechanical simulations from the design of experiment and after creating response surfaces using the ANOVA analysis. R² press coefficients, predictive correlation coefficient, are calculated for each metamodels and are from 0.977 to 0.996 for the thermal metamodels and 0.811 for the mechanical metamodel. The predictive correlation coefficient is calculated using a cross validation "leave-one-out" in order to measure the predictive quality of the metamodel [START_REF] Leh | Optimisation du dimensionnement d'un réservoir composite type IV pour stockage très haute pression d'hydrogène[END_REF]. Predictive correlation coefficients obtained in this study show the good quality of metamodels.

Thermal metamodels for the innovative mold for water heating are presented Eq. 1 and Eq. 2. Thermal metamodels for the innovative mold for oil heating are presented Eq. 3 and Eq. 4. The mechanical metamodel for the innovative mold is presented Eq. 5 which is independent of the chosen fluid.

t water = 450.6 + 18.51A -10.65B + 0.97A² + 0.31B² + 0.28C² + 0.04D² -0.04A 3 -0.003B 3 -0.15AB -0.21AC -0.04AD + 0.24BC

T water = -0.59 -0.229A + 0.076B + 0.153D + 0.015A² -0.002B² + 0.003D² + 0.007AC -0.015AD -0.002BC + 0.003BD -0.006CD

(2)

t oil = 2273 + 141A -107B -180C -8.06D + 1.33B² + 6.3C² -2.18AB -1.68AC -0.45AD + 3.67BC + 0.36BD + 0.78CD (3)
∆T oil = 1.73 -0.59A + 0.14D + 0.054A² -0.0036C² -0.0086AB -0.029AD -0.009BD -0.0045CD (4)

u max = 15.86 -5.13D + 0.615D² + 0.022E² -0.019D 3 -0.0002E 3 -0.051DE (5)
After conducting thermomechanical optimizations using genetic algorithm on the metamodels presented Eq. 1 to Eq. 5, optimal geometrical parameters presented Table 4 are obtained. 3a for the massive reference model and Fig. 3b for the optimized innovative mold. The shortest heating time is 430 s for oil heating, temperature fields observed at this time are presented Fig. 3c for the massive reference model and Fig. 3d for the optimized innovative mold. Optimization highlights the importance of taking into account technological aspects related to heating technologies for the mold design. Indeed, nature of the fluid, channels geometry, length of the fluid network and the pump associated with the thermoregulator are parameters having influence on pressure drop and therefore on the convection coefficient. Optimization on the rectangular channel model, shows that the constrain on the temperature difference of the molding surface is not influential. Thus, the geometric optimization tends to minimize the mass to be heated of the structure, while ensuring a significant convective exchange (by limiting the pressure losses) and ensuring sufficient mechanical stiffness. Therefore, transition from a water heater to an oil heater causes an increase in channels dimensions in order to limit pressure drop, requiring a steel thickness for a mechanical stiffness. Gains obtained on heating times after optimization of rectangular channel models compared to the massive reference model are 57% and 88% respectively for water heating and for oil heating.

Conclusion

This work was carried out for a mold and for heating technologies fixed in order to study the influence of main geometrical parameters related to thermal performance of molds. To improve thermal performances of molds, a thermomechanical optimization, which take into account technological parameters related to the circulation of the fluid, was carried out on a concept of low inertia mold with rectangular heating channels and a numerical confrontation was done with a massive reference mold with circular heating channels.

It appears that it is possible to achieve gains in terms of heating time from 57 to 88% regarding to a massive conventional mold, while controlling the temperature field on the cavity mold surface and while having sufficient rigidity. For this, it is essential when designing mold to take into account the technological aspects related to the heating system (fluid, pump, pressure drop) in order to find the optimum between: minimization of steel thickness in order to reduce heated mass, adequate channel geometry for a large convective coefficient and sufficient steel thickness to obtain a sufficient molding surface temperature field and stiffness. These heating rate gains are significant on an industrial scale since repeated on the heating and cooling phases for a very large number of cycles.

To conclude, this study confirms the strong interest of mold with low thermal inertia for obtaining better thermal performance in terms of heating rate, the need to take into account the heating technologies during the design of the mold and the interest of rectangular heating channels mold concept to promote the reduction of the mass to be heated and the control of the temperature field at the surface of the composite.

Figure 1 .

 1 Figure 1. Finite element mold models (a) conventional: massive and (b) innovative: low thermal inertia with rectangular heating channels

Figure 2 .

 2 Figure 2. Comparison of heating times between the reference model and the optimized innovative model for water heating and oil heating

Figure 3 .

 3 Figure 3. Temperature field comparison between the reference model and the optimized innovative model for water heating (a, b) and for oil heating (c, d)

Table 1 .

 1 presented in Table1. Material thermal properties are considered constant over the temperature range of the study and are presented in Table2. Mold characteristics

	Characteristics		Units
	Mold dimensions	570 x 570 x 110	mm 3
	Mold materials	Stainless steel (1.2311)	-
	Heating power	12	kW
	Heat transfer fluid	Water	-
	Fluid volume	16	l

Table 2 .

 2 Thermal properties of materials

	Stainless steel
	1.2311 [12]

Table 3 .

 3 Optimization variables

		Geometric characteristics	min	max
	A	Steel thickness	2 mm	15 mm
	B	Half channel width	5 mm	10 mm
	C	Channel height	2 mm	10 mm
	D	Distance between channels	5 mm	40 mm

Table 4 .

 4 Optimal geometrical parameters

		Parameter	Water	Oil
	Circular channel mold Reference	Steel thickness Channel diameter Distance between channels	16 mm 8 mm 16 mm
		A	2.5 mm	5 mm
	Rectangular channel mold	B	18 mm	35 mm
	Optimized	C	2 mm	4.5 mm
		D	5 mm	5 mm
	Results on the innovative optimized model for water heating are: heating time of 388 s, temperature
	difference on cavity mold surface of 0.64°C and maximum vertical displacement of 6.72 μm. For oil
	heating, results are: heating time of 430 s, temperature difference of cavity mold surface of 8.73°C. The
	comparison of heating times between the reference model and optimized innovative models for water
	heating and oil heating is presented Fig. 2. After optimization, the shortest heating time is 388 s for
	water heating. Temperature fields observed at this time are presented Fig.

J. Collomb, P. Balland, P. Francescato, D. Leh, Y. Gardet, P. Saffré

Acknowledgments

This research is supported by National Association of Research and Technology (ANRT).


Stef Graillat
email: stef.graillat@lip6.fr

Accurate simple zeros of polynomials in floating point arithmetic

Keywords: zeros of polynomials, Newton's method, condition number, floating point arithmetic AMS Subject Classifications: 12D10, 30C10, 30C15, 26C10

In the paper, we examine the behavior of the Newton's method in floating point arithmetic for the computation of a simple zero of a polynomial. We allow an extended precision (twice the working precision) in the computation of the residual. We prove that, for a sufficient number of iteration, the zero is as accurate as if computed in twice the working precision. We provides numerical experiments confirming this.

Introduction and notation

The key to compute an accurate solution to a nonlinear equation is the accurate evaluation of the function in use. In this paper, our purpose is to compute accurate simple zeros of univariate polynomials relying on Newton's method. To reach this aim, we need to focus on two important things:

• explaining what we mean by "accurate solution";

• having an accurate polynomial evalution algorithm to compute the residual in the Newton's iteration.

Let us explain now what we mean by "accurate solution". Let x be the computed solution of a problem (P ) whose exact solution is x. Suppose that the computations have been done with a t-bit floating point arithmetic. We will say the x is as accurate as if computed with twice the working precision if | x -x| |x| ≤ eps + Ceps 2 cond(P ). (1.1) where C is a moderate constant, eps = 2 -t , | • | is a norm on the space of the solution and cond(P ) is the condition number of the problem (P ). In the right-hand side of inequality (1.1), the second term reflects the computation in twice the working precision and the first one the rounding into the working precision. Relation (1.1) is what we called the compensated rule of thumb, the classic rule of thumb being [5, p.9] | x -x| |x| ≤ Ceps cond(P ).

Throughout the paper, we assume to work with a floating point arithmetic adhering to IEEE 754 floating point standard [START_REF]IEEE Standard for Binary Floating-Point Arithmetic, ANSI/IEEE Standard[END_REF]. We assume that neither overflow nor underflow occur. The set of floating point numbers is denoted by F and the relative rounding error by eps. For IEEE 754 double precision we have eps = 2 -53 and for IEEE 754 single precision eps = 2 -24 .

We denote by fl(•) the result of a floating point computation, where all operations inside parentheses are done in floating point working precision. Floating point operations in IEEE 754 satisfy [START_REF] Nicholas | Accuracy and stability of numerical algorithms[END_REF] fl(a

• b) = (a • b)(1 + ε) = for • = {+, -, •, /} and |ε| ≤ eps.
This implies that

|a • b -fl(a • b)| ≤ eps|a • b| and |a • b -fl(a • b)| ≤ eps| fl(a • b)| for • = {+, -, •, /}.
We use standard notation for error estimations. The quantities γ n are defined as usual [START_REF] Nicholas | Accuracy and stability of numerical algorithms[END_REF] by

γ n := neps 1 -neps for n ∈ N,
where we implicitly assume that neps ≤ 1.

The rest of the paper is organized as follows. In Section 2, we recall some results on Horner scheme, error-free transformations and the Compensated Horner scheme. In Section 3, we present the condition number of a simple zero. In Section 4, we present the Newton's method for root-finding using the Compensated Horner scheme to compute the residual. In Section 5, we give some numerical experiments. Finally, we conclude by giving some hints about future work.

Accurate polynomial evaluation

In this section, we first recall the Horner scheme as well as give an error bound. We then recall the classic error-free transformations. We use these transformations for a Compensated Horner scheme which gives a result as accurate as if computed by the classic Horner scheme using twice the working precision and then rounded to the working precision.

Classic Horner Scheme

The classic method for evaluating a polynomial

p(x) = n i=0 a i x i
is the Horner scheme which consists in the following algorithm. s n = a n for i = n -1 : -1 : 0

s i = s i+1 • x + a i end res = s 0
A forward error bound is (see [5, p.95]):

|p(x) -Horner(p, x)| ≤ γ 2n n i=0 |a i ||x| i = γ 2n p(|x|) (2.2)
where p(x) = n i=0 |a i |x i .

Error-free transformations (EFT)

One can notice that a • b ∈ R and fl(a • b) ∈ F but in general we do not have a • b ∈ F.

It is known that for the basic operations +, -, •, the approximation error of a floating point operation is still a floating point number (see for example [START_REF] Dekker | A floating-point technique for extending the available precision[END_REF]):

x = fl(a ± b) ⇒ a ± b = x + y with y ∈ F, x = fl(a • b) ⇒ a • b = x + y with y ∈ F, (2.3)
where no underflow is assumed for multiplication. These are error-free transformations of the pair (a, b) into the pair (x, y). Fortunately, the quantities x and y in (2.3) can be computed exactly in floating point arithmetic. For the algorithms, we use Matlab-like notations. For addition, we can use the following algorithm by Knuth [START_REF] Knuth | The Art of Computer Programming[END_REF]Thm B. p.236].

Algorithm 2.2 (Knuth [START_REF] Knuth | The Art of Computer Programming[END_REF]). Error-free transformation of the sum of two floating point numbers

function [x, y] = TwoSum(a, b) x = fl(a + b) z = fl(x -a) y = fl((a -(x -z)) + (b -z))
For the error-free transformation of a product, we first need to split the input argument into two parts. Let p be given by eps = 2 -p and define s = ⌈p/2⌉. For example, if the working precision is IEEE 754 double precision, then p = 53 and s = 27. The following algorithm by Dekker [START_REF] Dekker | A floating-point technique for extending the available precision[END_REF] splits a floating point number a ∈ F into two parts x and y such that a = x + y and x and y nonoverlapping with |y| ≤ |x|. Algorithm 2.3 (Dekker [3]). Error-free split of a floating point number into two part function

[x, y] = Split(a, b) factor = fl(2 s + 1) c = fl(factor • a) x = fl(c -(c -a)) y = fl(a -x)
With this function, an algorithm from Veltkamp (see [START_REF] Dekker | A floating-point technique for extending the available precision[END_REF]) makes it possible to compute an error-free transformation for the product of two floating point numbers. This algorithm returns two floating point numbers x and y such that

a • b = x + y with x = fl(a • b).
Algorithm 2.4 (Veltkamp [START_REF] Dekker | A floating-point technique for extending the available precision[END_REF]). Error-free transformation of the product of two floating point numbers function

[x, y] = TwoProduct(a, b) x = fl(a • b) [a 1 , a 2 ] = Split(a) [b 1 , b 2 ] = Split(b) y = fl(a 2 • b 2 -(((x -a 1 • b 1 ) -a 2 • b 1 ) -a 1 • b 2 ))
The following theorem summarizes the properties of algorithms TwoSum and TwoProduct.

a • b = x + y, x = fl(a • b), |y| ≤ eps|x|, |y| ≤ eps|a • b|. (2.5)
The algorithm TwoProduct requires 17 flops.

We present now an error-free transformation for the polynomial evaluation with Horner scheme.

Algorithm 2.5 (Graillat, Langlois and Louvet [START_REF] Graillat | Compensated Horner scheme[END_REF]). Error-free transformation for the Horner scheme

function [h, p π , p σ ] = EFTHorner(p, x) s n = a n for i = n -1 : -1 : 0 [p i , π i ] = TwoProduct(s i+1 , x) [s i , σ i ] = TwoSum(p i , a i ) Let π i be the coefficient of degree i in p π Let σ i be the coefficient of degree i in p σ end h = s 0
The next theorem proves that Algorithm 2.5 is an error-free transformation.

Theorem 2.2 (Graillat, Langlois and Louvet [START_REF] Graillat | Compensated Horner scheme[END_REF]). Let p(x) = n i=0 a i x i be a polynomial of degree n with floating point coefficients, and let x be a floating point value. Let [h, p π , p σ ] = EFTHorner(p, x) (Algorithm 2.5). Then i) the floating point evaluation h = Horner(p, x) and ii) two polynomials p π and p σ of degree n -1 with floating point coefficients,

satisfies p(x) = h + (p π + p σ )(x). (2.6)
Algorithm 2.5 requires 23n flops.

Compensated Horner Scheme

From Theorem 2.2, the global forward error affecting the floating point evaluation of p at x according to the Horner scheme is

e(x) = p(x) -Horner(p, x) = (p π + p σ )(x).
The coefficients of these polynomials are exactly computed by Algorithm 2.5, together with Horner(p, x). Indeed, if [h, p π , p σ ] = EFTHorner(p, x), then p π and p σ are two exactly representable polynomials. The key to increase the accuracy of the computed result is to compute an approximate of the global error e(x) in working precision, and then to compute a corrected result res = fl(Horner(p, x) + e(x)).

We say that c = fl(e(x)) is a corrective term for Horner(p, x). The corrected result res is expected to be more accurate than the first result Horner(p, x).

Our aim is now to compute the corrective term c = fl ((p π + p σ )(x)). For that we evaluate the polynomial whose coefficients are those of p π + p σ rounded to the nearest floating point value. This process is described by Algorithm 2.6.

Algorithm 2.6. Evaluation of the sum of two polynomials.

function res = HornerSum(p, q, x) r n = fl(a n + b n ) for i = n -1 : -1 : 0 r i = fl(r i+1 • x + (a i + b i )) end
We can now describe the Compensated Horner Scheme. The following theorem proves that the result of a polynomial evaluation computed with the Compensated Horner scheme (2.7) is as accurate as if computed by the classic Horner scheme using twice the working precision and then rounded to the working precision.

Theorem 2.3 (Graillat, Langlois and Louvet [START_REF] Graillat | Compensated Horner scheme[END_REF]). Given a polynomial p = n i=0 a i x i of degree n with floating point coefficients, and x a floating point value. We consider the result CompHorner(p, x) computed by Algorithm 2.7. Then,

|CompHorner(p, x) -p(x)| ≤ eps|p(x)| + γ 2 2n p(x).
(2.7)

The Algorithm CompHorner requires 26n + 3 flops.

Condition number for root finding

Given a problem, we want to know how to measure the difficulty of solving it. This will be done via the notion of condition number. Roughly speaking, the condition number measures the sensitivity of the solution to perturbation in the data. Here is the classic definition for the condition number of root finding for simple roots.

Definition 3.1. Let p(z) = n i=0 a i z i be a polynomial of degree n and x be a simple zero of p. The condition number of x is defined by

cond(p, x) = lim ε→0 sup |∆x| ε|x| : |∆a i | ≤ ε|a i | .
In the previous definition, ∆x represents the variation of the zero x when the polynomial is perturbed by a polynomial ∆p(z) = n i=0 ∆a i z i . It means that x + ∆x is a zero of p + ∆p.

The following theorem gives an explicit formula to compute the condition number.

Theorem 3.1 (Chaitin-Chatelin and Frayssé [START_REF] Chaitin-Chatelin | Lectures on finite precision computations. Software, Environments, and Tools[END_REF]). Let p be a polynomial of degree n and x be a simple zero of p. The condition number of x is given by

cond(p, x) = p(|x|) |x||p ′ (x)| .
Proof. We recall the proof done in [START_REF] Chaitin-Chatelin | Lectures on finite precision computations. Software, Environments, and Tools[END_REF]. Let p(z) = n i=0 a i z i and consider the map

ϕ : C n+1 -→ C, (a 0 , . . . , a n ) T -→ z such that p(z) = 0, z simple.
From the definition of ϕ, it follows that p(ϕ(p)) = 0. The chain rule gives

∂ϕ ∂p i (p)p ′ (z) + z i = 0, i = 0 : n.
This implies that ϕ ′ (p) = -1 p ′ (z) z T . Using Taylor expansion, we obtain

x + ∆x = x - 1 p ′ (x) x T ∆p + O( ∆p 2 ). (3.8) From Definition 3.1, it follows that cond(p, x) = 1 |x||p ′ (x)| sup |x T ∆p|, |∆a i | ≤ |a i | . It is clear now that cond(p, x) = |x| T |p| |x||p ′ (x)| = p(|x|) |x||p ′ (x)| .

Accurate Newton's method

In this section, we present the analysis of [START_REF] Tisseur | Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems[END_REF] for the Newton's method in floating point arithmetic (see also [START_REF] Daniel | Adaptive multiprecision path tracking[END_REF]). We specialize the result from [START_REF] Tisseur | Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems[END_REF] in the case of an univariate polynomial with a simple root. We use the Compensated Horner scheme to accuratly compute the residual. In that case, we show that the computed result (an approximation of the simple root of the polynomial) is as accurate as if computed with twice the working precision via the classic Newton's method and then rounded back to the working precision.

General results on Newton's method

In [START_REF] Tisseur | Newton's method in floating point arithmetic and iterative refinement of generalized eigenvalue problems[END_REF], F. Tisseur provided a comprehensive analysis of the Newton's method in floating point arithmetic. We recall hereafter her analysis and her results (see also [5, chap.25]). Let F : R m → R m be continuously differentiable on R m and J the Jacobian matrix (∂F i /∂x j ) of F . We assume that J is Lipschitz continuous with constant β in R m , that is,

J(w) -J(v) ≤ β w -v for all v, w ∈ R m ,
where • denotes any vector norm and the corresponding operator norm. We denote by κ(J) = J J -1 the condition number of the matrix J. We aim at solving the system of nonlinear equations F (x) = 0 by Newton's method:

J(x i )(x i+1 -x i ) = -F (x i ), i ≥ 0, (4.9)
for a given point x 0 . Equation (4.9) can be rewritten as

Solve J(x i )d i = -F (x i ), x i+1 = x i + d i .
Due to rounding errors in floating point arithmetic, we have

x i+1 = x i -(J( x i ) + E i ) -1 (F ( x i ) + e i ) + ε i ,
where

• e i is the error made during the computation of the residual F (x i ),

• E i is the error when forming J( x i ) and solving the linear system for d i ,

• ε i is the error made when adding d i to x i .

We assume that F ( x i ) is computed with extended precision eps ′ and then rounded back to working precision eps, and that d i , x i are computed at precision eps. Hence we assume that there exists a function ψ depending on F , x i , eps and eps ′ such that e i ≤ eps F ( x i ) + ψ(F, x i , eps, eps ′ ).

We assume that the error E i satisfies

E i ≤ epsϕ(F, x i , n, eps).
For the error ε i , we have ε i ≤ eps( x i + d i ).

Theorem 4.1 (Tisseur [9,Cor. 2.3]). Assume that there is an x such that F (x) = 0 and J = J(x) is nonsingular and satisfies eps κ(J) ≤ 1/8 where κ(J) = J J -1 denotes the condition number of the matrix J. Assume also that for ϕ, eps J( x i ) -1 ϕ(F, x i , n, eps) ≤ 1/8 for all i.

Then, for all x 0 such that β J -1 x 0 -x ≤ 1/8, Newton's method in floating point arithmetic generates a sequence of { x i } whose normwise relative error decreases until the first i for which

x i+1 -x x ≈ eps + J -1
x ψ(F, x i , eps, eps ′ ). (4.10)

Newton's method for polynomials

Let p be a given polynomial with simple zeros. We apply the Newton's method with F (x) = p(x) and so J(x) = p ′ (x). The classic Newton's method is Algorithm 4.1.

Algorithm 4.1. Classic Newton's method

x 0 = ξ x i+1 = x i -p(x i ) p ′ (x i )
Hereafter, we use the compensated Horner scheme to evaluate the residual p(x) in order to get a result as accurate as if computed in twice the working precision. We also assume that we already know that the root we are looking for belongs to [a, b] with a, b ∈ R. We also define

β = max x∈[a,b] |p ′ (x)|.
The accurate Newton's method is Algorithm 4.2.

Algorithm 4.2. Accurate Newton's method

x 0 = ξ x i+1 = x i -CompHorner(p,x i ) p ′ (x i )
In that case, using notation of

4.3

sub:newton, we have eps ′ = eps 2 and ψ(F, x i , eps, eps ′ ) = γ 2 2n p(x) thanks to Theorem 2.3. Moreover, using Theorem 4.1 and Theorem 3.1, it holds the following theorem. Theorem 4.2. Assume that there is an x such that p(x) = 0 and p ′ (x) = 0 is not too small. Assume also that eps cond(p, x) ≤ 1/8 for all i.

Then, for all x 0 such that β|p ′ (x) The theorem means that if we begin not too far from the simple root, the Newton's method gives an approximation of the root as accurate as if computed with twice the working precision.

The use of an accurate polynomial evaluation algorithm is essential. Indeed, if we use the classic Horner scheme, then, at the end of the iteration, we only have

|x i+1 -x| |x| ≈ γ 2n cond(p, x). ( 4
.12)

Numerical experiments

All our experiments are preformed using the IEEE-754 double precision with Matlab 7. When needed, we use the Symbolic Math Toolbox to accuratly compute the roots of polynomials (in order to compute the relative forward error).

We test the Newton's iterations on the expanded form of the polynomial p n (x) = (x -1) n -10 -8 for n = 1 : 40. The condition number cond(p n , x) where x is the root 1 + 10 -8/n varies from 10 4 to 10 22 .

Figure 1 shows the relative accuracy | x-x|/|x| where x the exact root and x is the computed value by the two algorithms 4.1 and 4.2. We also plot the a priori error estimation (4.12) and (4.11).

As we can see in Figure 1, the accurate Newton's iteration exhibit the expected behavior, that is to say, the compensated rule of thumb. As long as the condition number is less that 10 15 , the accurate Newton's iteration produce results with full precision (forward relative error of the order of 10 -16 . For condition numbers greater thant 10 15 , the accurary decreases. In the paper, we have proved that the Newton's iteration makes it possible to compute a good approximation of a simple root; by good approximation, we mean as accurate as if computed with twice the working precision. We only dealt with simple roots. If the root has multiplicity m > 1, one can use the modified Newton's iteration as follows. Algorithm 6.1. Modified Newton's method

x 0 = ξ x i+1 = x i -m p(x i ) p ′ (x i )
A future work will be to see if we can get the same kind of results than for simple roots when we already know the multiplicty of the root.

Algorithm 2 . 1 .

 21 Polynomial evaluation with Horner's scheme function res = Horner(p, x)

Theorem 2 . 1 (

 21 Ogita, Rump and Oishi[START_REF] Ogita | Accurate sum and dot product[END_REF]). Let a, b ∈ F and let x, y ∈ F such that [x, y] = TwoSum(a, b) (Algorithm 2.2). Then,a + b = x + y, x = fl(a + b), |y| ≤ eps|x|, |y| ≤ eps|a + b|.(2.4)The algorithm TwoSum requires 6 flops. Let a, b ∈ F and let x, y ∈ F such that [x, y] = TwoProduct(a, b) (Algorithm 2.4). Then,

Algorithm 2 . 7 (

 27 Graillat, Langlois and Louvet [4]). Compensated Horner scheme function res = CompHorner(p, x) [h, p π , p σ ] = EFTHorner(p, x) c = HornerSum(p π , p σ , x) res = fl(h + c)

Figure 1 :

 1 Figure 1: Accuracy of the classic Newton iteration and the accurate Newton iteration