Marc-Olivier Déplaude

Minimising dietary risk: the French association of salt producers and the manufacturing of ignorance

Keywords: Risk, dietary risk, social amplification of risk framework, ignorance, food industry, repertoires of collective action

Minimising dietary risk: the French association of salt producers and the manufacturing of ignorance

Introduction

In this article I aim to contribute to the literature on 'the social amplification of risk' [START_REF] Kasperson | The social amplification of risk: A conceptual framework[END_REF]. In contrast to the initial formulation of this theory and the numerous works it has inspired, my approach consists in making actors' strategies the core focus of my analysis and, more specifically, in examining how far they are prepared to go to minimise or conceal certain risks, and to avoid amplifying them. To broaden the application of the social amplification framework I draw on research in the field of Science and Technology Studies that has focussed in the 'social production of ignorance' [START_REF] Kleinman | Dying bees and the social production of ignorance[END_REF]. This research examines the social processes and strategies that lead to certain problems being ignored or minimised so that they are confined within limited or marginal spaces. I test this approach to the amplification of risk framework by studying the strategies employed by the French association of salt producers (Comité des Salines de France -the Comité) to counter official public health advice that French citizens should reduce their salt intake. I argue that these strategies are part of a repertoire of action available to industrial organisations which want to counteract adverse assessements of the risk posed by their products to human health.

Risk, ignorance and food

The social amplification and attenuation of risk

Kasperson and his colleagues developed the 'social amplification of risk framework' to account for their observation that: even risk events with minor physical consequences often elicit strong public concern and produce extraordinarily severe social impacts, at levels unanticipated by conventional risk analysis' (Kasperson et al., 1988, p. 177).

Drawing on communication theories, Kasperson and his colleagues suggested that an initial risk signal is modified every time it moves from one receiver/transmitter to another, and that being diffused in this manner, it can be amplified or reduced. They noted that the mass media played an important role in this process, as did certain organisations such as 'multinational corporations, business associations, and government agencies' (Kasperson & Kasperson, 1996, p. 98). They argued that the ways in which risk assessment was modified depended on the properties of each of these social different spheres. For example: Large corporations develop markedly different kinds of organizational cultures that shape powerfully their ability to identify and assess the risks of their activities and products and to determine if and how the risks will be communicated to other social institutions and publics. (Kasperson & Kasperson, 1996, p. 98) Proponents of this framework argued that these processes can have a very significant economic impact on certain sectors of activity, and cause certain products or technologies to be lastingly stigmatised in the public space.

This theory has been very influential in risk studies and a number of risk researchers have used it to explore the ways in which risk assessments are modified and in particular the role of the media in such changes [START_REF] Henderson | The role of the media in the construction and presentation of food risks[END_REF]. Some of these researchers have also examined the logic underpinning the ways in which public and private agencies respond to and shape risk assessments as they attempt to predict people's reactions or to avoid blame and the ways in which this can influence the amplification of risk [START_REF] Burgess | The making of the risk-centred society and the limits of social risk research[END_REF][START_REF] Torny | Food risk facing consumers[END_REF].

However there are important limitations to the the social amplification of risk framework. Firstly, the work of Kasperson and his colleagues is based on the assumption that there is 'an external risk signal that is altered in the communication process between an initial receiver/transmitter and subsequent receivers/transmitters' (Rayner, 1988, p. 202). This implies that risk communication starts with a 'real' risk, as assessed by competent experts, which is then distorted by amplification or attenutation. And that such distortions have undesirable effects, such as 'marring and compromising the potential benefits to society from economic growth and technological change' (Kasperson & Kasperson, 1996, p. 100). The focus of Kasperson and his colleagues is on the risk experts, policy makers and risk communication specialists, who need to accurately communicate risk information and minimise distortion with all its potentially harmful effects. Secondly, the framework focusses on the mechanic of communications, the movement of the risk message from one transmitting station to the next, and does not really explore the role of those transmitting the messages. For example, while the distorting effect of mass media is noted there is little consideration of the role of journalists and how and why they report information in the ways they do. Researchers using the amplification framework tend to explain the modification of risk messages largely in terms of cultural or structural factors and not in terms of the actions of those transmitting the messages (see [START_REF] Renn | The Social Amplification/attenuation of Risk Framework: Application to climate change[END_REF][START_REF] Zinn | Towards a better understanding of risk taking: Key concepts, dimensions and perspectives[END_REF]. Thirdly, in both the initial conceptualisation of the framework and in the research it stimulated the emphasis has been on amplification rather than attenuation processes. This is probably because researchers have been mainly interested in issues which have been the subject of fierce public debate (genetically modified organisms, mad cow's disease, Asian flu, etc.), and in which risk has been 'amplified' rather than in risks which are ignored or rarely discussed in the public space.

In this article, I intend to deal with the limitation of the social amplification of risk framework, by taking better account of the strategies used by organisations and those working for them and of the dynamics which might lead to certain risks being minimised or concealed. Rather than analysing risk amplification or reduction as mechanical processes, I will examine strategies which aim to make certain risks more visible or, on the contrary, to conceal them. Anticipating the negative consequences that greater visibility of certain risks might have on their activities, organisations and individuals who have an economic stake in a risk assessment can make considerable efforts 'to interrupt a social amplification of risk process' (Busby et al., 2009, p. 298). For this reason, I feel it is useful to combine works from the field of risk studies with current research on the social production of ignorance.

Manufacturing ignorance

Over the last ten years there has been a growing literature on ignorance. As McGoey (2012) recently underlined, this issue is not entirely new to sociology. Several authors, from different research traditions, have looked at similar subjects, such as amnesia and forgetfulness in institutions [START_REF] Douglas | How institutions think[END_REF], the uneven circulation of information in organisations [START_REF] Crozier | Actors and systems: The politics of collective action[END_REF][START_REF] Vaughan | The dark side of organizations: Mistake, misconduct, and disaster[END_REF], the strategic uses of knowledge in public arenas [START_REF] Hilgartner | Science on stage: Expert advice as public drama[END_REF] or the processes of confinement and invisibilisation of certain social problems [START_REF] Bachrach | Two faces of power[END_REF][START_REF] Gilbert | Defining social problems: Tensions between discreet compromise and publicity[END_REF]. However, it is within the framework of Science and Technology Studies that ignorance has become the central subject of an increasing number of works, with some social scientists using a neologism, 'agnotology' [START_REF] Proctor | Agnotology: A missing term to describe the cultural production of ignorance (and its study)[END_REF] to designate this field of research. Much of this research is concerned with the question of 'undone science' [START_REF] Hess | Alternative pathways in science and industry: Activism, innovation, and the environment in an era of globalization[END_REF][START_REF] Frickel | Undone science: Charting social movement and civil society challenges to research agenda setting[END_REF], that is: the kinds of research that get systematically left unfunded, ignored or sidelined, but [are] recognized by [other] actors as being worthy of serious consideration (Kleinman & Suryanarayanan, 2013, p. 5).

Kleinman and Suryanarayanan tried to identify and describe the social mechanisms that explain why certain types of knowledge are not produced or are not considered worthy. They identified a number of factors:

• The specific logic of individual academic disciplines. Each discipline tends to attach importance and value to certain subjects, this means that other potentially ones are neglected by that discipline; • The rules in force in institutional arenas of expertise. They are embedded in the protocols and instruments used in those arenas so that certain types of knowledge are systematically undervalued or disregarded [START_REF] Frickel | Untangling ignorance in environmental risk assessment[END_REF][START_REF] Henry | Militer pour le statu quo [Fighting for the status quo[END_REF]. For example, when studying the toxicological testing carried out in New Orleans following hurricane Katrina, [START_REF] Frickel | Hurricane Katrina, contamination, and the unintended organization of ignorance[END_REF] showed that said tests were not 'neutral technologies' and that they contributed towards highlighting certain types of risk and concealing others -such as those which resulted, for example, from the presence of low doses of toxins. Knowledge and ignorance are thus produced together. • The uneven social and spatial distribution of certain health and environmental risks. Marginalised populations are very unlikely to complain about damage to their health or their environment, and thus to make broader audiences aware of them. Such risks therefore tend to be disregarded [START_REF] Allen | Uneasy alchemy: Citizens and experts in Louisiana's chemical corridor dispute[END_REF][START_REF] Auyero | Flammable: Environmental suffering in an Argentine shantytown[END_REF][START_REF] Blum | Love Canal revisited: Race, class, and gender in environmental activism[END_REF].

The first group of studies, several of which can be included in the 'new political sociology of science' (Frickel & Moore, 2005), focuses on the structural and institutional mechanisms which produce ignorance, noting that the creation of such ignorance is often an unintentional result of said mechanisms. Although researchers in this first group do note that organisations and individuals with an economic stake can use the creation of ignorance to promote their interests, this is not the main focus of their research.

A second group of studies focuses on the efforts that certain actors have made to deny (or at least play down) any health or environmental risks. In studying the deliberate strategies which manufacture ignorance, these researchers have concentrated on the production of toxic substances such as tobacco [START_REF] Proctor | Golden holocaust: Origins of the cigarette catastrophe and the case for abolition[END_REF], chemicalS [START_REF] Markowitz | Deceit and denial: The deadly politics of industrial pollution[END_REF][START_REF] Ross | The Polluters: The making of our chemically altered environment[END_REF]Vogel, 2013) and asbestos [START_REF] Mcculloch | Defending the indefensible: The global asbestos industry and its fight for survival[END_REF]. [START_REF] Mcgoey | The logic of strategic ignorance[END_REF] coined the phrase 'strategic ignorance' to describe practices which consist in mobilising the unknowns of a given situation. Researchers who have examined this aspect of the creation of ignorance, some of whom have been interested in taking political action to uncover and denounce [START_REF] Michaels | Doubt is their product : How industry's assault on science threatens your health[END_REF][START_REF] Oreskes | Merchants of doubt : How a handful of scientists obscured the truth on issues from tobacco smoke to global warming[END_REF], have concentrated on collecting examples and empirical proof to support their theories. Few have tried to develop a more theoretical understanding based on systematic description and categorisation of the strategies designed to manufacture ignorance. Rayner's article (2012) is one of the rare attempts in doing this. Rayner starts by looking at how certain institutions manage what he calls 'uncomfortable knowledge', that is knowledge which might threaten the way they operate. Drawing on the anthropological works of [START_REF] Douglas | Purity and danger: An analysis of concepts of pollution and taboo[END_REF] and [START_REF] Evans-Pritchard | The Nuer[END_REF], Rayner argues that the survival of any institution or organisation relies on certain forms of knowledge being excluded: because they threaten to undermine key organizational arrangements or the ability of institutions to pursue their goals. (Rayner, 2012, p. 108) To avoid such a knowledge being used against them, managers in these organisations mobilise various more or less conscious strategies. Rayner identifies four types: strategies of denial, dismissal, diversion (or decoy) and displacement. Rayner defines denial as 'a refusal to acknowledge or engage with information' (Rayner, 2012, p. 113). He makes a distinction between these strategies and those of dismissal, which recognise 'the existence of information, and may involve some minimal engagement up to the point of rebutting it as erroneous or irrelevant' (Rayner, 2012, p. 113). In this article, I will adopt a broader approach and use denial to refer to both denial and dismissal as these are two alternative ways of disregarding a problem.

There are limitations of Rayner's approach. The first, which he acknowledges, is the temporary and incomplete nature of the list of strategies analysed, which as I have already noted could be grouped together differently. The other stems from his methodological approach; he builds his analysis based on examples of his experience as an expert in various public action programmes, especially in the field of the environment. This approach does not indicate how the same organisation can successively or simultaneously mobilise several types of strategy, or how the latter interconnect with one another. It is possible that these strategies are components of the same repertoire.. My aim in this article is to contribute towards the sociological study of risk, using recent works on the social production of ignorance to develop the social amplification of risk framework. But it also aims to add to this research by highlighting the existence of a repertoire of action that is common to numerous industry players.

Methods

In this article I draw on data from a case study of an association of salt manufacturers -the Comité des Salines de France. The aim of this study was to carry out an exhaustive analysis of the strategies that an association of manufacturers could use to minimise and conceal health risks relating to the activities or products of its members. As a case study design involves collecting a large volume of data on a limited subject, it enabled me to relate practices to specific social contexts and thus to adopt an interpretative understanding of social action, by carefully reconstituting the participants' strategies and reasons for acting, rather than using a priori models of behaviour (Ragin & Becker, 1992;[START_REF] Yin | Case study research: Design and methods[END_REF].

I chose to study the Comité des Salines de France, because it is a business association which has for some considerable time (more than three decades) used a wide range of strategies to try to undermine official advice about dietary salt and to promote a positive image of salt. The first stage of the study involved nine semi-structured interviews with Comité managers and with those who have had dealings with them (salt producers, experts and researchers and senior civil servants from the French Ministry of Health). These interviews gave me access to three private archives, which allowed me to gain a very precise overview of the Comité's practices. To begin with, a former manager at a cooperative of salt producers allowed me to examine almost all of the Comité's annual reports from 1974 to 2008. Averaging between twenty and thirty pages, these reports provide economic and technical data on the French salt sector and on Comité's main activities for the year in question. A former senior civil servant from the French ministry of health then allowed me to consult archives pertaining to the salt industry. These archives contained various letters from the Comité to the ministry of health and to the French National Institute of Health and Medical Research . Finally, an Institute of Health researcher who was sued for defamation by the Comité in 2007, sent me the documents he had collected while preparing his defence (Comité press files and releases, letters, business reports, etc.). I made use of two other sources to complete the data from these archives: as far as possible, I collected all the articles, documents and brochures written by Comité members or with the financial or documentary support of the Comité or its members (these documents were held in various university libraries); and I went through several press documents (including the staff newsletter for Salins' French subsidiary) and various national newspapers and magazines (including some medical journals) over a period ranging roughly from the mid-1970s to now. I also transcribed all reports relating to salt which were shown on early evening news programmes or as part of specialist programmes shown on the six main national channels, and which were available for that period from the French National Institute of Audiovisual Medias.

I gathered these different sets of data to reconstitute, as fully as possible, Comité strategies aimed at different targets (doctors, journalists, teachers, general public). This was also a way of using cross-cutting sources and of convincingly recontextualizing the idealistic image that the Comité paints of itself. In the 'grounded theory' spirit [START_REF] Glaser | The discovery of grounded theory[END_REF], I inductively analysed the data to allow the emergence of an analysis which would evolve certain pre-existing theoretical frameworks. I use the Comité as an example to show how strategies which produce ignorance are part of a repertoire of action common to many industry actors; this is a new contribution to both the literature on the social amplification of risk and to the literature on the social production of ignorance.

This research was conducted in accordance with the ethical guidelines laid down by the British Sociological Association (2002). When sociologists examine controversial issues, it is sometimes difficult for them to defend certain positions without being suspected or accused of bias towards one of the parties, and thus of harming certain actors. Through the simple fact of describing certain social practices, and thus objectifying them, sociologists can harm the interests of those who would prefer to keep their actions, or the reasons for their actions, hidden from as many people as possible. Whilst it is difficult to prevent such effects, which are inherent to social science research, it is certainly possible to avoid causing harm to specific persons. In this study I have therefore been careful to maintain the anonymity of people who have not already been named in the public space. But even more importantly, as mentioned above, by choosing a methodology based on an interpretative understanding of social action, I was able to accurately reconstitute the reasons behind the actors' actions. As far as this research is concerned, it was not a question of separating the 'heroes' from the 'villains', but of understanding how, within a given context, the representatives of an economic sector acted to defend their own interests.

Findings

A business association under challenge

The Comité was founded in the 1920s and is a very small organisation. From the 1970s onwards it had a maximum of two permanent employees; a 'general manager' and a secretary. As salt is relatively cheap, the Comité represents a modest economic sector which employed less than 3,000 persons in the early 1980s and approximately 2,000 at the time of writing. It is dominated by the Salins group, which currently employs 1,500 people (700 in France), with a turnover of less than 300 million euros. The group mainly produces crystallised salt for human consumption (the most profitable), agriculture, gritting roads and industrial chemicals. It owns the La Baleine TM brand which it uses to market the majority of its household salt. In France the Solvay group is the second largest salt producer.It is one of the world's leading chemical groups, and it produces brine for its chemical installations, and, in lesser quantities, crystallised salt for human consumption, via Cérébos, which it bought in 1967. In 2002, Cérébos became a subsidiary of the European Salt Company (ESCO), founded by Solvay and German group K+S, and then owned by K+S only. Salins and Solvay (then K+S) overwhelmingly dominate CSF's board of directors: since the beginning of the 1970s, the board has been mainly composed of Salins and Solvay executives and has always been chaired by a Salins director.

From the end of the 1970s, the Comité was facing a major challenge, with the development in France and in other western countries of new public health nutritional education campaigns stressing the need to moderate consumption of certain foodstuffs and nutrients, such as salt, which was implicated in the development of high blood pressure and cardiovascular diseases. In 1981, the Comité Français d'Education à la Santé (French committee for health education) began to promote the idea that 'healthy eating' meant having a balanced and varied diet, and 'reducing the levels of sugar, fat and salt in traditional recipes' [START_REF] Cfes | Présentation de la campagne 'Santé et plaisir de la table[END_REF]. Consumers were encouraged not to unthinkly add salt to all of their food. The messages were widely disseminated to the general population through new communication techniques, such as audiovisual media [START_REF] Berlivet | Uneasy prevention: The problematic modernization of health education in France after 1975[END_REF]. After a decline of these campaigns in the 1990s, new public health information campaigns were organised in the 2000s, mainly as part of the fight against obesity.

Faced with stagnating salt sales as from the end of the 1970s, the Salins group was particularly sensitive to these campaigns, especially as salt for human consumption was sold at a much higher price than that used for chemicals or gritting. Through the Comité began to make efforts to preserve salt's reputation in the fields of health and nutrition.

In 1977, the Comité decided to closely monitor the debates which were developing on how salt affects health. In 1983 it founded a 'public relations' work group made up of Salins and Solvay executives. Its purpose was to ensure that 'the image of table salt is not tarnished, as has notably been the case in the United States and Great Britain' [START_REF] Csf | Rapport du délégué général[END_REF].

Indirect communication and self-dissimulation

With the help of a public relations agency, the Comité decided to concentrate its efforts on three categories of professionals, who it expected to pass on its messages to a wider public: journalists, healthcare professionals and teachers. The choice of indirect communication can be interpreted in two ways. Firstly, it is less expensive than major advertising campaigns in the mainstream media. The Comité has very limited financial resources (in 2011 its annual budget was 300,000 €), and its members probably prefer to allocate their advertising budgets to the promotion of their own products. This explains why over the last forty years the Comité only organised one advertising campaign on French television -in 1990-1991. Secondly, the Comité hoped that these three categories of professionals would not only relay its messages to a wider audience, but that they would also lend their strength of authority, whilst at the same time blurring and dissimulating any connection with economic interests. With the same objective in mind, in 1983 the Comité also founded a Centre de Recherche sur les Utilisations du Sel, a centre for research on the uses of salt). In fact this research centre employs no researchers at all; it merely serves as an acronym that the Comité can use to disseminate documents which seem to be scientific, without appearing to be their author. Once again, this is a case of the Comité dissimulating or, even better, hiding its economic interests from the audiences targeted by the documents in question.

The Comité preferred method for changing journalists' opinions is to send them press kits and press releases, or to talk to them individually. To access healthcare professionals, the Comité's general manager was able to take part in congresses for dieticians and to arrange the publication of notes or articles in nutrition journals, including two in the Cahiers de Nutrition et de Diététique, a journal published by a learned society and aimed at academics and practitioners [START_REF] Crus | Le sel alimentaire [Dietary Salt][END_REF][START_REF] Moinier | Le sel fluoré: de la production à la consommation [Fluoridated salt: From consumption to production[END_REF]. Since 2008, the Comité has also been publishing a quarterly e-newsletter presenting recent scientific studies on salt and health. Finally, to influence school teachers and their pupils, the Comité offered pedagogical kits (fact sheets, slides, games) which were mainly distributed to primary and junior schools. For example, in its 1985 business report, the Comité's executive director explained:

A salt information sheet was created in 1984. It is in two parts. The first part describes the main aspects of the salt economy. The second provides educational information on salt production methods and its uses…It [this sheet] will allow primary and secondary school teachers and pupils […] to learn about this ordinary but essential product. (CSF, 1985, p. 19). The Comité's approach was reflected in that adopted by the two major salt manufacturers, Salins and Solvay who have provided documentary or financial support for several books on the history, production techniques and uses of salt (see for example, [START_REF] Colas | [END_REF][START_REF] Moinier | Sel et société[END_REF].

The Comité used a number of techniques to enhance the impact of its 'salt is good for you' message. It used Salins' documentation department, which houses a collection of books and works on the history and uses of salt. It employed a public relations agency whose main tasks were to monitor the press, to help develop documents aimed at outside audiences and to ensure their effective distribution. Since 1987, the Comité has worked with medical specialist Tilman Drüeke, a nephrologist and director of a research unit in a prestigious Paris hospital. Drüeke's research led him to emphasise the complexity of the nutritional factors which caused high blood pressure and thus to put into perspective the notion that salt in food is bad for health. His research stance was in line with the Comité's interests, and it invited him to make regular summaries of medical literature on the subject of 'salt and health', to monitor work carried out by the World Health Organization and to attend medical congresses on issues of interest to salt producers. In exchange, the Comité made payments to his hospital to fund research projects.

Using such resources, the Comité managed to compile a large volume of data which it then used to deny or play down the danger of salt for health, or to divert attention away from the issue.

Strategies of denial

In my analysis of the documents developed and disseminated by the Comité I found a second set of strategies, which involved making a selective use of scientific literature and bringing into play the uncertainty evident in debates taking place within academic circles. I will refer to these as strategies of denial, their main objective being to deny the fact that salt in food is a danger to health, or, at the very least, to deny that the dangers are such that they need to be made public. Strategies of denial thus consist in denying that a given problem is a problem worthy of attention, whatever the relevant existing data may be.

In order to minimise the role that salt plays in the development of high blood pressure, the Comité argued that the salt intake of French population was lower than what was commonly stated, and that it is generally at levels with which the kidneys can cope. Furthermore, blood pressure depended on many other factors, such as excess weight or genetic predispositions. The advantage of these arguments is that they do not place responsibility for the development of high blood pressure on salt and hence on manufacturers who encourage its consumption.

However, the Comité could not deny that research existed in the danger of excess salt consumption, which have led the public health agencies to recommend that the whole French population reduced its salt intake. The Comité sought to undermine this by highlighting the uncertainty of current research and academic opinion arguing that preventive action aimed at the whole population rather than just at-risk groups was premature. The Comité. In other words, the CSF was not content to simply assert that the question of how salt affects health was subject to debate -it actually spotlighted such debates about the uncertainty of research findings. One way in which it did this was to organise counter-conferences, events that brought together French and foreign experts that were sympathetic to the Comité . For example, in March 2001, the Agence Française de Sécurité Sanitaire sur les Aliments ( the French Food Safety Agency) set up a working group to examine ways of reducing the salt content in processed foods. Its conclusions were made public by the deputy minister for health in January 2002, during an international 'salt and health' conference organised by Food Safety Agency. To counter these initiatives and muddy the water by making the case that there was no scientific consensus on the risks of eating salt the Comité organised a counterconference in September 2001, and another one the same day as the Food Safety Agency 'salt and health' conference.

A similar approach is evident in the quarterly e-newsletter published by the Comité which from 2008 onwards has highlighted the abundance and diversity of scientific works on the links between salt and health. The repeated message in the newsletter is that the relationships between salt and health are complex, that there is on-going scientific debate that had not to date been conclusive: 'The literature monitoring carried out in early 2009 has taught us a great deal. First and foremost, it shows that the 'salt and human health' debate continues to fascinate researchers and clinicians, and that it is far from closed' (CSF, 2009, p. 1).

The Comité approach has been to maintain doubt, or, more specifically, of accomplishing a 'work of confusion' [START_REF] Auyero | The social production of toxic uncertainty[END_REF] which consisted in disseminating into the public space data which was presented as being contradictory. Such confusion was maintained through the complementary use of diversion strategies.

Creating diversions

The Comité's diversionary strategies consisted in drawing attention to the benefits of salt and to the virtuous practices of the salt industry. Its 1990-1991 television campaign sought to link salt with the pleasures of eating. Commenting on the television advert, which showed a plate of vegetables which look decidedly tired until they were freshened up with a sprinkling of salt, the Comité's general manager told his members that:

It was a deliberate choice to look at salt from a pleasure standpoint, in order to reassure those who some people have been trying to make feel guilty for using the salt shaker. [START_REF] Csf | Rapport du délégué général[END_REF] Furthermore, in the various documents the Comité distributed since the 1980s, particularly those targeting teachers and schoolchildren, it relentlessly stressed the fact that salt was a natural product and was vital to health, that salt had a long history, and through its many different uses made a major contribution towards the comfort of modern life. These arguments also underpinned the various Comité publications targeted at the general public. Whilst the publications reminded readers that they should not eat too much salt, this advice tended to be underplayed compared to the panegyric about salt's thousand-year history and the long list of its supposed benefits.

From the last 1980s, the Comité linked its narrative on the benefits of salt to the use of salt as a means of fighting deficiencies in iodine and fluorine within the French population. In 1952, public health authorities authorised salt producers to add small quantities of iodine to salt reduce the risk of goiter, a serious thyroid disorder. In 1985 the French government also authorised manufacturers to add fluoride as a means of reducing dental decay. Public health agencies considered salt as a good medium for such supplements as most of the population ate salt in small quantities.

In 1980s Salins and Solvay were already selling salt with iodine and in the late 1980s started selling salt which had both iodine and fluoride. This had a double benefit: not only was salt with iodine and fluorine more profitable it was also a way of promoting the health benefits of salt. Through the Journées Nationales de la Diététique (national diet weekly) in Marseille in 1988, the Comité's general manager set out the health benefits of salt with iodine and fluoride supplements. The Comité also organised publicity of the use and benefits of salt as a medium for health supplements and health promotion by taking part in panels about this subject under the auspices of the Journées Nationales de la Diététique in 1991, and of the Entretiens de Bichat (a training event for general medical practitioners) in 1994. In 1990 in partnership with the Union Française pour la Santé Bucco-dentaire (French union for oral hygiene) the French government organised a large-scale publicity campaign, stressing the benefits of fluoride supplements in salt. The campaign distributed 100,000 posters with the slogan 'change your salt' to dentists for display in their waiting rooms, together with several 100,000 leaflets.

From the mid-1990s, the Comité collaborated with the United Nations International Children's Fund (UNICEF) in a campaign for the universal iodisation of salt, aimed mainly at southern-hemisphere countries. Salins' marketing department initially developed this initiative with with the Centre International de l'Enfance (international childhood centre), and then with UNICEF. In 1995, the following text was printed on the side of 'La Baleine TM 's salt dispensers: La Baleine, sea salt and our children's growth. LA BALEINE sea salt does more than add flavour to food. With its unique know-how and high technology, LA BALEINE sea salt optimises our intake of iodine -an element which is vital to children's growth -and provides the fluoride which is needed to prevent tooth decay. Protecting what is most fragile: childhood. In many countries, children are not lucky enough to have an available source of iodine and fluoride. Which is why LA BALEINE sea salt is working with the Centre International de l'Enfance (CIE -international childhood centre) whose aim is to sustainably improve the health and development of children throughout the world. ('La Baleine partenaire… ', 1995) The salt with iodine and fluoride supplements enabled the Comité and its members to promote themselves as acting in the public interest and to reverse salt's negative image by linking it to a virtuous cause, that of helping vulnerable children in the southern hemisphere. At the same time, Salins has become involved in activities designed to protect the environment.

With such strategies, the Comité and Salins were trying not just to promote the benefits of salt and their sense of responsibility, but also to make people forget the negative aspects of salt -in other words, to create diversions. These diversion tactics served as ways of reassuring consumers about the benefits of eating salt. As the Comité's general manager noted with satisfaction in 1993, 'as a medium for iodine and fluoride, salt offers a reassuring image' [START_REF] Csf | Rapport du délégué général[END_REF]. Such diversion strategies served to build a façade of respectability, which helped conceal the manufacturers'self-interest and the potentially harmful nature of salt consumption. Such diversionary strategies were a form of dissimulation.

Udermining and intimidating opponents

From 2000, salt was again the subject of criticism by public health and nutritional experts who highlighted the 'hidden salt content' of foods and salt's role in cardiovascular disease and mortality. These experts also contested the Comité and its salt was a deadly poison, like tobacco,which was defamatory and untrue. However Meneton successfully defended himself in court using Comité documents to show similarities between the Comité's public relations strategies and those used by tobacco companies.

Discussion

The strategies used by the Comité to defend a positive image of salt and to undermine official dietary advice are similar to those used by other organisations facing with similar challenges. As Meneton successfully argued in court the defensive activities of the Comité were similar to those used by the tobacco companies to defend their interests. In his study of US tobacco companies, [START_REF] Proctor | Golden holocaust: Origins of the cigarette catastrophe and the case for abolition[END_REF] describes in great detail strategies of denial and diversion similar to those that I have analysed here. Markowitz and Rosner (2002, p. 64-107) have shown how certain chemical companies work to produce reassuring images of their products, for example by linking them to the medical world or to young children. In the same fashion, Ferreira has noted that 'dispelling consumers' risk perceptions is primarily attempted by means of directing attention to certain contexts, and hence, placing others at the edge or in opposition' (Ferreira, 2006, p. 857). Other studies, for example by [START_REF] Sismondo | Corporate disguises in medical science: Dodging the interest repertoire[END_REF] on the pharmaceutical industry, have highlighted the 'corporate disguises' of certain manufacturers and their use of 'key opinion leaders' to promote their products. Whilst the Comité's strategies were not exact replicas of those used by these organisations, and their financial resources were far more limited, those used by the Comité are not dissimilar. When manufacturers are trying to defend their reputation or that of their products, in terms of health or environmental risk, everything takes place as if there were not an infinite number of ways of achieving this, but rather a pre-established set of tools and modes of action that these organisations or companies can use.

Using [START_REF] Tilly | From mobilization to revolution[END_REF] study of collective action, I would argue that this preestablished set of modes of action constitutes a repertoire of collective action. For Tilly, the notion of a repertoire of collective action implies a relative stability in the modes of action that are used by a given category of actors to defend an interest or a cause: collective action usually takes well-defined forms already familiar to the participants, in the same sense that most of an era's art takes on of a small number of established forms. (Tilly, 1978, p. 143) For manufacturers, the range of actions available to them to promote a positive image of their products or to create confusion is always very limited, whatever resources they have. They do not systematically invent new forms of action, though this does not exclude, from time to time, innovations or adaptations which might lead to a gradual change in the repertoire content.

The existence of a repertoire of action available to organisations who are trying to defend their reputation or that of their products implies there is some circulation of information about this repertoire. Such circulation is probably helped by factors relating to the morphology of companies and associations of manufacturers, such as the multiple activity or internationalisation of firms, cross-shareholdings, or the existence of closely entwined multi-sector and/or transnational employer organisations. The Comité and its members are not isolated. Since the beginning of the 1980s the Comité has maintained regular relations with counterparts in other European countries, within the framework of a joint organisation founded in 1957, and also with the Salt Institute -the association of North-American salt producers. Similarly, Salins was owned by the American group Morton between 1996 and 2006, and Solvay, one of the world's leading chemical groups, has subsidiaries in other countries. These elements encourage the executives of large companies and employer organisations to maintain sociability networks beyond sectorial or national frameworks, thus promoting the circulation of knowledge and know-how (for example, since 1962 the Salt Institute has organised 'international symposia' on salt, bringing together researchers from academic institutions and corporate executives). The same is true of the many national and international advisory bodies, home to manufacturers' numerous representatives, especially in relation to environmental and health risks (Boudia & Jas, 2014). Finally, consultants and communication agencies -widely used by manufacturers and employer organisationsprobably play a major role in the circulation of public relations know-how and practices. Oreskes and Conway (2012) have recently substantiated this point by showing that in the United States, manufacturers from different sectors have often used the same communication agencies and the same experts on matters as diverse as tobacco, acid rain, the ozone hole and global warming.

It is common for manufacturers and employers' associations to use strategies designed to minimise or divert the public's attention away from certain risks. These strategies are part of everyday practices. The findings in this article support my intial argument that the social amplification of risk framework must be modified so as to take better account of participants' strategies, and to ensure that greater attention is paid to the practices of minimising or concealing certain risks.

Conclusion

In this article I have identified and described some of the principal strategies employed by manufacturers or their associations to defend the image of certain products when their health or environmental safety is challenged. There are certainly activities which were not evident in the case studied here, such as the displacement activities identified by Rayner, which consist in substituting: management of a representation of a problem […] for a management of the represented object or activity. (Rayner, 2012, p. 113) My aim in this article was not to identify all possible strategies, a task which would require a different type of research, but rather to show how such strategies could be simultaneously used by a single organisation, and to what extent they are part of the same repertoire of collective action into which other actors can delve. In this sense, the Comité was helping, at its own level, to update and perpetuate this repertoire.

I found through my analysis that some of the strategies are probably more effective than others. Diversion strategies are also evident in food marketing practices, and more particularly in packaging. As Cochoy clearly noted, packaging plays the dual role of projection and occultation since it prevents any direct access to the products and is a medium of displaying information controlled by the producers [START_REF] Cochoy | Sociologie du packaging ou l'âne de Buridan face au marché[END_REF]Cochoy & Grandclément-Jaffy, 2005). Despite ever more detailed regulations defining the content and form of the information which must appear on packaging, manufacturers systematically manage to highlight some of their products' attributes whilst concealing others. We can therefore support the hypothesis that whilst diversion strategies are not as spectacular as direct attacks on scientists, they are far more widespread and are probably more effective: in helping to disseminate contradictory messages into the public space, they probably make a far greater contribution to concealing or minimising certain environmental or health risks.

The example of the Comité is exemplary in the sense that it is a very small organisation. This suggests that even with very limited resources, many manufacturers' associations are in a position to accomplish the work that the Comité has been doing since the beginning of the 1980s. There are dozens of such 'micro-organisations' in the agrifood sector. The multiple messages disseminated by these organisations are certainly not entirely foreign to the coexistence of multiple and contradictory requirements in the public space. This 'nutritional cacophony' [START_REF] Fischler | L'homnivore. Le goût, la cuisine et le corps[END_REF], which nutritionists regularly lament, nevertheless constitutes one of the latter's preferred arguments for rejecting any nutritional policy which appears to be too proactive.

I have shown in this article the extent to which the seeds of confusion are sown by certain economic actors and employers' organisations in relation to food-related risks. I have argued that the objectification of these practices of minimizing or deliberately concealing risks is a challenge which is at least as important for the social study of risk as the analysis of processes of social risk amplification.

members claims that salt contributed to health. They accused the Comité of misinforming French people about salt's adverse effects on health, and of influencing official nutritional advice on salt consumption.

Faced with such criticism, the Comité continued to use its existing range of countermeasures to defend salt but also mobilised new strategies designed to undermine and intimidate its opponents. It used stigma reversal strategies, that is strategies which consisted in denying the stigma and deflecting back onto those who were trying to attach it to the salt manufacturers. The Comité asserted that the scandal did not lie in 'hidden salt' and in lobbying by the salt industry, but in the unfounded accusations made against salt manufacturers. The Comité accused those trying to stigmatise salt as dangerous -researchers, journalists, and consumer associations -conspiring in an 'antisalt lobby'. The Comité argued that this lobby selected only research and experts who supported their anti-salt cause and that any consensus over the harmful nature of salt was illusionary. For example the Comité's general manager in a letter to the French Director General for Health in 2003 stated that :

The purpose of the conference organised in January 2002 [ the Salt and Health conference organised by French Food Safety Agency], attended by a majority of reductionist experts (including members of the CASH group [a group established by British researchers in 1996 to campaign for a reduction in the amount of salt in processed foods -Consensus Action on Salt & Health]), was to create the impression of a consensus by keeping away experts who were more cautious about the opportunity for a universal reduction in sodium intake.

The Comité was accusing its opponents of using the practices which they accused the Comité of using. The Comité was willing to make such allegation not only about the collective action of its critics but also of individual critics. Such individual and personal attacks were evident in the Comité's focus on Pierre Meneton, a researcher from the French National Institute of Health and Medical Research who had a high media profile and who was critical of agrifood industries. The attacks came in three forms. In their press releases the Comité questioned Meneton's competency and impartiality. They attempted to silence him by complaining to his superiors. In March 2001, the Comité's general manager sent a thick file to the Institute's managing director to draw his attention to Meneton's 'wild anti-salt imaginings'. Sending managing director a reminder of the fact that the Comité had helped to fund the Institute's research, the Comité's general manager asked him whether the institute had 'a disciplinary committee which might reprimand' Meneton. In April 2006, following publication of another interview with Meneton in a news magazine, the Comité's general manager again wrote to the Institute's managing director, asking him 'to impose appropriate sanctions on this extravagant researcher'.

When these demands were not met, the Comité opted for another approach , using his latest interview a defamatory and suing Meneton in court . The Comité also joined up with Salins' French subsidiary to bring civil proceedings against the director and a journalist of the magazine, for having illustrated the interview with a reproduction of a La Baleine TM salt box with a label similar to those found on cigarette packets stating that 'salt kills'. According to the plaintiffs, the interview and its accompanying illustration amounted to the assertion that salt producers had tried to hide the fact that


A resource modality for RAII

Guillaume Combette (CNRS, ENS Lyon)

Guillaume.Combette@ens-lyon.fr Guillaume Munch-Maccagnoni (Inria, LS2N CNRS) Guillaume.Munch-Maccagnoni@inria.fr 31st May 2018

RAII and move semantics

Stroustrup's "Resource acquisition is initialisation" idiom (RAII, [START_REF] Stroustrup | The design and evolution of C++[END_REF]) attaches destructors to types in C++, called whenever the lifetime of a variable ends, either by the end of its scope being reached, by an exception being raised, or by a control operator (return, break) being called. It is used in C++ to ensure the basic exception-safety guarantee [START_REF] Stroustrup | Exception Safety: Concepts and Techniques[END_REF]. Unlike finalizers called by a tracing garbage collector, destructors are called at fixed and predictable times. RAII allows a form of resource management, for ensuring for instance that dynamically-allocated memory is always freed by the end of a scope. It is also used to ensure that locks are always freed, connections are always closed, etc. Doing so amounts to treat locks and connections as resources. Thus, a main point of RAII is that destructors may perform effects.

The extension of C++ with move semantics [START_REF] Hinnant | A Proposal to Add Move Semantics Support to the C++ Language[END_REF] allowed to express the moving of non-copiable resources. Moving a resource alters its lifetime: it can change the order in which destructors are called, or transfer the duty of calling the destructor to a different scope. Notably it allowed the definition of a non-copiable smart pointer for automatic resource management (unique_ptr) expressing ownership.

Baker (1994a,b, 1995) has proposed a synthesis of the notion of resource from systems programming with that of resources from linear logic [START_REF] Girard | Linear Logic[END_REF]. Arguably, it contained an early description of move semantics (it mentioned in particular the compatibility of moving with C++-style destructors). Although these articles described many of the ideas behind the resource management model of the C++11 (Stroustrup, Sutter, and Dos Reis, 2015) and Rust (Anderson, Bergstrom, Goregaokar, Matthews, McAllister, Moffitt, and Sapin, 2016) languages, they appear in advance of their time and rarely mentioned. In this presentation, we substantiate a link between C++-style destructors and linear logic.

A resource modality for RAII

We consider L any distributive symmetric monoidal closed category (such as in particular any standard model of linear logic). For any ∈ L, there is a monad -⊕ . It has been noticed in [START_REF] Hasegawa | Semantics of linear continuation-passing in call-by-name[END_REF]) that this monad lacks in general a strength, and therefore cannot be used to model exceptions like in a cartesian setting [START_REF] Moggi | Notions of computation and monads[END_REF]. Intuitively, the operations bind and raise (or throw) need to dispose of variables in their context.

The main idea to model exceptions in L is to consider the slice category L∕ where is the monoidal unit. We recall that the slice category C∕ of a category C for ∈ C has objects ( , ) for ∈ C and ∈ C( , ), and morphisms those in C that preserve the second component. In particular, ( , id ) is terminal, and so L∕ is affine. When an object ∈ L interprets a type and ∈ L( , ) interprets a derivation, we think of ( , ) as another type obtained by attaching the destructor to the type .

We are more generally interested in the case where we are given a strong monad ( , , , ) on L. We consider the slice category L∕ and think of objects ( , ∶ → ) as destructors that may perform an effect. We recall the following result attributed to Street1 : Proposition 1. For any monoid in L with multiplication ∈ L( ⊗ , ) and unit ∈ L( , ), the slice category L∕ has a monoidal structure with unit = ( , ) and tensor ( , )⊗ ( , ′ ) = ( ⊗ , • ⊗ ′ ). The forgetful functor ∶ L∕ → L is strict monoidal. Now, the object has a monoid structure given by • , ∶ ⊗ → and ∶ → . Thus, L∕ has a monoidal structure and strict monoidal forgetful functor ∶ L∕ → L. L∕ has a terminal object ( , id ). Notice that if L is symmetric, the symmetry does not necessarily lift to a symmetry on L∕ . This is the case though whenever is commutative. Otherwise, there is a definite order in the application of destructors: the destructor of ⊗ first calls the destructor of and then the destructor of .

We notice that the functor has a right adjoint if and only if L has the products -× (as is the case in any model of multiplicative-additive intuitionistic linear logic). One therefore has a monoidal adjunction L∕

/ / ⊥ L o o
, giving rise to a resource modality = on L in the sense of [START_REF] Melliès | Categorical semantics of linear logic[END_REF]. When L has finite products, this adjunction has the structure of a (non-commutative) linear call-by-push-value model [START_REF] Curien | A Theory of Effects and Resources: Adjunction Models and Polarised Calculi[END_REF]. In particular, its deductive system given by the oblique morphisms of the adjunction (Munch-Maccagnoni, 2014), still expresses multiplicative-additive intuitionistic linear logic, though with fewer identities between derivations, reflecting the presence of an evaluation order. The deductive system includes a symmetry ⊗ ⊢ ⊗ found in

L( ⊗ , ⊗ ) ≅ L∕ ( ⊗ , ( ⊗ )) ,
in words, moving resources is available as an effectful operation.

In this setting, we study the monad E = (-⊕ ) on L and strength-like maps , ∶ ⊗ E → E( ⊗ ) defined for ∈ L∕ and ∈ L.

Resource management modes as polarities

We will conclude with considerations of programming language design following from the analogy: smart pointer ∼ resource modality which is suggested by Chirimar, Gunter, and Riecke, 1996 (a resource modality for a referencecounted garbage collection) and the previous section (a resource modality for unique_ptr).

We propose to extend it into an analogy: resource management mode ∼ polarity where the notion of polarities [START_REF] Girard | A new constructive logic: Classical logic[END_REF][START_REF] Girard | On the Unity of Logic[END_REF] suggests a way of mixing different resource management modes as kinds in a functional programming language, presented recently in a companion article (Munch-Maccagnoni, 2018).

https://mathoverflow.net/a/229371


Joseph Morlier
email: joseph.morlier@isae-supaero.fr

Aniello Basile

Ankit Chiplunkar

Miguel Charlotte

An EGO-like Optimization Framework for Sensor Placement Optimization in Modal Analysis

In aircraft design, Ground/Flight Vibration Tests (GVT/FVT) are conducted to extract aircraft's modal parameters (natural frequencies, damping ratios and mode shapes) also known as the modal basis. The main problem in aircraft modal identification is the large number of sensors needed, which increases operational time and costs. The goal of this paper is to minimize the number of sensors by optimizing their locations in order to reconstruct a truncated modal basis of N mode shapes with a high level of accuracy in the reconstruction. There are several methods to solve Sensors Placement Optimization (SPO) problems, but for this case an original approach has been established based on an iterative process for mode shapes reconstruction through an adaptive Kriging Metamodeling approach so-called EGO-SPO. The main idea in this publication is to solve an optimization problem where the sensors locations are variables and the objective function is defined by maximizing the trace of criteria so-called AutoMAC. The results on a 2D wing demonstrate a reduction of sensors by 30% using our EGO-SPO strategy.

Introduction

Experimental Modal Analysis (EMA) is a common experimental methodology to estimate the dynamic characteristics of a structure. The identification process consists of estimating the modal parameters from a set of Frequency Response Functions (FRF). Sensor Placement Optimization (SPO) is a common problem encountered in many engineering applications, that has led to the development of several techniques [START_REF] Kammer | Sensor placement for on-orbit modal identification and correlation of large space structures[END_REF][START_REF] Minwoo | Optimal Sensor Placement for Modal Identification of Bridge Systems Considering Number of Sensing Nodes[END_REF][START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF]. The SPO techniques have been applied to various mechanical, aerospace, and structural systems for designing the best sensing locations, which are used to estimate modal parameters based on vibration responses [START_REF] Yuan | An approach to optimal sensor placement for vibration tests on large structures[END_REF][START_REF] Ewins | Modal Testing: Theory, Practice and Applications[END_REF].

Vibration tests on large structures such as aircrafts are however a costly and time consuming work. Typically, in aircraft design this involves a set of 500 accelerometers that, in addition to the difficulty to place them where it is appropriate, also add masses and may modify significantly then the real dynamic characteristics of the structure. In particular, the position of these vibration sensors influences the modal identification quality of structures, which for a given number of sensors is usually estimated in terms of correlation between the natural modes using the Modal Assurance Criterion (MAC) [START_REF] Allemang | The modal assurance criterion-twenty years of use and abuse[END_REF].

In the past, several works have already tried to tackle the SPO problem by various methods. Thus, Jung et al [START_REF] Jung | Sensor placement optimization for structural modal identification of flexible structures using genetic algorithm[END_REF] have recently proposed a discrete-type optimization problem using Genetic Algorithm (GA). This latter was formulated by defining the sensor positions and the MAC as the design variables and the objective function, respectively. Schulze et al [START_REF] Schulze | Optimal sensor placement for modal testing on wind turbines[END_REF] have also introduced a GA approach but with a weighted off-diagonal criterion. For wind turbine identification, this approach notably yields the sensor configuration with the highest quality. Tong et al [START_REF] Tong | Optimal sensor placement for mode shapes using improved simulated annealing[END_REF] have presented an improved simulated annealing (SA) algorithm to solve the sensor placement problem. The proposed method has been tested on a numerical slab model that consists of two hundred sensor location candidates using three types of objective functions: the determinant of the Fisher Information Matrix (FIM), the MAC and the Mean Square Error (MSE) of mode shapes. It may also be worth mentioning that optimal sensor placement is also a current trend in Structural Health Monitoring (SHM). Yi et al. has developed innovative algorithms for optimizing system sensor arrays in civil engineering [START_REF] Yi | A modified monkey algorithm for optimal sensor placement in structural health monitoring[END_REF][START_REF] Yi | Optimal sensor placement for structural health monitoring based on multiple optimization strategies[END_REF].

Besides the foregoing approaches, we have discussed in our previous work [START_REF] Morlier | Original statistical approach for the reliability in modal parameters estimation[END_REF] of the metamodeling capability of Kriging for mode shapes reconstruction. As a different application, this paper aims now at extending our original approach to the SPO aims. Recently a global optimization strategy so called Efficient Global Optimization (EGO) has been developed using the Kriging's variance information [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] and its (improved) use is here of interest as an alternative to the previous approaches. In short, Kriging or Gaussian Process regression is a method of interpolation for which the interpolated values are modeled by a Gaussian Process governed by a prior covariance [START_REF] Matheron | Principles of Geostatistics[END_REF]. More recently it has been extended on bridge structures using an improved version of kriging, called robust kriging (RK) [START_REF] Simon | Mode shape expansion for lively pedestrian bridges through Kriging[END_REF]. The basic idea of kriging is to predict the value of a function at a given point by computing a weighted average of the known values of the function in the neighborhood of the point.

The aforementioned EGO method becomes popular as it is an adaptive sequential method that balances local and global search, i.e. balance exploitation of some metamodels and exploration of the design space [START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF]. These metamodels, or surrogate models (also called experts) are notably mathematical functions tuned for approximating a black-box functions. An enhanced version of EGO was first introduced in the pioneering work of Forrester and Jones [START_REF] Forrester | Global optimization of deceptive functions with sparse sampling[END_REF] where they succeed in optimizing the geometry of a passive vibration isolating truss. Besides, the machine learning community has generalized Kriging's theory into a versatile computational framework called GPML [START_REF] Rasmussen | Gaussian Processes for machine learning[END_REF]. This methodology is well known in machine learning/engineering optimization for reaching a global optimum at a fixed budget even at in a high dimensional constrained optimization problem [START_REF] Bouhlel | Efficient global optimization for high-dimensional constrained problems by using the Kriging models combined with the partial least squares method[END_REF]. It also has been validated in aerodynamic shape optimization [START_REF] Bartoli | Improvement of efficient global optimization with application to aircraft wing design[END_REF][START_REF] Bartoli | An adaptive optimization strategy based on mixture of experts for wing aerodynamic design optimization[END_REF] using a modified EGO algorithm based on Mixture of Experts (MOE). Regarding the present work, we derive the fixed budget approach, normally used to limit costly computer experiments, to a sensor placement optimization problem. More precisely, we want to automate the sensor placement methodology using an adaptive strategy derived from EGO for mode shapes reconstruction (a basis of N modes). Starting from few sensor locations (initial design of experiments), the method iteratively adds new sensors at positions that tend to maximize the function trace(MAC). MAC is widely used for comparing mode shapes which is normally computed between the selected mode shapes and targeted mode shapes. In a supervised approach the targeted modes can be an analytical or FEM based. In an unsupervised methodology, the AUTOMAC can be used [START_REF] Allemang | The modal assurance criterion-twenty years of use and abuse[END_REF]. An equivalent objective function is to minimize norm(MAC -I), where I is the identity matrix. The EGO approach consists in using an analytical criterion, so-called Expected Improvement, to minimize/maximize an objective function.

The paper will be divided into three main parts. We first introduce theoretical background for modal analysis and metamodeling. Then the EGO-SPO original strategy will be presented step by step. A supervised test case will permit to validate EGO-SPO in 1D. Finally, two test cases based on aircraft wings will be introduced: the first one in 1D, the second one in 2D to reveal the ability of Kriging based method for mode shapes reconstruction. (2)

Modal parameter estimation is a special case of system identification where the a priori model of the system is known to be in the form of modal parameters. The identification process considers a set of Frequency Response Functions (FRF) as input. Identification of the FRF by means of spectral analysis is usually considered wellinvestigated. A unified matrix frequency-based approach can be found for instance in [START_REF] Allemang | A unified matrix polynomial approach to modal identification[END_REF].

We must precise however here that this paper is not dealing with errors of FRF estimation (H1, H2 etc…) [START_REF] Shin | Fundamentals of signal processing for sound and vibration engineers[END_REF] and so with the choice of optimal FRF estimator. The current approach in modal identification involves using numerical techniques to separate the contributions of individual modes of vibration in measurements such as FRF. Each term of the FRF matrix in Equation ( 2) can be represented in terms of pole location and a mode shape:

) ( ) ( ) ( * * k k k k j r j r j H l w l w w - + - = (3)
In Equation (3), k r is the (n by n) shape for the k th mode (* designates complex conjugates); k l is the pole value for mode k = Given that this work is dedicated to find the optimal spatial sensor locations while focusing only mode shapes reconstruction techniques, one major hypothesis for this aim is that the mechanical system under study must be linear. Then the only inputs for our methodology are the estimated mode shapes (also call modal vectors, or again eigen-vectors) that could be stored in a modal basis (or again a set of eigen-vectors associated to eigen-values).

Practically, it can be viewed as the response of displacement sensors giving the out-of-plane information at a certain frequency for a unitary amplitude excitation. The MAC is calculated as the normalized scalar product of the two sets of vectors {ϕ A } and {ϕ B }. The resulting scalars are arranged into the MAC matrix:

𝑀𝐴𝐶 𝑟, 𝑞 = @ A B C @ D E 6 @ A B C @ A B @ D E C @ D E (4)
In Equation ( 4), the matrix component MAC(r,q) is computed between mode r of the first family and mode q of the second family. MAC values oscillate between 0 and 1, the unitary value meaning a perfect correlation. In practice, a value greater to 0.9 is commonly recognized as acceptable to establish the correspondence between two mode shapes. The symbols A and B represent the two sets of mode shapes which usually refer respectively to simulation (analytical or FEA) and experimental data; the MAC matrix gives so quantitatively a good idea of the closeness between two families of mode shapes.

Stubbs and Park [START_REF] Stubbs | and Sikorski Optimal sensor placement for mode shapes via Shannon's sampling theorem[END_REF] have extended the Whittaker-Nyquist-Kotelnikov-Shannon sampling theorem to spatial data for avoiding the well-known problem called "aliasing". In figure 1, we can see that 9-grid points measurement are not enough precise to reconstruct the (3,1) mode shape. This example illustrates the importance of sensor placement when the natural modes of flexible structure are extracted by experiment using a limited number of vibration sensors. In this case, there are insufficient measured degrees-of-freedom in order to discriminate between the different modes. The only solution is to measure more degree-of-freedom. However, the MAC can tell us whether there are enough measurement points. This is done using a version of the MAC called the AutoMAC in which a set of modes are correlated with themselves.

Metamodeling for mode shapes reconstruction

In this paper, we will concentrate in particular on some approaches which are demonstrated to be very successful at finding global solutions of black-box global unconstrained optimization problems. The main idea behind these methods is to iteratively construct metamodels to approximate the black-box functions (globally) and use them to search for optimal solutions. Complementary reviews can be found in [START_REF] Vu | Surrogate-based methods for black-box optimization[END_REF][START_REF] Queipo | Surrogate-based analysis and optimization[END_REF]. A common algorithm for metamodel-based methods is as follows:

Plate length (cm) Plate width (cm) Phase 1 is commonly referred to as sampling (also called design of experiments, DOE) [START_REF] Sacks | Design and analysis of computer experiments[END_REF]. Its purpose is to find a set of points uniformly spread over the domain, so that, if we evaluate the function at these points, we might obtain a global representation of its range. Latin Hypercube Sampling (LHS) is the most popular method to perform sampling [START_REF] Morris | Exploratory designs for computational experiments[END_REF].

In Phase 2, various models can be used to approximate black-box functions. In the next section, we will introduce some of the most popular models, including polynomials, radial basis functions (RBF) and kriging.

Phase 3 is the crucial step in this common algorithm. Given the information from the current metamodel, we need to choose which point(s) should be evaluated in the subsequent step (so called enriching point). The most common strategy is to select the next point for evaluation as the one that maximizes (or minimizes) the merit function (also known as cost function).

Definition of the chosen metamodeling techniques

In this section, we choose to introduce the chosen metamodeling techniques and explain them using in few basic steps. Three models will be studied here: Polynomial Approximation (PA), Radial Basis Function (RBF) and Kriging.

A polynomial approximation of 𝑓 of order m can be written, in the one-variable case, as: 𝑓 𝑥, 𝑚, 𝑤 = 𝑤 I 𝑥 I J IKL [START_REF] Ewins | Modal Testing: Theory, Practice and Applications[END_REF] We want to estimate the weight vector w = {w 0 , …, w m } T in Equation ( 5) through the solution of y = Φw, where Φ is the Vandermonde's matrix. The maximum likelihood estimate of w is w = Φ + y, where Φ + =(Φ T Φ) -1 Φ T is the Moore-Penrose pseudo-inverse matrix. This gives an estimation of w at chosen order m. Obviously, the problem can be extended to more variables, while considering this general equation:

𝑓 𝑥 = 𝑤 I 𝜓 I O P IK4 (6)
In Equation ( 6), the ψ are picked from a basis truncated at order m. For example, with m = 2,

ψ (i) ∈ {1, x 1 , x 2 , x 1 x 2 , x 1 2 , x 2 2 }.
As an alternative, a more interesting metamodel can be used. The radial basis function approximation is given by:

𝑓 𝑥 = 𝑤 S 𝜓 = 𝑤 I 𝜓 O T IK4 𝑥 -𝑐 (I) (7)
In Equation [START_REF] Allemang | The modal assurance criterion-twenty years of use and abuse[END_REF], c (i) denotes the i th of the n c basis function centers and ψ is the n c -vector containing the values of the basis functions ψ evaluated at the Euclidean distances between the prediction site x and the centers c (i) of the basis functions. The classical basis function used is the cubic one ψ(r)= r 3 . Whether we choose a set of parametric basis functions or fixed ones, w is easy to estimate, via interpolation condition. Albeit the equation is linear in terms of the basis function weights w, yet the predictor 𝑓 can express highly nonlinear responses. It is easy to see that one of the conditions of obtaining a unique solution is that the system must be square, that is n c = n. Simplifications arise if the bases actually coincide with the data points, that is c (i) = x (i) , which leads to the equation Ψw = y, where Ψ is the Gram matrix, defined as Ψ I,1 = 𝜓 x (I) -x (1) for i,j = 1,…, n. The fundamental step of the parameter estimation process is to calculate w = Ψ -1 y. Of course, the choice of the basis functions can have important effects on the metamodel prediction [START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF].

Finally, we will present a more versatile metamodel so called Kriging. Originally developed for geostatistic purposes, Kriging is a method of interpolation for which the interpolated values are modeled by a Gaussian process governed by prior covariances [START_REF] Matheron | Principles of Geostatistics[END_REF]. It assumes that the spatial variation of an attribute is neither totally random, nor deterministic. Today, it is widely used in other domain of engineering and often called Gaussian Process Regression (GPR). This method uses a linear combination of all sampling values, their weights determined by their distances from the interpolation point. This requires therefore some knowledge about the relation between the distance and the covariance to be described. The Kriging method is merely built by injecting a basis function in Equation ( 7) defined as:

𝜓 (I) = 𝑒𝑥𝑝 - 𝜃 1 ] 1K4 𝑥 1 (I) -𝑥 1 ^_ ( 8
)
In equation ( 8), the vector θ allows to vary the width of the basis function from variable to variable. This equation ( 8) is also known as Standard Exponential (SE) kernel. Interested reader can find the detailed procedure used in [START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF]. The exponent p can vary for each dimension in x. We denote this by using a set of random vectors Y = {Y(x 1 ), …, Y(x n )} T . The random variables are correlated with each other with the Kriging basis function ψ (i) , from which it's possible to construct the correlation matrix Ψ and the covariance matrix cov(Y,Y)=σ 2 Ψ of size n. This assumed correlation between the sample data reflects our expectation that an engineering function will behave in a certain way. The correlations depend on the absolute distance between the sample points and on the parameters p and θ. For example, with a very low value of p there is no immediate correlation between the points. The parameter θ is a width parameter that affects how far a sample point's influence extends. A low θ means that there is high correlation between points; a high θ means the contrary. We can also consider θ as a measure of how 'active' is the function that we are approximating. The main problem is then to choose p and θ. A solution can be maximizing the likelihood of y using classical optimizer (gradient based, genetic algorithms etc…) [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF][START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF][START_REF] Forrester | Global optimization of deceptive functions with sparse sampling[END_REF][START_REF] Rasmussen | Gaussian Processes for machine learning[END_REF].

We can then compute the prediction and so-called predict function using the Forrester's toolbox [START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF]:

𝑦 𝑥 = 𝜇 + 𝜑 S 𝜓 d4 𝑦 -𝐼𝜇 (9)
The parameter 𝜇 (mean) can be defined as:

𝜇 = g C h ij k g C h ij g ( 10
)
The associated variance 𝑠 n as:

𝑠 n 𝑥 = 𝜎 n 1-𝜑 S 𝜓 d4 𝜑 + (4dg C h ij @)n g C h ij g ( 11
)
where

𝜎 n = (kdgp) C h ij (kdgp) O .

Illustrative example on a clamped plate (supervised case)

Let us precise our problem of mode shapes reconstruction. The design space belong to R 3 and can be defined as (x,y,Z). Here (x,y) represents the sensor location, and Z is the local mode shape value. The goal is to learn from this database of examples the true mode shape basis. Thus, the estimation of mode shapes reconstruction accuracy will increase together with the sampling density. Of course, the complexity will grow when taking into account triaxial sensors but this is not in the scope of this paper. The performance of our algorithm is defined by the number of evaluations needed (so called fixed budget) until an acceptable (global) solution is found. One stopping criterion can be the maximum number of sensors. The objective is to optimize the locations of sensors for the purpose of making the most accurate predictions of the mode shapes at unmeasured locations. We use a MATLAB based Finite Element model (FEM) of a plate, an eigenvalue solver for modal analysis. The presented test case is a clamped plate. The Fixed-Free plate has one clamped edge and three free ones (CFFF). It has been done to build a supervised data. Thus, we can compute standard prediction criteria such as Root Mean Square Error (RMSE). This is a frequently used measure of the differences between values predicted by a model or an estimator 𝑦 and the values actually observed y. The RMSE expresses the average model prediction error in units of the variable of interest. It is calculated as: [START_REF] Yi | Optimal sensor placement for structural health monitoring based on multiple optimization strategies[END_REF] The used geometrical and material properties are: length=width= 0.8m, height= 0.01m, E = 210e9 Pa, nu = 0.33, rho = 7700. A dedicated mesh is used to obtain the truncated reduced basis {Φ 1 ,Φ 2 ,..,Φ N }. In practice we use the nine first bending modes from mode (1,1) to mode [START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF][START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF]. Each classical reconstruction method has been compared with these supervised data. Three types of sampling have been used (regular grid, random grid, and pseudo random grid). Appendix A illustrates the result for a regular grid of 24 sensors with the mode [START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF][START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF]. The pseudo-random grid is made for a half part with uniformly spaced sensors while sensors are added randomly for the other half part. Table 1 This preliminary study confirms that Kriging is very well adapted to the mode shape reconstruction problem because it gives in each case the lowest RMSE.

𝑅𝑀𝑆𝐸 = 4 O 𝑦 1 -𝑦 1 n O 1K4

Sensor Placement Optimization inspired from Efficient Global Optimization

EGO

As already mentioned, the EGO algorithm [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] is an adaptive optimization method based on kriging. An initial design of experiment is used to build a first metamodel. In the present study, the MATLAB function, lhsdesign, was used to obtain the LHS points with the default criterion (maximin). This criterion maximizes the minimal distance between points: it proceeds by iteratively generating a number of LHS samples at random and chooses the best one based on the criterion maximin. To generate the experimental design (sample), the number of sample points is specified. A new point that maximizes a criterion is chosen as optimizer candidate at each iteration. The criterion uses a tradeoff between the metamodel value and the conditional variance. Then the new point is evaluated using the original model and the metamodel is re-learnt on the extended design of experiment. The original criteria use the Kriging variance information to maximize the Expected Improvement (so called EI).

The EGO algorithm tends to explore the design space while trying to exploit the region with the global optimum. Given equations 6 and 7, EI is given by:

𝐸 𝐼 𝑥 = 𝑦 JIO -𝑦 𝑥 𝛷 k uvw dk x y x + 𝑠𝜑 k uvw dk x y x 𝑖𝑓 𝑠 > 0 0 𝑖𝑓 𝑠 = 0 ( 13
)
Here Φ . and 𝜑 . are the cumulative distribution function and probability density function respectively. In many situations maximizing 𝐸 𝐼 x will prove to be the best way to finding the global optimum [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]. In the equation [START_REF] Tong | Optimal sensor placement for mode shapes using improved simulated annealing[END_REF] it is possible to see two terms separated by '+'. The first one is the exploitation term and is based more on the improvement at x, while the second one is the exploration term and is based more on the amount of the error. As advocated by the literature, one of the strong points of EGO is the balance between exploration and exploitation. But one can give more importance to one of these terms (depending on the problem), simply by adding a "weight" (Hybrid Criteria in [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF]). In particular, sometimes the EI can present very strong peak. Sassena et al [START_REF] Sasena | Metamodeling Sampling Criteria in a Global Optimization Framework[END_REF] proposed a smoother criterion so-called WB2 often preferred to EI.

EGO like Strategy: EGO-SPO

In this section, we explain our strategy by introducing how the sensor placement optimization problem for modal basis reconstruction (involving N modes) is solved step-by-step. It can be seen using Equation ( 4) that the orthogonality between the modal bases increases as the off-diagonal values decrease to zero while the diagonal terms approach to unity. That is, the modal identification of flexible structure by experiment becomes more accurate as the sensor placement produces the MAC closer to the above-mentioned ideal one.

Here, the MAC is always a function of x, the sensor set locations. In the general (unsupervised) case, we only have a set of experimental measurements X, so we use compute AutoMAC as:

𝐴𝑢𝑡𝑜𝑀𝐴𝐶 𝑟, 𝑞 = @ € B C @ € E 6 @ € B C @ € B @ € E C @ € E ( 14
)
We will solve two equivalent unconstrained bounded optimization problems solved using EGO. The equation ( 15) introduces the maximization of trace(AutoMAC) for blind reconstruction:

x 𝑚𝑎𝑥 trace AutoMAC(x)

with respect to x ∈ D [START_REF] Matheron | Principles of Geostatistics[END_REF] Where x is the sensor set locations, D is the geometry of the structure under study, I is the identity matrix of size N.

The following equation gives an equivalent minimization:

x 𝑚𝑖𝑛 norm AutoMAC(x) -I with respect to x ∈ D [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] In the supervised example of the next paragraph, we will use standard MAC to compare analytical mode shapes (modal basis A) to reconstructed mode shapes (modal basis B). It will be then possible to compute RMSE and control the robustness of our methodology.

EGO-SPO strategy consists in a two steps metamodels construction (Figure 2): first we build the mode shapes Mk(.) from an initial DoE, then we compute a prediction of the objective function mk(.) according to Algorithm 1. It can be noticed that one stopping criterion can also be a maximal number of sensor (at fixed budget). Both objective functions have been tested and the results are very close for each test case. We use in this paragraph a supervised approach in order to compare the results of our reconstruction methodology to analytical results. In this case since we have the analytical solution AutoMAC is not needed. The mode shapes of a cantilever beam are given by the analytical formula:

𝜑 O 𝑥 = 𝐴 O 𝑐𝑜𝑠ℎ 𝑚 O 𝑥 -𝑐𝑜𝑠 𝑚 O 𝑥 - yIO-J w ˜dyIO J w Tšy- J w ˜7™šy J w ˜𝑠𝑖𝑛ℎ 𝑚 O 𝑥 -𝑠𝑖𝑛 𝑚 O 𝑥 (17)
Here 𝐴 O is a constant (It can be used to normalize the mode shape), 𝑚 O = 2𝑛 -1 𝜋/2𝐿, L is the length of the beam and n is the number of the mode.

Figure 3 shows the iterations that will lead to add a new sensor. The figure 3b shows the construction of the function "prediction of trace(MAC)", in which only five initial points have been used. Then, evaluating the EI at each iteration (each reconstruction) a new point is added, where EI is maximal, and the function is updated (figure 3a). The new sensor will be located where the "prediction of trace(MAC)" is maximal (figure 3b).

(

Figure 3. Inside the strategy. On the left, the EI that determines the updating. On the right, the updating of prediction of trace(MAC).

In the figure 4, convergence of the objective function is shown for three different LHS initial DOE. All the sensors (maximum budget) have been added and trace(MAC) is maximized (equal to N, the modal basis size).

( Since we are in supervised approach we can compute the RMSE. For all constructed metamodels they achieve the fixed tolerance (1E-3) while trace(MAC) is maximized after 20,18 and 16 iterations respectively for initial DOE of 5,10,15 respectively. It becomes clear from Figure 4 that the final reconstruction is DOE-dependent. With the larger initial DOE, the RMSE is lower for each reconstruction and the convergence of the objective function is faster. These conclusions are important for the next chapter. We choose to analyze only bending modes around the Z axis (Figure 5). The studied wing is a cantilever beam with three concentrated masses that represent the engine and two fuel cells of the Embraer EMB 120 Brasilia from [START_REF] Venkataraman | Blending Classical and Practical Instruction in Structural Mechanics Using COMSOL[END_REF]. It can be found schematically in the following figure: We used a test case with a modal basis of size N=12 with maximum 30 sensors. The AutoMAC is used instead of MAC for blind reconstruction using trace(MAC) as an objective function. On figure 6, we can see that the convergence is fast: only 7 additional sensors are needed (no need to reach the maximum budget). Figure 7 shows only qualitatively the Kriging reconstruction, obtained using an initial LHS design of experiment and adding 7 points optimizing the objective function trace(MAC) through the EGO-SPO strategy, is quite good. Using this approach, we are optimizing the placement of the sensors for all the desired mode-shapes at once. Qualitatively, it is also relevant to underline the difference between a Kriging and a linear reconstruction of mode shapes. Figure 8 shows this difference, where the total number of points is 25, distributed in a regular grid for the linear reconstruction, while for the Kriging reconstruction they have been placed with the EGO-SPO strategy.

(a) (b) In this case the EGO-SPO strategy is capable of good reconstruction even at high frequency (mode 8 on figure 8b) with few sensors needed ( 22) compared to standard strategy such as linear reconstruction and Kriging (figure 8a) 4.2 2D Wing (Trapezoidal)

In order to demonstrate EGO-SPO gains on a more realistic example, we choose to analyze a 2D Wing using the MATLAB COFE. The wingspan is 4m length. We use Al_2024-T3 materials. The total number of nodes in the FE model is approximately 600 to ensure convergence on the highest natural frequency (9th mode). The mode shapes have been normalized hereafter. We aim at reconstructing the modal basis (N=9) with at most 25 sensors.

The AutoMAC is used instead of MAC for blind reconstruction. In this example, we choose to minimize norm(MAC -I) starting from an initial DOE of size 15 (Figure 9). The previous supervised test cases help us to tune our strategy. The figure 10 shows the performance of EGO-SPO strategy, comparing reconstructions of 9 th mode shape of the wing. Each reconstruction is compared to the High-Fidelity model (HF) that practically comes from FEM data and is shown on the left wing. Kriging reconstruction without the optimization strategy needs 36 sensors (regular grid) to reach a good level of reconstruction, compared to HF data, while the linear reconstruction with 36 sensors does not give a good result. This sensor placement strategy allows a reduction of 30.56% of sensors for the entire modal basis with only 25 sensors (15 initial regular DOE + 10 added sensors). The results of the EGO-SPO strategy is illustrated on the figure 14 in Appendix B. The complexity here, is closed from an industrial test case as the chosen modal basis exhibits bending modes, torsion modes, plus some coupling modes (bending & torsion).

Conclusion

In this paper, we developed a new strategy so called EGO-SPO dedicated to modal analysis based on an adaptive Kriging Metamodeling approach. The quality of modal identification is classically estimated using the MAC (or AutoMAC) criterion. We first proposed a totally supervised numerical experiments in order to calibrate the methodology. Then we proposed some "blind" modal basis reconstruction on 1D and 2D wing structures. The performance of the reconstruction outperforms existing methodology (linear reconstruction, RBF, ordinary Kriging). The gain is almost 30% less sensor for the same level of reconstruction on the trapezoidal example. The developed MATLAB software can be used both for benchmarking (supervised mode), sensor placement strategy using FEM (calibration) or experimental data (FVT, GVT). Of course, this methodology is sensitive to the initial DOE, we propose to work at fixed sensors budget will help the vibration engineer to make the best choice for his experimental test rig. Our future work will be dedicated to aeroelastic calibration of a full aircraft. Co-Kriging will be used to build a model with two kinds of data. For example, it will permit to merge lots of low fidelity FE model denoted as cheap data with few expensive data (experimental data or high fidelity models) in the same metamodel.

2. Theoretical background 2 . 1 . 1 )

 211 From EMA to mode shape estimation The equations of motion for a vibrating structure are commonly derived by applying Newton's second law:𝑀𝑥 𝑡 + 𝐶𝑥 𝑡 + 𝐾𝑥 𝑡 = 𝑓 𝑡 (The excitation forces f(t) and responses x(t) are functions of time (t) while M, C, K are the mass, damping and stiffness constants respectively. The equivalent frequency domain form of the dynamic model of equation in (1) can be represented in terms of transfer function 𝐻 𝑗𝜔 as:

  natural frequency and k z = damping ratio of mode k.

Figure 1 :

 1 Figure 1 : (3,1) mode of vibrating plate plus regular grid distribution of sensors in white circles (a) and The cubic interpolation which shows a spatial aliasing in mode shape reconstruction (b). A regular grid of 9 sensors (white circles) permits only to reconstruct the (1,1).

Phase 1 (

 1 design): Let k := 0. Select and evaluate a set x 0 of starting points. While stopping criteria are not met: Phase 2 (model): From the data {(x,f(x)) |x ∈ Sk}), construct a metamodel mk(.) that approximates the black-box function Phase 3 (search): Use mk(.) to search for a new point x to evaluate. Evaluate the new chosen point, update the data set Sk. Assign k := k + 1. Algorithm 1. Surrogate based optimization

Figure 2 .

 2 Figure 2. Steps used for the EGO-SPO strategy

Figure 4 .

 4 Figure 4. Convergence of objective function maximization (to 10) with a maximum number of 20 sensors (a) Initial DOE: LHS with 5 points. (b) Initial DOE: LHS with 10 points. (c). Initial DOE: LHS with 15 points

4 .

 4 Blind reconstruction: Results and discussion 4.1 1D wing: Embraer EMB 120 BrasiliaIn this paragraph, we analyze a 1D Wing but with operational constraints (fuel cells and engine mass are notably considered). To extract the modal data, we used a MATLAB toolbox compatible with NASTRAN inputs called CoFE[START_REF] Ricciardi | Nonlinear Aeroelastic Scaled-Model Design[END_REF]. CoFE is an open-source program for structural analysis and design. It supports linear statics, vibration eigenpair (eigenvalues and eigenvectors), and buckling eigenpair analysis. Since the code uses NASTRAN formatted input, commercial preprocessing tools (e.g. PATRAN) it can be used to create or check the bulk data inputs. The mode shapes have here been normalized.

Figure 5 .

 5 Figure 5. Schematization of the Wing, with Engine and fuel cells length (8m).

Figure 6 .

 6 Figure 6. Convergence of the objective function trace(MAC), from mode 1 to 12. Initial DOE: LHS with 15 points. Max number of sensors: 30.

Figure 7 .

 7 Figure 7. Reconstruction of mode shapes, from 1 to 8. Initial DOE: LHS with 15 points. Added sensors: 7.

Figure 8 .

 8 Figure 8. Comparison between Linear Reconstruction and Kriging. (a) first 4 modes, (b) zoom on mode 8 to clearly see the difference between classical linear interpolation and Kriging reconstruction

Figure 9 .

 9 Figure 9. Convergence of the objective function norm(MAC -I), from mode 1 to 9. Initial DOE: Regular Grid with 15 points. Maximum number of sensors: 25.

Figure 10 .

 10 Figure 10. (a) Comparison between HF FEA results (left) and a Linear Reconstruction with a regular grid of 36 sensors (right), for 9 th Mode Shape. (b) Comparison between HF FEA results (left) and a Kriging Reconstruction with a regular grid of 36 sensors (right), for 9 th Mode Shape. (c) Comparison between HF FEA results (left) and EGO-SPO strategy (right) The black dots represent the initial DOE (regular grid, 15 sensors). The yellow ones represent 10 added sensors. This sensor placement strategy allows a reduction of 30.56% of sensors for the entire modal basis.

Figure 12

 12 Figure 12. R 2 for increasing number of Sensors, from mode (1, 1) to (3, 3), regular grid, CFFF plate.

Figure 13 .

 13 Figure 13. RMSE for increasing number of Sensors, from mode (1, 1) to[START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF][START_REF] Minwoo | Optimal sensor configuration for flexible structures with multi-dimensional mode shapes[END_REF], regular grid, CFFF plate.

Table 2

 2 resumes the results of the presented metamodeling techniques (averaged 10 times).

			REGULAR GRID	RANDOM GRID	PSEUDO RANDOM GRID
		CFFF PLATE	Average Max RMSE	Average Mean RMSE	Average Max RMSE	Average Mean RMSE	Average Max RMSE	Average Mean RMSE
		RSM cubic	1,4624	0,4024	3,0812	0,7632	34,3243	6,9853
	METHOD	RBF cubic Kriging (from [15])	0,2091 0,2051	0,0741 0,0699	0,1731 0,1500	0,1026 0,0707	0,4333 0,2449	0,1364 0,0963
		Kriging (ordinary)	0,3757	0,1520	0,3341	0,2007	0,4450	0,2026

. Summary of RMSE for each reconstruction method for the entire modal basis (9 first bending modes of the CFFF plate). Kriging from

[START_REF] Forrester | Engineering design via surrogate modelling: a practical guide[END_REF]

exhibits the lowest RMSE from all the selected methodology.

Acknowledgements

The authors really want to thank Claire Meyer from Dassault aviation for given us industrial inputs and Nathalie Bartoli from ONERA for her advice on the developed methodology. (a) (b) (c)

Appendix A

The figures 11, 12 and 13 illustrate the CFFF plate. This is supervised example, RMSE and R 2 can be computed using a true (converged mesh) finite element model. The coefficient of determination, denoted 𝑅 n

. It provides a measure of how well observed outcomes are replicated by the model, based on the proportion of total variation of outcomes explained by the model. R 2 can be calculated as:

.

Here SS reg is the regression sum of squares, SS res is the residual sum of squares, SS T is the total sum of squares. R 2 is a measure of how much of the variance in y is explained by the model, f. As an example, the reconstruction of 9 first bending modes of the 2D trapezoidal wing has been inserted in the figure 14. The chosen modal basis exhibits bending, torsion, plus some coupling mode (bending and torsion).