Nicolas Sutton-Charani

Abdelhak Imoussaten

Sébastien Harispe

Jacky Montmain

Evidential Bagging: Combining Heterogeneous Classifiers in the Belief Functions Framework

Keywords: belief functions, information fusion, bagging, supervised learning

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Introduction

As the amount of learning algorithms and methodologies in the literature has reached a point where it is almost impossible to stay up to date on all of them, many users or even researchers tend to use them as black boxes, without focusing much on their understanding or interpretation, often setting model parameters to default values. Beside some obvious computational time differences between them, it has been proven that there is no optimal learning algorithm in the sense that most models are optimal for certain types of learning data and have advantages and drawbacks [START_REF] Wolpert | In: The Supervised Learning No-Free-Lunch Theorems[END_REF]. Indeed, dataset dimensions, attribute types, variance and noise make each learning dataset more suited to some learning algorithms than others. To illustrate this fact, six standard learning algorithms have been applied to six benchmark datasets and the resulting mean accuracies (i.e. the correct predictions rate) from 1000 10-fold cross validation simulations are presented in Figure 1. We can easily observe some disparities between the different types of classifier (decision trees, SVM, etc) used on those six datasets. For example, neural network seems to be one of the less accurate classifier on almost all datasets except for "Balance scale" where it is the most accurate one. We also can observe that decision trees and neural network have the highest accuracy variances. Ensemble learning methodologies consist in the learning of many classifiers from the same initial dataset. In that context, classifiers are then aggregated, or the predictions they provide (their outputs) are combined in order to get final predictions. The resulting classifier is usually more accurate and robust [START_REF] Qu | Bucket learning: Improving model quality through enhancing local patterns[END_REF][START_REF] Zhou | Ensemble Methods: Foundations and Algorithms[END_REF][START_REF] Polikar | Ensemble based systems in decision making[END_REF]. One particular ensemble method, called bagging or bootstrap aggregating, has an additional advantage: it tends to avoid over-fitting [START_REF] Zhou | Ensemble Methods: Foundations and Algorithms[END_REF]. Bagging uses some re-sampling methods in order to decrease the dependency of classifiers on the learning data, i.e. to decrease the learning data's bias and variance. Unlike other ensemble methods as Bucket-of-models [START_REF] Qu | Bucket learning: Improving model quality through enhancing local patterns[END_REF][START_REF] Džeroski | Is combining classifiers with stacking better than selecting the best one?[END_REF], bagging usually involves the simultaneous and multiple use of a single algorithm, to further combine their predictions with a simple vote procedure.

This problem can be seen as an information fusion problem if we consider the trained classifiers as information sources and their predictions as the information, or evidence, to fuse. Traditionally, in bagging methods classifiers' outputs are combined through a vote procedure as they usually involve the same type of classifiers. With that approach, fusing the same type of classifiers should indeed lead to a uniform weighting.

In the evidential framework, researchers have proposed some evidential bagging methods, handling uncertain data and the combination procedure is also performed between the same type of classifiers [START_REF] Vannoorenberghe | On aggregating belief decision trees[END_REF][START_REF] Xu | Evidential calibration of binary svm classifiers[END_REF][START_REF] Ma | Bagging likelihood-based belief decision trees[END_REF]. To the best of our knowledge, no work has been proposed that combine heterogeneous types of classifier with belief functions.

In this paper we present a bagging method that involves the fusion of different types of classifiers' outputs, or predictions. We propose to take into account the classifiers' outputs reliability during the aggregation step within the formalism of the belief functions theory [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF][START_REF] Shafer | A Mathematical Theory of Evidence[END_REF]. This choice is motivated by its generalization power (including probabilistic and possibilistic cases) and the flexible tools it provides. In addition many approaches have been developed in different contexts, especially for information fusion problems when we have evidence about the sources reliability [START_REF] Denoeux | Optimal object association in the Dempster-Shafer framework[END_REF]. In this work, we chose to use predictive performance of the classifiers as reliability evidences. Then the classifiers' probabilistic outputs and their reliabilities are used to build one predictive belief function per classifier. Finally, a suitable combination procedure is used to merge those belief functions into a global predictive mass function.

After defining our classification formalism, recalling bagging basis, the theory of belief functions with some basic fusion tools are presented in Section 2; our evidential bagging model is described in Section 3 experiments are provided on benchmark datasets in Section 4 and finally results and perspectives are discussed in Section 5.

Related works and positioning

In this section, first the general classification formalism is given, then a succinct overview of Ensemble learning methods is presented; finally Bagging is more precisely described.

Classification formalism

Starting from a dataset D containing N learning examples (x, y) i=1,...,N , classification tasks aim at learning a model f able to predict the class label y * of any new unlabeled example from its attribute (or feature) values x such that y * = f (x). The attributes X = (X 1 , . . . , X J ) take their values in X = X 1 × • • • × X J , the class Y in a finite set Ω. Spaces X j can be categorical or numerical.

D =    x 1 , y 1 . . . x N , y N    =    x 1 1 . . . x J 1 y 1 . . . . . . . . . x 1 N . . . x J N y N    .
Samples are assumed to be i.i.d. but in practice data are often noised, casual correlations can randomly occur and some outliers or very rare examples can occur even in very small datasets. In order to discount those outliers' influence and to reduce bias, bootstrapping techniques [START_REF] Efron | Bootstrap methods: Another look at the jackknife[END_REF] can be applied. By doing so, overfitting can be decreased without the need of large datasets but the different bootstrapped subsamples' results (i.e. predictions) have to be aggregated in a conservative way so that the most uncertain predictions can be weakened.

In this context, a classifier is a function c : X → Ω, the notation c(x) corresponds to the prediction obtained by c from attribute values x ∈ X .

Ensemble Learning methods

As almost each learning algorithm is optimal for specific types of learning data and problems [START_REF] Wolpert | In: The Supervised Learning No-Free-Lunch Theorems[END_REF], many researchers do not confine in single classifiers and tend to use many of them when dealing with real data. In this context, Ensemble Learning methods have become very common tools to improve classifiers performance (in term of accuracy and robustness) by using predictive algorithms' heterogeneity and resampling techniques to avoid overfitting. In fact, Ensemble Learning has become a whole research field [START_REF] Polikar | Ensemble based systems in decision making[END_REF][START_REF] Zhou | Ensemble Methods: Foundations and Algorithms[END_REF] where we can distinguish three main types of methodology: boosting, stacking and bagging. Boosting [START_REF] Freund | A decision-theoretic generalization of on-line learning and an application to boosting[END_REF] is an iterative procedure where the classifier is re-learnt in order to better classify misclassified examples. AdaBoost is the most famous boosting algorithm [START_REF] Schapire | Explaining AdaBoost[END_REF].

Stacking [START_REF] Wolpert | Stacked generalization[END_REF] focuses on the classifiers outputs aggregation step with the learning of an aggregation algorithm. Bagging, or Bootstrap-Aggregation, is based on resampling and was first presented by Breiman [START_REF] Breiman | Random Forests[END_REF] with the well known Random forest algorithm which consists in the learning of several decision trees from different bootstrapped subsamples and in their aggregation through a vote procedure. In this paper, a Bagging methodology is proposed based on a bucket of diverse learning algorithms with an aggregation step formalized in the evidential framework and involving belief functions generated from the single classifiers' predictions and evaluations.

Bagging

In most applications, getting a dataset truly representative of the actual population is a complex task as sampling often implies noise intrusion and thus bias and variance creation [START_REF] Cortes | In: Sample Selection Bias Correction Theory[END_REF]. As a matter of fact, in most learning datasets, some rare examples of the reality can be over-represented which often leads to overfitted models. During the learning process, those examples should not be given too much weight as they can be considered outliers. It is for that matter that Bagging involves resampling, in order to discount that kind of bias [START_REF] Efron | Bootstrap methods: Another look at the jackknife[END_REF].

Nevertheless, in most bagging approaches, the same learning algorithm is used to fill the bag because of the heterogeneity of the classifiers' natural outputs (scores, class probability, etc). During the aggregation step, trainable combiners have shown some asymptotically optimal properties [START_REF] Duin | The combining classifier: to train or not to train? Object recognition supported by user interaction for service robots[END_REF] but involves a common training set for all single classifiers. In this work we use a combiner learnt on such a validation set which will be also used for the single classifiers evaluations.

Theory of Belief Functions

The theory of belief functions, also known as Evidence or Dempster-Shafer theory was first presented by A. Dempster in 1967 in a statistical context [START_REF] Dempster | Upper and lower probabilities induced by a multivalued mapping[END_REF] as an attempt to reconcile frequentists and Bayesians. Dempster laid the foundation of a mathematical theory that deals with uncertainty in a much more general framework than standard probability theory and which handles aleatory (or objective) uncertainties and epistemic (or subjective) ones.

Even if some evidential works are related to statistical and classification contexts, many researchers use belief functions as a convenient framework for information fusion problems. As a matter of fact, fusion problems occur in many contexts, even in the classification ones. In the evidential framework two levels are considered: the credal and the decision ones. The credal level is dedicated to the uncertainty representation whereas the decision step uses the credal one to make decision.

Credal level Let Y be an uncertain quantity whose value y lies in a finite set Ω called the frame of discernment. Definition 1. A mass function m regarding Y 's value y ∈ Ω is a function defined on the set of subsets 1 of Ω, which is usually written 2 Ω or P(Ω) and called the powerset, with its values in [0, 1] and verifying

B∈2 Ω m(B) = 1
The quantity m(∅) is usually fixed to 0 in many cases. From m, two uncertainty measures can be defined about y, the belief function Bel and the plausibility function P l, which express the information contained in m in different ways, more conservatively for Bel than for P l. Since there is 1 to 1 correspondancies between m, Bel and P l, belief functions can indifferently refer to a mass function m or its corresponding inferior uncertainty measure Bel.

Decision level In a practical matter, the decision step of many problems is often handled by transforming belief functions into pignistic probabilities [START_REF] Smets | The transferable belief model[END_REF] (cf. Definition 2) to further consider the most probable event in the pignistic sense.

Definition 2. The pignistic probability distribution attached to a mass function m is defined by:

∀ω ∈ Ω, BetP (ω) = A⊆Ω|ω∈A m(A) |A| . (1)
Sources reliability and evidential Fusion. Information fusion problems aim at combining different informative contents coming from different sources. In the evidential framework, sources are represented by mass functions. Three main concepts must be taken into account when combining evidential sources: dependence, reliability and conflict. The first combination method proposed in the belief function theory was Dempster's conjunctive combination rule. Nevertheless, this rule handles conflict in a way that has been criticized [START_REF] Yager | On the dempster-shafer framework and new combination rules[END_REF][START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized bayesian[END_REF]. Moreover, this rule requires independence between sources.

To take more conveniently into account the conflict and the dependence between sources, several combination rules have been proposed [START_REF] Smets | Belief functions: the disjunctive rule of combination and the generalized bayesian[END_REF][START_REF] Yager | On the dempster-shafer framework and new combination rules[END_REF][START_REF] Dubois | Representation and combination of uncertainty with belief functions and possibility measures[END_REF]. To avoid dependence hypothesis, some works [START_REF] Denoeux | Optimal object association in the Dempster-Shafer framework[END_REF][START_REF] Florea | Adaptative combination rule and proportional conflict redistribution rule for information fusion[END_REF] use the average operator to combine beliefs functions, which is in line with voting procedures. In this paper we chose this operator as a basis.

Evidential Bagging

This section presents the three evidential generative models defined in this paper, as well as the fusion approach proposed to aggregate the predictions provided by the different single classifiers. Finally the general scheme of our approach is given.

Generative models

Even if the different learning algorithms may provide outputs of different structures, most data science software enable a calibration step in order to get a probability distribution on class labels from those outputs. Those probabilities express an uncertainty which should be integrated in any bagging approach that presents an uncertainty focus. Nevertheless, those uncertainty measures are computed on the learning data and are therefore subject to over-fitting. Better estimators of classifiers' performance can be computed on other dataset. In this paper we compute single classifiers' accuracy on such a separate dataset which we recall as the validation set.

If we consider the single classifiers as information sources, and their probabilist outputs as the uncertain information to merge, one way to evaluate the classifiers reliability is to evaluate their accuracy on the validation set. Our approach consists in discounting the classifiers' outputs according to their reliabilities. To this aim, belief functions are generated in order to enrich the probabilist outputs of the single classifiers. Those belief functions are based both on class probabilist predictions, and accuracies computed on validation sets. They are finally merged into a final belief function and its pignistic probability gives class predictions. Proposed models are introduced hereafter:

1. simple discounting (EBagSD): using this model we consider that the less accurate classifiers should be the most discounted during the aggregation step. Otherwise stated, we consider that the reliability of a classifier c is a function of its global accuracy denoted acc c (computed on the validation set). To this aim, the mass function associated to c is defined as ∀i ∈ {1, . . . , N }, ∀ω ∈ Ω:

m c i ({ω}) = P c i (ω) × acc c m c i (Ω) = 1 -acc c . ( 2
)
where P c i (ω) stands for the probability of the class label ω provided by the classifier c on the example x i .

2. class dependent model (EBagCD): some classifiers are more accurate in predicting some specific class labels. Such accuracy variations can be observed by analysing the confusion matrix associated to each classifier. We therefore propose to take advantage of this information to spread uncertainty about classification involving specific class labels from Ω. Using this model the following definitions of mass functions are considered; ∀c ∈ {1, . . . , C}, ∀i ∈ {1, . . . , N }, ∀ω ∈ Ω:

m c i ({ω}) = P c i (ω)× P c (Y = y i |c(x i ) = y i ) m c i (Ω) = P c (Y = y i |c(x i ) = y i ) . (3)
Note that for a given classifier c, P c (Y |c(X)) is estimated by using the confusion matrix of the classifier c computed on the validation set as follows:

P c (Y = y i |c(x i ) = y i ) = |(x,y)∈X ×Ω : {c(xi)=yi}∩{Y =yi}| |(x,y)∈X ×Ω : {c(x)=yi}| .
3. contextual model (EBagcon): we consider two types of regions of X : one containing instances often misclassified and its complementary. In addition, we consider that not all classifiers will misclassify instances of the same regions of X . We are therefore interested by the definition of a contextual model that will consider an estimation of the single classifiers' misclassification risk or probability of each examples. Formally, to estimate this risk, that is both dependent on the classifier and the processed instance, we consider that ∀c ∈ {1, . . . , C}, we have mis c : X → [0, 1] a function used to assess the misclassification risk of a classifier c regarding a given x i ∈ X whose predicted label is y i , i.e. y i = c(x i ); Intuitively, higher is the estimated risk, lower the confidence on the provided class will be. Based on this function, we consider the following definition of the mass function, ∀c ∈ {1, . . . , C}, ∀i ∈ {1, . . . , N }, ∀ω ∈ Ω:

m c i ({ω}) = P c i (ω)× (1 -mis c (x i )) m c i (Ω \ {y i }) = mis c (x i ) . ( 4
)
We consider that mis c is obtained using a binary classifier (SV M in our case) that will be trained using examples defined by the error and success classifications of c during the validation phase with a regression approach.

Otherwise stated the classifier predicts a numerical output in [0, 1] standing for the misclassifications observed in the validation set (1 for misclassified examples and 0 for well classified ones).

Whereas generative models EBag SD and EBag CD are based on the single classifiers global accuracy, model EBag con uses their local misclassification risk to discount their predictions. Therefore, the focal elements of models EBag SD and EBag CD are the class labels and the frame of discernment Ω. Model EBag con considers the class labels and their complementary sets -intuitively, if a single classifier misclassifies an instance, our belief should be focused on its prediction's complementary.

Combination and prediction

Classifiers' mass combination: As evidential independence is needed to apply Dempster's combination rule and disjunctive semantic has no sense in case of different classifiers predicting different class labels, the aggregation step of our model was done by averaging the mass functions generated by the single classifiers. Actually, in the evidential bagging contents, most authors have combined the classifiers' outputs through a vote procedure or with the average operator applied to their class probabilities because of the dependencies between classifiers [START_REF] François | In: Bagging Improves Uncertainty Representation in Evidential Pattern Classification[END_REF][START_REF] Vannoorenberghe | On aggregating belief decision trees[END_REF][START_REF] Xu | Evidential combination of pedestrian detectors[END_REF][START_REF] Ma | Bagging likelihood-based belief decision trees[END_REF].

Final prediction: At the decision or prediction step, we chose the most likely class labels in the pignistic sense, i.e. the average belief function (computed from all single classifiers outputs) is transformed into its pignistic probability and then the most likely class label is predicted.

prediction(m c i ) = argmax ω∈Ω BetP c i ({ω}).

Global procedure

Considering a set of classifiers, and a set of labelled data, the global procedure of our evidential bagging model is composed of four main steps illustrated in Fig. 2: 1 Training of each classifier. This step is used to estimate the best parameters of each classifiers based on training data. A set of trained classifiers, i.e. tuned models, is obtained.

2 Belief function generation through a post-analysis of training performance evaluations: the step used to compute data that will be used to define the mass functions. The treatments applied for each model vary; all rely on the analysis of the single classifiers' performance on the validation set. For the simple discounting model (Eq. 2), the accuracy of each classifier is computed on the validation set. For the class dependent model (Eq. 3) the confusion matrix is computed on the validation set. The conditional probabilities that will be used to define the mass functions are estimated based on the confusion matrix. Finally, using the contextual model (Eq. 4) the misclassification risk is estimated by training a SVM classifier in the aim of distinguishing cases for which the classifier fails to provide a good classification.

3 Mass combination: given an evidential generative model, the various evidential predictions provided by the classifiers are combined to output a single global mass function with the average operator.

4 Prediction (or decision) step: the global mass function is transformed into its corresponding pignistic probability in order to predict the most probable class labels.

Experiments

In this section, we evaluate our evidential bagging approach on several UCI1 , Kaggle2 and KEEL3 benchmark datasets. Number of examples, attributes and class labels of those datasets are summarized in Table 1.

The considered learning algorithms were: decision tree ('tree' ), random forest ('forest' ), support vector machine ('SVM' ), linear discriminant analysis ('lda' ) and naive Bayes classifier ('bayes' ). The implementation was handled in R with the following functions (and packages)rpart (rpart), randomForest (randomForest), svm (e1071), nnet (nnet), naiveBayes (e1071) and lda (MASS). All the learning algorithms were implemented in R with their default parameters. For each dataset, 100 10-fold cross validation procedures were implemented for different bagging methods involving different aggregation approaches:

-voteBag: simple vote procedure from the precise single classifiers' predictions -P mean : averaging of the single classifiers' probabilistic predictions and prediction of most probable class label -EBag SD : simple discounting model (see Eq. ( 2)) -EBag CD : class dependent model (see Eq. ( 3)) -EBag con : contextual model (see Eq. ( 4))

For each fold, a learning set and a validation set were built as follows: a quarter of the examples contained in the nine other folds were randomly selected as the validation set; the remaining examples were then used to build the learning set. The accuracy means and standard deviations results are summarised in Table 2. T-tests were performed between the voteBag models and the most accurate ones, bold accuracies stand for the significantly highest ones. R implementations of tested methods, links to datasets, as well as complete technical details about the evaluation are provided at https://github.com/lgi2p/evidentialBagging. Globally, there is no systematically and significantly outperformance between models even if EBag con seems to be the most accurate model. This is not too surprising as the general spirit of classifiers is to link attributes (i.e. context) and class labels, whereas inferring global reliability or treating class labels nonsymmetrically is not at the basis of standard classification tasks. EBag CD model is the less accurate and robust one for the datasets (Breast Tissue, E.Coli and Glass) containing the more class labels (i.e. for K ≥ 6). This suggests that expressing single classifiers' reliability on specific class labels has a sense for limited number of labels.

As in any bagging methods, the number of bagged classifiers has an impact on the resulting classifier's accuracy, bigger bags implying higher accuracies. In Figure 3, the accuracies over 100 10-fold cross validations are represented for different bag sizes (1 to 100 per learning algorithm) on the 'Tic tac toe' and 'Glass' datasets. For the dataset 'Tic tac toe', the most accurate model is EBag CD whereas it is EBag con for the dataset 'Glass'. This corroborates the fact that our class-dependent model (EBag CD ) is more accurate on dataset containing few class labels (2 for 'Tic tac toe', 6 for 'Glass'). As enhanced by Table 2, for datasets containing many class labels, our contextual model (EBag con ) is preferable.

Conclusions and perspectives

A general bagging approach has been proposed that involves belief mass generation from each classifiers and evidential fusion between classifiers' evidential predictions. Different aspects of the classifiers (global accuracy, confusion matrix or local misclassification risk) can be separately evaluated on a validation set and used to generate those belief functions. Experiments on benchmark datasets show encouraging results especially for the model EBag con , which is based on a misclassification risk learnt on a separated validation set and depending on attributes values. That approach can refer to the concepts of taking into account the learning context or learning the local context (before the actual learning process) and should be further studied.

A more complete study of this evidential bagging approach should include a global sensitivity analysis over the type of classifiers to bag, the combination rule (between single classifiers' evidential predictions) and the prediction, or decision step. It is noticeable that considering the pignistic transform after averaging the mass functions is equivalent to some straightforward probabilistic modelling. Nevertheless, since probabilities are some particular belief functions, the evidential framework provides many tools in term of fusion and decision for any future extension. In some recent works [START_REF] Denoeux | Maximum likelihood estimation from uncertain data in the belief function framework[END_REF][START_REF] Sutton-Charani | Learning decision trees from uncertain data with an evidential em approach[END_REF], likelihood-based tools have been presented that could represent an alternative to pignistic transformation. From an optimisation point of view, some clustering over class labels should improve the EBag CD model (i.e. the class dependent one). Moreover, ideas behind the three presented evidential generative models could be used to define a single model. In addition, in this paper, one of the used single classifier is based on a bagging approach: the random forest. By doing so, we actually made a second order bagging (or bagging of classifiers bags). To go further on this aspect, mathematical properties of bagging approaches should be taken into account in order to solve one pragmatic dilemma: should we make larger bags or should we nest single classifiers bags?

Fig. 1 .

 1 Fig. 1. Learning algorithms mean accuracies on 1000 simulations with default parameters tuning in R.

Fig

  Fig. 2. Global process

Fig. 3 .

 3 Fig. 3. Bag size effect

Table 1 .

 1 . 2. Global process Number of examples (N ), attributes (J) and class labels (K) of several benchmark datasets from UCI, Kaggle and KEEL

	dataset	N	J	K
	Balance scale	625	3	3
	Banana	5300	2	2
	Banknote	1372	4	2
	Breast tissue	106	9	6
	Contraceptive method 1473	9	3
	E.coli	336	5	8
	Glass	214	9	6
	Iris	150	4	3
	Mammographic	830	5	2
	Nursery	12958	8	4
	Occupancy	8143	6	2
	Pima	768	8	2
	Satimage	6435	36	6
	Tic tac toe	958	9	2
	Titanic	2201	3	2
	Wine	178	13	3

Table 2 .

 2 Mean accuracies on 30 10-fold cross validation procedures

	dataset	voteBag	Pmean	EBag SD	EBag CD	EBagcon
	Balance scale	0.767	0.765	0.765	0.759	0.764
	Banana	0.759	0.808	0.816	0.820	0.830
	Banknote	0.895	0.904	0.912	0.923	0.918
	Breast tissue	0.519	0.519	0.524	0.512	0.538
	Contraceptive method 0.558	0.559	0.560	0.559	0.561
	E. Coli	0.864	0.863	0.864	0.658	0.866
	Glass	0.689	0.688	0.690	0.675	0.700
	Iris	0.959	0.959	0.959	0.960	0.959
	Mammographic	0.833	0.834	0.834	0.835	0.836
	Nursery	0.957	0.969	0.970	0.968	0.970
	Occupancy	0.983	0.983	0.983	0.984	0.984
	Pima	0.766	0.765	0.765	0.712	0.766
	Satimage	0.881	0.873	0.875	0.880	0.881
	Tic tac toe	0.810	0.838	0.846	0.887	0.851
	Titanic	0.782	0.783	0.783	0.782	0.783
	Wine	0.978	0.976	0.977	0.978	0.978

The expressions B ⊆ Ω and B ∈

Ω are equivalent.

https://archive.ics.uci.edu/ml/datasets.html

https://www.kaggle.com/datasets

http://sci2s.ugr.es/keel/category.php?cat=clas


Adrien Simon

Daphné Berthier

Marie-Pierre Deffarges

Mathieu Venin

Florian Lacroix

Yoann Tendron

Stéphane Méo

Incorporation of fillers in elastomeric composites: impact of the process on the thermomechanical properties

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Dispersion in solvent

Mechanical behaviour

 A fourth monomer is incorporated to enhance the crosslinking action, named « Cure Site Monomer » (CSM)  FKM exhibits good mechanical, chemical and thermal properties

Previous works [1] led to the development of a new formula on the basis of FKM blend reinforced with Polyhedral Oligomeric Silesesquioxanes (POSS). The main objective consists in maintaining thermomechanical properties while changing the process scale. To this end, two new specimen geometries of vulcanized rubber are studied : thin films obtained by coating at laboratory scale through masterbatch and sheets obtained through rubber mill and thermocompression.

[1] D.

Formula of the terpolymer FKM Possible interactions with a POSS (X=1) [2]

Chemical structure of POSS

Processes and geometries

Fluorinated

  Impact of a new manufacturing way (tube by extrusion)  Comparison between the three final products (thin film-sheet-tube) Upgrade of the formula  Incorporation of POSS with new functionnal groups  Characterisation of materials after chemical and/or thermal aging  Impact of the process and of the specimen geometries on the macroscopic mechanical characterisation (by tensile test : underestimation of the materials properties on thin films), on the glass transition and on the cross linking reaction (on activation energy and activation temperature)  Presence of volatile compounds in thin films but not in sheets (TGA results)  No impact on degradation behaviour Thin films is a cost efficient way to provide thermal informations before sheet fabrication Perspectives Incorporation of fillers in elastomeric composites: impact of the process on the thermomechanical properties Adrien Simon, Daphné Berthier 2 , Marie-Pierre Deffarges 1 , Mathieu Venin 1 , Florian Lacroix 1 , Yohan Tendron 2 , Stéphane Méo 1 Same degradation trends  For thin films, a slightly weight loss, is supposed to be associated to evaporation of volatile compounds (around 130°C)  The glass transition (for raw and vulcanized samples), the activation temperature and the reaction area of the crosslinking reaction

de Mécanique Gabriel Lamé, Université de Tours, Université d'Orléans, INSA Centre Val de Loire, Polytech Tours, 7 avenue Marcel Dassault BP40, 37004 Tours, France 2 SAFRAN AEROSYSTEMS, 20 Avenue Georges Pompidou, 37600 Loches, France * In partnership with the LRCCP (Laboratoire de Recherche et de Contrôle du Caoutchouc et des Plastiques) Conclusions Polyhedral Oligomeric SilSesquioxanes (POSS)

	X : functionnal group
	with X ≥ 1
	Length Si-Si 0.5nm
	Length R-R 1.5nm

X = 1 1 Laboratoire R : non reactive group with R ≥ 1  Smallest silica particles (1 to 3nm)  Hybrid chemical composition  Role of functionnal groups (X) : chemical interaction with polymers


R Coltat
email: rémicoltat@univ-rennes1.fr

Y Branquet

P Gautier

H Campos Rodriguez

M Poujol

E Pelleter

S Mcclenaghan

G Manatschal

P Boulvais

Unravelling the root zone of ultramafic-hosted black smokers-like hydrothermalism from an Alpine analog

Mid-Ocean Ridges host various types of hydrothermal systems including high-T black-smokers found in ultramafic rocks exhumed along slow spreading ridges. These systems are mostly described in two dimensions as their exposure on the present-day seafloor lacks the vertical dimension. One way to understand these systems at depth is to study their fossilized equivalents preserved on-land. Such observation can be done in the Platta nappe, Switzerland, where a

Introduction

Mid-Ocean Ridges (MORs) are places where deformation-assisted fluid-rock interactions play an important role in elemental mobility (Rona and Scott, 1993). At (ultra)slow-spreading ridges where mantle exhumation is favored by the presence of detachment faults (i.e. oceanic core complexes; Tucholke et al., 1998), these interactions lead to serpentinization of peridotite, venting of hydrothermal fluids with related mineralization, and ophicalcitization.

Active ultramafic-hosted black-smoker systems (UHS) are characterized by a Cu-Co-Zn-Au- (Ni) enrichment and high contents in CH 4 and H 2 in the venting fluid [START_REF] Charlou | Geochemistry of high H2 and CH4 vent fluids issuing from ultramafic rocks at the Rainbow hydrothermal field (36°14[END_REF]. At the Mid-Atlantic Ridge (MAR), these UHS have been described for two decades through mineralogical (Hannington et al., 1995;Marques et al., 2006), geochemical (Herzig et al., 1998;[START_REF] Douville | The rainbow vent fluids (36°14'N, MAR): the influence of ultramafic rocks and phase separation on trace metal content in Mid-Atlantic Ridge hydrothermal fluids[END_REF] and tectonic studies (McCaig et al., 2007;McCaig et al., 2010). However, due to restricted access to the seafloor, the understanding of their hydrothermal plumbing system, especially the root zone, (i.e. the mineralized portion below the seafloor) remains limited and speculative (Fouquet et al., 2010) (Fig. 1).

Ocean-Continent Transitions (OCTs) display similarities with oceanic realms: mantle exhumation along detachment faults [START_REF] Boillot | Tectonic denudation of the upper mantle along passive margins; a model based on drilling results (ODP Leg 103, western Galicia margin[END_REF], presence of mafic magmatism and high geothermal gradients. Hence, fossil OCTs found in orogenic belts are promising targets to study oceanic hydrothermal processes (e.g. [START_REF] Alt | Normal faulting and evolution of fluid discharge in a Jurassic seafloor ultramafic-hosted hydrothermal system[END_REF]. Here we present the geometric, mineralogical, geochemical and geochronological characteristics of a serpentinite-hosted hydrothermal system now exposed in the Platta nappe (Fig. 2,3). We propose that this system, active during the Jurassic, represents the root zone of a UHS similar to those found along the MAR.

Geological setting

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The Platta nappe (Grisons, SE Switzerland) exposes the Jurassic OCT of the Adriatic plate and belongs to the South Pennine Alpine units (Fig. 2A). Jurassic mantle exhumation was accommodated along detachments faults (Froitzheim and Manatschal, 1996) and accompanied by mafic magmatism (Fig. 2B; [START_REF] Desmurs | Onset of magmatic accretion within a magma-poor rifted margin: a case study from the Platta ocean-continent transition, eastern Switzerland[END_REF] dated at 161± 1 Ma (U-Pb on zircon; Schaltegger et al., 2002). Evidence of hydrothermal circulation is recorded through alteration of mafic rocks (e.g. rodingitization and epidotization), formation of ultramafic-hosted mineralized systems (e.g. Marmorera-Cotschen, Fig. 3) and ophicalcitization [START_REF] Coltat | Syntectonic carbonation during syn-magmatic mantle exhumation at an ocean-continent transition[END_REF]. The end of the rifting phase is marked by deposition of the Radiolarian Chert Formation dated between 166 and 147 Ma [START_REF] Bill | Radiolarite ages in Alpine-Mediterranean ophiolites: Constraints on the oceanic spreading and the Tethys-Atlantic connection[END_REF]. During the Cretaceous, tectonic inversion triggered the subduction of the Tethyan realm and subsequent continental collision resulting in thickening through W-NWvergent nappe stacking (D1 phase, Froitzheim et al., 1994). The Platta nappe was only affected by a low-temperature prehnite-pumpellyite metamorphism (< 300°C; Dunnoyer de [START_REF] Dunoyer De Segonzac | Diagénèse et métamorphisme des argiles dans le Rhétien Sud-Alpin et Austro-alpin (Lombardie et Grisons)[END_REF]Ferreiro-Mahlmänn, 1994). During the Cenozoic, N-S-oriented shortening (D3 phase) resulted in the formation of large-scale E-W folds and N-vergent thrusts [START_REF] Epin | Evolution morpho-tectonique et magmatique polyphasée des marges ultradistales pauvres en magma : la transition océan-continent fossile de l'Err et de la Platta (SE Suisse) et comparaison avec des analogues actuels[END_REF].

Sampling and methods

The serpentinite-hosted mineralization is exposed along the Marmorera-Cotschen cliff (~600m-high; Fig. 3A) where detailed structural mapping and petro-structural analyses have been carried out. The Snake Pit outcrop is close to the paleo-detachment plane (Fig. 3B). A total of 75 thin sections and 9 polished mounts have been prepared from samples representative of the various hydrothermal assemblages: host serpentinites, Fe-Ca-silicates, semi-massive sulfides, Cu-rich massive sulfides (Cu-rich MS) and ophicalcites.

Results

Geometry of the mineralized zones

The Marmorera Cotschen Hydrothermal Systems (MCHS) is hosted in the serpentinized footwall of a top-to-the-W Jurassic detachment that juxtaposes basalts onto serpentinites [START_REF] Epin | Evolution morpho-tectonique et magmatique polyphasée des marges ultradistales pauvres en magma : la transition océan-continent fossile de l'Err et de la Platta (SE Suisse) et comparaison avec des analogues actuels[END_REF]; the basalts being altered to a greenschist assemblage at their base (Fig. 3A,B). The mineralization occurs along the cliff as sporadic thin oxidized reddish corridors and patches (Fig. 3A,B). It reaches its maximum width at the gossan structure (i.e. oxidized zone after primary sulfides) of Cotschen (Fig. 3B). Mineralization is locally spatially associated with highly altered

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This article is protected by copyright. All rights reserved mafic dyke-like intrusions oriented sub-parallel to the detachment (Fig. 3C-F). Moving away from the intrusions, host serpentinites show a petrographic zonation: Cu-rich massive to semi-massive sulfides evolve toward Fe-Ca-silicates stockwork and pervasive replacement and then to barren serpentinite (Fig. 3E,4). Close to the detachment, the mineralized serpentinite is crosscut by calcite veins.

Alpine deformation is marked by an almost flat N-vergent major thrust (i.e. D3 phase) responsible for the steepening of the former Jurassic detachment and nearby mineralization (Fig. 3A). Minor N-vergent faults occur within the mineralized bodies (Fig. 3F).

Mineralogy and geochemistry of MCHS

Detailed paragenetic sequences are presented in Table S1-1 and summarized in Table 1.

See also [START_REF] Dietrich | Die sulfidischen Vereezungen in den Oberhalbsteiner Serpentiniten[END_REF]. Chemical composition of selected mineralized phases (EPMA analyses) is given in Table S2-1 and S2-2. We first describe the sequence of the mineralization in the most mineralized zones, then the one of the Snake Pit outcrop, close to the detachment plane.

Lizardite is the serpentine polymorph forming serpentinite. Small magnetite and pentlandite grains (< 100µm) occur along former olivine grain boundaries or within serpentine domains, respectively (Fig. 4A). Magnetite is commonly found in replacement after Cr-spinel. Pentlandite is characterized by high Ni (35.84 to 36.14 wt.%) and low Co (1.66 to 1.80 wt.%) contents (with Co/Ni ratio ~0.05, Fig. 5A). Magnetite displays high Si contents (0.43 to 1.26 wt.%; Fig. 5B).

Toward the mineralized zones, lizardite is progressively replaced by Fe-antigorite/greenalite.

Serpentinites are crosscut by Fe-Ca-silicates veins comprised of ilvaite, hydro-andradite and Fediopside (Di 21 -Di 84 ), associated with fibrous Fe-antigorite/greenalite, defining a stockwork structure (Fig. 4B-D). Locally, the amount of Fe-Ca-silicates increases drastically and they massively replace serpentinites along metasomatic bands (Fig. 4E). Fe-Ca-silicates can be associated with magnetite, pyrrhotite, chalcopyrite, pentlandite, sphalerite and isocubanite (Fig. 4B, D, andF). Textural relationships attest for synchronous crystallization of Fe-Ca silicates and sulfides. Locally, magnetite associated with ilvaite displays growth or dendrite-like textures indicative of contemporaneity (Fig. 4G, S1-1).

The proportion of magnetite and sulfides increases toward the most mineralized zones, defining an assemblage of semi-massive sulfides. It consists of millimetric to centimetric veins made of

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This article is protected by copyright. All rights reserved pyrrhotite and magnetite in textural equilibrium, minor chalcopyrite and pentlandite and Feantigorite/greenalite at their rims (Fig. 4H). Pentlandite has low Ni (21.3 to 24.2 wt.%) and low Co (2.9 to 3.6 wt.%) contents (with Co/Ni ratio ~0.15, Fig. 5A). Pyrrhotite displays intermediate Ni contents (average ~2800 ppm, Fig. 5C). Secondary oxidation minerals are pyrite/marcasite, violarite, bornite and covellite (Fig. S1-1).

In the most mineralized zones, serpentinites are almost completely replaced by a Cu-Fe-Ni-Co-Znrich assemblage of euhedral pyrrhotite, magnetite, pentlandite, Fe-diopside and minor small sphalerite enclosed within chalcopyrite (from 20% up to 50% of the mineralization, Cu-rich MS) displaying isocubanite exsolutions. All grains are in apparent textural equilibrium (Fig. 4I).

Pentlandite displays low Ni (24.9 to 25.7 wt.%) and high Co (14.1 to 15.3 wt.%) contents (with Co/Ni ratio ~0.58, Fig. 5A). Pyrrhotite has low Ni contents (<2000 ppm, average ~720 ppm).

Close to the detachment (i.e. Snake Pit), Fe-Ca-silicates are absent. The mineralization consists of disseminated euhedral pyrite grains or aggregates with minor chalcopyrite, pyrrhotite, pentlandite and sphalerite (Fig. 6B). Pentlandite displays high Ni (33 to 33.8 wt.%) and low Co (1.4 to 1.6 wt.%) contents (with Co/Ni ratio ~0.05, Fig. 5A). Pyrrhotite has high Ni contents (>3000 ppm, average about 4700 ppm, Fig. 5C). Steeply-dipping N-S calcite veins crosscut the mineralized serpentinite (Fig. 6A). These carbonates display δ 13 C values from 0.94 to 2.35‰ (Table S3-1, Fig. 6C; full range of the MCHS from -1.28 to 2.35‰) and δ 18 O values from 11.6 to 13.1‰ (full range of the MCHS from 11.5 to 13.1‰). Calcite veins from Snake Pit yield a Jurassic U-Pb age of 144 ± 13 Ma (Fig. 6D).

Discussion 1) Preservation of the MCHS during the Alpine overprint

A recurrent problem when dealing with rift-related pre-Alpine structures in the Alps is to assess the Alpine tectonic and metamorphic overprint. At Cotschen, the Jurassic detachment and the nearby mineralization associated with dyke-like intrusions were tilted in the footwall of a Nvergent D3 thrust. Because they show roughly the same orientation as the detachment, the intrusions presently in "dyke-like" position were probably sills intruded near-parallel to the detachment and schistose serpentinites. Despite tilting, the intrusions retained their planar attitude,

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This article is protected by copyright. All rights reserved and are crosscut by only a few faults with metric offsets (Figs. 3C,D,F), indicating that the system underwent limited internal strain.

In addition, several petrographic and isotopic features show that the system is preserved. First, calcite veins crosscutting the mineralized serpentinite display carbon isotope signatures typical of present-day marine ophicalcites forming at the seafloor (Ribeiro da Costa et al., 2008;[START_REF] Bach | Carbonate veins trace seawater circulation during exhumation and uplift of mantle rock: Results from ODP Leg 209[END_REF]; their δ 18 O around 12‰ is diagnostic of carbonation at temperatures around 130°C (temperatures already reported along neighboring Jurassic detachment; [START_REF] Coltat | Syntectonic carbonation during syn-magmatic mantle exhumation at an ocean-continent transition[END_REF].

Finally, these carbonates yield a Jurassic U-Pb age (Fig. 6D).

2) The MCHS: a fossil root zone of UHS

The Cu-Fe-Ni-Co-Zn mineralization previously reported in mafic-ultramafic settings worldwide have been attributed to either a magmatic (i.e. orthomagmatic Ni-Cu-PGE and Fe-Ti-V deposits with cumulate and mat textures in mafic-ultramafic intrusions, Foose, 1985;Song et al., 2003) or to a hydrothermal origin (Foose et al., 1985;Thalhammer et al., 1986;Maslennikov et al., 2017). In the present case, the transformation of lizardite into Fe-antigorite/greenalite, the stockwork structures and the metasomatic replacement of serpentinites by ore phases and Fe-Casilicates are diagnostic of epigenetic mineralization attesting for a hydrothermal origin.

The MCHS presents similarities with the UHS at the MAR (Table 2). Both hydrothermal systems share a geometry implying a detachment fault juxtaposing basalts onto serpentinites (McCaig et al., 2007). Mineralogical similarities are numerous: Cu-Fe-Ni-Co-Zn mineralization of the MCHS is typical of UHS (Fouquet et al., 2010); the replacement textures observed in the Curich MS are expected to occur in the highly altered zones close to seafloor (Marques et al., 2006); the Co enrichment of Cu-rich MS pentlandite is a feature shared with pentlandite from active systems (Mozgova et al., 1996;[START_REF] Borodaev | Pentlandite and Co-enriched pentlandite as characteristic minerals of modern hydrothermal sulphide mounds hosted by serpentinized ultramafic rocks (Mid-Atlantic Ridge)[END_REF].

The presence of mafic intrusions and the lack of hydrothermal chimneys suggest that the MCHS represents the deep levels of an UHS. This is consistent with the limited size of the mineralized bodies when compared to sulfide lenses observed at UHS seafloor. Also, the mineralogical diversity recorded at the MCHS (i.e. Fe-Ca-silicates together with ore minerals) has never been reported for UHS where intensive steatization is rather observed. Similarly, the high and low proportions of magnetite and sphalerite, respectively, slightly contrast with the mineralogical assemblage described for UHS (see Fouquet et al., 2010 for a review, Table 2). We therefore

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This article is protected by copyright. All rights reserved propose that the geometric and mineralogical features of the MCHS are symptomatic of UHS root zones.

3) Conceptual model

A conceptual model for the formation of UHS based on this study is proposed in figure 7.

During mantle exhumation, downward infiltration of seawater induced pervasive serpentinization of the peridotites (Fig. 7A). The resulting fluids were likely focused all the way to the seafloor along the detachment. The inception and cooling of gabbroic bodies and associated dyke and sill networks at depth likely triggered hydrothermal circulation. Fluids circulated along faults cutting through basalts and into the permeable serpentinite basement where they interacted with rocks (Marques et al., 2006;McCaig et al., 2007) (Fig. 7A). The resulting acidic, reduced and hot black smoker-type fluids reacted with serpentinization-derived fluids to form the mineralogical assemblage reported at the MCHS (Fig. 7A). Upward fluid circulation was enhanced along maficultramafic interfaces; mafic intrusions acted as drains. In the deepest parts, Fe-Ca-silicates crystallized together with ore minerals. In the upper part, close to the detachment (Snake Pit), these silicates were no longer stable, possibly because of the circulation of oxidized seawater that increased f O2 and enhanced pyrite stabilization.

On the deepest parts, the mineralogical zonation from barren serpentinite to Fe-Ca-silicates to semi-massive sulfides and then to Cu-rich MS (Figs. 3E,7B) has never been reported before this study. The ilvaite-andradite-diopside assemblage has been described in low-grade metamorphic serpentinites (Agata and Agachi, 1995) and in oceanic ultramafic settings (Gaggero et al., 1995). Gaggero et al. (1995) suggested that this assemblage precipitated from early serpentinizationderived fluids. Gustafson (1974) showed that the ilvaite-hedenbergite-andradite-magnetite assemblage is stable at low f O2 (10 -25 to 10 -32 ), high-T (425±75°C) conditions without strong changes if considering diopside instead of hedenbergite. In parallel, the semi-massive sulfides and Cu-rich MS mineralogy suggests precipitation from acidic, reduced and hot (>350°C) fluids, which would have probably interacted with mafic rocks (McCaig et al., 2007) as well as ultramafic rocks to explain Ni enrichment. These conditions are compatible with those forming Fe-Ca-silicates as the formation of hydrated minerals (e.g. ilvaite, hydroandradite and actinolite) decreases the pH of the fluid [START_REF] Allen | Compositional controls on vents fluids from ultramafichosted hydrothermal systems at mid-ocean ridges: an experimental study at 400°C, 500 bars[END_REF]. In addition, CH 4 -and H 2 -rich venting fluids reported for UHS [START_REF] Charlou | Geochemistry of high H2 and CH4 vent fluids issuing from ultramafic rocks at the Rainbow hydrothermal field (36°14[END_REF] suggest that concomitant in-depth serpentinization

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This article is protected by copyright. All rights reserved occurred during black-smoker-related hydrothermal activity. We thus propose that the horizontal zonation observed at depth results from increasing fluid/rock ratios from the outer zones to the stockwork zone then to the most mineralized zones.

Finally, the cooling of the system is recorded through Late Jurassic hydrothermal carbonation (~130°C) close to the detachment plane. Previous studies have emphasized that carbonation could derive from the mixing between serpentinization-derived uprising fluids (along the detachment) and seawater close to the seafloor (Kelley et al., 2001;Ludwig et al., 2006). The δ 13 C values of the MCHS ophicalcites are similar to those reported at the Lost-City hydrothermal field suggesting that the same mechanism could have caused carbonate precipitation in the MCHS.

Conclusions

The distribution of mineralized assemblages across the Marmorera-Cotschen Hydrothermal System records the subsurface evolution of an ultramafic-hosted hydrothermal system, from a Cu-Fe-Ni-Co-Zn-rich assemblage and associated Fe-Ca-silicates in the deepest zones to a sulfide-rich, silicate-free mineralization close to the detachment plane. Carbonation marked the end of the hydrothermal circulation. The MCHS formed in a hyper-extended margin. This suggests that black smoker-type hydrothermal activity is not restricted to oceanic domains but can also occur in Ocean-Continent Transitions, which should lead us to reconsider metal resources available in such geodynamic settings.

Figure captions

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This article is protected by copyright. All rights reserved MAR (Mozgova et al., 1996). The increase in intensity of the hydrothermal alteration intensity increase is marked an increase of the Co/Ni ratio. The loss of Ni in pentlandite from semi-massive sulfides likely reflects the alteration of pentlandite into violarite (Nickel et al., 1974). (B) Si (ppm) vs. Fe (wt. %) diagram for magnetite from Marmorera and magnetite from the Rainbow hydrothermal field (this study). Compared to syn-serpentinization magnetite (green squares, 4000<Si<13000 ppm), syn-mineralization magnetite (all other squares) has lower Si contents (< 6200 ppm, average around 1900 ppm). (C) Frequency histogram of Ni content (ppm) for pyrrhotite from the MCHS. The mean composition of several pyrrhotite grains from different hydrothermal assemblages of Rainbow (MAR) is also shown (Marques et al., 2006). A negative correlation between the Ni content of pyrrhotite and the intensity of the hydrothermal alteration is visible in both the active (i.e. Rainbow) and fossil (i.e. MCHS) systems.

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This article is protected by copyright. All rights reserved mineralogical assemblage reported in this study. (B) Schematic view of the mineralogical zonation occurring on the margin of mafic dykes, ranging from barren serpentinite to an envelope of Fe-Casilicates stockwork, to semi-massive sulfides and ultimately Cu-rich MS. The transition between fresh lizardite-rich serpentinite and the Fe-Ca-silicates stockwork is marked by the replacement of lizardite by a Fe-rich serpentine (likely Fe-antigorite). The overall zonation likely reflects an increase of the fluid/rock ratio from the outer parts (i.e. barren serpentinite) to the inner part (i.e.

Cu-rich MS) of the system.

Table captions

Table 1. Synthetic paragenetic sequence of the mineralization observed at Marmorera-Cotschen (left) compared to the one observed at the Snake Pit outcrop (right). Fe-Ca-silicates (underlined in the Marmorera-Cotschen sequence, left) are absent at Snake Pit, pyrite and calcite (in bold underlined in the Snake pit sequence) are absent at Marmorera-Corschen. See text and supplementary data for details.

Figure 1 .

 1 Figure 1. (A) Simplified genetic model proposed for the formation of active ultramafic-hosted black-smokers deposits (e.g. Rainbow-type systems) along slow-spreading ridges showing the speculated fluid pathway (after McCaig et al., 2007). Seawater percolates through fractures and faults crosscutting the mafic cover and circulates through the permeable serpentinite basement where it gets heated in the vicinity of the gabbroic intrusions before rising up in the serpentinized footwall. The fluid thus discharges the metals by mixing with cold seawater. NVZ=Neo Volcanic Zone. (B) Seafloor expression of hydrothermal venting activity at the Rainbow field (2200 m.b.s.l., photo from Ifremer).

Figure 2 .

 2 Figure 2. (A) Simplified map of the main paleogeographic units of the Central and Western Alps and the Apennines. (B) Geological map of the Platta nappe (modified after Schaltegger et al., 2002) showing the Marmorera Cotschen area (black inset). (C) Reconstructed cross section shows mantle exhumation beneath the Adriatic plate during the Jurassic WNW-ESE-oriented opening of the Alpine Tethys Ocean (after Pinto et al., 2015). Mantle exhumation recorded in the Platta nappe was accommodated by detachment faults and was accompanied by mafic magmatism (gabbros, dolerites and basalts). As a consequence of a low magmatic budget, large parts of the seafloor expose mantle rocks without the establishment of a Penrose-type oceanic crust.

Figure 3 .

 3 Figure 3. (A) General view of the cliff of the Marmorera-Cotschen hydrothermal system. The mineralization is distributed almost continuously from the Marmorera Lake (bottom of the picture, 1684m above sea level (m.a.s.)) up to Cotschen (2308 m.a.s.) exposing about 600m of vertical relief. (B) Line-drawing of the picture displayed in (A) At most places, the serpentinite-hosted mineralization occurs as thin corridors (up to several meters) while the main gossan has a width of 90 meters. (C) Relationship between dyke-like south-dipping mafic intrusions and mineralization in the upper gossan. Mineralization is hosted in the serpentinite on the southern flank of each intrusion, resulting in a repeated S-N spatial sequence of mineralized serpentinite, the mafic intrusion and barren serpentinite (grassy areas). (D) Actinolite-rich (Act) alteration at a mafic intrusion -mineralized serpentinite interface in the upper gossan. (E) On the margin of mafic Accepted Article

Figure 4 .

 4 Figure 4. Micrographs illustrating the mineralogical evolution from the barren serpentinites to the Cu-rich MS (i.e. approaching the mafic intrusions). (A) Barren serpentinite exhibiting early small (< 50µm) magnetite (Mt I) and pentlandite (Pn I) grains along serpentine (Srp I) grain boundaries and within serpentine domains, respectively. Location in figure 3E. (B) Ilvaite (Ilv) and Fediopside (Dp) veins enclosing chalcopyrite (Cp) and magnetite (Mt II) grains from a Fe-Casilicates stockwork structure. Some magnetite grains are also found in the serpentine groundmass without specific relationship to Fe-Ca-silicates. (C) Typical Fe-Ca-silicates stockwork structure cutting through serpentinites. (D) Zoom on the internal composition of the stockwork shown in figure 4C. It consists of ilvaite (Ilv) and minor Fe-diopside (Dp) veins. The serpentine groundmass contains small disseminated grains or aggregates of sphalerite and pentlandite (Pn + Sp). Smaller sulfide grains (< 25 µm, not visible on that scale) are also present in the Fe-Ca-silicates veins, which suggests that sulfides formed before the silicate stockwork. Spinel (Spn) relics persist in the serpentine groundmass. (E) Extensive metasomatic replacement of serpentinite by Fe-Ca-silicates and associated sulfides and oxides. (F) Sulfide aggregate made of sphalerite (Sp), chalcopyrite (Cp) with isocubanite (Is) exsolutions, and magnetite (Mt II). The aggregate is surrounded by an assemblage of magnetite, hydroandradite (Adr) and ilvaite (Ilv). White arrows point to dendriticlike intergrowths of magnetite and ilvaite, suggesting a concomitant crystallization of the two phases. (G) Growth texture attesting for a synchronous crystallization of magnetite (Mt II) and ilvaite (Ilv). Chalcopyrite (Cp) is locally present within ilvaite. (H) Semi-massive sulfide made of magnetite (Mt II) and pyrrhotite (Po) veins associated with a newly formed serpentine phase (Srp II, likely Fe-antigorite or greenalite). Location in figure 3E. (I) Cu-rich massive sulfides with an assemblage made of chalcopyrite (Cp) with isocubanite (Is) exsolutions, pyrrhotite (Po), magnetite (Mt II), pentlandite (Pn II) and Fe-diopside (Dp) enclosing serpentine clasts (Srp II). Location in figure 3E.

Figure 5 .

 5 Figure 5. Elemental discriminant diagrams for pentlandite (A) and magnetite (B) grains from the MCHS and present-day occurrences at the MAR. (A) Co vs. Ni (wt. %) diagram for pentlandite from the MCHS plotted together with Co-rich pentlandite from seafloor Cu-rich chimneys at the

Figure 6 .

 6 Figure 6. Features associated with carbonation in the serpentinites right beneath the Jurassic detachment (Snake Pit and Marmorera outcrops, located in figure 3A and B). (A) Mineralized serpentinite crosscut by sub-vertical calcite veins. (B) Pyrite (Py) + chalcopyrite (Cp) aggregate hosted in serpentine (Srp) and crosscut by late calcite (Cc). (C) δ 13 C vs. δ 18 O isotopic diagram of ophicalcites from the MCHS and present-day marine ophicalcites from the 15°20'N Fracture Zone at MAR (Bach et al., 2011). δ 13 C signatures of the ophicalcites from the MCHS overlap with those from the MAR. In contrast, δ 18 O signatures of the ophicalcites from the MCHS are much lower than those from the MAR, reflecting different temperatures of formation for the two systems. (D) Tera-Wasserburg diagram for calcite grains from sample Cot16_49 at Snake Pit (LA-ICP-MS U-Pb analyses). The analyses are discordant and lie on a linear trend with a lower intercept date at 144 ± 13 Ma. Ellipsoids are displayed with a confidence of ± 2σ.

Figure 7 .

 7 Figure 7. (A) Genetic model for the formation of the MCHS before carbonation hydrothermal stage. During mantle exhumation along a detachment, seawater percolates downward through the faulted mafic hangingwall and the damaged serpentinite footwall where it leaches metals (the exact sources need further investigation). The heat transferred from cooling of shallow gabbroic bodies triggers hydrothermal convection. The fluids get heated and rise in the damage zone of the detachment likely up to the surface where metal deposition from the fluids form UHS. Fluid circulation is guided by lithological interfaces such as the contacts between mafic intrusions and serpentinites. Synchronously, at greater depth, active serpentinization of peridotite releases fluids which, on their way to the surface, likely mix with black smoker-type fluids, forming the

  Fig. 3F

  Fig. 4I Fig. 4I

  Fig. 4H Fig. 4H Cu-rich MS Cu-rich MS to semi-massive sulf. to semi-massive sulf.

Table 2 .

 2 Main characteristics of present-day and fossil oceanic hydrothermal systems

	(C)
	Asthenospheric	Upper mantle	Basalts,	Lower crust	Middle crust	Upper crust	Pre-rift sediments	Early-rift to
	mantle		dykes, gabbro					post-rift sediments

Acknowledgments

The authors would like to thank J. Langlade for help with microprobe analyses, X. Lecoz for the preparation of thin sections. We acknowledge Andrew Martin and one anonymous reviewer for their constructive reviews. This study was founded through CNRS-INSU and University of Rennes 1 "Défis Scientifiques" grants (accorded to P. Boulvais and R. Coltat).

Accepted Article

This article is protected by copyright. All rights reserved Firstova, A., Stepanova, T., Cherkashov, G., Goncharov, A., and Babaeva, S., 2016. Compositions and formation of gabbro-peridotite hosted seafloor massive sulfide deposits from the Ashadze-1 hydrothermal field, Mid-Atlantic Ridge: Minerals, 19, 20pp, doi:10.3390/min6010019 Foose, M. P., 1985. The setting of a magmatic sulfide occurrence in a dismembered ophiolite, southwest Oregon: U.S. Geol. Surv. Bull., 1626A Foose, M. P., Economou, M., and Panayiotou, A., 1985. Compositional and mineralogic constraints on the genesis of ophiolitic hosted nickel mineralization in the Pevkos area, Limassol forest, Cyprus: Mineralium Deposita, 20, 234-240 Fouquet, Y., et al., 2008. Serpentine Cruise -ultramafic hosted hydrothermal deposits on the Mid-Atlantic Ridge: First submersible studies on Ashadze 1 and 2, Logatchev 2 and Krasnov vents fields: InterRidge News, 17, 16-21 Fouquet, Y., et al., 2010. Geodiversity of hydrothermal processes along the Mid-Atlantic Ridge and ultramafic-hosted mineralization: a new type of oceanic Cu-Zn-Co-Au volcanogenic massive sulphide deposit: Geophysical Monograph Series, 188, 321-367 Froitzheim, N., Schmid, S. M., and Conti, P., 1994. Repeated change from crustal shortening to orogen-parallel extension in the Austroalpine units of Graubünden: Eclogae Geologicae Helvetiae, 87/2, 559-612, doi.org/10.5169/seals-167471 Froitzheim, N., and Manatschal, G., 1996. Kinematics of Jurassic faulting, mantle exhumation, and passive-margin formation in the Austroalpine and Penninic nappes (eastern Switzerland): GSA Bulletin, 108, 1120-1133 Gaggero, L., Cortesogno, L., and Gazzoti, M., 1997. Data Report: Oxides, sulphides and associated phases in veins and hydrothermally altered peridotitic rocks. In: Proceedings of the Ocean Drilling Program (Karson, J.A., Cannat, M., Miller, D.J., and Elthon, D.). Scientific Results, 153, 523-529 Goodfellow, W. D., and Franklin, J. M., 1993. Geology, mineralogy, and chemistry of sedimenthosted clastic massive sulfides in shallow cores, Middle Valley, Northern Juan de Fuca Ridge: Economic Geology, 88, 2037-2068 Accepted Article This article is protected by copyright. All rights reserved Gustafson, W. I., 1974. The stability of andradite, hedenbergite and related minerals in the system Ca-Fe-Si-O-OH: Journal of Petrology, 15, 455-96 Hannington, M. D., Tivey, M. K., Larocque, A. C. L., Petersen, S., and Rona, P. A., 1995. The occurrence of gold in sulfide deposits of the TAG hydrothermal field, Mid-Atlantic Ridge: Can. Mineral., 33, 1285-1310. Herzig, P.M., Petersen, S., and Hannington, M., 1998. Geochemistry and sulfur-isotopic composition of the TAG hydrothermal mound, Mid-Atlantic Ridge, 26°N': Proceedings of the Ocean Drilling Program, Scientific Results, 158, 47-70 Kelley, D. S., et al., 2001. An off-axis hydrothermal vent field near the Mid-Atlantic Ridge at 30°N: Nature, 412, 145-149. Ludwig, K. A., Kelley, D. S., Butterfield, D. A., Nelson, B. K., and Früh-Green, G., 2006. Formation and evolution of carbonate chimneys at the Lost City hydrothermal field: Geochimica et Cosmochimica Acta, 70, 3625-3645, doi:10.1016/j.gca.2006.04.016 Marques, A.F.A., Barriga, F., Chavagnac, V., and Fouquet, Y., 2006. Mineralogy, geoschemistry and Nd isotope composition of the Rainbow hydrothermal field, Mid-Atlantic Ridge: Mineralium Deposita, 41, 52-67, DOI 10.1007/s00126-005-0040-8 Maslennikov, V. V., et al., 2017. Chimneys in Paleozoic massive sulfide mounds of the Urals VMS deposits: Mineral and trace element comparison with modern black, grey, white and clear smokers: Ore Geology Reviews, 85, 64-106, doi:10.1016/j.oregeorev.2016.09.012 McCaig, A.M., Cliff, R.A., Escartin, J., Fallick, A.E., and McLeod, C.J., 2007. Oceanic detachment faults focus large volumes of black smoker fluids: Geology, 35, 935-938, doi:10.1130/G23657A McCaig, A.M., Delacour, A., Fallick, A.E., Castelain, T., and Gretchen, F.G., 2010. Detachment fault control on hydrothermal circulation systems: interpreting the subsurface beneath the TAG hydrothermal field using the isotopic and geological evolution of oceanic core complexes in the Atlantic: Geophysical Monograph Series, 188, 207-239

Accepted Article

This article is protected by copyright. All rights reserved Mozgova, N. N., Krasnov, S. G., Batuyev, B. N., Borodaev, Y. S., Efimov, A. V., Markov, V. F., and Stepanova, T., V., 1996. The first report of cobalt pentlandite from a Mid-Atlantic Rige hydrothermal deposit: The Canadian Mineralogist, 34, 23-28. Nehlig, P., and Marquez, L., 1998. Feeder zones of massive sulfide deposits: Constraints from Bent Hill, Juan de Fuca Ridge -ODP Leg 169: Water-rock interaction, 751-754 Nickel, E. H., Ross, J. R., and Thornber, M. R., 1974. The supergene alteration of pyrrhotitepentlandite ore at Kambalda, Western Australia: Economic Geology, 69, 93-107. Ondreas, H., Cannat, M., Fouquet, Y., and Normand, A., 2012. Geological context and vents morphology of the ultramafic-hosted Ashadze hydrothermal areas (Mid-Atlantic Ridge 13°N): Geochemistry, Geophysics, Geosystems, 13, 20pp, doi:10.1029/2012GC004433 Pinto,V. H. G., Manatschal, G., Karpoff, A. M., and Viana, A., 2015. Tracing mantle-reacted fluids in magma-poor rifted margins: the example of Alpine Tethyan rifted margins: Geochemistry, Geophysics, Geosystems, 16, 3271-3308, doi:10.1002/2015GC005830. Ribeiro da Costa, I., Barriga, F. J. A. S., and Taylor, R. N., 2008. Late seafloor carbonate precipitation in serpentinites from the Rainbow and Saldanha sites (Mid-Atlantic Ridge): Eur. J. Mineral., 20, 173-181, doi:10.1127/0935-1221/2008/0020-1803 Rona, P. A., and Scott, S. D., 1993. A special issue on sea-floor hydrothermal mineralizations: new perspectives: Economic Geology, 88, Schaltegger, U., Desmurs, L., Manatschal, G., Müntener, O., Meier, M., Frank, M., and Bernoulli, D., 2002. The transition from rifting to sea-floor spreading within a magma-poor rifted margin: field and isotopic constraints: Terra Nova, 14, 156-162, doi:10.1046/j.1365-3121.2002.00406.x Song, X.-Y., Zhou, M.-F., Cao, Z.-M., Sun, M., and Wang, Y.-L., 2003. Ni-Cu-(PGE) magmatic deposits in the Yangliuping area, Permian Emeishan igneous province, SW China: Mineralium Deposita, 38, 831-943, doi:10.1007/s00126-003-0362-3 Thalhammer, O., Stumpfl, E. F., and Panayiotou, A., 1986. Postmagmatic, hydrothermal origin of sulfide and arsenide mineralizations at Limassol forest, Cyprus: Mineralium Deposita, 21, 95-105

Accepted Article

This article is protected by copyright. All rights reserved Tucholke, B. E., Lin, J., and Kleinrock, M. C., 1998. Megamullions and mullion structure defining oceanic metamorphic core complexes on the Mid-Atlantic Ridge: Journal of Geophysical Research, 103,[9857][9858][9859][9860][9861][9862][9863][9864][9865][9866]

Data Availability Statement

The data that support the findings of this study are provided in the supplementary material.

Supplementary material

Supporting information includes the caption for figure S1-1.

Tables S1, S2 and S3 can be found in files Table S1-1.pdf, Table S1-2.pdf, Table S2-1.pdf, Table S2-2.pdf, Table S3-1.pdf, Table S3-2.pdf and Table S3-3.pdf.

Accepted Article

Marmorera -Cotschen

Mineralization stage & Fe-Ca-silicates Supergene alteration Table 1. Synthetic paragenetic sequence of the mineralization observed at Marmorera-Cotschen (left) compared to the one observed at the Snake Pit outcrop (right). Fe-Ca-silicates (underlined in the Marmorera-Cotschen sequence, left) are absent at Snake Pit, pyrite and calcite (in bold underlined in the Snake pit sequence) are absent at Marmorera-Corschen. See text and supplementary data for details. This article is protected by copyright. All rights reserved

Accepted Article

This article is protected by copyright. All rights reserved [START_REF] Charlou | Geochemistry of high H2 and CH4 vent fluids issuing from ultramafic rocks at the Rainbow hydrothermal field (36°14[END_REF][START_REF] Bortnikov | The Rainbow serpentinite-hosted hydrothermal field, Mid-Atlantic Ridge, 36°14[END_REF]Fouquet et al., 2010;Evrard et al., 2015;Fouquet et al., 2008Fouquet et al., 2010;Fouquet et al., 2008;Ondreas et al., 2012;Firstova et al., 2016Fouquet et al, 2010;Herzig et al., 1998;Rona and Scott, 1993;Hannington et al., 1995 Nehlig andMarquez, 1998;Goodfellow andFranklin, 1993 Dietrich, 1972; This study

In bold are displayed the hydrothermal phases which differ from present-day systems

The elements displayed in the mineralization type are given by modal abundance


Vermot Des Roches

Guillaume

Etienne Balmes

Thierry Chancelier

MDRE: AN EFFICIENT EXPANSION TOOL TO PERFORM MODEL UPDATING FROM SQUEAL MEASUREMENTS

Keywords: Expansion, model reduction, model updating, parametric study, test/model correlation

In brake FEM, model updating is often needed to improve the model accuracy and well describe problematic phenomena such as the squeal. To avoid performing a full model updating which is often time consuming, the use of the Minimum Dynamic Residual Expansion method is proposed to help building the updating strategy. The procedure proposed in this paper is evaluated on a disc brake system, using experimental measurements and the nominal model as input data. From experimental squeal measurements, two shapes are extracted and expanded on the current model. The evaluation of the residual error of model shows areas where the model is wrong and guides through the definition of sensitive parameters which need to be updated. Once the model is parameterized, a model reduction strategy is proposed for further computations to be performed in a time compatible with industrial processes. A parametric study is then achieved: the expansion is computed for all the combinations of the chosen parameters. It is finally possible to navigate through the expansion results for all the parameters, evaluate the evolution of the model accuracy and extract the best combination which improves the model representability.

TECHNICAL PAPER

In brake models, it is difficult to be confident in the a priori relevance of some parameters. Model updating is thus often needed to properly represent the real system. In previous works, a full model updating method has been proposed [START_REF] Martin | Review of model updating processes used for brake components[END_REF][START_REF] Martin | Updating and design sensitivity processes applied to drum brake squeal analysis[END_REF]. To limit model bias in the updating result, the strategy was to update component alone first (geometry and material properties) and then to iteratively update isolated couplings in sub-models. This strategy is really relevant at the component level and for some couplings but can be time consuming (many measurements are needed) and some parameters cannot be isolated in substructures (sliding contacts, shoes/back-plate, pad/bracket…) A complementary updating procedure is thus needed. In [START_REF] Martin | Squeal measurement using operational deflection shape. Quality assessment and analysis improvement using FEM expansion[END_REF], the Minimum Dynamic Residual Expansion (MDRE) [START_REF] Balmes | Review and Evaluation of Shape Expansion Methods[END_REF][START_REF] Bobillot | Solving minimum dynamic residual expansion and using results for error localization[END_REF], was first proposed to improve the interpretation of experimental shapes measured with low sensor density, typically with accelerometers. In presence of higher number of sensors, this expansion technique can also be used to highlight areas where model error exists [START_REF] Martin | Squeal measurement with 3D Scanning Laser Doppler Vibrometer: handling of the time varying system behavior and analysis improvement using FEM expansion[END_REF][START_REF] Martin | Squeal measurement using operational deflection shape. Quality assessment and analysis improvement using FEM expansion[END_REF]. It is proposed in this paper to go further with two steps: first with a model parameterization of the areas pointed out by the MDRE result and secondly by updating the parameters. First, the system used as illustration is presented with a description of both numerical and experimental input data. Then, MDRE expansion theory is summarized and expansion results on the test case are presented. Next, the implementation of the parametric model is developed and precisions are provided on how it can be used in the expansion process. Focus is also made on calculation times. Finally, results of the parametric study are shown and perspectives are provided to go further.

TEST CASE PRESENTATION

When dealing with squeal phenomenon, some parameters are really sensitive to braking conditions (especially the braking pressure) and need to be addressed to obtain a model representative enough to propose design countermeasures. Thus experimental data must be exploited to correlate and improve the model. Using in-squeal ODS correlation provides a good way of updating a brake model in specific operating conditions.

Two main techniques are available to perform the ODS measurements: using accelerometers which can be measured synchronously but have a low spatial resolution or using 3D Scanning Laser Doppler Vibrometer (3DSLDV) measurements which allow a higher spatial density but lead to sequential measurements. Using sequential measurements is a difficulty previously addressed in [START_REF] Martin | Squeal measurement using operational deflection shape. Quality assessment and analysis improvement using FEM expansion[END_REF][START_REF] Martin | Squeal measurement with 3D Scanning Laser Doppler Vibrometer: handling of the time varying system behavior and analysis improvement using FEM expansion[END_REF] because the squealing system cannot be considered as time invariant. Results from previous papers, with experiments on both drum and disc brakes, showed that the shapes associated with squeal instabilities are expected and actually found to be dominated by the combination of two real modes. Despite frequency shifts, two real shapes dominate the response and have variations that are coherent with changes due to the wheel position.

Here a disc brake is used as illustration. The disc brake is measured in squealing conditions on test bench shown in Figure 1 left using the Polytec 3DSLDV PSV500. The test wireframe is constructed in two parts: one in the direct view to measure the disc, a part of the caliper and a part of the anchor bracket and one through the mirror to measure the remaining parts of the caliper and the anchor bracket. It is composed of 619 points, leading to 1857 measured channels (XYZ directions). The test wireframe is overlaid with the FEM in Figure 1 right. To extract the two main shapes; three reference mono-axial accelerometers were used: one on the caliper, one on the anchor bracket and one on the arm. From sequential time measurements, the procedure presented in [START_REF] Martin | Squeal measurement using operational deflection shape. Quality assessment and analysis improvement using FEM expansion[END_REF][START_REF] Martin | Squeal measurement with 3D Scanning Laser Doppler Vibrometer: handling of the time varying system behavior and analysis improvement using FEM expansion[END_REF] was used to extract two main real shapes around 4250Hz, where squeal occurs. The two main shapes are shown in Figure 2. The first shape shows mainly a deformation of the bracket. The second shape shows another deformation of the bracket with a deformation of the disc.

On the modelling side, the brake assembly is simulated with finite elements. All components are in contact friction-interaction. A steady-sliding state in brake actuation conditions is computed to obtain contact and friction statuses from which a tangent model is derived (linearization) for eigenvalue analysis. The model used in this study is a quality checked (mesh convergence, steady sliding state consistency ...) state-of-the-art model featuring 3.000.000 DOF.

MINIMUM DYNAMIC RESIDUAL EXPANSION

Dealing with modal correlation, the most widely used technique is the Modal Assurance Criterion MAC [START_REF] Allemang | The modal assurance criterion (MAC): twenty years of use and abuse[END_REF] for shape pairing and frequency correspondence for paired shapes. This methodology has proven its efficiency, but misses the capability to guide the model updating procedure. When the correlation result is poor: it is difficult to localize the modelling errors from a MAC result only. Moreover, the MAC shows irregularity when modes switch and thus cannot be efficiently used in an optimization procedure [START_REF] Martin | Méthodes de corrélation calcul/essai pour l'analyse du crissement[END_REF].

The MDRE algorithm [START_REF] Balmes | Review and Evaluation of Shape Expansion Methods[END_REF] combines test and model to estimate test shapes on the full FEM. This process is called expansion. The first interest is to complement 3D-SLDV measurement, which despite their high resolution cannot estimate motion on hidden parts or interfaces between components as shown in the overlay in Figure 1 right. Estimating the shape everywhere may ease proposing modification to impact the squeal phenomenon. The second interest is to introduce a measure highlighting areas where the FEM may have errors.

Given a FEM shape 𝜙 , one seeks to define a cost function describing a first objective that is its consistence with a test shape {𝑦 } measured at sensors and a second objective in the fact that this shape is close to being a solution of the model.

The observation of the FEM shape at sensors is described by an observation matrix [𝑐]. One can thus define an observation or test error using the classical Euclidian norm of the different between the measured shape and the observation of the FEM shape

𝜖 = [𝑐] 𝜙 -{𝑦 } [𝑐] 𝜙 -{𝑦 } (1)
If the model is somewhat correct, it is also expected that the FEM shape respects equilibrium equations. For a modeshape 𝑍(𝜔) 𝜙 should be zero, for a forced response 𝑍(𝜔) 𝜙 -𝐹(𝜔) should be zero, … If the shape is not a FEM solution, one can define a mechanical unbalance residual. For example {𝑅 (𝜔)} = 𝑍(𝜔) 𝜙 will be the residual for a modeshape. Since this residual is a load, one needs to associate an energy measure to properly quantify its size. The classic approach is to introduce a static response induced by the unbalanced forces

{𝑅 (𝜔)} = [𝐾] {𝑅 (𝜔)} = [𝐾] 𝑍(𝜔) 𝜙

(2) and to compute the strain energy of this shape as equilibrium or model error

𝜖 = {𝑅 (𝜔)} [𝐾]{𝑅 (𝜔)} (3)
If the model is perfectly correct and matches the test, both errors should be zero. In practice, one defines a multi-objective cost function, associated with an error weighting 𝛾 as

J(γ, 𝜙 ) = ϵ (Z, 𝜙 ) + γ ϵ (y , 𝜙 ) (4)
which combines the observation 𝜖 and equilibrium 𝜖 errors. For any value of the weighting 𝛾, the minimization problem (4) can be put in matrix form (see [START_REF] Balmes | Review and Evaluation of Shape Expansion Methods[END_REF][START_REF] Martin | Méthodes de corrélation calcul/essai pour l'analyse du crissement[END_REF]) allowing the computation of an optimal expanded shape 𝜙 (𝛾) minimizing the objective function.

The expanded shape 𝜙 (𝛾) provides the best compromise between the respect of the measured shape and the respect of the model equilibrium, the balance being controlled by 𝛾.

In practice, full resolution of MDRE is generally not accessible [START_REF] Balmes | Review and Evaluation of Shape Expansion Methods[END_REF] so that the reduction proposed in [START_REF] Martin | Méthodes de corrélation calcul/essai pour l'analyse du crissement[END_REF] is a major contribution. The FEM is reduced on a basis combining the static responses to unit loads at sensors, for the result to be at least as good as the static expansion, and the free modes of the structure (considering the friction coefficient equals to zero), for the expansion to be exact if the model matches the measurements perfectly. This family of shapes is orthonormalized with respect to mass and stiffness matrices, leading to the basis

[𝑇] = [𝛷] [𝛷 ]
(5) with [𝛷] the free mode shapes and [𝛷 ] the part of the static response to unit loads at sensors that is orthogonal to the free modes (this part will be called enrichment later).

Furthermore, this model reduction can be useful if the expansion is used in combination with an updating procedure, to speed up the parametric studies. It also allows a quick evaluation of the MDRE result for several values of the parameter 𝛾.

Rather than absolute test and model errors, relative errors are introduced to ease interpretation of the role of parameter 𝛾. The relative test error is defined using the Euclidian norm of the test shape to normalize

𝜖 = 𝜖 ‖𝑦 ‖ (6)
and the relative model error is normalized with the strain energy of the expanded shape

𝜖 = 𝜖 𝜙 [𝐾] 𝜙 (7)
Exploiting the reduction basis topology (5), the DOFs 𝑞 can be decomposed in those linked to the free modes 𝑞 and those linked to the enrichment 𝑞 . In this basis, the stiffness matrix is diagonal with first the mode pulsations 𝜔 and then the pseudo-pulsations 𝜔 . From this property, the model error can be decomposed in a part linked to the free modes and a part linked to the enrichment

𝜖 = {𝑅 } | ⋱ ω ⋱ {𝑅 } | + {𝑅 } | ⋱ ω ⋱ {𝑅 } | (8)
Several expansions with an increasing 𝛾 from 1 to 1e10 have been performed on the disc brake test case. The evolution of the relative test and model errors is shown in Figure 3. For a very low value of 𝛾 = 1, the relative model error is really low but the expanded shape does not match the measurements. When increasing the value of 𝛾, the test error decreases with at first an increasing model error linked to the enrichment (up to 𝛾 = 1𝑒5). Then, the test error continues to decrease but with a strong increase of the model error linked to the free modes: to represent the measurement well, the free modes are used but do not satisfy the mechanical equilibrium well. The free mode subspace seems thus relevant but there are errors on the associated frequencies.

In the following sections, an intermediate 𝛾 value of 1e+06 is chosen to compute the expansion: at this value, the relative error of model is distributed between a participation of the modes and a participation of the enrichment, and the relative error of measurement is acceptable (<10%). The expanded mode shapes corresponding to this 𝛾 are show in Figure 4.

The first one is dominated by the deformation of the right column and the left side of the caliper, with a small level of deformation of the disc. The second shape is mostly a deformation of the whole caliper and again the right column, plus a deformation of the disc. The expansion can also be used to analyse the model quality. Indeed, since the model error is a strain energy (see (3)), it is possible to visualize its repartition in the model elements and thus highlight areas where the mechanical equilibrium is poor. This model error can be split in two (see [START_REF] Martin | Méthodes de corrélation calcul/essai pour l'analyse du crissement[END_REF]): the part related to the free mode shapes, and the part related to the enrichment shapes.

Figure 5 shows the model error part related the free mode shapes. It is located close to the abutments between the pads and the bracket: surfaces A corresponds to contacts in the tangential direction of the disc and surfaces B in the radial directions. Figure 6 shows that the model error part related to the enrichment shapes is mostly located near the sensor: the model is forced to follow the test shape which contains measurement errors leading to residual loads close to the sensors. The poor readability of this plot illustrates the motivation for splitting energy contributions in two parts. A major interest of contact definition with stiffness evolution is the ability to use multi-model reduction [START_REF] Hammami | Intégration de modèles de jonctions dissipatives dans la conception vibratoire de structures amorties[END_REF] for very fast computations. The full model is only computed for some snapshots corresponding to sets of parameter values {𝛼 }. The associated collection of vectors (mode shapes of each snapshot) is orthonormalized with respect to the mass [𝑀] and the most rigid stiffness matrix 𝐾 𝛼 (highest value for each parameter 𝛼 ) to generate a reduction basis 𝑇 = [𝑇(𝛼 ), 𝑇(𝛼 ), … , 𝑇(𝛼 )] (10) Using a standard Ritz-Galerkin procedure, the reduced model takes the form

([𝑀 ]𝑠 + [𝐾 (𝛼 )]){𝑞′} = [T ] {𝐹(𝑠)} (11)

with [𝑀 ] = [T ] [𝑀][T ] and [𝐾 (𝛼 )] = [T ] [𝐾(𝛼 )][T ]

The choice of the snapshots points and normalization remains open. A pertinent strategy is to generate snapshots corresponding to the nominal configuration and the sequence of configurations with one of the parameters set to its minimal value. Nevertheless, the number of required snapshots can become large and time consuming as these correspond to full model results. Iterative subspace enrichment techniques could be explored in such case. In practice sensitivity analyses allow discarding irrelevant parameters, and it usually remains acceptable to compute a few tens of full points in extreme cases.

For the considered test case, the 4 junctions identified in the previous section use nominal contact stiffness densities 100 times higher than the component material representative stiffness. The nominal junctions 𝐾 are in saturated state, i.e. stiffer than the components interfaces. In practice the effective junction stiffness may be lower than its saturated value accounting for contact imprecisions (effective surface, effective friction, microopening/sliding, ...). The chosen parameters lumps these effects by spreading the stiffness density factors 𝛼 between values 1 (nominal saturated) and 1e-8 (released). The snapshots will thus correspond to a combination of 5 configurations:

𝛼 𝛼 𝛼 𝛼 1 1 1 1 1e-8 1 1 1 1 1e-8 1 1 1 1 1e-8 1 1 1 1
1e-8 For the model parameters to be accessible in the expansion process, the reduction basis used for expansion, previously described in (5), becomes

[𝑇] = [𝑇 ] [𝛷 ]
(12) It is then possible to compute in this subspace the expansion result 𝜙 (γ, α ) minimizing J(γ, α , 𝜙 ) = ϵ (Z, α , 𝜙 ) + γ ϵ (y , 𝜙 ) (13) for any configuration 𝛼 and any weighting γ.

For a fixed weighting 𝛾, updated model parameters 𝛼 are those which minimize the relative model error defined in [START_REF] Allemang | The modal assurance criterion (MAC): twenty years of use and abuse[END_REF]. The influence of 𝛾 on 𝛼 that will be considered in future work.

The gain of using a reduced model is both in terms of time (cost of 5 computations for hundreds of points in the design space) and memory (storing the reduced modeshapes associated with all points is practical when storing their full version would requires gigabytes). To highlight the computational gain in, the full model features 3.000.000 DOF and the computation of modes up to 12 kHz takes about half an hour. The enrichment with loads at the 1857 sensors and the orthonormalisation of all the sets of shapes takes about 2 hours. Obtaining a parametric model compatible with the MDRE process thus takes about 5 hours. This parameterized model features 2181 generalized DOFs and it takes about 1 second to get the expansion results for any design point (γ, α ).

PARAMETER STUDY ON THE APPLICATION TEST CASE

This paper presents preliminary results where a single stiffness factor 𝛼 identically applied to the 4 junctions. Computation is performed between 𝛼 = 10 and 𝛼 = 10 (no evolution outside) and Figure 7 shows the evolution of mode frequencies with the parameter. When the stiffness increases, modal frequencies rise up and several shape crossings pass through 4250Hz (the squeal occurrence and thus expansion frequency). To get the optimal parameter, the relative model error is shown on Figure 9. The same behavior occurs with the second shape with two minima reached at K/Kmax=8e-7 (local minimum) and K/Kmax=5e-5 (global minimum), whose expanded shapes are shown in Figure 11. One again sees either a global mode or the same local lower pad mode. Future investigations will address the following issues: the optimum parameter differs for the two test shapes so a consistent value must be found combining an objective on both shapes; the 4 parameters should be considered independently. Decoupling normal and tangential stiffness in the contact parameterization is also possible. The crossing shapes in the S curves (Figure 7), probably corresponding to localized deformations shown in Figure 10 and Figure 11, must be analyzed using modal observations [START_REF] Martin | Méthodes de corrélation calcul/essai pour l'analyse du crissement[END_REF]. Finally, it would be interesting to look at the evolution of test/analysis correlation with updating using the MAC matrix.

CONCLUSION

This paper summarized the theory behind Minimum Dynamic Residual Expansion MDRE and showed it to be applicable to an industrial brake squeal problem. The expansion gives access to an estimate of the limit cycle shapes on the full set of DOF including hidden parts and junctions and it further gives model error evaluations that can be used to localize problems in the model based on the squeal shapes only. The evaluation of model error was used to define pertinent junction parameters that were later considered in an updating loop that did not show the usual irregularities that plague MAC based objective criteria.

While the theory behind MDRE has been known for decades, the combination of model reduction techniques proposed here was crucial to allow parametric studies in a reasonable time. The need to work on criteria for interpretation was illustrated by the distinction between model error on free and enrichment shapes, which was key to allow the clear localization of errors on the springs between the anchor bracket and pads. For the updating part, the need to consider more independent parameters and to introduce multi-shape objective criteria was also pointed out.

Other topics for future work are the consideration of relations between parametrization and bias in the updated parameters, understanding expanded shape errors in the presence of incorrect models for a given test wireframe density, and obviously the introduction of strategies that could hint on the impact of squeal mitigation measures.

Figure 1 :

 1 Figure 1 : Test case. Left : disc-brake measured on test bench. Right : Measurement wireframe on top of the FEM.

Figure 2 :

 2 Figure 2 : Visualization of the two main shapes extracted from all the sequential time measurement with the previously developed method

Figure 3 :

 3 Figure 3 : Evolution of relative model and test error with 𝛾. Left: first main shape, right: second.

Figure 4 :

 4 Figure 4 : Expansion result: first (left) and second (right) experimental shape for 𝛾 = 10

Figure 5 :

 5 Figure 5 : Model error on the free mode shapes These model error maps were used to define a relevant parameterization of the model to perform model updating: contact stiffness between the two pads and the calliper at these 4 locations.

Figure 6 :

 6 Figure 6 : Model error on the enrichment shapes PARAMETRIZATION AND MULTI-MODEL REDUCTION

Figure 7 :

 7 Figure 7 : Mode frequency evolution with the parameter evolution Figure 8 presents the parametric stability diagram (or root locus) for the study. This highlights the strong sensitivity of modal damping, thus confirming the interest for the parameters pointed by the initial MDRE analysis on squeal stability.

Figure 8 :

 8 Figure 8 : Damping and frequency evolution of complex modes with the parameter evolution

Figure 9 :

 9 Figure 9 : Relative error of model in function of the stiffness parameter for the first shape (left) and the second shape (right)𝛾 = 10 For the first shape, two minima are reached at K/Kmax=6e-6 (global minimum) and K/Kmax=3e-5 (local minimum). Expanded shapes corresponding to these parameters are shown in Figure 10. The first is global while the second is a local bending of the wear indicator / edge of lower pad.

Figure 10 :

 10 Figure 10 : Expansion of the first shape for the parameters K/Kmax=6e-6 and K/Kmax=3e-5

Figure 11 :

 11 Figure 11 : Expansion of the second shape for the parameters K/Kmax=8e-7 and K/Kmax=5e-5The existence of minima pointing to local unmeasured motion is the indication that local modes of this part may exist. Adding more measurement points would either remove the local minimum of the objective function or confirm the influence of that local mode on the squeal event near 4250 Hz. Vibration of the wear indicated was actually considered in design studies.