est destinée au dépôt

Le bassin de la Saône

Nous avons sélectionné le bassin de la Saône comme terrain d'application. C'est une rivière de 482 km de long qui se conforte progressivement par l'apport de nombreux affluents. Le réseau hydrographique de 9 000 km s'étend sur un bassin versant de près de 30 000 km².

RESULTATS ET CONCLUSION

L'adaptation du cadre conceptuel à la problématique

Pour ce travail, le cadre DPSIR est décliné de façon à s'adapter d'une part à la problématique de l'impact de l'occupation du sol sur la qualité des cours d'eau et d'autre part à l'état des connaissances sur le bassin et à la disponibilité des données. Cela conduit à la définition de choix méthodologiques. Les indicateurs concernant la restauration des cours d'eau sont dans un premier temps déduits de la cartographie d'occupation du sol par des potentialités de restauration.

Les différents indicateurs seront mis en relation dans des modèles spatialisés permettant ainsi de quantifier l'impact multi-échelles des différentes occupations du sol sur la qualité des cours d'eau.

  Evaluation de l'influence de l'occupation du sol sur la qualité des cours d'eau en utilisant le cadre DPSIR : application sur le bassin de la Saône Nathalie Lalande, Flavie Cernesson, Aurélia Decherf, Marie-George Tournoud To cite this version: Nathalie Lalande, Flavie Cernesson, Aurélia Decherf, Marie-George Tournoud. Evaluation de l'influence de l'occupation du sol sur la qualité des cours d'eau en utilisant le cadre DPSIR : application sur le bassin de la Saône. Recherches et actions au service des fleuves et grandes rivières IS.Rivers 2012, Jun 2012, Lyon, France. 3 p. hal-00802347

Evaluation de l'influence de l'occupation du sol sur la qualité des cours d'eau en utilisant le cadre DPSIR : Application sur le bassin de la Saône

  The aim of our paper is to present the adaptation of DPSIR conceptual framework ("driving forces, pressures, state, impact, responses") promoted by the European Environment Agency to structure the information and the way to build indicators. Drivers and pressures indicators are estimated by land use data. State indicators and impacts indicators are respectively estimated by IBGN and EQR-IBGN values. Responses indicators are characterized by a riverside restoration potentiality. The study case is the Saone catchment because of its hydrogeographical characteristics (area of 30 000km ² and 9000km of rivers) and because of the diversity of human activities in the basin.Les cours d'eau sont des systèmes fragiles et complexes dont le fonctionnement peut être altéré par les activités anthropiques sur la ressource en eau elle-même et sur les territoires à proximité. La gestion des cours d'eau doit donc intégrer la complexité du fonctionnement des rivières tout en s'efforçant d'atteindre les objectifs réglementaires ambitieux des cadres législatifs. De nombreux acteurs de la gestion de l'eau interagissent en fonction de leurs objectifs ainsi que des échelles auxquelles ils opèrent, du niveau local au niveau national. A ces enjeux s'ajoute un facteur social dont l'importance s'est accrue au cours de ces dernières années. La littérature identifie clairement l'occupation du sol comme une variable clé de l'estimation des pressions sur la qualité des cours d'eau qu'elles soient positives ou négatives. Ces pressions varient dans le temps et dans l'espace(Allan 2004, Johnson and[START_REF] Johnson | Recent developments in landscape approaches for the study of aquatic ecosystems[END_REF]. Evaluer les impacts multi échelle de l'occupation du sol sur la qualité des cours d'eau est donc une question majeure pour les gestionnaires de la ressource en eau. Pour y répondre, les connaissances nécessaires et les outils développés nécessitent de nombreuses données décrivant l'occupation du sol, la qualité des cours d'eau et les actions de restauration menées sur les cours d'eau. Ces données se caractérisent par une grande hétérogénéité et une grande complexité, tant par leur forme propre (observations, enquêtes, dires d'experts, résultats de cartographie ou de modélisation…) que par leurs structures spatiale et temporelle (série chronologique, information datée comme les cartographies et les enquêtes).

	Evaluation of land use influence on rivers water quality
	using the DPSIR framework: Application on the Saone
	catchment
	Nathalie Lalande 1 , Flavie Cernesson 2 , Aurélia Decherf 1 , Marie
	George Tournoud 3
	1 Cemagref/Irstea -UMR TETIS Maison de la Télédétection 34093 Montpellier
	cedex 05 (contact : nathalie.lalande@teledetection.fr). 2 AgroParisTech -UMR TETIS 3 Université Montpellier 2, UMR 5569 Hydrosciences
	Les drivers et les pressions sont estimés à partir des données d'occupation du sol et les données d'états et d'impacts sont estimées par les valeurs d'IBGN et de l'EQR-IBGN, les réponses sont exprimées en termes de potentialités de restauration. Le bassin de la Saône a été choisi comme bassin d'application, en raison de ses caractéristiques hydrogéographiques (superficie de 30 000km² et 9000km de cours d'eau) et de la diversité des activités anthropiques présentes. ABSTRACT Rivers' functioning can be altered by human activities. Guided by ambitious and rigorous legislative frameworks, rivers management involves many actors, each with specific objectives depending on the scales at which they operate. The literature clearly shows land use as a key variable in the estimation of positive and negative pressures. Assess multi-scale impacts of land use on river water quality is a major issue for water managers and scientist. 2 METHODE ET MATERIEL 2.1 Le cadre conceptuel DPSIR

RÉSUMÉ

Les cours d'eau sont des systèmes fragiles et complexes dont le fonctionnement peut être altéré par les activités anthropiques. Guidée par des cadres législatifs stricts et ambitieux, la gestion des cours d'eau implique de nombreux acteurs ayant chacun des objectifs propres en fonction des échelles auxquelles ils opèrent. La littérature identifie clairement l'occupation du sol comme une variable clé de l'estimation des pressions qu'elles soient positives ou négatives. Evaluer les impacts multi échelle de l'occupation du sol sur la qualité des cours d'eau est une question majeure pour les gestionnaires et les scientifiques. L'objectif de notre communication est de présenter l'adaptation du cadre conceptuel DPSIR (« driving forces, pressures, state, impact, responses ») de l'Agence Européenne de l'environnement pour structurer les informations et la construction des indicateurs.

MOTS CLES

DPSIR; Environmental indicators; Land use; Spatial indicators, Spatial models

1 INTRODUCTION

Pour appréhender la complexité de la problématique, nous avons adopté le cadre conceptuel DPSIR (« driving forces drivers, pressures, state, impacts, responses ») construit par l'Agence Européenne de l'Environnement en 1998, pour le suivi de la mise en oeuvre des politiques publiques et favoriser la communication interdisciplinaire (E.E.A. 1999) comme cadre d'analyse. Le cadre DPSIR décompose les problématiques environnementales en cinq composantes. Les drivers correspondent aux activités humaines socio-économiques (industrie, agriculture, urbanisation, transport…). Ces activités humaines génèrent des pressions sur le milieu aquatique. Ces pressions de nature et d'intensité variables altèrent l'état du cours d'eau. Ces pressions se traduisent par un certain nombre d'impacts sur les différents composants du système étudié : impacts environnementaux, économiques et sociaux. Pour faire face à ces impacts, la société élabore des réponses (programmes d'actions, programmes de mesures, actions de restauration, décrets, lois,…). L'utilisation du cadre DPSIR est basée sur la construction d'indicateurs environnementaux pertinents permettant d'expliciter les relations entre les différents compartiments du système.

  La Saône présente une grande diversité de contextes naturels et de pressions anthropiques. Le climat continental est soumis à des influences méridionales dans la partie sud du bassin. Le sol est essentiellement calcaire excepté dans la partie amont vosgienne. Rivière navigable sur la majeure partie de son cours, elle est modifiée par la présence de nombreux ouvrages de navigation et d'ingénierie lourde. Avec une population de 2,6 millions d'habitants, le bassin est peu urbanisé et les foyers industriels sont localisés à proximité des principales agglomérations: Lyon, Dijon, Besançon,… Le bassin est marqué par une tradition viticole forte en rive droite (terroirs Beaujolais et Bourguignon). L'élevage domine sur le haut du bassin, tandis qu'en rive gauche et sur la basse vallée, on trouve principalement de la céréaliculture et du maraîchage. La gestion de l'eau s'appuie sur une dynamique territoriale forte coordonnée par l'établissement public territorial du bassin (EPTB) Saône et Doubs. 80% du bassin est soumis à des plans de gestion sous forme de contrats de milieux ou de SAGE.

  Les drivers sont synthétisées par l'occupation du sol qui est décrite selon une nomenclature hiérarchisée représentant les grands types d'activités humaines et d'environnements naturels : zones artificialisées, zones agricoles, zones naturelles et semi-naturelles et surfaces en eau. Les pressions diffuses sont estimées pour chaque type d'activité anthropique et de contexte naturel par le biais d'indicateurs d'occupation du sol. L'hypothèse sous-jacente est que l'on peut associer un « forfait filtre » ou un « forfait pollution » à chaque type d'occupation du sol dans une emprise géographique donnée. Les pressions ne seront donc pas déduites au travers de calculs de flux issus d'observations ou de modèles mécanistes. L'état des cours d'eau est défini selon les paramètres biologiques et physicochimiques prescrits par la DCE et mesurés sur les réseaux nationaux de suivi de la qualité des cours d'eau. Nous avons sélectionné l'IBGN (indice biologique global normalisé). Les impacts sont considérés uniquement du point de vue environnemental. Leur estimation se fait par comparaison des valeurs des paramètres d'état d'une situation de référence. Ainsi, l'impact correspondant à l'IBGN est l'EQR-IBGN. Enfin les réponses de la société sont restreintes aux actions de restauration sur les cours d'eau.La construction des indicateurs passent par la compréhension et la quantification des différents phénomènes impliqués ainsi que celles de leurs dynamiques spatiales et temporelles. Les indicateurs d'état et d'impacts sont basés sur les mesures d'IBGN et de l'EQR IBGN. Ces données issues des observations locales et ponctuelles présentent une forte disparité spatiotemporelle. Il y a donc nécessité d'analyser leur représentativité spatiale et d'autre part l'évolution temporelle pour pouvoir construire les indicateurs.Les caractéristiques spatiales des indicateurs de pressions construits à partir des données de drivers varient en fonction des processus impliqués. La bibliographie précise deux niveaux d'influence de l'occupation du sol sur les cours d'eau : le bassin versant qui caractérise les pressions sur l'ensemble du réseau hydrographique et les corridors qui modulent localement ces grandes tendances. Les données d'occupation du sol utilisées auront donc une finesse spatiale et sémantique adaptée à chaque niveau. Nous utilisons ainsi les données Corine Land Cover au niveau du bassin versant. Pour les corridors rivulaires, nous avons donc construit une cartographie à très haute résolution spatiale de l'occupation du sol dans les corridors rivulaires du bassin de la Saône selon une nomenclature adaptée aux objectifs de construction d'indicateurs de pressions à partir de données d'occupation du sol. La littérature recommande pour la construction des indicateurs d'occupation du sol dans les corridors des emprises qui dépendent du type d'activité ou d'environnement. Nous avons ainsi construit des indicateurs de pressions urbaines et industrielles sur des emprises de 100 m à 150 m et de pressions agricoles sur des emprises de 300 m à 400 m. La modélisation des rôles de l'occupation du sol naturelle ou semi naturelle est réalisée sur une emprise de 10 m à 50 m. On construit également des indicateurs de contact pour estimer les effets de "barrière protectrice" ainsi que des indicateurs paysagers traduisant la continuité/fragmentation de ces occupations du sol le long des cours d'eau.

	3.2 La construction des indicateurs


Son Dinh

Nguyen

Frédéric Vignat
email: frederic.vignat@g-scop.inpg.fr

Daniel Brissaud
email: daniel.brissaud@g-scop.inpg.fr

Taking into account geometrical variation effect on product performance

Keywords: life-cycle engineering, manufacturing, product geometrical deviations, product lifecycle engineering, geometrical variation effect, product performance

Due to rapid development of technology and strict competition in the context of global and concurrent economy, the requirements of customers such as quality, reliability, sustainability and cost of products are more and more high and tight. Thus, satisfaction of those is an important key of product designers. However, the product designers work mainly on the nominal model of the product within a CAD/CAM system. These models can only represent nominal information about the product and have no ability to deal with various kinds of deviations during the product life cycle. These deviations can make the designed product not to meet fully the requirements of the customers. Thus, it is necessary to take into account the geometrical deviations to simulate the 'real' performances of the product. In this paper, we propose a method to integrate the geometrical deviations of the product into 'real' performance simulations. As a result, the product designers can generate the performance of the expected population of 'real' products. They can verify that the product they are designing will have 'real' performances satisfying the requirements of customers and users.

Frédéric Vignat obtained his PhD from the Grenoble Institute of Technology in 2005. He is actually an Associate Professor at the University of Grenoble (Grenoble-INP). He is working in the Laboratory G-SCOP of the Grenoble University in the area of integrated design. His research interests include manufacturing simulation, product tolerancing and life cycle engineering. He has published more than 20 papers in these areas in international journals and conferences. He mentored many Masters thesis and two PhD theses.

Daniel Brissaud has been a Professor of Engineering Design and Eco-Design at the University of Grenoble (Grenoble-INP) since 1998. He is the Head of the Cluster of Research 'Management and Organisation of Production Systems and Innovation' for the Rhône-Alpes area. He is a member of the European Manufacturing and Innovation Research Association 'EMIRAcle', and CIRP Fellow. He was the Head of the Doctoral studies in Industrial Organisations and Production Systems. His scientific interests are eco-design, environmental assessment, life cycle engineering, clean technologies, product-service systems design, integrated design and requirement engineering. He has authored more than 60 papers in international journals and books and mentored 16 PhD theses.

Introduction

Owing to the development of information technology, computers are becoming a useful and effective tool supporting most of the engineering activities in product design and manufacturing. Most of the product designers create a numerical model of a product by a number of computer aided tools, such as CAD/CAM, engineering simulation, product data management (PDM) and product life cycle management (PLM). However, the product model created in this environment is a nominal representation that does not take into account variations generated along the product life cycle, such as geometrical deviations generated by manufacturing and assembly processes.

The product must, in fact, pass through many stages of its life cycle before arriving in the hand of the customer and of the users. Each part making up the product is manufactured from raw material by the manufacturing processes (forging, cutting, grinding, etc.). Geometrical deviations on each part are generated and accumulated over the successive set-up of the multistage manufacturing process due to imperfections of material, tooling and machine. The parts with deviations are then assembled at the assembly stage. The surfaces' deviations of each part have not only an influence on the assemblability but are also accumulated on the product in the assembly stage. The product geometry is, therefore, different from the nominal one at the end of the manufacturing and assembly stages.

The current product modelling technology presented above is, unable to take into account these deviations. Most of the simulations to predict the behaviour of the product (kinematics, dynamics, resistance, fluid flow, etc.) are carried out based on the nominal model of the product. Thus, the result of the performance simulation of the designed product can be considered as nominal and consequently different from the real one. The risk is then that the designed product does not meet fully the requirements of the customers and users. In this case, the product-process design has to be considered as unsuitable or at least not to be robust.

There are, therefore, important issues that require to be considered:

• How to model the geometrical deviations of a product generated throughout its life cycle?

• How to manage the causes and consequences of these deviations in the design stage?

• How to simulate the performance with geometrical deviations of the product? Some answers exist today in the academic research for each stage of the product life cycle separately. There is, however, no complete answer for all stages yet. Thus, we propose, in this paper, a method that allows to model the geometrical deviations of the product throughout all stages of its life cycle and integrating these deviations into the performance simulation of the product. The aim is to predict the real performance of the product taking into account the geometrical deviations. As a result, product designers can verify that the product they are designing will have real performances satisfying the requirements of customers and users.

2 Literature review

Manufacturing stage

Many researches for modelling geometrical deviations generated during manufacturing stage have been done. They can be classified into two different approaches. The first one is based on the concept of state space model. The models to describe dimensional variation propagation along multistage machining processes using this approach are proposed by [START_REF] Zhou | State Space modeling of dimensional variation propagation in multistage machining process using differential motion vectors[END_REF] and [START_REF] Huang | Part dimensional error and its propagation modeling in multi-operational machining processes[END_REF]. A state vector x(k) is used to describe workpiece deviations at kth stage. These deviations are accumulated and transformed onto the workpiece during previous stages (1 ... k -1) of the multistage machining process. The set of state vectors x(k) describes the workpiece deviations relative to nominal one resulting of the whole machining processes. This model provides a quantitative relationship between the fixture locator errors and the final workpiece geometrical error and has great potential to be applied to fault diagnosis and process design evaluation for complicated machining processes.

The second approach is based on the concept of small displacement torsor (SDT) that is proposed by [START_REF] Bourdet | The concept of the small displacement tosor in metrology[END_REF]. The method based on the SDT approach to perform 3D manufacturing tolerancing for mechanical parts is, firstly, proposed by [START_REF] Villeneuve | Tolerancing for manufacturing: a three-dimensional model[END_REF]. Thereafter, a model of manufactured parts (MMP) based on the SDT for simulating and storing the manufacturing defects in 3D has been developed by [START_REF] Vignat | Simulation of the manufacturing process, generation of a model of the manufactured parts[END_REF]. It permits to collect the deviations generated during a virtual manufacturing process. The defects generated by a machining process are considered to be the result of two independent phenomena: the positioning and the machining deviations accumulated over the successive set-ups. The positioning deviation is the deviation of the nominal part relative to the nominal machine. The positioning operation of the part on the part-holder is realised by a set of hierarchically organised elementary connections. The manufactured deviations of surface relative to its nominal position in MMP are expressed by the parameters of the SDT. [START_REF] Tichadou | 3D geometrical manufacturing simulation: compared approaches between integrated CAD/CAM systems and small displacement torsor models[END_REF] proposed a graph representation of the manufacturing process. This graph model the successive set-up and for each set-up the positioning surface and their hierarchy and the machined surfaces. This graph makes it possible to highlight the influential paths. They propose then two analysis methods. The first one is based on the concept of SDT. The second one is based on the use of CAD software in which they model a manufacturing process with defects. They then virtually measure the realised part and check its conformity.

Assembly stage

A product is made up of parts assembled by the way of connections. Each part has already passes through the manufacturing stage where geometrical deviations have been generated. Then the product passes through assembly stage of its life cycle. Assembly stage of the product life cycle is an essential stage, and it obviously brings its share of deviations to the product.

In general, models for mechanical assembly can be categorised into two different approaches. The first one is based on the state space model. Some models representing this approach are proposed by [START_REF] Mantripragada | Modeling and controlling variation propagation in mechanical assemblies[END_REF] and [START_REF] Huang | Stream-of-variation modelling -Part I: a generic three-dimensional variation model for rigid-body assembly in single station assembly processes[END_REF] with stream-of-variation model (SOVA) for 3D rigid assemblies dimensional variation propagation analysis in multi-station processes. The deviations accumulated at ith assembly station are described by a vector 1 () .

t n Xi R × ∈
The state space model of a multistage assembly process is represented by equation (1).

() ( 1 ) ( 1 ) () () () () () () () X iA i X i B i U iW i Yi CiXi Vi =- -+ + ⎧ ⎨ =+ ⎩ (1)
where U(i) ∈ R m(i)×1 is the fixture/part deviation contribution from station i; Y(i) ∈ R q(i)×1 is the measurement obtained on station i; W(i) and V(i) are mutually independent noise and A(i), B(i) and C(i) are transformation matrix.

The second approach is based on the SDT concept. [START_REF] Bourdet | Geometrical behavior laws for computer aided tolerancing[END_REF] introduced the geometrical behaviour laws based on the SDT for modelling geometrical deviations in the mechanism. Thereafter, [START_REF] Thiebaut | Contribution à la définition d'un moyen unifié de gestion de la géométrie réaliste basée sur le calcul des lois de comportement des mécanismes[END_REF] proposed a model to analyse part deviation in assembly. The positioning variation of the part relative to its nominal position in the global coordinate system is expressed by equation (2).

(

) ( ) ( )

(/) / / / (/) AA B A DA R E A S T S S E S B DB R =+ ++ (2)
where

D(A/R)
is the variation of part A relative to its nominal position in global coordinate system.

E(A/S A

) is the variation of surface S A of the part A relative to its nominal position.

T(S A /S B ) is the variation of link between the surface S A of the part A and the surface S B of the part B.

E(S A /B) is the variation of surface S B of the part B relative to its nominal position.

D(B/R)

is the variation of part B relative to its initial nominal position in global coordinate system.

A linear system of equations is created from the contribution of each connection between part A and part B. The positioning variation of part A is determined by resolution of the linear system of equations based on Gauss-elimination method.

The main principle of the models proposed by [START_REF] Mantripragada | Modeling and controlling variation propagation in mechanical assemblies[END_REF], [START_REF] Huang | Stream-of-variation modelling -Part I: a generic three-dimensional variation model for rigid-body assembly in single station assembly processes[END_REF] and [START_REF] Thiebaut | Contribution à la définition d'un moyen unifié de gestion de la géométrie réaliste basée sur le calcul des lois de comportement des mécanismes[END_REF] is to model the variation of the part at each stage along the assembly process. However, they do not link the geometrical deviations of the surfaces of the parts to the parameters of the manufacturing process. These parameters are those of the MMP [START_REF] Vignat | Simulation of the manufacturing process, generation of a model of the manufactured parts[END_REF] that can be measured from the manufacturing process as proposed by [START_REF] Tichadou | 3-D manufacturing dispersions: two experimental applications[END_REF]. That is the reason why the authors have already proposed a model using the SDT in order to model geometrical deviations generated during the manufacturing stage and accumulated in the assembly stage.

Design stage

The geometrical deviations generated and accumulated during the product life cycle affect the performance of the product. It is thus necessary to manage their causes and consequences at the design stage in order to reduce their effect on the product performance all along its life cycle. [START_REF] Parkinson | Robust mechanical design using engineering models[END_REF] proposed to use engineering models for developing robust design in order to reduce the variance of the design under variation of sources, such as manufacturing operations, variation in material properties, and operating environment. [START_REF] Yu | Design for robustness based on manufacturing variation patterns[END_REF] addressed the impact of the manufacturing errors on the performance of the product. He defined the manufacturing variation pattern (MVP) to represent the manufacturing characteristics and investigated its effects on the performance of the product. [START_REF] Zhu | Performance distribution analysis and robust design[END_REF] presented the theory that offers the analytical and geometrical description of the performance sensitivity distribution of a product in the variation space. The theory can be applied to find the robust design less sensitive to the dimensional variation due to manufacturing errors or product wear. [START_REF] Liu | Relative entropy based method for probabilistic sensitivity analysis in engineering design[END_REF] proposed a new probabilistic sensitivity analysis (PSA) approach for design under uncertainty based on the concept of relative entropy. This approach allows to provide the valuable information about the impact of the design variables on the performance of the product and whole range or a partial range of the performance distribution. [START_REF] Bruyère | Statistical tolerance analysis of bevel gear by tooth contact analysis and Monte Carlo simulation[END_REF] proposed a statistical approach in order to evaluate the impact of the geometrical variations on the angular rotational velocity between two bevel gears. Monte-Carlo simulation method is used to consider the geometrical behaviour simulation and tooth contact analysis. [START_REF] Yin | Efficient random field uncertainty propagation in design using multiscale analysis[END_REF] proposed to integrate material and manufacturing process uncertainties in the design in order to consider their impacts on the performance of the product. They developed a procedure for uncertainty propagation from the material random field to the end product performance based on the product finite-element mesh.

These studies examine the impact of the variation sources or the geometrical variations on the product performance variation. Most of the relationship approximations between the performance and the design variables are only based on the nominal model of the product. Thus we propose, in this paper, a method that allows to model the geometrical deviations of product during its life cycle and integrating them into the performance simulation. The aim is to manage the geometrical variability throughout the product life cycle and their impacts on the performance.

Integration of geometrical deviations into the performance simulation of a product

The numerical model of product created in CAD/CAM systems is usually used to approximate the practical performance of the product. The approximation can make performance of the designed product less precise in comparison with the customers' requirements because there are many variation sources that affect the product performance in the real environment.

Figure 1 The overview of method (see online version for colours) [START_REF] Kimura | Modeling, evaluation and design of product quality under disturbances throughout the total product life cycle[END_REF] mentioned the existence of disturbance in the total product life cycle, such as material property defects, manufacturing process errors, assembling process inaccuracy and mal-operations in product usage stages, etc. They may influence the product quality and functional failure. He also proposed a reliability design method to predict product behaviour under these disturbances and to verify product functionality throughout the product life cycle. However, this method only relates to functional failure analysis not to geometrical variability analysis. Thus, we propose, in this paper, a method that allows to model the geometrical deviations and to integrate them into the performance simulation. An overview of the method is described by Figure 1. It is separated in two distinct branches. The first one consists in the geometrical deviation model generated in the manufacturing and assembly stage, and the second one includes the simulation of the performance of the product taking into account these deviations.

The geometrical deviations generated and accumulated during the manufacturing and assembly processes are modelled by the MMP and the model of assembled part (MAP). These models are presented in the next chapter. Monte-Carlo simulation method is used to evaluate the geometrical variations of the product. The relationship between the product performance and the parameters of the variation sources is established by using the design of experiment method. The distribution of the population of the product performance is then calculated in order to verify of the performance variation in comparison with the customers requirements.

Geometrical deviation model of a product

The geometrical deviation model of the product during the manufacturing and assembly stage has been presented in detail by [START_REF] Nguyen | Geometrical deviation model of product throughout its life cycle[END_REF]. For manufacturing stage, the deviation of surface j of manufactured part i, manufactured in set-up Sj, relative to its nominal position is considered to be the result of two independent phenomena: the positioning and the machining deviations accumulated over the successive set-ups, as expressed by equation (3). ,, ,

ii i i j j P PS j P S j P TT T =- + (3)
, i j Sj P T is a SDT modelling the deviation of the machined surface j realised in set-up Sj relative to the nominal machine. This torsor merges deviations of the surface swept by the tool and cutting local deformations.

, i Sj P T is a SDT modelling the positioning deviation of workpiece i in set-up Sj.

For assembly stage, the geometrical deviations accumulated on the product are modelled by the MAP based on the gap torsor concept proposed by [START_REF] Bourdet | The concept of the small displacement tosor in metrology[END_REF]. The geometrical deviation of surface j of part i relative to the global coordinate system is expressed by equation (4). T is a SDT modelling the positioning deviations of part i relative to its nominal position in the global coordinate system of the product. The determination of this torsor is based on a set of elementary connection between part i and others, as explained in detail by [START_REF] Nguyen | Geometrical deviation model of product throughout its life cycle[END_REF].

In conclusion, the whole geometrical deviations of all the surfaces of the product relative to their nominal positions in the global coordinate system of product will be collected by a set of SDT , .

i j P P T

Simulation of product deviation generation

In order to integrate geometrical deviations into the simulation of the product performance, it is necessary to have a good image of the real production. There are many methods to study geometrical deviations variation, such as method of moments for estimating its characteristics as mean, variance, median, etc.; bootstrap for approximating the distribution of the population; first/second order reliability method (FORM/SORM) to consider the geometrical reliability of the product [START_REF] Ballu | Geometrical reliability of overconstrained mechanisms with gaps[END_REF]; Monte-Carlo simulation methods to study systems with a large number of coupled degrees of freedom, such as fluids, disordered materials, strongly coupled solids, and cellular structures. In the current case, Monte-Carlo simulation method can thus be considered as the most effective tool to give an overall image based on the relationship between geometrical deviations and parameters of variation sources (parameters representing part-holder quality, machining/tool defects, etc.) established by the geometrical deviation model. It will be used to predict the amount of variations generated and accumulated throughout the product life cycle. This is the second step of the proposed method. It is to simulate geometrical deviations by using a Monte-Carlo method to generate an image of the real production with geometrical deviations. To this aim, the population of products is virtually manufactured by the following steps:

Step 1 Define the probability distribution type or the variation zone of the input variables

The input variables are the parameters of the torsors representing the quality of the manufacturing fixtures and the capabilities of the machine tool. The determination of the distribution of these variables is based on experimental measurements or manufacturing process knowledge [START_REF] Tichadou | 3-D manufacturing dispersions: two experimental applications[END_REF]. The strategies to mathematically bound variation of the input parameters are presented by [START_REF] Nejad | Simulation of the geometrical defects of manufacturing[END_REF]. In the case of a plane, the torsor T Sj,HiSj modelling the deviations of the part-holder surface is expressed by equation ( 5). (5)

The variation zone of each parameter can mathematically be defined by equation ( 6) where max and min value for each parameter comes from measurement or process knowledge.

⎧ ≤≤ ⎪ ⎪ ≤≤ ⎨ ⎪ ≤≤ ⎪ ⎩ (6)
Step 2 Generate randomly a set of input values according to their distributions and variation zone

The set of input values is randomly generated by cellular automata algorithm in relation with the defined distributions and variations zone. The cellular automata have been defined by [START_REF] Wolfram | Statistical mechanics of cellular automata[END_REF]. It is a simple mathematical idealisation of natural systems. This algorithm is available within mathematica software and that is the reason why it has been chosen for this work.

Step

Calculate the values of the determined components (lr, lt) of the link deviation torsor of the MMP

The determination is based on the non-penetration constraints and positioning functions. The deviation torsor , ii j P P T of the surface j of the manufactured part i is calculated by replacing the determined parameters of the deviation link torsor.

In the case of floating contact, the determined components are randomly chosen in an interval satisfying the non-penetration constraints. For a cylinder-cylinder connection [see Figure 2(a)], the components of the link torsor are expressed by equation ( 7). The determined components must comply with the non-penetration conditions expressed in equation ( 8). Thus, for their determination, the parameters are randomly generated within the intervals expressed in equation ( 9) and are accepted if they comply with (8) or rejected in the case of non-compliance.

⎧ -+ - + + ≥ ⎪ ⎪ ⎨ ⎪ -+ + + - ≥ ⎪ ⎩ (8) ( ) ( ) 12 12 12 12 2 2 Sj Sj Sj Sj RR lrx H RR lry H ltx R R lty R R - ⎧ ≤ ⎪ ⎪ ⎪ - ⎪ ≤ ⎨ ⎪ ≤- ⎪ ⎪ ≤- ⎪ ⎩ (9)
where R 1 , R 2 and H are the radius of cylinder and the height of cylindrical link respectively.

In the case of slipping contact, the determined components are calculated by maximising the positioning function subject to non-penetration constraints. For the planplan connection [see Figure 2

⎧ --+ ≥ ⎪ ⎪ ⎪ -++ ≥ ⎪ ⎨ ⎪ -+ ≥ ⎪ ⎪ ⎪ ++ ≥ ⎩
Step 4 Calculate the surface deviations of each manufactured part of the product based on the MMPs This is simply done by addition of the deviation value to the machining value.

Step 5 Assemble the manufactured part of the product based on verification of the CA During this step, the determined components of the deviation link torsor are calculated based on verification of the assembly constraint (CA). The determination is realised as for

Step 3.

Step 6 Calculate the surface deviations of the product in the global coordinate system

The surface deviations of the product in the global coordinate system are determined by replacing all input parameters into equation ( 4). All parameters of the model are collected in the end of this step. Then the procedure will be restarted from Step 1 until the desired number of product is reached.

A set of m virtual products is then produced by Monte-Carlo simulation method. All values of parameters of the model are collected in order to be used for simulation of product performance.

Integration of geometrical deviations into product performance simulation

The next step is to integrate these deviations into the simulation of product performance. However, it is difficult to integrate all of them into the simulation of the m products, m possibly being one million and each simulation lasting sometime hours (three hours for the computational fluid dynamics (CFD) simulation treated in the next example). Thus a strategy based on design of experiment method to reduce calculation time is proposed in this paper. This method uses factorial design to determine the relationship between the performance and the product geometrical deviations parameters.

Factorial design

Factorial design is usually used to understand the effect of two or more independent variables upon a single dependent variable. It is used to study the relationship between the product performance and the geometrical deviations. In order to initiate this method, it is necessary to create an experimental table, called design matrix, which includes factors and their levels. These factors can be the non-geometrical or geometrical deviations parameters. The overview of this method is shown in Figure 3. are defined based on expert knowledge. Then the number of levels for these factors has to be defined depending on a compromise between the desired precision and the calculation time. The value of each level of each factor is calculated according to its range of variation. The k n values of the n parameters are then gathered in a design matrix P, as given in equation ( 12). In the simple case, the function performance is known and the performance value is obtained by the value of each parameter p ij of each line of the design matrix into analytical formula. In the complex case, simulation tools as FEA, CFD, etc., need to be used to find the product performance. Thus a set of k n deviated models of the product corresponding to the geometrical parameters p ij has to be created in the CAD system. Each model is used to simulate the product performance in order to determine the response vector R corresponding to the design matrix P, as given in equation ( 13).

{ } 12 ,, , n T k Rr r r = … (13)
The relationship between the product performance and the selected factors is established by the linear or non-linear regression model, as given in equation ( 14).

( )

123 ,,,, n Performance f p p p p = … (14)
For example, in the case of n factors and two levels, the function f describing the relationship between the performance and the factors is established from the result of 2 n simulations, as given in equation (15).

ˆ.

fp β ε =+ (15)
where p = {p 1 , p 2 , p 3 , …, p n } t is a vector of the n factors β is a coefficient vector of the model, as calculated by equation ( 16)

( ) 1 ˆ.. . tt PP PR β - = (16)
i is a residual vector, as calculated by equation ( 17).

m i i ER r m ε = == ∑ (17)
The performance of the population of products will be generated by replacing the value of the selected factors {p 1 , p 2 , p 3 , …, p n }, collected from the Monte-Carlo simulation results, into equation ( 14). The collection of performance data can then be analysed by using statistic tools.

Case studies

In previous chapters, a method to generate an image of the real performance of the product has been proposed. In order to illustrate this method, an example of centrifugal pump design will be presented in this section.

A centrifugal pump design

The numerical model of the centrifugal pump is created in the CAD/CAM software (see Figure 4). In this case, the performance studied is the pump flowrate.

Geometrical deviation model of the centrifugal pump

In order to manufacture the pump, manufacturing processes, assembly processes and associated resources are selected according to requirements of customers and cost. The geometrical deviation model of this centrifugal pump during manufacturing and assembly stage is generated by the method as presented in the previous chapter. For example, the shaft of the pump is realised by a turning process on a lathe machine (shown in Figure 5). The surface deviations of the planar head of the shaft relative to its nominal position are described by the torsor (18). (18) In this torsor, the parameters rx 4,1 , rx 4,8 , ry 4,1 , ry 4,8 , tz 4,1 , tz 4,9 represent the machining inaccuracy and the parameters rx 4,8S1 , ry 4,8S1 , tz 4,9S1 represent the deviations of the fixture surfaces in Set-up 1. The parameters lrx 4,8S1 , lry 4,8S1 , ltz 4,8S1 represent the link between the fixture surfaces and the surfaces of workpiece in Set-up 1.

The manufactured parts of the pump are then assembled according to the selected assembly process (shown in Figure 6). The geometrical deviations of all surfaces of the pump are described by torsors in the MAP. In conclusion, the geometrical deviations of all surfaces of the pump relative to its nominal position in the global coordination system can be modelled according to the selected the manufacturing and assembly processes and the associated resources.

Monte-Carlo method for geometrical deviation simulation

The Monte-Carlo simulation method is applied to calculate the values of the geometrical deviations of the pump. The input variables are the parameters of the torsors that represent the machining inaccuracy and the deviation of the fixture surface, such as rx i,j , ry i,j , tx i,j , ty i,j , tz i,j for the manufacturing operation of the surface j of the part i and rx i,jSk , ry i,jSk , tx i,jSk , ty i,jSk , tz i,jSk for the surface j of the fixture in the set-up Sk. As explained in the previous chapter, the input variables are considered as independent. The probability distributions and the variation range need to be determined. A uniform distribution is chosen in this case. The variation ranges are determined according to the associated resources. For example, the variation ranges of the input parameters for the manufacturing of the shaft are shown in Table 1. One million of pump is virtually manufactured and assembled with deviations by using Monte-Carlo simulation method, as presented above chapter. Then the product designer can use the result to verify any geometrical requirements and to check the assemblability of the pump. For example, the designer can estimate the distribution of the gap between the impeller and the casing of the pump which has to remain positive to avoid contact between impeller and casing. The distribution of the gap is shown in Figure 7(a)

(mean μ = 5.7571 mm, standard deviation σ = 0.01976 mm). The variation range of the gap according to 6σ standard is from 5.6978 mm to 5.8164 mm. The designer can also know the distribution of positioning deviation of each part of the pump. For example, the distribution of the gap between the plan of the impeller and the plan of the volute back casing is shown in Figure 7(b). The distribution of the impeller's centre (green colour) and the distribution of the casing's centre (cyan colour) according to two perpendicular axes X and Y are shown in Figure 7(b) and 7(c) respectively.

Design of experiments

The relationship between the pump flowrate and the geometrical deviations parameters is determined by the factorial design presented above chapter. Many parameters of the geometrical deviations affect the pump flowrate. It is impossible to integrate all of them into the flowrate simulation. Thus the expert knowledge is used to filter the geometrical deviations parameters that have a strong influence on the pump flowrate. In this example, the gap between the top surface of the impeller and the casing of the pump is selected based on the experiment of [START_REF] Engin | Performance characteristics of a centrifugal pump impeller with running tip clearance pumping liquid-solid mixtures[END_REF]. The translation of the impeller relative to two perpendicular axes X, Y in the global coordination system is also chosen based on the study of [START_REF] Baun | Effect of relative impeller-to-volute position on hydraulic efficiency and static radial force distribution in a circular volute centrifugal pump[END_REF]. Then the value of these three parameters level is calculated from the one million of virtually manufactured pump according to the deviation of the surfaces of the casing and the impeller relative to the global coordinate system of the pump. The value of these deviations is coming from the previous step of the method.

The levels of each factor are then chosen from the data collection of these three factors according to the 6σ standard, i.e.:

• high level at +3 σ (1)

• medium level at μ (0) (average value of the parameter)

• low level at -3 σ (-1).

From the selected factors and levels (see Table 2), 27 deviated models of the pump are created in the CAD software. These are 27 deviated models are used to simulate the pump performance, such as the velocity of fluid, mass flowrate, etc., by using the CFD software. The results concerning the mass flowrate are shown in Table 3. The relationship between the pump flowrate and the selected parameters is determined by the factorial design method. The mass flowrate Q is a function of the gap between the impeller and the casing of the pump (Gap) and the translation of the impeller relative to two perpendicular axes (Tx, Ty), as given in equation ( 19). The product designer then calculates the flowrate of the 1 million pumps based on the values of the Gap, Ty and Ty found in the above session and equation ( 19). The distribution for the one million of pumps is shown in Figure 8(d). The mean and standard deviation of the mass flowrate of the pump are equal to 157.127 kg/s and 1.572 kg/s respectively. As a result, the product designer can verify the real mass flowrate of the pump and the satisfaction of customers requirements. Furthermore, he can ensure that the design of the pump plans to deliver to customers is robust.

Conclusions

Today, the product designers work on the numerical model of the product within a CAD system. This model can only represent nominal product information and most of the simulations to predict the behaviour and performance is carried out on this model. Therefore, it limits the ability to deal with geometrical variations generated during the product life cycle. These variations can make the designed product not to meet fully the requirements of the customers and the users. This paper proposes a method that allows to model the geometrical deviations of the surfaces of the product generated throughout its life cycle and to integrate them into the simulation of the 'real' performance of the product. This method also allows to analyse robustness based on the performance variation. It can thus overcome the weak points of the current product modelling technology by taking into account the geometrical deviations of the product in the performance simulation.

In future work, we will develop this method to deal with the identification of the root cause of performance variation and thus help to make the designed product reliable and robust.

Figure 2

 2 Figure 2 Floating cylindrical contact, (a) floating cylindrical contact (b) slipping planar contact (see online version for colours)

Figure 3

 3 Figure 3 Integration of geometrical deviations into product performance simulation(see online version for colours)

Figure 4

 4 Figure 4 The parts of designed centrifugal pump (see online version for colours)

Figure 5

 5 Figure 5 Manufacturing process for the shaft of pump (see online version for colours)

Figure 6

 6 Figure 6 Assembly graph of the pump (see online version for colours)

Figure 7 Figure 7

 77 Figure 7 Monte-Carlo simulation results (see online version for colours)

Figure 8

 8 Figure 8 shows the associated response surfaces representing the dependence between the mass flowrate of the pump and the factors Gap, Tx [Figure 8(a)], the factors Gap, Ty [Figure 8(b)] and the factors Tx, Ty [Figure 8(c)].

Figure 8

 8 Figure 8 Response surfaces and distribution of the mass flowrate (see online version for colours)

  (b)], the components of the link deviation torsor are expressed by equation (10).The determined components (lr, lt) are calculated by the procedure (11).

	, HiSj P m i T	{ OXYZ ,,, ⎧⎫ Sj Sj Sj Sj Sj Sj lrx Ultx lry Ulty Ulrz ltz ⎪⎪ ⎪⎪ = ⎨⎬ ⎪⎪ ⎩⎭ ⎪⎪	}	(10)
	Maximise : -	ltz	Sj
	Subject to :
	22 Sj ba lrx	lry	Sj	ltz	Sj	0
	22 Sj ba lrx	lry	Sj	ltz	Sj	0( 1 1 )
	22 Sj ba lrx	lry	Sj	ltz	Sj	0
	22 Sj ba lrx	lry	Sj	ltz	Sj	0

Table 1

 1 Variation range of the input variables

		Set-up 1
		Fixture
	Plane	rx 4,9S1 , ry 4,9S1 variation range 0.0001
	4,9S1	tz 4,9S1 variation range 0.02
	Cylinder 4,8S1	rx 4,8S1 , ry 4,8S1 variation range 0.0002 Tx 4,9S1 , ty 4,8S1 variation range 0.01
		radius ra 4,8S1 variation range 0.01
		Machining
	Plan	rx 4,5 , ry 4,5 variation range 0.0005
	4,5	tz 4,5 variation range 0.01
	Cylinder 4,4	Rx 4,4 , ry 4,4 variation range 0.00025 tx 44, , ty 4,4 variation range 0.01
		radius ra 4,4 variation range 0.025
	Plan	Rx 4,7 , ry 4,7 variation range 0.00015
	4,7	tz 4,7 variation range 0.02

Table 2

 2 The value of the selected parameters

				Gap (mm)		Tx (mm)			Ty (mm)
	High level (1)		5.8164			0.08497			0.09186
	Medium level (0)		5.7571			0.00295			-0.0027
	Low level (-1)		5.6978			-0.07907		-0.09726
	Table 3	The results of the flowrate simulation
	No.	Gap	Tx	Ty	Q (g/s)	No.	Gap	Tx	Ty	Q (g/s)
	1	-1	1	1	157975	15	0	0	-1	152688
	2	-1	1	0	157942	16	0	-1	1	158973
	3	-1	1	-1	154046	17	0	-1	0	159758
	4	-1	0	1	159899	18	0	-1	-1	154705
	5	-1	0	0	159219	19	1	1	1	153587
	6	-1	0	-1	161851	20	1	1	0	158190
	7	-1	-1	1	163015	21	1	1	-1	156502
	8	-1	-1	0	158858	22	1	0	1	164443
	9	-1	-1	-1	155472	23	1	0	0	158696
	10	0	1	1	155809	24	1	0	-1	154314
	11	0	1	0	158052	25	1	-1	1	145164
	12	0	1	-1	140433	26	1	-1	0	150938
	13	0	0	1	157969	27	1	-1	-1	143754
	14	0	0	0	156666