Gabriela Medina-Oliva
email: gabriela.medina-oliva@univ-lorraine.fr

Alexandre Voisin
email: alexandre.voisin@univ-lorraine.fr

Maxime Monnin
email: maxime.monnin@predict.frflavien.peysson@predict.fr

Flavien Peysson

Jean-Baptiste Leger
email: jean-baptiste.leger@predict.fr

Prognostics Assessment Using Fleet-wide Ontology

Large complex systems, such as power plants, ships and aircraft, are composed of multiple systems, subsystems and components. When they are considered as embedded in operating systems such as a fleet, mission readiness and maintenance management issues are raised. PHM (Prognostics and Health Management) plays a key role in controlling the performance level of such systems, at least on the basis of adapted PHM strategies and system developments. Moreover considering a fleet implies to provide managers and engineers a relevant synthesis of information and to keep this information updated in terms of the global health of the fleet as well as the current status of their maintenance efforts. In order to achieve PHM at a fleet level, it is thus necessary to manage relevant knowledge arising from both modeling and monitoring of the fleet. In that way, this paper presents a knowledge structuring scheme based on ontologies for fleet PHM management applied to marine domain, with emphasis on prognostics modeling.

INTRODUCTION

Nowadays, due to the high competitiveness, industrial enterprises need to aim at higher performances, i.e. higher quality of products/services, lower costs, sustainability, etc. [START_REF] Kleindorfer | Sustainable operations management[END_REF]. In that way, the importance of maintenance has increased due to its key role on improving system availability, performance efficiency, products quality, etc. [START_REF] Alsyouf | The role of maintenance in improving companies' productivity and profitability[END_REF]. These requirements promote the evolution of maintenance strategies from a "fail and fix" to "predict and prevent" approach. This new vision is supported by condition-based/Prognostics and Health Management (PHM) maintenance. Despite this proactive approach, failures still occur. This could be explained since prognostics involved the prediction of the future which is uncertain [START_REF] Provan | Prognosis and Condition-Based Monitoring: an open systems architecture[END_REF]. Furthermore the whole acquisition and treatment algorithms could fail leading sometimes to some errors such as false alarms or nondetections [START_REF] Barros | A maintenance policy for two-unit parallel systems based on imperfect monitoring information[END_REF].

Implementing a proactive approach at a system level requires the consideration of failure rates of different equipment built on different technologies (mechanical, electrical, electronic or software natures) [START_REF] Verma | A systemic approach to integrated E-maintenance of large engineering plants[END_REF] whose behaviour can vary all along the different phases of their lifecycle [START_REF] Bonissone | Predicting the Best Unit within a Fleet: Prognostics Capabilities Enabled by Peer Learning, Fuzzy Similarity, and Evolutionary Design Process[END_REF]. However to improve PHM processes for large and complex systems such as power plants, ships and aircrafts, one possible approach is to take advantage of the fleet dimension. This dimension can provide knowledge and data to improve diagnostic and prognostics models.

A fleet shall be viewed as a set of systems, sub-systems and equipment. In this paper, the naval domain is addressed. Hence, in the following an unit of a fleet will be considered as a system (e.g. ship), a sub-system (e.g. propulsion or electric power generation) or equipment (e.g. diesel engine, shaft...) depending on the nature of the study. To be in accordance with the need of improving PHM at the fleet level, an original methodology is proposed in this paper wherein individual knowledge (of each unit) is capitalized for reuse purpose in order to improve PHM activities such as prognostics. To take advantage of the individual knowledge at the fleet level, a semantic model is proposed for the PHM activities in the naval domain. Such a semantic model enables to reuse particular data, such as maintenance history, reliability analysis, failure analysis, data analysis at a fleet level in order to provide knowledge. As data become available, prognostics models could benefit from more contextual information.

PHM AT EQUIPMENT AND SYSTEM LEVEL

PHM activities can give warning about failure events before they occur, reduce the life cycle cost of a product by decreasing inspection costs, downtime and inventory (Pecht, 2008), (Vichare and Pecht, 2006).

The prognostics process consists on treating, via algorithms, a set of input information to produce a future estimation. Mathematical models are used for the extrapolation of value of the degradation indicator. These mathematical models need as an input [START_REF] Voisin | Generic prognosis model for proactive maintenance decision support: Application to pre-industrial e-Maintenance test bed[END_REF]:

• Past data: feedback about past failures on the system, as well as historic data about the evolution of degradation indicators under different circumstances (mission, environment, etc.). This data allows to identify the characteristics and parameters of the prognostics model.

• Current data: on-line data in order to provide the values in real-time of the monitored indicators (variables). This data is captured and must be treated and analyzed. This data warn maintenance engineers about the current state of the unit and it should be used to feed/adjust the current prognostics model.

• Future data: information and/or hypothesis about the future usage of the unit should be provided such as the missions, the operational context, future maintenance interventions, etc. As mentioned in [START_REF] Peysson | A generic prognostics methodology using damage trajectory models[END_REF] the prognostics of a complex systems (S) is described by three levels (1):

Ρ Ε Μ = , , S (1)
M is the mission that defines the use of the system during a time period; E is the environment that represents the area where the mission is accomplished and where the process evolves and P is the process that gives the necessary means to accomplish the mission.

The process is decomposed according to different resources that are monitored. This set of information/data allows to refine the evolution of degradation indicators in different situations and simulations. While analyzing these inputs, different sources of uncertainty will appear such as measurement and sensor errors, future load and usage uncertainty, prediction under conditions that are different from training data and so on. However uncertainty could be reduced when more data becomes available (Pecht, 2008). In these cases the notion of fleet becomes very interesting. It can provide more capitalized data and information coming from other members of the fleet. for the improvement/development of the prognostics models

The following section presents a review about the use of the fleet notion in the PHM domain.

PHM VS. FLEET-WIDE APPROACH

Fleet integrated PHM review

A fleet generally refers to a gathering of group of ships and the term is extended also to any kind of vehicle (e.g. trains, aircrafts, or cars). For industrial systems, the term fleet designs a set of assets or production lines. In general, a fleet refers to the whole assets of an owner's systems. Hence, the fleet here is only an abstract point of view to consider a set of objects for a specific purpose (e.g. a unit maintenance planning), for a given time (e.g. before the end of the current mission). Indeed, the fleet can be viewed as a population consisting of a finite set of objects (individuals) on which a study is ongoing. In this context, a fleet is generally a subset of the real fleet under consideration, i.e. a sub fleet related to the aim of the study. Individuals making up the fleet/sub fleet may be, as needed, systems themselves [START_REF] Bonissone | Predicting the Best Unit within a Fleet: Prognostics Capabilities Enabled by Peer Learning, Fuzzy Similarity, and Evolutionary Design Process[END_REF], [START_REF] Patrick | Integrated Software Platform for Fleet Data Analysis, Enhanced Diagnostics, and Safe Transition to Prognostics for Helicopter Component CBM[END_REF], subsystems or equipment [START_REF] Umiliacchi | Predictive Maintenance of railway subsystems using an Ontology based modelling approach[END_REF]. In the following, systems, sub-systems or equipment constituting the fleet, according to the study purpose, will be referred to as units.

In fact, fleet's units must share some characteristics that enable to group them together according to a specific purpose. These common characteristics may be of technical, operational or contextual nature (Monnin et al. 2011a). They allow to put data or information related to all the fleet units on the same benchmark in order to bring out pertinent results for monitoring, diagnostics, prognostics or maintenance decision making. Common characteristics among units allow to define three types of fleet composition: identical, similar or heterogeneous fleets.

Based on these three types of fleet, some relevant works are reviewed below:

• Fleet composed of identical units: When considering maintenance operator's point of view, fleet management aims at making decisions that affect asset life extension and performance, operational costs and future planning [START_REF] Wheeler | A survey of health management user objectives related to diagnostic and prognostics metrics[END_REF], [START_REF] Bonissone | Predicting the Best Unit within a Fleet: Prognostics Capabilities Enabled by Peer Learning, Fuzzy Similarity, and Evolutionary Design Process[END_REF], [START_REF] Williams | Fleet analysis and planning using CBM+ open architecture[END_REF]. In [START_REF] Patrick | Integrated Software Platform for Fleet Data Analysis, Enhanced Diagnostics, and Safe Transition to Prognostics for Helicopter Component CBM[END_REF], the authors notice that thresholds indicative of condition indicators limits could be derived from statistical studies of fleet wide behaviors and known cases of faults. [START_REF] Reymonet | Ontology Based Information Retrieval: an application to automotive diagnosis[END_REF] propose to apply to the failed system the technical solution corresponding to a similar incident already solved with a comparable asset. [START_REF] Wang | A similarity-based prognostics approach for Remaining Useful Life estimation of engineered systems[END_REF] present a similarity-based approach for estimating the Remaining Useful Life (RUL) in prognostics using data from a fleet composed by the same type of units. Nevertheless, knowledge derived from the fleet arises from the same kind of units. In a domain where customized units are common, these approaches may give poor results.

• Fleet composed of similar units: the fact of comparing similar units has rarely been addressed as a whole in the literature. In that sense, [START_REF] Umiliacchi | Predictive Maintenance of railway subsystems using an Ontology based modelling approach[END_REF] show the importance of having a standard format for the diagnostic data in order to facilitate their understanding across several subsystems and trains within a railway fleet.

• Fleet composed of heterogeneous units: to fully exploit the knowledge issue of the fleet dimension, we propose in this paper to consider the heterogeneous units that compose the fleet level for decision making (e.g. prognostics, maintenance purposes). Situations (i.e. prognostics results, signal evolution) issued from an historical data of a fleet of heterogeneous units are searched based on some similar characteristics to the units in study (e.g. current situation under investigation) (Monnin et al., 2011a), [START_REF] Monnin | Fleet historical cases for predictive maintenance[END_REF]. The originality of the proposed method is to enlarge the search to heterogeneous units (and not identical ones) where the similarity will be defined online by the user, according to the results of the search, in order to find relevant information to be reused.

But how could be used the fleet dimension to improve PHM processes?

One of the industrial realities is the lack of capitalization of knowledge and model reuse which represents high costs and efforts for the enterprises (Weber et al., 2011) (Medina-Oliva et al., 2012). In some fields such as the naval one, units are very customized leading to heterogeneous units. These facts limits mainly:

• Historical data exploitation for identical units. Due to the exposition of industrials systems to different and uncertain missions and environmental conditions.

• Knowledge capitalization about the evolution of the degradation trajectory of an unit under an identical context (identical mission and environment). To tackle this issue the fleet dimension could provide enough information and data about diagnostic and prognostic models. In that sense, when searching nonidentical units but similar ones a higher volume of data becomes available to reduce diagnostic/prognostics models uncertainty. This data could be obtained through the identification of "similar contexts" or "similar individuals". For example, in the naval field, a technical similarity for diesel engines, which are critical equipment for propulsion and power generation, could be the membership to "4strokes engines" or "high speed engines" features.

The objective of our proposition is to create an iterative investigation process that will allow to define a sub-fleet. The sub-fleet is defined by grouping a set of units (i.e. systems, sub-systems or equipment) based on "similar characteristics". Figure 1 presents the main steps of this process. When prognostics models are implemented on a new unit or when the unit is merged in new operational conditions, the prognostics must be first characterized. Characterization consists on the description of the unit to be prognosticated (e.g. type, age, usage, operating environment) as well as data on which an analysis will be carried out (Figure 1-A). To guide this process, the fleetwide application proposes different criteria based on the technical features of units as well as on the mission and on the environment description. According to this, a targeted population is defined within the whole fleet (Figure 1-B). Within the targeted population, the potential similar prognostics models and data concerning the evolution of the degradation indicators are investigated in order to complete current knowledge for prognostics (Figure 1-C). In an iterative process of steps B and C, the targeted population can be refined if results (i.e. potential similar degradation indicator data) are too far from the current situation or conversely if targeted fleet points out too many similar situations. Then, the prognosticated unit benefits from the past analysis results (Figure 1-D) for building the corresponding prognostics model. Using this original approach could bring several benefits as presented in (Monnin et al., 2011a), [START_REF] Monnin | Fleet historical cases for predictive maintenance[END_REF], [START_REF] Peysson | New approaches for ships fleet-wide management and naval mission prognostics[END_REF].

Toward this goal, the knowledge corresponding to the fleet domain must be well formalized and structured in order to facilitate the manipulation of the multidimensional aspects of the fleet and the heterogeneous data among all fleet units' databases

Prognostics characterization

Available information

Target population definition

Sub fleet data/information analysis

Sub-fleet characterization

Nowadays, fleet managers/engineers query separately ship databases for identical units to obtain data/information. However, units (e.g. diesel engine) in the naval domain are very specific and customized. This fact leads to dispose of few identical units. For this reason the proposed methodology leads to search non-identical units but similar ones. Furthermore, another issue emerges. At a fleet level, engineers must treat different databases of different units.

These databases might be heterogeneous, in the sense that the database structure might be different, they might have different names for the primary keys and foreign keys, etc.

To tackle these issues, a common semantic becomes necessary (Figure 2). A semantic model provides a high level definition of terms that is common to all databases allowing to query them. It allows to define characteristics of similarities among units and contexts. For instance, to define common characteristics in the technical, operational and contextual domains.

In the following sections a semantic model specifying the similar characteristics necessary to obtain data/information to perform prognostics is presented. The semantic model is specified by means of ontologies. An ontology explicits formal specifications of knowledge in a domain by defining the terms (vocabulary) and relations among them [START_REF] Gruber | Ontology. In: the Encyclopedia of Database Systems[END_REF]. To represent knowledge and to explicit semantic (vocabulary and relations), the ontology is coded in Web Ontology Language (OWL) supported by Protégé * ontology editor. OWL is composed of classes, properties of the classes and instances. These elements are explained as follow:

• Classes describe concepts in the domain. In PHM domain, an example of classes could be "equipments" or "degradation indicator".

• Properties of the classes describe the attributes of the concepts. For example, the class equipment has a property "is monitored by" the class degradation indicator. The property "is monitored by" link the class "equipment" with the class "degradation indicator".

• Subclasses represent concepts that are more specific than the superclass (mother class). When a superclass has a subclass, it means that they are linked by a * http://protege.stanford.edu/ subsumption relation, i.e. "is a" relation, allowing a taxonomy to be defined. Hence, a hierarchy of classes is established, from general classes to specific ones.

• Instances are the set of specific individuals of classes. For example, the engine Baudouin 12M26.2P2-002 is a specific individual that is part of the class "equipment". OWL allows to establish taxonomies. This capability is a useful for example to represent systems, subsystems and equipments. Moreover, OWL provides inference capabilities with plugged reasoners. Inference is based on open-world reasoning. Explicit and manually constructed classes that belong to taxonomy constitute an asserted hierarchy. But thanks to OWL reasoners, an inferred hierarchy is automatically computed allowing to emerge new knowledge. For example if an engine has an internal electrical degradation, the ontology could induce that it is an electrical engine. Moreover, OWL reasoners shall performed consistency checking. Hence, one shall guarantee that the ontology has been built correctly in the sense that no syntactic and inconsistency remain in the ontology. For example, if a fuel-engine is associated to an electrical degradation, an inconsistency will be point out by the reasoner.

Ontologies seem to be a suitable modeling method to provide common semantic and to query heterogeneous databases. Some of the capabilities provided consist on: sharing common understanding of the structure of information among people or software agents, making domain assumptions explicit, defining concepts and knowledge (i.e. a high speed engine 1000rpm, subjectivity is limited) and making domain inferences to obtain nonexplicit knowledge [START_REF] Noy | Ontology development 101: A guide to creating your first ontology[END_REF].

Figure 3: Typical architecture of Fleet-wide PHM system, (Monnin, 2011a) The proposed approach based on PHM ontology is not a goal itself. This ontology is a support for PHM software applications through the KASEM software platform (Leger, 2004), (Monnin, 2011c) Figure 3.

In the next section, the formalization of an ontology to support prognostics activities in the naval domain is proposed.

Ontology-based PHM assessment

Ontology-based fleet-wide PHM

In the section 2.1. key elements to perform prognostics were identified. These elements are technical characteristics of the system/sub-system/equipment, degradation modes, degradation indicators, the mission and the environment.

A semantic model to these elements will be provided in order to obtain formalized knowledge that allows the definition of "similarities" among these units. Based on (Monnin, 2011a) different contexts are defined: a technical context (e.g. characteristics of the system/subsystem/equipment), a dysfunctional context (e.g. degradation modes), an operational context (e.g. mission and environment), a service context (e.g. usage of units) and an application context (e.g. degradation indicators). For a graphical representation of the ontology for these contexts, classes are represented as ovals and relations are represented as links between the classes. Once the ontology model is formalized comparisons of heterogeneous units shall be performed on the basis of context similarity.

Technical context

One might think that the definition of every model of equipment could be enough to take advantage of the fleet dimension. But this modeling choice narrows the manageability to strictly identical units. For this reason, a technical context is proposed. The technical context integrates the technical features and characteristics of the system/sub-system/equipment. This model allows the comparison of heterogeneous units for instance when seeking relevant characteristics such as "4-strokes engines" or "direct injection engines". To model the technical context, "Equipment" (Figure 4) classes are specified as well as "Properties" classes (Figure 5) which define all their features. Hence, units with similar technical properties could be clustered according to their technical properties such as the power output, the rotation speed, the number of cylinders, etc. (Figure 5) in order to retrieve data/information.

DieselEngine GasEngine

Dysfunctional context

The dysfunctional context takes into account the information about the degradation modes on the units. It considers the generic degradation modes. Generic degradation modes are taken from the standard (IEC 60812, 2006). Classes include electrical degradation modes, mechanical degradation modes, hydraulic degradation modes, etc. Degradation modes are linked to units (Figure 6).

Furthermore, in this context, it is considered that one degradation mode could be caused by another degradation mode. In that sense, this context allows to describe information about the causality chain of degradation modes that produced an undesirable event. This knowledge is very valuable to retrieve information/data for troubleshooting and corrective maintenance issues. This modeling choice allows to explore the main causes of similar degradation modes. For example to explore common causes of pumps failures regardless of the use of the pumps.

Operational context

Even if units are identical, the operational conditions lead to different units' behaviors. For this reason, one could need to cluster information/data according the operational and service contexts. The operational context integrates the operational conditions to which the units are exposed to. As explained in section 2, for prognostics the mission (M) and the environment (E) are considered. The operational conditions are given by the mission to be performed for units as well as the environment that surround them (Figure 7). In the naval domain, the mission is a sequence of dated tasks in a geographical area (e.g. Port of call mission) [START_REF] Peysson | A generic prognostics methodology using damage trajectory models[END_REF]. Hence, similar missions on similar units could be compared (Figure 7). On the other side, the environment takes into account the weather conditions, the chemical composition of water (pH, salinity…), the pressure, water turbulence, etc. (Figure 7) which might impact degradation mechanisms and units' functioning behavior.

The mission and the environment could affect equipment, sub-system and system performances. For this reason the mission was formalized at different abstraction levels (system, sub-system, equipment).

Service context

The service context deals with the usage of units. Even when units are similar they are exposed to different usages according to the corresponding mission tasks. Hence a service context is formalized in order to differentiate behaviors of the evolution of degradation indicators. In that sense, usage could be divided according to the operating steps, operating phases and the configuration of units (Figure 8).

In order words, degradation behavior can be analyzed according to different abstraction levels.

Application context

The application context is related to the needs of PHM optimization. Within the application context, the optimization aims the capitalization of knowledge to perform health assessment. Health assessment deals with the definition of indicators such as functional, dysfunctional and environmental indicators at different levels as well as the treatment (processing) of these indicators, etc (Figure 9). This context enables data/model retrieval of the monitored unit with its corresponding context defined in the ontology.

The ontology-based knowledge formalization provides the basis to capitalize knowledge with contextual information.

In that sense, one might define similar units, i.e. high speed engines with similar missions and under similar context in order to compare signal evolution of indicators. The next section illustrates the benefits of the proposed methodology within the PHM context.

Prognostic-model retrieval from fleets composed of heterogeneous units

The proposed approach could be very useful for PHM processes for new units. In the case of new equipment there's neither degradation indicators defined nor historical data about their evolution. But when disposing of an ontology-based approach, knowledge and data could be gathered from heterogeneous units and contexts. To illustrate the proposed methodology, let's consider finding a prognostics model for the degradation trajectory of a diesel engine within a costal surveillance mission in the south Atlantic for a new ship. In this example, we suppose the new ship is propelled by a diesel engine Baudouin 8M26SRP.

For the purpose of this example, the fleet is limited to diesel engines. Ten engines are considered and briefly presented in Table 1. The table shows an extract of the technical features of the engines.

To tackle the fleet dimension three possibilities are shown:

1. Consideration of identical fuel engines to the Baudouin 8M26SRP engine 2. Consideration of all fuel engines composing the fleet 3. Consideration of the heterogeneous units composing the fleet using the ontology-based approach 1. Consideration of identical fuel engines to Baudouin 8M26SRP The first step to capitalize data from the fleet dimension would be to consider identical units to Baudouin 8M26SRP. Nevertheless, if the same kind of units is considered, then any unit would match to the results as shown in Table 1. For these reason, other approaches should be investigated. 2. Consideration of all fuel engines composing the fleet Another approach to take advantage of the fleet dimension would be to consider all the fuel engines composing the fleet. This approach allows the capitalization of data. However, the fleet is composed of a wide variety of different engines. If one take a look to Table 1, it is possible to notice the fleet is composed of big size, high power, two-stroke engines such as the. Wärtsilä RT-FLEX82T engine and of other engines with lower output power and four-stroke cycles such as the Wärtsilä 12V38 engine. These units are very different and thus their degradation behavior is very different as well. For this reason considering all the fuel engines might not be appropriated to study the degradation behavior of the Baudouin 8M26SRP engine. 3. Consideration of the heterogeneous units composing the fleet using the ontology-based approach As mentioned previously, the developed application based on ontologies guides the definition of "similar characteristics" in order to define the sub-fleet of units to be used for prognostics purposes (Figure 1-A). To guide the sub-fleet definition some questions will be asked, automatically by the application, based on the relations between the different contexts defined in the ontology. One question concerns the application context, if one seeks a degradation indicator, a treatment or mainly a prognostics model. For this example, a prognostics model is searched. Then another question deals with the technical context. The application poses questions about the type of unit (i.e. engine, thruster, pump…), the application domain (i.e. marine, land, aeronautics, etc.) and the subsystem (i.e. propulsion and power generation). For this example, an engine located on a ship (marine) for propulsion purposes is sought.

In the ontology the definition and relations of prognostics have been established. For this reason, another question is asked about the mission and the environment of the engine (operational context). The mission in this example is costal surveillance mission (Figure 1-B). One can start searching data found in the sub-fleet definition (Figure 1-C).

Afterward, the application proposes to keep finding other similarities criteria for example those related to the geographical area (Figure 1-A). It proposes several choices such as the south Atlantic, north Atlantic, Indian, south Pacific oceans, etc. For this example the environment is located in the south Atlantic. The subfleet evolves automatically when other criteria are chosen.

The ontology embeds that hot oceans impact the performances of the cooling systems. For this reason, another question is asked about the type of cooling system such as high and low temperature separated circuits, sea water cooling, dry water cooling system or desalination systems (technical context). The answer, in this case, for the Baudouin 8M26SRP engine is a high and low temperature separated circuits. The sub-fleet is shown automatically. Three similar equipments with similar contexts are found: Baudouin 12M26P1FR, Wärtsilä 12V64 engine and ISOTTA vl1716hpcr (Figure 1-D).

Figure 10 shows the updated degradation prognostics models of these three engines. The time horizon represents the mission that the engines have already experienced. Capitalizing this information allows to build a prognostics model for the Baudouin 8M26SRP engine (Figure 13-orange line). To build the estimated degradation trajectory, histograms for each monitored time issues from retrieved trajectories are built. Then the estimated trajectory is computed as the mean on each of the histograms. The orange line shows the estimate of the degradation trajectory of the Baudouin 8M26SRP engine for the given mission. However, more sophisticated methods could be used to estimate the degradation trajectory such as proposed by [START_REF] Liu | Similarity based method for manufacturing process performance prediction and diagnosis[END_REF] and (Wang et al., 2012).

CONCLUSION

Fleet-wide PHM requires knowledge-based system that is able to handle contextual information. Prognosis and maintenance decision making processes are improved by means of semantic modeling that deals with concepts definition and description. In this paper, a knowledge model based on ontologies is proposed. Contextual information is structured by means of specific contexts. These contexts allow to consider fleet unit similarities and heterogeneities. Data of the monitored unit are considered within their context and enhance the identification of the corresponding health condition.

From a prognosis point of view, the analysis of a degradation variable evolution could improve the prognostics model and precision can be improved through the capitalized data.

The fleet knowledge model has been structured for a marine application. The resulting ontology has been integrated in the KASEM industrial PHM platform. Some experimentation has already been done however; other experimentations should be tested to show the feasibility and the added value of this methodology. This paper arises some perspectives related to the manipulation of the uncertainty of prognosis (Petch, 2008), as well as the definition of the trajectory model based on retrieved models [START_REF] Liu | Similarity based method for manufacturing process performance prediction and diagnosis[END_REF] and (Wang et al., 2012).

Figure 1 .

 1 Figure 1. Main steps for fleet case re-use

Figure 2 .

 2 Figure 2. Semantic level to query heterogeneous databases 4. PROPOSITION OF ONTOLOGY FOR FLEET-WIDE SEMANTIC 4.1. Ontology for providing semantic

FigureFigure 5 :

 5 Figure 4: Part of the equipment taxonomy

Figure 6 :Figure 7 :

 67 Figure 6: Part of the dysfunctional analysis on units

Figure 8 :

 8 Figure 8: Part of the service context

Figure 10 .

 10 Figure 10. Degradation trajectory estimation for the Baudouin 8M26SRP engine based on similar units behaviors for a costal surveillance mission

Equipment hasIndicators/ refersToEquipment Installation hasIndicators/ refersToInstallation Ship hasIndicators/ refersToShip Health Condition Equipment Indicator Installation Indicator Ship Indicator Environmental Indicator Dysfunctional Indicator Functional Indicator Environmental Indicator Dysfunctional Indicator Functional Indicator Environmental Indicator Dysfunctional Indicator Functional Indicator IsComposedby/compose IsComposedby/compose Processing isComputedBy/ computes isComputedBy/ computes isComputedBy/ computes

	ID	Engine Ref	Output power (kW)	Nb. of Cylinder	Confi-guration	Engine Speed (rpm)	Type of cooling system	Engine cycle	Tag related to the ontology
	Eqpt1	Wärtsilä 12V38	8 700	12	V	600	Dry water cooling system Four-stroke engine	Fuel engine
	Eqpt2	Wärtsilä 12V38	8 700	12	V	600	Dry water cooling system Four-stroke engine	Fuel engine
	Eqpt3 Baudouin 6M26SRP1	331	6	L	1800	Sea water cooling	Four-stroke engine	Fuel engine
	Eqpt4	Man V8-1200	883	8	V	2300	Desalination systems	Four-stroke engine	Fuel engine
	Eqpt5	Man V8-1200	883	8	V	2300	Desalination systems	Four-stroke engine	Fuel engine
	Eqpt6 Wärtsilä RT-flex50	13 960	8	L	124	Sea water cooling system Two-stroke engine	Fuel engine
	Eqpt7	Wärtsilä RT-FLEX82T	40 680	9	L	80	Sea water cooling system Two-stroke engine	Fuel engine
	Eqpt8	Wärtsilä 12V64	23280	12	V	400	High and low temperature separated circuits	Four-stroke engine	Fuel engine
	Eqpt9 ISOTTA vl1716hpcr	2750	16	V	2100	High and low temperature separated circuits	Four-stroke engine	Fuel engine
	Eqpt10 Baudoin 12M26P1FR	357.94	12	V	1800	High and low temperature separated circuits	Four-stroke engine	Fuel engine

Figure 9: Part of the application context

Table 1 :

 1 Extract of engine fleet technical features stored in the data bases

BIOGRAPHIES

Dr Gabriela MEDINA-OLIVA is a Research Engineer at CRAN. She received his PhD from the University of Nancy and her M.S. in Reliability and Risk Analysis from the University of Las Palmas Gran Canarias in Spain. She has experience in maintenance within the oil industry. Moreover, she has worked in the formalization of knowledge with probabilistic tools for maintenance strategies assessment. Her current work focuses on fleetwide health management. He is member of the French DIAG 21 Association (www.diag21.com) and of the PHM Society.

Dr Alexandre VOISIN was born in Metz, France, in 1969. He obtained an engineering degree in Electrical Engineering form the EFREI in 1992 and he received his Ph. D degree in Electrical Engineering from the Université de Lorraine. He is currently assistant professor at the Université de Lorraine. His primary research were in the field of fuzzy logic and information processing where he applied these techniques to subjective evaluation in the area of car seat comfort.


Bernard Yannou
email: bernard.yannou@ecp.fr

Nadège Troussier
email: nadege.troussier@utc.fr

Alaa Chateauneuf
email: alaa.chateauneuf@polytech.univ-bpclermont.fr

Nassim Boudaoud
email: nassim.boudaoud@utc.fr

Dominique Scaravetti
email: dominique.scaravetti@bordeaux.ensam.fr

Dimensioning a product in preliminary design through different exploration techniques

Keywords: design exploration, robust design, design for reliability, constraint programming, Pareto solutions, embodiment design, parametric design, design optimization

Introduction and overview of embodiment design methods

The exploration of design parametric dimensioning under uncertainty can be made in using [START_REF] Suh | Axiomatic Design: Advances and Applications[END_REF] fuzzy sets theory applied to design engineering [START_REF] Antonsson | Imprecision in Engineering Design[END_REF], (2) Monte Carlo simulations (i.e. generating feasible design points) followed by an exploration within the feasible design space (see for example [START_REF] Stump | The ARL Trade Space Visualizer -An Engineering Decision-Making Tool[END_REF]) and (3) constraint programming techniques [START_REF] Yannou | Use of Constraint Programming for Design[END_REF]. We have already experimented that constraint programming techniques may be convenient (as soon as they are well tuned) to quickly result in an encompassing approximation of the design space (see [START_REF] Yannou | Truss dimensioning with an uncertainty reduction paradigm[END_REF]). In addition, a further Monte Carlo sampling within this approximate space has been shown (see (Yannou et al., 2005)) to be very efficient to straightforwardly obtain a good ratio of feasible design points (i.e. respecting the different design constraints and specifications). An exploration of a set of feasible design points is a valuable alternative and even may be preferred to the use of a global objective function (or a global preference aggregation function) because in preliminary design the weighting of the different objectives may be very subjective. This is why we believe that exploration lets the designer to be more opportunistic and even to acquire a better comprehension of the potential of its parameterized design. Lastly, techniques for generating and exploring the Pareto optimal solutions are evoked. On the example of the truss structure, it is shown that the design issue is already well constrained and that a tradeoff is useful to loosen somewhat some specifications in order to keep a sufficient degree of freedom to tackle robust design and design for reliability.

The multi-disciplinary design uses multi-objective optimization algorithms to find the best design that fulfills the numerous performances to be reached. These performances are evaluated using numerous disciplinary theories. It is often difficult to take the different theories into account in the unique formalism of an optimization approach. We propose to use a metamodel (or surrogate model) [START_REF] Papalambros | The optimization paradigm in engineering design: promises and challenges[END_REF] that enables to formalize multi-disciplinary knowledge and to evaluate several performances using just one kind of mathematical formulation. To illustrate the elaboration of a metamodel, the design of experiments method and an identification method (least mean square method) are used to identify predefined mathematical functions that link design parameters to performances.

Once these functions are validated, they represent with a unique form several physical phenomena and can be used to evaluate, approximately but very quickly, the behavior of the design that can be evaluated. This evaluation can be made according to different criteria such as robustness or reliability.

As defined by Taguchi, a robust design is a design of a product so that its functionality varies minimally despite of disturbing factor influences, which can be associated to environment factors, usage factors or technical factors such as design parameters… The aim is not to find the most performing design according to the set of performances to be reached, but to guarantee that the higher level of performing is guaranteed whatever the perturbations on the definition, process, usage or environment parameters are. The use of desirability functions to formalize the optimization goal and the signal/noise ratio in design of experiments enables to find on the truss structure the most robust solution.

The design for reliability aims to find the optimal solution that fulfills a given reliability condition. The fluctuation of loads, the variability of material properties and the uncertainties regarding the analysis models, contribute to make the performance of the optimal design different from the expected one. In this sense, the optimization process has a large effect on the structural safety and/or reliability. However, the safety factor approach cannot ensure the required safety level, as they do not explicitly consider the probability of failure regarding some performance criteria. In other words, the optimal design resulting from deterministic optimization procedures does not necessarily ensure the required reliability level. The design for reliability allows us to consider the safety margin evolution, leading to the settlement of the best compromise between the life-cycle cost and the required reliability. This task is further complicated due to the inherent nondeterministic nature of the input information. For this reason, many analysis methods have been developed to deal with the statistical nature of data. The process efficiency is mandatory to deal with realistic engineering problems [START_REF] Kharmanda | Efficient reliability-based design optimization using a hybrid space with application to finite element analysis[END_REF]; the metamodels can thus be very helpful in achieving the reliability-based optimal design with a reasonable computation effort. The solution obtained on the reliability basis is rather robust as the uncertain parameters are penalized during the design process, compared to a greater commitment of the well-controlled parameters. Practically, the design problem is formulated as a minimization of the cost function under some prescribed reliability targets (Aoues and Chateauneuf, 2007, in Press).

For the robust design and design for reliability approaches, the truss structure example has been made more complex in adding two performances: fundamental frequency and section area.

This paper can be considered as a brief survey of the main approaches that can be followed during the embodiment design stage of a product. Its purpose is, on a practical example of the parametric dimensioning of a truss structure, to concurrently apply these approaches so as to figure out that it is worthy to apply them successively in an ever refining embodiment design process. The paper is structured into five more sections: a presentation of the truss structure dimensioning issue, one section for each of the three approaches and a section of concluding remarks.

The truss structure dimensioning design issue: A first modelling

Our case study consists of dimensioning the two members of the truss structure shown in Figure 1. This problem was originally proposed by [START_REF] Wood | Computations with imprecise parameters in engineering design: background and theory[END_REF] to compute imprecise performance parameters from imprecise design parameters via fuzzy sets theory. This example has also been used by [START_REF] Scott | Using Indifference Points in Engineering Decisions[END_REF] in a different parameterized form to select an optimal Pareto solution that could not be selected via a linear aggregation function using importance weights. For this example, we use the exact parameterization and initial design variable ranges of the truss structure described by [START_REF] Wood | Computations with imprecise parameters in engineering design: background and theory[END_REF]), but we have chosen the more complex design constraints and performance parameters used by [START_REF] Scott | Using Indifference Points in Engineering Decisions[END_REF]. The requirement is to design a mechanical structure supporting an overhanging vertical load at a distance L from the wall with a minimal mass. One possible configuration (see Figure 1) consists in a two-member pin-jointed bracket with a horizontal member (CD) and a compression member (AB) attached to the wall at an angle of sixty degrees. The common pin is located at two thirds of L from the wall. Both members have rectangular cross sections: w AB t for (AB) and w CD t for (CD), w standing for width and t for thickness. Additional design decisions have been made: the material of both members is steel, and we impose w CD = w AB -0.025. The designer has to make decisions for the values of the following design parameters: t, w AB and L. Moreover, the specification of the overhanging load W is imprecise at the beginning of the design process, varying from 15-20 kN; consequently, W is treated as a fourth design variable.

The two mechanical constraints to satisfy are:

the maximum bending stress, b , in member (CD) must be less than or equal to the allowable bending limit, r (here 225 MPa for steel).

the compression force F AB in member (AB) must be less then or equal to the buckling limit F b .

The maximum bending stress, b , is located at point B and is given by the following formulas involving W CD , the weight of member (CD):

025 . 0 with 6 2 2 AB CD CD CD CD CD b w w tL gw W t w W W L (1)
The compression force in member (AB) is given by the following formulas involving W AB , the weight of member (AB):

L L tL gw W W W W W W F AB AB AB AB CD AB CD AB 9 3 4 and with 2 2 3 3 2 3 2 9 2 2 (2)
The buckling limit in member (AB) is given as: (3)

The performance variables are the mass M of the structure (to be minimized) and the safety factor, s, i.e., the amount of over-dimensioning beyond the satisfaction of the two mechanical constraints. The mass M is given by:

CD AB W W M (4)
The safety factor of the truss structure s is the minimum between the safety factor below the allowable bending limit, b , namely, s , and the safety factor below the buckling limit, F b , namely, s F , which is expressed as:

F AB b F b r s s s F F s s , min , , (5)
The two mechanical constraints may be merely expressed by: 1 , 1

Exploration of design parametric dimensioning under uncertainty

Design space exploration during embodiment design is an active research field. It consists in exploring the relationships between the choices of design parameter values and the performance variable values. This exploration provides the designer with a deep understanding of the potential of the given design concept that is studied, in comparison of a direct optimization of an objective function (function of the performance values). Often, a design space exploration is performed within a "design under uncertainty" process which is a process of dimensioning the product in progressively and consciously narrowing the domains (of allowable values) of design parameters while respecting the specifications on performance variables and constraints (like the aforementioned mechanical constraints). Three family of techniques are used to support the uncertainty reduction forwardly (from parameter domains to performance domains) and backwardly (from performances to parameter domains):

Fuzzy sets theory in design engineering (see [START_REF] Antonsson | Imprecision in Engineering Design[END_REF], not detailed here), Probabilistic techniques: most of these approaches consist in generating a number of feasible design points (complying with the constraints) and apply graphical postprocessing to visualize correlations between variables, the Pareto frontier or a preference structure among the design solutions (see [START_REF] Stump | The ARL Trade Space Visualizer -An Engineering Decision-Making Tool[END_REF]). The generation of feasible design points is often a statistical (Monte Carlo) generation of potential candidates sampled within initial variable domains, followed by the checking of constraints, which may become inefficient if the design problem is highly constrained since a majority of candidates that are generated do not belong to the (small) feasible solution space.

Constraint Programming (CP) techniques. They are not only well adapted to a numerical exploration of dimensional values for both performance variables and design parameters. In addition, they are also adapted to topological explorations of design concepts or architectures under constraints and, by the way, to the consideration (and exploration) different configurations of system functioning constraints (such as external load conditions) depending of the considered lifecycle stages (like flight phases, in case of (Scaravetti et al., 2006b)). Scaravetti et al (Scaravetti et al., 2006a) have studied the way that CP techniques modelled under the CSP formalism (Constraint Satisfaction Problem, which is generally the case for CP techniques) must be used within an industrial deployment of a design project: which constrained variables, which constraints, evaluation of several alternatives, choice within component catalogs).

With Constraint Programming (CP) over reals, performance variables and design parameters are modelled as intervals of allowable values. These constrained variables may be equated to uniform distributions of values in probabilistic modelling. CP techniques consist of sophisticated evolutions of interval analysis or interval arithmetics (see [START_REF] Moore | Methods and Applications of Interval Analysis[END_REF])) applied on a set of analytical constraints. Starting from a set of initial domains for the constrained variables and from a set of mathematical constraints linking the variables, different CP consistency or filtering techniques (such as Hull, Box, weak-3B or 3B, see for instance [START_REF] Benhamou | Revising Hull and Box Consistency[END_REF] and [START_REF] Yannou | Use of Constraint Programming for Design[END_REF]) try to contract as much as their consistency degree allows the variable domains so as to eliminate infeasible values. This domain contraction stage is called the filtering stage. One tries to result in the most tightened Cartesian product of intervals, ensuring at any moment that any feasible solution is kept inside. This last important property refers to the completeness property and guarantees that the contraction process results in an outer design space approximation.

In the second stage, the mechanism of domain splitting (bisection for instance) is recursively applied in parallel with the filtering mechanism. A search tree is built until a stopping criterion (e.g., width of the domains, number of solutions) is reached. This branch-and-prune algorithm allows pruning out large parts of the design space whenever a domain is found to be empty. At the end of the process, the design space is approximated by a number of elementary and disjointed Cartesian products of small intervals, denoted as boxes. The resulting hull of boxes provides the designer with valuable information about the potential values remaining for any design variable at this stage. Finally, a graphical representation of this collection of n-dimensional boxes (n being the number of constrained design variables) is easy and convenient for obtaining good pictures of the resulting design space (by 2D or 3D projections on pairs or triplets of design variables).

Figure 2 illustrates the four outer approximations of the design space that we can consider in a CP computation process, namely:

The initial domains;

The filtered domains after the uncertainty reduction propagation has been made for the first time;

The hull of boxes, i.e., the projection on variable domains of the collection of boxes that have not been considered inconsistent after the domain splitting process (with no guarantee of any actual solution inside); and

The collection of boxes itself.

It is obvious that, these four outer approximations of the design space are ordered in an increasing rank of refinement.

We have already showed in [START_REF] Yannou | Truss dimensioning with an uncertainty reduction paradigm[END_REF]) that the graphical representations of the collections of boxes could be meaningful for the designer(s) to perform relevant analyses of variable correlations and tendencies and making good decisions in a multi-stages "design under uncertainty" process. This is a first important utility of CP techniques in preliminary parametric dimensioning. Two stages of such a dimensioning process are illustrated in Table 1 with two cases:

-The case of a "not so constrained" design problem, which means that the initial domain is not large compared to the effective solution space. This is the situation of Case #1 of the specification constraints on the truss structure (the safety factor is just constrained to be greater or equal than 1).

-The case of a "highly constrained" design problem, which means that the initial domains are much larger than the actual solution space. This is the case of Case #2 of the specification constraints on the truss structure.

Note that in both cases, the ever more constrained problem is performed throughout constraints on performances. It means that the designer(s) can, here, really design in a functional manner, starting from the need and propagating consequences towards the solution (the means). This interesting facility is here permitted by the back-propagation properties of the CP filtering mechanisms. Another interesting case of domain reductions by CP techniques may be seen on the design exploration of an aircraft air conditioning system by Scaravetti et al (Scaravetti et al., 2006b). The design problem consists in optimizing the internal structures of the heat exchangers while satisfying the functioning constraints imposed by the system environment. The design variables are geometric and structural variables (lengths, surface types and pass number in the exchangers) but also thermodynamic variables (pressures, mass flow rates, temperatures). The performance variables are linked to efficiencies, mass, drag induced. After having sufficiently explored the design space of our truss structure and after several domain (and uncertainty) reductions, the designer(s) has converged toward a small design space of interest. It is time now to use probabilistic techniques to result in a cloud of feasible design points that one could apprehend one by one. We can perform a brute Monte Carlo simulation in sampling 100,000 design points by random trials within the initial domains of the design parameters (provided in Figure 2.left). After the checking of mechanical constraints, we have obtained (see also (Yannou et al., 2005)) 15,000 feasible design points in the "not so constrained" case (case #1) versus only 4 in the "highly constrained" case (case #2). These feasible and unfeasible design points have been represented in Figure3-left to figure out the very low ratio of feasible design points. It is then problematic when the designer wants to finely explore the design space since the design space is not dense enough to display a continuous variation of performances. A second utility of CP computation techniques has then been proposed by Yannou et al in (Yannou et al., 2005): the ratio of feasible design solutions is much more efficient when sampling inside the collection of boxes (obtained from a primary CP computation). In that case #2, the number of feasible design points has been 9,500 instead of 4 (Yannou et al., 2005), a great gain in efficiency. [START_REF] Messac | The Normalized Normal Constraint Method for Generating the Pareto Frontier[END_REF]) is the locus of non-dominated solutions, a dominated solution being a solution (design point) for which at least one solution (design point) is better for any of the performances. Highlighting the subset of Pareto solutions among the feasible design points is then very meaningful for designers. In [START_REF] Stump | The ARL Trade Space Visualizer -An Engineering Decision-Making Tool[END_REF] and in (Scaravetti et al., 2006b), the authors have developed graphical interactive tools in informing the designer on the performance values of a clicked optimal Pareto solution (the click is made in a performance plane or space) and in highlighting the corresponding design parameter values (made in a -designparameter plane or space). The weighting factor must be identified by the designers. It is well known that, in case of a convex Pareto frontier, making varying results in running all along the Pareto frontier (see Figure 4). But, in case of a concave Pareto frontier, some portions are not covered by an optimization process and, consequently, some Pareto solutions practically become non-eligible. This non trivial issue may be solved in using more sophisticated forms of the objective functions (see Scott and al for instance [START_REF] Scott | Using Indifference Points in Engineering Decisions[END_REF]).

Design parameters

Robust design

The aim of this section is to use a robust design approach as described by Taguchi [START_REF] Fowlkes | Engineering Methods for Robust Product Design: Using Taguchi Methods in Technology and Product Development[END_REF] on the truss example. The first section has presented how to explore the design space and to identify the space of decision among the values of the different design parameters in order to reach fixed specifications or performances. Then, knowing the ranges of parameters that enable to reach the specifications, several criteria can be formalized to focus not only on a convenient design, but also on the best design.

The criteria that we propose to use is the robustness, i.e. we search for the design solution that guaranties the level of performances whatever the variabilities on the design parameters are. That is to say that, the design parameters are chosen in order to have the performance levels the less sensitive as possible to the variabilities of design parameters, process variables and environment factors. These uncontrolled variabilities (for instance due to tolerances) should affect the less as possible the levels of performances. The problem is not to suppress or control the variabilities but to minimize their effects on performances.

Two different scales of design parameter values are used. The first level (concerned with the first section of this paper) of the scale is dealing with the variation of design parameters (different values of each design parameters that should be chosen to define the design). The second level is concerning with the variabilities around a design parameter value. These variabilities are small compared to the possible variation of the design parameter values but it is very important to be able to take these variabilities into account for decision making in the whole design process (Crossland et al., 2003, Ullman and[START_REF] Ullman | Taxonomy for Classifying Engineering Decision Problems and Support Systems[END_REF].

In the present section, we will take some design parameter values, identified for the truss example in the first part of the paper as convenient values, to reach the specifications in terms of performances. Around these values some variabilities are considered to take into account some uncertainties on design parameters, process variables and environment factors. Then, a robust design approach is used to propose the most robust design in the design space under uncertainties.

Specificities of the truss example for robust design

Managing the quality of the truss design requires ensuring the reachable performances levels in a constrained and uncertain context. Even if a common practice is to find the first convenient design that enables to reach the functional specifications, numerous research works are conducted to improve the design methodologies in order to be able to find the best design instead of a good design. N. Suh with Axiomatic Design [START_REF] Suh | Axiomatic Design: Advances and Applications[END_REF] is one of the first authors who proposes an approach that enables to guide design activities based on two axioms that evaluate the design all along the design process. Quality Function Deployment is especially appropriated to evaluate the quality of the product all along the design process. However, we will use the vocabulary introduced by the informational spaces proposed by N. Suh ( in the following description of the robust design approach. Designing is then a transformation between the variables through these different spaces. Even if the role of history and knowledge is not highlighted by this model, it will be used to formalize the truss problem in order to use a robustness criterion for its design evaluation [START_REF] El-Haik | Axiomatic Quality: Integrating Axiomatic Design with Six-Sigma, Reliability, and Quality Engineering[END_REF]. On the truss structure the performances or functional requirements initially considered are M , s , F s , s and the design parameters are t , AB w , L , W as described in Figure 5. Other design parameters are added in the problem formulation in order to have more sensitive performances. The two more design parameters that we consider in this section are: h , the distance between the top and the bottom of the truss, and that parameterizes the position of the joint linking the two members of the truss along the horizontal member.

The Taguchi robust design approach

Axiomatic Design provides two axioms (the independence and minimal information axioms) that enable to evaluate the quality of the design [START_REF] Suh | Axiomatic Design: Advances and Applications[END_REF]. If these axioms present a great interest for new innovative design on "uncoupled" systems, this design context is not usual when complex products are improved from a design to another one and the constraints imply that the axioms cannot be respected in practice. For instance, on the previous truss design formulation, the two axioms are not respected by the formulation used in engineering design. Then, under specific conditions, a multiobjective optimization formulation can be formulated in order to find the solution that provides the best level of performances. However, the design solution obtained is often sensitive to design parameters variabilities on design parameters due to the process (uncertainties managed as tolerances), the environment (uncertainties not managed but to support such as temperature, humidity, etc) but also due to desired variabilities such as users preferences, contexts of use of the product, etc. For instance the best solution for the truss structure previously defined is very sensitive to several variabilities such as illustrated in Figure 6. In this figure, the performance s is provided with respect of the h and t design parameters.

In this case, the robust design approach enables us to find the best solution and the most robust to design parameters variabilities. Robust design has been introduced by G. Taguchi and is defined in [START_REF] Flowkes | Engineering methods for robust product design: using Taguchi methods in technology and product development[END_REF]. The usual design approach aims at finding a design solution that reaches functional specifications and minimizing the uncertainties on design parameters. If some design parameters can be defined accurately, some tolerances always exit and perturbing factors can always be considered. Then, the robust design approach proposes to take into account the variabilities in the performances evaluation instead of avoiding it. Taguchi's proposition is based on the design of experiment (DoE) method to evaluate the sensitivity of the performances taking uncertainties into account. These uncertainties or variabilities are introduced as noise factors in the Taguchi table of the design of experiment as a small variability around each value of the design parameters. The impact of these variabilities around the design parameters values can then be analyzed and the most robust solution can be obtained by maximizing the signal to noise ratio. The design of experiment considered on the truss example is provided in Table 2. The mechanical laws are a bit more sophisticated than those formulated by formulas ( 1) to (6) and are not provided here for reasons of brevity. These laws are used to evaluate the truss performances and fulfill the performances evaluation (the two last columns of Table 2).

For each line of the design of experiment, a set of design parameters values are considered and a level of variability around the fixed value is also provided.

Then the response surfaces (fitted metamodel) are built for each performance and the data of robustness level is added on the response surfaces as a color level. For instance, Figure 7 provides the representation of a performance (here M and s) with respect to two design parameters (h and t) and taking into account the level of the signal to noise due to the variabilities on (the higher the color on the surface response, the higher the signal to ratio).

Table 2 The first lines of the table of experiments used to identify a linear model with interactions

t wab L W h Noise on M s -1 -1 -1 -1 -1 -1 142 141 -1 -1 -1 -1 -1 1 503,3 504,3 -1 -1 -1 -1 1 -1 507 506 -1 -1 -1 -1 1 1 657,1 658,1 -1 -1 -1 1 -1 -1 142 141 -1 -1 -1 1 -1 1 503,3 504,3 -1 -1 -1 1 1 -1 507 506 -1 -1 -1 1 1 1 657,1 658,1 -1 -1 1 -1 -1 -1 189,3 188,3 -1 -1 1 -1 -1 1 671 672 -1 -1 1 -1 1 -1 553,1 552,1 -1 -1 1 -1 1 1 794,9 795,9 -1 -1 1 1 -1 -1 189,3 188,3 -1 -1 1 1 -1 1 671 672 -1 -1 1 1 1 -1 553,1 552,1 -1 -1 1 1 1 1 794,9 795,9 -1 1 -1 -1 -1 -1 979,8 978,8 -1 1 -1 -1 -1 1 2154 2155 -1 1 -1 -1 1 -1 2166 2165 -1 1 -1 -1 1 1 2654 2655 -1 1 -1 1 -1 -1 979,8 978,8 -1 1 -1 1 -1 1 2154 2155 -1 1 -1 1 1 -1 2166 2165 -1 1 -1 1 1 1 2654 2655 -1 1 1 -1 -1 -1 1306 1305 -1 1 1 -1 -1 1 2872 2873 -1 1 1 -1 1 -1 2489 2488 -1 1 1 -1 1 1 3275 3276 -1 1 1 1 -1 -1 1306 1305 -1 1 1 1 -1 1 2872 2873 -1 1 1 1 1 -1 2489 2488 -1 1 1 1 1 1 3275 3276 1 -1 -1 -1 -1 -1 354,9 353,9 1 -1 -1 -1 -1 1 1258 1259 1 -1 -1 -1 1 -1 1267 1266 1 -1 -1 -1 1 1 1643 1644 1 -1 -1 1 -1 -1 354,9 353,9 1 -1 -1 1 -1 1 1258 1259 1 -1 -1 1 1 -1 1267 1266 1 -1 -1 1 1 1 1643 1644 1 -1 1 -1 -1 -1 473,2 472,2 1 -1 1 -1 -1 1 1678 1679 1 -1 1 -1 1 -1 1383 1382 1 -1 1 -1 1 1 1987 1988 1 -1 1 1 -1 -1 473,2 472,2 1 -1 1 1 -1 1 1678 1679 1 -1 1 1 1 -1 1383 1382 1 -1 1 1 1 1 1987 1988 1 1 -1 -1 -1 -1 2450 2449 1 1 -1 -1 -1 1 5385 5386 1 1 -1 -1 1 -1 5415 5414 1 1 -1 -1 1 1 6635 6636 1 1 -1 1 -1 -1 2450 2449 1 1 -1 1 -1 1 5385 5386 1 1 -1 1 1 -1 5415 5414 1 1 -1 1 1 1 6635 6636 1 1 1 -1 -1 -1 3266 3265 1 1 1 -1 -1 1 7180 7181 1 1 1 -1 1 -1 6222 6221 1 1 1 -1 1 1 8187 8188 1 1 1 1 -1 -1 3266 3265 1 1 1 1 -1 1 7180 7181 1 1 1 1 1 -1 6222 6221 1 1 1 1 1 1 8187 8188
Then, the most robust solution is found (given in Table 3) by maximizing the signal to noise ratio for the performances under consideration. The choice of the performance(s) to guaranty has to be done. In Table 2, we can find the most robust design solution considering the robustness of s performance, taking the variabilities on design parameter into account. Table 3 The best solution with the robustness criteria on s, with variabilities on

t w AB L W h M s
Robust design 0.1 0.13 3.42 17115 1.5 0.5

If the most robust design solution is searched, then combination functions can be used, such as the desirability (or usability) functions to determine a global robustness. The most robust solution can be:

the one that provides the less sensitive global desirability taking design parameters variabilities into account, the solution that maximizes the global desirability level (or minimizes a global loss function calculated on the whole responses), with a desirability level (or a loss level) associated to each signal to noise ratio provided by the impact of the variabilities on each performance.

Results and interests of robust design for uncertainty management in embodiment design

In order to make a decision under uncertainties in an engineering design context, and in particular in the embodiment design stage (defined in [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF]), the robustness analysis can be used. The results provided, using a response surface with a color grid to evaluate the signal to noise ratio, enables the designer to choose not only the best design solution in the design space but also the less sensitive one to the design parameters variabilities due to uncertainties. The robustness can be studied performance by performance or considering a set of performances to guaranty.

Design for reliability

The design of structures requires the verification of a certain number of rules resulting from the knowledge of physics and mechanical experience of designers and constructors. These rules come from the necessity to limit the loading effects such as stresses and displacements. Each rule represents an elementary event and the occurrence of several events leads to a failure scenario. In addition to the deterministic variables d k to be used in the system control and optimization, the uncertainties are modeled by stochastic variables affecting the failure scenario. The knowledge of these variables is not, at best, more than statistical information and we admit a representation in the form of random variables (noted X i whose realizations are x i ). For a given design rule, the basic random variables are defined by their probability distribution with some expected parameters.

Reliability analysis

The safety is defined as the state where the structure is able to fulfill all the functioning requirements: mechanical and serviceability, for which it is designed. To evaluate the failure probability with respect to a chosen failure scenario, a performance function G(x i ,d k ) is defined by the condition of good operation of the structure. The limit between the state of failure G(x i ,d k ) ≤ 0 and the state of safety G(x i ,d k ) > 0 is known as the limit state surface G(x i ,d k ) = 0 (see Figure 8).

In the First Order Reliability Method, the reliability level is defined by an invariant reliability index , which is evaluated by solving the constrained optimization problem: 0 ,

) ( ) ( min 2 k i i j i j i d x G : constraint the under x T x T V (8)
where V( ) is the distance between the median point and the failure subspace in the normalized space and T i ( ) is an appropriate probabilistic transformation; the solution of this problem is called the design point. At the first order approximation, the failure probability P f is given as a function of the reliability index:

) ( 0 , Pr k i f d x G P (9)
where Pr [.] is the probability operator and (.) is the standard Gaussian cumulated function.

Reliability-based design optimization

The Reliability-Based Design Optimization (RBDO) aims at searching for the best compromise between cost reduction and reliability assurance, by taking the system uncertainties into account; therefore, the RBDO leads to economical and safe design. It offers a good alternative to the safety factor approach, which is based on deterministic considerations and cannot take account for reduction of safety margins during the optimization procedure. In RBDO models, there are two kinds of variables:

The design variables d k which are the deterministic variables to be defined in order to optimize the design. They represent the control parameters of the mechanical system (e.g. dimensions, materials, loads) and of the probabilistic model (e.g. means and standard deviations of the random variables).

The random variables x i which represent the structural uncertainties, identified by probabilistic distributions. These variables can be related to geometrical dimensions, material characteristics and applied external loading.

Basically, the RBDO aims at minimizing the total expected cost C T (see Figure 9) which is given in terms of initial manufacturing and construction costs C c and direct failure cost Due to difficulties in the failure cost estimation C f (especially when dealing with human lives), the direct use of the above equation is not that easy. A practical formulation consists in minimizing the initial cost under the constraint of satisfying a target safety level β t :

C f : k i f f k c T d x P C d C C , (10)
U k k L k t i k k c d d d d x d to subject d C k , : min (11)
where d k L and d k U are respectively the lower and upper bounds of the k th design variable. This formulation represents two embedded optimization problems. The outer one concerns the search for optimal design variables to minimize the cost and the inner one concerns the evaluation of the reliability index in the space of random variables. The coupling between the optimization and reliability problems is a complex task and leads to a very high calculation cost. The major difficulty lies in the evaluation of the structural reliability, which is carried out by a particular optimization procedure. In the random variable space, the reliability analysis implies a large number of mechanical calls, where in the design variable space, the search procedure modifies the structural configuration and hence requires the re-evaluation of the reliability level at each iteration. For this reason, the solution of these two problems (optimization and reliability) requires very important computation resources that seriously reduces the applicability of this approach. where f Y is the yield stress, F b is the buckling load of member AB, b is the bending stress at point B and F AB is the normal force in member AB (detailed expressions are given in section 2 of the paper). For a target reliability t = 2 (corresponding to a failure probability of 1%), the reliability-based optimization problem is written: where 1 and 2 are the reliability indexes related to G 1 and G 2 , respectively. In this example, the uncertainties are related to the applied load W, the material strength f Y and the truss length L, where the coefficients of variation are respectively 15%, 8% and 2%.

Application to truss structure

Figure 10 shows the reliability index evolution for the two limit states, in terms of the design variables w AB and t. For the required safety level 2, the safe design space is reduced to points above this level. The search for the minimum weight in this subspace leads to the optimal solution given by: w AB =0.091 m and t=0.10 m, corresponding to a global mass of 4180 kg. At this solution, the bending limit state is observed as the most critical one. This result also indicates a safety factor of 1.38 for this limit state.

Concluding remarks: Three complementary approaches for embodiment design

The studied approaches of parametric design exploration have shown to be complementary and interactive. The exploration of the design space allows us to define the region of potentially interesting feasible solutions, which can be efficiently used for further investigations. The use of metamodels becomes thus precise as the search region is shrunk. These metamodels can then be used for multi-disciplinary design considering several performance objectives and constraints. As the design solution is almost described, it becomes necessary to carry out a design for reliability in order to take parameter and functioning fluctuations into account as well as uncertainties. Finally, the whole process allows us to reach a cost-based robust and reliable design. The application to a simple truss illustrates the advantages and difficulties in the different stages of the proposed process. The three considered successive design stages of design exploration, robust design and design for reliability are more and more sophisticated since they need more and more modeling information to provide a result. This is the reason why they must be successively used within a design process. Indeed, applying one of the three approaches allows to quickly figure out inadequacies with performance specifications or initial allowable bounds of design parameters and then to backtrack or to refine the design issue before passing to the next stage or approach. First, the design space is more and more precisely defined like in a Toyota-like set-based approach of design under uncertaintysee [START_REF] Ward | Set-based concurrent engineering and Toyota[END_REF][START_REF] Finch | A set-based system for eliminating infeasible designs in engineering problems dominated by uncertainty[END_REF]) -. Second, we clearly show that the successive optimal designs obtained by the three categories of methods are notably different, but that the optimal point obtained in a given approach is used to explore its surroundings in the next approach. This paper is just an illustration of this progressive and ever complexified preliminary parametric design process.

Table 4 summarizes that the best considered designs are significantly different in the three design stages. The comparison of the different methods is not that easy, as each one is based on specific assumptions. For example, the reliable design indicates that the safety factor should be reduced to 1.38 instead of 1.506; this leads to enlarge the design exploration space. However, if the designer requires higher reliability levels, the safety factor is automatically reduced at the optimal design.

Figure 1

 1 Figure 1The parameterization of the truss structure

Figure 2

 2 Figure 2 Initial interval domains of design parameters and performance variables for the trussstructure problem (left). The four successive outer approximations of the design space available after a Constraint Programming (CP) computation (right).

  of the feasible design points may be eligible as the best or preferred solution: this is the set of Pareto optimal solutions. This set of Pareto solutions is represented as a curve in the performance space: this is the Pareto frontier. By definition, a Pareto frontier (see

Figure 3

 3 Figure 3 Representation of feasible and unfeasible design points in the performance space for case #2 with a sampling of 100,000 design points within two different sets of domains X=[t,w,L,W].

Figure 4

 4 Figure 4 An optimization of an objective function (function of s and M), making varying the parameter of linear combination, describes the all Pareto frontier.

Figure 5

 5 Figure 5The new parameters of the truss structure

Figure 6

 6 Figure6An example of a response surface which provides s, one of the performances {FRs} with respect to some sensitive design parameters of {DPs}, t and

Figure 7

 7 Figure 7 Example of a response surfaces on which robustness data are added (right part of figure)

Figure 8

 8 Figure 8 Parameter joint distribution and failure probability

Figure 9

 9 Figure 9 Expected total cost in terms of failure probability

Figure 10

 10 Figure 10 Reliability index evolution in terms of truss dimensions

Table 1

 1 CP computation of the truss structure -considering two series of specification constraints

	Specif. traints cons-	projection L w t AB , ,	projection M W L , ,	projection s M W , ,	Hull of boxes
						0.0621,	0.1
							0.0654,	0.13
						3,	4
	Case #1				15000,	2	0000
	s	1				2077.9,	6	. 300	9
						1,	2	.	567

Table 4

 4 The best solutions found during the stages of design exploration, robust and reliable design

		t	w AB	L	W	h	M	s
	Opportunistic
	Design exploration	0.0995	0.0927 3.01 16570		3190	1.506
	(given in
	(Yannou and
	Hamdi, 2004))
	Robust design	0.10	0.13	3.42 17115 1.5	0.5
	Reliable design 0.10	0.091	4.00 15385		4180	1.38

He received a M. Sc. (received a Ph. D. (1994