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

Jean-Pierre Nadeau
email: jean-pierre-nadeau@bordeaux.ensam.fr

Jérôme Pailhès

Patrick Sebastian
email: patrick.sebastian@bordeaux.ensam.fr

Jérôme Pailhes
email: jerome.pailhes@bordeaux.ensam.fr

Structuring of embodiment design problem based on the product lifecycle

Keywords: embodiment design, structuring elements, design problem analysis, decision support system, layout, product life-cycle requirements, CSP

Early stages of the design process are often based on designers' experience; assumptions and irreversible decisions restricting the solution space are taken. It is difficult to take into account simultaneously every requirement imposed by the different phases of the product life-cycle. In this paper, a method is proposed to perform the analysis of the embodiment design problem. It facilitates the search of the indispensable elements, suitable for structuring the preliminary design phase. Our approach is performed for the relevant life-cycle situations of the product. A four level analysis (need, functions, organic structure and physical behaviours) is proposed. The set of structuring elements allow the design problem definition as a Constraint Satisfaction Problem (CSP).

Introduction

At the preliminary design stage, the decisions being taken influence up to 70% of the life-cycle cost (Figure 1). Several methods have been proposed to structure the global design process [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF][START_REF] Dieter | Engineering Design -A Materials and Processing Approach[END_REF][START_REF] Ullman | The Mechanical Design Process[END_REF][START_REF] Aoussat | La Pertinence en Innovation: Nécessité d'un Approche Plurielle[END_REF][START_REF] Cavailles | Méthodes de Management de Programme[END_REF]. On the other hand, there are few suitable tools to guarantee the relevance of the embodiment solutions [START_REF] Yannou | Préconception de Produits[END_REF]. At this design process stage, the following dilemma appears to be of major importance: designers should be lowly constrained in order to be creative [START_REF] Matthews | The introduction of a design heuristics extraction method[END_REF] and have to be constrained so that the decisions being taken verify the requirements of the design problem. Thus, it is necessary to have parsimonious but precise information [START_REF] Sebastian | Approximation et Intégration de Modèles Pour L'aide à la Décision en Génie des Procédés[END_REF] in order to be able to initiate the formalisation of the solutions and, on the other hand, to be able to make valid decisions engaging the durability of the choices. The systematic approach [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF] is rather widespread; however difficulties can arise in the early stages of the classical process of product design. They are mainly related to the simultaneous integration of the requirements stemming from various life-cycle stages, and to decisions that designers have to take early in the design process [START_REF] Chakrabarti | Towards a decision-support framework for the embodiment phase of mechanical design[END_REF].

First of all, several relevant concepts are rising from conceptual design. Designers have to perform a first choice among these concepts. In order to limit the duration of the design process and to decrease risks, they tend to focus very early towards a solution they can handle. Thus, interesting alternatives might be eliminated, because they are unknown, unused, unusual or non-evaluated.

Then, starting from a chosen concept, embodiment design aims to define rough arrangements and structural dimensions of the indented product, in accordance with technical, economic and aesthetic considerations [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF]. Designers have to identify key parameters relevant for this definition. Even if there are few parameters and standard elements for a simple mechanism, there is however a great number of possible combinations. Then, designers must make choices [START_REF] Hicks | An integrated modelling environment for the embodiment of mechanical systems[END_REF], based on their own experiences. They operate on a try-corrections mode.

So, design iterations cannot be avoided between the conceptual and the embodiment design phases, before reaching design solutions [START_REF] Ashby | Choix des Matériaux en Conception Mécanique[END_REF][START_REF] Hicks | An integrated modelling environment for the embodiment of mechanical systems[END_REF]. This iterative process appears clearly in many models of the literature [START_REF] Tate | A design process roadmap as a general tool for structuring and supporting design activities[END_REF]. Moreover, the embodiment solutions are depending on these initial decisions, which greatly limit the field of design investigations.

Another difficulty in an integrated engineering context is the confrontation, as soon as possible, between the different points of view of the design actors and technical skills (technical data, marketing data, company identity, environmental data, ergonomics, cultural and symbolic aspects). Constraints, specific to the product life-cycle phases, must be also taken into account.

More to the point, at a stage where the knowledge is uncertain, most of existing computer-aided tools are based on models requiring the complete geometrical definition of the product. In order to evaluate a design alternative, the performances of several configurations should be estimated rather than a particular solution performance, which might be, moreover, evaluated too late within the design process.

Constraint-based embodiment design problem modelling

A decision support system for the embodiment design phase has been developed to overcome these difficulties, by coupling a numeric CSP (constraint satisfaction problem) solver and a knowledge base [START_REF] Fischer | Qualitative constraints in integrated design[END_REF]. An approach has been proposed in order to investigate the performances of these tools within known design processes [START_REF] Scaravetti | Structuring functions and constraints formulation; for an enhanced embodiment design[END_REF]. It starts from relevant concepts and leads to feasible embodiment solutions. Figure 2 shows the stages and job-status of embodiment design enhanced by the use of the decision support system for a mechanical device design [START_REF] Scaravetti | Aided decision-making for an embodiment design problem[END_REF]:

• this process starts from two relevant working structures

• an analysis phase leads to identify the structuring characteristics of the embodiment design problem

• once the model is written, the CSP solver gives the solutions (a solution is a combination of structuring parameters, which check all the listed constraints)

• finally, the embodiment solutions are sorted, using an objective function. The use of this decision support system supposes the transcription of the design problem as a set of constraints (equality, inequality, logical rule). The CSP solver finds out combinations of variables satisfying all the constraints. This approach avoids the a priori decision-making and also shuns dismissing solutions; indeed causality and sequentiality during calculations are avoided.

Instead of starting with one retained concept, several concepts can be described by a set of specific constraints. The choice between several concepts is performed through the numerical processing of the constraint satisfaction problem.

All the conditions that determine the product characteristics are also translated into constraints: geometry, functional requirements criteria and physical behaviours for the relevant life-cycle stages, standard elements, technical skill rules, manufacturing constraint, cost evaluations. The variables can belong to continuous domains; thus, no variable is a priori fixed and the potentialities related to the concepts are preserved. The variables can also be discrete, enumerated and tabulated (catalogues).

The difficulty inherent in this approach is to correctly formalise the knowledge base (set of constraints) so that this base is coherent, non-redundant and complete. Designers require a method for capturing the information in the form of knowledge [START_REF] Theodosiou | Information models of layout constraints for product life-cycle management: a solid-modelling approach[END_REF].

We propose an analysis and structuring methodology of the design problem, in order to facilitate the formulation of constraints by designers: this approach is based on a four level analysis (need, functional, structural, physical) leading to determine the relevant parameters of design (like a designer does using his expertise) and to the identification of structuring characteristics and functions, translated later into constraints.

During the first steps of a design project, it is necessary to reduce the extent of the design problem. Designers do not take into account every requirement of the design problem and only some functions are selected. Their reasoning is based on structuring functions that initiate the design process and leads to the first validated product architecture.

In the same way, necessary and sufficient elements are identified in order to limit the embodiment design problem complexity. These elements are called structuring functions and criteria. On the contrary, the systematic approach [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF] looks for the exhaustiveness. The purpose of design methodologies is to guide designers and specify a design process with stages and milestones. We do not propose a new design method, but a method to select these structuring elements useful for the design embodiment phase.

A four level approach for the design problem analysis and structuring

The proposed methodology for the identification of structuring elements recovers information in functional performance specifications, physical behaviours analysis and technical skill rules. This methodology is based on classical tools performing need analysis, functional and structural analysis. Functional analysis (AFNOR, 1990(AFNOR, -1991) ) is usually used in an industrial context. The regulation documentation necessary for quality certifications and for the reliability, availability, safety assessments are based on functional analysis. It is also useful for tasks identification in project management. Thus, our approach based on classical tools does not necessitate new competences.

The search for structuring elements is performed by an approach based on a four level problem analysis: need, functional, organic, and physical (Figure 3). For each step, a selection method of structuring elements is proposed. Need analysis is related to a customer interested in the product. The customer can be:

• the end user of the product to be designed; the internal functions do not concern him and some criteria will not be a part of his qualification criteria.

• a company department, an applicant, a control service, a certifier: he is interested in the criteria related to internal functions and to expected physical behaviours.

They will be expressed during the next steps.

D. Scaravetti, J-P. Nadeau, J. Pailhès and P. Sebastian

Need expression

A rational and efficient design method starts with an exhaustive formulation of the expressed or implicit user's need (AFNOR, 1990(AFNOR, -1991)). At this stage, the customer often formulates its need in marketing and economic terms; the expected results and time limits are also expressed.

The functional analysis translates and integrates the need expression for the end user, the customer and the designer [START_REF] Prudhomme | Integrating into the design process the needs of those involved in the product life-cycle[END_REF]. However, the need expressed on a functional form, does not prejudge of the architecture of the system to be designed. So, the solutions field is not restricted.

The global function is always structuring, but may not be the most structuring one.

Product life-cycle analysis

Figure 4 enumerates the life-cycle situations related to the stages of the product life-cycle. The functional performance specifications expressed by the customer defines expected features of the product. In most cases, it does not systematically mention the life-cycle situations of the product. In that case, it is necessary to determine these situations from the expressed functions. Then, they are organised into a hierarchy from the hierarchical organisation of the functions. These reference life-cycle situations for the customer, initially expressed or not, will be a part of structuring situations.

New life-cycle situations emerge due to the evolution of the political, economic, industrial contexts, like end of life and recycling for example. These evolutions make prematurely obsolete the products, which did not integrate during their design the constraints connected to these life-cycle situations. These new requirements become inescapable and are imposed for example by standards.

Qualification criteria for customer

The customer has to express the relevance criteria of the design. These criteria will qualify the proposed design.

Table 1 enumerates qualification criteria by categories. It indicates the used variables and criteria, as well as the way of writing the constraints.

• The technological functional criteria concern the performances expected by customer and safety requirements. The structural criteria concern the constitution of the product, its resistance to the environment (atmosphere, vibrations, corrosion), its complexity. The product complexity (PC) can be defined as PC = Σ j × F j : j is the level of the functional decomposition and F j is the number of functions at this level [START_REF] Bashir | Estimating design complexity[END_REF]. The used functional description is coherent with the technical organisation chart used in the step #3 (see Figure 5): these functions correspond to the functions of every functional block. PC will be calculated later.

A limit value can be fixed thanks to a repository and previous projects experience.

• The various costs are enumerated in the economic criteria.

• The indicators of environmental impact and of sustainable development are also integrated in the qualification criteria. Several specialised indicators exist. More globally, the eco-efficiency indicator [START_REF] Janin | Démarche D'éco-Conception en Entreprise -Un Enjeu: Construire la Cohérence Entre Outils et Processus[END_REF][START_REF] Verfaillie | Measuring eco-efficiency -a guide to reporting company performance[END_REF] compares the economic performance and the environmental performance. It is a relative measurement of the environmental impact. Such indicators can be: Mass of sold product or net sales per

• Energy consumption • Material consumption •
Greenhouse gas emissions

• The qualification criteria relevant for the company and marketing concerns development times, design risks and product perception.

The project duration can be estimated from the product complexity PC [START_REF] Bashir | Estimating design complexity[END_REF]:

Time (hours) = A × PC D 0.85
• 'A' is a constant depending on the company size and on the information circulation. It can vary from 30 for a small firm with a good communication, up to 150 for a big company with an average communication [START_REF] Ullman | The Mechanical Design Process[END_REF].

• 'D' is related to project difficulty (1: little difficult, using known technologies; 2: difficult, few new technologies; 3: very difficult). [START_REF] Larson | Managing design processes: a risk assessment approach[END_REF] and, [START_REF] Sarbacker | A framework for evaluating risk in innovative product development[END_REF] give indications onto the design risks and their consequences, from the designer point of view; the consequences are always related to the costs and project duration. The index of risk priority (IRP) allows to quantify the risks connected to a system or to a component [START_REF] Landy | AMDEC Guide Pratique[END_REF]. The risk is defined as a magnitude with two dimensions (characterising one dreaded event): The probability of occurrence and the gravity of the consequences (defined in terms of loss of performance or financial loss) [START_REF] Grenouilleau | Analyse des risques opérationnels liés à la gestion des éléments de rechange d'un système spatial complexe en phase de conception[END_REF]. (2003) Legend: Origin of the constraint limit: (1) Functional performance specifications;

(2) Standards and regulations;

(3) R&D choice; (4) Marketing and (5) Financial and temporal management of project.

Relevance indicators

From the relevant concepts, several feasible architectures can be proposed. The relevance indicators give elements to designers to make choices and take decisions.

The adaptable or negotiable or quantifiable qualification criteria (for customer) are then translated into relevance indicators.

Relevance indicators are structuring elements, integrated within the model and calculated. They are suitable to calculate performance indicators (see Section 5.4) and they highlight closeness or distance compared to a reference. They are integrated into objective function(s), useful to sort the embodiment solutions.

Functional approach (Step #2)

When external functional analysis is realised for all life-cycle situations, functions and external environments are exhaustively enumerated. We wish to retain the relevant life-cycle situations, organise the functions into a hierarchy and extract those, which are structuring for the problem of preliminary design.

Choosing relevant life-cycle situations

First of all, the life-cycle situations deducted from the customer's functional specifications are relevant. They are explicitly expressed or deducted from the satisfaction of the global function or from the functional specifications.

After the external functional analysis, risky life-cycle situations are selected. They concern safety (reliability, maintainability, availability). The more important ones are situations badly mastered and identified thanks to experience feedbacks. Life-cycle situations where product or environment can be modified must be selected. For example: 'prolonged stop', 'functioning in aggressive or polluting environment' or any situations susceptible to modify the state of the product. The situation 'maintenance' is necessary to ensure the good functioning of the product.

'Design' and 'manufacturing' also retained life-cycle situations, because designers take into account constraints related to manufacturing possibilities (state of the art, technical skills rules).

All these relevant life-cycle situations influence steps #2 and #4 of the structuring and analysis of the design problem. Indeed, these two steps are realised only for these situations.

Structuring functions

Among the functions stemming from functional analysis, the structuring ones are enumerated into Table 2. The constraints can be expressed in an explicit way when a particular industry is targeted. An example concerning fluids cooling is developed in [START_REF] Bouchama | Flash evaporation, modelling and constraint formulation[END_REF]. For the structuring functions, assessment criteria and variables are expressed. An assessment criterion allows to appreciate the way a function is filled or a constraint is satisfied (AFNOR, 1990(AFNOR, -1991)). The level and the flexibility of a criterion can be translated on a constraint form.

A redundancy can appear between these assessment criteria and qualification criteria for customers. The redundancy is useful to guarantee the exhaustiveness of the identification of the structuring elements.

• Functions with no flexibility are identified thanks to functional performance specifications when assessment criteria are enumerated (level, flexibility, limit …).

Functions with low or nil level flexibility are retained as structuring functions.

• Risky functions (on the technical level) are identified thanks to already known problems, negative feedbacks, etc. For that purpose, it is necessary to capitalise knowledge and to know how to reuse it. According to [START_REF] Ullman | The Mechanical Design Process[END_REF], when knowledge is weak they are critical functions. A risky function can result from a precision or an exactness imposed by the customer.

A function that was never realised might be risky. Risk analysis takes into account the novelty, complexity, arisen events and gravity of the consequences related to the product design [START_REF] Leroy | Le Risque Technologique[END_REF].

A functional risk might occur when the energy is badly managed, entailing a dysfunction [START_REF] Desroches | Aspects conceptuels des risques[END_REF].

• Imperative constraint functions are imposed by the external environments (technology, market, situation and choices of the company, etc.). On one hand, there are constraint functions (AFNOR, 1990(AFNOR, -1991) ) concerning system behaviour or performances (legal regulations, imposed materials or solutions, etc.). On the other hand, some structural constraints do not concern directly the realisation of a function (time limits, supplying, etc.).

• Technical skill rules are identified through functional analysis, for the life-cycle situations 'design' and/or 'manufacturing' and/or control and/or maintenance. For example, these rules intervene as soon as that the company has to manufacture 'with the potential of the company':

• Know-how rules of the company. Design rules, calculus rules, manufacturing rules, cost or duration estimation rules, etc.

• A manufacturer has technical reports, capitalising the know-how and defining criteria to be respected. For a car manufacturer for example: ground clearance, protection of the electric wires, seat command (ergonomics), design recommendations for recycling, etc.

•

Guidelines for the environmental integration are used by some manufacturers [START_REF] Janin | Démarche D'éco-Conception en Entreprise -Un Enjeu: Construire la Cohérence Entre Outils et Processus[END_REF]. They enumerate the banned or advised materials, instructions of conception in order to facilitate their dismantling, their recycling, etc.

•

Manufacturing rules connected to the company know-how, induce design constraints (curvature radius of pipes, minimum crankcases thicknesses, etc.).

• Global design rules. Embodiment design rules [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF] are concerned with form choices appropriated to manufacturing processes or facilitating the assembly.

• Technical skill rules indicate the type of tool or material to be used, in order to obtain for example, a surface specification or a tolerance.

• Technical skill rules related to the application of professional codes. These dimensioning codes are related to a specific product or industry sector: CODAP (pressure vessels), EUROCODE, BAEL (civil engineering), PS (seismic), RCCM, FEM, CM66, ASME, NEIGE, VENT, etc.

The codes impose the way to design, the banned solutions, the manufacturing modes, the controls and maintenance to be made. They also impose the calculation rules connected to the materials being used and prescribed manufacturing processes. In some cases, the physical models are supplied. This type of constraint is raised in the stage #4 (physical approach). These enumerated structuring functions generate constraints (equality, inequality, logical rules), intervals or enumerations among which variables will be chosen.

Relevance indicators, from the designer's point of view

Technical rules may impose causality between the variables, that is to say, a procedure or an algorithm to treat the design problem. Designer criteria might not be expressed in the functional performance specifications. These intermediate elements are useful to compare some design alternatives. Designer criteria are relevance indicators:

• Relevant criterion in the realisation of a technical function. The designer may add parameters which are not expressed by the customer, but allow him to qualify a function.

• The functioning risk can be estimated by the distance compared to a reference (standard, rule, experience feedback, test on prototype) with the performance indicator.

Once identified the relevance indicators stemming from the qualification criteria for the customer and the designer's relevance indicators, we compare them with a reference. A performance indicator highlights relevance indicator's closeness or distance compared to a reference.

A performance indicator must be minimised.

• Ip = I/I ref, if the performance corresponds to the distance compared to the reference.

In that case, it is connected, for instance, with a risk of operation.

• Ip = I ref/I, if the performance corresponds to the closeness compared to the reference. In that case, it is connected, for instance, with an optimisation process.

The reference (I ref) can be:

• Fixed. An evaluated element, imposed by the customer, by standards, etc. It is a fixed reference, intrinsic to the problem and independent from the context. If it can be expressed, it is preferable to use it.

A risk can be evaluated by knowing a reference. For example, the distance of a parameter can be quantified by comparison to a standard. The degree of change can also be quantified. In certain cases of re-design, minimising changes is needed.

• Relative. The minimal or maximal value of a relevance indicator among a set of results: For a given configuration and a given context, the solutions can be compared with each other regarding the indicator.

So, the performance indicators allow:

• The comparison regarding a customer reference: it necessitates to overcome the calculation of the indicator corresponding to this reference.

• The comparison of the solutions between them. If a criterion cannot be defined completely, it is however necessary to identify relevant parameters relative to the criterion.

A comparison between concepts can be [START_REF] Ullman | The Mechanical Design Process[END_REF]:

• absolute: each concept is compared to objectives using criteria

• relative: concepts are compared by using measurements defined by the criteria.

Organic approach (Step #3)

The product concept is defined in a first technical organisation chart by assembling functional blocks (Figure 5). However, only the first levels define the product at the conceptual stage. Beyond the first levels, other choices are realised, during the embodiment design phase.

This chart allows to identify and to define elements in the Table 3. Structuring organic elements

Structuring elements Characterisation

To be chosen For example, a domestic vacuum cleaner with a disposable bag can be decomposed in three main functional blocs (pipe, fan, bag) and two external environments (air, dust).

The standard elements are to be chosen or dimensioned:

• The standard components to be chosen are components ready-to-use belonging to catalogues. Their choice is made on relevant parameters, relative to the functions they perform.

They impose their intrinsic constraints (dimensional constraints, interface constraints, energy supply, etc.): these variables are identified in the problem, but they can take only standard values. On the other hand, some constraints are associated to various life-cycle situations, not directly connected to the functions they fulfil, and given by the manufacturers (constraints of use, warehousing or transport, etc.). It is also necessary to investigate their sensitivity to harmful factors.

• The standard elements to be dimensioned are concepts already mastered by the company (known functional blocks, where only some parameters are adjusted).

Dimensioning rules exist for these typical elements; they arise from the company knowledge, from technical books or from professional dimensioning standards (CODAP, TEMA, etc.).

• Design of non-defined elements cannot be avoided and are thus structuring. They have to be dimensioned.

The design risk is estimated [START_REF] Landy | AMDEC Guide Pratique[END_REF] and choices have to be made according to the anticipated risk. The badly known alternatives are dismissed.

• The relations between the system and the external environments are enumerated.

The interface constraints express the compatibility between the elements or between one element and its environments. Some interface constraints may be already identified at the previous step.

Various types of interfaces are [START_REF] Cavailles | Méthodes de Management de Programme[END_REF]:

• functional interface (compatibility of mechanical links, electric or electronic links): connections, fixation; assembly; tensions, amperages; signals (input, output, form); computer interface (protocols, languages)

• physical interface, compatibility between characteristics of nearby constituents: electromagnetic, geometrical compatibility (encumbrance, assembly …)

For the interfaces, the connection between two elements must be checked and the conservation laws of fluxes must be satisfied.

Physical approach (Step #4)

The model of the design problem has to contain a description of the relevant physical behaviours. In order to enumerate them, the structuring variables and behaviour of the system have to be investigated in an exhaustive way. For that purpose, functional block diagrams (FBD) and substances-fields graph (SFG) are used. Table 4 summarises the structuring elements related to the physical approach. The FBD is a representation of the path of the functional fluxes through a system corresponding to a life-cycle situation. These fluxes connect the external environments to the system or the system components together (internal fluxes). The designer has to choose the level of the system representation. Indeed, FBD is suitable to describe the whole system or only a functional block. At a detailed level, its use may be complex.

A FBD corresponds to each design solution.

Completing each function induces a flux [START_REF] Ullman | The Mechanical Design Process[END_REF] and fluxes may be of various nature: energy, material, information, user esteem. The functional flux path corresponding to the functional conditions can be visualised using the FBD. The main functional fluxes pass through structuring elements.

The contact graph lists exhaustively the connections between functional blocks. The distant actions are estimated using a systematic questionnaire. The effects of gravity, mobile parts (effects of inertia or centrifugal), non-material fields (sounds, ultrasounds, optics, electric, radiation, magnetic, etc.) are successively examined.

However, the FBD does not identify the interactions between the transported materials and the induced effects. The FBD solely describes the functional fluxes passing through the components. Moreover, only the useful functional components appear within the FBD, whereas the effects quoted above are not taken into account. The physical behaviours are described more precisely with substances-fields graph (SFG) [START_REF] Savransky | Engineering of Creativity: Introduction to TRIZ Methodology of Inventive Problem Solving[END_REF].

SFG is suitable to integrate the transported materials or present materials into the system, and to take into account the generated fluxes and the effects induced by these fluxes. A systematic list of generated fluxes and induced effects has been proposed [START_REF] Mal'in | Méthodes d'Aide à L'INnovation (Methods for Innovation Aid)[END_REF]. TRIZ theory differentiates the useful or insufficient functions and the unwanted effects (harmful effects).

Figure 6 displays the evolution from a given FBD (partial) to a SFG. These graphs concern the functions 'to suck up and store the dust' for the domestic vacuum cleaner. We found the functional blocks identified in the previous step (Figure 7). These actions are characterised by physical behaviours. For example, the generated air flux is related to a pressure gradient (state-variable of the air) allowing the transport of the dust.

These physical behaviours are structuring for the design problem. Tables (like Table 1), give an exhaustive vision of fluxes generated by the gradients of state-variables and their induced effects. These tables are available during the definition of the graphs with the software tool MAL' IN (2003) (online help).

These graphs have to be interpreted to determine the constraints of the design problem. The functional fluxes are achieved through physical phenomena. At this stage, designers have to choose, among the set of identified phenomena, the physical behaviours that can be neglected. These choices are subjective and appeal to the designers' experience.

The relevant and dominating physical behaviours are highlighted by this approach. Each physical flow is related to one or several constraints:

• Fields are translated by continuity conditions between the substances. The conditions of the flux passage are identified through the physical phenomena corresponding to interaction between substances and determine the boundary conditions.

• Fluxes are interpreted by behaviour laws or/and physical conservation laws. Fluxes can convey material or energy. Balances must be written (mass conservation, energy conservation, etc.).

Some fluxes result from gradients of state-variables such as temperature or pressure. These fluxes may induce non-wished effects such as pressure losses, dilation, creep, deformation, clogging, etc. This phenomena identification process gives the opportunity to express the durability of components. Complementary material criteria related to mechanical functions (rigidity, elasticity, etc.) or to specific functions (thermal, chemical …) may result from this process. System behaviours may be also determined using professional codes (step #2). In these codes, the calculation rules and physical models may be supplied to the designer: the physical behaviours are translated in a parsimonious way in agreement with the dimensioning needs of the embodiment design.

The physical phenomena identified below lead to the definition of relevant variables. In the vacuum cleaner example, the harmful action 'bag clogging' is related to a physical behaviour described using the relevant variables: bag porosity, bag volume.

Recapitulation of structuring elements of the product life-cycle

The constraints specific to certain phases of the product life-cycle initiate the identification approach of structuring elements during the preliminary design:

• Structuring life-cycle situations of the product are identified (step #1). During steps #2 and #4, tools are deployed only for these situations of the life-cycle.

• Specific constraints corresponding to certain life-cycle situations are displayed through the following structuring elements: • Constraints specific to life-cycle situations of the standard elements are identified (transport, use, etc.). They are supplied by the manufacturers.

All the items listed above take into account the constraints related to the life-cycle of the product in the definition of the embodiment design problem. The set of solutions satisfying all these constraints defines the field of solutions.

Conclusion

Based on functional analysis, a method has been proposed to help designers to define and enumerate the structuring elements in the early phases of the design process. This method is involved in the preliminary analysis of a design problem and leads to a set of structuring functions, criteria and components. These elements are those that are necessary and sufficient for the preliminary design phase.

After performing the life-cycle analysis, an analysis divided into four stages has to be carried out:

• need analysis

• functional analysis

• organic analysis • physics analysis.

Several tables defining an exhaustive enumeration of structuring elements have been proposed. The quality of the functional and physical analysis determines the performances of the results obtained by this method.

An example of application for a mechanical device (see [START_REF] Scaravetti | Aided decision-making for an embodiment design problem[END_REF]) illustrates this method and has been involved in the aid to choice (choice between several concepts of solution and choice of components) and in the search of feasible embodiments. An improvement of the design process duration was observed thanks to the reduction of the iterations. The choices carried out also facilitate the detail design. Finally, the choice performed within the field of embodiment solutions based on an objective function, maximises the performances expected by the customer (physical behaviour, cost).

These works have been involved in a global project aiming to develop decision-making tools for preliminary design:

• the software MAL'IN (Methods for Innovation Aid) is being developed in our laboratory

• the software CONSTRAINT EXPLORER within the framework of the RNTL project CO 2 (national network of software technologies: Design and Constraint Programming).

This project CO 2 gathers Dassault Aviation, CRIL Technology, TREFLE and LIPSI laboratories (interested in design) and artificial intelligence laboratories: LINA and LIP6.

Figure 1

 1 Figure 1 Evolution of committed costs during design process

Figure 2

 2 Figure 2 Different job status of the enhanced embodiment design phase

Figure 3

 3 Figure 3 Stages of the method and tools for identification of the structuring elements

Figure 4

 4 Figure 4 Product life-cycle and the corresponding life-cycle situations

Figure 5

 5 Figure 5 Technical organisation chart

Figure 6 Figure 7

 67 Figure 6 From FBD (6(a)) to SFG (6(b)), for a vacuum cleaner

Table 1

 1 Qualification criteria for customer and characterisation

			Constraint
	Qualification	Characterisation:	(quantitative	Origin	Qualitative
	criteria:	variables and criteria	evaluation)	x,y, ..	evaluation	Ref
	Technological criteria
	Functional criteria
	Expected	Efficiency\power\physi	Parameter ∈	(1)
	performances	cal parameters (force,	[x,y]
		speed, temperature, pressure …)	Parameter∈ {x, y, z}
		Productivity (number of actions per time unit)	Parameter > x
	Regulation	Criteria based on	Parameter	(2)
		standards	∈ [x, y]
			Parameter < x
	Minimal life	MTTF: mean time to	MTTF > x	(1)		Ligeron and
	expectancy	(first) failure				Lyonnet (1992)
	Reliability	MTBF: mean time	MTBF > x	(1)		Ligeron and
		between failures				Lyonnet (1992)
	Availability	Indicator of operational	1 > D op > x	(1)		Monchy (2000)
		availability:
		D op = MTBF/MTBF +
		MTTR
	Maintainability	MTTR: mean time to	MTTR > x	(1)		Ligeron and
		repair rate of standard				Lyonnet (1992)
		components
		Ease of cleaning			Yes/no
	Security	FMECA criteria:	Criticality < x (3)		Landy (2002)
		Criticality = severity				Garin (1994)
		level×proba.
		occurrence× proba. not
		detection
		Certification	Criterion < x	(2)
		requirements
	Robustness	S/N (signal/noise)	S/N maxi	(3)		Taguchi (2003)
	Autonomy	Energetic (operation	Autonomy > x	(1)
		life)
		Do not necessitate			Yes\no
		human intervention
	Structural criteria
	Mass	Total mass, maximum	Mass < x	(1)
		(or minimum)
	Encumbrance	Minimum dimensions	Dimension > x (1) (2)
		(or maximum)
		Volume coefficient of	1 < VPC < x	(1) (2)		CEN, European
		the packaging:				Committee for
		VCP = parallelopiped				Standardisation
		packaging				(2000)
		volume/contents
		volume

Table 1

 1 Qualification criteria for customer and characterisation (continued)

			Constraint
	Qualification	Characterisation	(quantitative	Origin	Qualitative
	criteria	:variables and criteria	evaluation)	x,y, ..	evaluation Ref
	Structural criteria
	Resistance to	Maxi level of	Temperature <	(1)
	the environment	temperature\pressure\hu	x…
		midity\salinity\vibratio
		ns (amplitude,
		frequency) …
	Complexity	Product complexity:	PC < PC ref .	(3)	Bashir and
		PC = Σ j × F j …	PC < PC min		Thompson (1999)
	Economic criteria
	Industrialisation	Costs	Cost < x	(5)
	cost	equipments + machines
		\workforce
	Production cost Cost machine	Cost < x	(5)
		functioning\production
		unit\workforce
	Ownership cost Energy consumption	Cost < x	(1) (4)	Millet (2003)
		Life-cycle cost	Consumption
		(acquisition, operation,	< x
		maintenance)	Energetic
			contents < x
	Maintenance	Operation cost\parts	Cost < x	(1) (4)
	cost
	Recycling cost Dismantling	Cost <x	(2)
		costs\treatment …
	Objective cost Product cost	Cost < x	Value
		(components, materials,		analysis
		manufacturing)
	Investment	Necessary	Cost < x	(5)
	capacity	investment\possible
		investment
	Recyclability	% total recyclable mass %mass < %ma	(2)	Removable
			xi		compatible
		Irreversibility of the			Yes\No
		connections
		(removable\irremovable)
		Compatibility of materials (yes\no) Number of materials	Material number<x
	Sustainable	Eco-efficiency	Indicator < x	(2)	Janin (2000)
	development	indicator = product value/environmental impact			Verfaillie and Bidwell (2000)

Table 1

 1 Qualification criteria for customer and characterisation (continued)

			Constraint
	Qualification	Characterisation:	(quantitative	Origin	Qualitative
	criteria	variables and criteria	evaluation)	x,y, ..	evaluation Ref
	Economic criteria
	Environmental	Power consumption	Consumption	(1) (4)
	impact (in the		< x
	various life-cycle phases : manufacturing, distribution, of life …) use, end	Water and material consumption Resources impact indicator pollutants, waste: Emissions of gas and	Consumption < x RII < x Emission < x	(2) (2)	Brent and Visser (2004) Millet (2003)
		impact indicator
		'greenhouse effect':
		kg of CO 2
		equivalents emitted
		impact indicator
		'destruction ozone
		layer': kg of CFC11
		equivalents emitted
		Heat generation: heat	Flux < x	(2)
		flux
		Humidity generation:	Flux < x	(2)
		flux of material
	Qualification criteria for company, marketing
	Project	Time (h) = A×PC×D 0.85 Duration < x	(5)	Bashir and
	duration				Thompson
		% new elements	% new		(1999)
		(cf technical	elements < x
		organisation chart)
	Risks: design,	IRP =	IRP < x	(5)	Landy (2002)
	costs, time	gravity×occurrence×vali
		dation	Overrun < x
		Possible overruns (costs,
		time limits)
	Startup	Necessary	Time < x	(1) (4)	Qualificati
		time\qualification			on yes\no
		Temperature setting … Time < x
	Environmental	Level of generated noise Noise level < x (2)
	impact (nuisances)	Generation of smell Visual nuisance			Yes\no Yes\no
	User	Associated variables	Variable < x		Dore et al.
	perception

Table 2

 2 Different types of structuring functions

	Types of structuring functions
	Function related to
	qualification criteria
	for customer
	Function with no
	flexibility
	Risky function	Already known problems, negative feedbacks
		Function never realised
		Precision or an exactness imposed by the customer
		Energy management
	Imperative constraint	System behaviour or performances
	function	Standard or regulation to respect: technical or industrial normative
		documents related to person or material protection.
		Interchangeability
		Prohibition or imposition of the principle solution:
		industrial context, usable technologies
		qualified concept (Aeronautics; chemical and
		pharmaceutical industries)
		industrial property
		Known risk
		Environmental criteria (Janin, 2000):
		legal requirements, standards : impact on environment
		(release, pollution, waste, noise, consumption of resources,
		recyclability), re-use, reprocessing, stocking
		demountability, with the aim of the materials separation
		(materials, connections)
		Environmental costs: equipments, backfitting, materials (type and
		number of different materials), assemblies, resumption of
		packagings
		Material:
		imposition: food, ageing, resistance (mechanical, chemical,
		thermal, etc.)
		Cost
		Structural constraints
		Limit time
		Material:
		unavailability
		prohibition (danger, standards…), in the product or in the
		process
		dependence material/process (if process is imposed)
		supply, supplier
		Manufacturing functions (manufacturability, facility of assembly,
		treatment of surface, etc.)

Table 4

 4 Structuring elements related to the physical approach

	Structuring element	Characterisation
	Key elements, in which the
	main functional flux passes
	through
	Relevant and dominating	Conservation laws
	physical behaviour	of mass
		of energy (heat, mechanical or deformation
		energy, etc.)
		of the impulse
		of electric charge
		Boundary conditions
		Induced relations of coupling
		pressure losses
		models of contact, friction
		electromagnetic interaction
		deformation, fastening, creep
		clogging etc.
	Durability of components
	Calculation rules imposed by
	the professional codes
	Material criteria	Mechanical functions (rigidity, elasticity, resistance to
		fatigue, wear, shocks, etc.)
		Specific functions: thermal, chemical (resistance to
		corrosion, etc.), electric, optical, magnetic, acoustic, etc.

  • design: qualification criteria for company, technical skill rules, imperative constraint functions • manufacturing: technical skill rules, economical criteria • supply of components and possible materials: imperative constraint functions • marketing: economical criteria, marketing criteria • use (situation often the most described by the customer and the functional specifications): performance criteria, structural and economical criteria, imperative constraint functions, physical behaviours • control, maintenance: economical criteria, technical skill rules • beneficiation, recycling: economical and marketing criteria, imperative constraint functions.


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

Patrick Sébastian
email: patrick.sebastian@bordeaux.ensam.fr

Design space exploration in embodiment design: an application to the design of aircraft air conditioners

Keywords: design space exploration, embodiment design, genetic algorithm, trade-off, nondominated solutions, Pareto, decision support, bootstrap, aeronautics

The embodiment design of aeronautic systems proves to be a difficult design phase due to the variability in the system environment; the design is often constrained by atmospheric conditions. These atmospheric conditions appear to be highly variable according to the flight phases of the aircraft.

The difficulty when designing air-conditioning systems for civil aircrafts is also inherent to the complexity of the coupling of the non linear physical phenomena inside of these systems (multi-physics, multi-scaling). Therefore, the design space appears to be very broad and quite difficult to explore. Embodiment design choices are relating to continuous and discrete design variables while the system effectiveness is extremely sensitive to most of these design variables. There is a lack of tools to support the investigation of the design exploration space and designer decisions at early stages of the design process. In this paper, a method is proposed to generate feasible embodiments and manage the compromise between various design requirements.

A digital tool based on the meta-heuristic of Genetic Algorithms (Gas) has been developed to investigate the design problem. The selection of non-dominated solutions (Pareto) is used to identify relevant values for design variables and to facilitate choices among the design solutions of the air conditioning system.

Biographical notes: Dominique Scaravetti has been an Associate Professor at the TREFLE Laboratory of the Arts et Métiers ParisTech Graduate School of Engineering since 2006. He received a PhD (2004) in Mechanical Engineering from Ecole Nationale Supérieure d'Arts et Métiers (ENSAM). Before that, he had been teaching at ENSAM since 1997. His research interests are centred on decision aid during preliminary design phases.

Patrick Sebastian has been an Associate Professor at the Bordeaux Sciences University since 1993. He received a PhD in Mechanics in 1992. He has been a mechanical engineer (with a degree from ENSAM) since 1989. His research interests are centred on knowledge integration and the resolution of constraint solving problems.

Embodiment design of complex systems

Research context

Before the detail design phase, the embodiment design phase leads to the definition of a design configuration where the main dimensions and components are chosen [START_REF] Pahl | Engineering Design: A Systematic Approach[END_REF]. This design phase remains challenging in the context of industrial design departments. Indeed, at a stage where the knowledge is uncertain, most of the existing computer-aided tools are not suitable because they are based on models requiring the complete geometrical definition of the product. Designers do not have tools to support them in making decisions. The decisions regarding the choice of concepts are based on the assessment of performances [START_REF] O'sullivan | Constraint-aided conceptual design[END_REF]. Therefore, many a priori choices are performed according to the expertise of the designer or of the company; these choices hide a great part of the potential solution space.

Embodiment design problems can be naturally expressed as mixed Constraint Satisfaction Problems (CSPs) [START_REF] Scaravetti | Formulation préalable d'un problème de conception, pour l'aide à la décision en conception préliminaire[END_REF][START_REF] Thornton | The use of constraint-based design knowledge to improve the search for feasible designs[END_REF], which cannot be solved using classical mechanical simulation tools. In order to overcome this difficulty, embodiment design problems are tackled using several solving strategies.

An overview of these solving strategies is presented by [START_REF] Antonsson | Formal Engineering Design Synthesis[END_REF]. The existing tools are mainly concerned with genetic algorithm, evolutionary programming, agent-based systems and ensemblist methods [START_REF] Chenouard | Solving an air conditioning system problem in an embodiment design context using constraint satisfaction techniques[END_REF]. These tools are developed for application relating to structural configuration in mechanics, microsystem and robotic synthesis, and chemical processes [START_REF] Hugget | Global optimization of a dryer by using neural networks and genetic algorithms[END_REF].

Application: airconditioning system design

This work is carried out in cooperation with Dassault Aviation, which is developing decision support tools for preliminary design.

Airconditioning systems for civil aircraft are made of several complex components which interact to regulate the air temperature and pressure in the cabin [START_REF] Pérez-Grande | Optimization of a commercial aircraft environmental control system[END_REF]. The system considered in this paper (see Figure 1) is a Joule-Brayton cycle-based airconditioning system (bootstrap). It is mainly made up of two cross-flow plate-type heat exchangers (main and precooling exchangers), one turbine and one compressor coupled together. Nozzles, valves and pipes are used to manage the air flow between these elements. The air flowing through the cabin comes from the main compressor of one of the turbo reactors of the aircraft. This air flow (main air flow) goes through a precooling heat exchanger, a turbo machine (compressor, turbine, coupling shaft) and the main heat exchanger. Heat extracted from the main air flow is transferred to a ram air flow taken from the frontal surface of the aircraft. Due to the aircraft velocity, the ram air flows through the precooling and main heat exchangers. The temperature of the main air flow is regulated by taking cool air from the outlet of the precooling heat exchanger (secondary air flow) and mixing it with cold air flow from the outlet of the turbine. The secondary air flow regulation is performed by a valve.

The two circuits (main air and ram air) cross each other in alternate layers inside the exchangers. Exchange surfaces are made of stacked fins. For each air circuit, the fins have to be chosen among 48 different standard shapes [START_REF] Kays | Compact Heat Exchangers[END_REF]. Since there are two exchangers, and two air circuits per exchanger, there are 5.3 million (48 4 ) possible alternatives according to the choice of heat-transfer surfaces types.

Thanks to the discrete nature of some Design Variables (DVs) and to the discernment precision of the others, the number of solution principles of the design problem can be assessed. For example, the dimensions of the inlet sections of the two heat exchangers along two axes belong to continuous domains. However, they are related to discernment precision (minimum distance between two values of the variables regarded as different values). The domains of these dimensions range from one 0.01 m to 0.5 m. Their discernment precision has been defined by designers at 0.01 m. Thus, the number of combinations of distinct domains for these variables is equal to 2500 (50²).

The size of the global bootstrap design problem has been assessed at approximately 1.04•10 15 design configurations when considering all of the possible value combinations of the DVs. However, not every design configuration corresponds to a design solution. Not every design configuration fits the design requirements and some design configurations do not work for some system environment conditions.

Moreover, the air-conditioning system has to supply air at constant temperature and pressure to the aircraft cabin. Therefore, the design of the bootstrap must take into account different flight phases corresponding to different altitudes and the relative speed of the aircraft. The temperature, pressure and relative speed of the air strongly vary between these flight points.

Embodiment design difficulties and proposal

Due to the complex interactions of the components, air-conditioning designers have low support to guide them in the process of determining the more efficient fin types inside the heat exchangers. Manual (as opposed to digital) solving processes require knowing the values of the DVs. Therefore, bootstrap designers usually give a priori values for the heat exchanger efficiencies, which is equivalent to considering one configuration of a heat exchanger without considering the variability of the design problem.

Designers frequently develop spreadsheets dedicated to specific issues, but few tools are available to assist them during this design phase, especially because the product behaviour modelling has to take into account complex combinations of physical phenomena. Moreover, design solution validation is performed as soon as all of the architectural choices are made. It is usually processed by developing a digital simulation code of an air-conditioning system and by testing and comparing the performances of a small number of configurations. Finally, the dimensioning of the system is performed for the most critical life cycle stage, the prevailing situation. Using a trial and error mode, designers converge towards a solution which is not necessarily the most powerful one, or which does not necessarily correspond to the best trade-off between the various flight phases.

The approach proposed in this paper leads to the identification of the best design compromises between various design objectives, whatever the life cycle stage. It consists of:

• the expression of the design problem as a set of constraints

• the exploration of the whole design space using Genetic Algorithms (GAs)

• the assessment of their performances using criteria relating to the design objectives

• the reduction of the solution space using the Pareto front: it allows the identification of solutions which are the best compromise in simultaneously satisfying the design objectives, without favouring one objective.

Until the end of the embodiment design phase, no potential solutions have been dismissed by any choice of the designer. The results of the approach proposed in this paper allow decision support at this stage of the design process.

2 Design problem modelling

Constraint modelling

In order to avoid the usual trial and error iterative design process and to not dismiss a solution because of a priori choices, we propose to simultaneously take all the constraints into consideration, without any hierarchy or preliminary choices, by modelling the problem as a set of constraints.

The relations linking the variables (equalities, inequalities, logical rules) and their domains of evaluation are the constraints of the design problem. Variables can be real numbers, integers or enumerated values. Parameter values of standard components can be described using discrete variables. Domain use is well adapted to the uncertain knowledge inherent to the embodiment design phase.

The variables which define the product architecture to be designed (dimension, standard component characteristics, materials, number of elements, etc.) are the DVs. The assessment of a solution is performed by using the criteria relating to the design objectives (Scaravetti et al., 2006b): the resolution allows the evaluation of various working structures meeting the working conditions of the system. These solutions may be numerous and may thus need to be classified. In order to objectify the choices among these solutions, criteria which express the performance of each design configuration are used.

Consequently, a design problem is described by a set of DVs, criteria and constraints. A solution corresponds to a set of values -for all DVs and criteria -which meets all the stated constraints. These values define a particular design configuration.

Modelling of the design problem of the air-conditioning system

A methodology has been proposed to identify the structuring characteristics of the embodiment design problem, which must be translated into constraints (Scaravetti et al., 2005a). The knowledge base of the air-conditioning system consists of:

• twenty-three thermodynamic state variables (pressures, mass flow rates, temperatures, etc.)

• fourteen geometric and structural variables (lengths, surface types, pass number, etc.)

• eight criteria (outputs, efficiencies, etc.) which define the performance or which are used to qualify the quality of a design configuration

• twenty-four auxiliary variables which are defined as functions which correlate their definitions with the 45 preceding variables.

The global model is constituted by 69 constraints. Constraints relating to physics have been derived from energy, momentum or mass flow conservation laws. Due to the complexity of the physical models, which indicates strongly coupled physical phenomena, model reduction methods have been applied (Scaravetti et al., 2006b). The model of the design problem also includes the functional performance specifications and the description of standard components. The aircraft manufacturer expresses technical skills and manufacturing rules. The DVs are:

• the type of fin (integer value ranging from 1 to 48)

• the dimensions of each exchanger (from 0.1 m to 0.5 m, with a discernment precision of 0.01 m)

• the diameter of the bypass pipes.

The performance criteria are:

• the mass of the heat exchangers

• the total volume of the two exchangers

• the drag force induced on the aircraft by the air-conditioning system, which can be negative when the system is providing thrust.

The system has to deliver air at constant temperature and pressure to the cabin whatever the flight phase (ground, lift, economic cruise, economic cruise for long haul flights, descent, etc.). The four main flight phases (see Table 1) have been included in the model of the design problem and investigated in this paper.

Generation of design configurations using a genetic algorithm

A first investigation of the design search space was carried out using a CSP solver (Scaravetti et al., 2006a;[START_REF] Sébastian | The embodiment design constraint satisfaction problem of the bootstrap facing interval analysis and genetic algorithm based decision support tools[END_REF]. Only one flight point was investigated at this stage. More to the point, the bootstrap design problem was solved by considering only six kinds of fins inside the heat exchangers, whereas in this paper, 48 different types of fins and four different flight points have been considered. In this paper, several different life cycle situations (flight points) are taken into account, leading to a much more complex design space. Some solutions appear to be effective regarding several life cycle situations, whereas some others are only effective for one life cycle situation. Our approach has been extended by discriminating solutions according to DV values, in order to highlight relevant combinations of DVs. The most relevant solutions belong to the Pareto front and are effective for several flight points.

In the following paragraphs, the computing times of every exploration job have been limited to one hour of Central Processing Unit (CPU) time, in order to facilitate the comparison of the results. A GA-based design code has been developed to improve the solving performance of the bootstrap design problem process. The GA heuristic is a metaheuristic with a large scope of interest and applications [START_REF] Diveux | Horizontal axis wind turbine systems: optimization using genetic algorithms[END_REF][START_REF] Hugget | Global optimization of a dryer by using neural networks and genetic algorithms[END_REF]. It is based on the mutation, crossover and selection of individuals from an evolving population. Evolving generations tend to improve an optimisation criterion (the fitness function) by mixing individual genes and generating random individuals to search the complete exploration space of solutions. This heuristic method is involved in many different types of mechanical design tools. However, in industrial conditions, it is mainly used as an optimisation tool for the management of several simulation codes. Simulation codes simulate the behaviour of different parts of the mechanical system, whereas the GA-based optimisation tool supervises the system architecture and the design performance criteria.

The design code proposed in this paper is used to solve a design CSP, rather than optimising a design problem. This means that the design code conducts a search for sets of solutions rather than optimised solutions. This is performed using an optimisation criterion guaranteeing the satisfaction of every constraint of the design problem. Every constraint 'C i ' of the problem is related to a value 'SAT(C i )': if C i is satisfied, SAT(C i ) = 1, else SAT(C i ) = 0. The fitness function optimised by the GA is described in Equation ( 1):

( )

1 1 . 2 n i i SAT C SAT = + = ∏ (1)
The solutions considered in the following paragraphs satisfy every constraint of the problem (SAT = 1). The population size has been fixed to 100 individuals (candidate solutions) and the crossover probability is 0.5.

Due to the extreme sensitivity of the system functioning to the DV values (chiefly the selection of the heat exchanger fins), the mutation probability of the GA proves to be a key parameter of the solving process. A pure random search-based algorithm leads to relevant results (the design space is almost completely explored) but the computation times are high [START_REF] Scaravetti | Generation of design spaces and trade off between feasible embodiments: the bootstrap design problem[END_REF]. As a consequence, we take into consideration several values of the mutation probability, ranging from low values to very high values. In this paper, three different mutation probabilities are considered. The variable Mp equals 0.1 (Mp1), 0.3 (Mp3) or 0.6 (Mp6). Figure 2 highlights solutions minimising the performance criteria for each Mp. This selection was obtained by computing the set of nondominated solutions according to a Pareto method (see Section 4.1).

Figure 2 displays all the nondominated sets of solution spaces obtained for the three values of the mutation probability. Every solution is represented according to three performance criteria: the global mass and the volume of the heat exchangers, and the drag force induced. Figure 3 shows the same three sets on a 3D space. We note that the reading of the projections is easier. Each mutation probability leads to a different solution space, which performances are quite different. In the continuation of the paper, only the value Mp = 0.3 is considered in order to limit the size of the data exploitation and to better show the use of visualisation of relevant DV values.

Design space exploration

Once the investigation of the design configurations has been performed, we aim to support the decision of the designers by guiding the exploitation of the results. This means:

• guiding the choice of DVs among the variables of the problem by finding the relevant DVs or the relevant combinations of variables

• supporting decisions by reducing the solution space and finding suitable solutions.

Solution space reduction

The types of fins and the arrangement of the stacked plates in the heat exchangers have a decisive influence on the mass and the volume of the system. They also affect the efficiencies of the exchangers and therefore the air-conditioning efficiency of the system. Moreover, some of the air feeding the conditioning systems is scooped off the surface of the aircraft using a scoop. This air bleeding induces a drag force at the surface of the aircraft, which is detrimental to the global performance of the aircraft. This drag force has to be minimised. Therefore, a compromise has to be found between these antagonistic requirements.

The solution classification has been investigated with an objective function (Scaravetti et al., 2005a), but a weight factor-based method raises the problem of the value assignment of the weight factors.

Every Pareto solution is nondominated [START_REF] Goldberg | Genetic Algorithms in Search, Optimization and Machine Learning[END_REF] and the front definition aims at minimising the performance criteria values. The selected solutions correspond to a trade-off between disparate and conflicting design performances [START_REF] Hamdi | Exploration in the preliminary mechanical design of trade-offs between automotive architecture constraints and aggregate noise performances[END_REF]. A solution is nondominated if no other solution relates to the best values for all criteria (mass, volume and drag induced in our case).

The reduction of the solution space is quite efficient. Table 2 displays the sizes of the solution spaces and Pareto sets. This selection process drops the number of solutions from 2634 solutions to 35 solutions (Mp = 0.3). Less than 1% of the complete solution space (considering all Mp's) belongs to the front (nondominated solutions). Figures 4 to 6 display the solutions in the performance space and locate the Pareto solutions matching the performance criteria pair by pair. A solution minimising two performance criteria may be detrimental to the third performance criterion. For instance, the solution corresponding to a mass of 13.8 kg is the Pareto solution according to the masses and volumes of the heat exchanger, but is detrimental to the drag force. The Pareto set is located at the frontier of the solution space when displaying the solutions according to mass versus drag and mass versus volume (Figures 4 and5). The reduction of the solution space and its visualisation within the performance space facilitates the choice between feasible design configurations.

Design space exploration in embodiment design

Design space exploration and decision support

Despite the reduction of the solution space, design decisions remain difficult. To overcome this difficulty, we propose to supply designers with an overview of the design space: the solutions are displayed according to combined DVs to assist designers in making choices among them.

Figures 7 and8 display the solutions (for Mp = 0.3) in the design space, according to the fin types of the two heat exchangers (main and precooling exchanger). Figure 7 concerns the main air circuit and Figure 8, the ram air circuit. We can observe in Figures 7 and8 that there are few relevant fin combinations. In spite of 2634 solutions, the design space is not much filled. Thus, several solutions use the same fin combinations.

Designers can easily identify the relevant combinations for each circuit and the fin types which do not match any solution. For example, type 7 is relevant for the precooling exchanger on the ram air circuit as many embodiment design solutions use this surface type; many of them are nondominated solutions (see Figure 8).

Moreover, the identification of efficient surfaces is facilitated by highlighting the Pareto solutions and the solutions compatible with several life cycle stages (called 'X-lcs' in Figures 7 to 9). The Pareto solutions are the design solutions managing the best compromise between the requirements (minimising the drag, mass and volume of exchangers). These solutions are shown in Figures 7 to 9 to put emphasis on the optimal associations of fin types. Even if Pareto solutions and solutions functioning in many life cycle stages do not match, we can see that they use the same fin combinations; consequently, they differ in the values of other DVs. Some fin types are well adapted to the definition of design solutions, but they are different for each circuit:

• For the main exchanger, fin type 41 is the most appropriate for the main air circuit and fin type 45 appears to be well adapted to the ram air circuit.

• For the precooling exchanger, fin types 12 and 42 may be used in the main air circuit, whereas fin type 7 is more adapted to the ram air circuit.

The intersection of these configurations of fins jointly belongs to the Pareto set and to the X-lcs set; for example, fin type 41 (main exchanger) is related to type 12 (precooling exchanger) for the main air circuit (see Figure 9). Finally, some types of fins do not match any design solution and, therefore, seem to be irrelevant for the bootstrap application. For instance, whatever the air circuit being considered, types 17 to 19 are never involved in the functioning of the main heat exchanger.

These observations are confirmed by Figure 9, which superposes, on the solutions selected for Mp = 0.3, the Pareto set obtained for all the solutions (all Mp's, i.e., 5636 solutions). The Pareto set for Mp = 0.3 matches the Pareto set for every value of Mp (all Mp's).

The preceding analysis cannot guide designers to the optimal solution of the bootstrap design problem. However, this investigation supports designer decisions by suggesting relevant guidelines and avoiding some irrelevant choices. These visualisations show that few fin types are well adapted to one particular air circuit and one particular exchanger. In particular, it supports designers in the process of choosing the fins. This choice highly influences the system functioning and is difficult to know a priori. Besides, we can see that the relevant types are not the same ones for the two circuits, which are functioning under different conditions.

Finally, a small number of DV configurations meets the whole of the design constraints for the various flight phases. This consideration corroborates the difficulty of finding design solutions suitable for the complete bootstrap life cycle; this difficulty appears to be the bottleneck of the bootstrap design process.

Conclusions

In this paper, the design of an aircraft air-conditioning system has been discussed. This air-conditioning system is based on a Joule-Brayton cycle, including a turbo engine and two heat exchangers. Designers are faced with the complexity and the coupling of physical phenomena inside this system. More to the point, the system must run during four flight phases corresponding to different pressures and temperatures of the air outside of the aircraft. Due to the complexity and the large size of the design exploration space, classical design processes are based on the investigation of a small number of design configurations.

In this paper, a wide design space (1.04•10 15 design configurations) has been explored without considering a priori choices. This digital approach is based on GAs. Based on the selection, mutation and crossing of design solution principles, the exploration is performed by improving the satisfaction of the constraints of the design problem. The design problem is regarded as a CSP and a design solution must satisfy every constraint of the design problem. This approach has proved to be influenced by some genetic parameters, mainly the mutation probability. Due to the coupling of the physical phenomena inside the air-conditioning system, the solving of the CSP appears to converge towards numerous solutions for high values of the mutation probability (30% to 60%). Despite the fact that the computing process converges towards 5636 feasible embodiment design solutions (1 hour of calculation), a few configurations proved to be adapted to the complete design problem; namely, a design problem facing several life cycle situations (several flight points).

Such an approach may be relevant to designing complex industrial systems or when the life cycle of the system is faced with varying situations. It supports designers in the process of determining feasible embodiments: designers can make decisions based on the identification of relevant or irrelevant values of the DVs. These decisions are also based on the management of the compromise between the design requirements. However, it has been observed that optimal solutions according to one particular life cycle situation may be unrelated to optimal solutions according to several life cycle situations.

Our perspectives concern the development of new operators inside GAs suitable for improving the robustness of the design solutions and for improving the performance of the solver with regard to the CSP paradigm. The CSP paradigm aims at separating knowledge (design constraints) and programming (solving methods).

Figure 1

 1 Figure 1 Air conditioning system components and environment

Figure 2 Figure 3

 23 Figure 2 Pareto sets of solution spaces obtained for each mutation probability, displayed among performance criteria pair by pair

Figure 4 Figure 5 Figure 6

 456 Figure 4 Performance space (volume versus drag force) and Pareto set

Figure 7

 7 Figure 7 Design space and visualisation of relevant designer variable values (surface type) for main air circuit (see online version for colours)

Figure 8 Figure 9

 89 Figure 8 Design space and visualisation of relevant design variable values (surface type) for ram air circuit (see online version for colours)

Table 1

 1 Flight phase characteristics

	Aircraft speed	Altitude	Atm. pressure	Atm. temperature
	Mach number	M	Pa (10 4 )	K
	0.8	11 000	2.272	216.7
	0.5	3000	7.028	268.7
	0.4	2000	7.967	275.2
	0.6	5000	5.418	255.7

Table 2

 2 Number of solutions according to the mutation probability

		Mp 1	Mp 3	Mp 6	All Mp's
	Nb solutions	2325	2634	677	5636
	Nb sol. Pareto set	29	35	28	37
		1.25%	1.33%	4 .13%	0.66%