Mehdi El

Jérôme Pailhes
email: j.pailhes@i2m.u-bordeaux1.fr

Nicolas Perry
email: n.perry@i2m.u-bordeaux1.fr

Comparison of different Multiple-criteria decision analysis methods in the context of conceptual design: application to the development of a solar collector structure

Keywords: Multi-criteria decision aid methods, Selection methods, Aggregative methods, Conceptual design, Consistency

. In this work, an analysis of six MCDA methods (weighed sum, weighted product, Kim & Lin, compromise programming, TOPSIS, and ELECTRE I) was conducted. Our analysis was performed via an industrial case of solar collector structure development. The objective is to define the most appropriate MCDA methods in term of three criteria: (i) the consistency of the results, (ii) the ease of understanding and, (iii) the adaptation of the decision type. The results show that TOPSIS is the most consistent MCDA method in our case.

1-Introduction

Decision making is an inherent task in the product development process. It can be broken into (i) conceptual design decisions, when decision makers have to choose between several design concepts, and (ii) embodiment design decisions, when designers have to optimize design variables for the selected concepts. In this paper, we treat only the problem of conceptual design decisions. Decision support was first proposed by Nobel Prize winner Herbert Simon in the nineteen sixties [START_REF] Simon | The new science of management decision[END_REF]. In a broad sense, decision making is conducted in four steps: (i) identifying the problem (ii) generating design alternatives, (iii) evaluating design alternatives via evaluation schemes, and (iv) selecting the best alternative. The research community in decision support methods usually recognize that the most critical step in decision making process is how to choose among a given number of design alternatives (step (iv)). The ability of decision makers to make the best choice is strongly conditioned by two factors: (i) having a clear definition of design objectives and requirements, and (ii) being able to evaluate or predict the performance of the proposed alternatives. However, uncertainties and vagueness are inherent in engineering design and they characterize both design objectives and alternatives evaluation schemes. This is particularly true in the first steps of conceptual design. The decisions taken at this design phase have the greatest influence on overall product life cycle cost. In order to minimize risk and reduce the cost of regret in later processes, companies often opt for a least commitment and late design decisions during product development process. This usually implies the development of several design concepts in parallel to finally settle down the optimal solution. Indeed this strategy is effective to minimize risk. It is very important to adopt appropriate methodologies and theories to structure and ease decision making process. This can significantly reduce product development lead-time as well as the amount of human and material resources involved in the development process. As shown in Fig. 1, multi-criteria decision making process can be decomposed into three principal steps: (i) an observation step, when data are collected about each alternative, (ii) an interpretation step, when decision makers express their preferences for each design criterion on the basis on data collected in the observation step, and (iii) a results analysis step, when decision makers combine interpretation step outcomes for different design criteria in order to determine the best alternative(s). Because of the great difficulty for decision makers to intuitively follow this process, especially to combine information in appropriate ways (step 3 of decision making process), MCDA methods can be used to aid in multi-criteria decision making. The preference expression mode used in step (ii) can have an important influence on decision making results [START_REF] Guitouni | Tentative guidelines to help choosing an appropriate MCDA method[END_REF]. The different expression modes used by the MCDA methods are: direct rating, pairwise comparisons, and lotteries. Each mode has its advantages and disadvantages. There are three factors to consider when choosing the preference expression mode: (i) it must be adapted to the type of information available on the design alternatives (step 1 in decision making process), (ii) decision makers must feel comfortable when using it, and (iii) the type of yielded information must be adapted to the decision situation. Various MCDA methods are proposed and adopted by researchers and engineers to support design decision-making in engineering design. Choosing the most suitable MCDA method to compare multiple alternatives is a critical issue because these methods may yield different results for the same problem [START_REF] Zanakis | Multi-attribute decision making: A simulation comparison of select methods[END_REF][START_REF] Guitouni | Tentative guidelines to help choosing an appropriate MCDA method[END_REF][START_REF] Triantaphyllou | Development and evaluation of five fuzzy multiattribute decision-making methods[END_REF]. There are two families of MCDA methods: (i) multi-criteria selection methods, where interpretation step results are taken into account simultaneously to compare alternatives, and (ii) aggregative methods, where interpretation step results are aggregated into a single variable called performance index (PI) that reflect the performance of the design alternatives. In the next section, the MCDA methods analyzed in our study are presented. After that, the industrial case study is presented and examined. Then, in Section 4, the objectives of the study and the proposed approach are presented. Finally, the results and the interpretations are presented and recommendations are given at the conclusion.

2-Presentation of MCDA methods

In this work, five aggregative methods (weighed sum, weighted product, Kim & Lin, compromise programming, and TOPSIS) and one selection method (ELECTRE I) were considered. The weighted sum is the most widely used aggregation method. The weights assignment reflects the proportional importance of the different aggregated variables. However, the major disadvantage of this method is that it doesn't satisfy the principle of annihilation, which is generally not acceptable in design decision problems. The performance index is expressed by:

(1)

With

The weighted product (WP) provides a conservative aggregative method that satisfies the principle of annihilation. However, the meaning of weights is less intuitive than weighted sum. They reflect the exponential relative importance, and not proportional between variables. The performance index is expressed by:

(2)

With

Kim & Lin [START_REF] Kim | Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions[END_REF] is the most conservative aggregative method. It also satisfies the principle of annihilation. However, the biggest disadvantage is that it does not allow any kind of compensation. Its use is thus very limited in engineering design. The performance index is expressed by:

(3)

TOPSIS (Technique for Order Preference by Similarity to

Ideal Solution) is a compensatory method that was developed by Hwang (Hwang and Yoon [START_REF] Chen | Fuzzy multiple attribute decision making: methods and applications[END_REF]) and widely applied by other researchers (Deng et al. [START_REF] Deng | Inter-company comparison using modified TOPSIS with objective weights[END_REF]; Tsaur et al. [START_REF] Tsaur | The evaluation of airline service quality by fuzzy MCDM[END_REF]; Opricovic and Tzeng [START_REF] Opricovic | Multicriteria planning of post-earthquake sustainable reconstruction[END_REF]; Cheng et al. [START_REF] Cheng | An integrated multi-criteria decision analysis and inexact mixed integer linear programming approach for solid waste management[END_REF]; Montanari [START_REF] Montanari | Environmental efficiency analysis for fresnel thermo-power plants[END_REF]; Tong et al. [START_REF] Tong | Dynamic multiple responses by ideal solution analysis[END_REF]; Tzeng et al. [START_REF] Tzeng | Multi-criteria analysis of alternative fuel buses for public transportation[END_REF], etc.). In conceptual design, TOPSIS has been used in a very limited way [START_REF] Lin | Using AHP and TOPSIS approaches in customer-driven product design process[END_REF]. The basic principle of TOPSIS is that the best alternative should have the shortest distance from the ideal solution and the farthest distance from the negative-ideal solution. The TOPSIS procedure consists of the following steps: (i) calculate the normalized decision matrix, (ii) calculate the weighted normalized decision matrix, (iii) determine the ideal and negative-ideal solution, (iv) calculate the separation measures from the ideal and negative-ideal solution (respectively and ) by using the n-dimensional Euclidean distance, (v) calculate the relative closeness to the ideal solution, and (iii) calculate the performance index of each solution by the following formula:

(4)
The separation measures from the ideal and negative-ideal solution are calculated by the following formula: -3-

With : The minimum interpretation value for criterion j : The maximum interpretation value for criterion j

Compromise Programming (CP) was developed by Yu [START_REF] Zeleney | Compromise programming[END_REF] and Zeleny [START_REF] Yu | Multiple criteria decision-making: concepts, techniques and extensions[END_REF]. The basic principle of CP is that the best alternative should have the shortest distance from the ideal solution. The performance index is expressed by: ( 7)

With

Where and have the same meaning as in TOPSIS. The parameter p defines the desired type of distance. In this study, the CP method has been applied twice by using the parameter p equal to 1 and 2. The ELECTRE I is an outranking method that is widely used by decision makers in many fields. It was first developed by Benayoun et al. [START_REF] Zeleney | Compromise programming[END_REF]. The ELECTRE I procedure consists of the following steps: (i) define a concordance index for each pair of alternatives, it represents the sum of weights of attributes for which alternative A is better than B, (ii) define a discordance index for each pair of alternatives. It denotes the absolute difference of this pair of attributes divided by the maximum difference over all pairs, (iii) establish threshold values for the two indices, (iv) generate the set of alternatives that is not outranked by any other alternative. In order to obtain an overall ranking of the alternatives, many threshold values are used in our study and each alternative is ranked according to how many times it belong to the set of non-outranked alternatives.

3-Presentation of industrial case

A previous work has been performed within a design department developing a collector for a solar thermal power plant with Fresnel mirrors. The main function of the solar collector is to concentrate and redirect solar radiation into the absorber tubes to heat up the transfer medium in absorber tubes. The recovered heat is then used to generate high pressure steam which drives a turbine to produce electricity. The solar collector is composed of a reflecting glass and a metal structure, whose function is to give and maintain reflecting glass shape (Fig. 2). A support solution is performed between the reflecting surface and the metal structure to ensure the connection between the two. Solar collector is driven by a rotation movement in order to pursue the movement of the sun. In order to maintain a good thermal efficiency of the plant, a high reflection performance is required from the collector. This implies that the elastic deformation of the structure must remain as low as possible. A high level of precision must also be considered when manufacturing the collector.

The technology of solar thermal power plant with Fresnel mirrors is relatively recent and thus the historical background is limited. This induces a real difficulty when evaluating design alternatives. In addition, in order to reduce logistics cost, the concept of a movable factory has been adopted by the company. This choice induces directly and indirectly a set of particular constraints on the solar collector design. The interaction product/manufacturing process in this context is not well understood by development engineers. The implementation site also has a strong influence on collector design: climatic data, market data, etc. The difficulty is that the choice of implementation sites is not made and there is a variety of possible implementation sites where the conditions may vary significantly. In this study, we treat only the preliminary design of the metal structure (see Fig. 2). Three main solutions have been proposed for the metal structure: truss structure (S1), sandwich structure (S2) and tube structure (S3). Compared to the tube and truss structure, the sandwich structure is the most mature solution. Its behavior is well characterized by the company due to the many prototypes that have been made. By contrast, truss and tube structures are new solutions for the company and there are much less mature than the sandwich solution. As the product development progresses, the behavior models used become more and more accurate especially due to prototypes testing. Rectifications are introduced to the three solutions throughout the development process. The criteria taken into account in this study are: elastic deflection (C1), angular elastic deformation (C2), raw material cost (C3), durability (C4), development time (C5), ease of industrialization (C6), adaptation to the mirror mounting solution (C7), adaptation to the guidance solution (C8). The development process is composed of three major milestones. At each milestone, a set of tasks is planned in order to investigate the performance of each concept. These actions generally consist in: numerical analysis and prototyping/tests. At the end of each milestone, an assessment grid is established to synthesize investigation works of designers.

4-Objectives and approach

In the present study, various MCDA methods were tested and analyzed based on the conceptual design phase of a solar collector of a thermodynamic solar power plant. The objective is to evaluate the appropriateness of MCDA methods in our context.

The first step of our study is to evaluate weights for the different criteria. It is difficult and risky to directly assign weights to criteria because they are from different natures. In our study, pairwise comparison was used because it is more appropriate to this case. Using the semantic scale in Table .1, the judgment matrix was constructed. The results are given in Table . 2. The matrix normalization method [START_REF] Lebel | Prise de décision multi critères[END_REF] was then used to calculate criteria weights from the judgment matrix. In order to limit the inconsistencies that occur when performing pairwise comparisons, the Consistency Ratio (CR) proposed by Saaty [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF] was used as a guidance to check for consistency. As recommended by Saaty [START_REF] Saaty | A scaling method for priorities in hierarchical structures[END_REF], a value of 0,1 was used as threshold for CR.

After the evaluation of criteria weights, the next step is the evaluation of alternatives against each criterion (the interpretation step). A direct rating was used to evaluate each alternative against each criterion. This preference expression mode was used because it is simple and it yields a cardinal rating. Pairwise comparison is more suitable to deal with qualitative criteria and uncertainty in data. However, it yields ordinal evaluation of alternatives which is not suitable to use with aggregative methods. In order to help decision makers to assign interpretation values, the semantic scale in Table . 3 was used. The results of the interpretation step are presented in Fig. 2.

Based on the interpretation results (Table . 2) and criteria weights obtained by pairwise comparisons, different MCDA methods were used to rank alternatives. From this survey, MCDA methods were evaluated in terms of three criteria: (i) the consistency of results, (ii) the ease of understanding and, (iii) the adaptation the decision type.

The consistency of results

The consistency of results is very important criterion of MCDA methods. It measures the closeness of the result given by the MCDA method to what really corresponds to decision makers' preferences. In our study, this criterion was evaluated by comparing the ranking results given by MCDA methods to the intuitive ranking addressed by decision makers

The ease of understanding

This aspect measures the effort and the time required for the decision makers to understand the MCDA method (assumptions, tradeoffs, and procedures of method). The less time and effort required to understanding the method, the better decision makers will use the method and the more effective the method will be [START_REF] Mahmoud | Comparison of different multicriteria evaluation methods for the Red Bluff diversion dam[END_REF]. Experienced decision makers generally prefer simple, more transparent methods [START_REF] Hobbs | Does choice of multicriteria method matter? An experiment in water resource planning[END_REF]. The MCDA methods were tested by three persons participating in the development project. The methods were then ranked on a scale of 1 to 5 (1 if the method is very difficult to understand and 5 if the method is very easy to understand).

Adaptation to the decision type

It is important that the type of decision given by the MCDA method corresponds to the type of decision expected by decision makers. For example, if the decision makers want to get a cardinal ranking of alternatives, then an outranking method is not appropriate.

5-Results and interpretation

As shown in Fig. 4, clear discrepancies are observed in the ranking of alternatives between compensatory and non compensatory aggregation methods. All the compensatory aggregation methods (weighted sum, TOPSIS, compromise programming with p=1, 2) yield the following ranking: A1A3A4. On the other hand, the two non-compensatory strategies (Kim & Lin and weighed product) yield the following ranking: A3A1A4. ELECTRE I yields the same ranking as compensatory aggregative methods.

Apart from TOPSIS and Kim & Lin aggregation methods, the performance indexes obtained by the other aggregation methods are very close together. It is thus difficult and risky to draw a conclusion based on these results since a small error in interpretation values could reverse the ranking. Kim & Lin aggregation is not suitable for our case because it does not provide any kind of compensation.

According to the interpretation results (Fig. 3), the truss alternative highly satisfies the criterion of raw material cost which is the most important criterion. In contrast, because of its low maturity, it collects low interpretation values (Fig. 3) for other criteria, in particular for the criterion 'ease of industrialization' which also has a very strong importance in our case. Therefore truss alternative obtains a low performance index by conservative aggregative methods. But the important fact that should be considered here is that the truss structure is a novel and immature alternative compared with the others. The potential of improvement is thus high. That is why this alternative was particularly attractive for decision makers. This statement was intuitively felt by decision makers and it was highlighted strongly by TOPSIS and little by the other compensatory aggregative methods. For the other two solutions, the ranking yielded by compensatory aggregative methods is also coherent with the preferences of decision makers. From this observation we can estimate the consistency of evaluated MCDA methods in our case.

It can be remarked that TOPSIS intensifies the relative importance between alternatives compared to other compensatory aggregative methods. This intensification effect can be attributed to two factors: (i) the two reference points (ideal and negative-ideal solutions) are obtained according to the alternatives tested (relation ( 5) and ( 6)), and (ii) this method uses an Euclidian distance to calculate the closeness of alternatives to the reference points.

For the criterion of 'ease of understanding', WS and Kim & Lin were the easiest methods to understand, followed by WP, CP, TOPSIS and finally ELECTRE I which was particularly difficult to understand by decision makers who participate in the survey. Table 4 gives more details about the evaluation of MCDA methods against this criterion.

For the criterion of 'adaptation of the decision type', the most adapted decision type in our case is the cardinal ranking. Ordinal ranking (yielded by ELECTRE I) is not suitable because decision makers need to know how much better an alternative is over the others in order to be able to make a decision. Aside from ELECTRE I, all the other methods respect this criterion since they yield a cardinal ranking.

Another disadvantage of the ELECTRE I method is that it evaluates a criterion even if it has a weight equal to zero. This can mislead the decision maker, since the discordance calculation does not consider the value of the weights. The use of ELECTRE I is clearly not suitable in our case.

6-Conclusion

A comparative study of different MCDA methods was performed in this paper based on an industrial case study of a solar structure development. The study focused on the two main steps of decision making: the interpretation step and result analysis step (Fig. 1). Even if pairwise comparison seems to be more suitable to deal with qualitative criteria, the direct rating was privileged in the interpretation step because it allows having cardinal interpretation values which are much more suitable to use with aggregative methods. Many MCDA methods were then used to rank alternatives. It was found that the ranking differs between conservative and nonconservative methods. Generally it was found that the results yielded by compensatory aggregative methods are more consistent.

TOPSIS was the most consistent with decision makers' preferences because the results yielded by this method respect very well the decision makers' preferences. However, in the final design phases, it could be more appropriate to use non compensatory aggregation methods because the potential of improvement is low and decision makers cannot afford to take risks.

A further investigation should be done in a future works in order to evaluate more efficiently the performance of the MCDA methods. For example, a sensitivity analysis could be performed in order to evaluate the robustness of these methods with respect to uncertainties in interpretation values or criteria weights. Further industrial cases of product development could also be considered.

Fig. 1 .

 1 Fig. 1. Schematization of the decision making process.

  The results are discussed by different actors participating in the development process. The development team decides then what concepts should be eliminated and what actions must be done for the next milestone. In this study, we treat the decision making problem at the first milestone.

Fig. 2 .

 2 Fig. 2. Schematization of solar collector structure.

Table 2 .

 2 Evaluation of criteria weights by pairwise comparisons

		C1	C2	C3	C4	C5	C6	C7	C8
	C1		0,14	0,13	0,20	0,20	0,13	0,17	0,50
	C2			0,20	0,50	3,00	0,20	0,33	5,00
	C3				3,00	7,00	4,00	5,00	8,00
	C4					2,00	0,25	3,00	5,00
	C5						0,17	0,50	4,00
	C6							5,00	7,00
	C7								5,00
	C8
	Final criteria weight	0,02	0,08	0,38	0,12	0,05	0,25	0,09	0,02

Table 5 .

 5 Evaluation of MCDA methods against the three criteria.

		Consistency	Ease of	Adaptation of the
			understand	decision type
	WS	Average	5	Good
	WP	Poor	4	Good
	Kim & Lin	Poor	5	Good
	CP (p=1)	Average	3	Good
	CP (p=2)	Average	3	Good
	TOPSIS	Good	2	Good
	ELECTRE I	Poor	1	Poor


Yoann Le Diagon
email: yoann.le-diagon@ensam.eun.perry@i2m.u-bordeaux1.fr

Nicolas Perry

Stéphane Pompidou
email: s.pompidou@i2m.u-bordeaux1.fr

Reidson Pereira Gouvinhas
email: reidson@ct.ufrn.br

Concept -Improve -Ingegraf Integration of end-of-life options as a design criterion in methods and tools for ecodesign

Keywords: Ecodesign, design for X, life cycle assessment, end-of-life, wind turbine

Ecodesigning a product consists (amongst other things) in assessing what its environmental impacts will be throughout its life (that is to say from its design phase to its end of life), in order to limit them. Some tools and methods exist to (eco)design a product, just like methods that assess its environmental impacts (more often, a posteriori). But it is now well accepted that these are the early design decisions that will initiate the greatest consequences on the product's end-of-life options and their impacts. Thus, the present work aims at analysing traditional design tools, so as to integrate end-of-life possibilities in the form of recommendations for the design step. This proposal will be illustrated by means of a wind turbine design.

1-Purpose of the work

Ecodesign is an alternative approach to design a product with at least the same functional level, but (amongst other things) using less materials or fewer energy in the manufacturing process, favouring the use of higher renewable resources, optimising design for limiting maintenance operations, and improving product end-of-life treatments. The goal of this paper is to show how traditional design tools can be used for ecodesigning a product. More specifically, it aims at integrating end-of-life options from the early stages of the product's design, to limit the environmental impacts due to this life phase. A product's environmental impacts can quite easily be estimated a posteriori by a lifecycle assessment (LCA), even if some materials, constituents, processes or life stages are less known than others. On the contrary, it is more difficult to assess its impacts a priori in the early design phase, when the product definition is low-detailed. However, it is well known that the early to preliminary design decisions will initiate the greatest consequences in terms of impacts due to its end-oflife options, both environmentally and economically [START_REF] Fabrycky | Life-cycle cost and economic analysis[END_REF]. Indeed, some raw materials become strategic because of their decreasing availability: more and more constituents must be easily recycled. Lastly in a legislative point of view, regulations get stricter about the environmental impacts of a product. As a consequence, integrating ecodesign approaches becomes a strategic action for companies. Besides, they improve their image for consumers who seek more and more ecofriendly products. In this framework, the present analysis aims at integrating end-of-life into previous studies that were rather focused on the implementation of the ecodesign approach, and its benefits on the development of tools for ecodesign. This paper will start with a quick state-of-the-art of ecodesign approaches. The next section presents the product development cycle. Then, end of life and recycling criteria are introduced as key characteristics to evaluate. The fifth section describes the proposal linking end-of-life options into the product development cycle. Finally, the last section applies the proposal to a wind turbine design case study, before concluding.

2-Ecodesign framework

2.1-Overview

Ecodesign approaches can be applied to any product, that is to say a good or a service. However in the present analysis, we will mainly focus on the design of mechanical systems. First of all, let us claim that assessing a product life cycle does not mean following an ecodesign approach. Even if LCA gives the most realistic environmental impacts' assessment for the designed product, this method is only practicable on a fully designed product. Indeed, the product definition (structure, material and manufacturing process) and its envis-aged life scenarios (logistic, use phase and end-of-life option) must be as complete as possible in order to get the best assessment. So, how can this tool be used at the early design stages? To fill the gap between environmental assessment and analyses led by classical design tools, we propose to enrich design methods with criteria and systematic questions. In spite of the potential benefits of the implementation of ecodesign, the method is not used yet in all design cases. On the one hand, collecting data is particularly difficult because the product lifecycle steps should be precisely known, including all stakeholders involved (i.e. raw materials suppliers, manufacturing stakeholders, transporters, consumers and recycling companies). On the other hand, some existing tools (see below) can be difficult to use and to implement for designing the product. Actually, the main barrier is the lack of know-how in the product design steps.

2.2-Ecodesign tools

Ecodesign tools have been developed to ease and guide societies, team project and designers to implement associated approaches and processes. They especially help companies in assessing the environmental impacts of their products, and in defining the main improvement axes to (re)design them. More than 150 ecodesign and communication tools already exist for the design process; strategic tools can also be added to the list [BB1]. But they often turned out not suitable for SME companies [START_REF] Gouvinhas | The Utilization of Ecodesign Practices within Brazilian SME Companies[END_REF], as they are too complicated to be useful for their daily work [START_REF] Gouvinhas | The use of design methods within production machinery companies[END_REF]. Among ecodesign tools [START_REF] Vallet | How ecodesign tools are really used. Requirements list for a context-related ecodesign tool[END_REF], two types stand out: assessment and recommendation tools.

2.2.1-Assessment tools

Both qualitative and quantitative assessment tools can be taken in consideration [START_REF] Bellini | Écoconception : état de l'art des outils disponibles[END_REF]:

 Qualitative assessment tools include streamlined life cycle assessment and complete monocriteria-LCA, matrix approaches as MET (material cycle, energy use and toxic emissions) [START_REF] Brezet | UNEP IE Cleaner Production Network[END_REF], MECO (materials, energy, chemicals, others), ERPA (environmentally responsible product assessment).  Quantitative assessment tools include all mono-criteria analyses (e.g. water footprint or carbon footprint) and multi-criteria LCA; they lead to an accurate evaluation of the product's environmental weak points.

2.2.2-Recommendation tools

In order to help design teams make decision [G1,GB1], all checklists, standards, guidelines and substances lists (e.g. REACH regulation and RoHS directive) can be used [L1]. Moreover, design for X [T1] (where X is for recycling, disassembly, environment [AZ1], reuse, remanufacturing or sustainability) are suitable in this case [START_REF] Rio | Toward proactive (eco)design process: modeling information transformations among designers activities[END_REF].

3-Formalisation of the product development cycle

In order to introduce end-of-life options from the design phase (cf. figure 1), that is to say either in classical design or ecodesign tools (depending on the available data at each step), the product development cycle has to be detailed. As defined by Pahl et al. [START_REF] Pahl | Engineering design: a systematic approach[END_REF], it can be divided into the following main steps.

3.1-The early design

The early design phase is mainly based on the customer's needs. In this way, the product's major function is translated by designers in a technical requirements list that also takes into account the constraints related to the market, a business strategy, technology, legislation, etc. (see also figure 1.a)

3.2-The preliminary design

The preliminary design usually involves the two following steps:  conceptual design that aims at defining the functional structure. Many concepts are developed to answer the needs previously listed, then compared to assess issues that could stem from every choice. A solution principle is thus selected;  architectural design that consists in assessing the previous proposal. The most suitable structure from technical and economic points of view is chosen, including components (shapes and materials), pre-sized or nonstandard constituents, etc.

Detailed design

How end-of-life can be integrated in the product design?

Preliminary design

Industrialisation ?

3.3-The detail design

In the detail design step, the selected architecture is mocked up by CAD, optimised and finally approved. A definition file containing all sized parts and assembly drawings is provided (the definitive layout).

3.4-The industrialisation

Lastly, the industrialisation phase aims at choosing the best manufacturing process, optimising the supply chain, drafting a quality plan, and eventually implementing a prototype to be tested.

4-Formalisation of the product's lifecycle and focus on its end-of-life

The product lifecycle is usually divided into the following main consecutive steps, as sketched in figure 1.b.

4.1-Raw materials production

This first phase mainly consists in the raw materials production; they can either be first extraction or recycled ones (usually called secondary-raw materials).

4.2-Manufacturing

The second step includes all manufacturing operations; that is to say, amongst others, raw materials transformation, production operations previously chosen for the industrial process, thermal and mechanical treatments, and packaging.

4.3-Transportation and logistics

This step includes all transport and logistics issues in relation with the product throughout its lifecycle. Consequently, these operations do not only concern the distribution phase (i.e. between the manufacturing and the purchasing sites) as suggested by figure 1.b, but can actually be found between all the other phases.

4.1-Use and maintenance

This phase of the lifecycle focuses on the period during which the product is used, i.e. between its purchase and its end-of-life (e.g. when it becomes a cause of failure). This step also includes all the maintenance operations needed for increasing its lifespan.

4.2-The end-of-life

When the product becomes unused, it is generally more suitable to talk about its end-of-life phase, rather than to consider it as a waste. Indeed, this last expression would imply an absolute loss of value, while several ways exist to extend the life of either the whole product (and thus, preserve part of the worth) or only some parts (components, constituents or constitutive materials).

4.2.1-Ways of treatment of an end-of-life product

The five main options to treat a life-ending (sub-)product are listed below, from the most valuable to the less profitable solution.

 First of all, the reuse option mainly consists in the recovery of the whole product. It can be reused for the function it was designed, or another one. This end-of-life treatment is clearly the best in terms of maximisation of the product value.  The second way is the remanufacturing process. It more often consists in recovering parts that will be reinserted in a new product, during a later manufacturing process. However, this option is possible if previous design choices assure an easy disassembly. Only then, parts may be modified, adapted, or simply reassembled in a second generation product; their value is preserved.  The recycling is the third possible way of treatment. It mainly consists in extracting strategic materials from the waste flow, and regenerating them in new ones, usually called secondary raw materials. They are more ecofriendly, and theoretically have the potential of other recycling loops. However, this reprocessing path requires firstly the preparation of the waste; it includes collect, depollution, cleaning, dismantling, part and/or material identification, sorting, etc. Early design options would obviously ease the product disassembly, the accessibility to the (sub-)parts and then, the materials separation. With respect to the recycling process (specific for each nature of material), performance indicators will be varied as a function of the dismantling quality. Again, early design choices thus impact the recycling ratio (e.g. ease/difficulty to reach target materials), the efficiency of the recycling process (e.g. downcycling likelihood due to a bad separability), and then, the availability of good secondary-raw materials for new manufacturing processes.  The fourth end-of-life option mainly comes down to an energy recovery by burning the product (manifestly likened to a waste).  Lastly, the worst end-of-life option consists in landfilling and burying the inert waste that couldn't be treated by one of the previous ways.

4.2.2-Environmental impacts of the end-of-life

A product inevitably impacts the environment almost continuously, from the extraction of the raw materials, and all along its use phase (energy used for its working, associated substances rejected in water or in the air, etc.). But even after, environment remains impacted by the landfilled parts (e.g. land occupation or constituent toxicity), or by the reprocessing or recycling treatments (e.g. energy consumption, supplies needed for the process, etc.). However, as previously mentioned, design choices may assure an easier disassembly of the product, a better materials separability, or a greater reintegration of the secondary-raw materials in a next manufacturing process. As a consequence, we propose to assess how the end-of-life step could be integrated as a design criterion, while controlling potential transfers of pollution.

5-The proposal

The goal of this part is to display, for each design step, (i) the tools that can be used and (ii) the questions to ask for integrating end-of-life options as design criteria, in order to provide indicators which could help the design phase. Both aim at improving the performance of the end-of-life product. But before that, it is important to differentiate, in terms of scale of treatment, the management of the production waste (e.g. unused constituents, off-cuts, rejected parts, etc.), and the end-of-life products. Moreover, there is no added value for a production waste. Thus, efforts have to be kept to minimize production wastes in an ecodesign context. Our analysis will be linked to the three main steps of the design phase.

5.1-Early design

It has been previously seen that the earlier the end-of-life is taken into account in the design process, the better the economic and environmental impacts of the associated valorisations are. However at the early design step, the product is still an idea, or at best a customer's requirement (cf. § 3.1-); in all cases, available data are extremely limited. Despite that, questions about the strategic position of the manufacturer (i.e. a company) must be taken into consideration. A first level of thought must be led about existing regulations, and legislation to come. Some regulations already limit the use of materials and chemicals, or will require a minimum recycling rate at the end-of-life. As an example, this first analysis of legal texts will force designers and manufacturers to limit the use of hazardous substances that will be difficult to treat at a later stage. The second focus for reflection is related to the company's management strategy. Usually, the company favours some end-of-life options (e.g. recycling or thermal recovery) for internal processes (i.e. production waste) and external ones (i.e. when user feels/reckons the product reaches its end-oflife). This will enable to choose the most suitable materials and constituents to fit in the more easily treatment sectors. Both questions directly influence the design possibilities. For example, if the company gives priority to recycling (and so to recycling rates), then mass and volume will be key design criteria. Giving priority to reuse, the design will have to be modular, and each sub-system will only fulfil one function. In this case, number and nature of functions, and then number of sub-parts to disassemble will be the key parameter. In order to integrate end of life criteria at this step, the work focuses on functional analysis tools: tools called (in a mundane way) octopus diagram (or functional interactor diagram) and bull chart, technical specifications, SADT (structured analysis and design technique), FAST (functional analysis system technique) and FBD (function block diagram), etc. The functional interactor diagram enables to find all external interactors to the product for all its living phases. Analysing end-of-life emphasis the functions linked with this step. Then, technical specifications allow matching functions with criteria (e.g. a recycling rate required by regulations). In the context of an eco-redesign, FAST diagram allows putting emphasis on the materials choice regarding the end-of-life (e.g. recycled or recyclable material, strategic material, chemicals and assembly methods). The BFD, which depicts flows and links between components, enable to determinate sub-systems and their imbrication. Lastly, only two end-of-life indicators can guide designers: a composition indicator (if design is monitored by legislation) and a separability/disassembility indicator which can fix objectives in terms of accessibility and ease for the associated end-of-life options.

5.2-Preliminary design

In the present part and the next one, we will only focus on the recycling option because of the lack of knowledge and available data, as already mentioned. Let us remind that the preliminary design step firstly consists in translating technical specifications in a concept, then in an architecture with pre-sized components (cf. § 3.2-). Taking into account end-of-life options from this step, requires looking at two main parameters: the assembly (in order to anticipate the components separation) and the nature of materials (that will have to be recycled). Thus, the present analysis focuses on several tools: CAD, simulation and materials choice tools, decision matrix, physical rules, etc. Firstly, design must be managed to take into consideration an easy disassembly, of parts or components, but also, with a view to recycling, of materials. Disassembling difficulty is tested by assessing means and time needed; degree of linkage standardisation and accessibility to the joint (which is important for both end-of-life and maintenance operations) are deduced. Indeed, a FMECA (failure mode, effects and criticality analysis) can highlight critical components which can easily be repairable or replaceable. The second reflexion axis relates to the assembly process because welding or riveting for example, can introduce filler material which can be a pollutant for the recycling process. Then, CAD tools and kinematic drawings enable to give a first evaluation of the separability performance. Otherwise, analysing the used materials is a requirement to improve the end-of-life treatment. Obviously, they are mainly chosen in accordance with their mechanical properties; however, it is important to evaluate the recycled ones, depending on the treatment industry. Another key point is the identification of constituents. Indeed, in order to choose the better way of treatment, materials have to be easily identifiable, particularly for high-value materials. Lastly, materials can have an incidence for technical intervention (i.e. maintenance or lubrication), so these choice criteria must be taken into consideration at the same level. A decision matrix is suitable to compare them. Previous indicators also guide designers and become more and more accurate, especially thanks to the increase of available data due to the knowledge of the product. Another indicator assesses the pollution due to the recycling process, that is to say chemicals or materials which will have a negative or hazardous impact on the process. Quantification of this indicator will depend on the recycling option.

5.3-Detailed design

One of the goals of the detailed design is to optimize shapes and materials, for example with CAD tools (cf. § 3.3-). At this stage, the involved tools are overall the same as in the previous one.

A new way to optimise end-of-life lays in searching alternative materials with (at less) the same mechanical properties that would balance environmental performances (e.g. with CES software). Another reflexion must be led about the choice between different grades of material. Indeed, some can be more or less desirable depending on the recycling process and the quality of the recycled material. In this sense, one must take a close look on surface treatments; as an example, tinplating or copper plating reduce the mechanical performances of a recycled steel [START_REF] Castro | A thermodynamic approach to the compatibility of materials combinations for recycling[END_REF]. Coatings have to be taken into consideration for the same reason. Moreover, it is essential to compare all the possible alternatives of assembly. With an eye to recycling, joints easier to disassemble are more desirable. On the contrary, if a rigid linkage is required, the main problem remains the polluting effect of one constituent on the other in the recycling process, and the degradation of the material (i.e. downcycling); a less efficient recycling way could be chosen. Lastly, other constituents' properties can be taken into account. As examples, ferromagnetic materials can be easily sorted in a waste flow, just like materials of different density floating in a fluid. Use of such materials will ease their separation and orientation in the recycling process.

In an economic point of view, it is essential to know if the ratio (by mass or volume) recyclable constituent/total waste is likely to be profitable to recycling industries [START_REF] Reuter | Opportunities and limits of recycling: A dynamic-model-based analysis[END_REF].

6-Case study: wind system

This part aims at showing how end-of-life option can be integrated in the design of a wind turbine.

6.1-Legislative framework

The present work matches with two improvement points identified by the French Ministry of Ecology, Sustainable Development and Energy in June 2012. The first one gives priority on ecodesign approaches and the research on alternative materials in place of rare or strategic ones. The second point forces designer to use ecodesign approaches not only throughout the main mechanical system lifecycle, but also to all annex components and structural parts (e.g. foundations, etc.).

6.2-Material composition and treatment performances

6.2.1-Material composition study

This part aims at studying the arrangement of a horizontalaxis wind turbine (HAWT), with a 85m high tower and a 90m in diameter rotor. Mass of components are classified in table 1.

Figure 2 details the contribution of these components to its overall cost. These data are also classified in table 2 to highlight the economic impacts of the two possible end-of-life scenarios we propose for the wind system. The first one is an assumption of what could be considered as end-of-life options; the second one is an alternative version that favours the reuse of some parts instead of their only recycling. This leads us to ask three questions. First, blades are made of glass reinforced plastic. But to date, there is no industrial solution to recycle such composites. Consequently, the end-of-life option currently led is the thermal recovery, although blades are high added value components (about 20% of overall cost of the HAWT).

Secondly, the electrical components remain difficult to recycle. Indeed, generator includes a copper-steel mix difficult to treat. As previously explained, copper is a pollutant for steel in terms of quality of the secondary raw material. Lastly, foundation is a huge buried structure. Consequently, it also has to be dismantled when the HAWT reaches its endof-life. Otherwise, a repowering could be planned with the same foundation. These three points have so to be taken into account from the design stage in order to (i) anticipate the components' end of life, and (ii) avoid a loss of the added value with a treatment which wouldn't be effective enough.

6.2.1-Treatment performances

Designers now have to focus on the loss of added value due to the choices of end-of-life treatments. Indeed, components of a wind system represent about 90% of the overall cost.

In order to highlight the loss of value in the end-of-life phase, two scenarios are proposed: Scenario 1: Only electrical components are reused; Scenario 2: Electrical components, gearbox and main shaft are reused. In both approaches, rotor blades and nacelle housing are burned; non-quoted materials are supposed to be recycled.

The table 2 compares these scenarios in terms of allocation of value. In this economic-only point of view, we have already mentioned that recycling is more attractive: it leads to a new material, when burning option is just elimination, admittedly with an energy gain, but leading to wastes difficult to make more attractive. Firstly as already mentioned, the treatment of composite parts lead to a 24% loss of the overall value (even if energy can be recovered). Secondly, since gearbox and main shaft can be reused, 20% of the system value could be preserved. As a conclusion, to enhance end-of-life performances, two points have to be studied: (i) develop recycling methods for composites, or substitute composite parts in the blades design, and (ii) better control mechanical fatigue of some components in order to reuse them.

6.3-Integration of the recycling option as design criteria

Since we assume to favour recycling option, consequences of the preliminary structural choices must now be studied, and connections between choices, end-of-life indicators and tools must be underlined. Our analysis deals with six main structural choices for a HAWT: nacelle rotation, blades rotation, number and material of blades, drive system (with or without gearbox), rotor size and tower size. The table 3 displays a first proposition to show how end-of-life indicators can be linked with a design step. Thus, it is important to note that preliminary design is the step where end of life criteria are the more likely to be taken into account. Before, it is difficult to constraint some functions by end-of-life options, even if a wide range of hypothesises can be adopted regarding recycling rates, material composition and facility for disassembly, which will all be affined in preliminary design and optimized in detailed design, in order to balance production gain and end-of-life performances.

7-Conclusions. Application and limits

The purpose of this paper is to analyse how end-of-life options can be integrated as a criterion in the design process.

For that, we propose to integrate questions about end of life or indicators in each design steps. It enables to see what step is the more suitable to consider end-of-life issues and to also highlight the role that tools can play to improve the end-oflife performances of a product. These examples of very simple reflexions can be led in all design cases. Currently, the main limit is the case per case quantification of indicators, and the study of how they can be precisely be integrated into tools (e.g. how can designers link a recycling rate to FMECA? To what extent can indicators influence the use of tools?) These questions deserve to be solved because, as shown in the wind system example, the improvement of the design by the consideration of end-of-life issue, may preserve a lot of the invested value. Indeed, taking into account the analysis of the end-of-life scenarios, 20% of invested value could be reused in order to save the investment, rather than recycling parts.

8-References

[AV1] D. Ancona and J.

a.Figure 1 :

 1 Figure 1: Integration of the end-of-life options as a design criterion in the product development process

  end-of-life options as a design criterion in methods and tools for ecodesign Copyright of IDMME -Virtual Paper number -5-Concept -Improve -Ingegraf

Table 1 :

 1 Thus, designer have to wonder if it is better profitable to extract it (if it is not required), or if it is well enough sized for fixing a new wind system. Material composition of the studied HAWT [AV1]

						Glass
						reinf.
	Components	Concrete	Steel	Alu.	Copper	plastic
	Hub		18 t
	Blades		1 t			19 t
	Gearbox		25 t
	Generator		13 t		7 t
	Frame,
	machinery
	and shell		21 t	2 t	1 t	1 t
	Tower	3 t	145 t
	Foundation	480 t	2 t

Table 2 :

 2 Economic-based analysis of two possible end-of-life scenarios for a wind turbine (ER: energy recovery)[E1]. Data are the contribution of the wind turbine components to the system overall cost[E1]. The second scenario (optimisation of the first one) favours the reuse of some parts instead of their recycling (data marked with an asterisk)

	Structural

choices Consequences for recycling Involved indicators Decision tools Early design Preliminary design Detailed design Nacelle rota- tion

		Yaw drives reuse/recycling Yaw gears disassembly Yaw system constituents Maintainability	Disassembly Composition Separability	Characterisation of functions	FMECA Simulation
	Blades orien-tation	Hub size Hub complexity Maintainability and reuse of pitch cylinders Blades disassembly	Disassembly Composition Economic viability		FMECA Simulation
	Number of blades and constituents	Quantity of material to treat Disassembly Hub complexity	Disassembly Composition Separability Economic viability Pollutant	functions Client demand (personal choices) Characterisation of	FMECA Simulation	Optimisation (production gain)
	Driving sys-tem	Quantity of material to treat Number of components Maintainability	Disassembly Separability Composition	Strategic position-ing of the company	FMECA Simulation
	Rotor size	Quantity and nature of materials to treat Size of the main frame (dimensioning) Size of the tower (dimensioning)	Composition Disassembly Economic viability	Legislation Characterisation of functions	Simulation	Optimisation (production gain)
	Tower size	Type of material	Composition Disassembly Economic viability Pollutant	Legislation Characterisation of functions	Simulation	Optimisation (production gain)

Table 3 :

 3 Impacts of structual choices on the only recycling

  McVeigh, "Wind Turbine -Materials and Manufacturing Fact Sheet." Princeton Energy Resources International, 29-Aug-2001. [AZ1] H. Alhomsi and P. Zwolinski, "The use of DfE rules during the conceptual design phase of a product to give a quantitative environmental evaluation to designers," in Proceedings of the 19th CIRP Design Conference, Cranfield (United Kingdom), 2009.

[BB1] H. Baumann, F. Boons, and A. Bragd, "Mapping the green product development field: engineering, policy and business perspectives," J. Clean. Prod., vol. 10, no. 5, pp. 409-425, Oct. 2002.


Etienne Balmes
email: balmes@sdtools.com

Integrating damping and non-linearities in a vibration design process

Keywords: damping, non-linearity, model reduction, vibration design ω j = mode frequency

Classical vibration design uses modes and transfer functions generated with the superposition principle to allow the verification of design objectives. If redesign is needed, one optimizes mass and stiffness in order to modify the transfer until the specification is met. Integrating damping and non-linearities in the optimization of detailed industrial models is however still considered a major difficulty, even though the physical mechanisms are well known. Approaches to handle viscoelastic damping and time domain modal damping are thus discussed. Distributed non-linearities, such as contact and friction, are becoming accessible to transient simulation, but lead to responses where modes are no longer defined. It is however illustrated that operational deflection shapes, associated with a singular value decomposition of the response, give similar information. Finally, a fundamental aspect of non-linear vibration simulation is the volume of output and the associated numerical cost. Model reduction is a key ingredient of practical approaches and a perspective on related issues is given.

NOMENCLATURE

INTRODUCTION

Vibration design is often critical for reasons of safety (flutter, seismic response, high cycle fatigue, ...), performance or comfort (acoustic, squeal, ...). As shown in figure 1, classical vibration design is very closely linked to the notion of transfer function. One specifies an input that characterizes excitation in terms of spatial and frequency/time content. One then describes the relation between inputs and response in the form of computed or measured transfers. Under the assumption of linearity, the superposition principle is used to obtain responses which are checked against specifications. If redesign is needed, one optimizes properties in order to modify the transfer until the specification is met. This standard process has many constraints and limitations • The definition of loads is often a problem. Getting from a measured load to a specification load opens the door to overdesign but is necessary to demonstrate the verification of objectives. In the case of multiple random sources, the combination of excitations poses questions as to specific realizations and statistical analysis of responses. Tools such as spectral radii are seldom used. When loads are associated with component coupling as discussed by [START_REF] Corus | Force identification using a statically completed reduced model deriving from tests[END_REF], it is often difficult to measure loads and justify that they are not specific to a given design.

• While mass and stiffness design processes are relatively well mastered, integration of damping is rarely considered, in great part because predictive modeling of damping is often an issue. Advances in this area will be the focus of the first section.

• If the system is non-linear, transfers do not exist and principle of superposition, which drastically reduces the cost of linear system computations, is lost. A safe process is the statistical analysis of many simulations or tests, and model reduction is a key ingredient needed to make the associated cost acceptable. Extracting information from the time responses is another objective that will be discussed.

• Objective functions are generally evaluated at the system level as shown in Figure 2. Redesign of the component based on the system response is often quite costly. Response surface methodologies are not necessarily suited for system behavior showing many modes. Specifically tailored model reduction methods will thus be shown to be of interest.

Damping modeling

Physical mechanisms leading to dissipation are well known

• Viscoelasticity corresponds to the case where stresses depend on the history of strains rather than their instant value, see [START_REF] Nashif | Damped & non-linear vibration design[END_REF]. Delays associated with this behavior is a source of dissipation of many materials and polymers in particular. The highest levels of damping are found for materials close to their glass transition temperature. In damping applications, one thus almost always has a significant dependence of moduli to temperature and frequency. For low amplitudes this dependence is accounted for using frequency/temperature nomograms. For higher loads (suspension blocs, ...), non-linear viscoelastic effects (Paine, Mullins, ...) must be accounted for.

• Friction describes tangential loads occurring when contacting surfaces slide with respect to each other. One can distinguish physically simple but locally detailed models (Signorini/Coulomb, ...) and locally rich models (Dahl, ...) which are often used at the macroscopic level. These mechanisms are non-linear and will be discussed in the next section.

• Fluid viscosity, piezoelectricity, thermo-elasticity, Eddie currents, micromechanical behavior (dislocations, microscopic plastic deformations, ...) are other sources of dissipation that have also been used in vibration damping devices.

All these effects can be modeled and documented cases of good test/analysis correlation exist. The main difficulty limiting their use is their integration into models, with enough detail to allow precise optimization of components, and low enough cost for realistic evaluation.

Taking the case of viscoelastic damping which is commonly used in many applications (see for example the review given in [Rao, 2003]), the behavior is well described by a complex modulus that depends on frequency and temperature

E(ω, T ) = E(ωα(T )) = E ′ (ωα(T ))(1 + iη(ωα(T ))) (1)
where the temperature/frequency equivalence property, states that the dependence on ω and T is related to a single reduced frequency ωα(T ). E ′ the real part of the modulus is called the storage modulus and corresponds to the elastic behavior. η the ratio between the real and imaginary parts, is called the loss factor and gives a measure of the energy dissipated over one cycle. Integrating such behavior into full order model leads to forced frequency responses of the form

[Z(ω, T )] (N×N) {q(ω, T )} (N) = [F(ω)] (N) (2)
where the number N of DOF q typically needs to be large to allow a correct representation of geometrical detail of the viscoelastic treatments, and the dynamic stiffness Z can be decomposed as a sum of constant matrices

[Z(E i , ω)] = -Mω 2 + K e + iK ei + iωC + ∑ k E k (ω, T, σ 0 ) K vk (E 0 ) E 0 (3)
For realistic model sizes, in the 10 5 to 10 7 DOF range, classical direct frequency response leads to prohibitive computation costs. On the optimistic basis of 1 mn per frequency point for an industrial model, one cannot envision a study with 100 frequencies and 500 design points (300 days). The results shown in figure 3a are thus only accessible through model reduction methods discussed in particular in [START_REF] Balmes | Model validation for heavily damped structures. application to a windshield joint[END_REF]. The case shown, also illustrates the non-trivial evolution of responses with viscoelastic design parameters. Performing optimization, thus typically requires many detailed evaluations. Figure 3b, gives another illustration of the need to perform parametric analysis. For the case of an isolated mode, the figure shows iso-damping curves within the modulus/loss factor plane. Within this plane, material selection will define complex moduli, which are shown as colored curves. For each material, temperature changes will correspond to a change of modulus along the curve, thus leading to a performance change. It should be noted that a change of thickness will typically modify the target modulus. For example a shear layer has a stiffness that is inversely proportional to thickness. Thus dividing the thickness by two will double the equivalent stiffness. In the figure, the high loss blue material could thus be used at a lower thickness and thus become the optimal solution.

A major difficulty of the complex modulus approach is that it does not have a time domain equivalent and is thus not compatible with non-linearities that need to be described in the time domain. Typical practice is the use of global or piecewise Rayleigh models which assume a viscous damping matrix that is a linear combination of mass and stiffness. Using the coordinate system associated with elastic normal modes φ j leads the Modal Strain Energy [START_REF] Rogers | The modal strain energy finite element method and its application to damped laminated beams[END_REF] approximation where the modal damping ratio is given by

2ζ j ω j = ∑ k α k φ T j [M k ]φ j + β e φ T j [K k ]φ j (4)
with Rayleigh parameters α k and β k that need to be adjusted by some form of optimization. For global Rayleigh damping, the Rayleigh parameters are constand for all elements and one has a modal damping ratio given by

ζ j = α 1 ω j + β ω j (5)
with a mass contribution decreasing with frequency and a stiffness contribution varying linearly with frequency, as shown in figure 4a. The reality is more often a distribution of damping ratio within a band that does not increase with frequency. Rayleigh damping is thus fundamentally only relevant for fairly narrow frequency bands. In piece-wise Rayleigh, multiple linear lines can be created with the damping of each mode depending on the energy fraction in the associated component.

In the modal basis, the traditional practice is to use modal damping where the viscous damping matrix is diagonal of the form 2ζ j ω j , where the damping ratio ζ j are derived from test or based on design rules. Going back to the physical basis the modal diagonal matrix becomes full and unusable for transient analysis. [START_REF] Bianchi | Using modal damping for full model transient analysis. application to pantograph/catenary vibration[END_REF] proposed to use the mass orthogonality condition, from which one can derive

[Φ] -1 = [Φ] T [M] and [Φ T ] -1 = [M][Φ]
to obtain a closed form solution of the modal damping matrix in the physical domain

[C] = [MΦ] N×N m \ 2ζ j ω j \ N m ×N m [MΦ] T N m ×N (6)
While computing [C] is not acceptable, computing the product [C]{u} at each time step as a series of three matrix vector products is quite acceptable. Figure 4, illustrates how this can generate a flat modal damping ratio for a low frequency band and resume with the traditionnal Rayleigh behavior only in the higher range. The difference on associated transfer functions clearly illustrates the major impact of this change.

NON-LINEARITIES AND FEATURE EXTRACTION

When considering transient non-linear responses, multiple applications come to mind. Rapid transients (crash, press forming, ...), one then rarely talks about vibration because the characteristics of the system undergo major changes and the total type is typically relatively small compared to the time characteristics of modes of interest.

Multi-body simulations (vehicle dynamics, engine dynamics, global dynamics of rotating machinery, ...) get closer to vibration problems. The need to represent component flexibility is however typically limited (the first few flexible modes of a rotating body) and the representation of interfaces is generally simplified. For example, one may consider a fairly complex hydrodynamic behavior in a bearing but the flexibility of the bearing will generally be ignored. The number of Degree of Freedom in the model is most often below the thousand and distributed non-linearities are rarely treated.

A last class of non-linear problems, which will be the focus of the discussion made here, considers vibration problems without geometric non-linearity but with distributed non-linearities (non-linear contact, friction, squeeze film, ...). Sample applications are the study of brake squeal by [START_REF] Vermot | Benchmarking signorini and exponential contact laws for an industrial train brake squeal application[END_REF], assembled joints by [START_REF] Gaul | Nonlinear vibration damping of structures with bolted joints[END_REF], systems with gaps by [START_REF] Thénint | Definition of a linear equivalent model for a nonlinear system with impacts[END_REF]. The distributed nature of the non-linearity implies that detailed models, model reduction, and adjustments of time integration solvers are often needed, as discussed for example by [Vermot Des Roches, 2010] and [Thénint, 2011]. The state of the art is that such adjustments can be made so that parametric studies based on time transients are becoming accessible for a fairly wide range of problems.

When dealing with linear vibrations, mode shapes and frequencies are typically exploited to orient design. One thus identifies specific resonances in a response and orients the redesign to modify their properties. Extracting shape information from non-linear responses is thus a key question.

A first classical form of feature extraction is the case of non-linear response with coarse frequency content. Nonlinear harmonic responses and non-linear modes, see [START_REF] Kerschen | Past, present and future of nonlinear system identification in structural dynamics[END_REF], Sinou, 2008], are thus studied by introducing solvers that can make use of the coarse content. Examples of such solvers are harmonic balance, or MAN [START_REF] Cochelin | Méthode asymptotique numérique[END_REF]. These approaches are not always relevant (for example if the excitation is broadband) and typically require the implementation of specialized solvers that are not easily manipulated. It thus is quite interesting to obtain similar shape extraction results for standard time response transients. The notion of Operational Deflection Shapes, often used to extract shapes from experiments, can often be useful.

For a nearly periodic response, one can thus extract a block of responses Q i j with DOFs as rows and time steps as columns. The Singular Value Decomposition (SVD)

[Q i j ] N×Nt = ∑ j U j N×1 σ j V j T 1×Nt (7)
with possible mass or strain energy weighting, then gives, as illustrated in figure 5, left singular vectors U i that correspond to shapes, singular values that give an amplitude and right singular vectors V T i that correspond to a time dependence. The case of a gap non-linearity under broadband excitation treated by [Thénint, 2011] provides another illustration non-linear feature extraction. In this case, [START_REF] Thénint | Definition of a linear equivalent model for a nonlinear system with impacts[END_REF] showed that non-linear excitation amplitude and linear variable contact damping/stiffness led to very similar trends in the evolution of Power Spectral Densities (PSD, shown in figure 6). Furthermore, the evolution of first singular vector associated with the main peak in the PSD is clearly similar to that of the first model of the equivalent linear model.

PARAMETRIC STUDIES, LONG TRANSIENTS, AND MODEL REDUCTION

When running simulations for problems that require optimization for vibration performance, it is often critical to represent the geometry in significant detail to allow proper reproduction of the modes and associated loads in the areas that are being designed. Model sizes in the 0,1 to 10 million DOF range are thus the norm. Design implies the comparison of multiple configurations. Running tens to thousands of configuration is thus a clear objective. Furthermore, the number of cases of interest tends to grow as a power of the number of parameters. For frequency responses thousands of points are typically needed, for time simulations the number of steps is typically multiple orders of magnitude larger.

Taking a simple case of a 1e6 DOF model, with 100e3 time steps and 5 design points, storing results in double precision requires close to 4 TB. A fundamental issue is thus to optimize feature extraction to reduce the pure volume of data but also possibly to allow the introduction of optimized solvers.

An interesting branch of research calls this issue the curse of dimensionality (see [START_REF] González | Recent advances on the use of separated representations[END_REF]). The configuration parameters are then considered as additional dimensions and solutions are searched using variable separation

{u(x, y, z, ω, p 1 , ...p k )} = N ∑ j=1 F j 1 (x) × . . . × F j k (p k ) (8)
leading to Proper Orthogonal Decomposition (POD) or Proper Generalized Decomposition (PGD). The relation with the a posteriori decomposition (7) of full responses should be noted.

For vibration problems, one is however interested in spectra, so that some responses need to be known over the full time length, and shapes, so that some responses need to be known over the full spatial detail of the FEM model. Applying general purpose variable separation on space and time/frequency is thus difficult.

Classical Component Mode Synthesis (CMS) methods are however well established methods that perform variable separation on space and time/frequency coordinates. Assuming fixed basis reduction [Balmes, 1996], considers a similar When dealing with modal frequencies, curve veering is a critical issue illustrated in figure 7 for the sensitivity of an unstable brake squeal mode to the modulus of the disk. The transition from stable to unstable is a behavior that is typically difficult to represent using smooth representations as the PGD (8). In the CMS, this is not a difficulty because a reduced model ( 10) is solved. A good way to phrase the difference is to say that PGD generates meta models of the solution, while CMS generates a low cost meta-model of equations to be solved. A critical aspect of the costs associated with CMS methods is the fact that the memory needed to store the subspace T can be significantly lowered by considering subspaces for multiple components as illustrated in figure 8a. This principle is the basis of multi-frontal sparse solvers used by all finite element software and Automated Multi-Level Substructuring (AMLS, see [START_REF] Gao | An implementation and evaluation of the amls method for sparse eigenvalue problems[END_REF]) which is a generalization to dynamics. When CMS methods were developed in the early 80's the need was to lower computational costs and the base proposition was that the reduction subspace for each component should be independent of information on other components. When the number of DOFs at the interface between two components becomes large, the associated matrix topology (illustrated in 8b for a brake model) tends to present large bands of non-zero terms which can drastically limit the interest. [START_REF] Vermot | Using component modes in a system design process[END_REF] proposed to use traces of system modes as reduction bases. This approach is thus particularly well suited for applications, such as brake squeal, where transient analysis needs to be performed with detailed description of non-linearities. One keeps DOFs q i close to the non-linearity explicitly (for example the pad shown right in figure 9), and reduces other DOFs q c (the disk and associated components) by assuming a linear combination of the trace of assembled system modes {q} ≈

[Φ Total ] c 0 0 I q Rc q i (11)

Figure 8c clearly shows that this significantly lowers the interface band, while preserving exact computations for linear computations in the frequency domain. For design studies it is often desirable to be able to analyze the impact of components integrated within a full system model (see figure 2). A simple generalization of the reduction proposed above is to build for each component a basis combining free component modes and the trace of assembled system modes

T C = [Φ C,Free ] [Φ Total ] C Orth ( 12
)
This reduction, detailed in [START_REF] Vermot | Using component modes in a system design process[END_REF], provides the clear advantage of using modal amplitudes of components as explicit degrees of freedom. Changing component mode frequencies or damping ratio then becomes a simple modification of the stiffness or viscous damping matrices.

CONCLUSION

The combination of physically representative models of viscoelastic damping and contact/friction models with appropriate model reduction methods was shown to open the way for accurate and yet accessible numerical simulations of these phenomena. These first steps pave the way for numerically oriented design processes, which will require the choice of features considered for the optimization and processing of enormous volumes of results to orient design. Rethinking computations flows in this light, rather than in the traditional validation process, where only a few computations are used, is thus seen as an interesting challenge for the coming years.

  load E i = complex modulus of i th viscoelastic material Greek Symbols ω = frequency α(T ) = viscoelastic reduced frequency shift [Φ] = matrix of real (elastic) modes

Figure 1 -

 1 Figure 1 -Classical vibration design process.

Figure 2 -

 2 Figure 2 -V design cycle. Objectives (modes, transients, ...) are evaluated at the system level while design is done at component level.

Figure 3 -

 3 Figure 3 -Left: evolution of a transfer function a s a function of a design stiffness (damped windshield). Right : iso-performance curve in the modulus/loss factor plane, with material laws for a multiple possible materials.

  Figure 4 -a) Global Rayleigh, b) Piece-wise Rayleigh with low frequency modal damping, c) Transfers with and without modal damping [Bianchi et al., 2010]

Figure 6 -

 6 Figure 5 -SVD of a non-linear squeal cycle. a) principle, b) sample shape for a realistic model

  reduction {u(x, y, z, , ω, p 1 , ...p k )} = N ∑ j=1 [T (x, y, z)]{q R (ω, p 1 , ...p k )} (9) and transient or frequency domain resolution of reduced equations of the form T T [Z(ω, p 1 , ...p k )]T {q R } = [T ] T {F(ω, p 1 , ...p k )} (10)

Figure 7 -

 7 Figure 7 -Evolution of complex pole with a disk parameter, [Vermot Des Roches, 2010]

Figure 8

 8 Figure 8 -a) Decomposition topology for multiple components. b) Stiffness matrix topology for a Craig Bampton reduction. c) Matrix topolology for CMT.

Figure 9 -

 9 Figure 9 -Decomposition of DOFs in reduced and kept areas.

RESPONSIBILITY NOTICE

The author(s) is (are) the only responsible for the printed material included in this paper.