Feng Zhang
email: feng.zhang@g-scop.inpg.fr

Peggy Zwolinski
email: peggy.zwolinski@g-scop.inpg.fr

OPTIMIZED NAVIGATION SYSTEM FOR ECO DESIGN MANAGEMENT

Keywords: Eco design, Strategy decision support, Quantifiable, Continuously improvement, Integration

With the contribution of active scientific studies and worldwide promotion, "environmental-friendly" concept have been gradually understood and accepted by the public. They are now not only one of the major concerns of the governments in the world, but also, as an issue of systematic engineering, have been embedded in all industries. Following the development of environment-concern, to have an intelligent "environmental strategy" program, which could analyze the needs of industrial competitiveness and social responsibility, identify the adaptive actions and make project plan of integrated solutions, is now becoming extremely important. This paper introduces a brand new optimized green navigation system. "TArget-hitting Navigation Kit", named as TA.N.K also, working on the analysis of actual situation, is proposed to support the decision on any environment-concern actions in the enterprises. TA.N.K, involving all the activities and functions in a unique and sharable information flow, could ensure the sharability and readability of preparatory studies, avoid the misunderstanding among different functions and it also provides the possibility of continuous improvement.

Introduction

Awareness of the fragility of our natural capital and the needs to design cleaner industrial systems show the environmental concern as a must. To curb and prevent degradation of the planet, society must change the logic of production and consumption. In order to formulate new socially and culturally acceptable proposals, "environment-concern" mechanism which indicates a framework at industrial level, is aiming to connect "technically possible" to "ecologically necessary". [START_REF] Manzini | Limits and Possibilities of Eco-design[END_REF] With the great contribution from numerous scientific studies and strong promotion on environment, the conception of environmental-concern mechanism has been gradually accepted, understood and focused on by the government, the industry and the public.

A "Mintel" (released in 2009) study showed that the green market outperformed the US economy as a whole in 2009 and grew by over 40% from 2004 to 2009. And there are over 370 eco label systems in 214 countries and 25 industry sectors in the world for multi-criteria environmental certificate. [2] These systems could directly provide a panorama for proving that environmental concern conception is now becoming an important part of global economy and it has been embedded into the "blood" of design activities.

On legislation, the industry should face a stricter legislative environment which is under accomplishment over time. Some countries and country unions (such as European Union) have developed or are developing their environmental laws to accelerate the environmental progress and educate the public to take care of the environment. Legislative framework including directives like RoHS, REACH, WEEE, ERP and Battery etc., which is developed by European Union, could be considered as a great example, and this law system is being copied into some different local versions by other countries.

Additional and also important, enterprises should face a "greener" public. According to a Mintel study in March, 2010, [START_REF]GLOBAL ECOLABEL MONITOR[END_REF] more than a third of US consumers now say they are willing to pay a premium for eco-friendly products, and in some cases this is even higher, for example 53% of US consumers would be willing to pay a premium for a greener television, according to the Consumer Electronics Association. Another example from UK, according to a 2009 Carbon Trust study, 44% of UK consumers want more information on what companies are doing to be green. Final customers focus on environmental issues and also, logically, in Business to Business mode, downstream enterprises need the environmental proof for constructing their green offers (Product/Solution) to answer the requirements of their final customers.

To answer these above-mentioned requirements and enhance the competence of competition, the industry is acting. Till now, it's very difficult to find out an enterprise without any sustainable strategies. Thus, the environment-concern mechanism can be defined as a schematic solution to find a great balance among the technical constraints, environment-concern, and economic requirements to the products or services (Brezet et al, 2007). The actions of environment-concern program or "eco design" should be tailored to the specifics of the companies and their products and be taken into account in all indicators of the value chain. These environmental approaches rely on multi-criteria approaches, multi-steps and multi-actors, which must be integrated into a global structure of piloting the process and balancing the conflicts. An intelligent and strong environmental strategy and related professional action program seem extremely important.

In this paper, by analyzing the actual industrial situation and some definition process inside of enterprises, a new green navigation system has been proposed for integrating all activities of enterprise and optimizing related strategy definition process.

The analysis of worldwide industry environmental practices

As mentioned earlier, in order to answer the external requirements and profit some advantages from green strategy [START_REF] Kemp | Strategies for eco-efficiency innovation[END_REF], the industry is acting. A mass of eco programs have been lunched to declare the environmental characteristics or improve the green performance. The following table (Table 1) presents the survey results of environment-concern program from 33 electric and electronic enterprises. The resource of this table came from the inter-standards, the program presentation and promotion document or the sustainable report of each enterprise. The environmental concern programs are classed in two principal groups by final objective: declaration of environmental impacts and certification of the green performance. For judging the green performances of product/service, there are comparative criteria (compare the new product with a selected reference) and check list with special threshold (such as consumed value, avoidances, life time etc.). All principal environmental indicators have been focused on by the initial activities of enterprise and some enterprises even have developed a higher class environment-concern program to update the strategy to improve the environmental performance, Eco labels system (Check list 02, 04, 06 and 09) in EIZO as an example. The analysis of those practices leads to different considerations about the ability of enterprises to share new environmental studies, to choose the right strategies and identify the improvement roadmap.

Considerations on the sharability to preparatory studies

The industry is facing a complex environment, so its analysis (Include the law, directive, custom requirement, marketing potential value/direction and the competition needs) should be prepared for contributing to individual strategy of environment-concern or "Eco design". But typically, these analysis and those reports are to be processed by different specialists (either in same sustainable department or dispersed in different functions), and the reports and the experiences could not be totally shared among them because of the different structures, approaches, methods and habits. Furthermore, how to compare the results of all these mandatory action requirements, how to define their priorities, and how to continuously modify these results and priorities in future circumstance has to be defined. For improving this part, the industry needs an evaluation system which could:

1 make the analysis more readable 1 make the analysis more sharable 1 completely transfer the evaluation results in contributing to the strategy decision 1 make the analysis more comparable 1 allow easier continuous modification on the analysis of future results

Considerations on the way to select environmental strategies

The environment-concern program is not only the affairs of our planet, but it is also necessary to contribute to the competition. While achieving similar objectives, the implementing measures could be diversified. The enterprise could straightly focus on the environmental performance via some initial eco innovations, or just answer the mandatory external requirements. For selecting the right implementing measures, the enterprise needs:

1 the efficient implementing measures to adapt the needs of the enterprise 1 to open the door to the scientific studies to enrich the experience and available solution 1 the right formula to avoid the negative impact to existing process 1 good cost performance 1 to make difference with competitors

Considerations on the way to improve environmental strategies

The circumstance keeps changing with time. So the existing strategy could not be always valid to guide the actual activities without any update. Moreover, the changes happen with numerous indicators. So the enterprise needs a mechanism which could:

1 continuously monitor the change of external requirements 1 continuously analyze the internal situation 1 compare the actual situation with previous 1 synthesize the multi-indicators 1 define the priorities of the changes 1 refine the strategy or program in a right way at the right time 1 destroy the actual program at a minimum level

Summary for requirements to better consider environmental strategies

In order to resolve these above-mentioned, a schematic framework needs to be proposed to improve the offer's environmental performance and optimize the enterprise's green approach:

1 Provide a unique, quantifiable and sharable evaluation system 1 Improve the efficiency of information flow within all enterprise functions (MARKETING, Purchasing, R&D, etc.) by speaking "a same language" 1 Provide the quantifiable decision system 1 Involve the scientific research 1 Ensure the continuous modification and update 3 Optimized Green Navigation system -"TArget-hitting" Navigation Kit (TA.N.K)

General Architecture

The original intention is to continuously improve all aspects of environmental issues with a quantified and unified evaluation flow. Our TA.N.K system proposes a quantifiable audit platform to evaluate all environment-concern actions and related enterprise achievement status. By analyzing these audited scores, especially the score's gap between the external requirements and the internal situation [START_REF] Moez | Quels outils de contrôle pour décliner les stratégies environnementales[END_REF] [10], this TA.N.K supports the enterprise to set up an adaptive strategy and the prioritized action plan to reduce the gap. So the gap reducing could be considered as a "motor" of enterprise environmental actions of this navigation system. [START_REF] Azzone | Exploiting green strategy for competitive advantage[END_REF] If it exists a visible gap (the external scores is much bigger than internal) of some environmental issues, which means the enterprise should make a move to respond the external requirements, the adaptive programs will be created to achieve the requirement, meanwhile to reduce the related gap. For the needs of leadership or competition, some advanced program should be created to increase the gap (the internal scores are more and more big than external) of some issues which have not been focused on by extern. Cooperating with the marketing education and promotion plan, such issue's external scores will be increased which will finally re-reduce the gap.

In sum, the TA.N.K will continuously monitor the situation and modify the scores till the gap between the external and internal situation is small enough which leads the related action could be considered as out of the focusing area.

As mentioned earlier, the environment-concern program is a global framework which needs to involve all functions of the enterprise from top management to every project team [START_REF] Lamberton | Sustainability accounting-a brief history and conceptual framework[END_REF] [START_REF] Moez | Quels outils de contrôle pour décliner les stratégies environnementales[END_REF]. Our optimized green navigation system classifies all related activities and related functions to propose a general architecture which includes 6 sub-systems: With six basic sub-systems, analysis of the external requirem (EValuation System), the evaluat and corporate focusing area defi table. These outputs are transferr selected actions and fix the fina table are transferred to APS (Ap roadmap. And then, this progra monitored by IOS (Internal Obse independent monitoring report, IO report it to EVS. At last, the EVS and reports to SDS to end the pro Throughout the process, the s sub-system in order to the uniform 3.2The instruction of each sub-sy EVS sub-system, a scientifi responds to external requiremen quantifiable and continuous impro comparing with the company's ow actions for strategy preparation. is divided into two sub-working m "Translate" module provide quantifiable evaluation, to sort, environmental-concern actions. different formula, method or s indicators" [START_REF] Lagerstedt | Functional and environmental factors in early phases of product development-Eco Functional Matrix[END_REF] [START_REF] Al-Tuwaijri | The relations among environmental disclosure, environmental performance, and economic performance: a simultaneous equations approach[END_REF]. Certainly, th actions.

All follow-up priority defini translated indicators of the ana system tific Evaluation System, is the first part of TA ents. The ultimate goal is to establish an evaluat provement solution that assesses all environment-co own internal status, this EVS sub-system could co To achieve the above approach needs, EVS sub g modules -"Translate" module and "Evaluation" m des a unified platform, to ensure continuous im rt, re-combine and quantify each collected analys . Through this module, we could translate all a r structure into the unique and standardized for this Translate module can also be directly used in inition and the action plan of entire navigation sy nalysis to environmental-concern actions. There 29 Mars-1er avril 2011 on Kit" d as follow: When all een translated by EVS to the technical criteria nd related actual status inal release (modify the list and related status plementing details and nistration System) and status report and IOS on's internal status and ort from EOS and IOS) cision-making for each TA.N.K system which uation system, using a concern actions. After confirm the prioritized ub-system in the macro " module. improvement and the lysis report of typical l analysis results with formula by "common each new analysis of system depend on the erefore, the quality of translation is the key point. The global situation which involv needs, the competition, the fea be enough simple, straight and needs of environment-concern considered as the reference:

-Indicator G -"gravity" This indicator is define actions for certain enter enterprise, not only for graded by 10 hierarchical -Indicator F -"feasibility" This indicator is defined concern actions. The industrialization and the according to the classific hierarchical levels, from 1 -Indicator V -"visibility"

This indicator is defin actions. The "visibility" including the execution innovation and competiti hierarchical levels, from 1

For these environment-conce the above three indicators ( environment-concern actions.

Logically, according to the translating the internal situation three major indicators have to be he selected indicators of this module must be a olves the environmental impacts, cost pressu feasibility and the society impacts. And also, th nd easy to compare. This navigation system, comb n issues, defines following three major indicat ined to demonstrate the importance of certain e terprises. This indicator focuses on the total pro r just one product. According to this importance cal levels, from 1 to 10.

ty" ed to demonstrate the implementing feasibility of e "feasibility" classifies the technical feasibi he cost to achieve the required actions. As for the fication of feasibility, the performance is graded an m 1 to 10.

"

fined to demonstrate the popularity of certain y" grades the marketing and public awareness lev n of the directive, Law, the common industrial titive promotion etc. Normally, the performance m 1 to 10.

cern actions, each indicator will be given a score. T ) will be considered as the overall he need of comparison, a similar approach shou on of enterprise for each environment-concern ac be considered as references:

ment" ed to demonstrate the level of achievement for c rk with previous indicator, the achievement level w m 1 to 10. For example, this indicator could be calc r percentage of turnover.

ntability" ed to demonstrate the implementing feasibility in ther the actual organization, supply chain, technic ss could be easy to achieve the objective. It also a e a global and complete view. To work with pre raded also by 10 hierarchical levels, from 1 to 10.

nication" d to demonstrate the communication level of each e ign is always not good enough", this indicator will nvironment-concern action has been promoted to pu rformance comparing with competitors. As all the raded also by 10 hierarchical levels, from 1 to 10. All following up decisions a depend on the future analysis on reports need to and must be regist "Evaluation" module of EVS environment-concern action to n Indicator P which is calculated b and make an order of the actio logically, the order of the gap c problems. The general mechanism 12342356789A3BC542 Meanwhile, the Business Fore guide out the potential important precocious program.

Based on these approaches a summarizes a "selected list" and system. SDS sub-system, an intelligen strategy and the future updates, c sub-system, accepting the draft r financial situation, market, R&D re-audits the priority of certain en released program dashboard inclu action list, will be transferred to n

The difference between the a context and technical feasibility, image, future development, econo F APS sub-system, a harmonize implementing measures for each are always hundreds of differen multi-steps programs etc.) for the considered as a project which wi the optimal and robust measures most complex formula, but it nee IMECA Le Mont Dore -2 7 s and the creation of program of this navigation on these six indicators. It is important that all proofs istered for future discussion.

VS sub-system provides a strategy recommendatio next sub-system. This proposition depends on the d by the above mentioned 6 indicators. The genera tions with the gap between external scores and i could be considered as the order of the severene ism is explained as in follow equation:

89A3BC5427D1E F 4C7B42 4C7B42
orecast Model and the linear regression are involve ant environmental issues which will be as a referen s and considering the enterprise's threshold value, nd proposes a draft environment-concern recomme gent Strategy Decision System, who plays a final d s, could be considered as the "CPU" of entire nav ft recommendation of EVS as an input, and taking D competences, stakeholder and corporate positi environment-concern actions and modifies the "sel cluding the general goal, the constraints, the requir o next sub-system as a "constitutional law". e analysis in EVS and in SDS is that EVS focuse ity, but SDS, at a strategic height, is more incline onomic situation and corporate culture etc.

Figure 4 the schematic process of SDS sub-system ized Application System, is a professional part to c ch selected environment-concern action. Like any o ent implementing measures (a sole program, a gr the same objectives or requirements. And also, the will involve multi-functional departments. So APS res which is not mandatory from the most difficu needs to select the most fitted one, who adapts the 29 Mars-1er avril 2011 ion system will totally ofs and related original tion including selected the priority indicatoreral aim is to filter out d internal scores. And eness of environmental lved into this module to rence to identify some e, the EVS sub-system mendation to next subl decisive role of initial avigation system. SDS ing account of current sitioning, etc., [START_REF] Richard | L'adoption d'une stratégie de développement durable: un générateur de ressources et de compétences organisationnelles, Une analyse ressource based[END_REF] [START_REF] Jean Carner Stead | Eco-enterprise strategy : Standing for sustainability[END_REF] selected list". The final uirements and selected uses on the impersonal lined to the company's o construct the detailed y other activities, there group of programs or the program should be PS is going to find out icult technology or the the actual situation and punctually (directly or by multi-steps) responds to the requirements of the enterprise with the lowest cost and few impact to existing process and habits. So the major role of APS is to decide the implementing measure and to plan a program which could embed these measures into the existing processes. According to this approach, APS sub-system in the macro is divided into two sub-working modules -"Selection" module and "Planning" module.

Involved with the scientific contribution and knowledge database, "selection" module gives out some default proposals through analyzing the "major characteristics" of each implementing measure relating with the "selected list" from SDS. There are three "major characteristics" defined:

-Execution timing: it describes the estimated finish time of each measure. As default, the scores could be classified as an index (Low/Medium/High) or as a real value. -Execution technology: it describes the technology or technical level required by this measure.

As default, the scores could be classified as an index (Low/Medium/High) or as a real description -Execution cost: it describes the estimated cost for each measure. This cost includes the technical cost, human resource, and the cost for building new process. As default, the scores could be classified as an index (Low/Medium/High) or as a real value.

-Execution influence: it describes the final influence by executing this measure. It evaluates the influence on marketing and competition, the leadership, the sustainability and the technical advantage etc. As default, the scores could be classified as an index (Low/Medium/High) or as a real description.

For each selected environment-concern action, the "selection" module creates or imports (from an expertise database) all available implementing measures, and evaluates them by three above characteristics. According to the score of SDS or EVS, the selection module decides a measure by following some general rules:

-If there is a great gap between indicator "G" and indicator "A" of the related environment concern action, a fastest measure is proposed. -If there is a great gap between indicator "V" and indicator "C" of the related environment concern action, a measure requires low technology is proposed. -If there is a great gap between indicator "F" and indicator "I" " of the related environment concern action, a measure with a larger influence is proposed. -If the selected action is to create the leadership, a measure with a larger influence is also proposed.

Meanwhile, with the constraint of budget, the characteristic "Execution cost" should also be focused on in parallel.

This general mechanism ensures the enterprise could professionally identify the multi-steps program at the beginning (the first step is to quickly respond to the strictest requirements and the following-up will focus on the improvement to accelerate the reduction of the gap).

Combining all selected implementing measures, the "selection" module proposes a global program with general roadmap and schedule plan. These results are translated into "planning" module to build the details.

"Planning" module of APS is to combine and develop the executing details for each selected implementing measure. This module collects and takes a summary of the executing requirements, which elaborates the details of 4Ws ("When","Who", "What"and "Way") during the entire process. And then, this module needs to embed these requirements into the existing process with minimum and inevitable impact. Finally, a specific implementation dashboard with detail is constructed for implementation.

For these details, APS's planning module should also define the KPIs (in process and as final deliverables) which are controlled, monitored and easy to be achieved. And then the corresponding templates, tools, and the official implementation instructions should be released to guide the process and each measure. This module must ensure that the final deliverable dossiers can be directly converted to as the feedback of EVS sub-system which could continuously modify the internal status of enterprise. ADS sub-system, a professional Administration System, is the real implementation system of this navigation system. It is responsible and accountable to ensure sustainable program to meet the goals defined by APS. ADS could be considered as a giant project. There are some typical activities that could be proposed to improve the program quality:

1. The multi hierarchical steering committees: based on actual needs, the committees are necessary while discussing and analyzing the current problems encountered during the implementation, the feedback for re-assigning the resource, resolving the conflicts and risks and motivating the group to achieve the goal on time. ADS sub-system set up three different committees for such general proposition:

The committee at enterprise level -harmonizing with another program and verifying the status / roadmap

The committee at branch level -verifying the resource and controlling execution situation The committee or specialist at project level -process control and support 2. Internal/External promotion: To ensure the most efficient implementation, some internal promotion campaign is essential to each enterprise. These campaigns involve two major areas: communicating with top management area to attract the attention and get adequate resource, and promoting to related departments to make sure that all requirements and information have been delivered to targeted individuals. 4. KPI daily monitoring: frequently monitor KPI's status to help to reallocate the resource. EOS sub-system, a continuous External environment's Observation System, is to, based on the change of external demands and environmental hot points, promptly evaluate and update EVS results. EOS monitoring includes:

1. Follow-up of existing external changes (laws and regulations, customer demand, market attention) 2. Follow-up of the change of executable environmental actions 3. Follow-up of hot issues 4. Thorough tracking of environment-concern actions selected by EVS and SDS sub-systems EOS will separate or disperse assesses, and analyze the above detection range. Basing on the common "translate" module of EVS, the monitoring results will feedback to EVS to modify the scores of external status.

-If the action is selected to answer the external requirements (there is a big gap of the scores of indicators), EOS continuously monitors the external change and re-modifies the objective. -If the action is selected to create the leadership (without a visible gap or the external scores are not significant), EOS continuously monitors the external change to verify if the objective has been achieved. IOS sub-system, a continuous Internal Observation System, aiming at continuous improvement, is the second feedback system for assessing the change of internal status. Following up the process of environment-concern program, IOS continuously analyzes the internal status and reports it to EVS sub-system (harmonized with "Translate" module) to modify the scores of internal part. IOS also monitors and analyzes independently the risks, conflicts and difficult issues during the implementation and reports them to ADS or APS to improve and update the program plan, and even pass to SDS to update the strategy.

Conclusion

In this paper, basing on the analysis of actual situation, a new optimized green navigation system -"TArget-hitting Navigation Kit", named as TA.N.K also, has been proposed to support the decision of any environment-concern acts into the enterprise. TA.N.K system involves all activities and functions under a simply, unique and sharable information flow. It ensures the sharability and readability of each study, avoid the misunderstanding of the severeness between different functions and it also provides the possibility of continuous evaluation and improvement.

Figure 2 the arc

  Figure 2 the arc

  module, each indicator has a score (from 1 to 10) is considered as the overall assessment 29 Mars-1er avril 2011 e able to integrate the sure, the marketing the indicators should mbined with the actual ators which could be environment-concern product/service line of ce, the performance is f certain environmentsibility in large-scale the previous indicator, and summarized by 10 environment-concern vel of certain actions ial solution / potential e is graded also by 10 e. The multiplication of ll assessment of these hould also be used in action. The following r certain environmentl was graded also by 10 alculated by percentage inside enterprise. This nical competences and o analyzes the relevant previous indicator, the h environment-concern ill evaluate whether the public. It also analyzes he other indicators, the 10). The multiplication nt to the inter-situation 12ème Colloque National AIP PRIM

3 .

 3 Internal training: environment-concern program will make the environment criteria become one of the major indicators of product design. Therefore, the training on the background, the details of program and the related competences development becomes very important. The training can motivate the enthusiasm of the relevant employees, set up a great implementing environment and augment the anticipation of project resource to ensure efficient operation.

Table 1 .

 1 The situation analysis of environment-concern program in EE enterprises


O Mahdaoui

P Laure

J.-F Agassant

Numerical Investigations of Polyester Coextrusion Instabilities

Keywords: Coextrusion, interface instability, rheology, Spatio-temporal analysis

Interfacial instabilities occurring in the coextrusion process of molten polymers have been widely studied. The theoretical work based on the stability of 2D Poiseuille multilayer flows (invariant along the flow motion) have pointed out the convective nature of this instability (it is either amplified or damped in the flow direction). This behaviour has been confirmed by experimental observations. As this instability is purely elastic (the Reynold number in coextrusion process is very small), it is necessary to introduce a viscoelastic constitutive equation for the polymers. Comparisons between experimental observations and theoretical approaches based on the Giesekus or White-Metzner models give good agreements for laboratory simple devices. However, the applicability of this approach for industrial coextrusion process remains an important challenge, because one has to deal with the influence of complex geometries, varying temperature and more realistic constitutive equations. As it is still very costly in computational time to perform 3D instationary computations for viscoelastic fluids, a methodology to analyse the appearance of interfacial instabilities in a complex geometry is proposed within this paper. First, a stationary 3D computation is performed in the real geometry for a single polymer. This computation allows to find zones in which the flow motion is essentially 2D, isotherm and with maximum values of the shear rate and, thus, the Weissenberg number. Instationary 2D finite element computations for two layer flows and a multi-mode differential viscoelastic model are then performed in these zones. Spatio-temporal analyses give information on the spatial behaviour of interfacial disturbances. This methodology is applied to the coextrusion of two polyesters for which Preprint submitted to Journal of Non-Newtonian

Introduction

The coextrusion process consists in extruding different polymers in the same die in order to get a multi-layer product which combines the properties (mechanical, optical, adhesion, barrier) of the different layers. It allows for example to substitute lacquer for steel protection in food packaging industry. Here, the coextruded film is composed of two layers, the bottom one providing good adhesion properties on the steel sheet substrate, and the upper one good printing property. The steel coated sheet will then be deep-drawn to shape for example a beverage can. The two polymer layers are constituted of various polyester patented formulations which present very different rheological properties. For some processing conditions, coextrusion instabilities are observed between the two layers. They induce unacceptable optical properties for the final steel coated product (figure 1).

Coextrusion instabilities have been experimentally investigated for a long time. Three kinds of experimental studies have been performed.

-In the first approach, the transition between stable and unstable processing conditions is investigated by examining the aspect of the extrudate obtained with different couples of commercial polymers (Han and coworkers [START_REF] Han | Rheological investigation of interfacial instability in two layer flat-film coextrusion[END_REF][START_REF] Han | Studies on multilayer film coextrusion I. The rheology of flat film coextrusion[END_REF][START_REF] Han | Studies on multilayer film coextrusion II. Interfacial instability in flat film coextrusion[END_REF], Antukar et al. [START_REF] Anturkar | Estimation of critical stability parameters by asymptotic analysis in multilayer extrusion[END_REF], Valette et al. [START_REF] Valette | Investigation of the interfacial instabilities in the coextrusion flow of polyethylene and polystyrene[END_REF]). The authors related stable and unstable processing conditions to thickness ratio as well as viscosity and first normal stress difference ratios at the interface.

-In the second approach, the coextrusion flow and the development of instabilities have been examined in a very long die between parallel plates: after stopping the rotation velocity of the two extruders, the die is quickly cooled down, then dismounted and the interface between the two incompatible polymers is carefully analysed [START_REF] Valette | Experimental investigations of the development of interfacial instabilities in two-layer coextrusion dies[END_REF]. The instability is not visible at die entrance, but then develops and amplifies along the die length, pointing out its convective character.

-In the third approach, the instability is studied by looking at its development inside a partially transparent die for slightly unstable processing conditions, that is to say near the transition between stable and unstable coextrusion conditions. In Khomami and coworkers papers [START_REF] Wilson | An experimental investigation of interfacial instabilities in multilayer flow of viscoelastic fluids. Part I. Incompatible polymer systems[END_REF][START_REF] Wilson | An experimental investigation of interfacial instabilities in multilayer flow of viscoelastic fluids. II. Elastic and nonlinear effects in incompatible polymer systems[END_REF][START_REF] Wilson | An experimental investigation of interfacial instabilities in multilayer flow of viscoelastic fluids. III. Compatible polymer systems[END_REF][START_REF] Khomami | Experimental studies of interfacial instabilities in multilayer pressure-driven flow of polymeric melts[END_REF][START_REF] Khomami | An experimental/theoretical investigation of interfacial instabilities in superposed pressure-driven channel flow of Newtonian and well characterized viscoelastic fluids. part I: linear stability and encapsulation effects[END_REF] evolution of the interface deviation along the die is observed through four optical windows for various wave numbers of the interface defect and various layer thickness ratios (or flow rate ratios). In Valette et al. paper [START_REF] Valette | Convective linear stability analysis of two-layer coextrusionflow for molten polymers[END_REF] the same phenomenon has been observed, but the lateral walls of the channel die are now totally transparent, which allows a precise analysis of the progressive development of the instability. Interface perturbations are generated by adding small amplitude periodic oscillations of the flow rate at die entrance. These perturbations grow or decay as they travel downstream, pointing clearly out the convective nature of the interfacial instability [START_REF] Valette | Convective instabilities in coextrusion process[END_REF].

From a theoretical point of view, the occurrence of such instabilities has been largely studied by means of a linear stability analysis of the two-layer Couette or Poiseuille flow. In such analysis, one studies the growth or decay rate of periodic and arbitrary small perturbations using the Gasters transformation [START_REF] Gaster | A note on the relation between temporally-increasing and spatially-increasing disturbances in hydrodynamic stability[END_REF]. In the twenty past years, stability analysis have been carried out for different viscoelastic constitutive equations (see for example [START_REF] Chen | Interfacial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids[END_REF][START_REF] Su | Interfacial stability of multilayer viscoelastic fluids in slit and converging channel dies geometries[END_REF][START_REF] Su | Purely elastic interfacial instabilities in surperposed flow of polymeric fluids[END_REF][START_REF] Laure | Linear stability of multilayer plane Poiseuille flows of Oldroyd-B fluid[END_REF]). All these studies point out that instability occurs even at very low Reynolds number and is mainly governed by the viscosity ratio, the elasticity ratio and the thickness ratio of the two layers of polymers.

More recently, authors use more realistic constitutive equations to compare linear stability analysis results with experiments. Ganpule and Khomami [START_REF] Ganpule | The effect of transient viscoelastic properties on interfacial instabilities in superposed pressure driven channel flows[END_REF][START_REF] Ganpule | An investigation of interfacial instabilities in the superposed channel flow of viscoelastic fluids[END_REF] use a 4-mode Giesekus constitutive equation, as Valette et al. [START_REF] Valette | Experimental investigations of the development of interfacial instabilities in two-layer coextrusion dies[END_REF] use a White-Metzner constitutive equation. There is a quantitative agreement with experimental observations both in terms of critical flow rate ratio and spatial amplification of interfacial disturbances.

For the first time Valette et al. [START_REF] Valette | Convective instabilities in coextrusion process[END_REF] computed numerically the behaviour in time and space of an interfacial perturbation generated at the inlet of a Poiseuille flow of two coextruded Maxwell fluids of high viscosity. The perturbation moves as a wave packet in the flow direction with a velocity different from that of the interface and is spatially damped or amplified according to parameter values (viscosity, elasticity and flow rate ratios). Due to the high viscosity of both polymers, the Reynolds number is close to zero and the driving parameter of the instability is the Weissenberg number which balances the first normal stress difference and the shear stress. As usual, theses previous non dimensionless numbers are defined thanks to characteristic length and velocity of apparatus by taking

Re = ρ U ref L ref η ; W e = λ U ref L ref ,
where η, ρ and λ are respectively the viscosity, the density and the relaxation time of fluid. For more complex fluids for which the relaxation time and viscosity are shear dependant, new expressions of theses non dimensionless numbers are given at the end of the next section. This numerical analysis of the effect of a small perturbation was then compared to the linear stability analysis of the two-layer Poiseuille flow and a good agreement was found. Time-dependent viscoelastic computations involve two major difficulties [START_REF] Demay | Computation of 2D viscoelastic flows for differential constitutive equation[END_REF][START_REF] Baaijens | An iterative solver for the DEVSS/DG method with application to smooth and non-smooth flows of the Upper Convected Maxwell fluid[END_REF]. First, the non-linearity and the hybrid type of constitutive equations (elliptic and hyperbolic) make the convergence of the iterative algorithm difficult especially with geometrical or boundary conditions singularities. The second difficulty is the time-dependent computation of the interface. For these reasons, time dependent viscoelastic computations of multi-layer flow have been performed with rather simple rheological models, at moderate Weissenberg numbers and in simple geometries [START_REF] Valette | Convective instabilities in coextrusion process[END_REF][START_REF] Yamaguchi | An arbitrary Lagrangian-Eulerian approach for simulating viscoelastic fluids[END_REF][START_REF] Zatloukal | Numerical simulation of polymer coextrusion flows: a criterion for detection of ?wave? interfacial instability onset[END_REF]. The competition between surface tension force and elastic stress for mono-layer flows is also explored in [START_REF] Bousfield | Non-linear analysis of the surface tension driven breakup of viscoelastic filaments[END_REF][START_REF] Langouche | Rheological characterisation of a highdensity polyethylene with a multi-mode differential viscoelastic model and numerical simulation of transient elongational recovery experiments[END_REF].

The aim of this paper is to perform a numerical study of the propagation of an interface perturbation in real die geometries and with realistic constitutive equations for the molten polymers. First we use a 3D finite element computation of the whole coat-hanger die geometry (with only one polymer). The velocity is maximal at the die inlet and at the die exit and it is supposed that instabilities will originate in these flow regions where the Weissenberg number is a maximum. In addition the flow pattern is 2D in these two flow regions. So we develop a 2D multilayer viscoelastic model with a precise capturing method of the interface. Then we apply this method to the inlet zone and the exit zone geometry of the coat-hanger die for various initial instability frequencies and for various processing conditions where defects are (or are not) experimentally observed. Finally we test some modifications of the geometry and of the processing conditions.

Presentation of coextrusion process and polymers

We consider a geometry composed of three zones (figure 2): the inlet zone is a feeding channel with a narrow width and an important thickness; then there is a distribution channel with a complex geometry (called coat hanger geometry) which allows to distribute regularly the polymer flow through the die width (350 mm). The third broad zone is of uniform thin thickness (1 mm).

The two coextruded polymers (PET1 and PET2) are polyester based patented formulations. They present very different rheological properties which have been measured using a cone and plate RMS 800 rheometer. Dynamic rheology measurements give the complex modulus (usually called G ′ and G ′′ ) for various temperatures. By assuming the Cox-Merz rule, viscosity η (the norm (G ′ /ω) 2 + (G"/ω) 2 of complex viscosity)) and relaxation time (λ = G ′ /ωG ′′ ) are deduced. The bottom layer (PET1) in contact with the steel sheet has good adhesive properties and marked viscoelastic behaviour. The upper layer (PET2) is classical polyester with a quasi-Newtonian behaviour. Figure 3 shows the master curve for the viscosity of both polymers at 260 o C and figure 4 the master curve for the relaxation time at the same temperature. The activation energy of both polymers is very different: E = 38 kJ/mole for PET1 and E = 61 kJ/mole for PET2.

This makes this coextrusion situation particularly interesting. At the reference temperature (260 o C) PET1 is much more viscous than PET2 within the extrusion processing range (typically 10 to 100 s -1 ) but both viscosities become equivalent at higher rates. When decreasing the temperature the viscosity of PET2 increases much more than the viscosity of PET1 and so the gap between viscosities of both polymers decreases and the shear rate for which the viscosities are equivalent decreases towards the range of processing conditions. Following the literature ( [START_REF] Han | Rheological investigation of interfacial instability in two layer flat-film coextrusion[END_REF][START_REF] Han | Studies on multilayer film coextrusion I. The rheology of flat film coextrusion[END_REF][START_REF] Han | Studies on multilayer film coextrusion II. Interfacial instability in flat film coextrusion[END_REF][START_REF] Anturkar | Estimation of critical stability parameters by asymptotic analysis in multilayer extrusion[END_REF]) one may believe a strong influence of temperature and flow rate on the defect appearance. Figure 4 shows that the mean relaxation time is much more important for PET1 than for PET2 in the extrusion processing range.

As a consequence PET2 may be considered at a first glance as a Newtonian fluid ( = 582 P a.s at the reference temperature 260 o C).

PET1 behaviour is fitted with two constitutive equations:

-A Carreau-Yasuda equation for the viscosity will be used for the 3D computation:

η( γ, T ) = η 0 exp E R 1 T - 1 T 0 1 + (λ γ) a m-1 a (1)
where η 0 is the Newtonian plateau viscosity at the reference temperature T 0 (η 0 = 4791 P a.s at 260 o C), γ is the second invariant of the rate of strain tensor,

γ = 2 i,j ǫ2 i,j ,
R is the universal gas constant, λ (0.34), a (0.42) and m (0.48) are fitting parameters.

-A multimode Johnson-Segalman model [START_REF] Johnson | A model for viscoelastic fluid behavior which allows nonaffine deformation[END_REF] will be used for the 2D time dependent computations:

σ = -pI + 2η s ǫ(u) + τ ; τ = i τ i (2)
τ i + λ i (1 - b 2 ) δτ i δt + b 2
Dτ i Dt = 2η i ǫ(u) (3)
η s is a solvent viscosity (η s = 466 P a.s), b is a parameter which balances the upper convected derivative and the Jauman derivative of the extrastress tensor τ i [START_REF] Johnson | A model for viscoelastic fluid behavior which allows non-affine deformation[END_REF] (b = 0.15), p is the isotropic part of the stress tensor σ and ǫ(u) is the rate of deformation tensor. η i and λ i are deduced from the dynamic rheology measurements using Marquadt-Levenberg method (table 1). Six modes allow to obtain a good agreement with dynamic measurements [START_REF] Boukellal | Evaluation of a tubebased constitutive equation using conventional and planar elongation flow optical rheometers[END_REF].

The figures 3 and 4 shows that the viscosity and relaxation time of PET1 are shear dependant. Therefore the Reynold number and the Weissenberg number can be defined by describing this dependency by a power law ( η = η 0 | γ| m-1 and λ = λ 0 | γ| n-1 for example). In this way, one gets for a die with a width W and a thickness h subjected to a flow rate Q, the following expressions

Re = ρ Q 2-m h m-1 W m-2 η 0 ; W e = λ 0 Q W h 2 n (4)

Coextrusion experimental measurements

Several experiments have been performed at Arcelor Mittal facilities in Thionville (France). The temperature regulation of both single screw extruders (diameter 25mm and 30mm respectively) and their rotation velocities have been adjusted in such a way to obtain a constant temperature of 260 o C at die outlet. The die temperature has been also fixed to 260 o C. The rotation velocity of the two extruders has been varied and, after stabilization of the process, the quality of the coextruded sample has been quantified from 0 (good quality) to 5 (bad quality). In table 2 experimental observations are reported. A pressure sensor is put at the output of each extruder and measurements are made with and without the coat hanger die for various flow rates. In this way the pressure loss inside coextrusion die can be evaluated. We have kept samples for which the defects are not too important and correspond to rather regular waves. We assume that such disturbances could be computed more easily and exhibit a linear spatial growth rate.

3D monolayer temperature dependent computation in the extrusion die

To analyse the flow pattern, we perform a 3D monolayer calculation of the polymer flow in the coat hanger die using the finite element software REM3D c dedicated to polymer processing applications in injection and extrusion [START_REF] Silva | Three-dimensional injection molding simulation[END_REF]. We consider the flow of PET1 with a purely viscous temperature dependent behaviour using the CarreauYasuda law [START_REF] Han | Rheological investigation of interfacial instability in two layer flat-film coextrusion[END_REF]. An anisotropic mesh with 55739 elements and 315314 nodes is used to mesh the computational domain depicted in figure 2. It is refined in the thickness and coarse in the two other directions [START_REF] Gruau | Coupez, 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric[END_REF]. We solve iteratively the mass and force balance equations (incompressible) and the energy balance equation till convergence. The computation is made for a flow rate Q = 1.10 cm 3 /s and the computed pressure loss is 3.4 MP a. This latter value is consistent with experimental measurement pressure loss (3.5MP a).

Figure 5 shows the velocity distribution in the die. The velocity component in the transverse direction is negligible except in the coat hanger region which distributes the polymer throughout the width of the die (figure 5b). As a consequence, the flow may be considered as 2D both in the inlet and in the outlet regions of the die. It is possible to compute a crude evaluation of the Reynolds and Weissenberg numbers in this two zones for both polymers (see relation ( 4)). The Reynolds number is around 10 -5 for both fluids and both zones. For PET1, the Weissenberg number is around 1 in inlet zone and larger in the outlet zone. It increases with the shear rate (or the mean flow).

Figure 6 shows the mean temperature field (throughout the die gap) when the initial temperature of the polymer is 281 o C and the temperature of the die is 260 o C. One observes an important temperature decrease within the die, but the flow may be considered as isothermal both in the inlet and the outlet regions.

We assume that introducing a viscoelastic constitutive equation will provide the same conclusions for the velocity field and the temperature field. As a consequence we will develop 2D dynamic isothermal viscoelastic computations in these two flow regions in order to analyse if perturbations are damped or amplified.

2D multilayer viscoelastic isothermal computation

We apply the finite element viscoelastic computation method, already developed by Silva et al. [START_REF] Silva | Viscoelastic compressible flow and application in 3D injection modeling simulation[END_REF][START_REF] Silva | A new three-dimensional mixed finite element for direct numerical simulation of compressible viscoelastic flows with moving free surfaces[END_REF], to the multi-mode Johnson-Segalman model [START_REF] Han | Studies on multilayer film coextrusion I. The rheology of flat film coextrusion[END_REF][START_REF] Han | Studies on multilayer film coextrusion II. Interfacial instability in flat film coextrusion[END_REF]. The numerical method is explained for one layer and for one mode. First the dynamic equilibrium and mass balance equations are solved by a splitting method which writes when neglecting mass and inertia forces:

∇ . [2η s ǫ(u n+1 )] -∇p n+1 I = ∇ . τ n (5) ∇ . u n+1 = 0
At time step (n + 1) one solves equations ( 5) with a known value of the extra-stress τ n at time step n. Then one solves the constitutive equation ( 6) to determine τ n+1 :

τ n+1 + λ (1 - b 2 ) δτ n+1 δt + b 2
Dτ n+1 Dt = 2η ǫ(u n+1 ) (6)
An advantage of decoupling is that the global problem, with a large number of unknowns, and therefore large-consuming memory, is broken up into smaller systems: a Stokes incompressible problem perturbed by an elastic term (treated as a known body force term) and the non-linear constitutive equations that allow the determination of τ . Furthermore, each problem can be treated taking into account its mathematical type: the Stokes problem ( 5) is elliptic and linear, whereas the computation of extra stress tensor with the constitutive equation ( 6) involves non-linearities and convection. Disadvantages are only related to the need of a marching time scheme for both calculations, which could lead to time instabilities if not carefully carried out.

Consequently the computational time is rather important as we have to deal with two constraints on the time step ∆t:

∆t < 1/ max ( γ) and ∆t < min

i (λ i ) (7)
Finally, the time step chosen in the sequel is ∆t = .005. Finite element method is used to solve the above equations. For the Stokes equation ( 5), we choose a linear interpolation for velocity and pressure with bubble enrichment for velocity, the P + 1 /P 1 element, also referred as the MINI-element [START_REF] Arnold | A stable finite element for the Stokes equations[END_REF]. Note also that a solvent viscosity is included in the proposed formulation [START_REF] Valette | Investigation of the interfacial instabilities in the coextrusion flow of polyethylene and polystyrene[END_REF]. Some authors can completely eliminate it, by using stabilization methods like the EVSS or the DEVSS formulations. In our approach, solvent viscosity can be very small because the bubble stabilization procedure is approaching the DEVSS method of Guénette and Fortin [START_REF] Guénette | A new finite element method for computing viscoelastic flows[END_REF].

The evolution equation ( 6) is solved through a Space-Time Discontinuous Galerkin method. Basically, Space-Time means that it allows a simultaneous resolution in space and in time (x, t) and the computational domain is (Ω×]0, t e [), where t e is the duration of the process studied; Discontinuous Galerkin means that we choose discontinuous interpolations both in space (low order) and in time (higher order). On the element K, extra stress is interpolated by functions constant and discontinuous in space, P 0(K), and polynomial (degree n) and discontinuous in time (P n(]t i , t i+1 [)). The case n = 0 corresponds to the classical discontinuous Galerkin method [START_REF] Cockburn | The local discontinuous Galerkin method for time-dependent convection-diffusion system[END_REF][START_REF] Fortin | Ten Years Using Discontinuous Galerkin Method for Polymer Processing Problems[END_REF].

For the bi-layer coextrusion considered here (figure 8), the bottom layer Ω 1 is viscoelastic and the upper layer Ω 2 is purely Newtonian and direct solutions of equations ( 5) are computed. In addition to classical sticking conditions along the die walls, we assume at the interface Γ the continuity of the velocity, the non-miscibility and the continuity of the normal stress components (neglecting surface tension) :

u 1 = u 2 ; u . n = 0 ; σ 1 . n = σ 2 . n ( 8
)
where n is the normal at the interface. Two flow rates are imposed at the inlet (one for each polymer) and a free surface condition at die exit. The first conditions correspond to a Dirichlet conditions as the imposed velocity is chosen such that it satisfies the condition

Γ u i • n dΓ = Q i
in each layer, n being the normal to die inlet. The second condition is the classical Neuman boundary condition, that is

σ i • n = 0.
n being here the normal to die outlet.

In order to calculate a moving free surface, we have adopted an Eulerian approach. Let us introduce I the characteristic function of the first layer domain defined by:

I(x, t) = 1 if x ∈ Ω 1 0 if x / ∈ Ω1 (9)
In this way, the rheological behaviour of both fluids depends on this characteristic function and the finite element formulation satisfies automatically the velocity and normal stress component continuity at the interface. The non-miscibility (kinematic) condition which gives the evolution of each phase with respect to flow motion corresponds to a transport equation for the characteristic function [START_REF] Fortin | Computation of stationary interface between generalized Newtonian fluids[END_REF]:

∂I ∂t + u . ∇I = 0 (10)
Therefore, the characteristic function itself becomes an unknown function which can be approximated using a finite element technique. In practical, an approximation I h constant by element (P 0 approximation) has been used:

I h | K = I K = |K| ∩ |Ω 1 | |K| . ( 11
)
It is simply the ratio in each element K between the surface occupied by the phase 1 and the total surface of element K. This hyperbolic equation is also solved by a space-time Galerkin Discontinuous method [START_REF] Batkam | A space-time discontinuous Galerkin method for convection and diffusion in injection moulding[END_REF]. In order to improve the precision, refined anisotropic meshes have been used around the interface (see figure 7) [START_REF] Gruau | Coupez, 3D tetrahedral, unstructured and anisotropic mesh generation with adaptation to natural and multidomain metric[END_REF].

Post-processor tools allow to visualize and analyse precisely the evolution of the interface and are described in the sequel. First, the thickness of the bottom layer at each abscissa x i is given by:

h(x i ) = I(y, x i ) dy = K I K l K (x i ) ( 12
)
where l K (x i ) is the length of the vertical line (x i , y) which belongs to element K (figure 7). For a more comprehensive view of disturbance evolution, a xt diagram is plotted. This diagram shows the isovalue of relative perturbation in the (x, t)) plane. Computations are made for a large number of time steps (typically ten time the forcing period) and the mean value of interface is estimated for each abscissa x i :

h m (x i ) = Nt j=1 h(x i , t j )/N t ( 13
)
and therefore the relative perturbation writes:

∂h(x i , t j ) = h(x i , t j ) -h m (x i ) h m (x i ) ; (14)
As shown in figures 9 and 10, travelling wave gives isovalues parallel to a line satisfying the relation

2πt/λ t -2πx/λ x = Cte, (15)
λ t and λ x being the temporal and spatial periods of the travelling wave. Spatial evolution is represented by color evolution along x for each time t. Finally, the numerical value of growth or decay rate of the disturbance is computed by tracing the exponential envelope. The spatial and temporal periods of the progressive wave are computed with a two dimensional Fourier analysis.

2D isothermal Dynamic stability analysis

For purely viscous fluids, the driving force for instability development is the Reynolds number. Due to the high viscosity of both polymers, the Reynolds number is very small (of the order 10 -5 ) and Mahdaoui showed [START_REF] Mahdaoui | Étude numérique des Instabilités dans le procédé de coextrusion des polyesters[END_REF] that it was unable to develop instabilities. In the present situation the driving force is the Weissenberg number which is near zero for PET2 but may be of the order of one for PET1. Figures 8 present the stationary interfaces for both inlet (figure 8a) and outlet (figure 8b) zones for the case N o 4 of table 1. In the inlet region, the interface between both fluids stabilizes on a short distance (less than one gap) and then remains flat. The same situation is observed in the outlet region. The location of the interface changes only in the convergent part of the die and stabilizes again.

Starting from the stable interface location (Figures 8a and8b), a periodic disturbance is applied on the first layer [START_REF] Mahdaoui | Étude numérique des Instabilités dans le procédé de coextrusion des polyesters[END_REF]:

Q 1 = Q 10 1 + A sin 2π T t ( 16
)
where the amplitude of the perturbation A = 0.1 and T is the forcing period which may correspond to experimental flow fluctuations related to the screw rotation of each extruder. This would correspond to forcing frequencies between 0.3 and 1.5 Hz and we have chosen to check two frequencies 0.5 Hz and 1 Hz. Diagrams xt show the appearance of a travelling wave. Moreover the computations of the growth rate are made when they can be fitted by an exponential law ( that means that it does not correspond to a non linear saturation). The sign of this coefficient gives the stability. For more complex situation, we only check if the perturbation is bigger at the exit than at the entrance.

Numerical simulations in the inlet zone

The inlet zone corresponds to a rectangular channel of size 110 × 10 (aspect ratio ∼ .091). Figures 9 show the evolution of the perturbation along the inlet zone of the die for 0.5 Hz forcing frequency and experimental case number 4 (see table 1). It is found that the perturbation has the form of travelling wave having a damping amplitude and a temporal dependency corresponding to the forcing frequency. The spatial periodicity is 23 mm and the decay rate of the disturbance is 0.007. The values for the other cases are reported in table 3 and show that the perturbation is always damped. The forcing frequency is always recovered expect for the case N o 2. In this case, the spatial disturbance associated to forcing frequency is too large with respect to die length. Therefore the system selects a smaller spatial periodicity that induces a slight difference between the measured and imposed temporal frequencies.

Numerical simulation in the outlet zone

The outlet zone corresponds to two successive rectangular channels (60 × 2 and 17 × .9) connected by a convergent geometry with an angle of 15 o (see figure 8b). Figures 10 show the evolution of the perturbation along this outlet zone for 0.5 Hz forcing frequency and experimental case number 4 (see table 1). Here again, the perturbation has the form of a travelling wave but it is amplified. The spatial evolution for 1 Hz forcing frequency is also plotted in figure 11. The situation is less unstable but this picture could help the discussion and the comparisons when processing conditions will be changed. The xt diagram shows two areas before and after the convergent with different spatial periodicities (see figures 10). Figure 10a shows that the perturbation is amplified in the first flow region and not damped by the convergent part and the final flow region.

The two dimensional Fourier analysis and the damping (or growth) rate of the perturbation is computed before the convergent part. The final die geometry is too short to allow calculating an accurate damping or growth rate factor in this region. In the case depicted in figures 10, the forcing frequency is recovered . There are various situations for which the growth rate is difficult to evaluate:

-For case N o 1 and forcing frequency 0.5 Hz, the perturbation is damped in the first part and it is amplified in the convergent part.

-For case N o 1 and forcing frequency 1 Hz, several frequencies are activated and it is difficult to compute a growth rate as the disturbance is subjected to nonlinear effects. Nevertheless the perturbation remains important at the die exit and the configuration is labelled as unstable.

-For case N o 4 and forcing frequency 1 Hz, the perturbation is first spatially amplified, then is damped just before the convergent part. The final part does not seem to change the amplitude of the perturbation.

Finally the conclusions of this section are the following :

-The perturbation is only amplified in the outlet zone.

-The numerical computations are rather in good agreement with experiments: stable samples (case 2 and 3) reported in -The small frequencies are most dangerous for this system.

-The influence of the convergent geometry is not clear: it reduces the amplitude of perturbations for case 4 (forcing frequency 1 Hz) but amplified it in case 1 (forcing frequency 0.5 Hz).

7. Other Processing Conditions (Geometry, Flow rate, Temperature)

In this section, we analyse the influence of processing conditions and die outlet geometry on the appearance of perturbation. For a fixed thickness ratio (which is imposed by the target sheet application), it is possible to modify both flow rates as well as the temperature of one or both polymers in order to prevent the appearance of wavy interface. Case N o 4 in the outlet zone is considered for different flow rates and different temperatures.

First, the flow rates of both polymers are multiplied by two which means that the shear rate and viscosity ratios change. Figure 12 depicts the spatial evolution of the disturbance for the two forcing frequencies 0.5 et 1Hz. In both cases, the travelling waves have a spatial amplification. It is most important for 1 Hz (the growth rate is around 0.02) than 0.5 Hz (growth rate ∼ 0.015). Note that other spatial frequencies are observed which characterizes a nonlinear behaviour or saturation. In that case we are faced to two contradictory phenomena: increasing the flow rates and so the shear rates will decrease the viscosity ratio between the two polymers (see figures 2) and this is supposed to stabilize the process, but at the same time the Weissenberg number which drives the development of the instability will increase.

The computations of section 6 are made with a temperature of 260 o C. If the temperature is increased to 280 o C, the viscosity ratio is increased because the activation energy are different (38 KJ/mol for the PET1 and 61 KJ/mol for the PET2). In the same time the relaxation time of PET1 (the distribution of relaxation time of table 1) decreases and also the Weissenberg number.

Figures 13 show spatially amplified travelling waves for both forcing frequencies. For 0.5 Hz, the growth rate is 0.01, whereas it is 0.006 for 1 Hz.

They are smaller than those computed for 260 o C (see table 4). Here again, increasing the Weissenberg number seems to influence more the stability of process than increasing the viscosity ratio.

We investigate the influence of the convergent angle between the two slit flows on the amplitude of the perturbation. Three geometries have been considered with 0 o (parallel slit die), 15 o (previous slit die) and 30 o .

For a slit of uniform thickness (0 o angle), figure 14 shows the evolution of perturbations along the flow direction for the forcing frequencies 0.5 and 1 Hz. The generated travelling waves are maintained for frequency 1 Hz and amplified for 0.5 Hz. Therefore without the convergent geometry, the perturbation is maintained or amplified whereas it was slightly damped for forcing frequency 1 Hz and amplified for 0.5 Hz for the angle 15 o which was previously investigated.

For the angle 30 o , figures 15 show the evolution of perturbations along the flow motion for the forcing frequencies 0.5 and 1 Hz. As previously the initial perturbation is amplified for 0.5 Hz and maintained for 1 Hz. The change of spatial periodicity after the convergent geometry is pointed out clearly. It is more important than for an angle of 15 o . Then, the convergent geometry modifies the spatial periodicity of perturbation without amplification. Khomami and Wilson [START_REF] Khomami | An experimental investigation of interfacial instability in superposed flow of viscoelastic fluids in a converging/diverging channel geometry[END_REF] studied experimentally interfacial instabilities in both diverging and converging channels. They found that it is less amplified in converging channel. Theoretical analysis of Hinch and coworkers [START_REF] Hinch | The instability mechanism for two elastic liquids being co-extruded[END_REF] predicts the stabilizing effect of converging channel flows and a destabilizing effect for diverging ones. In this industrial set-up , we do not observed any significant influence of the shape of the convergent geometry on the development of instabilities.

Conclusions

Dynamic stability analysis of the interface between two layers in the coextrusion process of thermoplastic polymers has been performed. It points out the development of progressive waves along the die geometry. For viscoelastic fluids, the driving force is the Weissenberg number which is much higher than the Reynolds number. The results show that the amplitude of the progressive waves may increase or damp towards the die exit depending on the processing parameters, the die geometry and the rheology of the polymer. The results are in agreement with experiments done on industrial coextrusion device. Nevertheless they require further experiments and the test of more advanced constitutive equations to become an useful tool for optimizing the processing parameters (temperatures, output rates, ...), the die geometry and then adjusting the rheology of the coextruded polymers. 4: Fourier Analysis and growth rate of the perturbation in the outlet zone; the growth or damped rates are only given when the envelope of perturbation can be fitted by an exponential law.

X t

  i

Figure 1 :

 1 Figure 1: Characteristic defect on an industrial coextrusion line

Figure 2 :Figure 3 :Figure 4 :

 234 Figure 2: View from the top and through the thickness of the coat-hanger geometry: (a) 1/2 die and the three different areas ; (b) view in the thickness direction along the symmetry plane

Figure 5 :

 5 Figure 5: Mid-plane velocity field (cm/s) in the flow direction (a) and in the transverse direction (b) (the flow rate is 1.10 cm 3 /s) and density ρ = 1.33 kg/dm 3 .

Figure 6 :

 6 Figure 6: Mean temperature distribution in the die: the inlet temperature is 281 o C, die temperature is 260 o C; same flow rate as in figure 5

Figure 7 :

 7 Figure 7: Isovalue of the characteristic function and mesh refining. Computation of the straight line l K which belongs to the element K

Figure 8 :Figure 9 :

 89 Figure 8: Calculation of stationary interface : (a) inlet zone; (b) outlet zone. (Q PET1 = 1.99 cm 3 /s; Q PET2 = 3.72 cm 3 /s, case 4)

Figure 10 :

 10 Figure 10: Relative perturbation for a forcing frequency of 0.5 Hz in the outlet zone; processing conditions correspond to the unstable experimental case 4: (a) Spatial evolution for various time steps; (b) xt diagram

  table 2 correspond to negative growth rate of the perturbations. The most unstable sample (case 1) reported in table 2 is also reported as unstable for both frequencies in table 4. Sample 4 which is reported as slightly unstable in table 2 is reported as stable for hight frequency (1 Hz) and unstable for low frequency (0.5 Hz) in table 4.

Table 1 :

 1 Relaxation spectrum for PET1 at the reference temperature 260 o C.

		1	2	3	4	5	6
	η i	457.64 614.97 770.55 681.61 202 25.23
	λ i (s) 0.04	0.14	0.51	1.84 5.44 20.37
	case PET1 (rpm) PET 2 (rpm) Experimental Observations
	1	20	90			2
	2	50	25			0
	3	50	50			0
	4	50	90			1

Table 2 :

 2 Coextrusion observations: the quality of coextruded product has been marked from 0 (good quality) to 5 (very bad quality).

	Case	1/λ t (Hz)	λ x (mm)	Growth rate
		1 Hz 0.5 Hz 1 Hz 0.5 Hz	1 Hz	0.5 Hz
	1	1.0	0 .50	12.23	6.80	-0.037 (S) -0.018 (S)
	2	0.8	0.40	23.90	27.04	-0.009 (S) -0.004 (S)
	3	1.0	0.50	9.05	18.20	-0.044 (S) -0.019 (S)
	4	1.0	0.50	11.32	22.63	-0.035 (S) -0.007 (S)

Table 3 :

 3 Fourier Analysis and growth rate of the perturbation in the inlet zone

	Case	1/λ t (Hz)	λ x (mm)	Growth rate
		1 Hz 0.5 Hz 1 Hz 0.5 Hz	1 Hz	0.5 Hz
	1	1.45	0.68	18.81	8.06	U	U
	2	1.00	0.52	3.52	7.05	-0.10 (S)	-0.028 (S)
	3	1.00	0.48	4.70	9.40	-0.043 (S)	-0.002 (S)
	4	1.06	0.50	6.27	14.7	S	0.022 (U)
	Table

Acknowledgements

O. Mahdaoui has received a support from Arcelor-Mittal. The authors also thank A. Merveille from Arcelor-Mittal for his helpful comments and remarks.


Eric Angel

Romain Campigotto

LIMOS Christian Laforest
email: christian.laforest@isima.fr

A New Lower Bound on the Independence Number of Graphs ✩

Keywords: independence number, graphs, variance, expectation, vertex cover

). We obtain our bound by using the Bhatia-Davis inequality applied with analytical results (minimum, maximum, expectation and variance) of an algorithm for the vertex cover problem.

Introduction

The independence number of a graph is one of the most fundamental and well-studied graph parameters [START_REF] Lozin | Foreword: Special Issue on Stability in Graphs and Related Topics[END_REF]. In view of its computational hardness [START_REF] Håstad | Clique is Hard to Approximate Within n 1-ǫ[END_REF], various bounds have been proposed [START_REF] Bossecker | Interpolating between Bounds on the Independence Number[END_REF][START_REF] Cartwright | The Maximum Independent Sets of de Bruijn Graphs of Diameter 3[END_REF][START_REF] Chung | The Average Distance and the Independence Number[END_REF][START_REF] Mubayi | On the Independence Number of the Erdős-Rényi and Projective Norm Graphs and a Related Hypergraph[END_REF][START_REF] Selkow | A Probabilistic Lower Bound on the Independence Number of Graphs[END_REF]. Obtaining such bounds is an important and hard challenge. The first one was given independently by Y. Caro and V. Wei [START_REF] Caro | New Results on the Independence Number[END_REF][START_REF] Wei | A Lower Bound on the Stability Number of a Simple Graph[END_REF]. The most recent is due to J. Harant [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF].

In this paper, we propose a new lower bound for this parameter. We prove that our bound is often strictly better than the ones recently proposed in [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF] and [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF]. To obtain it, we use the Bhatia-Davis inequality (see [START_REF] Bhatia | A Better Bound on the Variance[END_REF]) applied with the parameters capturing the behavior of an algorithm to compute the wellknown vertex cover problem. Sect. 2 contains these preliminary analytical results. In Sect. 3, we prove our lower bound and compare it.

Definitions, Notations, Problems

Graphs G = (V, E) considered throughout this paper are undirected, finite, simple and unweighted. We denote by n the number of vertices (n = |V |). For any vertex u ∈ V , we denote by N (u) the set of neighbors of u (i.e. the set of vertices sharing an edge with u), d(u) = |N (u)| the degree of vertex u and ∆ = max{d(u) | u ∈ V } the maximum degree of G.

The Vertex Cover Problem. The vertex cover problem is a well-known classical NP-complete optimization graph problem [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF]. A cover C of G is a subset of vertices such that every edge contains (or is covered by) at least one vertex of C, that is C ⊆ V and ∀e = uv ∈ E, one has u ∈ C or v ∈ C (or both). The vertex cover problem is to find a cover of minimum size.

The Independence Number of a Graph. A set of vertices I ⊆ V in a graph G is called independent if the graph induced by vertices of I in G contains no edge. The independence number α of G is the largest cardinality |I| among all independent sets I of G.

Preliminary Results

We focus in this section on the vertex cover problem. More precisely, we analyze a simple algorithm, that we call LL: for any graph G, we express the size of the best and the worst solutions it can produce; we also express the expectation and the variance of the sizes. These results will be used in Sect. 3 to express our main result, namely our lower bound on independence number of any graph (thanks to the Bhatia-Davis inequality).

Labeling of Nodes, Left and Right Neighbors. In a labeled graph, denoted by G = (V, L, E), the n vertices of G are labeled by a given function L (called labeling) such that each vertex u ∈ V has a unique label L(u) ∈ {1, . . . , n} (L is a bijection between V and {1, . . . , n}). We denote by L(G) the set of all possible labelings L for a graph G = (V, E). Given a labeled graph G = (V, L, E) and a vertex u ∈ V , v is called a right neighbor (resp. left neighbor ) of u if v ∈ N (u) and if v has a label larger (resp. smaller) than u.

The algorithm LL can be described as follows on any labeled graph G = (V, L, E). For each vertex u ∈ V , u is added to the cover if it has at least one right neighbor.

It is easy to see that LL constructs a cover for any graph G, given any labeling L of G. Of course, the solution (and size) it returns strongly depends on L.

Property 1 ([2]). For any graph G:

1. There exists L * ∈ L(G) such that LL returns an optimal solution on G = (V, L * , E). 2. LL cannot construct a cover of size greater than nc (where is c the number of connected components of G).

With equiprobability on the labelings of G with n vertices, we have:

(a) The probability that a vertex u of G is in the solution is 1 -1 d(u)+1 .

(b) The expected size of the solution produced by LL on G is

E [LL(G)] = n - u∈V 1 d(u) + 1 . (1)
All these properties are easy and are proved in [START_REF] Angel | Analysis and Comparison of Three Algorithms for the Vertex Cover Problem on Large Graphs with Low Memory Capacities[END_REF]. To be complete, we recall the ideas. Let L * labeling first the vertices of any optimal vertex cover C * and after all the other vertices. Clearly LL applied on L * returns C * . For the second property, suppose that c = 1. In any labeling, the last labeled vertex will not be chosen, producing a solution of size at most n -1 (generalization to any c is trivial).

The probability 1 -1 d(u)+1 is the complement of the result shown in the famous proof of Y. Caro and V. Wei for the Turan's theorem (see [START_REF] Alon | The Probabilistic Method[END_REF]), since the complement of any independent set is a vertex cover (and vice-versa). The expectation E [LL(G)] is then easily obtained from the previous probability and the linearity of the expectation.

We complete these results by the variance of LL in what follows.

Theorem 1. Let G = (V, E) be any graph. By considering the n! labelings of L(G) with equiprobability assumption, the variance of LL is

V [LL(G)] = u∈V d(u) (d(u) + 1) 2 -2 uv∈E 1 (d(u) + 1)(d(v) + 1) + 2 uv ∈E d uv (d(u) + 1)(d(v) + 1)(2 + d(u) + d(v) -d uv ) , (2)
with d uv = |{w ∈ V | uw ∈ E ∧ wv ∈ E}|.
Proof. We can express V [LL(G)] by a sum of covariances (for more details, see e.g. Chap. 9 of [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]). Let C be a cover returned by algorithm LL. For each vertex u ∈ V , we associate a random variable ½ u such that ½ u = 1 if u ∈ C and 0 otherwise. Let X = u∈V ½ u be the random variable giving the size of C.

We have

V [X] = u∈V Cov(½ u , ½ u ) + 2 {u,v}⊆V u<v Cov(½ u , ½ v ) , (3)
where u < v means that we count only once Cov(½ u , ½ v ) in the sum;

Cov(½ u , ½ v ) = E [½ u ½ v ] -E [½ u ] • E [½ v ] is the covariance of random variables ½ u and ½ v . With Property 1, we get E [½ u ] = 1 - 1 d(u) + 1 . (4)
Now, by using (4), we can give closed formulas for each term of (3).

First, as we have

Cov(½ u , ½ u ) = E ½ 2 u -E [½ u ] 2 = E [½ u ] -E [½ u ] 2 , we obtain Cov(½ u , ½ u ) = 1 - 1 d(u) + 1 -1 - 1 d(u) + 1 2 = d(u) (d(u) + 1) 2 . (5)
Now, for Cov(½ u , ½ v ), we distinguish the case where uv ∈ E and the one where uv ∈ E.

When uv ∈ E, at least u or v is in the solution. So ½ u ½ v = 1 if the "second" vertex is also in the solution. If L(u) < L(v) (resp. L(v) < L(u)) then u (resp. v) is in the solution and v (resp. u) is not in the solution with probability 1 d(v)+1 (resp.
½ u ½ v = 1) with probability 1 -( 1 d(u)+1 + 1 d(v)+1 ) (L(u) < L(v)
and L(v) < L(u) are two disjoints cases). As ½ u ½ v = 0 or 1, we get

E [½ u ½ v ] = 1 - 1 d(u) + 1 - 1 d(v) + 1
.

Thus, if uv ∈ E, Cov(½ u , ½ v ) = 1 - 1 d(u) + 1 - 1 d(v) + 1 -1 - 1 d(u) + 1 1 - 1 d(v) + 1 = -1 (d(u) + 1)(d(v) + 1) . (6)
We treat now the case where uv ∈ E. We prove later that

E [½ u ½ v ] = 1 - (d(u) + d(v) -d uv + 1)(d(u) + d(v) + 2) (d(u) + 1)(d(v) + 1)(d(u) + d(v) -d uv + 2) . (7)
Before making the proof of [START_REF] Cartwright | The Maximum Independent Sets of de Bruijn Graphs of Diameter 3[END_REF], suppose first that the result is correct. In this case, when uv ∈ E, we have

Cov(½ u , ½ v ) = 1 - (d(u) + d(v) -d uv + 1)(d(u) + d(v) + 2) (d(u) + 1)(d(v) + 1)(d(u) + d(v) -d uv + 2) -1 - 1 d(u) + 1 1 - 1 d(v) + 1 = d uv (d(u) + 1)(d(v) + 1)(d(u) + d(v) -d uv + 2) . (8)
Therefore, by replacing the expressions of covariances of ( 5), ( 6) and ( 8) in the formula (3), we obtain (2). Hence, to complete the proof of Theorem 1, we just need now to prove [START_REF] Cartwright | The Maximum Independent Sets of de Bruijn Graphs of Diameter 3[END_REF], with the hypothesis that uv ∈ E.

By definition here, E [½ u ½ v ] = P [u ∈ C ∧ v ∈ C] (
where C designates a so- lution produced by LL). By decomposing the sub-cases of the complementary probability, this expectation is also equal to

1 -(P [u ∈ C] + P [v ∈ C] -P [u ∈ C ∧ v ∈ C]) .
We already saw that P

[u ∈ C] = 1 d(u)+1 and similarly P [v ∈ C] = 1 d(v)+1 . Let us compute now P [u ∈ C ∧ v ∈ C].
We consider the case where L(v) < L(u) (the other is symmetric). To get the result, we just need to consider the relative numberings of u, v and their d(u)+d(v)-d uv neighbors (the labels of the others vertices play no role here on the fact that u or v be in C). As L(v) < L(u), u must have the highest of these d(u) + d(v)d uv + 2 labelings (otherwise, u or v is in C). Choose d(v) + 1 of them, fix v at the first chosen position (i.e. it gets the highest label), and assign the remaining d(v) neighbors of v arbitrarily to the remaining d(v) chosen positions. Finally, assign the d(u)d uv vertices (which are neighbors of u but not of v) arbitrarily in the remaining d(u)d uv positions. This gives

d(u) + d(v) -d uv + 1 d(v) + 1 • d(v)! • (d(u) -d uv )! = (d(u) + d(v) -d uv + 1)! d(v) + 1 cases. Replacing d(u) by d(v) gives the symmetric result when L(v) > L(u).
Hence, dividing by the (d(u) + d(v)d uv + 2)! possible (restricted) labelings, we obtain

P [u ∈ C ∧ v ∈ C] = 1 (d(u) + d(v) -d uv + 2) 1 d(u) + 1 + 1 d(v) + 1 = d(u) + d(v) + 2 (d(u) + d(v) -d uv + 2)(d(u) + 1)(d(v) + 1))
.

Combining all these results, we get

E [½ u ½ v ] that is 1 - 1 d(u) + 1 - 1 d(v) + 1 + d(u) + d(v) + 2 (d(u) + d(v) -d uv + 2)(d(u) + 1)(d(v) + 1))
.

Easy simplifications lead to [START_REF] Cartwright | The Maximum Independent Sets of de Bruijn Graphs of Diameter 3[END_REF] and ends the proof of Theorem 1.

A New Lower Bound for the Independence Number

We can now use the result of R. Bhatia and C. Davis [START_REF] Bhatia | A Better Bound on the Variance[END_REF] linking the variance ϑ, the expected value µ, the minimum m and the maximum M of any random variable (with bounded distribution) by the formula ϑ ≤ (Mµ)(µm).

Transforming this formula we get m ≤ µ -ϑ M-µ . With the help of Property 1, we replace ϑ by V [LL(G)], µ by E [LL(G)], m by OP T (the size of the optimal vertex cover of G) and M by nc to get

OP T ≤ E [LL(G)] - V [LL(G)] n -c -E [LL(G)]
.

Now, as algorithm LL constructs a cover C of G = (V, E), it is well known (and not difficult to prove) that I = V -C is an independent set of G. Hence, α = n -OP T . Using this with the previous inequality, we obtain our lower bound on α:

α ≥ n -E [LL(G)] - V [LL(G)] n -c -E [LL(G)] . (9)
Note that this bound can be calculated for any graph, except when nc = E [LL(G)]. In this case, this means that the average size of covers constructed by LL is the worst one, of size n-c. As LL can construct the optimal one, this means that

OP T = E [LL(G)] = n -c (otherwise, we would have E [LL(G)] < n -c).
As α = n -OP T , and as OP T = nc here, our lower bound becomes:

if

E [LL(G)] = n -c then α ≥ n -(n -c) = c . (10)
With this additional case [START_REF] Garey | Computers and Intractability: A Guide to the Theory of NP-Completeness[END_REF], our bound can be computed in polynomial time for any graph G (c is the number of connected components of G and E [LL(G)] can be computed with the formula of Property 1 and V [LL(G)] with the formula of Theorem 1).

In the following of this section, we compare our lower bound to the recent one of [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF] by J. Harant and to a previous one [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF] proposed by J. Harant and I. Schiermeyer. We prove analytically in Sect. 3.1 that on at least one infinite family of graphs (the wheel graphs with even number of vertices), the difference between our bound and the one of [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF] tends to infinity when the number of vertices grows, meaning that on this family our bound gets better when n grows. In Sect. 3.2, we compare the bounds by calculating them numerically on hundreds of graphs. We show that our bound is better on random graphs and random bipartite graphs.

Comparison of Lower Bounds on Independence Number on Wheel Graphs

In this section, we compare our lower bound [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF] to the one of Harant in [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF] in a family of graphs, namely the wheels. The wheel graph W (n) has n vertices and is composed of a cycle of n -1 vertices and an additional "central" vertex directly connected to all the n -1 other ones. We prove in this section that when n (restricted here to even n for simplifications) tends to infinity, our lower bound minus the Harant's lower bounds tends to infinity, proving that our lower bound is greater and thus better on these graphs.

To do that, we need the bound of J. Harant. Here is the exact formulation of his theorem as it can be founded in [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF].

Theorem 2 (Theorem 2 in [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF]). Let G = (V, E) be a finite, simple, connected and non-complete graph on n ≥ 3 vertices, where |E| ≥ n. Moreover, let α ≤ n 2 be the independence number, ∆ be the maximum degree of G, n j be the number of vertices of degree j in G and

x(j) = j(j + 1) j(j + 1) -2 2 j + 1 -(∆ -j) n ∆ + • • • + 2 j + 1 -1 n j+1 + 2n j j + 1 + • • • + 2n 1 2 -2
for j ∈ {2, . . . , ∆}. Then, there is a unique j 0 ∈ {2, . . . , ∆} such that 0 ≤

x(j 0 ) < n ∆ + • • • + n j0 and α ≥ x(j 0 ) + n j0+1 + 2n j0+2 + • • • + (∆ -j 0 )n ∆ 2 + 1 .
To obtain the following results, we used Maple c software since these are tedious elementary formal calculations that are done easily and safely by a computer.

Calculation of the Bound of Harant H(2p) on W (2p)

. We can note that Theorem 2 can be applied here since all the hypothesis are satisfied. To compute the bound of Harant on W (n), that we note H(n), we need to detail parameters used in the theorem. To simplify the calculations, we restrict ourselves to the case where n in even, and we write n = 2p. In W (2p), ∆ = n -1 = 2p -1, there are n -1 vertices of degree 3 and one vertex of degree n -1. Thus, n 3 = 2p -1, n ∆ = n 2p-1 = 1 and all the other n i are 0.

The next step is to find the index j 0 . We claim that j 0 = p + 2. To prove that, we just have to show that 0 ≤ x(j

0 ) < n ∆ + • • • + n j0 . As j 0 > 3 (when p is large enough), n ∆ + • • • + n j0 = n ∆ = 1. We get x(j 0 ) = 1 2 (7+p)(p+2)
p 2 +5p+4 . Clearly, x(j 0 ) ≥ 0. We have to prove that x(j 0 ) < 1. This is true iff (7 + p)(p + 2) -2p 2 -10p -8 < 0. After simplifications, this is equivalent to -p 2p + 6 < 0. When p is large enough, this is clearly true.

Hence, the (unique) index j 0 of Theorem 2 is p + 2. We can now write the formula of the lower bound H(2p) of Harant on W (2p):

H(2p) = (x(j 0 ) + 2p -1 -p -2) 2 + 1 = 1 4 7p + 6 + 9p 2 + 2p 3 p 2 + 5p + 4 .
Calculation of our Bound U (2p). We do the same kind of work to get the lower bound given by our method in [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]. To do that, we need to calculate the expectation E [LL(W (2P ))], noted here Exp(2p) to simplify, and the variance on W (2p), noted here V ar(2p). We use here again the fact that ∆ = n -1 = 2p -1, and that there is one vertex of degree n -1 and n -1 vertices of degree 3. After calculations with (1) and Theorem 1 and reorganizations and several simplifications (using Maple c again), we get

Exp(2p) = 3p 2 + 1 4 - 1 2p and V ar(2p) = (2p) 2 112 + 2p 24 + 3 2(2p) - 1 (2p) 2 - 185 336
.

Using (9), we can now express our lower bound, noted U (2p):

U (2p) = 2p -Exp(2p) - V ar(2p) n -1 -Exp(2p) = 2 21 (6p -11)p p -2 .
Comparison of the Two Bounds H(2p) and U (2p). After simplifications, we obtain

U (2p) -H(2p) = 1 84 6p 4 +47p 3 -17p 2 -184p+252 (p-2)(p 2 +5p+4)
. From that, we get lim p→+∞ (U (2p) -H(2p)) = +∞. This proves that there is an infinite number of graphs for which our bound is better than Harant's one and that the difference increases.

Numerical Comparisons of Lower Bounds on Independence Number on

Random Graphs In Sect. 3.1, we proved by analytical arguments that our lower bound is strictly better than the one of [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF] and that the difference tends to infinity for at least a particular family of graphs. On others graphs, like cycles, it can be shown that the lower bound of Harant is better than ours.

As studying, even restricted class of graphs, is a long and hard analytical task, we decided to compare the bounds numerically on several graphs. Bounds Compared. We compared our bound with the ones of [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF] (see Theorem 2) and [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF] that has an analytical expression similar to Theorem 2. We do not include it here but we can note that the hypothesis given by the authors to apply it is only that G must be connected and non-complete. No other hypothesis is required. At the opposite, Theorem 2 is valid only if the independence number α is less than or equal to n 2 . We must keep this point in mind when we make calculations on random graphs, since we cannot be sure that this constraint is satisfied a priori.

We programed these three bounds using Maple c . We calculated the integer values of these bounds (for example, the bound of Theorem 2 calculated is

x(j0)+nj 0 +1+2nj 0 +2+•••+(∆-j0)n∆ 2 + 1 ).
Results Obtained. We calculated the bounds on random graphs produced by Maple c (library RandomGraphs) using the option connected to be sure to work on connected graphs. As we cannot be sure that α ≤ n 2 , we computed Theorem 2: if the index j 0 exists then we calculated the final value; otherwise, we cannot use the formula. Note that even if we obtain a result, we are not sure that it is valid (because maybe α > n 2 and thus we are not in the validity conditions).

For each experiment, we randomly generated an integer n in a range R n (typically [START_REF] Bossecker | Interpolating between Bounds on the Independence Number[END_REF]250]) and a probability p in a range R p (e.g. [0.05, 0.95]). Then we generated a connected random graph G with n vertices and probability p for each possible edge1 . We computed the three bounds on G and compared them.

We evaluated more than 1500, with different ranges of vertices (but mostly [START_REF] Bossecker | Interpolating between Bounds on the Independence Number[END_REF]250]) and different ranges of probabilities R p ([0.05, 0.95] but also "dense" graphs [0.5, 0.95] and "sparse" graphs [0.05, 0.15]). On these 1500 calculations, the bound of Theorem 2 "failed" 3 times (no parameter j 0 found). In 561 cases, our bound is the same than the best of [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF] and [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF]. In 935 cases, our bound is better than the best of the two others by one unit and in 4 cases, our bound is one unit worst than the best of the two others.

Thus, in 99.73% cases, our bound is at least as good as the others and in 62.33%, it is strictly better.

We describe now other numerical comparisons on random bipartite graphs. For that, we generated randomly two integers a and b in a range [START_REF] Bossecker | Interpolating between Bounds on the Independence Number[END_REF]125] and a probability p in a given range. Then we generated random bipartite connected graphs with one part V 1 of the bi-partition having a vertices and the second one V 2 having b vertices (hence, the whole graphs contains between 10 and 250 vertices); edges (between V 1 and V 2 only) are randomly generated with probability p. Here we only compared our bound with the one of [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF] (in too many cases, Theorem 2 could not be computed, probably because of the α parameter). We did 200 comparisons here. In 100% of the cases, our bound is at least as good as the other. In 89.5%, our bound is strictly better. The maximum of difference between the two bounds is 63 units (on one experiment). In average, our bound in better by 6.14 units.

We also made computations on wheel graphs analytically studied in Sect. 3.1. They confirm our analysis (even on wheels with an odd number of vertices).

Conclusion

In a first part of this paper, we gave an analytical formula expressing the variance of an algorithm (called LL) for the vertex cover problem. As the others parameters (min., max. and average sizes) were already been calculated in [START_REF] Angel | Analysis and Comparison of Three Algorithms for the Vertex Cover Problem on Large Graphs with Low Memory Capacities[END_REF], we refine the knowledge of the behavior of LL. We can note that all these bounds are exact and can be polynomially calculated on any graph G. It is an important result in itself since we showed in [START_REF] Angel | Analysis and Comparison of Three Algorithms for the Vertex Cover Problem on Large Graphs with Low Memory Capacities[END_REF] that LL is adapted to treat huge graphs. We also experimented LL in [START_REF] Angel | Implementation and Comparison of Heuristics for the Vertex Cover Problem on Huge Graphs[END_REF] and shown that it is able to construct a solution on graphs of up to 100.10 9 vertices and edges in around 7 hours on a laptop. For such huge graphs, the precise knowledge of the statistical behavior of the algorithm is important.

In a second part, we used these results to propose a new lower bound on the independence number of a graph. We compared our bound to the recent ones proposed in [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF] and [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF]. We showed that for an infinite number of graphs, our bounds is strictly better. We also compared it numerically on hundreds of random graphs.

From these results, we can conclude that: our bound can always be calculated, for any graph, without restrictions (this is not the case of the ones of [START_REF] Harant | A Lower Bound on Independence Number of a Graph in Terms of Degrees[END_REF] and [START_REF] Harant | A Lower Bound on Independence in Terms of Degrees[END_REF]); for many graphs, our bounds is at least as good as the two others and often strictly better.

A future work is to improve our lower bound by applying the Bhatia-Davis inequality with other algorithms for which we can express (exactly or approximatively) min., max., average sizes and variance.

1 d

 1 (u)+1 ): see Property 1. Hence, both vertices u and v are in the solution (and

Maple c generates a random tree and then generates random edges with probability p.

✩ Work partially supported by the French Agency for Research under the DEFIS project ANR-09-EMER-010 TODO.