Julie Auberger
email: julie.auberger@inrae.fr

Geneviève Gésan-Guiziou

Caroline Malnoë

Guilhem Rostain

Magalie Weber

Hervé Guillemin

Stéphane Dervaux

Caroline Pénicaud

On "The role of emerging economies in global food security"

Keywords: LCI tool, interoperability, food engineering

1-Rationale and objective Food systems are the main contributors to environmental impacts (e.g. climate change, water scarcity, land use). Meanwhile, half humanity suffers either from lack of food or from low-quality diets. Drastic changes in diets and in food production are necessary both to decrease environmental impacts of food and to improve global human health (Willet et al., 2019). To accompany or pilot these changes, data on food are necessary. In France, databases such as Ciqual and Agribalyse provide information on nutritional composition and environmental impacts, respectively, of standard food products. However, they neither cover the huge variability of products in the food sector nor help to assess new products or the new ways of production that are needed. The DataSusFood project aims to develop a modular IT system to organize, store and share data related to food engineering, and to assess the composition, nutritional and sensory properties, and environmental impacts of food products. Initially, researchers will be the main users of the IT system, but the data and knowledge produced will be available in open access. The IT system of DataSusFood relies on two tools: the PO² ontology developed by INRAE (French national research institute for agriculture, alimentation and environment) and MEANS-InOut, developed by INRAE and Cirad (French agricultural research and international cooperation organization working for the sustainable development of tropical and Mediterranean regions). The PO² ontology [START_REF] Ibanescu | PO² -a process and observation ontology in food science. Application to dairy gels[END_REF] organizes all relevant information about food and bio-product engineering; its core model is embodied in the PO²-BaGaTel database that capitalizes data on production processes and product characteristics. MEANS-InOut [START_REF] Auberger | MEANS-InOut: user-friendly software to generate LCIs of farming systems[END_REF]) is a userfriendly web app that helps users to apply life cycle assessment (LCA) by building life cycle inventories (LCI) from data that describe the system studied. MEANS-InOut is currently operational to generate LCI of agricultural production systems at farm gate. In the IT system of DataSusFood, the role of MEANS-InOut is to support the calculation of environmental data about food and bioproducts. The objective of this presentation is to describe how MEANS-InOut has been adapted to food and bio-product engineering, in relation to PO² ontology and PO²-BaGaTel database.

2-Approach and methodology

The DataSusFood project organizes collaboration between methodological questions and development of MEANS-InOut. First, two scientists in food and bio-product engineering defined user needs. Next, a larger group of food-engineering scientists and LCA practitioners at INRAE reviewed, enriched and validated these needs. The main users targeted are food and bio-products engineering scientists, who are specialists of the systems and processes they want to assess; secondary targeted users are people who want to assess food processes without being familiar of food engineering. The needs were thus defined for users who want to gather detailed information about processes, and to be guided for LCA modelling. The users' needs were translated into requirements of MEANS-InOut. Two requirements fall under "general functionalities" of software. MEANS-InOut should (1) be userfriendly for food scientists, rather than for LCA practitioners; (2) connect LCA data to other tools, especially PO² ontology and PO²-BaGaTel database, in order to capitalize knowledge on food and bio-product engineering. The objective is to store in PO²-BaGatel database the data describing the process, the associated Life Cycle Inventory and the LCA results: the values of environmental impact indicators related to a functional unit. Three requirements are related to the implementation of LCA of food and bio-products processes. MEANS-InOut should: (3) cover a wide range of production processes and of food and bio-products; (4) allow all information relevant for LCA of food and bioproducts and/or their production processes to be collected to create life cycle inventories of good quality; and (5) address several questions about environmental impacts of food and bio-product engineering, such as identifying hotspots, comparing processes for the same product, comparing products. The first phase of developing MEANS-InOut for food and bio-product engineering is completed. MEANS-InOut has been presented to food scientists regularly during the development phase to verify that the tool is adapted to their needs. Food scientists have tested it, which has revealed modeling errors and omissions. These errors have been corrected before the delivery of MEANS-InOut to users.

3-Results and discussion

We describe the main features of MEANS-InOut developed to meet the requirements for food and bio-product engineering (Figure 1). In MEANS-InOut, users will be able to study food and bioproduct transformation at the process scale, by describing the foreground system. To represent a process under study, users draw a process-flow diagram. This is a usual way of process description for food and bio-product engineering scientists, (requirement 1). The process-flow diagram defines all steps of the process, including cleaning and transport. A generic framework was developed to describe the steps (requirement 3). Each box in the diagram generates input forms to guide data collection for process flows: ingredient and product flows, energy, water, cooling practices, packaging, equipment, consumables, detergents, waste or water treatment, pollutant emissions and transport (type and distance). The forms are built at a fine level of detail, for the process study. These detailed forms ensure that users do not forget important input, product, waste, or pollutant flows. Mass-balance checks are performed between inputs (ingredients and product under process, coming from previous step) and outputs (product under process going to further step, coproduct going out of the process, losses, waste, and emissions) for each step of a given process to reduce the risk of dataentry errors (requirements 1 and 4). A reference vocabulary (e.g. names of ingredients, products, processes, steps) comes from the PO² ontology (requirement 2). This reference vocabulary is used to assist data entry using drop-down menus and to specify the products and processes under study. An export function generates LCI files from the information collected by users. The creation by MEANS-InOut of these process or product Life cycle inventories is based on an embedded mapping between the resources mobilized by the process (ingredients, energy, machines, waste treatment...) and LCI from ecoinvent and Agribalyse databases (requirement 1). This ensures the coverage of background system. Depending on their objective, users can create a process LCI or product and coproducts LCI, and for the latter, several allocation methods are possible: mass on a dry or wet basis, volume, protein, fat, energy or economic. Process LCI and products LCI are structured as a combination of unit operations and ingredients, which allows identifying the hotspots of the process (requirement 5). The LCI files generated with MEANS-InOut are ready to be imported in LCA software, which is required 12-14 October 2022, Lima, Peru (hybrid conference) 3 to calculate environmental impact indicators. The developments are not yet fully adapted to represent a food supply chain, as data collection forms have been designed for the fine description of the processes. The consistency of the forms with other steps of a value chain such as retail has not been evaluated. A further adaptation of MEANS-InOut to the whole food supply chain is thus needed. It should be based on concepts to be added in PO2ontology, which describe the elements of the supply chain.

Web services are under development to exchange data on processes (process description, LCIs, LCIAs), first from MEANS-InOut to the PO²-BaGaTel database. We also plan to create a link between MEANS-InOut and OpenLCA, to launch directly from the interface of MEANS-InOut the module for calculating the indicators of environmental impacts of the studied system (requirement 2). Being able to export LCA data from MEANS-InOut to the PO²-BaGaTel database and connect it to other dimensions of food products will be an achievement for the DataSusFood project.

Figure 1 : Main functions of MEANS-InOut for food and bio-product processing, link to PO²-BagaTel database will not be available at first delivery but after further developments.

The generic framework for collection forms allows describing processes with flexibility. Users can describe systems with more or less details, more or less unit operations aggregated in steps and test different process scenarios. However, in this generic framework, default or reference values for specific processes such as energy consumption for sterilization, or the quantity of water needed to clean a membrane are not available. The absence of default values is intended to make MEANS-InOut more flexible to use for food and bio-product engineering specialists, to study all types of processes, including innovative processes or processes for which few references are available. But this absence of default values may also make MEANS-InOut more difficult to use for those who are not specialists of their studied process, or users who study food engineering within a system, when food engineering is not the core issue. The possibility of querying PO²-BaGaTel from MEANS-InOut, which is intended in further developments, will be helpful to overcome this limitation: knowledge and data available on food and processes in PO²-BaGaTel will be accessible to MEANS-InOut users in order to perform LCA with adequate system descriptions and data.

PO² ontology defines both a data structuration to describe the food and bio-products production processes and the vocabulary associated. It is a reference in INRAE for food and food processes description. Process representation and data collection forms in MEANS-InOut are based on PO² data structuration and vocabulary. Using PO² data structuration ensures that MEANS-InOut data format for food and bio-product processing is validated by the scientific community, and is shared with INRAE food and bio-products engineering scientists. It also ensures the connection with PO²-BaGaTel database and the capitalization of the data generated with MEANS-InOut. However, MEANS-InOut collects data that are specific for LCA calculation (e.g.: functional unit, factor used to calculate allocations) or documentation (e.g.: PEF quality notation, temporal boundaries of the process). It can also be data important in environmental assessment but initially not identified as essential when describing a transformation process, such as data about cleaning of equipment (detergent types, quantity of water involved).

Therefore, new concepts have been added in the PO2 ontology to allow the description of these data specific to LCA. Thus the intended inter-operability between PO2 ontology and MEANS-InOut has not only given a structure to the latter but also it has enriched PO2 ontology.

4-Conclusion and perspectives

MEANS-InOut aims to become an international reference tool for environmental assessment in the agri-food sector. It was able to calculate agricultural production LCIs at farm gate. The described new developments of MEANS-InOut allow to calculate food or bio-products LCIs at food-plant output gate. In the next years, further developments will be pursued to model food chains and perform economic assessment at the food-chain scale, allowing environmental and economic assessment of complete value chains from farm to fork. By studying agricultural systems in MEANS-InOut or addressing methodological questions, users have identified new needs, which have led to improvements in the tool for agricultural systems. We therefore look forward for the use of MEANS-InOut for the study of food engineering systems to arise needs or to improve existing functionalities. Access to and use of MEANS-InOut (available in English and French) is open to anyone, free of charges for six months. Further use is subject to the terms of a service contract.

  13 th International Conference on Life Cycle Assessment of Food 2022 (LCA Foods 2022) On "The role of emerging economies in global food security"

th International Conference on Life Cycle Assessment of Food 2022 (LCA Foods 2022) On "The role of emerging economies in global food security"12-14 October 2022, Lima, Peru (hybrid conference)

Acknowledgement

The authors thank the French National Agency for Research (ANR), for funding the DataSusFood Project (grant number ANR-19-DATA-0016), and Michael Corson for reading this article and providing most helpful language corrections.


Damien Vandembroucq

Thierry Deschamps

Camille Coussa

Antoine Perriot

Etienne Barthel

Bernard Champagnon

Christine Martinet

Density hardening plasticity and mechanical aging of silica glass under pressure: A Raman spectroscopic study

(Dated: May 13, 2009) In addition of a flow, plastic deformation of structural glasses (in particular amorphous silica) is characterized by a permanent densification. Raman spectroscopic estimators are shown to give a full account of the plastic behavior of silica under pressure. While the permanent densification of silica has been widely discussed in terms of amorphous-amorphous transition, from a plasticity point of view, the evolution of the residual densification with the maximum pressure of a pressure cycle can be discussed as a density hardening phenomenon. In the framework of such a mechanical aging effect, we propose that the glass structure could be labelled by the maximum pressure experienced by the glass and that the saturation of densification could be associated with the densest packing of tetrahedra only linked by their vertices.

I. INTRODUCTION

The behavior of amorphous silica under high pressure has been intensively studied in the last decades 1,2,3,4,5,6,7 and has recently motivated an increasing amount of numerical studies 8,9,10,11,12,13,14,15,16,17 . Silica appears to be elastic up to around 10 GPa and to exhibit a plastic behavior at higher pressure. Two features can be emphasized at that level: i) in the elastic regime, the compressibility exhibits a surprising non monotonous evolution with a maximum at around 2-3 GPa 1,18,19 ii) when unloading from the plastic regime a permanent densification up to 20 % can be observed 1,2,3,4,5,6,7 . No clear evidence of alteration of the tetrahedral short range order in this unloaded state has been observed 5,20 .

Above 25 GPa, a change from 4-fold to 6-fold coordination is observed 5 . This 6-fold amorphous seems not to be quenchable at zero pressure. When unloading downto zero pressure, no trace of 6-fold coordination is obtained. Performing X-ray Raman scattering experiments on oxygen K-edge, Lin et al 21 observed a reversible electronic bonding transition between 10 and 25 GPa. The latter was attributed to a fourfold quartz-like to a sixfold stishovitelike change of configuration of silica glass. For P > 25 GPa the densification process saturates and after unloading to ambient, the density level is the one obtained with a maximum pressure P ≃ 25 GPa 18,19 .

Questions remain about the nature of the densified phase and the mechanism of densification. Recent studies 17 have proposed the existence of an "activated" five-fold coordination at high pressure allowing reorganization toward a denser tetrahedral network. In former studies, in analogy with amorphous ice, Lacks 12 has proposed a first order transition between 2 different tetrahedral amorphous phases of silica. This transition would be kinetically hindered at room temperature. This idea of poly-amorphism has received a lot of attention 7,14,15,16,22,23,24,25,26 . At this stage it is important to separate the known transition at very HP between a 4-fold amorphous silica and a 6-fold amorphous silica involving a change in the short-range order 5,9 from an additional hypothetical transition at lower P between 2 different amorphous phases of tetrahedral silica and involving the medium range order 27 .

The most recent numerical 17 and experimental 7 works as well as the existence of a continuous range of densities for amorphous silica 6 after return to ambient pressure seem to rule out this idea of transition between 2 different forms of amorphous tetrahedral silica. However, the original observation of Lacks remains of interest: performing molecular dynamics simulations driven in volume 11 , he noted discontinuities in the pressure signal associated with local pressure induced mechanical instabilities. The latter are reminiscent of the shear induced mechanical instabilities previously identified in flowing liquids 28 . Note that similar localized transitions are widely believed to be the main mechanism of shear plasticity of amorphous materials 29 and can be associated to the vanishing of one eigenvalue of the Hessian matrix of the interatomic potential 30,31 .

In parallel to a structural study, it may be worth considering the pressure induced densification process in silica according to a mechanical perspective. In particular, in the Raman spectroscopic measurements to be presented below, we will use cycles of pressure of increasing maxima. This protocol will help us to discriminate in the spectral patterns, modifications due to the reversible elastic deformation of the network from other ones due to plastic structural reorganizations.

We first present the experimental methods i.e. Raman measurements of silica submitted to pressure cycles and analyze their results in terms of mechanical behavior. Two series of experiments are discussed. In series A Raman measurements are performed in situ during the loading and unloading stages of successive pressure cycles. In series B Raman measurements are performed ex situ at ambient pressure before and after pressure cycles of increasing maximum pressure.

The results of these spectroscopic measurements are presented in the next section and discussed in the framework of continuum plasticity. The first series of in situ experiments gives a nice illustration of (densification) plasticity in the context of a silica glass while the second series of experiments allows us to follow the evolution of permanent densification vs the pressure maximum of the cycle i.e. the density hardening behavior of silica.

We finally give a discussion, first in terms of mechanical behavior, then in terms of amorphous structure. These results are of primary importance in the description of the mechanical properties of silica. While silica is daily used as a calibration sample for nano-indentation measurements, it appears that the mechanical behavior of this material is not fully described. It has been shown recently 32 that the constitutive laws available [START_REF] Shorey | Proc SPIE[END_REF][START_REF] Xin | Proc. SPIE[END_REF] which do not take hardening into account fail to fully describe the densification process induced by an indentation test. The above data can be used to include hardening in a simple constitutive law of silica [START_REF] Kermouche | [END_REF] , which gives a precise account for this behavior.

Beyond their interest in terms of mechanical behavior, these results can also be discussed in terms of amorphous structure. In particular, as detailed below, densification can be regarded as a typical glassy phenomenon, resulting from a mechanical aging process. Following this perspective, the amorphous densified structure would be associated with the quench of the structure at high pressure and could be labelled by a fictive pressure in the very same acception as a fictive temperature can be used to label a structure obtained by thermal aging.

II. EXPERIMENTAL METHODS

Bars of amorphous silica (Saint-Gobain Quartz IDD) are shattered into pieces. Splinters of characteristic length of 10 µm are submitted to cycles of pressure in a "Sidoine" Diamond Anvil Cell (DAC) with a maximum pressure in the range [1-25 GPa]. Raman spectra are collected with a Renishaw RM 1000 micro-spectrometer with a Ar + -ion 514 nm, 50 mW laser excitation). A small piece of ruby is introduced together with the silica splinter in order to monitor the pressure level using the shift of the R1-luminescence band.

Two series of experiments are presented.

• In series A in situ measurements of the Raman spectrum are collected all along the compression cycle. Three cycles are presented, The first one with P max = 7.3 GPa lies in the elastic domain, te third cycle consists of a compression up to P max = 18 GPa followed by a decompression at 1 GPa, finally the last cycle consists of a compression up to P max = 16 GPa and a direct return to ambient pressure induced by the breakdown of a diamond.

Methanol is used as a pressure transmitting fluid in this series of experiments (quasi-hydrostatic conditions). A detailed presentation of these in situ experiments can be found in 36 .

• In series B Raman measurements are performedex situ at ambient pressure before and after each cycle of pressure. A series of compression cycles is presented where the pressure maximum is increasing from 9 GPa for the first cycle to 25 GPa for the last cycle. In this case, the pressure transmitting fluid is a mix similar to 5:1 methanol-ethanol, which ensures hydrostaticity up to 16 to 20 GPa 37 . The Raman spectrum is measured before loading and after unloading, the diamond-anvil-cell being emptied of the transmitting fluid.

Figure 1 contrasts the initial spectrum of a sample (plain line) and that obtained after a 18 GPa hydrostatic loading (red line).

Between 200 and 750 cm -1 we can identify a main band at 440 cm -1 . This band is intense, and affected by the densification process: the band gets narrower and FIG. 1: Raman spectra obtained with an amorphous silica sample before (plain line) and after (dotted line) a 18 GPa hydrostatic loading.

is shifted to higher wave numbers. This band has originally been attributed to the symmetric stretching mode of bridging oxygens between two Si atoms and its evolution under densification to the decrease of the intertetrahedral angles Si-O-Si 38 . Recent determinations of the Raman spectrum from first principles 39 have established that this broad band originates more likely from bending motions of oxygen atoms that do not belong to small rings.

The defect lines D 1 and D 2 , at 492 and 605 cm -1 , are respectively attributed to the breathing modes of the four-membered and three-membered rings 13,40,41 . Their area ratio was previously used in literature as an indicator for the variation of the "fictive temperature" 42,43,44 , which is associated with a slight change of density. The effect of pressure seems to be better accounted for by the shift of the D 2 line. As discussed by Polsky et al 45 , this may be due to pressure induced variations of the Raman cross section. The D 2 line is of particular interest since it has almost no overlap with the main band. Sugiura et al. 46 correlated the position of the D 2 line with the ratio of the sample density ρ to its initial density ρ 0 . The residual density evaluated through this relation accounts for both irreversible and elastic densification due to residual elastic strain. However, several works 46,47,48 evidenced that the D 2 line position is only marginally sensitive to residual elastic strains. This correlation was recently used to probe the densification gradient surrounding a plastic imprint in silica obtained by indentation 32 . 1 before plasticity sets in again. The elastic limit has thus evolved under loading from P Y 0 to P Y 1 and P Y 2 . The knowledge of this density hardening behavior (evolution of the limit elastic pressure with density at zero pressure) is necessary to give a proper modelling of plasticity of glasses.

III. RESULTS AND INTERPRETATION

We describe in the following the experimental results and give an interpretation in the framework of the elastoplastic response of continuous media.

A. Continuum mechanics

We first recall briefly the formalism of elasto-plasticity. Fig. 2 shows the hydrostatic stress as a function of the volumetric strain of a medium submitted to pressure. Below a threshold value of pressure P Y 0 the material remains fully elastic and the volumetric deformation is reversible. When increasing the pressure beyond threshold, plasticity sets in and an irreversible deformation adds up to the elastic reversible deformation. When unloading from P Y 1 > P Y 0 , the material behaves elastically and only the elastic part of deformation is recovererd. Loading again to a higher pressure P Y 2 > P Y 1 , we observe the same phenomenology with a crucial difference: the onset of plasticity has increased from P Y 0 to P Y 1 . In other words, the mechanical behavior depends on the history of mechanical loading. The material has experienced hardening [START_REF] Khan | Continuum theory of plasticity[END_REF] which can be regarded as a mechanical aging. Such a behavior is standard for metal shear plasticity and usually results at the structural level from the entanglement or the pinning of dislocations by impurities [START_REF] Cottrell | Dislocations and plastic flow in crystals[END_REF] . Metals however do not exhibit any volumetric plastic deformation (dislocation motion is a volume conserving mech- anism). However, irreversible changes of density are familiar in soil mechanics and granular materials (dilatancy effect) [START_REF] Jaeger | [END_REF] .

The typical mechanical behavior of a material like silica glass can be summarized as follows. Before any loading, the material is elastic up to a threshold which depends both on pressure and shear. Using hydrostatic pressure p and equivalent shear stress τ as coordinates, this elastic threshold corresponds to a continuous curve intersecting the two axis. When reaching threshold, if the stress is increased, plasticity sets in and the elastic limit is convected to the maximum value of stress experienced by the material. At macroscopic scale, one tries to characterize this hardening behavior, relating the evolution of densification with the pressure maximum. At microscopic scale, in the present case of silica, in absence of a microscopic mechanism as well defined as the motion of dislocation, a pending question remains to identify and understand a structural signature of density hardening.

B. Elasto-plastic behavior of silica under pressure

In this section we present the results of in situ raman measurements (series A). Two patterns of the Raman spectra measured in situ during the pressure cycles are discussed: i) the shift of the main band ω M ; i) the shift of the D 2 line ω D2 .

On Fig. 3 and4 we show the evolution of the position of these two bands for a cycle of pressure up to P max = 7.3 GPa (black symbols). We observe a full reversibility between loading and unloading. This result is consistent with the usual estimate P c ≃ 10 GPa for the onset of permanent densification in silica. The evolution of these indicators is also plotted on the same figures for the second series of cycles up to P max = 17.9 GPa. We now see that below P c ≃ 10 GPa, the same elastic be- havior as before is recovered; then a change of slope on loading is noticeable at least for the position of the main band; then the unloading curve does not reproduce the initial loading one and there appears a permanent change of the spectrum at ambient pressure; when loading again, one follows the very last unloading curve. These two curves appear to be very similar to the ideal case of hardening plasticity depicted in Fig. 2. This spectroscopic study allows us to closely follow the elastic-plastic behavior of silica under pressure.

We now discuss the evolution of permanent densification after cycles of increasing pressure. Even if the signal-to-noise ratio is less favorable for the determination of the D 2 line than for the main band (see Fig. 3 vs Fig. 4), we choose the former to estimate the permanent densification. As discussed above, the main reason for this choice is that in contrast to the main band, the shift of the D 2 line appears to be rather independent of the elastic stress. This allows us to use it a density probe in samples affected by residual elastic stress. Because of the microscopic size of the silica samples used in the diamond anvil cell, a quantitative calibration could not be performed. Densification can be obtained on millimetric samples but require high temperature treatments 6 and it is far from being obvious that the medium range order is directly comparable to the one obtained under high pressure at ambient temperature. Following Ref. 32 , the densification was estimated using the empirical relation:

∆ω D2 ω D2 ≃ ∆ρ ρ 0.14 , (1)
which was extracted from the experimental data of Sugiura et al. 46 obtained on shock wave experiments. This calibration step is thus only approximative.

The results are summarized on Fig. 5. We thus ob-tain the evolution of the permanent densification with the pressure maximum P Y of each pressure cycle. We obtain a continuous range of increasing densities with an apparent saturation. As shown on Fig. 5, this evolution can be approached by a sigmoidal curve. Comparable results were obtained recently on window glass using an octahedral multi-anvil apparatus 52 . In mechanical terms, this allows us to give a quantitative account of the density hardening behavior of silica. In the space of stresses, we have obtained the evolution of the maximum pressure P Y below which the material remains fully elastic as a function of the glass density.

IV. DISCUSSION

Raman spectroscopic estimators have been shownn to give a full account of the density hardening behavior of silica under pressure. Raman scattering measurements during a loading/unloading pressure cycle closely reproduce the elastoplastic behavior usually observed in stress/strain curves: full reversibility below a limit stress, appearance of a residual deformation with an elastic unloading for larger stresses. A crucial difference obviously lies in the fact that in the present study the evolution of hydrostatic pressure vs density is considered instead of shear stress vs shear strain as in usual metal plasticity. From a mechanical point of view, the detailed analysis of the density increase with the maximum pressure of the cycle allowed us to study the evolution of the residual densification with the elastic limit pressure P Y . Such a knowledge is of crucial importance for the determination of constitutive equation modelling the plastic behavior of silica. It was shown in Ref. 32 that when restricting the plastic criterion to perfect plasticity (i.e. assuming no hardening effect) it was not possible to account for the permanent densification of amorphous silica around a plastic imprint induced by indentation. Conversely, as shown in Ref. [START_REF] Kermouche | [END_REF] the assumption of an elliptic plastic criterion coupling shear stress and pressure together with the data of density hardening extracted from the present experiments allow to describe successfully this phenomenon of indentation induced densification.

From a physical/structural point of view, the present results suggest that the description of densified silica need not the hypothesis of an amorphous-amorphous transition between two types of tetrahedral networks. A simpler and alternative scenario consists of pressure induced reorganizations of the amorphous network allowing a more efficient packing of tetrahedra remaining linked by their vertices only. Such a scenario does not exclude the occurence of 5-fold or 6-fold coordinated silica in the plastic regime at high pressure. However, the latter would correspond to intermediate states between two amorphous tetrahedral structures. This occurrence of 5-fold or 6-fold coordination would thus simply denote the necessity of cutting and rebonding between the two structures. The denser structure would thus be quenched when pressure decreases down to ambient conditions. In that sense, the final structure could be labelled by the maximum pressure the material experienced. In this context of mechanical aging, the latter pressure could be thought of as a "fictive" pressure in the same acception as the fictive temperature in a more classical thermal aging experiment. Note that these denser structures may be affected by internal stresses due to the succession of localized reorganizations. More generally, these results indicate that the structure and the density of a densified sample of vitreous silica will depend crucially on the particular path it has followed in the plane pressure/temperature : it is likely that the medium range sructure of densified silica is not fully characterized by the only density parameter.

Looking finally at orders of magnitude for the density of the various phases of silica, we observe that the density of the stable crystalline equivalent at zero pressure (quartz) is 2.65 while the density of the metastable coesite (stable at 2-3 GPa) is 3.01. The density of fused silica being 2.2, a 20% increase gives 2.64 i.e values close to quartz but well below coesite. In analogy with hexagonal and random close packing, an interesting question could thus be whether the maximum observed density of amorphous silica corresponds to any geometrical maximal packing of tetrahedra bonded by their vertices only i.e. the maximum density of the continuous random network. To our knowledge, though the packing of space by tetrahedra, or ellipsoids has been considered [START_REF] Binder | Glassy materials and disordered solids[END_REF][START_REF] Donev | [END_REF] , this question has not been discussed yet.

V. ACKNOWLEDGMENTS

FIG. 2 :

 2 FIG.2: Sketch of a typical pressure/density curve expected in elastoplasticity. A reversible elastic behavior is obtained up to the elastic limit pressure P Y 0 . When loading above P Y 0 , plasticity sets in and the (elastic) unloading from P Y 1 is characterized by a residual densification. A subsequent loading at P Y 2 > P Y 1 reproduces the previous unloading curve up to P Y 1 before plasticity sets in again. The elastic limit has thus evolved under loading from P Y 0 to P Y 1 and P Y 2 . The knowledge of this density hardening behavior (evolution of the limit elastic pressure with density at zero pressure) is necessary to give a proper modelling of plasticity of glasses.

3 FIG. 3 :

 33 FIG.3: Raman shift of the main band ωM in the elasticplastic regime. Successive pressure loading and unloading cycles are depicted. Filled and empty symbols correspond to loading and unloading curves respectively. For a pressure maximum Pmax = 7.3 GPa, a full reversibility is obtained (elastic behavior). For a larger maximum pressure Pmax = 18 GPa, a residual shift is obtained after unloading (plastic behavior). An additional loading at Pmax = 16 GPa reproduces the last unloading curve, indicating the increase of the elastic limit under loading (hardening).

3 FIG. 4 :

 34 FIG.4: Raman shift of D2 line ωD2 in the elastic-plastic regime (same conditions as above).

FIG. 5 :

 5 FIG.5: Evolution of the residual densification vs the maximum pressure of the pressure cycle (symbols). The dashed line is an indicative sigmoidal curve corresponding to a 18% maximum densification.

The authors acknowledge useful discussions with G. Kermouche, W. Kob and S. Ispas-Crouzet. They would like to thank CECOMO and High Pressure facilities of University Lyon1. This work has been supported by the ANR "plastiglass" contract n o ANR-05-BLAN-0367-01.


N Mitton

-E Fleury -I

Guérin Lassous

S Tixeuil

Self-stabilization in self-organized Multihop Wireless Networks *

Keywords: multihop wireless networks, clusterization, self-stabilization, scalability, density

In large scale multihop wireless networks, flat architectures are not scalable. In order to overcome this major drawback, clusterization is introduced to support selforganization and to enable hierarchical routing. When dealing with multihop wireless networks, the robustness is a main issue due to the dynamicity of such networks. Several algorithms have been designed for the clustering process. As far as we know, very few studies check the robustness feature of their clustering protocols.

In this paper, we show that a clustering algorithm, that seems to present good properties of robustness, is selfstabilizing. We propose several enhancements to reduce the stabilization time and to improve stability. The use of a Directed Acyclic Graph ensures that the self-stabilizing properties always hold regardless of the underlying topology. These extra criterion are tested by simulations.

Introduction

Ad hoc networks or wireless sensor networks (wireless multihop networks) are composed of devices that communicate via wireless interfaces. They require no fixed infrastructure and no human intervention. Both are strongly based on self-organization and self-stabilization. Even if every mobile can move everywhere, and thus can disappear or appear in the network at any time, the network manages the changes in topology and provides the connectivity between any pair of terminals. If the current wireless cards allow the communication between mobiles that are in communication range, a routing protocol is required to provide the full connectivity of the network. As there are no dedicated devices in the network, all mobiles are potential routers. Such networks have become very popular due to their ease of use. Their applications range from the network extension when cabling is not possible or too expensive to spontaneous networks in case of natural disasters where the infrastructure has been totally destroyed by going through the monitoring and the collect of data with wireless sensor networks.

Due to the dynamics of such networks (devices mobility and/or instability of the wireless medium), routing protocols for fixed networks are not adapted. Ad hoc routing protocols proposed in the MANET working group at IETF1 are all flat routing protocols. It means that there is no hierarchy and all terminals have the same role. If flat protocols are quite effective on small and medium networks, they are not suitable on large scale networks due to bandwidth and processing overhead. Hierarchical routing seems to be more adapted to such large networks. It often relies on a specific partition of the network, called clustering: the devices are gathered into clusters according to some criteria and specific routing protocols are used within and between the clusters. In addition to its scalability, such an organization presents numerous advantages such as synchronizing mobiles in a cluster or attributing new services zones. Many algorithms have been designed for the clustering step. As mentioned in [START_REF] Rajaraman | Topology control and routing in ad hoc networks: a survey[END_REF], "one measure of robustness of the topology is given by the maximum number of nodes that need to change their topology information as a result of a movement of a node". As far as we know, very few studies check the robustness feature of their clustering protocols. Moreover, when it is the case, the evaluation is driven by simulations and never by a theoretical approach.

In this article, we apply self-stabilization principles over a clustering protocol proposed in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF] and which presents good properties of robustness. With a theoretical approach, which can be applied to several clustering schemes, we show that, under some assumptions, the algorithm is selfstabilizing. We also improve the robustness by adding extra-advanced features and we show that the resulting algorithm is still self-stabilizing. These properties are further validated by simulations. A brief state-of-the-art on clustering algorithms in multihop wireless networks is given in Section 2. The description of the chosen clustering algorithm is done in Section 3. In Section 4, we provide a formal analysis, showing that this algorithm is self-stabilizing. We also discuss some improvements for robustness in this section. The properties of the proposed algorithm and the different improvements are evaluated by simulations in Section 5.

State-of-the-art

Flat routing protocols (like the classical reactive or proactive protocols) are not really suitable for large wireless multihop networks. Indeed, such routing protocols become ineffective on a large scale because of bandwidth (flooding of control messages) and processing (routing table computation) overhead. One solution to solve this scalability problem is to introduce a hierarchical routing by gathering geographically close nodes into clusters [START_REF] Krishna | A cluster based approach for routing in dynamic networks[END_REF]. Many techniques for clusters formation and cluster-heads selection have been proposed. All solutions aim to identify subsets of nodes within the network and to bind each of them to an unique leader. Some solutions try to gather nodes into homogeneous clusters by using either an identity criteria (e.g., the lowest identity [START_REF] Basu | A mobility based metric for clustering in mobile ad hoc networks[END_REF]) or a fixed connectivity criteria (for instance maximum degree [START_REF] Chen | Clustering and routing in mobile wireless networks[END_REF][START_REF] Nikaein | DDR-distributed dynamic routing algorithm for mobile ad hoc networks[END_REF], k-hops clusters [START_REF] Fernandess | K-clustering in wireless ad hoc networks[END_REF]) or a value computed from different metrics (as connectivity and identity criteria in max-min d-cluster [START_REF] Amis | Max-min d-cluster formation in wireless ad hoc networks[END_REF], [START_REF] Chatterjee | Wca: A weight based distributed clustering algorithm for mobile ad hoc networks[END_REF]). To maintain the clusters structure, most of the solutions try to keep a fixed cluster diameter [START_REF] Amis | Max-min d-cluster formation in wireless ad hoc networks[END_REF], a fixed cluster radius [START_REF] Lin | A clustering technique for large multihop mobile wireless networks[END_REF] or a constant number of nodes in the clusters [START_REF] Ramanathan | Hierarchicallyorganized, multihop mobile wireless networks for qualityof-service support[END_REF]. These solutions are not adapted to large multihop wireless networks. First, a small modification in the network topology (due to the mobility of one node for instance) often implies new computations to build the new clusters and to elect the cluster-heads. Moreover, building and maintaining clusters with a constant feature (like the diameter or the number of nodes) may generate a significant number of useless clusters. For instance, why separating a set of nodes that can communicate just because it can not fit into one cluster since the constant feature is not respected? In [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF], a density criteria is proposed. This metric allows to limit the exchanged traffic generated while clusters are rebuilt and the nodes' tables updated. Therefore it presents good properties of robustness. In this paper, we show that this algorithm is self-stabilizing and its robustness can be enhanced with some extra rules.

Density driven clustering algorithm

The model. The system is composed of a set V of nodes and each node has an unique identifier. Each node p can communicate with a subset N p ⊆ V of nodes determined by the range of the radio signal; N p is called the neighborhood of node p. Note that p does not belong to N p (p / ∈ N p ). We assume that communication capability is bidirectional: q ∈ N p iff p ∈ N q . Define N 1 p = N p and for i > 1,

N i p = N i-1 p ∪ { r | (∃q ∈ N i-1 p , r ∈ N q ) } (let's call N i p
the i-neighborhood of p). We assume that the distribution of nodes is sparse: there is some known constant δ such that for any node p, |N p | ≤ δ. Note that a control on density can be done by adjusting their communication range and/or powering off nodes in areas that are too dense.

The density metric criteria. The notion of density, firstly introduced in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF], characterizes the relative importance of a node in the network and in its 1-neighborhood. The underlying idea is that if some nodes move in N p , changes will affect the microscopic view of node p (for instance its degree |N p | will change) but its macroscopic view will not drastically change since globally the network does not change and its N p globally remains the same. The density will smooth changes down in N p by considering the ratio between the number of links and the number of nodes in the 1-neighborhood. The definition is the following:

Definition 1 The density of a node p ∈ V is d p = | {e = (v, w) ∈ E s.t. w ∈ {p} ∪ N p and v ∈ N p } | |N p |
Cluster-heads selection and clusters formation. Due to space limitations, we describe the heuristic process informally. The algorithm and its analysis are more detailed in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF]. Each node watches its neighborhood. If this one changes at each step, the node is considered as too mobile to be integrated into a cluster and does not participate to the clustering algorithm (or it may cause instability into the cluster formation). Otherwise, it locally computes its density value and regularly broadcasts it to all its 1-neighbors. Each node is thus able to compare its density value to its 1neighbors' and decide by itself whether it joins one of them (the one with the highest density value) or it wins in its 1neighborhood and elects itself as a cluster-head. If there are some joint winners, the smallest identity is used to decide between them. In this way, two neighbors can not be both cluster-heads. If node p has joined node w, we will say that w is node p's parent. A node's parent can also have joined another node and so on. The cluster-head will be the node which has elected itself as its own cluster-head. A cluster can then extend itself until it reaches a cluster frontier. As each node joins a node within its 1-neighborhood, the cluster can also be seen as a directed tree where the cluster-head is the root (that we will denote as clustering tree in the following).

Features. This metric has been studied in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF] with both simulations and a stochastic analysis. It outlined that the number of cluster-heads computed with this metric is bounded and decreases when the nodes intensity (number of nodes per surface unit) increases. This is an advantage because if many nodes are in the communication range of each other, there is no need to separate them into different clusters as they can hear each other. Moreover, this heuristic has revealed to be more stable towards nodes mobility than other metrics, like the degree and the max-min metrics [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF].

Self-stabilization

In this section we study the self-stabilizing properties of the density-driven clustering algorithm presented in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF]. We follow the same assumptions and principles as the ones given in [START_REF] Herman | A distributed tdma slot assignment algorithm for wireless sensor networks[END_REF]. Due to space limitations, we briefly describe these points. We consider that the algorithm stabilizes when each node knows to which cluster it belongs, i.e. when it knows its cluster-head's identity. The stabilization time is thus related to the depth of the clustering trees.

Hypothesis. We assume that the implementation of CSMA/CA satisfies the following point: there exists a constant τ > 0 such that the probability of a frame transmission without collision is at least τ (this corresponds to typical assumptions for multi-access channels [START_REF] Bertsekas | Data Networks[END_REF]; the independence of τ for different frame transmissions indicates that we assume a memoryless probability distribution in a Markov model).

Notation. We describe algorithms using the notation of guarded assignment statements: G → S represents a guarded assignment, where G is a predicate of the local variables of a node, and S is an assignment to local variables of the node. If predicate G (called the guard) holds, then assignment S is executed, otherwise S is skipped. Some guards can be event predicates that hold upon the event of receiving a message. We assume that all such guarded assignments execute atomically when a message is received. At any system state, where a given guard G holds, we say that G is enabled at that state.

Execution Semantics. The life of computing at every node consists of the infinite repetition of evaluating its guarded actions. We assume that every action is executed within a constant time finding a guard and executing its corresponding assignment or skipping it when the guard is false. Generally, we suppose that when a node executes its program, all statements with true guards are executed within a constant time (done, for example, in round-robin order). Shared Variable Propagation. Some variables of nodes are designated as shared variables. Following the scheme presented in [START_REF] Herman | A distributed tdma slot assignment algorithm for wireless sensor networks[END_REF], nodes periodically transmit the values of their shared variables, based on a timed discipline. Beyond periodic retransmission, an assignment to a shared variable causes peremptory transmission: if a statement G → S assigns a shared variable, then we suppose that there is a transformation of the statement into a computation that slows execution down so that it does not exceed some desired rate, and also provides randomization to avoid collision in messages that carry the shared variable values. One technique for implementing such G → S is presented in [START_REF] Herman | A distributed tdma slot assignment algorithm for wireless sensor networks[END_REF]. In the remaining of the section, we assume that nodes use this scheme to learn N p and N 2 p . This is because the topology of the networks under considering is dynamic.

Constant Height DAG Construction

In the chosen clustering algorithm, as in every clustering algorithm using the node identity as last decision, the worst case is encountered (i) when every node has the same deciding value, i.e. in our case the density value, and ii when nodes' identifiers are unique in the network and badly distributed. In such a case, the algorithm builds only one cluster which may have a diameter as big as the diameter of the network. This may cause scalability problems because the stabilization time is likely to depend on this diameter. Moreover, it is obvious that building such a cluster is useless as we could have used the network without clusters instead. To overcome this drawback, it can be useful to give nodes smaller names (also named colors), from a constant space of names, in a way which ensures that names are locally unique. A DAG (Directed Acyclic Graph) can be constructed by using these identifiers and by orienting edges between neighbors from the higher identifier to the lower one.

Our constant height DAG construction is based on the randomized technique described in [START_REF] Herman | A distributed tdma slot assignment algorithm for wireless sensor networks[END_REF], but uses a much smaller name-space γ (|γ| is equal to δ 6 in [START_REF] Herman | A distributed tdma slot assignment algorithm for wireless sensor networks[END_REF], while δ 2 or even δ is sufficient in our case). Let Id p be a shared variable that belongs to the domain γ; variable Id p is the name of node p. Another variable is used to collect the names of neighboring nodes: Cids p = { u Id q | q ∈ N p }, where u Id q refers to the cache copy of the shared variable Id q at node p. Let random(S) choose with uniform probability some element of set S. Node p uses the following function to compute Id p :

newId(Id p ) = u Id p if u Id p ∈ Cids p random(γ \ Cids p ) otherwise
The algorithm for constant height DAG construction is the following: N1: true → Id p := newId(Id p ) Theorem 1 Algorithm N1 self-stabilizes with probability 1 in an expected constant time to a DAG which height is at most |γ| + 1.

The proof of this theorem is similar to the one in [START_REF] Herman | A distributed tdma slot assignment algorithm for wireless sensor networks[END_REF]. There are two competing motivations for tuning the parameter γ. On one hand, a large value of |γ| decreases the expected convergence time of N1. On the other hand, a small value of |γ| decreases the DAG's height, and thus the expected convergence time of subsequent algorithms.

Density-driven Clusters Construction

Each node p maintains two shared variables, denoted by d p and H(p). d p denotes the density value of node p given in Definition 1. H(p) denotes the cluster-head chosen by p. We define ≺ as a binary total order such that p ≺ q if and only if d p < d q or (d p = d q ) ∧ (Id q < Id p ). Let max ≺ denote the maximum function associated to this total order. When a node p computes the result of ≺ or max ≺ , it uses the cached values of its neighborhood (assuming u Id p = Id p and u d p = d p ).

We now define the clusterHead choice function: Proof: After an expected constant time, each node p has a correct view of its neighborhood at distance two. Then, after R1 is executed, the density d p of p is correct.

clusterHead = Id p if ∀q ∈ N p , q ≺ p, H(max ≺ {q ∈ N p }) otherwise.
2

Lemma 2 Each node p has a correct cluster-head value H(p) within an expected constant time.

Proof: Assume that all nodes have correct density values (this is true after an expected constant time by Lemma 1). After the shared variable d p has been communicated without collision to all nodes in N p (this occurs in an expected constant time), each node has a correct cache value of all density values in its neighborhood. We now consider the DAG induced by the ≺ relation (thereafter denoted by DAG ≺ ). In an expected constant time, the roots of DAG ≺ have a correct cluster-head value (that is their own identifier). Now assume that all nodes up to distance n from the roots of the DAG ≺ have a correct cluster-head value. When R2 is executed on nodes at distance n + 1 from the roots of DAG ≺ , those nodes get a correct cluster-head value (because the cluster-head is deterministically determined (i) by the density and local topology -which are fixed -and (ii) by the cluster-head values of nodes at distance up to n from the roots of the DAG ≺ ). By induction, the time needed for stabilization is proportional to the height of the DAG ≺ .

We now prove that the height of the DAG ≺ is bounded by a constant value. Node identifiers are bounded by a constant γ. The number of edges in the neighborhood at distance one is bounded by δ 2 , the number of neighbors at distance one is bounded by δ, so the number of possible values for the density function is at most δ 3 . Overall, the name-space of values in the DAG ≺ is γδ 3 , which is bounded by a constant. As a result, the height of the DAG ≺ is also bounded by a constant.

The algorithm stabilizes in an expected time proportional to the height of the DAG ≺ , and the height of the DAG ≺ is constant, so the expected time for stabilization is also constant. 2

Improving Stability

We improve the stability of the algorithm by adding some selection criterion. First, when two nodes compete for being cluster-heads (they have the same density value), the winner will be, first, the one which was cluster-head before (if it exists), then the one with the lowest DAG Id (as defined in Section 4). This scheme adds stability into clusters organization by limiting clusters reconstruction. Clusterheads remain cluster-heads as long as possible. It is a good property since the only cluster-heads' role is to give an identity to the clusters. This refinement preserves the structure of our stabilization proof, since it is equivalent to define the total order relation ≺ as p ≺ q if and only if

(d p < d q ) or (d p = d q ) ∧ (H(q) = Id q ) ∧ (H(p) = Id p ) or (d p = d q ) ∧ (H(p) = Id p ) ∧ (H(q) = Id q ) ∧ (Id q < Id p ).
In addition, the height of the new DAG ≺ is similar to the height of the previous one.

Second, if a node p is a 1-neighbor of two different cluster-heads u and v (which are not directly linked), it will initiate a fusion between u and v's clusters: if p has chosen v as cluster-head, that means that u ≺ v and that v will remain cluster-head unlike u. This ensures that (i) a cluster-head is not too off-centered in its own cluster, (ii) a cluster has at least a diameter of two, and (iii) that two cluster-heads are distant of at least three hops. Again, this refinement preserves our stabilization proof, since it is sufficient to use the alternative clusterHead function:

clusterHead =    Id p if (∀q ∈ N p , q ≺ p)∧ (∀q ∈ N 2 p s.t.H(q) = Id q =⇒ q ≺ p) H(max ≺ {q ∈ N p }) otherwise
The condition for being a cluster-head thus becomes "I am locally maximal (in the sense of ≺) and any cluster-head in my 2-neighborhood is smaller than me (and they should not remain cluster-heads)". The remaining of the algorithm (and thus proof) is the same.

Simulations

As mentioned in Section 4, we suppose that there exists a constant τ > 0 such that the probability of a frame transmission without collision is at least τ . Yet, we can suppose that in a bounded time ∆(τ ), each node is able to locally broadcast one frame and then receive all packets sent by its 1-neighbors. Such a ∆(τ ) time unit is called a step, during which each node can receive each packet of all its 1-neighbors. After one step, each node can discover its 1-neighbors. After two steps, each node can discover its 2-neighbors and then compute its density. After only three steps, each node knows its parent. Then, the number of steps required to discover its cluster-head identity directly depends on the distance in number of hops from the node to its cluster-head in the clustering tree and is bounded by the depth of this tree.

We performed simulations in order to evaluate the performance of the proposed heuristic and estimate the importance of the introduction of the DAG. Nodes are randomly deployed using a Poisson process with different intensity levels λ (λ corresponds to the mean number of nodes per surface unit) in a 1 × 1 square with various transmission ranges R varying from 0.05 to 0.1. Each given statistic is the average over 1000 simulations. All these results are fully described in [START_REF] Mitton | Selfstabilization in self-organized multi-hops wireless networks[END_REF].

To build the DAG, each node randomly chooses a DAG Id between 0 and δ 2 where δ is the maximum node's degree in the graph as defined in Section 3. For this, each node randomly chooses a DAG Id and then compares it to its neighbors'ones. If DAG Ids are the same, the node with the smallest "normal" Id chooses another DAG Id and so on until every node has a different DAG Id than the ones of its 1-neighbors. For simulations on a grid and on a random geometry topology with λ equal to 1000, the number of steps required to build the DAG does not take a lot of time since it only requires two steps on average, whatever R is. Therefore, building the DAG is not costly.

We measure the following criterion: number of clusterheads per surface unit, clustering tree length (also in order to evaluate time of stabilization as they are proportional) and cluster-head eccentricity. Table 1 shows these criterion for λ = 1000 and different values of R. (Note that results are given f or λ = 1000 but simulations have shown that the algorithm behavior the same way for different values of λ.)

We note e(H(u)/C) = max v∈C(u) (d(H(u), v)) in number of hops, the eccentricity of the cluster-head of node u inside its cluster, where d(u, v) is the minimum number of hops to reach v from u. Whatever the transmission radius is (and so the node's degree), we can note that the mean cluster-head eccentricity and tree length do not vary too much. This confirms our assumption that the transmission of the clusterhead identity can be expected within a constant and low time. At last, let's note that in the cases where nodes and node's Id are homogeneously and randomly distributed, the use of the DAG does not bring much help. This is due to the fact that in such a nodes distribution, a node uses very rarely the Id to choose its parent because density values are well-distributed and scarcely equal.

We now consider a scenario where nodes are distributed over a grid with Ids increasing from left to right and from the bottom to the top. All interior nodes will have the same value density and the only criteria to select a cluster-head is the Id. As the nodes' Ids are not well distributed, all nodes will finally join the same head. Table 2 shows the obtained results in this case. One can note that the DAG construction is very useful in such a case as it allows to drastically reduce the number of steps needed before stabilization. Figure 1 shows an example of clusters organization obtained for a radius equal to R = 0.05 (One color per cluster). Clusterheads appear in blue. On Figure 1(a) DAG is implemented and several clusters are created. On Figure 1(b), the DAG is not implemented and only one cluster is created.

We have also tested the extra criterion given in Section 4.3 to improve the stability. Due to space limitation, we will not present all the results. We performed simulations where nodes move randomly at a randomly chosen speed during 15 minutes. We computed the percentage of clusterheads which remained cluster-heads after each 2 seconds. For a node mobility between 0 to 1.6m/s (for pedestrians) the percentage of cluster-heads reelection is about 82% with our improvement rules and 78 % without. For a mobility from 0 to 10m/s (for cars) this percentage is 31 % with the new rules against 25 % without. Thus, our improvements are useful in terms of cluster-head stability.

Conclusion

In this paper, we have addressed several issues concerning the self-stabilization on the clustering process in multihop wireless network. We have proved that the clustering algorithm based on the density criteria, defined in [START_REF] Mitton | Self-organization in large scale ad hoc networks[END_REF] is self-stabilizing. We have proposed different enhancements to reduce the stabilization time and to improve stability of the cluster-heads. Note that our contribution regarding the self-stabilization could be applied to several clustering metrics as for instance the node's degree ( [START_REF] Chen | Clustering and routing in mobile wireless networks[END_REF][START_REF] Nikaein | DDR-distributed dynamic routing algorithm for mobile ad hoc networks[END_REF]).

In the future, several possible extensions of this work are open to investigation. It could be interesting to derive sharp bounds on the stabilization as a function of the mobility, e.g., speed of the nodes, mobility model, frequency of links failure, etc. Based on these bounds, we also plan to study hierarchical self-stabilizing algorithms. Finally, we also want to consider energy constraints in the stabilization algorithm and we are investigating energy-efficient organization algorithms.

1

 1 The cluster-head algorithm runs as follows: R1: true → d p := density R2: true → H(p) := clusterHead Lemma Each node p has a correct density value d p within an expected constant time.

Figure 1 .

 1 Figure 1. Clustering example in grid with R = 0.05.

Table 1 .

 1 Clusters features on a random geometric graph for λ = 1000.

		R = 0.05	R = 0.08	R = 0.1
		With DAG No DAG With DAG No DAG With DAG No DAG
	# clusters	61.0	61.4	19.2	19.5	11.7	11.7
	ẽ(H(u)/C(u))	2.6	2.6	3.1	3.1	3.2	3.2
	average tree length	2.7	2.7	3.3	3.3	3.5	3.5
		R = 0.05	R = 0.08	R = 0.1
		With DAG No DAG With DAG No DAG With DAG No DAG
	# clusters	52.8	1.0	29.3	1.0	18.5	1.0
	ẽ(H(u)/C(u))	3.4	29.1	4.1	19.1	3.6	6.5
	average tree length	3.7	83.4	4.7	100.5	4.5	32.1

Table 2 .

 2 Clusters characteristics on a grid for λ = 1000.

http://www.ietf.org/html.charters/manet-charter.html

* This work is supported by the FNS of the French Ministry of Research through the FRAGILE project [7] of the ACI sécurité et informatique.