domain: sde shs

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Executive summary

The data source book aims at gathering all the metadata needed in the labs for future modeling and baseline set up. This deliverable and metadata structure is derived from the D5.2 on Data Management Plan (DMP) and is therefore coherent with Talanoa Water's DMP. The aim of the DMP is that data in the project follows the principles of FAIR : Findability, Accessibility, Interoperability, and Reusability.

For each of the 6 water labs a summary of the principles of modeling is first presented to argue why the data collected is needed. Then metadata includes 20 variables that describes what, how and why the data has been collected and precise operational elements such as data openness.

In this first version, the priority has been given to metadata collection and structuring. Data sharing will follow in early 2022. The metadata and database must be seen as evolutionary throughout the project as both the science and the participation of TALANOA Water labs will be developed during the project and might identify the need for -or produce -new datasets.

Actualized versions of the metadata per lab are available at TALANOA_Sourcebook_Metadata -Google Sheets.

The data that are public and can be shared will be uploaded in Zenodo in the near future. This deliverable objective is to gather and structure all the input metadata of the labs. This deliverable includes both a data sourcebook and a description of the data challenges and uses for each lab.

One of the important requirement is that the data produced follows the principles of FAIR :

Findability, Accessibility, Interoperability, and Reusability. This is one of the main aims of the Data Management Plan.

The deliverable must also be understood as a state of the art in data identificiation, characterization and collection at the end of 2021 rather than a finalized version. If labs have already identified their main modeling strategy and consequent data needs, the process proposed by TALANOA Water which implies an important "field work" means continuous, or at least an extra year of, specification of objectives, of scenarios and strategies or processes to represent with models; and with each new or better (with participation) defined strategies or scenarios new data needs will emerge and will add to the present metadata sourcebook.

There are three main reasons for trying to harmonize the data :

-The objective to run the WAPOR accounting approach.

-The objective to have similar workflows and understanding of the modular approaches even if they are different in each lab.

-The objective to ease understanding of data in each lab for anyone in the project and above to encourage collaborations across labs and even after the project ends.

The collection and generation of data will inform and catalyze the objectives of TALANOA-WATER across its three pillars: Talanoa Water Dialogue, Actionable Socio-Hydrology Science, and Water Laboratories. By adopting transformational adaptation strategies to water scarcity under climate change, the project contributes to its IWRM (Integrated Water Resources Management) objectives of social equity, economic efficiency and environmental sustainability. Concepts from both pillars, the Talanoa Water Dialogue and Socio-Hydrology Science, will empirically feed into the outputs of all six pilot laboratories.

Input Data Characteristics

This deliverable is aimed at gathering raw or harmonized data. TALANOA-WATER will build on both freely accessible existing data and data with restricted access. This is why the metadata represents the core of the contribution of this deliverable. The database (data uploaded to the database) is the remaining. To organize the metadata across lab and project we characterize data according to 20 optional variables.

Categories in the metadata are presented on OTHER : Other data, that might be specific to each lab and required according the models or processes that will be represented

The remaining of the deliverable is the metadata for each lab. Actualized versions of this metadata are available at TALANOA_Sourcebook_Metadata -Google Sheets.

Generic data & modeling

Two modeling approaches are at a higher scale than the labs :

-The WAPOR accounting approach -The macro-economic modeling approach a. WAPOR data

To perform the rapid water accounting plus (WA+) modeling approach, designed by IHE Delft with its partners FAO and IWMI, the FAO WaPOR database (WaPOR v2.0), along with other open-access global databases, will be utilized. In places where WaPOR data is not available (i.e. Europe), the Lebanese Lab can provide the ET data for water accounting using the HSEB model (Jaafar et.al, 2022, forthcoming). Work on the WaPOR database will include deriving remote sensing data, preparing the data for computations of monthly water balance for WA+ purposes, and performing quality checks such as comparing WaPOR precipitation data with in situ observations, as well as Actual Evapotranspiration (Eta) with other remotely sensed Eta estimates.

The FAO WaPOR database (2009-2021) is publicly accessible and uses satellite data that allows the monitoring of agricultural water productivity at different scales and available for Africa and Near East.

For the purpose of running the WA+ tool, data derived from WaPOR will incorporate: monthly actual evapotranspiration and interception, dekadal interception, monthly reference evapotranspiration, daily and monthly precipitation, as well as yearly land cover classification.

The WA+ modeling approach will also include global data derived from several databases such as monthly 2018) to get a regionalized database at the NUTS2-1 level. For the fishery sector we use information from the Regional Dependency on Fisheries report (EU, 2007) and for forestry data from the Global Forest model [START_REF] Di Fulvio | Spatially explicit assessment of roundwood and logging residues availability and costs for the EU28[END_REF].

Sub-national domestic demand and trade with other regions within the country are computed using the so-called Simple Locations Quotients (SLQs) [START_REF] Bonfiglio | Evaluating Implications of Agricultural Policies in a Rural Region through a CGE Analysis[END_REF][START_REF] Bonfiglio | Assessing the Behaviour of Non-Survey Methods for Constructing Regional Input-Output Tables through a Monte Carlo Simulation[END_REF], Miller and Blair, 2009). SLQs provide a measure of the sectoral specialization in the regional economy and starting from this it is possible to determine the domestic demand and aggregate demand for imports. Then a gravitational approach in line with Horridge and Wittwer ( 2010) is implemented to estimate the bilateral trade flows across sub-national regions. An exhaustive description of the methodology can be found in [START_REF] Bosello | A Sub-national CGE Model for the European Mediterranean Countries[END_REF].

In the macro model a special focus is dedicated to the European Mediterranean countries (Spain, France, High levels of soil salinity can be tolerated if salt-tolerant plants are grown (FAO 1992).

Technical feasibility of saline agriculture in Egypt (Moen, et. al., 2013, Heijden, et. al., 2014) needs to be established at two levels: conceptual and practical. At the conceptual level, simulation and demonstration models can establish the optimum spatial scales for the range of crops and application systems. This will account for the spatial limitations of the application system, the constraints imposed by sensing needs and capability, and the ability of simulation tools to accurately predict soil moisture, yield and total dry biomass of crops in salty soils. This stage must also determine if the diagnostic tools needed to determine the causes of particular crop responses are available and sufficiently accurate. Vegetation Index (SAVI), Brightness Index (BI), Salinity Index (SI) and Soil Salinity & Sodicity Index (SSSI)

have been used to develop the optical data (multi spectral) from Enhanced Thematic Mapper i.e.

LANDSAT ETM + using GIS and remote sensing.

Objectives:

 Optimum cropping pattern by using Geographic Information System and Remote sensing & socioeconomic constraints;

 Mapping water quality & salt affected lands; and Methodology:

"Model Farm" will be established at least at four test sites in four different command areas in order to achieve our proposed goal and objectives. These pilot areas are chosen, because they represent hotspots of soil and water salinity of the Nile Delta.

Soil sampling:

Bouaziz et al. ( 2011) method will be used for soil sample collection. Global positioning system (GPS) will be used for the exact coordinates for each composite soil sampling at the proposed test site. Secondary data from the published water quality reports, soil survey reports, soil atlas and maps for the project areas will be collected and used during interpretation.

Project area base map:

We will use the databases to portray the salt affected lands and post-classification processes in GIS & RS to develop project base map.

Satellite data processing:

LANDSAT ETM+ satellite images of high spatial resolution will be acquired for the test sites. The downloaded images will be geo-fixed to the UTM (Universal Transverse Mercator) coordinating system by the World Geodetic System -1984 (WGS-84). We will use the Enhanced Visualization Imaging i.e.

ENVI software to standardize and process all the remote sensing data.

Land suitability and capability

The Agriculture Land Evaluation System for arid and semi-arid regions (ALES-Arid) (Ismail et al., 2005) was developed as a software1 to estimate the agriculture land evaluation linked directly to its relational database and indirectly with a GIS through a loosely coupled strategy. The ALESarid-GIS software is used for evaluating land suitability and capability for producing different crops based on physical and chemical properties of soil, in addition to irrigation water quality. ALES-Arid was redeveloped as a dynamic link library (.dll) integrated with a GIS through the embedded coupling strategy on a common GIS user interface (ArcGIS desktop) entitled ALESarid-GIS as a GIS-based multicriteria decision analysis (GIS-MCDA) in order to assess the agricultural land capability and suitability for 28 activities (field crops, vegetable crops, forage crops and fruit trees). In addition, wheat and maize yield prediction (Abd El-Kawy et al., 2010), The evaluation is based on crop suitability affected by the environmental potentially at the site, such as the physical, chemical and fertility characteristics of the soil, irrigation water quality, and climatic conditions that represent the main factors affecting agricultural soil suitability and productivity in arid and semi-arid regions. Input data consists of soil physical properties, soil chemical properties, soil fertility properties, and finally climate data. ALESarid-GIS was designed to assess land suitability and capability for producing different crops based on both decision trees and maximum limitation tables.

The model retrieved the structured input data from the relational database and then temporally stored.

Input data consist of:

 Soil physical properties (soil texture, soil depth, available water, and soil permeability);

 Soil chemical properties (soil salinity, soil alkalinity, calcium carbonate content, gypsum content, cation exchange capacity, and soil reaction);

 Soil fertility properties (organic matter, available forms of N, P, and K);

 Irrigation water characteristics and qualities (water salinity and toxicity); and

 Climatic data present & future (means summer and winter temperature). The data inputs needed to run these models is described in the metadata file.

b. Metadata

Metadata for the egyptian lab are still under characterization and can be seen here:

TALANOA_Sourcebook_Metadata -Google Sheets
One of the specifity is that the majority of data is not open access and will not be uploaded because of data sharing restrictions resulted from the political situation arised due to the dilemma of the Ethiopian Dam. The coupling framework depicted in Figure 1 is designed to be replicable and flexible, capable of including alternative micro-, macro-economic and hydrologic models. The coupling framework can be integrated with standard agricultural microeconomic models including Expected Utility [START_REF] Von Neumann | Theory of games and economic behavior[END_REF], Linear Programming [START_REF] Paris | An Economic Interpretation of Linear Programming[END_REF], Positive Mathematical Programming [START_REF] Howitt | Positive Mathematical Programming[END_REF], Multi-criteria Decision Models [START_REF] Pereira | Demand and delivery simulation and multicriteria analysis for water saving[END_REF][START_REF] Sumpsi | On farmers' objectives: A multi-criteria approach[END_REF] and Positive Multi-Attribute Utility Programming models (Essenfelder et al., 2018;Gutiérrez-Martín and Gómez, 2011). Macroeconomic standard models that can be incorporated into the coupling framework include Computable General Equilibrium (CGE) (Hertel and Liu, 2016) and Input Output (IO) models [START_REF] Oosterhaven | A New Approach to Modeling the Impact of Disruptive Events[END_REF]. Ideally, macroeconomic models should be regionally-calibrated (NUTS 2 scale or similar) to increase the spatial disaggregation of the shares of value added, and accordingly of labor, capital, natural resources and land, and the accuracy of the coupling with microeconomic models [START_REF] Koks | Regional disaster impact analysis: comparing Input-Output and Computable General Equilibrium models[END_REF][START_REF] Koks | Regional disaster impact analysis: comparing Input-Output and Computable General Equilibrium models[END_REF]. Finally, a large pool of hydrologic models is available in the literature, but not all of them can be integrated into the coupling framework above. In the Italian Lab the choice has not already been made. To be compatible with economic models in the modeling framework proposed in this paper, hydrologic models must meet the following criteria: i) be spatially distributed (i.e. fully or semi-distributed models, to be capable of spatially representing different microeconomic agents); and ii) have a land management module (i.e. be capable of translating the crop portfolio choices taken by the different microeconomic agents into hydrological responses). A non-exhaustive list of models satisfying these criteria includes: the

b. Metadata

Table 1. HEAC metadatabase for the Italian Lab

c. Perspectives in terms of data collection, validation and data gaps

For macro-economic data see section on Global data.

Environmental data

Concerning the required environmental data to feed hydrological models the Italian lab can rely on substantial regional databases that ranges from hydrology (water availability and uses, quality) for surface and ground water, to meteorological and GIS layers data, with particular emphasis on Emilia Romagna Regional datasets, that include also outputs from numerical models such as groundwater model.

Such data shall be used to validate-calibrate selected hydrological models locally; over 60% of data can be retrieved through direct access to Regional web portals, while for the remaining ones a request shall be made to the data owners.

Better Availability is for core data necessary for hydrological modeling validation such as Terrain, land/soil/geology, hydrometeorological gauging stations( surface and ground water), while groundwater modeling inputs/outputs shall be requested directly to the Agency (or Regional office) that set up the model. While the detailed modeling will be determined with stakeholders and elaborated during the bilateral exchanges and in the science policy workshops, the general idea of modeling is presented in Figure 2. This figure also presents the articulation with the participation in the lab. The core of the modeling architecture will be formed by the agro-economic model and the hydrological model that will be coupled thanks to a platform such as, e.g., Openfluid (Fabre et al. 2010). There is still uncertainty on how the crop growth (agronomic processes) will be represented and if will be integrated into the economic model. The opportunity to resort to a macroeconomic model (see e.g. Figure 4 in the Spanish lab) is also still to be studied and contemplated with the Italian partners (CMCC) that master macroeconomic modeling. The microeconomic model will be an agricultural economic programming model (e.g. Graveline, 2016)

developed with GAMS software. The challenge for the economic modeling will be to integrate disruptive innovations whereas this type of modeling is known to be appropriated to handle marginal changes (increases in prices of outputs or inputs). This challenge will be dealt

Objectives:

We aim to assess the water resources situation in the Litani Basin from 2017 to 2020. Sustainable utilization of the Litani River water resources is critical and requires getting further insights on water availability, withdrawals, consumptive and non-consumptive uses, and the resulting services and benefits. For this purpose, we will run the rapid Water Accounting plus system (WA+), designed by IHE Delft with its partners FAO and IWMI using remote sensing derived data from FAO's WaPOR database (WaPOR v2.0)

along with other open-access databases. The WA+ framework is essentially required to complement the lack of routine water resources data collection while incorporating spatially distributed water consumption. Remarkably, our analysis will perform a basin-wide analysis (WA+ Resource Base) and further investigate the following:



Model scheme:

The WA+ methodology is based on the early work of Molden (1997), and is further developed by Karimi (2014) and Bastiaanssen (2015). We use the models and scripts that are created to calculate the water accounts available in GitHub under the water accounting account2 . The Rapid water accounting tool available in GitHub only provides the resource base sheet, which is the major output of water accounting+.

It is mainly used when we are interested in rapidly assessing the water resource situation in a basin or when all data needed to complete WA+ might not be available. The approach distinguishes green evapotranspiration (from rainfall) and incremental evapotranspiration (groundwater or water supplied The required data will be collected for the recent years (2017-2020). However, before using the data for the Water Accounting Plus, several quality checks will be performed including:

 comparing WaPOR precipitation data with in situ observations  comparing WaPOR ETa (Evapotranspiration) with other remotely sensed ETa estimates  assessing the difference between WaPOR-derived basin-scale water balance and total water storage from GRACE, the Gravity Recovery and Climate Experiment NASA mission that studies key changes in the planet's waters, ice sheets, and the solid Earth.

Economic Data

The Lebanese water lab has already collected most of the data needed for running the models since the beginning of the project. Economic data may be utilized in the socio-economic model that will be designed and developed by TALANOA consortium partners. The lab will potentially contribute to informing the transformational strategies of TALANOA-WATER by utilizing the economic data, along with the WA+ tool, to explore single or multiple socio-economic scenarios (e.g. policy scenario or climate change scenario). We aim to run micro-models that will expedite the assessment of the responses of users, as well as allow the evaluation of the possible socio-economic reprecussions of alternative transformational policies.

d. References

Karimi Below we provide an illustrative example of the protocol-based modular hierarchy that will be adopted in the Cega Water Lab (Figure 1). In this basin, policymakers and other relevant stakeholders are mostly concerned about the impacts of water conservation policies on agriculture, and the tradeoffs that emerge between environmental and economic impacts. Since the policy target is restore and conserving water bodies, a hydrological model will be necessary to empirically assess the repercussions of the policy on them. Also a microeconomic model will be necessary, since theoretical and empirical evidence shows that farmer's beliefs and perception of water scarcity is a key factor determining adaptive responses in humanwater systems, which in turn condition water use and other physical (e.g., land use) and economic impacts (Alam, 2015;[START_REF] Udmale | Farmers' perception of drought impacts, local adaptation and administrative mitigation measures in Maharashtra State, India[END_REF]. If changes are marginal or happen at a small scale, we can assume that input (e.g., labor, machinery) and output prices will not be affected by the policy ("small open economy" assumption) [START_REF] Schöb | Ecological Tax Reforms and the Environment: A Note[END_REF]. However, large-scale shifts in the crop portfolio, e.g., as a result of a region-or nation-wide policy, can lead to impacts on prices through feedbacks into the output of economic sectors at a regional and supra-regional scale (Hertel and Liu, 2016). As the economy transitions towards a new equilibrium, inputs and commodity prices, including those relevant for agriculture, will change, affecting in turn farmers' decisions. This calls for a macroeconomic model. Although the Cega is a relatively minor basin within the Douro River Basin and the Castile and León Region, it concentrates much of the horticultural production of the region, and thus any local policy can affect regional and even national markets. Thus, a macroeconomic module will be included as well.

The hierarchy will work as follows: initial states and forcings (scenarios), defined by unique combinations of a policy/policy mix (as per those agreed with stakeholders, a preliminary description of which is available in the Grant Agreement) that will force microeconomic models to assess users' responses.

Resulting impacts on land use, income, and/or utility (monetized, e.g., through compensating variation)

are adapted and imported into a macroeconomic modeling framework to calculate the economy-wide repercussions of the policy shock across sectors and regions [START_REF] Koks | Regional disaster impact analysis: comparing Input-Output and Computable General Equilibrium models[END_REF]. Macroeconomic models can feedback into microeconomic models, e.g., through price changes as predicted in Computable General Equilibrium models [START_REF] Hasegawa | Introducing detailed land-based mitigation measures into a computable general equilibrium model[END_REF][START_REF] Ronneberger | KLUM@GTAP: Introducing Biophysical Aspects of Land-Use Decisions into a Computable General Equilibrium Model a Coupling Experiment[END_REF]. Under a time-invariant setting (setting 1), this iterative process can be repeated until convergence is reached, at which point predictions are stable and consistent, and comparative statics are performed. Convergence can be assessed through a convergence test [START_REF] Hasegawa | Introducing detailed land-based mitigation measures into a computable general equilibrium model[END_REF][START_REF] Ronneberger | KLUM@GTAP: Introducing Biophysical Aspects of Land-Use Decisions into a Computable General Equilibrium Model a Coupling Experiment[END_REF]. Physical outputs (e.g., land use, water The Tunisian water lab will need more details data and information on how hydrological, social, and economic aspects will impact agriculture production and citizens welfare in water scarcity context accentuated by water quality (salinity) increasing problems (Ben Hammouda et al., 2021;Nefzaoui et al., 2021). This water salinity problem constraigning currently agriculture production will be a focus of the Water Lab of Jeffara to bring scientif and practical solutions helping farmers improving their economic profitability and the sustainability of their lands. To this end, we plan to extend the modelling effort to water salinity dynamics et its possible future trend over the existing aquifers in the studied plain.

For this reason, we propose to use the social, economic, and hydrological data to develop tailored thematic models and to couple them using a computing platform (WEAP for example) in a harmonized way. The ambition is to make available for the water lab a simulation platform that supports stakeholders dialogue

Conclusion & perspective

The metadata and databases must be seen as evolutionary throughout the project as both the science and the participation of TALANOA Water labs will be developed during the project and might identify the need for -or produce -new datasets. These will be updated in the metadata base and database with respect to guidelines and principles of the data management plan (D.5.2). A new version of the metadata will be made available and harmonized for about month 16.

  1. Introduction : Objective of the Data Collection ................................................................................................ 2. Input Data Characteristics ............................................................................................................................... 3. Generic data & modeling ................................................................................................................................-The WAPOR accounting approach ............................................................................................................... -The macro-economic modeling approach .................................................................................................... a. WAPOR data ............................................................................................................................................... b. The macro-economic modeling approach .................................................................................................... 4. Egyptian Water Lab. ...................................................................................................................................... a. Presentation of the modeling ambition in the lab .......................................................................................... b. Metadata .................................................................................................................................................. c. Perspectives in terms of data collection, validation and data gaps ............................................................. d. References: ................................................................................................................................................... 5. Italy -Po River basin .....................................................................................................................................a. Presentation of the modeling ambition in the lab .......................................................................................... b. Metadata .................................................................................................................................................. c. Perspectives in terms of data collection, validation and data gaps ............................................................. d. References ................................................................................................................................................ 6. France ........................................................................................................................................................... a. Presentation of the modeling ambition in the lab ...................................................................................... b. Metadata .................................................................................................................................................. c. Perspectives in terms of data collection, validation, and data gaps ............................................................ d. References ................................................................................................................................................ 7. Lebanon ........................................................................................................................................................ a. Presentation of the modeling ambition in the lab .......................................................................................... b. Metadata ...................................................................................................................................................... c. Perspectives in terms of data collection, validation, and data gaps ................................................................ d. References ................................................................................................................................................ 8. Spain ............................................................................................................................................................. a. Presentation of the modeling ambition in the lab ...................................................................................... 1. Introduction : Objective of the Data Collection The TALANOA Water project is structured in four thematic work packages WP1-4 (ENGAGE, DATA, MODELING, and LABORATORIES); this deliverable is the first in the WP2 DATA. It must be coherent with the D.5.2 Data Management Plan that sets operational characteristics and rules to manage, collect, store the data within TALANOA Water.

  Italy and Portugal) which are all dis-aggregated at the NUTS-2 sub-national level, the maximum geographical detail in the macro-economic model. Tunisia and Egypt have only national detail. Lebanon will need additional work because in the GTAP database it is inside a broader region called "rest of Western Asia" which includes also Iraq, Jordan, Palestinian Territory, Syria and Yemen.Egyptian Water Lab.4. Egyptian Water Lab. a. Presentation of the modeling ambition in the labThe Egyptian Lab. ativities will focus on the Nile Delta region. The cultivated area of the Nile Delta is 4.35 million feddan (1 feddan equals 4,200 m 2 or 0.42 ha), 93% of which are old dark lands of alluvial soils that have a texture which ranges from heavy clayey to clay. This area represents 71.6% of the total old alluvial land in Egypt (6 million feddan) and 55.5% of the total officially cultivated lands (8.6 million feddan). The Nile Delta has recently been suffering from a sharp decline in the area of fertile agricultural land, which is difficult to compensate as a result of salinity, water logging, and seawater intrusion in addition to pollution, nutrient depletion, and population encroachment by builing and other uses, and this degradation is subject to increase as a result of reducing the areas planted by rice(Abu-Zeid 1988, FAO 1992). In addition, soil salinity is caused by excessive accumulation of salts and is typically pronounced at the soil surface. Saline soil distribution is closely related to environmental factors such as climatic, geological, geochemical and hydrological conditions. Poor soil and water management, the intrusion of seawater, the use of slightly saline water (drainage or mixed water) for irrigation without proper management and agronomic practices, or the use of saline groundwater as the only source for irrigation are presently the main causes of salinization in Egypt(AbouKheira, 2005). The majority of salt-affected soils in Egypt are located in the northern-central part of the Nile Delta and in its eastern and western sides.Other affected areas include Wadi-El-Natroun, Tal El-Kebeer, the oases, many parts of the Nile Delta and Valley, and Fayoum province. Approximately, 0.9 million ha suffer from salinization problems in cultivated areas. Furthermore, 60 % of cultivated lands in northern Delta, 20% of the southern Delta and Middle Egypt, and 25% of the Upper Egypt regions are all salt-affected. Salinity of irrigation water can cause a build-up of salts in the root zone, particularly if the internal drainage of soils (or leaching), either due to rainfall or by applied irrigation, is inadequate. Proper irrigation management can avoid salt accumulation by providing adequate drainage to leach added salts from the affected soil root zone layers.

  On a practical level, the feasibility of crops & practices should be proven and demonstrated in field trials with specific on-farm irrigation water management component. These could account for and discuss with the other non technical constraints of famers or companies running agriculture. Indeed other constraints such as capital or human resources and know how are important to consider agricultural development. Accordingly, the databases will be used to depict salinity-affected lands and post-classification operations in GIS and RS (remote sensing) to develop maps for distributing saline water and lands affected by salinity, as well as optimum cropping pattern in the Nile Delta. 1. Mapping water quality and salt affected lands: Satellite Technologies (GIS & Remote sensing) are a convenient mean for decision support systems. Digitalimage processing is acquired by GIS & RS for brackish/saline materials (water and soils) demarcation for the location of "Optimum cropping pattern" for salt-affected lands and saline water. Improved mapping through remote sensing of highly reflective saline soil helps us for taking mitigation measures. Qualitative and quantitative salinity modelling is now possible due to the optical radar data. Thus various approaches like Aster Salinity Index (ASI for Agriculture), Normalized Difference Salinity Index (NDSI), Soil Adjusted

3.

  Irrigation management and crop production in salinity prone areas of the Nile Delta SALTMED model is a generic model that can be used for a variety of irrigation systems, soil types, crops and trees, water application strategies and different water qualities. The early version was successfully tested against field experimental data. The current version, SALTMED 2015, includes additional sub-models, crop growth according to heat units/degree days, crop rotations, nitrogen dynamics, soil temperature, dry matter and yield, subsurface irrigation, deficit irrigation including the Partial Root Drying, PRD, drainage flow to tile or open drains systems, presence of shallow groundwater, evapotranspiration (ET) using Penman-Monteith equation, with different options to obtain the canopy conductance. The current version allows up to 20 fields or treatments to run simultaneously (Ragab et. al., 2016).

  of the modeling ambition in the lab The Italian Lab is the Italian Po river basin located in the Emilia Romagna region. To explore alternative & transformative adaptation strategies to water management we develop the following modeling architecture.

Figure 1 .

 1 Figure 1 . Schematic representation of the five protocol connections that define the methodological steps to iteratively couple the micro, macro and hydro modules.

  Soil and Water Assessment Tool -SWAT(Arnold et al., 1998); Annualized Agricultural Non-Point Source Pollution Model -AnnAGNPS[START_REF] Young | AGNPS: A nonpoint-source pollution model for evaluating agricultural watersheds[END_REF]; Areal Nonpoint Source Watershed Environment Response Simulation -ANSWERS 2000 (Bouraoui and Dillaha, 2000); Agricultural Policy / Environmental eXtender Model -APEX (Gassman et al., 2009); US Army Corps of Engineers -Hydrologic Engineering Center -Hydrologic Modeling System -HEC-HMS (US Army Corps of Engineers, 2015); Soil and Water Integrated Model -SWIM[START_REF] Krysanova | Development of the ecohydrological model SWIM for regional impact studies and vulnerability assessment[END_REF].

  with a PhD dedicated to it. The hydrological model will be developed with the open source airGRiwrm approach (Dorchies et al. 2020).

Figure 2

 2 Figure 2 Preliminary vision of modeling for the French water lab & Articulation between modeling & participation

Figures

  Figures from 2009 to 2016 show that water availability in the Basin is only 800 cubic meters per capita per year. The groundwater resources are depleted by around 57.5 million cubic meters per year, and water storage is decreasing to about 50 million cubic meters per year. Most of the domestic and industrial water in the Basin is left untreated, with high levels of pollution resulting in non-recoverable water of about 469.5 million cubic meters per year, threatening public health and ecosystem functions.

  Assess the current water resources availability and utilization status in the Litani Basin  Track surface and groundwater fluxes  Assess inflows and outflows  Map net water generation and expenditure  Reflect on the quality of the WaPOR v2.0 data for WA+

  through irrigation). It also distinguishes ET (both rainfall and incremental) per land-use class. In addition, WA+ tracks surface and groundwater fluxes and assesses inflows and outflows. The WA+ framework will act as a reporting mechanism for water fluxes, flows, and stocks, that are summarized by means of the WA+ sheet. The assessment holds a potential contribution to better understanding the possible reprecussions of water productivity increases on water users, and provides an enabling environment for exploring the role of land use and land cover on consuming and producing water.

Figure 3

 3 Figure 3 Flow chart of the water accounting process and the data used to develop the water accounting for Litani River Basin

  withdrawals, water consumption) resulting from micro-and macro-economic simulations next feed hydrological simulations, along with climate scenarios (obtained from climate ensembles in CMIP6, 2021) that condition water availability. If hydrologic simulation outputs show an impact on physical constraints of relevance for economic agents (such as a new binding water availability constraint), the process continues with a new iteration to assess further adaptation responses in the human system. This iterative process can be repeated until convergence is reached(Essenfelder et al., 2018).

Figure 4

 4 Figure4The Cega Water Lab multi-system modular hierarchy

Figure 5 .

 5 Figure 5. Ambition of the socio-hydrology modelling for the Tunisian Water Lab

Table 1 :

 1

			Short name	value	comment
		Variable	of variable
		ID.	id.	[1; 2; 3...]	unique number
	Data	Category	cat.	See below
	summary	Type of data	type		these are the cat given by Hadi (first table; generic names
					i.e. geologocal map)
		Data name	name
		Variables	variables		main variables of interest
		Data Provider	owner	Specific/Own	e.g. Water agency / own elaboration (Partner)
				elaboration
		Time Start	time start		"-" if no time related info (e.g. GIS)
	Characteristics	Time End	time end		"-" if no time related info (e.g. GIS)
		Time Step	time step		"-" if no time related info (e.g. GIS)
		location	location		e.g river basin
		spatial	resolution		pixel, 8*8 km, municipality...
		resolution
		Data Collection	method		Specify which method was used to produce the data
	Method	Method			including the instruments that have been used or other
					source data. To be filled if "own elaboration" is
					previously mentioned
	Use	Data Use	use	e.g. hydrologic	Ideally specify the model, not only the WP. More than
		(WP,model)		modul	one is possible
		Open data	open	1/0	1=yes; 0=no
					If Open data = 1 then the data needs to be available on
					the data drive
		Data Use	restrictions	text	If Open data = 0. This clearly states why this data is not
		Restrictions			open data, if any, (ex. Private source; governmental data;
					etc.).
	Practical info	Collected	collected	1/0	This variable is to know whether the lab has already
					received/collected the data. This is to help monitor and
					guide the labs work.
		Data sharing	uploaded	1/0
		Link to the data	link		(URL)
		Data Format	format	e.g tiff, shp., csv.

Table 1 -

 1 Metadata characterization in the TALANOA Water project

	The category of data are:
	(i)	CLIMATE : climate data
	(ii)	HYDRO : hydro(geo)logic including dams related data
	(iii)	LAND USE : land use & agronomic data, that will help characterize the water uptake
	(iv)	ECON : economic datasets (micro & macroeconomics)
	(v)	GIS : GIS data
	(vi)

Table 1

 1 Abu-Zeid M. 1988. Egypt's policies to use low-quality water for irrilzation. Proc. Symp. Re-use of Lowquality Water for Irrigaticn. Water Research Centre, Cairo. Integrating GIS and modeling for agricultural land suitability evaluation at East Wadi El-Natrun, Egypt. Egyptian Journal of Soil Science, 45, 297-322. Moen, H., K. Roest & Kamstra, A. 2013. Feasibility of the Use of Brackish Groundwater in Integrated Aqua-Agriculture systems in Egypt. Study Report by Dienst Landelijk Gebied, ALTERA, Wageningen UR and IMARES, Wageningen UR. 46 pp plus annexes.

	Bouaziz, M.; Gloaguena, R.; Samirb, B. 2011. Remote mapping of susceptible areas to soil salinity, based
	on hyperspectral data and geochemical, in the Southern Part of Tunisia. Proc. SPIE 2011, 8174,
	doi:10.1117/12.898199. 3.

. HEAC metadatabase for the Egyptian Lab c. Perspectives in terms of data collection, validation and data gaps The collection of the data have started since the beginning of the project concerning physical & chemical soil properties. We plan to receive the first dataset by May -June 2022. Concerning economic data, we still do not know how the economic model will be designed and will collect the data when this will be fixed (April 2022). d. References: Abd El-Kawy, O. R.; H. A. Ismail; J. K. Rod and A. S. Suliman. 2010. A developed GIS-based land evaluation model for agricultural land suitability assessments in arid and semi-arid regions. Research Journal of Agriculture and Biological Sciences, 6(5): 589-599. Aboukheira, A. A. 2005. A study of trickle irrigation systems for irrigating some horticultural crops in Delta soils. Ph.D. thesis. Shebin El-Kom, Minufiya University, Egypt. Food and Agriculture Organization. 1992. The use of saline waters for crop production. Tech. Rep., Food and Agriculture Organization, Rome, Italy, 1992. Wageningen UR (University & Research centre), Alterra report 2526. 52 pp.; 3 fig.; 2 photos; 1 tab.; 5 ref. Ismail, H. A., Bahnassy, M. H. & Abd El-Kawy, O. R. 2005. Ragab, R., Choukr-Allah, R., Nghira, A. & Hirich, A. 2016. SALTMED Model and Its Application on Field Crops, Different Water and Field Management and Under Current and Future Climate Change. In: Choukr Allah, R., Ragab, R., Bouchaou, L., Barceló, D. (eds.) The Souss-Massa River Basin, Morocco. The Handbook of Environmental Chemistry. Springer: Berlin, pp. 1-48, doi:10.1007/698_2016_74.

  , P. 2014. Water Accounting Plus for Water Resources Reporting and River Basin Planning (PhD thesis). TU Delft, Delft, The Netherlands. Review of the accuracy of the remote sensing data. Hydrological Earth Systems Science, 19, 507-532. https://doi.org/10.5194/hess-19-507-2015 Molden, D. 1997. Accounting for Water Use and Productivity. Swim paper. Colombo, Sri Lanka. Retrieved from http://www.iwmi.cgiar.org/Publications/SWIM_Papers/PDFs/SWIM01.PDF Recent advances in the construction of protocol-based modular frameworks provide the backbone for the development of interdisciplinary modeling hierarchies that connect multiple systems through two-wayfeedbacks (multi-system hierarchy). Each module within the hierarchy can be populated with multiple models (multi-model ensemble) and combined with scenario discovery techniques that explore scenario uncertainty through varying initial states and forcings (e.g., climate change scenarios, policy scenarios) (CMIP6, 2021; ISIMIP, 2021). The result is a large database of simulations in which each simulation represents the economic and environmental performance under one specific scenario and modeling setting. This information can be used to identify futures where proposed policies meet or miss their objectives, explore potential tipping points, and inform the development of robust policies that show a satisfactory performance under most conceivable futures(Marchau et al., 2019).

	8. Spain
	a. Presentation of the modeling ambition in the lab

Karimi, P. & Bastiaanssen, W.G.M. (2015)

. Spatial evapotranspiration, rainfall and land use data in water accounting -Part 1:

Table 1 .

 1 HEAC metadatabase for the Tunisia Lab

	Simualtions
	Scenarios	Models
		Social	Information
		Economic
	Parameters		Maps	Data	Rules
		Hydrology

https://github.com/wateraccounting

Difficulties and slow response times are envisaged in collecting more granular information's on water quality, point discharges and withdrawals /diversions from permitting authorities and main users (that feed surface water balance schemes and support evaluations over expected environmental effects of discharges regimes); support form Regional or District Authority that may have already summarized such data (i.e. for water management masterplans) shall be requested to try and speed up missing data collection. A reasonable estimation for the collection of non-downloadable data is mid 2022.

d. References

Arnold, J.G., Srinivasan, R., Muttiah, R.S., Williams, J.R., 1998. Large Area Hydrologic Modeling and Assessment Part I: Model Development. J. Am. Water Resour. Assoc. 34,[73][74][75][76][77][78][79][80][81][82][83][84][85][86][87][88][89] https://doi.org/10.1111/j. 1752-1688[START_REF] Schöb | Ecological Tax Reforms and the Environment: A Note[END_REF].tb05962.x Bouraoui, F., Dillaha, T., 2000. ANSWERS-2000 experiments that assess the implications of alternative initial states and forcings based on a priori knowledge. Other modules and protocols will be developed and used, notably by including a timevariant/dynamic setting-i.e., instead of looking for convergence, information is carried forward in time.

Note that as we include additional modules and protocols, models and scenarios, uncertainties compound.

Far from being negative, this outcome is desirable: navigating deep uncertainty type 1 necessitates a thorough sampling of possible futures that goes beyond consolidative modeling and point predictions, so to identify robust policies that perform reasonably well in most conceivable situations [START_REF] Groves | Developing Key Indicators for Adaptive Water Planning[END_REF][START_REF] Haasnoot | Dynamic adaptive policy pathways: A method for crafting robust decisions for a deeply uncertain world[END_REF].

The data inputs needed to run this modeling hierarchy are described in the metadata file.

b. Metadata

The Metadata file is viewable here : TALANOA_Sourcebook_Metadata -Google Sheets c. Perspectives in terms of data collection, validation, and data gaps

The Cega Water Lab has already collected all the hard data necessary to run the modeling hierarchy presented above; as well as all the necessary remote sensing data to build the water accounting sheets of the Cega Basin, which have been already submitted to the Accounting Data coordinator (Hadi Jaafar, AUB). A preliminary version of the water consumption and biomass production simulations obtained processing the remote sensing data for the Cega Water Lab has been produced by AUB. Since the beginning of the TALANOA project, about 30% of the data needed was collected from official departments. Mainly, we collected most of the climatic data as well as hydrological, geological, and social ones. Economic data requires ground surveys as well as face-to-face contacts with local and regional administration and organizations, whichwhat requires trips to the study area and the availability of the budget (made available only on November 2021). To fill this gap, we plan to ensure a campaign of data collection in the Jeffara plain in the first trimester of 2022. The mission will target previous national and international projects dedicated to water management, transfer of technologies, and agriculture production in southern Tunisia.

d. References

Haddad, R., Nouiri, I., Alshihabi, O., Maßmann, J., Huber, M., Laghouane, A., Yahiaoui, H., and Tarhouni, J.


Bo Yang

Mohamed Ichchou
email: mohamed.ichchou@ec-lyon.fr

Christophe Droz
email: christophe.droz@inria.fr

Abdelmalek Zine
email: abdel-malek.zine@ec-lyon.fr

Wave Transmission and Reflection Analysis Based on the Three-dimensional Second Strain Gradient Theory

Keywords: Second strain gradient theory, Wave finite element method, Wave transmission, Wave reflection

The scattering of guided waves through a coupling region is a crucial information when studying waveguides. In this paper, the second strain gradient theory (SSG) is used to describe wave transmission and reflection in a three-dimensional micro-sized medium. First, the constitutive relation of 3D SSG model is derived while six quintic Hermite polynomial shape functions are used for the displacement field. Then Hamilton's principle is used for the weak formulation of the unit-cell's stiffness matrix finite element stiffness, mass matrices and force vector. Eventually the wave diffusion (i.e. including reflection and transmission coefficients) are computed and discussed for various coupling conditions.

Introduction

The dynamical properties of guided waves such as wave transmission and reflection have been widely studied over the past decade especially in the field of acoustics, earthquake and electromagnetic. Initially, studies focused on the guided waves interaction at interfaces between different macro-medias. But for the micro-sized structure with size effects, the micro-particles such as atoms with associated energy on the free surface of the structure has a significant influence on the structure's behavior. This energy related to surface atoms is called surface free energy which produces surface tension. The surface tension can not be ignored due to the very large ratio between the surface and the volume of structure. On the other hand, long-range or non-local interaction between micro-particles has also an indispensable effect on the micro-sized structure's dynamical behaviors [START_REF] Miller | Size-dependent elastic properties of nano-sized structural elements[END_REF][START_REF] Lim | Size-dependent nonlinear response of thin elastic films with nano-scale thickness[END_REF][START_REF] Rosi | On the validity range of strain-gradient elasticity: a mixed static-dynamic identification procedure[END_REF]. The wave propagation and diffusion in micro-medias can no longer be reasonably predicted by Classical Theory (CT) of continuum mechanics [START_REF] Yang | Dynamic analysis of second strain gradient elasticity through a wave finite element approach[END_REF].

Therefore, the non-classical continuum theories of elasticity that can interpret the properties of micro-sized structures have been proposed. Generally, these non-classical theories can be categorized into non-local elasticity theory [START_REF] Kroner | Elasticity theory of materials with long range cohesive forces[END_REF], micro-continuum theory [START_REF] Eringen | Simple microfluids[END_REF][START_REF] Eringen | Linear theory of micropolar elasticity[END_REF], surface elasticity theory [START_REF] Gurtin | A continuum theory of elastic material surfaces[END_REF] and strain gradient family [START_REF] Lazar | Dislocation in second strain gradient elasticity[END_REF]. The strain gradient family is composed of the couple stress theory, the first and second strain gradient theories and the modified couple stress theory. Mindlin established one of the strain gradient family called First Strain Gradient (SG) theory [START_REF] Mindlin | Micro-structure in linear elasticity[END_REF] in which the constitutive relations is composed of strain and the first gradient of strain. The atomic structure with the nearest and next nearest interactions between different particles is used to describe the SG theory in the framework of lattice spring model, but only in noncentro-symmetric materials [START_REF] Shodja | Calculation of the additional constants for fcc materials in second strain gradient elasticity: behavior of a nano-size bernoulli-euler beam with surface effects[END_REF]. In order to explore the properties of centro-symmetric materials, the Second Strain Gradient (SSG) theory [START_REF] Mindlin | Second gradient of strain and surface tension in linear elasticity[END_REF] was put forward, which offers a reasonable description of the strain and surface tension properties on the micro-structure's surface by introducing the high-order parameters. The constitutive relations in SSG theory is a function of strain, first gradient of strain and second gradient of strain. The connections between the SSG theory and lattice spring model with the nearest, next nearest and next-next nearest neighbor interactions for 1D structures can be confirmed through the Fourier series transform [START_REF] Yang | Dynamic analysis of second strain gradient elasticity through a wave finite element approach[END_REF][START_REF] Khakalo | Form ii of mindlin's second strain gradient theory of elasticity with a simplification: For materials and structures from nano-to macro-scales[END_REF].

On the other hand, in order to study the wave diffusion in complex structures, the numerical methods such as Spectral Finite Mlement (SFE) method [START_REF] Mahapatra | A spectral finite element for analysis of wave propagation in uniform composite tubes[END_REF] and Semi-Analytical Finite Element (SAFE) [START_REF] Innveden | Waveguide finite elements for curved structures[END_REF] can be used. In the past decade, the Wave Finte Element Method (WFEM) [START_REF] Mencik | Wave finite elements in guided elastodynamics with internal fluid[END_REF][START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF][START_REF] Zhao | Improved hermite finite element smoothing method for full-field strain measurement over arbitrary region of interest in digital image correlation[END_REF] has attracted many works. The advantage of WFEM is the convenient application in engineering field. Since it can be developed from the Finte Element Method (FEM) packages which allows the current element library and grid generation procedures to be applied for the modelling of different waveguide structures. In addition, WFEM can reduce a global periodic structure into a single substructure or unit cell based on the periodic structures theory [START_REF] Mead | A general theory of harmonic wave propagation in linear periodic systems with multiple coupling[END_REF][START_REF] Singh | Stochastic wave finite element quadratic formulation for periodic media: 1d and 2d[END_REF]. The resulting stiffness and mass matrices are post processed to offer the dynamic stiffness matrix. The dynamical properties of the periodic structure can be reflected through the spectral analysis of the unit cell [START_REF] Duhamel | Finite element analysis of the vibrations of waveguides and periodic structures[END_REF][START_REF] Ahsani | Diffusion based homogenization method for 1d wave propagation[END_REF]. The main objective of this work is, firstly, to calculate the multi mode propagation in a 3D periodic waveguide by SSG theory, and, secondly, to confirm the wave diffusion under a complex coupling condition. The article's structure is the following: in section 2, the constitutive relations of 3D micro-sized model are introduced in the SSG theory framework and the weak formulations including element stiffness, mass matrices and force vector are calculated. Afterwards, in section 3, free wave propagation characteristics are expressed by solving eigenvalue problems in the direct WFEM framework, diffusion matrix for a complex coupling condition are confirmed. In section 4, wave diffusion is introduced. Ultimately, some useful conclusions are presented in section 5.

A brief of second strain gradient elasticity

3D constitutive relations calculation

The strain energy density U composed of strain ε=sym(∇W), first gradient of strain ξ=∇∇W and second gradient of strain ζ=∇∇∇W in the SSG theory framework was put forward by Mindlin [START_REF] Mindlin | Second gradient of strain and surface tension in linear elasticity[END_REF], as below:

U = 1 2 λε ii ε jj + µε ij ε ij + a 1 ξ ijj ξ ikk + a 2 ξ iik ξ kjj + a 3 ξ iik ξ jjk + a 4 ξ ijk ξ ijk + a 5 ξ ijk ξ jki + b 1 ζ iijj ζ kkll + b 2 ζ ijkk ζ ijll + b 3 ζ iijk ζ jkll + b 4 ζ iijk ζ llkj + b 5 ζ iijk ζ lljk + b 6 ζ ijkl ζ ijkl + b 7 ζ ijkl ζ jkli + c 1 ε ii ζ jjkk + c 2 ε ij ζ ijkk + c 3 ε ij ζ kkij , (1)
where W is the displacement vector, symbol ∇ means the gradient operator, λ and µ represent the Lamé parameters which are related to the Young's modulus E, the Poisson's ratio ν and the shear modulus G [START_REF] Yang | Dynamic analysis of second strain gradient elasticity through a wave finite element approach[END_REF]. a i , b i and c i denote the higher order parameters [START_REF] Shodja | Calculation of the additional constants for fcc materials in second strain gradient elasticity: behavior of a nano-size bernoulli-euler beam with surface effects[END_REF] in SSG theory. The higher order parameters for Al and Cu are shown in Table 1. Based on the 3D elasticity theory, the vector of displacement field defined in the Cartesian coordinate system (x, y, z) is given as:

W(x, y, z, t) = u(x, y, z, t), v(x, y, z, t), w(x, y, z, t) T , (2)
where u, v and w are the the displacements along x, y and z direction.

The relations between strains and displacement components can be defined by introducing the vectors of first, second and third order derivatives of displacement components:

ε = Q 1 W, ξ = Q 2 W, ζ = Q 3 W, (3)
where

Q 1 =   ∂ 1 0 0 0 ∂ 3 0 0 ∂ 2 0 ∂ 3 0 ∂ 2 0 0 ∂ 3 ∂ 2 ∂ 1 ∂ 1   T , Q 2 =   t 1 0 0 0 t 1 0 0 0 t 1   ⊗ ∂ 11 ∂ 22 ∂ 33 2∂ 12 2∂ 13 2∂ 23 T , Q 3 =   t 2 0 0 0 t 2 0 0 0 t 2   ⊗ ∂ 111 ∂ 222 ∂ 333 3∂ 112 3∂ 113 3∂ 221 3∂ 223 3∂ 331 3∂ 332 6∂ 123
T , symbol ⊗ stands for the Kronecker product, t 1 with size 6 × 1 and t 2 with size 10 × 1 are the matrices whose element value is 1.

Then, the constitutive relations for 3D model by SSG theory can be defined as:

τ 1 = Lε + Cζ, τ 2 = Aξ, τ 3 = Bζ + C T ε, (4)
in which L is matrix including classical parameters, A, B and C are matrices including higher order parameters [START_REF] Torabi | Second strain gradient finite element analysis of vibratory nanostructures based on the three-dimensional elasticity theory[END_REF]. Finally, the strain energy density for SSG theory can be rewritten as the matrix form:

U = 1 2 ε T Lε + ξ T Aξ + ζ T Bζ + 2ζ T C T ε . (5)
Here should be noted that, Eq. 5 is the basic form of building a 3D model using partial differential equations (PDE) weak form in some commercial numerical simulation software (e.g., COMSOL).

Finite element discretization

After obtaining the strain energy density, the next step is to calculate the weak form including stiffness and mass matrices and force vector. The definition of node degree of freedoms (DOFs) for 1D and 3D Hermite elements. Firstly, in order to ensure the continuity of higher derivatives up to the second order between 1D elements, the six-term polynomial function is considered to interpolate the scalar field W 1 =u(x, t) inside a 1D element, as follows:

W 1 = 1 x x 2 x 3 x 4 x 5 s 0 s 1 s 2 s 3 s 4 s 5 T = xs. ( 6
)
The evaluation of the nodal displacement vector w (e)

1 of 1D element gives:

w (e) 1 = d 1 d 2 d 3 d 4 d 5 d 6 T s = ds, (7)
where ). Then, submitting Eq.7 into Eq.6, the displacement vector within the 1D element can be derived by employing the six quin-tic Hermite polynomial shape function and nodal displacement vector, as follows:

d 1 =(1 -l e l 2 e -l 3 e l 4 e -l 5
W 1 = xd -1 w (e) 1 = S(x)w (e) 1 , (8)
in which the shape function S(x) is written as:

S(x) = S 0 1 (x), S 1 1 (x), S 2 1 (x), S 0 2 (x), S 1 2 (x), S 2 2 (x) , (9)
where S 0 1 (x)= 5x

= [N 0 1 (y), S 1 1 (y), S 2 1 (y), S 0 2 (y), S 1 2 (y), S 2 2 (y)], S(z) = S(x)| x=z = [S 0 1 (z), S 1 1 (z), S 2 1 (z), S 0 2 (z), S 1 2 (z), S 2 2 (z)].
The shape function of hexahedral element:

S(x, y, z) = S 1 (x, y, z) ⊗ e 1 , S 2 (x, y, z) ⊗ e 2 , S 3 (x, y, z) ⊗ e 3 T . (10)
The element in S p (x, y, z) and e p (p = 1, 2, 3) are defined as:

S i(j,k,l) p (x, y, z) = S j i ′ (x)S k i ′′ (y)S l i ′′′ (z), e p = ϵ p1 ϵ p2 ϵ p3 , (11)
where i = 1, ..., 8. j, k, l = 0, 1, 2. S i(j,k,l) p (x, y, z) is associated with the DOFs ∂ j+k+l u i 1 /(∂x j ∂y k ∂z l ) of node i of the hexahedron element. i ′ , i ′′ , i ′′′ = 1, 2 relate to the node number in the corresponding 1D element and they take values of 1 or 2 if the coordinate value of node i is -l e or l e . The displacement vector W(x, y, z) within 3D element can be expressed as: W(x, y, z, t) =S(x, y, z)w (e) (t), [START_REF] Mahapatra | A spectral finite element for analysis of wave propagation in uniform composite tubes[END_REF] where w (e) =[(w ) T ] T (p = 1, 2, 3). Then, integrating the strain energy density over its volume to obtain the strain potential energy U as:

U = V UdV = 1 2 w (e)T V S T Q T 1 LQ 1 S + S T Q T 2 AQ 2 S + S T Q T 3 BQ 3 S + 2S T Q T 3 CQ 1 S dV w (e) . ( 13
) On the other hand, the beam kinetic energy is expressed as:

T = 1 2 V ∂W ∂t T ρ ∂W ∂t dV = 1 2 ∂w (e) ∂t T V S T ρS dV ∂w (e) ∂t , (14)
where ρ denotes the linear mass density, l 1 and l 2 are higher-order length-scale parameters. Meanwhile, the work done δW by external force can be expressed as:

δW = V δW T f V dV + S δW T f S dS = δw (e)T V S T f V dV + S S T f S dS , (15)
where f V is the volume force, f S means the face force. The 3D element stiffness, mass matrices and force vector can be confirmed as:

K (e) = V S T Q T 1 LQ 1 S + S T Q T 2 AQ 2 S + S T Q T 3 BQ 3 S + 2S T Q T 3 CQ 1 S dV, M (e) = V S T ρS dV, F (e) = V S T f V dV + S S T f S dS. ( 16
)
3 Diffusion analysis through wave finite element method This section is concerned with the characterization of coupling conditions between two semi-infinite periodic waveguides which are connected through an elastic coupling element at surfaces 1 and 2 as shown in Fig. 1. The reflection coefficients (R) and transmission coefficients (T) are confirmed through a complex coupling condition: the wave modes calculation in waveguides is based on the CT but SSG theory in coupling element (CT-SSG). The propagation constants Λ and eigenvectors Ψ u in a unit cell can be solved by direct Bloch formulation [START_REF] Duhamel | Finite element analysis of the vibrations of waveguides and periodic structures[END_REF][START_REF] Yang | Dynamic analysis of second strain gradient elasticity through a wave finite element approach[END_REF][START_REF] Droz | A reduced formulation for the free-wave propagation analysis in composite structures[END_REF] as:

L k L c L k+1 ^ŵ (k) ŵ(c) L 2 L1 ŵ(c) R 1
D RL (ω)Λ -1 + (D RR (ω) + (D LL (ω)) + D LR (ω)Λ Ψ u = 0, ( 17
)
where Λ=diag {λ j } j=1,...,2p , Ψ u = {ϕ j } j=1,...,2p which can be divided into Ψ + u = ϕ + j j=1,...,p and Ψ - u = ϕ - j j=p+1,...,2p , in which p=m for SSG, p=n for CT. The waves propagate to positive when |1/λ j | <1. The waves propagate to negative when |λ j | >1. Here, λ j take the form λ j =exp(-iκ j L (q) ), subscript q=c for coupling element, q=k, k+1 for unit cells. It should be noted that the spectral analysis method of the coupling element and unit cell k+1 is the same as that of unit cell k. The displacement field of coupling element (c) and unit cells (k, k+1) can be represented by the superposition of the eigenmodes: ŵ(q)± = Ψ (q)± u υ (q)± P (q)± , [START_REF] Torabi | Second strain gradient finite element analysis of vibratory nanostructures based on the three-dimensional elasticity theory[END_REF] where P (q)± is the amplitudes of wave modes. υ (q)± =diag exp ∓iκ (q) j x (0 ≤ x ≤ L (q) ), symbol q=c for coupling element, q=k, k+1 for unit cells. On the other hand, the force components of coupling element (c) and unit cells (k, k+1) can be expressed as: F(q)± = Ψ (q)± F υ (q)± P (q)± , [START_REF] Zhu | Wave transmission and reflection analysis through complex media based onthe second strain gradient theory[END_REF] in which

Ψ (q)± F =D (q) LL Ψ (q)± u +D (q) LR Ψ (q)± u diag exp ∓iκ (q)
j L (q) , symbol q=c for coupling element, q=k, k+1 for unit cells. The eigensolutions will be the same when waveguides 1 and 2 have the same cross-section and material. The state vector on the right side of surface 1, represented by R 1 :

ŵ(c) R1 F(c) R1 = Ψ (c)+ u P (c)+ + Ψ (c)- u P (c)- Ψ (c)+ F P (c)+ + Ψ (c)- F P (c)-. ( 20
)
In addition, the state vector on the left side of surface 2, represented by L 2 , is written as:

ŵ(c) L2 F(c) L2 = Ψ (c)+ u υ (c)+ | x=Lc P (c)+ + Ψ (c)- u υ (c)-| x=Lc P (c)- Ψ (c)+ F υ (c)+ | x=Lc P (c)+ + Ψ (c)- F υ (c)-| x=Lc P (c)-. ( 21
)
Combining Eq.20 and Eq.21, the relation between state vector on left and right side of coupling element will be conformed:

ŵ(c) L2 , F(c) L2 T = X (c) ŵ(c) R1 , F(c) R1 T , (22)
with

X (c) = Ψ (c)+ u υ (c)+ | x=Lc , Ψ (c)- u υ
(c)-| x=Lc Ψ (c)+ F υ (c)+ | x=Lc , Ψ (c)- F υ (c)-| x=Lc Ψ (c)+ u Ψ (c)- u Ψ (c)+ F Ψ (c)- F -1 . (23)
Next, assuming that the incident waves come from the infinity of waveguide 1 and there is no reflection from the end of waveguide 2. The initial boundary of waveguide 1 is also non-reflecting. The state vector on the left side of surface 1, represented by L 1 , is expressed as:

ŵ(k) L1 F(k) L1 = Ψ (k)+ u υ (k)+ | x=L k P (k)+ + Ψ (k)- u υ (k)-| x=L k P (k)- Ψ (k)+ F υ (k)+ | x=L k P (k)+ + Ψ (k)- F υ (k)-| x=L k P (k)-. (24)
The state vector on the right side of surface 2, represented by R 2 , is written as:

ŵ(k+1) R2 F(k+1) R2 = Ψ (k+1)+ u υ (k+1)+ | x=0 P (k+1)+ Ψ (k+1)+ F υ (k+1)+ | x=0 P (k+1)+ . (25)
Here should be noted that the size of state vector for coupling element is 2m × 1, but 2n × 1 for unit cells k and k+1. The higher order parts in state vector for coupling element is 2(m -n) × 1. In order to ensure the continuity on surfaces 1 and 2, defining new state vectors including higher order parts (2(m -n) × 1) for unit cells k and k+1:

ŵ * (k) L1 , F * (k) L1 T = ŵ(k) L1 , ŵ′ (k) L1 , F(k) L1 , F′ (k) L1 T , ŵ * (k+1) R2 , F * (k+1) R2 T = ŵ(k+1) R2 , ŵ′ (k+1)
R2 , F (k+1)
R2 , F′ (k+1)
R2 T , (26)
where

ŵ′ (k) L1 , F′ (k) L1 and ŵ′ (k+1) L1 , F′ (k+1)

L1

are unknown higher order displacements and forces vectors for unit cell k and k+1 respectively. The continuity on surfaces 1 and 2 is:

ŵ(c) R1 , F(c) R1 T = ŵ * (k) L1 , F * (k) L1 T , ŵ(c) L2 , F(c) L2 T = ŵ * (k+1) R2 , F * (k+1) R2 T . (27)
Combining Eq.22, Eq.26 and Eq.27, assume that higher order parts for k and k+1 are 0. Define P (k)+ =I, P (k)-=R, P (k+1)+ =T, the R and T can be conformed as:

G = X (c) H, (28)
where G=[Ψ

(k+1)+ u υ (k+1)+ | x=0 T, ŵ (k+1)

R2

,

Ψ (k+1)+ F υ (k+1)+ | x=0 T, 0] T , H=[Ψ (k)+ u υ (k)+ | x=L k I + Ψ (k)- u υ (k)-| x=L k R, ŵ(k) L1 , Ψ (k)+ F υ (k)+ | x=L k I + Ψ (k)- F υ (k)-| x=L k R, 0] T .

Numerical applications

Coupling element c In this part, the WFEM is applied to analyze the multi-mode diffusion. The unit cells k and k+1 with L k/k+1 = 50a 0 , L y = 300a 0 and L z = 300a 0 (a 0 is the lattice parameter) as presented in Fig. 2. The coupling element with L c = 100a 0 as shown in Fig. 2. Materials Aluminum (Al) and Copper (Cu) are used here. The Young's modulus E is 70 GPa for Al and 110 GPa for Cu, linear mass density ρ is 2.7 g/cm 3 for Al and 8.96 g/cm 3 for Cu. he damping lose factor η=1e -4 . Unit cells k and k+1 are meshed into 16 3D elements, coupling element is meshed 64 3D elements.

In order to study the wave diffusion under a complex coupling condition, as shown in Fig. of all, defining the the materials of coupling element, unit cell k and k+1 are Al, the diffusion model of coupling element is built by SSG theory with higher-order parameters, the diffusion models of unit cell k and k+1 are built by CT. The R and T coefficients can be calculated from Eq. 28 including bending, tension and torsion modes. As shown in Fig. 3, the black lines denote the WFEM results. The value of R representing the non-classical part of reflection is no longer 0, and the value of T representing the non-classical part of transmission is no longer 1. The influence of non-local interactions caused by higher-order parameters can be reflected by this model. On the other hand, an analytical method [START_REF] Zhu | Wave transmission and reflection analysis through complex media based onthe second strain gradient theory[END_REF], shown by the red lines, is used to valid the WFEM results for bending and tension. As we can see, for the reflection coefficient, the result obtained by WFEM is very close to the one by the analytical method at low frequency, but the results are different at high frequencies. For the transmission coefficient, the results by WFEM matches the results by analytical method well. addition, the joint influence of classical parameters (i. g., Young's modulus, Poisson's ratio and mass density) and higher-order parameters of material on diffusion is also a very meaningful study. Defining the material of coupling element is Cu, unit cell k and k+1 are Al, the diffusion model of coupling element is built by SSG theory with higher-order parameters, the diffusion models of unit cell k and k+1 are built by CT. The R and T coefficients including bending, tension and torsion modes can be illustrated by Eq. 28 as well. As shown in Fig. 4, the black lines denote the WFEM results and the red lines represent the results from an analytical method [START_REF] Zhu | Wave transmission and reflection analysis through complex media based onthe second strain gradient theory[END_REF]. The diffusion is different from the case presented in Fig. 3, this shows that the impedance mismatch is not only due to the non-local interactions caused by higher order parameters in the SSG theory model but also the local interactions caused by classical parameters.

Conclusions

In this paper, SSG theory is used for the multi-mode diffusion analysis within the WFEM framework. The diffusion is confirmed through two different cases. For the first case, the value of R representing the non-classical part of reflection is no longer 0, and the value of T representing the non-classical part of transmission is no longer 1. The influence of non-local interactions caused by higher-order parameters can be reflected by this model. The second case shows that the impedance mismatch is not only due to the non-local interactions caused by higher order parameters in the SSG theory model but also the local interactions caused by classical parameters such as Young's modulus, Poisson's ratio and mass density.

  e ), d 2 =(0 1 -2l e 3l 2 e -4l 3 e 5l 4 e ), d 3 =(0 0 2 -6l e 12l 2 e -20l 3 e ), d 4 =(1 l e l 2 e l 3 e l 4 e l 5 e ), d 5 =(0 1 2l e 3l 2 e 4l 3 e 5l 4 e ), d 6 =(0 0 2 6l e 12l 2 e 20l 3 e

Fig. 1 .

 1 Fig. 1. Two coupled waveguides through a coupling element (x is the local coordinate).

Fig. 2 .

 2 Fig. 2. Finite element model of two waveguides coupled by a coupling element.

  Reflection coefficients (R).

Fig. 3 .

 3 Fig. 3. Absolute values of diffusion coefficients (the materials of coupling element, unit cell k and k+1 are Al).

  Reflection coefficients (R).

Fig. 4 .

 4 Fig. 4. Absolute values of diffusion coefficients (the material of coupling element is Cu, unit cell k and k+1 are Al).

Table 1 .

 1 Higher order material parameters ai (eV / Å), bi (eV • Å), ci (eV / Å).

	Material a1	a2	a3	a4	a5	b1	b2	b3	b4	b5	b6	b7	c1	c2	c3
	Al	0.140 0.002 -0.008 0.096 0.258 0.792 0.064 -0.194 -0.001 0.001 16.156 48.529 0.504 0.357 0.178
	Cu 0.183 0.010 0.001 0.072 0.189 0.661 0.066 0.206 0.002 0.002 12.625 37.940 0.845 0.573 0.347

Acknowledgements

This work is supported by LabEx CeLyA (Centre Lyonnais d'Acoustique, ANR-10-LABX-0060) of Université de Lyon. The research of B. Yang is funded by the China Scholarship Council (CSC).


Ahmad Alali Alhouaij
email: ahmad.alali-alhouaij@g-scop.inpg.fr

Zineb Simeu-Abazi
email: zineb.simeu-abazi@g-scop.inpg.fr

MODULAR MODELING OF MAINTENANCE ACTIVITIES FOR MULTI-SITES STRUCTURE

Keywords: Distributed maintenance, Modelling, Petri Stochastic Nets (SPNs), Availability, Scheduling, Cost

This paper concerns the modelling of maintenance activities in a distributed context. In this work we study the particular case where the maintenance activities are executed by two workshops: a central maintenance workshop (CMW) and a mobile maintenance workshop (MMW). The (CMW) concerns the repairing process for the corrective maintenance and the (MMW) execute all preventive maintenance in several sites according to a defined scheduling. The aim is to take in account the resources (spare parts in the (MMW) and maintenance actions for a given operating budget. A modular approach for modelling a multi-site structure is proposed to achieve the aim of improving the availability of production sites while minimizing the cost of maintenance.

INTRODUCTION

The concept of Maintenance Workshop has developed in the sectors of activities where the availability of equipment is really important. A maintenance workshop (MW) supports not only the equipment repairing but also the preventive maintenance activities which related with the inspection and changing. In the distributed context, this workshop (MW) can be used to maintain equipment from different production sites. To reduce the failures occurrence, the (MW) must manage resources (operators, tools and spare parts), achieving the real-time of the equipment status in different sites and preventive maintenance activities plan. To improve the availability of production equipment and minimize maintenance costs it is necessary to determine the resources and wellplanned preventive actions. The resources determination related with the management of maintenance team as far as the tools and spare parts. To achieve the preventive action it is necessary to set a robust schedule, capable of absorbing the operating hazards and having a good security level.

In this distributed context, the maintenance activities are divided on these two following structures: -Central (CMW) which realizes the process of reparation -corrective maintenance; -Mobile (MMW) which carries out inspections and replacement on the various production sites. Fig. 1. illustrates the context of the study and highlights the relationship between the various sites [START_REF] Simeu-Abazi | Maintenance centralisée pour les systèmes de production multi-sites[END_REF].

As a result, this distributed context requires the development of new modelling techniques and scheduling maintenance tasks. The simulation models can take into account the interactions between different sites and determine the resources. More oriented research in the marine field focused on optimizing the availability of merchant ships by proposing the determination of spare parts with a limited cost investment [START_REF] Rustenburg | A system approach to budgetconstrained spare parts management[END_REF], [Lau et al. , 2006], [Keizers et al., 2001]. Others concern in spare parts determination in space [Caggiano and Muckstadt, 2000].

A methodological approach on the implementation maintenance which is centralized at the production systems has been developed recently. It relates more particularly with the design of a maintenance workshop within an integrated production system [START_REF] Abbou | Contribution à la mise en oeuvre d'une maintenance centralisée : conception et optimisation d'un atelier de maintenance[END_REF], [START_REF] Abbou | Atelier de maintenance intégré dans un système de production : conception et évaluation des performances[END_REF]. The (MW) is then integrated and dedicated to the maintenance of certain and specific equipment which includes repairing several phases (diagnosis, disassembly, repair itself depending on the type of failure, assembly and test). This paper focuses particularly in cases where one has a Central (CMW) and Mobile (MMW) that can involve making replacements at several sites according to a predefined scheduling. It is a question of dimensioning the various resources (spare parts in the Central or Mobile (MW) and the maintenance team). A modular approach for modelling of a multisite structure is proposed. The aim is to minimize the average cost by applying an appropriate inspection policy for the mobile workshop (MMW). After a brief presentation of the framework, the chosen modelling tool (Petri Nets) is presented. For proposing a case study, a model of maintenance workshops (central and mobile) is proposed and illustrated with some simulation results. Some perspectives on the directions of this work will be at the end of this article.

MODELING TOOL

Petri Nets are tools well appropriate to modelling and analysis of complex discrete events systems by introducing the competition and the synchronization. They were introduced by Carl Adam Petri in 1962 to describe the overall behaviour of complex systems and to carry out the qualitative analysis.

A (PNs) is composed of:

-An assembly seats which is representing various states of the system (represented by circles).

-A set of transitions explain the possibility of system state change (represented by rectangles).

-A series of oriented arcs link between places and transitions.

-Marks (or chips) are represented as black dots in places of constituent's network.

This tool has very descriptive power to represent and analyze the manners and behaviour of complex system, including the phenomena of synchronisation, parallelism, competition or the sharing of resources.

Later in 1978, Stochastic Petri Nets (SPNs) were introduced [Natkin, 80] and [Molloy,81]. They can calculate probabilities and statistics, and taking into account random events as the occurrence of failures.

Taking the immediate transitions (unstable transitions) and the transitions stochastic (stable or delayed), has identified an extension of the (SPNs) which is called Generalized SPNs (GSPNs).

Several modules (PNs) can be synchronized by external events. This dependence is achieved through the linkage between the various modules, which determine the transition firing. This is the global notion of Synchronized PNs (SyPNs) [Moalla,78].

In this article we have extended the concept of Synchronization events with other events from both the outside and inside the system. In this case it is called the Internal SyPNs (GS 2 PNs).

APPLICATION OF DISTRIBUTED CONTEXT

The objective is to propose an approach to modeling and performance evaluation of many of production sites with taking into account the interactions between these sites. The expected performance is more related with improving the availability of production sites, minimizing the number of failures and minimizing maintenance costs.

Taking into account the corrective actions is made through using Stochastic Petri Nets (SPNs). They can model systems subjected to random parameters such as failure or repairs. To take into account the preventive actions, the Generalised SPNs (GSPNs) are used. Indeed, it is necessary to take into account the immediate transitions (unstable) where the staying time is deemed null (Zero and a hierarchical analysis of the system which is decomposed as follows:

1. A generic model which represents the production sites SP i (i: number of the site).

A generic model which represents the central maintenance workshop (CMW). 3. A model representing the mobile

maintenance workshop with the associated routing (MMW). 4. A generic model of the stock of the mobile maintenance workshop (MMW).

Table 1 lists the various data which are used in the proposed models and their objectives. The decomposition of the system in several modules makes them better and simple in the analysis phase.

Type of system

Data

It is supported by GS 2 PNs.

To obtain the model of the whole system from different modules it is necessary to follow these steps: [START_REF] Sassine Ch | Intégration des politiques de maintenance dans les systèmes de production manufacturiers[END_REF]] :

1. Make a structural division of the system: a system of (n) production sites can be decomposed (n) cells. 2. A generic model is associated with each cell using GS 2 PNs. At each cell, synchronization signals are generated: places where they are active send signals that determine the transitions that receive them.

ASSUMPTIONS (HYPOTHESES)

Our work focuses on the application of centralized maintenance of multi-site production workshop which include a central (CMW), several production sites and a (MMW). To evaluate the performance of this complex system, we made some assumptions.

Assumptions on production sites.

1. All SP i are identical, using equipments with identical characteristics.

2. No resources are shared between production and maintenance sites. 3. If the equipment is failed, the production in the site is stopped. 4. The preventive maintenance (PM) is executed only if SP in the rest state and at the date of predefined occurrence of a PM. (Preventive Systematic maintenance).

Assumptions on the Central MW.

1. The failed equipment is repaired with the time duration corresponding to the average repairing rate (μ). 2. In the (CMW), there is always a sufficient stock of the repaired parts that are ready to be loaded.

Assumptions on the Mobile MW.

1. At each site, there is only a standard replacement for one single type of part. 2. When the number of parts in stock is higher than the security stock level, mobile MW continued its scheduled maintenance operation. 3. When the number of parts in stock is less than or equal to the security stock level, mobile MW follows a predefined scenarios. 4. The staying time at the site is characterized by the average replacement time.

THE GENERIC MODELS

The multi-site system is composed of various generic models which are based on GS 2 PNs.

• The production sites SP i : is a model that must take into account the internal workings of the site i but with also maintenance tasks.

• The central maintenance workshop (CMW)

is a model that must take into account the complete process of repairing in addition to the arrival rate of defective parts. • The mobile workshop (MMW) is a model that must take into account the storage capacity of the (MMW). • The maintenance policy is a model that must take account of corrective or preventive maintenance tasks. • The routing is a model that represents the planning of maintenance tasks of the following (MMW) with well defined scenarios.

The model of production sites.

It is a model that represents the various models of production site operation SP. It is synchronized by the scheduled preventive maintenance events (preventive maintenance and presence of (MMW) on the site).

The GS 2 PNs of this model (see Fig. After the average time of service associated to the transition T3, the system returns to its waiting position through P4 which sends the signal (!fus) corresponding to the end of the production cycle.

In the case of a date PM, SP stop its production cycles, it will run as PM. The presence of (MMW) allows launching the process of replacing (P10 instead of sending a signal-from beginning PM).

The staying length in this position P10 depends and the replacement average time associated with the transition T12. At the end of replacement a signal is generated by P11. An average waiting time (β ic ), is associated with the transition T6. It corresponds to the time needed to availability of resources (new parts in (MMW), tools and operators).

Throughout corrective maintenance, beginning signal of the corrective maintenance (!dmc) is transmitted. The duration of the CM corresponds the repairing average time (μ ci) , is associated with T7.

When the time is finished, a signal corresponding to the end of maintenance (!fmc) is sent.

The model for the production site consists of two representing cycles, the different modes of operation of the site:

In the production phase, it is following closed cycle (loop) : (P1 T1 P2 T2 P3 T3 P4 T4 P1).

In corrective maintenance phase the loop is : (P3 T5 P5 T6 P6 T7 P7 T8 P1), In the case of preventive maintenance the loop is : (T9 P8 T10 P9 T11 P10 T12 P11 T13 P1).

Note1:

It was considered that the failure can happened only during he running, not the state of rest.

Note 2:

Signals the beginning and end of maintenance are used for stock management of (MMW).

Model of the CMW.

The model of (CMW) is detailed in the thesis of ABBOU [START_REF] Abbou | Contribution à la mise en oeuvre d'une maintenance centralisée : conception et optimisation d'un atelier de maintenance[END_REF]. It consists of several items needed to repair failed equipment The non-appearance of the signal (pds, available place in stock) reflects the fact that the stock is zero.

Note 4:

Initially n b = n d

Maintenance Policies

The model of PNs consists of two parts, the policy of maintenance corrective CM (CMW), and policy maintenance preventive PM (MMW).

Model Corrective Maintenance CM.

It is a model consists of three places and three transitions as shown in Fig. 5. It is synchronized with the random occurrence of a failure issued by the model of the production site and the availability of resources. P1 represents the state of rest of the maintenance team. The corrective maintenance starts only if there is reported (?pas) and that (MMW) is present on the site (?asp). Note that the signal (pas) contains all the information on the type of failure and associated with SP. Throughout repair a signal is emitted by the place P2 indicating the beginning of maintenance (?dmc). After an average time of repairing (τc) associated with the transition T2, a signal of end of this activity (!fmc , the end of the CM) is issued.

Model Preventive Maintenance PM.

As a model of corrective maintenance, the model consists of three places and three transitions as it is shown in Fig. 6. The Model Preventive Maintenance is synchronized with the scheduled case of preventive actions. The place P1 represents the state rest of the maintenance team. The preventive maintenance doesn't starts up to the date of occurrence of an PM (?omp) according to a predefined planning and if (MMW) is present on the site (?asp).

!dmc= début de CM. !fmc= fin CM. ?pas= panne SP. ?asp= arriver au SP. ?omp= occurrence date de PM. ?asp= arriver au SP.

Fig. 6. Module of CM policy

When T3 is reached, the operation of replacement of failed equipment with a new one is launched by sending the signal (?dmp). The average replacement time (τp), is associated with the transition T4. The signal of end of this replacement procedure (!fmp) indicates that the SP has been sent within function of maintenance is available for other maintenance operations.

Model of overall maintenance.

To address the problem of conflict which can reach in a case of the preventive maintenance, we offer a single module which includes maintenance models CM and PM. Whenever the event occurs, the priority is given to corrective maintenance. Therefore, the displacement from (MMW) to SP (failure) has always given priority. In these models we take into account the condition of available part in stock because the displacement of (MMW) to the production site is already conditioned by the presence of non-defective parts.

Note 6:

At each signal the beginning of maintenance (dmc or dmp), a piece of the new parts stock is consumed, and each signal of end of maintenance (fmc, fmp) refers that there is a piece in the stock of defective parts is added.

Model of routing the MMW

It is a model that represents the sequence of actions of maintenance (corrective or preventive) to be performed by the mobile workshop. It is synchronized with the dates of occurrence of PM, the case of a failure and the level of stock. The parameters are taken into account in the evaluation phase are: -Distances between the sites -Storage capacity of the (MMW)

The (MMW) deals with the supply of spare parts (new or repaired) at the (CMW) but also related with different sites for replacement operations. These activities are defined by the following algorithm (Fig. 8.). It is considered here that there is always the number of repaired the stock out of (CMW), it is also sufficient to meet the need for (MMW) (its maximum capacity).

From the Fig. 8. You can save the algorithm following:

Function routing "Centralization of maintenance for n production sites" Initialization "MMW is the site of the CMW" n = finished, "the number of production sites" In P0, the (MMW) buys from the (CMW). without failure and if the level of stock exceeds the threshold (safety stock), routing is done according to predefined schedule. The occurrence of a failure is likely to require scheduling tasks.

Moving to production site Replacement of failed parts with new ones

In the absence of failure, the moving to the following site if the stock level is higher Ss …. If planning is robust presence from the (MMW) to the SP, it must correspond to the time interval where the SP is in a state of rest.

Note 8:

Each model PNs may be represented by a black box with input / output of signals associated synchronization and other parameters such as timers, which will be associated with the crossing of transitions.

THE SIMULATION MODELS

The best tool suited for the simulation of various models proposed is Matlab. Indeed, the toolbox allows State-flow modelling and simulating such complex systems. We will limit the simulation part to a single production site, taking into account its various modes of operation. Thus, the normal operation, stops for corrective maintenance, stops for systematic preventive maintenance are modelled and simulated.

To represent the behavior of the site as well as different modes of operation, it is necessary to build some functions that generate random events as the occurrence of a failure. The occurrence of a failure leads to the stopping of the site awaiting repair. In addition, the duration of repair depends on the failure (we take it as random), a random time is associated with duration of repair (Fig. 10.). consider, the model for this system was built under certain assumptions. The model based on GS 2 PNs can better understand the distributed maintenance process, which is the main theme of this study. The simulation of various workshops synchronized (in progress) can analyse and evaluate the performances of maintenance (failure rate, availability, cost, rate of return synthetic…). This work opens several perspectives. The first is to remove some of the assumptions in the modelling distributed maintenance system.

In the model proposed a single type of equipment is taken into account at each site (assumption1). This assumption restricts the field of distributed maintenance. It is necessary to take into account the diversity of equipment, and MW must be able to rehabilitate various types of equipment. The proposed generic models can be evolved from GS2PNs to the Colored GS2PNs (Co-GS2PNs). In addition, replacement of equipment can generate substantial costs, and then it is necessary to consider replacement multi-level equipment. In other words, it is important to take into account not only the replacement of the full equipment but only a section of the equipment.

Fig. 1 .

 1 Fig. 1. Relations between the different production sites and maintenance workshops.

  ) and stochastic transitions (stable or delayed). When marking valid at the time of immediate transitions and delayed transitions, priority is given to immediate crossings transitions. Exchanges between the different sites (production and maintenance) are achieved through signals synchronization. We use this case the GSPNs with Internal Synchronization where the different modules are synchronized by external or internal events. The tool GS 2 PNs is well adapted to the distributed context. The decomposition of the complete system in different module, allows a modular analysis for our system. It is a way we propose a representation Local

  Fig. 2. Model GS 2 PN for production SiteThe P1 place represents the state rests or the state of SP is ready to start one of its operation modes. If any PM is programmed, the production operation can be launched (crossing T1) with sending the signal (!dus) corresponding to beginning of production cycle to the place P2. . The operating state in production mode is represented by the place P3. After the average time of service associated to the transition T3, the system returns to its waiting position through P4 which sends the signal (!fus) corresponding to the end of the production cycle. In the case of a date PM, SP stop its production cycles, it will run as PM. The presence of (MMW) allows launching the process of replacing (P10 instead of sending a signal-from beginning PM). The staying length in this position P10 depends and the replacement average time associated with the transition T12. At the end of replacement a signal is generated by P11. An average waiting time (β ic ), is associated with the transition T6. It corresponds to the time needed to availability of resources (new parts in (MMW), tools and operators).

Fig. 3 .

 3 Fig. 3. Module of the central MW 6.3. The Stock of MMW It is a model that allows managing the new and failed parts to be presented in the (MMW). It is synchronized with the model representing the policies of this maintenance. The stock of (MMW) is represented by a GS 2 PNs with two states and two transitions as shown in (Fig. 4.). The place P1 sends a signal (!pds) which indicates that there is at least one part available in stock. ηj is the number of marks in place P1 which represents the maximum capacity of the (MMW) stock. The transition T2 is conditioned by the beginning of operation of corrective or preventive Maintenance. When T2 is crossed, a new part is gotout from stock and a place is getting free to receive a defective part by sending the signal (!pls) from the place P2. When the replacement is finished, depending on whether the maintenance is corrective or preventive, the signals (?fmc or ?fmp) validate the transition T1. The marking of P1 represents the number of parts in proper functioning (n b ), marking P2 represents the number of defective parts (n d ).

Fig

  Fig. 4. Module StockNote 3:The non-appearance of the signal (pds, available place in stock) reflects the fact that the stock is zero.

Fig. 5 .

 5 Fig. 5. Module of CM policy

  Fig. 7. Model ,GS 2 PN of Maintenance Note 5:In these models we take into account the condition of available part in stock because the displacement of (MMW) to the production site is already conditioned by the presence of non-defective parts.

Fig. 8 .

 8 Fig. 8. Flow chart of routing Note 7:

  Fig. 9. Module of (MMW) routing

Fig 10 .

 10 Fig 10. Generation a random function.

Fig. 11 .

 11 Fig. 11. Simulation Model for the production operation in each SP 8. CONCLUSION In this article, we proposed a model to take into account maintenance operation in the multii-sites context. To simplify the number of constraints to

Table 1 -

 1

Summary data and targets for each type of workshop 4. MODULAR MODELING: PRINCIPLE