Panayiotis M Papadopoulos
email: p.papadopoulos@cyi.ac.cy

Silvia Erba
email: silvia.erba@polimi.it

Peter Riederer
email: peter.riederer@cstb.fr

Ioanna Kyprianou
email: i.kyprianous@cyi.ac.cy

Frédéric Wurtz
email: frederic.wurtz@g2elab.grenoble-inp.fr

Mohammadreza Aghaei
email: mohammadreza.aghaei@ntnu.no

Salman Muhammad

Shahid

Benoit Delinchant
email: benoit.delinchant@g2elab.grenoble-inp.fr

Salvatore Carlucci
email: s.carlucci@cyi.ac.cy

Indoor Thermal Comfort Analysis for Developing Energy-Saving Strategies in Buildings

Keywords: Smart buildings, Thermal comfort, ASHRAE Likelihood of dissatisfaction, Energy savings

Humans spend most of their time indoors, whether in their place of residence or work, with large amounts of energy consumed to create comfortable living conditions. Buildings are, therefore, accountable for a considerable proportion of global energy demand; within them, heating, ventilation, and air-conditioning systems constitute major energy drains. Traditionally, these systems are controlled by conventional, mainly static set points, but research has shown that substantial energy savings can be achieved by applying adaptive ones. This work aims to showcase the lower energy consumption achievable when employing adaptive over static approaches, using empirical data from a non-residential living lab. Assessments of rational and adaptive thermal comfort indices over the energy used in HVAC systems are provided, and the energy-saving potential of adaptive thermal comfort models in the design of HVAC control algorithms is estimated. The findings of this work highlight that controlling indoor setpoint temperature according to the adaptive comfort model can achieve energy savings from 15% up to 33%, compared to the rational one, while providing a satisfactory thermal environment.

I. INTRODUCTION

Buildings, from their nature, are designed to provide a safe, secure, and pleasant environment for humans to live, work, study, etc. On the other hand, to serve their role, they require a vast amount of energy that can reach around 40% and 37% of the primary energy sector in the USA and Europe, respectively, according to [START_REF] Pérez-Lombard | A review on buildings energy consumption information[END_REF]. It is well known and recognized that indoor environment control (i.e., heating, cooling, and ventilation) dominates the annual energy use of a building (in both residential and commercial sectors) since in the USA, it constitutes more than 50% of the total primary energy in buildings [START_REF] Pérez-Lombard | A review on buildings energy consumption information[END_REF]. Nowadays, energy crises and climate change are the main challenges people are facing on a global scale, and buildings are among the main responsible. It has been seen in the press that Germany's energy minister called people to set their heating systems to lower temperatures [START_REF] Connolly | Germany approves limits on heating public buildings to save energy[END_REF]. In addition, on 29 April 2023, the Italian Government promulgated Law n. 34, which implements the Italian plan to "immediately make savings useful at the European level to prepare for possible interruptions in gas supplies from Russia". Among the measures envisaged a reduction of 1 ºC for the space heating of buildings, from 17 with plus or minus 2 ºC of tolerance for buildings used for industrial, craft, and similar activities, from 19 with plus or minus 2 ºC of tolerance for all other buildings. Furthermore, "the operating limits of the heating systems are reduced by 15 days as regards the operation period (postponing the start date by 8 days and bringing forward the end date by 7 days) and by 1 hour as regards the daily switchon time". The Law does not apply to hospitals and nursing homes, or rather "sensitive users" [3]. Moreover, France introduced public holidays during predicted extreme weather periods to compensate for the effects of gas shortage. Generally, indoor environmental conditions are regulated by heating, ventilation, and air-conditioning (HVAC) systems. Most HVAC systems emphasize thermal comfort by regulating the indoor air temperature, while more advanced ones offer control of indoor air quality (IAQ) by regulating relative humidity, CO2 concentration, and other variables for specified uses.

Indoor environmental quality (IEQ) refers to indoor environmental conditions in a building related to the health and comfort of its occupants. IEQ includes thermal, visual, and acoustic comfort, as well as indoor air quality. These aspects are all linked to each other and are strongly related to the energy performance of a building [4]- [START_REF] Mardaljevic | Daylight metrics and energy savings[END_REF]; therefore, they should be incorporated into the building design and its operation.

Thermal comfort is defined as "that condition of mind that expresses satisfaction with the thermal environment and is assessed by subjective evaluation" [START_REF]Standard 55 -Thermal Environmental Conditions for Human Occupancy[END_REF]. Conventionally, there are two main approaches reflected in standards that provide methods to assess the expected thermal response of people to given indoor hygro-thermal conditions. The first approach, usually referred to as the rational approach, is based on the assumption that a person's thermal satisfaction with respect to the thermal environment can be modeled with a steady-state energy balance of the human body. According to this approach, comfort levels are predicted based on laboratorybased experiments and people's ratings of the thermal condition to which they are exposed. The most common energy balance model is the one proposed by Fanger [START_REF] Fanger | Thermal comfort. Analysis and applications in environmental engineering[END_REF], which expresses thermal sensation as a function of four environmental variables (air temperature, mean radiant temperature, relative humidity, and airspeed) and two personal variables (metabolic activity and clothing level). The second approach, known as adaptive thermal comfort, acknowledges that humans actively manage their thermal environment to ensure comfort rather than being passive with regard to it. Thus, it is possible to think of thermal comfort as a self-regulating system that considers only the heat transfer between a person and their environment but also the physiological, behavioral, and psychological responses of the person as well as the control opportunities provided by the building's design and construction [START_REF] Hellwig | A framework for adopting adaptive thermal comfort principles in design and operation of buildings[END_REF]. The adaptive approach has been incorporated into two standards, the ASHRAE 55 and the EN and its adoption is restricted to "naturally ventilated buildings" and "buildings without a mechanic cooling system", respectively. However, an adaptive control algorithm (ACA) based on the principles of the adaptive comfort theory of Nicol and Humphreys was developed in the SCATs project and used to create a new control system for air-conditioned buildings [START_REF] Mccartney | Developing an adaptive control algorithm for Europe[END_REF].

More recently, following this tendency, research on the use of the two schools of thermal comfort models suggests that designing buildings according to comfort ranges recommended by adaptive models generally requires lower energy use as long as convenient, responsive, and effective means for occupants to improve their environment, like operable windows, are available [START_REF] Tällberg | Comparison of the energy saving potential of adaptive and controllable smart windows: A stateof-the-art review and simulation studies of thermochromic, photochromic and electrochromic technologies[END_REF]- [START_REF] Singh | Adaptive thermal comfort model for different climatic zones of North-East India[END_REF]. Likewise, HVAC system control strategies often rely on steady-state assumptions, resulting in oversized system sizing estimations to meet buildings' energy demands. Strategies to reduce the energy consumption of HVAC systems include mechanical interventions in the individual components of the systems, as well as smart-enabled approaches [START_REF] Barone | A new thermal comfort model based on physiological parameters for the smart design and control of energy-efficient HVAC systems[END_REF], [START_REF] Vakiloroaya | A review of different strategies for HVAC energy saving[END_REF]. The improvement of HVAC energy savings is an area with great potential since it accounts for approximately half of the energy use in residential and office buildings [START_REF] Yang | Thermal comfort and building energy consumption implications -A review[END_REF], [START_REF] Jung | Human-in-the-loop HVAC operations: A quantitative review on occupancy, comfort, and energy-efficiency dimensions[END_REF]. For instance, a simulation study estimated up to 60% energy savings from optimized use of HVAC in US office buildings [START_REF] Papadopoulos | Rethinking HVAC temperature setpoints in commercial buildings: The potential for zero-cost energy savings and comfort improvement in different climates[END_REF] and a retrofitted HVAC system in Hong Kong has shown an energy reduction of 50% while improving indoor air quality [START_REF] Che | Energy consumption, indoor thermal comfort and air quality in a commercial office with retrofitted heat, ventilation and air conditioning (HVAC) system[END_REF]. HVAC systems can affect all aspects of IEQ and provide health, satisfaction, and productivity for building occupants [START_REF] Guo | Technologies toward thermal comfort-based and energy-efficient HVAC systems: A review[END_REF]. Implications of the trade-offs between energy savings and occupant comfort were identified in early studies of HVAC control design with modified setpoint temperatures, where a static and adaptive strategy was implemented in Australian office buildings. The study aimed to improve comfort and increase energy savings, and, although daily HVAC electricity consumption was reduced similarly under both strategies, occupant discomfort also marked an increase on both occasions [START_REF] Roussac | A preliminary evaluation of two strategies for raising indoor air temperature setpoints in office buildings[END_REF]. This and subsequent studies argue that the theoretical and achievable potential for energy savings is considerable, without reducing satisfaction levels [START_REF] Che | Energy consumption, indoor thermal comfort and air quality in a commercial office with retrofitted heat, ventilation and air conditioning (HVAC) system[END_REF], [START_REF] Hoyt | Extending air temperature setpoints: Simulated energy savings and design considerations for new and retrofit buildings[END_REF]. However, the human factor needs to be accounted for, and this is where smart control systems, enriched with occupant feedback, are essential. Although it has been established that an appropriate HVAC system specification, operated under an adaptive approach, benefits both the comfort of occupants and the energy savings of buildings, difficulties still remain in translating this principle into design practice [START_REF] Hellwig | A framework for adopting adaptive thermal comfort principles in design and operation of buildings[END_REF].

The contribution of this paper is to provide a comparative study based on in-field data that shows the energy-saving potential of adopting adaptive thermal comfort models in the design of HVAC control algorithms. In particular, data from indoor environmental monitoring devices and the building management system installed in a living lab were gathered and analyzed to assess the thermal comfort over the HVAC energy use.

II. METHODOLOGY

This section presents the methodology followed to show the potential of energy savings by using the adaptive thermal comfort approach (i.e., ASHRAE adaptive comfort model) to define the setpoint temperature instead of the steady-state rational approach (i.e., Fanger's comfort model) in the development and design of building control strategies. The methodology involves the calculation of the predicted dissatisfaction level based on the aforementioned thermal comfort models using actual field data and the comparison of the results with respect to energy use. In the following sections, we present the mathematical formulation of the thermal comfort indices, the description of the site, and the description of the dataset used for the analysis.

A. Equations

The Predicted Percentage of Dissatisfied (PPD) is a thermal comfort index developed by Fanger model and proposed for application in mechanically conditioned buildings. PPD is expressed as a function of the Predicted Mean Vote (PMV) through the following equation (1):

𝑃𝑃𝐷 𝐹𝑎𝑛𝑔𝑒𝑟 𝐼𝑆𝑂 (𝑃𝑀𝑉) = 100 -95𝑒 -0.03353•𝑃𝑀𝑉 4 -0.2179•𝑃𝑀𝑉 2 ∈ (5, 100) (1)
The PMV predicts the mean value of the thermal sensation votes of a large group of people, and it is a function of the dry-bulb air temperature (ºC), mean radiant temperature (ºC), average air speed (m/s), relative humidity (%), metabolic activity (met), clothing insulation (clo) and external work (met). The calculation of the PPD and PMV is performed using pythermalcomfort, "a Python package that allows users to calculate the most common thermal comfort indices in compliance with the main thermal comfort standards" [START_REF] Tartarini | pythermalcomfort: A Python package for thermal comfort research[END_REF].

The ASHRAE Likelihood of Dissatisfaction (ALD) is developed by Carlucci et al. [START_REF] Carlucci | ASHRAE Likelihood of Dissatisfaction: A new right-here and right-now thermal comfort index for assessing the Likelihood of dissatisfaction according to the ASHRAE adaptive comfort model[END_REF] to assess the likelihood of dissatisfaction according to the ASHRAE adaptive comfort model. The expression to compute ALD is in equation ( 2):

𝐴𝐿𝐷(∆𝑇 𝑜𝑝 ) = 𝑃𝑃𝐷(∆𝑇 𝑜𝑝 ) = 𝑒 -3.057+0.419∆𝑇 𝑜𝑝 +0.007∆𝑇 𝑜𝑝 2 1+𝑒 -3.057+0.419∆𝑇𝑜𝑝+0.007∆𝑇 𝑜𝑝 2 ( 2
)
where ∆𝑇 𝑜𝑝 = |𝑇 𝑜𝑝,𝑖𝑛 -𝑇 𝑐,𝐴𝑆𝐻𝑅𝐴𝐸 | is the offset from the ASHRAE optimal operative temperature Tc,ASHRAE (°C), which is derived from the prevailing mean outdoor air temperature tpma(out) and that is calculated as equation (3): 𝑇 𝑐,𝐴𝑆𝐻𝑅𝐴𝐸 = 0.31 • 𝑡 𝑝𝑚𝑎(𝑜𝑢𝑡) + 17.8

(3)

For calculating the tpma(out), weather data from the Meteostat python library are used [START_REF] Lamprecht | Meteostat Python[END_REF] using the data from the nearest meteorological station to the pilot building for period under study, i.e., 17-30/03/2023.

B. Dataset description

As described in Section A, the calculation of the selected indoor thermal comfort indicators requires a collection of field measurements. For this, both legacy equipment (i.e., BMS) and indoor environmental monitoring devices (i.e., temperature, relative humidity, and IAQ data logger named Sphensors and built by the LSI LASTEM company) installed for the COLLECTiEF project were used to collect measurements for HVAC power use, and the indoor environmental quality. However, since some of the input information or data were difficult to get (e.g., require specialized equipment, user input), the following assumptions were considered for calculation purposes:

• The mean radiant temperature is assumed to be equal to the dry-bulb air temperature measured by the Sphensor devices with a time resolution of 1 minute, • Relative humidity (%) is assumed constant and equal to 60 %, • Metabolic activity is assumed equal to 1.0 met, corresponding to the "Seated, quiet" item of ASHRAE 55, • and clothing insulation is assumed equal to 0.6 clo corresponding to the 'casual' item of ASHRAE 55, • Average airspeed is considered constant, with v = 0.01 m/s, • External work is set to 0 met. The data collection period corresponds to the heating season, starting from 17/03/2023 and ending on 30/03/2023. The power use of the HVAC system is calculated at the room level as a function of the water inlet/outlet temperature and the water flow of the hydronic system installed on the ceiling of each room.

C. Pilot site description

The proposed methodology is evaluated in a living lab with advanced BMS. This living lab is situated in the Grenoble Electrical Engineering Lab (G2ELab), which is accommodated in the GreEn-ER building that stands for the Teaching and Research on Energy in Grenoble. Inaugurated in 2015, the GreEn-ER building brings in research, academia, and creativity in the domain of sustainable energy within an environment, having students and researchers under one roof. It is an energy-efficient building constructed in conformity with the French thermal regulation of 2012 and is managed under a public-private partnership. The purpose of a living lab is to realize cross-domain research while keeping humans in the loop where user engagement and feedback serve to perform research to support the energy transition [START_REF] Twum-Duah | A Comparison of Direct and Indirect Flexibilities on the Self-Consumption of an Office Building: The Case of Predis-MHI, a Smart Office Building[END_REF], [START_REF] Wurtz | Smart buildings' integrated in 'smart grids': A key challenge for the energy transition by using physical models and optimization with a 'human-in-the-loop' approach[END_REF].

The building is partially energy self-sufficient due to photovoltaic (PV) panels installed in the premises of the building (i.e., 183 kWp on the rooftop and 22 kWp on the roof of the bicycle stand for a total of 205 kWp). In addition, electric vehicle charging stations are also available within the building premises, which provide free charging at the expense of the building energy consumption (four plugs of 7 kW and four plugs of 22 kW). The total primary energy consumption of the building is less than 2200 MWh/year, corresponding to 110 kWh/m 2 [START_REF] Delinchant | GreEn-ER living lab : A green building with energy aware occupants[END_REF]. Fig. 1 presents the location of the living lab in the GreEn-ER building. The BMS of GreEn-ER is connected to a number of sensors. These sensors are implemented top-down, whereby they measure different variables at the building and room levels and trickle down to measuring energy consumption at the plug level. The living lab is a part of G2ELab, which consists of some offices, a classroom, and a laboratory (as illustrated in Fig. 1). Fig. 2 presents the top view of the living lab. BMS data are collected for the following variables of interest from all rooms using the pre-installed sensors.

• Room temperature (°C),

• HVAC thermal power consumption (W),

• Thermal energy consumption (kWh),

• Water flow in the HVAC circuit (m 3 /h),

• Supply and return water temperature in the circuit of each room (°C), • Ventilation damper opening (%) and room inlet temperatures (°C) • Plug-level electricity consumption (kWh),

• Lighting consumption (kWh), • CO2 concentration except in 4A013 (ppm), Note that the sampling period of the variables depends on the kind and the resolution of each sensor, but the typical interval is in the order of minutes.

The thermal condition can be modulated by modifying the air-temperature set points of the rooms (except 4A013 and 4A017). These room-level setpoints include: (a) CO2 concentration setpoint, (b) Hot temperature setpoint (implementable during winter season), and (c) Cold temperature setpoint (implementable during summer season). These setpoints can either be modified by the LCD screen installed in each room, or they can be changed by using the web interface of the BMS. However, the experiment requires hourly modification of setpoints done by an automatic code. Due to data privacy and cyber-security concerns, it is not allowed to give external researchers to G2ELab direct access to the BMS. Therefore, an intermediate platform is used to share the collected data and get the new setpoints for each room to outside certificated and identified users (typically partner researchers). Through an automated code, both actions have been executed on an hourly basis. The intermediate platform is known as SG-InterOp (acronym of Smart-Grid Inter-Operability). The objective of this platform is to do cross-sectional research on the inter-operability of the smart grid, ranging from (but not limited to) the electricity production from hydraulic turbines, energy storage, and energy flexibility at the end-user level [START_REF] Labonne | Méthodes et Outils pour la simulation et la validation expérimentale temps-réel des réseaux intelligents ou 'Smart Grids[END_REF]. Fig. 3 presents the schematic diagram of data sharing and automatic remote control via SG-InterOp. Besides the data of pre-installed sensors, the experiment also benefits from specific sensors with higher resolution. These sensors are commercially known as "Sphensors" and have a data retrieving resolution of 1 minute. The sensors are placed in each room at an optimal location, from where the data is transferred to the dedicated server of the experiment. Table I presents the surface area and variables measured by the Sphensors in each room. III. RESULTS This section presents the potential of the application of adaptive thermal comfort models to develop energy-saving control strategies without compromising indoor thermal comfort.

Fig. 4 presents a comparative analysis of the percentage of dissatisfaction (%) generated by the PPD and ALD indices with respect to HVAC power use per zone. Specifically, each dot corresponds to a specific time interval of 1 hour in which the calculated index, along with the instantaneous HVAC power use, is measured. The blue and orange lines show the linear regression of the data points for PPD and ALD indices, respectively. From the experimental data, it can be shown that the adaptive thermal comfort model, which is expressed by the ALD index, can provide additional energy savings or more room for energy flexibility without affecting the perceived thermal comfort of the building users. Specifically, it can be observed that with the same amount of HVAC power, we can achieve significantly reduced thermal dissatisfaction with the ALD with respect to the PPD since the orange line is for all rooms below the blue line. Table II shows the calculated percentage of indoor thermal comfort improvement per room by computing the minimum and maximum error between the blue and orange lines. Assuming a linear relationship between thermal comfort and power use, the adoption of the adaptive thermal comfort approach in the design of control strategies could provide energy savings estimated to be between 15% and 33% without compromising the thermal sensation perceived by the occupants.

IV. CONCLUSIONS

This work presents a comparison study between steady-state and adaptive thermal comfort models with respect to building energy use using in-field data. It is shown that, during winter season in Grenoble, France, adaptive comfort models can reduce significantly discomfort conditions than the steadystate ones, with same HVAC power use. This result implies significant energy savings by adopting adaptive temperature set points. Future works could be: (i) the thermal comfort assessment based on occupants' feedback (i.e., questionnaires) and furthermore, (ii) the testing and evaluation of an adaptive temperature set point in real conditions.

ACKNOWLEDGMENT

Fig. 1 .

 1 Fig.1. Living Lab situated in GreEn-ER building[START_REF] Twum-Duah | A Comparison of Direct and Indirect Flexibilities on the Self-Consumption of an Office Building: The Case of Predis-MHI, a Smart Office Building[END_REF]-[START_REF] Delinchant | GreEn-ER living lab : A green building with energy aware occupants[END_REF]

Fig. 2 .

 2 Fig. 2. The drawing of rooms in the Living Lab

Fig. 3 .

 3 Fig. 3. Schematic diagram of data sharing and reception of new setpoints using SG-InterOp

TABLE I .

 I THE VARIABLES MEASURED BY THE SPHENSORS IN EACH ROOM

	Id Room Id Type of	Surfac	Variables measured by
		Room	e area	Sphensors
			(m 2 )
	01 4A013	Laboratory 63,5	Sphensor 1: Room
				temperature, relative
				humidity, luminosity
				Sphensor 2: CO2
				Concentration, Volatile
				organic compound,
				Particulate Matter (1, 2.5, 4,
				10)
	02 4A014	Office	42,0	Room temperature, relative
				humidity, luminosity
	03 4A015	Office	25,0	Room temperature, relative
				humidity, luminosity
	04 4A016	Office	18,5	Room temperature, relative
				humidity, luminosity
	05 4A017	Office	19,0	Room temperature, relative
				humidity, luminosity
	06 4A018	Office	18,5	Room temperature, relative
				humidity, luminosity
	07 4A019	Office	27,5	Room temperature, relative
				humidity, luminosity
	08 4A020	Classroom 78,5	Sphensor 1: Room
				temperature, relative
				humidity, luminosity
				Sphensor 2: CO2
				Concentration, Volatile
				organic compound,
				Particulate Matter (1, 2.5, 4,
				10)

TABLE II .

 II PERCENTAGE OF THERMAL COMFORT IMPROVEMENT PER ROOM

	Room ID	Range of indoor thermal comfort
		improvement
	4A015	5-38%
	4A016	10-33%
	4A018	15-33%
	4A019	15-40%

This work has been supported by the COLLECTiEF project, which has received research funding from European Union's Horizon 2020 research and innovation program under Grant Agreement No. 101033683. Special thanks to the directors of the G2Elab, Benoit Delinchant, and Frederic Wurtz, for providing access to the pilot site. The views and opinions expressed are those of the authors and do not reflect the views of the European Union.


Cédric Doudard
email: cedric.doudard@ensieta.fr

Sylvain Calloch
email: sylvain.calloch@ensieta.fr

François Hild
email: francois.hild@lmt.ens-cachan.fr

Stéphane Roux
email: stephane.roux@lmt.ens-cachan.fr

Identification of heat source fields from infra-red thermography: Determination of 'self-heating' in a dual-phase steel by using a dog bone sample

Keywords: thermography, source identification, fatigue, self-heating

    Identification of heat source fields from infra-red thermography: Determination of 'self-heating' in a dual-phase steel by using a dog bone sample

Introduction

The measurement and analysis of temperature fields during mechanical tests are often used (i) to analyze a physical phenomenon (e.g., Portevin-Le Châtelier (PLC) bands in aluminum alloys [START_REF] Ranc | Some aspects of Portevin-Le Chatelier plastic instabilities investigated by infrared pyrometry[END_REF][START_REF] Louche | Thermal observations associated with the Portevin-Le Chatelier effect in an Al-Mg alloy[END_REF], phase transformations in shape memory alloys [START_REF] Balandraud | Effet du couplage thermomécanique sur la propagation d'un front de changement de phase[END_REF]), (ii) to validate hypotheses (e.g., self-heating localized in a structure to correlate with the region of fatigue initiation [START_REF] Charkaluk | Fatigue Design of Structures under Thermomechanical Loadings[END_REF]), (iii) to identify macroscopic parameters (e.g., velocity of PLC bands [START_REF] Ranc | Some aspects of Portevin-Le Chatelier plastic instabilities investigated by infrared pyrometry[END_REF][START_REF] Louche | Thermal observations associated with the Portevin-Le Chatelier effect in an Al-Mg alloy[END_REF], fatigue properties [START_REF] Luong | Metal Fatigue Limit Evaluation Using Infrared Thermography[END_REF][START_REF] Krapez | Application of lock-in thermography to rapid evaluation of fatigue limit in metals[END_REF][START_REF] Rosa | Thermographic methodology for rapid determination of the fatigue limit of materials and mechanical components[END_REF][START_REF] Liaw | Thermographic detection of fatigue damage of pressure vessel steels at 1000 Hz and 20 Hz[END_REF][START_REF] Morabito | Energy-analysis of fatigue damage by thermographic technique[END_REF][START_REF] Doudard | A probabilistic two-scale model for high cycle fatigue life predictions[END_REF][START_REF] Meneghetti | Analysis of the fatigue strength of a stainless steel based on the energy dissipation[END_REF][START_REF] Ummenhofer | On the use of infrared thermography for the analysis of fatigue damage processes in welded joints[END_REF], the scatter of fatigue limits [START_REF] Doudard | Identification of the scatter in high cycle fatigue from temperature measurements[END_REF], multiaxial fatigue criteria [START_REF] Doudard | Determination of an HCF criterion by thermal measurements under biaxial cyclic loading[END_REF][START_REF] Poncelet | Prediction of self-heating measurements under proportional and nonproportional multiaxial cyclic loadings[END_REF]), (iv) or to validate the thermodynamic framework of a macroscopic model [START_REF] Vincent | On the ability of some cyclic plasticity models to predict the evolution of stored energy in a type 304L stainless steel submitted to high cycle fatigue[END_REF]. As temperature variation is not totally intrinsic to the material behavior (it depends on the diffusion properties of the materials but also on thermal boundary conditions), the development of constitutive equations in a thermodynamic framework [START_REF] Germain | Continuum Thermodynamics[END_REF][START_REF] Lemaitre | Mechanics of Solid Materials[END_REF] requires the determination of the heat source field accompanying these phenomena (e.g., PLC bands, fatigue properties). The heat source field is identified by analyzing temperature fields obtained from infrared images. In this paper, a new method is proposed to carry out this identification.

The reconstruction of heat source maps from the measurements of 2D temperature fields is performed by solving the heat conduction equation. Many different approaches are available to handle this inverse problem (reconstruction of heat source fields). Due to the ill-posed nature of such problems, an introduction to regularizations and hypotheses about the heat source distribution is required. In the present paper, an analysis of the intrinsic dissipation during cyclic loadings is proposed. Several authors have worked on an estimation of high cycle fatigue properties based upon self-heating measurements [START_REF] Doudard | A probabilistic two-scale model for high cycle fatigue life predictions[END_REF]. The proposed test, hereafter called 'self-heating test', consists in observing thermal effects during cyclic loadings and is performed on a specimen with a constant cross-section. A change of the steady-state temperature occurs beyond a given stress amplitude that is close to the fatigue limit. A model [START_REF] Doudard | A probabilistic two-scale model for high cycle fatigue life predictions[END_REF] allows us, on the one hand, to describe thermal effects and, on the other hand, to relate the thermal effects to the fatigue properties. The application of this approach to a structure is possible but is not easily interpretable. For the classic test, an analysis of the mean temperature of the specimen is sufficient whereas an average analysis for a structure is seldom applicable [START_REF] Poncelet | Prediction of self-heating measurements under proportional and nonproportional multiaxial cyclic loadings[END_REF] so that a general study of the temperature field is necessary to identify the heat source field.

Different methods have been proposed to solve this problem in the studied case (determination of the intrinsic dissipation during cyclic loadings). Among these, Fourier techniques using spectral solutions of the heat conduction equation [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF] or special local least-squares fitting of the thermal signal [START_REF] Boulanger | Calorimetric Analysis of Dissipative and Thermoelastic Effects Associated with the Fatigue Behavior of Steels[END_REF] can be mentioned. The first method uses a spectral basis composed of the eigen functions that are compatible with the boundary conditions. In this paper a similar approach is proposed. It is based on a combination of two bases for the projection of the temperature field. The first one, which corresponds to eigen functions not taking into account the boundary conditions, describes the local variations. The second one describes the boundary conditions at the two ends of the specimen. This analysis is performed for the particular case of a 'dog bone' specimen. A one-dimensional (1D) thermal diffusion approach is sufficient for this study.

The paper is divided into three main sections. In the first, the thermodynamic framework and the local heat conduction equation are introduced. To define a 1D or a 0D thermal diffusion problem, this equation is averaged. In the second section, the proposed approach is presented and illustrated by the study of a 1D thermal diffusion problem. The same method can also be used in a 2D study. The third section deals with the validation of the proposed approach.

The aim is to identify the self-heating properties of the material by using a dog bone specimen.

Heat conduction equation

The thermodynamics of irreversible processes is used herein. The thermodynamic equilibrium state of a volume element is defined at each instant t by the current value of a set of state variables, namely, T , the temperature, ǫ, the strain tensor, or the elastic strain tensor ǫ e = ǫǫ p , with ǫ p the plastic strain tensor, and V k internal variables. Internal variables are chosen in accordance with the physical mechanisms within the studied material. Then, the first and second principles of thermodynamics must be checked for any evolution.

Local form

The local heat conduction equation is deduced from the local form of energy

conservation ρc Ṫ + div(q) = ∆ + r + ρT ∂ 2 Ψ ∂V k ∂T . Vk + ρT ∂ 2 Ψ ∂ǫ e ∂T : ǫe = s t (1)
where Ψ denotes the specific Helmholtz free energy, c = -T ∂ 2 Ψ/∂T 2 the specific heat capacity, ρ the mass density, q the heat flux vector, and s t the heat source that is broken down into three parts:

• intrinsic dissipation ∆ = Σ : . ǫ p -A k . . V k ≥ 0 with Σ = ρ ∂Ψ ∂ǫ e and A k = ρ ∂Ψ ∂V k ;
• external heat supply or density loss r;

• thermodynamic coupling source ρT ∂ 2 Ψ ∂V k ∂T Vk + ρT ∂ 2 Ψ ∂ǫ e ∂T : ǫe .
The heat conduction equation (1) allows us to relate the absorption heat capacity (ρc .

T ) to the losses by conduction (div(q)) and the heat sources.

Application to the self-heating test

In the case of a self heating test (i.e., the analysis of temperature fields during cyclic mechanical loadings), small temperature variations are observed.

Consequently, it is assumed that [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF]:

• c is independent of the internal state;

• the convection term included in the material time derivation is neglected (i.e., dT /dt = ∂T /∂t);

• the external heat supply is time-independent;

• the thermodynamic coupling source is reduced to the thermoelastic contribution (the other contributions are neglected).

The heat source evaluation is an ill-posed problem that is impossible to solve without any information on the heat source distribution. During a classic self-heating test (using a specimen with a constant cross-section) the stress is uniform at a mesoscopic scale, so that the mean heat source is uniform throughout the gauge volume. To identify the mean heat source, the inverse problem may be solved by considering the mean temperature throughout the gauge volume (0D thermal diffusion problem [START_REF] Doudard | A probabilistic two-scale model for high cycle fatigue life predictions[END_REF][START_REF] Boulanger | Calorimetric Analysis of Dissipative and Thermoelastic Effects Associated with the Fatigue Behavior of Steels[END_REF]). The regularizing effect of heat diffusion limits the thermal gradients so that this mean temperature is measured by a simple thermocouple. For a specimen with a varying crosssection, the stress varies throughout the gauge volume so that the mean heat source distribution is no longer uniform at the considered scale. However for the particular case of the specimen shown in Figure 1, stress essentially depends on height x. The inverse problem can be solved by considering the mean temperature for each cross-sectional area, S(x), so that the temperature field depends only on x (1D heat diffusion problem). To model local heat conduction, Fourier's law is chosen

q = -λ ′ gradT (2)
where λ ′ denotes the isotropic thermal conductivity. In the case of a 1D thermal diffusion problem the conduction losses can be deduced from an analysis of mean losses for a volume element S(x)dx

1 S(x)dx div(q)dV = φ long + φ lat , (3)
where φ long denotes the conduction losses for height x

φ long = - λ ′ S(x) ∂ ∂x ∂θ ∂x (x, t)S(x) , (4)
and φ lat denotes the heat exchange with the surrounding environment [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF][START_REF] De | La Transmission de la Chaleur[END_REF]

φ lat = 2h l (e + l) el θ = ρc τ ′ eq θ, (5)
with θ(x, t) = S △T dS, h l the heat exchange parameter between the specimen and the surrounding air, and τ ′ eq a characteristic time. Thickness, width and length are denoted by e, l and L, respectively.

The longitudinal boundary condition is written as

-λ ′ ∂θ ∂x (±L/2, t) = ±h 2 θ(±L/2, t), (6)
where h 2 is the heat-exchange parameter between the specimen and the grips of the testing machine. The characteristic length of the longitudinal losses is therefore defined by λ -1

2 = λ ′ /h 2 .
The heat conduction equation is written as a 1D thermal diffusion problem

ρc θ(x, t) + ρcθ(x, t) τ ′ eq -λ ′ ∂ 2 θ(x, t) ∂x 2 -λ ′ ∂θ(x, t) ∂x ∂S(x) ∂x 1 S(x) = S t (x, t), (7)
where S t (x, t) = S s t dS, with the following boundary condition

- ∂θ ∂x (±L/2, t) = ±λ 2 θ(±L/2, t). (8)
The heat exchanges with the surrounding air are low compared with the longitudinal losses. Consequently, the second term in the heat conduction equation is neglected.

In the case of a 0D thermal diffusion problem, a similar analysis of heat losses by conduction allows us to define a characteristic time, τ eq , given by

1 τ eq = 2h l (e + l) el + 2h M L , (9)
where h M is the heat exchange parameter between the grips and the specimen.

For a classic servo-hydraulic testing machine, τ eq is equal to several tens of seconds, and is low compared with τ ′ eq , that is equal to several hundreds of seconds. Lastly, the local heat conduction is written as a 0D thermal diffusion

problem ρc ˙ θ(t) + ρc θ(t) τ eq = S t (t), (10)
where S t (t) = V s t dV and θ(t) = V △T dV .

The following section deals with the proposed method to solve a 1D thermal diffusion problem (Equation ( 7)), namely, the identification of the heat source distribution.

3 Proposed method for identification of heat source distribution By using the above method, the present section aims at deducing the heat source distribution from infrared images. This method is based on the properties of the Fourier series. As proposed by Chrysochoos and Louche [START_REF] Chrysochoos | An infrared image processing to analyse the calorific effects accompanying strain localisation[END_REF],

the temperature distribution is expressed in the spectral basis composed of the eigen functions that are compatible with the boundary conditions. Consequently, the eigen frequencies depend on the heat exchange coefficients. It is therefore necessary to estimate these coefficients accurately. In this paper, an alternative method is proposed: an initial spectral basis disregarding boundary conditions is combined with additional components to describe the exchanges.

To present this method, a specimen with a constant cross-section is considered.

Spectral basis

For the considered basis, the 1D thermal diffusion problem is solved regardless of boundary losses at x = ±L/2. The heat sources being neglected, the local heat conduction equation becomes

. θ(x, t) + θ(x, t) τ ′ eq - λ ′ ρc ∂ 2 θ(x, t) ∂x 2 = 0. ( 11
)
If θ(x, t) = φ(t)w(x), Equation ( 11) reads

ρc λ ′ . φ(t) φ(t) + ρc λ ′ τ ′ eq = 1 w(x) ∂ 2 w(x) ∂x 2 = -δ. (12)
By choosing √ δ = w i = 2πi/L, the solution is expressed as

w(x) = i A i sin(w i x) + B i cos(w i x). ( 13
)
where A i and B i are heat-source-dependent parameters. Consequently the temperature is related to the thermal source by

θ(x, t) = i t 0 S t (x, s) ρc , cos(w i x) exp(-w ′ i (t -s))ds cos(w i x) + t 0 S t (x, s) ρc , sin(w i x) exp(-w ′ i (t -s))ds sin(w i x), (14)
where

a(x), b(x) = L 2 -L 2 a(x)b(x)dx and w ′ i = w 2 i λ ′ /ρc + 1/τ ′ eq .
With the considered basis and a uniform heat source S t0 , the mean steady state temperature variation is given by

θ = S t0 ρc τ ′ eq = S t0 ρc τ eq , (15)
where S t0 τ eq /ρc denotes the responses obtained with a 0D approach. This result shows the need to take into account the heat losses due to exchanges with the grips in order to estimate heat diffusion accurately. There are two ways of solving this problem. The first consists in defining eigen functions that are compatible with the boundary conditions. However, it is necessary to evaluate the heat exchange coefficients beforehand. The second consists in completing the initial basis. It is assumed that the boundary losses have a significant influence on the mean response but a weaker influence on local variation.

It is therefore proposed to complete the initial basis by a second degree polynomial. This choice allows us to account for arbitrary heat fluxes at both ends of the specimen x = ±L/2.

Polynomial basis

It is considered that θ(x) = ax 2 + bx + c. Leaving aside the losses due to exchanges with the surrounding air because they are lower than these caused by exchanges with the grips, the steady state heat local conduction equation is written as

St 0 = -2aλ ′ (16)
The boundary conditions are given by

-2a L 2 + b = λ 2 a L 2 4 -b L 2 + c , (17) 2a
L 2 + b = -λ 2 a L 2 4 + b L 2 + c . (18)
These equations allow us to relate the three parameters a, b and c to the uniform heat source and the heat exchange coefficients by

a = - St 0 2λ ′ , b = 0 , c = - aL λ 2 + a L 2 4 . (19)
In the considered case, the problem is symmetrical. Consequently b = 0. The proposed polynomial basis allows therefore the description of the 'mean' behavior. In the following section, a solution to the 1D thermal diffusion problem using the complete basis is proposed.

Orthogonal basis

The proposed basis to solve the 1D problem is not orthogonal because of the linear and quadratic polynomial terms. It is therefore proposed to subtract from them their projection over the Fourier series. The first order polynomial, Q(x) = bx, can easily be projected over the eigen functions to give

Q /F = k -bL(-1) k πk sin 2πkx L . (20)
The same computation can be performed for the second order polynomial P (x) = ax 2 , whose projection can be written

P /F = aL 2 12 + k aL 2 (-1) k π 2 k 2 cos 2πkx L . (21)
Consequently, the following orthogonal basis is chosen

U a = P -P /F , U b = Q -Q /F , U 0 = 1, U k = cos 2πkx L , V k = sin 2πkx L , (22)
where k > 0. The projection of the temperature signal onto the three first terms leads to the identification of a, b and c. θ ak and θ bk denote the pro-jection onto the other terms. Consequently, the change of the steady-state temperature variation is described by

θ = ax 2 + bx + c + k>0 θ ′ ak cos 2πkx L + θ ′ bk sin 2πkx L (23)
with θ ′ ak = θ ak -aL 2 (-1) k /π 2 k 2 and θ ′ bk = θ bk + bL(-1) k /πk. The heat source distribution is then related to the temperature distribution by

St 0 = -2aλ ′ , St ak = ρc w2 k θ ′ ak , St bk = ρc w2 k θ ′ bk ( 24
)
with wk = 2πk

L 2 λ ′ ρc .
The proposed basis allows us to take into account heat losses due to the exchange with the grips and is easy to implement (i.e., it is not necessary to identify different parameters beforehand). An illustration of the present approach with the analysis of self-heating measurements of a 'dog bone' specimen during cyclic loadings is proposed.

4 Self-heating properties in a 'dog bone' specimen

Self-heating test

The identification of high cycle fatigue (HCF) properties from traditional tests is time consuming and very expensive. Alternative experimental methods were developed to identify HCF characteristics [START_REF] Luong | Metal Fatigue Limit Evaluation Using Infrared Thermography[END_REF][START_REF] Rosa | Thermographic methodology for rapid determination of the fatigue limit of materials and mechanical components[END_REF][START_REF] Stromeyer | The determination of Fatigue Limits under Alternating Stress Conditions[END_REF][START_REF] Moore | Fatigue of Metals Under Repeated Stress[END_REF][START_REF] Lehr | Die Dauerfestigkeit, ihre Bedeutung für die Praxis und ihre kurzfristige Ermittlung mittels neuartiger Prüfmaschinen[END_REF][START_REF] Welter | Bending and tension-compression fatigue testing[END_REF] more rapidly. Among these, the self-heating test monitors the specimen temperature changes during cyclic loadings (usually a 0D approach is used to analyze tests). It consists in applying successive series of cycles with different increasing stress amplitudes on a specimen with a constant cross-section. For each stress amplitude, the change in the temperature variation, θ is recorded. The mean temperature becomes stable after several hundred cycles and equals θ depending on stress level Σ 0 and loading frequency f r . Figure 2 shows the self-heating curve for a dual-phase steel (i.e., the change of θ with the load amplitude). Beyond a given stress level the mean steady-state temperature starts to increase significantly.

This regime change corresponds to microplastic activation that induces an intrinsic dissipation.

To describe the thermal effects, a probabilistic approach was put forward. If it is assumed that HCF damage is caused by this microplastic activity, the proposed approach allows the description of the fatigue behavior within the same framework [START_REF] Doudard | Identification of the scatter in high cycle fatigue from temperature measurements[END_REF]. The dissipated energy density for a loading cycle D cycle is related to the stress amplitude by

D cycle = δ Σ m+2 0 , ( 25
)
where δ is a parameter depending on the material and m the Weibull modulus.

The mean temperature variation induced by this dissipation is determined by solving the local heat conduction equation (0D approach)

ρc ˙ θ(t) + ρc θ(t) τ eq = f r D cycle . (26)
The mean steady-state temperature is also related to the stress amplitude by

θ = f r δτ eq ρc Σ m+2 0 . (27)
Consequently the Weibull modulus can be identified from self-heating measurements [START_REF] Doudard | Identification of the scatter in high cycle fatigue from temperature measurements[END_REF] (see Figure 2). For the considered material, a value m = 12.5 is obtained.

In the following section, an analysis of self-heating measurements is proposed for a 'dog bone' specimen for which the 1D approach will be used.

'Dog bone' specimen

For a 'dog bone' specimen shown in Figure 1, the longitudinal stress depends only on the height x and the cross-sectional area is related to x by

S(x) = el 0   1 + 2R u l 0   1 -1 - x R u 2     , (28)
where x = 0 corresponds to the smaller cross-section. The stress amplitude is also given by

Σ 0 (x) = Σ 0M 1 + 2R u l 0 1 -1 - x R u 2 , (29)
where Σ 0M is the largest stress amplitude (i.e., for x = 0).

Consequently, the heat source reads

S t (x) = δf r Σ 0 (x) m+2 = δf r        Σ 0M 1 + 2R u l 0 1 -1 - x R u 2        m+2 , (30)
The self-heating test with a 'dog bone' specimen consists in applying one series of cycles with a constant stress amplitude. The heat source distribution is obtained from infrared images by solving the local heat conduction equation.

The change of the heat source with the stress amplitude is deduced directly.

This result can thus be used to identify the Weibull modulus m. It is proposed to validate this identification method of the Weibull modulus by comparing the values of m obtained from a 'dog bone' specimen and another value with a constant cross-section. However, it is necessary to develop the above identification method for the case of a specimen with a non-constant cross-section.

Application to a varying cross-section

The local heat conduction equation for a specimen with a non-constant crosssection is given by (1D approach)

ρc θ(x, t) -λ ′ ∂ 2 θ(x, t) ∂x 2 -λ ′ ∂θ(x, t) ∂x ∂S(x) ∂x 1 S(x) = S t (x, t). (31)
If the temperature field is expanded as follows

θ = ax 2 + bx + c + k>0 θ ′ ak cos (w k x) + θ ′ bk sin (w k x) , (32)
where w k = 2πk/L, Equation (31) becomes

-λ ′ 2a (1 + g(x)) -λ ′ b h(x) L +λ ′ k>0 w 2 k θ ′ ak cos(w k x) + θ ′ bk sin(w k x) +λ ′ k>0 w k θ ′ ak sin(w k x) h(x) L -w k θ ′ bk cos(w k x) h(x) L = S t (x), (33)
with

g(x) = ∂S(x) ∂x x S(x) and h(x) = ∂S(x) ∂x L S(x) . (34)
Consequently, the following relationships are obtained

St 0 = -2aλ ′ (1 + g 0 ) -λ ′ b h 0 L + λ ′   k>0 w k θ ′ ak h bk 2L -w k θ ′ bk h ak 2L   , (35)
St ak = λ ′ w 2 k θ ′ ak -2ag ak -b h ak L + j>0 w j 2L θ ′ aj (h bj-k + h bj+k ) -θ ′ bj (h aj+k + h aj-k )   , (36)
St bk = λ ′ w 2 k θ ′ bk -2ag bk -b h bk L + j>0 w j 2L θ ′ aj (h aj-k -h aj+k ) -θ ′ bj (h bj+k -h bj-k )   . ( 37
)
These three relationships allow us to relate the heat source distribution to the temperature distribution.

Despite all the precautions used for data acquisition, temperature measurements are noisy. In the following, it is proposed to determine the relevant harmonic number to achieve a good estimation of the heat source distribution.

Relevant harmonic number

With the considered basis, the larger the harmonic of the eigen basis, the smaller the signal-to-noise ratio (see Equation ( 14)). Consequently it is necessary to choose the dimension of the eigen basis, k lim . For a general study, it is difficult to give a unique answer. However, in the case considered, the general trend of the heat source distribution is given by Equation (30). It is thus possible to determine the relevant harmonic number to achieve a good estimation of the Weibull modulus m by considering a temperature field without noise.

This temperature distribution is obtained from FE simulations by considering λ ′ = 64 Wm -1 K -1 , λ 2 = 70 m -1 and m = 12. Figure 3a shows the change of the steady-state temperature variation with x obtained by this method.

From the temperature change, the proposed method allows the identification of the heat source distribution for different values of k lim (relevant number of harmonics), see Figure 3b. For each heat source distribution, the value of m is determined (Table 1).

Consequently, k lim = 3 is considered as sufficient for the following study. Table 1 Identified values of m from the heat source distribution for different values of k lim (considered number of harmonics). The reference is m = 12.

Experiment on a dual-phase steel

A self-heating test is carried out on a servohydraulic testing machine. This load-controlled test has a 320-MPa stress amplitude (when x = 0) and a 10-Hz frequency. The temperature field is monitored using an infrared camera (JADE IRFPA, resolution: 320 × 356 pixels, medium wave with a 3-5µm spectral range). Figure 4a shows the change of the steady-state temperature variation with x (average over 25 pixels in the width direction). By using the proposed method, the heat source distribution is identified (Figure 4b). Figure 5 shows the self-heating curve of the material considered deduced from the present analysis (i.e., the heat source distribution with a unique stress amplitude). A value of m = 12 is obtained. This value is in accordance with that obtained from traditional self-heating tests (i.e., m = 12.5, Figure 2). This result validates, on the one hand, the proposed identification method and, on the other hand, the use of a 'dog bone' specimen to identify the value of the Weibull modulus.

Conclusion

The method proposed herein to identify heat source distributions from infrared images is based on the use of a new basis for the projection of the temperature field. It is broken down into two parts. The first one, which corresponds to an eigen basis regardless of the boundary conditions, is used to describe the local variations of the temperature field. The second describes the boundary conditions at the two ends. Three advantages can be underlined. First, this approach is easy to implement. Second, it is not necessary to identify the heat exchange parameters. Third, it is possible to define a region of interest that is independent of the length of the specimen gauge part.

The validation of the proposed approach is performed by analyzing a selfheating test of a 'dog bone' sample made of dual-phase steel. The results with

Fig. 1 .

 1 Fig. 1. 'Dog bone' specimen with R u = 80 mm, and l 0 = 10 mm.

Fig. 2 .

 2 Fig. 2. Self-heating curve of a dual-phase steel. Change of the steady-state mean temperature with the stress amplitude (f r = 10Hz).

Fig. 3 .

 3 Fig. 3. -a-Change of the steady-state temperature variation with x (f r = 10 Hz and λ ′ = 64 Wm -1 K -1 ) obtained from a numerical simulation -b-Identified heat source distribution from temperature changes for different values of k lim . Dotted line: k lim = 1, dashed line: k lim = 2, solid line: k lim = 3.

Fig. 4 .

 4 Fig. 4. -a-Change of the steady-state temperature variation with x given by an infrared camera (f r = 10Hz) -b-Identified heat source distribution from temperature change with k lim = 3 and λ ′ = 64 Wm -1 K -1 .

Fig. 5 .

 5 Fig. 5. Self-heating curve of a dual-phase steel (heat source change with the stress amplitude (MPa)). Solid line: identified change (m = 12), Symbol +: experimental data.

this new method are in accordance with those determined from a traditional configuration (i.e., specimen with a constant cross-section). The proposed approach may be used to analyze the self-heating of structures and to identify their fatigue properties.


Rawan Ghnemat

Cyrille Bertelle
email: cyrille.bertelle@gmail.com

Gérard H E Duchamp

A METHODOLOGY FOR URBAN AND LAND-USE MANAGEMENT SIMULATION USING SPATIAL SELF-ORGANIZATION PROCESSES

Keywords: Land-use, self-organization, geographical information systems, agent-based modeling, automata with multiplicities, community detection, territorial intelligence

In this study, we develop a methodology based on computational intelligence concepts, for decision making tools using simulation of self-organized complex systems. Land-use management is considered here as the output of sustainable development strategies, dealing with the achievement of many objectives, interacting in a complex way, like environmental, economical and social objectives. The methodology presented here can be considered as a conceptual evolution in simulation processes from the simulations based on rule systems over geographical cellular automata toward the simulations involving self-organized agent-based systems over geographical information systems (GIS). Our methodology is based on self-organization patterns detection which emerge from spatial and behavioral systems. According to the complex systems modelling principles, we let the system evolve by itself with only partial control implemented here by an evolutive and selective process based on a fitness function over the whole system.

Introduction: Complexity Concept Approach for Land-Use Management

Land-use management, in regional or urban development, deals with Territorial Intelligence concepts. The word "Intelligence" has to be considered from its latin root, "Intelligere", which means understanding. So, our purpose is to understand the territorial management by the complex interaction of many kinds of phenomena. The description proposed here for territorial management is based on the Sustainable Development strategies which try to avoid naive solutions which may appear at first efficient to solve specific problems in short a term vision. Without understanding the complex implication of some naive solutions, the whole system can lead to irreversible evolutions, making it fail on long term development for social or environmental features. Sustainable development has mainly three objectives which are environmental, economical and social ones. The complexity of the interaction network between these objectives leads to the necessity to develop decision making tools based on simulation to help understand the system evolution on a spatial and a temporal multi-scale description.

The complexity analysis on which we focus our attention to develop the methodology is based on multi-description approaches : (i) the multi-scale modelling is here essential and an example of this need can be illustrated by the climatic change problem as a global Earth phenomenon which can induce modifications on urban management. (ii) The multi-actors modelling is also needed for resources management which is affected or concerned by farmers, economists and politicians. (iii) A Multi-disciplinary approach is also needed to take into account environment, economy and social dynamics, using multi-modelling methods, to mix differential and individual-based models. To conclude, we finally deal with an integrative description simulation which is based on very specific features related to complex systems modelling.

The complexity analysis that we have to integrate in our way of modelling must contain some conceptual functions based on emergent self-organized systems and their associated dynamical morphologies. We have to implement micro-macro interactions in multi-scale description which are generally the basis of the dynamic of the emergent organizations. By these functions, we must be able to understand evolution and adaptation properties of theses organizations. These evolutions generally lead to hierarchical structures of the organizations themselves, like in geographical systems where simulations may need descriptions from urban districts to regional or international area. Organization feed-back of these herarchical structures on the entities are also essential to understand the organization evolution (in multi-scale approaches) like, for example, how country laws can feed-back on the cities management.

Modelling and Simulation Approaches

We will present, in the first section, a short review of the main features concerning land-use management modelling. Then, in the following sections, we will develop our own approach, based on agent-based modelling for complex systems using automata with multiplicities.

Land-use systems modelling

Following the review proposed by Itzhak Benenson [START_REF] Benenson | Modeling Population Dynamics in the City: from a regional to a Multi-Agent Approach[END_REF], we can classify the different approaches of territorial systems modelling in two main classes: the "Black-box" macro-modeling and the individual-based micro-modeling.

The first category of approaches concerns mainly "stock and flow" descriptions of socio-economic indicators. The main first contributors generally mentionned, are I.S.Lowry [START_REF] Lowry | A model of metropolis[END_REF] and J.W. Forrester [START_REF] Forrester | Urban Dynamics[END_REF]. Lowry's model of urban system, applied to the city of Pittsburg, proposed some "integrated" model, defining a flow chart between the three main indicator classes: (i) the basic sector of industrial and business activities, (ii) the householder sector and (iii) the retail sector concerning the local population. This flow chart model already deals with a mile-square decomposition similar to spatial decomposition used later as an adaptation of cellular automata grid to geographical real space. The final output of the modeling process leads to a kind of socio-economic equilibrium state. This approach finds its limit because of its static description and dynamical models are essential to understand the city evolution. Forrester proposed a dynamical modeling based on the application of industrial dynamics on urban dynamics. His model is based on non spatial stock and flow models. Stocks are exchanged within a three income levels decomposition over housing, jobs and population. This model based on simple urban description was aimed at generating simulation and Forrester claims the benefit of computer simulation to understand the city evolution and how we can predict its evolution by the modification of guiding policies within the system.

The stock and flow models continue to be improved and to propose more and more details, including transportation subsystem or land market, for example. One of the most complete, called Integrated Urban Model (IUM) was proposed by Bertuglia et al. [START_REF] Bertuglia | An integrated urban model[END_REF]. The computational complexity increases with the accuracity of the description and finally avoids to obtain reasonable estimates of the parameters. These models are more representational tools than simulation tools [START_REF] Benenson | Modeling Population Dynamics in the City: from a regional to a Multi-Agent Approach[END_REF].

To buid efficient simulation models, the idea was to simplify the description, using a more global one facilitating the analytical description. From the inspiration of dynamics of population theory, some researchers proposed to build urban dynamic models from ecological modeling. The paradigm of prey-predators systems is then used to give efficient simulation tools to investigate the main feature allowing then to understand the global dynamics. For example, Dendrinos and Mulally [START_REF] Dendrinos | Urban Evolution: Studies in the Mathematical Ecology of the Cities[END_REF] use a prey-predator model, assuming that the increase of city population make decrease the economic status. The predator represents the urban population and the preys, the per capita income.

All the previous described models are based on top-down approaches to model the system dynamics. We first consider the whole phenomenon and we propose a way of how to split it in many sub-problems and then in stocks and flows or in different terms contituting the equational system. Another class of modeling is based on micro-modeling and bottom-up representation of the city as a collection of individual-based descriptions, behavioral rulebased descriptions and interaction systems. From this constructive approach, we want to obtain an emergent description of the whole system or of some sub-systems included in a hierarchical process. Two complementary methodologies can be used for that and we detail them in the following paragraphs.

The first methodology consists in generating a simulation where all the components, behaviors and rule-based, interact over a environment, perceiving and acting on it. The environment evolving is the support of emergent properties. The cellular automata modeling deals with this kind of simulations. The basic definition of cellular automata for urban or regional modeling, for exemple, consists in the decomposition of the city, region or any geographical area in a lattice of cells. Each cell is in some state which belongs to a finite set S. At each time step, the cells change their own state according to some transition rule based on their previous state and their neighbor cells. Many works based on cellular automata, have been developped for geographical systems and urban dynamics [START_REF] Batty | From cells to cities[END_REF][START_REF] Dubos-Paillard | Analyse de l'évolution urbaine par automate cellulaire. Le modèle SpaCelle[END_REF]. Extensions on environmental problems like water streaming are using these medels as efficient tools [START_REF] Langlois | RuiCells, automate cellulaire pour la simulation du ruissellement de surface[END_REF]. Cellular automata can be seen mainly as distributed tools to model diffusive penomena using rule-based systems. One of the first researchers in human sciences who proposed models based on diffusive rule-based systems is T. Hägerstrand in a very early period, during 50' [START_REF] Hagerstand | Innovation Diffusion as a spatial process[END_REF] but his work itself started to be diffused over the science community more than 15 years later, when the computer development became able to implement its model in realistic studies. One of the most famous cellular-based model for social modeling is du to T. Schelling [START_REF] Schelling | Dynamic Models of Segregation[END_REF], describing the segregation process. But with the implementation of this model, we face an important extension to cellular automata where we need to represent individuals moving after a simple deliberative process. The mixing of spatial data and cellular automata with autonomous entities, like agents, is here needed [START_REF] Daude | Introduction à la modélisation multi-agents des systèmes complexes en géographie[END_REF].

The second methodology to deal with emergent description in micromodelling, consists in completing the previous approach based on simulation, by introducing some computational processes which are able to detect emergent systems or organizations. The final goal of this method is then to be able to re-introduce these emergent systems or organizations inside the simulation and manage their evolutions and their interactions with the components of the system. The re-integration of the emergent systems, during the simulation, can be explicitely expressed like in the multiscale fluid flow simulation proposed by P. Tranouez [START_REF] Tranouez | Changing levels of description in a fluid flow simulation[END_REF] or it can be implicitly expressed using a self-controled process as we describe in the following, using genetic algorithms.

Basic agent-based concepts and complex systems modelling

In this section, we give the basis of the conceptual tools which allow to extend the reactive and diffusive grid cases behavior to more sophisticated entities, using agent-based models. We propose to model the agent behavior with automata with multiplicities which are powerful algebraic stuctures.

According to General System Theory [START_REF] Le Moigne | La modélisation des systèmes complexes[END_REF], a complex system is composed of entities in mutual interaction and interacting with the outside environment. A system has some characteristic properties which confer its structural aspects, as schematically described in part (a) of Figure 1: • The set elements or entities are in interactive dependance. The alteration of only one entity or one interaction reverberates on the whole system.

• A global organization emerges from interacting constitutive elements. This organization can be identified and carries its own autonomous behavior while it is in relation and dependance with its environment. The emergent organization possesses new properties that its own constitutive entities do not have.

• The global organization retro-acts over its constitutive components.

The interacting entities network as described in part (b) of Figure 1 leads each entity to perceive informations or actions from other entities or from the whole system. Each entity also acts on the interaction network or on the environment.

A well-adapted modeling consists in using an agent-based system representation which is composed of a set of entities called agent. Each entity perceives and acts on an environment, using an autonomous behaviour as described in part (c) of Figure 1.

To compute a simulation composed of such entities, we need to describe the behaviour of each agent. This one can be schematically described using internal states and transition processes between these states, as described in part (d) of Figure 1. So an automaton with multiplicities as described in the following section is well-adapted for the agent behavior modelling. Each transition is labelled by an input as the agent perception and an output as the agent action.

Automata-based modelling for agent behavior

An automaton with multiplicities is based on the fact that the output data of the automata belongs to a specific algebraic structure, a semiring, including real, complex, probabilistic, non commutative semantic outputs (transducers) [START_REF] Golan | Power algebras over semirings[END_REF][START_REF] Stanley | Enumerative combinatorics[END_REF]. In that way, we will be able to build effective operations on such automata, using the power of the algebraic structures of the output data. We are specifically able to describe automata by means of a matrix representation with all the power of the new (i.e. with semirings) linear algebra.

Definition 2.1 (Automaton with multiplicities)

An automaton with multiplicities over an alphabet Σ and a semiring K is the Urban Management Simulation Using Spatial Self-Organization Processes 7 5-uple (Σ, Q, I, T, F ) where

• Q = {S 1 , S 2 • • • S n }
is the finite set of state;

• I : Q → K is a function over the set of states, which associates to each initial state a value of K, called entry cost, and to non-initial state a zero value ;

• F : Q → K is a function over the set states, which associates to each final state a value of K, called final cost, and to non-final state a zero value;

• T is the transition function, that is T : Roughly it is the least structure which allows the matrix "calculus" with unit (one can think of a ring without the "minus" operation). The previous automata with multiplicities can be, equivalently, expressed by a matrix representation which is a triplet

Q × Σ × Q → K
• λ ∈ K 1×Q
which is a row-vector which coefficients are λ i = I(S i ),

• γ ∈ K Q×1 is a column-vector which coefficients are γ i = F (S i ),

• µ : Σ * → K Q×Q is a morphism of monoids (indeed K Q×Q is endowed with the product of matrices) such that the coefficient on the q i th row and q j th column of µ(a) is T (q i , a, q j ) Definition 2.3 (Automata-Based Agent Behavior)

We represent the agent behavior by automata with multiplicities (Σ, Q, I, T, F ) over a semiring K:

• The agent behavior is composed of a states set Q and of rule-based transitions between them. These transitions are represented by T ; I and F represent the initial and final transitions;

• Alphabet Σ corresponds to the agent perceptions set;

• The semiring K is the set of agent actions, sometimes associated to a probabilistic value which is the action realization probability (as defined in [START_REF] Ghnemat | Automata-Based Adaptive Behavior for Economic Modelling Using Game Theory[END_REF]).

Agent Behavior Metric Space

The main advantage of automata-based agent modelling is their efficient operators. We deal is this paragraph with an innovative way to define a behavioral semi-distance as the essential key of self-organization processes proposed later.

Definition 2.4 (Evaluation function for automata-based behavior) Let x be an agent whose behavior is defined by A, an automaton with multiplicities over the semiring K, we define the evaluation function e(x) by:

e(x) = V (A)
where V (A) stands for the stringing-up of all coefficients of (λ, µ, γ), the linear representation of A, defined in remark 2.2.

Definition 2.5 (Behavioral semi-distance)

Let x and y two agents and e(x) and e(y) their respective evaluations as described in the previous definition 2.4. We define d(x, y) a semi-distance or pseudometrics1 between the two agents x and y as d(x, y) = ||e(x)e(y)||.

The notation ||.|| standing for any vector norm.

Spatial and Behavioral Modeling Based on Community Detection

In this section, we give an operational definition of a community in terms of functional concepts dealing with complex and social modelling. To model such communities, we have to complete the concept of automata behavior with some spatial aspects and with some adaptive capabilities that genetic operators can allow to implement. With these concepts, we can model communities by evolutive population of these genetic spatial automata.

Definition 3.1 (Community operational definition)

A community is a system or an organization which is characterized by a spatial property, a behavior property and the interaction between both.

Example 3.2 In ecology, a community is a group of plants or animals living in a specific region, each interacting with the others.

The spatial patterns generated by Schelling models are some examples of communities and these spatial patterns are linked with the very specific behavioral rules implemented for each grid case. Our purpose here is to give a more generic processus which can be mixed with sophisticated agent behaviors. Using agent communication protocol, we can extend the diffusion process linked with cellular automata to more distant communications and interaction between spatial agents. In the following, we define this notion of spatial automata-based agents and then we develop the genetic operators allowing to transform automata with multiplicities to genetic automata. We will explain how the definition of adapted fitness will generate the detection processus.

Definition 3.3 (Spatial Automata-Based Agent)

A spatial automata-based agent is defined by its structural representation:

• An automaton with multiplicities corresponding to its behavior as a whole processus managing its perceptions and its actions over its environment. They include its communication capabilities and so its social behavior;

• A spatial location defined on some specific metric space.

Genetic operators on automata population

We consider, in the following, a population of automata with multiplicities which are each represented by a chromosome, following the genetic algorithm principles. We define the chromosome for each automata with multiplicities as the sequence of all matrices associated to each letter from the (linearly ordered) alphabet. The chromosomes are composed with alleles which are here the lines of the matrix [START_REF] Bertelle | Genetic algorithms on automata with multiplicities for adaptive agent behaviour in emergent organizations In SCI[END_REF].

In the following, genetic algorithms are going to generate new automata containing possibly new transitions from the ones included in the initial automata.

The genetic algorithm over the population of automata with multiplicities follows a reproduction iteration dividing in three steps [START_REF] Goldberg | Genetic Algorithms[END_REF]:

• Duplication: each automaton generates a clone of itself;

• Crossing-over: for each couple of automata, we consider a sequence of lines of their matrix. A permutation on the lines of this sequence is made between the analogue matrices of this couple of automata;

• Mutation: a line of each matrix is choosen at random and a sequence of new values is given for this line.

Finally, the whole genetic algorithm scheduling for a full process of reproduction over all the population of agents is the evolutionary algorithm:

1. For all couple of agents, two children are created by duplication, crossover and mutation mechanisms over their behavioral automata. The location of the children can be choosen in many ways: on the linear segment defined by the parents location or as the node of a square described by them and their parents (more details are given in [START_REF] Ghnemat | Self-organized communities modelling by genetic spatial automata[END_REF]);

2. The fitness for each automaton is computed;

3. For all 4-uple composed of parents and children, the performless agents, in term of fitness computed in the previous step, are suppressed. The two agents, still living, result from the evolution of the two initial parents.

Remark 3.4 The fitness is not defined at this level of abstract formulation, but it is defined corresponding to the context for which the automaton is a model, as we will do in the next section.

Adaptive processus to implement community detection

The community detection is based on a genetic algorithm over a population of spatial automata-based agents. The formation of the community is the result of the population evolution crossing by a selection process computed with the fitness function defined in the following.

For this computation, we deal with two distances defined on the set of agents. The first is the spatial distance associated to the agent spatial location and the second is the behavioral semi-distance defined in the definition 2.5.

Definition 3.5 Community clustering and detection fitness

Let V x a neighbourhood of the agent x, relatively to its spatial location. We define f (x) the agent fitness of the agent x as :

f (x) =        card(V x ) yi∈Vx d(x, y i ) 2 if yi∈Vx d(x, y i ) 2 = 0
∞ otherwise where d(x, y) is the behavioral semi-distance between the two agents x and y.

As such, the agent fitness f (x), increases when the automaton finds similar behaviors within its neighbourhood. On figure 2, we represent a population of automata where each automaton is a colored chain representing its chromosom. Automata with similar colored chain must be understood as similar behavioral automata. In the left part of the figure, we focus on one high fitness individual after computing its spatial neighbourhood and observing the behavioral similarity of all automata included in this neighbourhood. In the right part of the figure, composed of the same population, we focus on a low fitness individual, showing a behavior dissimilar to the other automata of its neighbourhood.

The genetic evolution of the spatial automata-based agents leads to a self-organization which creates a clustering of the agents set in such way that each cluster contains agents of similar behavior. During the evaluation process, genetic algorithms can be tuned such that individuals outside their communities be attracted by them. The center of the clusters, the size of the clusters and the behavior of the agents within the center of each cluster are the result of the overall genetic process which generates self-organization communities.

Conclusion and Perspective : Integrating

Self-Organization Simulations within Geographical Information Systems

As explained in the end of the section 2.1, social and environmental simulations can need to mix spatial data with autonomous entities, like agents. Both the agent programming and the spatial data available have been increased since the last decade. On the one hand, we present in the previous section some innovative approaches to deal with agent programming, fusing artificial intelligence, automata modelling, distributed computing, swarm intelligence and genetic algorithms to solve complex systems based phenomena. On the other hand, geographical information becomes a very wide and huge database following the impressive development of Geographical Information Systems (GIS). Land-use management has now to deal with a very huge amount of heterogeneous geographical, economical and social data which are expected to be used for relevant analysis allowing efficient decision making. The goal of our new methodology based on the community swarm optimization is to include the community detection as a agent-based self-organization processus inside GIS, including spatial adaptation.

It is an open way to implement intelligent tools inside decision making systems for urban or territorial management, respecting their complexity.

Figure 1 :

 1 Figure 1: Multi-scale complex system description: from global to individual models

Figure 2 :

 2 Figure 2: Fitness evaluation for community detection

see[START_REF] Bourbaki | Elements of Mathematics: General Topology[END_REF] ch IX