Gabriel Morgado

Bogdan Nowakowski

Annie Lemarchand

DSMC Simulations Of Turing Patterns In Concentrated Growing Systems

The control of self-organized structures such as Turing patterns is a challenge for chemical engineers. We propose a reaction-diffusion model that includes the reactive role of the solvent and admits Turing patterns of adaptable wavelength. Changing the concentration of the solute is sufficient to tune the wavelength. Particles dynamics simulations are successfully performed, illustrating wavelength control down to the submicrometric scale.

INTRODUCTION

The formation of a periodic spatial structure along an axis is a typical example of self-organisation encountered in biology in the plant kingdom as in the animal kingdom. Palm leaves, the segments of many insects or crustaceans, and the spine of vertebrates give some examples of structures resulting from axial segmentation. It is tempting to use models from biology in the context of chemical engineering, with the goal of designing periodic spatial structures by biomimetism. In particular, the model of morphogenesis introduced by Turing has the advantage of being compatible with a microscopic approach [1]. A model with a small number of elementary processes of reaction and diffusion type can be used to simulate submicrometric Turing patterns [2]. However, the wavelength of the structure is imposed by the dynamical parameters whereas an application to chemical engineering would require a flexible pattern with a tunable wavelength. We propose to control the value of the wavelength, simply by playing on the dilution of the system. Indeed, sufficiently far from the high dilution limit, the solvent can be explicitly involved in the reaction scheme, in such a way that varying the total concentration could be sufficient to tune the wavelength of Turing pattern [3]. Our goal is to show the possibility of controlling the wavelength of submicrometric Turing patterns by adapting the Direct Simulation Monte Carlo (DSMC) method to the simulation of concentrated solutions. DSMC has been introduced as a direct simulation of the Boltzmann equation in order to efficiently simulate the dynamics of rarefied gases with large Knudsen numbers for which the continuous description is not valid. The principles of DSMC extend over the entire range of validity of the Boltzmann equation. We apply the Bird simulation method to a reactive system said to be concentrated because of the relatively high proportion of the solute with respect to the solvent. Hence, the departure from dilute conditions is characterized by the ratio of the solute density to the total density. However, the total density is supposed to remain sufficiently low to ensure that dynamics is correctly described in the framework of the Boltzmann equation. When binary reactive collisions are involved, the kinetic Monte Carlo simulation method is straightforwardly extended to reactive systems, the transition rates being deduced from the reactive collision integrals of the corresponding Boltzmann equation. We use a trick to simulate ternary reactions that already proved to be satisfactory [4,5]: The trimolecular interaction is decomposed into a standard binary collision and reaction is accepted depending on the presence of the third reactive species in the neighborhood of the colliding pair. We refer to reference [6] for a detailed presentation of the adapted DSMC algorithm. As a result, the diffusion coefficients of the components and the rate constant of the ternary reaction depend on the total density and consequently the departure from dilute conditions, implying a scaling of the simulation results before the comparison with those of the macroscopic description.

In addition to simulating the Boltzmann equation, DSMC includes the description of the internal fluctuations. Using DSMC to simulate Turing patterns allows us to check the robustness of the spatial structure to fluctuations in small systems down to the nanoscale.

The paper is organized as follows. Section 2 is devoted to the presentation of the reaction-diffusion model including an explicit role for the solvent. Section 3 gives the specificities of DSMC algorithm used to reproduce a Turing pattern in a concentrated system. We give the results in section 4. Section 5 contains conclusion.

A MICROSCOPIC MODEL OF TURING PATTERNS WITH POSSIBLE SOLVENT ROLE

We consider the following reaction scheme involving an activator A with a diffusion coefficient D A , an inhibitor B with a larger diffusion coefficient D B > D A , and interaction with two reservoirs of chemical species or chemostats R 1 and R 2 which maintain the system far from equilibrium [7,8,9]:

A k 1 -→ R 1 (1) 2A + B k 2 -→ 3A (2) B k 3 ⇋ k ′ -3 R 2 (3)
This system is known to be associated with reaction-diffusion equations admitting spatial structures of Turing type.

As explained by Gierer and Meinhardt [10], stabilization of a Turing pattern requires local activation through the autocatalytic formation of the activator A and long-range inhibition through the fast-diffusive inhibitor B. We already proved at the macroscopic scale that considering the solvent S as an actual reactant in a concentrated system according to

A + S -→ R 1 (4) 2A + B -→ 3A (5) B + S ⇋ R 2 (6)
leads to a Turing pattern whose wavelength can be tuned simply by varying the total concentration C = A + B + S , e.g. the quantity of solvent S at constant A + B value [3]. In view of DSMC simulations, it is tempting to assign the role of reservoir particles to solvent particles. In particular, the first step given in Eq. ( 4) is replaced by A+S -→ 2S, which interestingly maintains the total number of particles constant. However, the third step of the scheme introduced in the framework of a macroscopic description cannot be straightforwardly used in particle dynamics simulations in which the solvent would play the role of the reservoir R 2 . Indeed, considering the step B + S ⇋ 2S would lead to the correct term proportional to -BS but to an erroneous term proportional to S 2 instead of a constant in the rate equation of species B. Eventually, the chemical scheme is modified as follows [6] A

+ S k S 1 -→ 2S (7) 2A + B k 2 -→ 3A (8) B + S k S 3 -→ 2S (9)              R k -3 -→ B S -→ R (10)
where the two last steps mean that B concentration is increased at the constant rate k -3 = k ′ -3 R while S concentration is decreased by the same amount to keep the total concentration C constant. It is worth noting that the total concentration C = A + B + S is conserved. The concentration A and B obey the partial differential equations:

∂A ∂t = -k 1 A 1 -A+B C + k 2 A 2 B + D A ∂ 2 A ∂x 2 (11)
∂B ∂t = k -3 -k 3 B 1 -A+B C -k 2 A 2 B + D B ∂ 2 B ∂x 2 (12)
where the relation of conservation has been used to eliminate S and the relations

k 1 = k S 1 C and k 3 = k S 3 C
between the rate constants introduced in Eqs. (1-3) and Eqs. (7-10) have been introduced. We straightforwardly verify that Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] converge towards the equations associated with the system not involving the solvent given in Eqs. (1-3) in the high dilution limit A+B C -→ 0. In a range of rate constants, the system given in Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] has three stationary states, two of which denoted by (A T , B T ) and (A 0 = 0, B 0 ) are stable in the homogeneous case. When the ratio of the diffusion coefficients, d = D B /D A , is sufficiently large and for appropriate rate constant values, the steady state (A T , B T ) is destabilized by inhomogeneous perturbations and replaced by a periodic spatial structure of Turing type. The steady state (A 0 = 0, B 0 ) remains stable and can be used to define a criterion evaluating the deviation from the dilution limit:

δ = A 0 + B 0 C (13)
which remains unchanged for given parameter values, whereas A + B may vary in space and time.

ADAPTATION OF DSMC ALGORITHM

The standard Direct Simulation Monte Carlo (DSMC) method [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] has to be adapted to the simulation of a concentrated solution including two species A and B with sufficiently different diffusion coefficients, D A and D B , and a solvent S. The three kinds of particles P = A, B, S are hard spheres and we use their radius r P as discriminating parameter. Collisions between three species are necessary to obtain different values of D A and D B . Even in the most concentrated case considered, S is in excess with respect to A and B and the collisions between A and B can be neglected with respect to the collisions between A and S and the collisions between B and S as regards diffusion. Qualitatively, we impose r B ≪ r S ≪ r A and S ≫ A + B to obtain D B > D A , as required to stabilize the Turing structure. The precise conditions are given in references [2,6]. The diffusion coefficient D P of species P = A, B obeys:

D P = 3 8(P + S )(r P + r S ) 2 k B T πm ( 14
)
where k B is Boltzmann constant, T is temperature, and m is the mass of the hard spheres. It is worth noting that D P depends on S , i.e. on the dilution of the system. Roughly speaking, one has D P ∝ δ, where δ is the deviation from the dilution limit defined in Eq. ( 13).

Another specificity of the system is the ternary reactive step given in Eq. ( 8) and associated with the autocatalytic formation of the activator A. We already address this issue in DSMC simulations of Schlögl model [5] and solved the problem as follows. A binary collision between A and B is performed with the usual condition on the relative velocity of the colliding pair leading to the formation of the short-lived complex AB. Then, the reaction between the complex AB and the nearest molecule A is accepted with a probability A/C, leading to the release of three A as expected. Hence, the rate constant k 2 obeys:

k 2 = 4(r A + r B ) 2 C πk B T m (15)
The specific expression of k 2 induces a dependence on C, i.e. on the deviation δ from the dilution limit. As for the diffusion coefficient D P with P = A, B, we have k 2 ∝ δ.

The ternary step leads to a cubic term in the reaction-diffusion equations given in Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] which is necessary for the system to have three steady states. This last condition is required to observe trigger waves propagating between two homogeneous stable steady states, one of which may be destabilized by inhomogeneous perturbations leading to a Turing pattern. We exploit this property in the choice of original boundary conditions inspired by the formation of the spine in vertebrate embryos and introduced here by biomimicry [9,13,14,3]. The great advantage of considering a propagating front in a growing system is to avoid fixed boundary conditions, known to perturb the wavelength of the Turing pattern. Indeed, using DSMC for zero-flux boundary conditions in a small system with a length slightly smaller than two wavelengths, transitions between Turing patterns developing over one and a half wavelength or two wavelengths can be observed [2]. This phenomenon could affect the estimation of the wavelength of the structure, whereas considering a Turing pattern developing between zero-flux boundary conditions at one end and a propagating wave front ensures the reproducibility of wavelength determination.

TURING PATTERNS WITH TUNABLE WAVELENGTH

We choose a step function as initial condition with the first 50 cells on the left prepared in the steady state (A T , B T ) and the remaining cells prepared in the steady state (A 0 , B 0 ). Typical profiles for A and B species deduced from the numerical integration of Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] and DSMC simulations are given in Fig. 1. A Turing structure with a well- ), for a given deviation δ = 0.05 from the dilution limit, at scaled time k 2 t = 6986. Snapshot of the profiles for species A (a) and B (b). The parameters take the following values:

k 1 k 2 = 2.9 × 10 4 , k 3 k 2 = 2.2 × 10 4 , k -3 k 2 = 8.8 × 10 6 , d = 10.
defined wavelength λ develops behind a wave front propagating to the right for both approaches. The main question is: Does λ vary as the deviation δ from the dilution limit or equivalently the total concentration C varies ? Before answering the question, we indirectly check the ability of DSMC simulations to properly reproduce the rate constant values for different deviations from the dilution limit. Figure 2 gives the variation of the steady state (A T , B T ) versus the deviation δ from the dilution limit. The prediction of the macroscopic approach based on Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] compares well with the spatially-averaged numbers N A T and N B T of particles A and B deduced from DSMC. The average is performed over four wavelengths. The simulation results are not displayed for δ = 0.01 due to the computation cost: In this case, the total number of particles per cell would typically reach 40400 for the chosen parameter values. We compare the variations of the scaled wavelength λ k 2 D A of Turing pattern versus δ deduced from Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] and DSMC results in Fig. 3. Indeed, the scaled wavelength depends on ratios of rate constants and diffusion coefficients. These ratios are set in all the simulations to the same values given in the caption of Fig. 1 and do not vary with the deviation δ from the dilution limit. Knowing that D A and k 2 are constant in the macroscopic approach, we introduce the scaled wavelength to isolate the intrinsic dependence on δ introduced by the role of the solvent in the chemical mechanism, regardless of the specific expressions of the parameters D A (Eq. ( 14)) and k 2 (Eq. ( 15)) in the microscopic approach. As shown in Fig. 3, the agreement between the microscopic and the macroscopic descriptions is very good, which validates that DSMC correctly reproduce the ratio of diffusion coefficients. Simulation results become more noisy as δ increases, in agreement with the loss of stability of Turing structure predicted by Eqs. [START_REF] Bird | Molecular Gas Dynamics and the Direct Simulation of Gas Flows[END_REF][START_REF] Alexander | [END_REF] [6]. DSMC results are obtained for δ = 0 by considering the chemical mechanism without explicit role of the solvent given in Eqs. (1-3). We have checked that, for δ = 0, the same result was obtained for all the considered values of S . Figure 4 gives the answer to the main question raised in this work. The wavelength of a Turing pattern may be tuned in particle dynamics simulations by varying the total concentration of solvent, i.e. the deviation δ from the dilution limit. Typically, for the chosen parameter values, the wavelength goes from 26 to 17 when δ goes from 0.02 to 0.04. Hence, increasing the deviation from the dilution limit by a factor 2 leads to a decrease of the wavelength by a factor greater than 1.5. The simulation results are given in arbitrary units. The conversion of the wavelength in nanometers is easy. For example, for δ = 0.05 and small molecules A and B in water S, we find a wavelength of the order of 200 nm [6]. Hence, DSMC reveals the possibility to tune the wavelength of Turing pattern on a submicrometric scale, simply by playing on the dilution of the system. The application to chemical engineering nevertheless requires the control of a chemical system whose kinetics explicitly involves the solvent, for example as made precise in the mechanism given in Eqs. (4)(5)(6).

CONCLUSION

We have reported on a reaction-diffusion model involving the reactive role of the solvent and proved that dilution could be used to control the wavelength of a Turing pattern. The direct simulation Monte Carlo (DSMC) method has been successfully extended to concentrated solutions. The simulation results show that it is possible to monitor Turing patterns at a submicrometric scale, simply by changing the total concentration: It is sufficient to increase concentration by a factor 2 to obtain a wavelength reduction by the same factor. The results give some indications to design a chemical scheme allowing experimental validation in the framework of chemical engineering.

FIGURE 1 .

 1 FIGURE 1. Concentration profiles deduced from the numerical solution of Eqs. (11,12) (dashed line) and number of particles per cell in simulations (solid line) versus scaled spatial coordinate x k 2 D A for Eqs. (7-10), for a given deviation δ = 0.05 from the dilution limit, at scaled time k 2 t = 6986. Snapshot of the profiles for species A (a) and B (b). The parameters take the following values: k 1 k 2 = 2.9 × 10 4 , k 3 k 2 = 2.2 × 10 4 , k -3 k 2 = 8.8 × 10 6 , d = 10.

FIGURE 2 .

 2 FIGURE 2. Macroscopic prediction of the steady state concentrations A T (solid line) and B T (dotted line) versus the deviation δ from the dilution limit. Circles (diamonds, resp.) give the mean number of A (B, resp.) particles per cell in the region of the simulated Turing pattern. For δ 0, the mechanism given in Eqs. (7-10) is used whereas the mechanism given in Eqs. (1-3) is used for δ = 0. The value of the other parameters is given in caption of Fig.1.

FIGURE 3 .

 3 FIGURE 3. Macroscopic prediction of the scaled wavelength of the Turing pattern (solid line) and scaled wavelength deduced from the simulations (circles) versus deviation δ from the dilution limit. For δ 0, the mechanism given in Eqs.(7)(8)(9)(10) is used whereas the mechanism given in Eqs. (1-3) is used for δ = 0. The value of the other parameters is given in caption of Fig.1.

FIGURE 4 .

 4 FIGURE 4. Same caption as in Fig.3without scaling of the wavelength.

ACKNOWLEDGMENTS

This publication is part of a project that has received funding from the European Unions Horizon 2020 (H2020-EU.1.3.4.) research and innovation program under the Marie Skłodowska-Curie Actions (MSCA-COFUND ID 711859) and from the Polish Ministry of Science and Higher Education for the implementation of an international cofinanced project.


J Lu

G Déplaude

Pierre Freton

Jean Jacques Gonzalez
email: gonzalez@laplace.univ-tlse.fr

P Joyeux

J-J Gonzalez

EXPERIMENTAL AND SIMULATION STUDIES ON THE VOLTAGE DROP OF ARC IN LOW-VOLTAGE CIRCUIT BREAKER

Keywords: Low-voltage circuit breaker, voltage drop, splitters, experiment, simulation

and research institutions in France

INTRODUCTION 1

The principle of LVCB consists to create a current The most theoretical approach studied and applied 9 in LVCB is the one proposed by Lindmayer [START_REF] Lindmayer | The process of arc splitting between metal plates in 205 low voltage arc chutes[END_REF]. This 10 method allows considering an additional drop volt-11 age through the resistivity in the layers. Another 12 approach consists to describe the sheath with a non-13 equilibrium approach [START_REF] Benilov | A model of the 214 cathode region of atmospheric pressure arc[END_REF]. Nevertheless this method 14 is difficult to implement for a real configuration with several splitters plates. Some authors [START_REF] Yang | Numerical analysis of 217 arc characteristics of splitting process considering 218 ferromagnetic plate in low-voltage arc chamber[END_REF] was analyzed and compared [START_REF] Shin | Arc 252 modeling to predict arc extinction in low-voltage 253 switching devices[END_REF] with the Lindmayer 32 approach [START_REF] Lindmayer | The process of arc splitting between metal plates in 205 low voltage arc chutes[END_REF]. The most complete study was proposed 33 by Iturregi et al. [START_REF] Iturregi | Electric arc in low-voltage circuit breakers: 257 Experiments and simulation[END_REF]. They designed a 3D model in

34

LVCB to analyze the arc behavior. Nevertheless inco-35 herencies appear in the comparison between one and 36 two splitters plate: the additional drop voltage due 37 to a second splitter is not present in the total volt-38 age evolution versus time. The method is not clearly 39 described as the authors used directly the @Ansys 40 module which includes the Lindmayer approach. In 41 this paper, the process of arc splitting is analyzed 42 by theoretical and experimental works. Experimental 43 tests have been realized using one and two splitters: 44 high speed camera and electrical measurements. For 45 the model we have implemented the Lindmayer ap-46 proach and used the same geometries of Iturregi et 47 al. [START_REF] Iturregi | Electric arc in low-voltage circuit breakers: 257 Experiments and simulation[END_REF].

SIMULATION RESULTS

98

We present the main results obtained with the 3D 99 model applying the Lindmayer approach. The arc 100 splitting behavior is shown through the current densi-101 ties in Figure 4 for one and two splitter plates. Figure simulate the sheath contribution. There is no symme-106 try plane in our simulation. In order to reduce the 107 calculation time, the arc is ignited 1.5mm front of 108 splitters plates. In case of one splitter: At t = 50 µs the arc starts 110 to bend and moves toward the splitters. Due to 111 the arc elongation the voltage increases progressively 112 from 61V at t=50 µs to its maximum value 94 V at 113 t=0.19ms. Then the voltage decreases. At t=0.19ms 114 the drop voltage is due to the effects of the arc elon-115 gation and to the additional voltage sheathes. After 116 time t=0.19ms, the current flows progressively into 117 the splitter leading to a diminution of the current 118 path and of the total voltage. The current path goes 119 into the splitter totally at t = 0.21 ms. At t = 0.25 120 ms there is only one path of current and the volt-121 age drop decreases to 81 V. The contribution of the 122 splitter sheath on the total voltage is around 30 V in 123 the following case. In case of two splitters, the pre-124 sented times are not the same; indeed the arc motion, 125 the splitting process time and behavior are different. 126 At t = 0.14 ms, the arc column bends around the 127 splitter plates, the arc voltage increases up to 76.6 V. 128 Compared to one splitter case, we can observe that 129 at t=0.15ms the arc begins to jump on the splitters. 130 mm for the splitter plate. There is a wire located in 149 front of the splitter to ignite the electrical arc. The 150 experimental setup is composed by a generator which 151 can product a prospective peak current up to 13 kA 152 with a maximum charging voltage of 600 V in AC, 153 a high-speed camera (Photron SA5), a differential 154 voltage probe and a Rogowski coil for the electrical 155 measurements. Figure 5 presents the current evolu-156 tion and the total voltages versus time for one and 157 two splitters. The first peak of voltage at t = 0.25 158 ms characterizes the arc ignition. After that, the arc 159 moves toward to the upper side of the chamber. Dur-160 ing the arc motion the voltage is nearly constant 60 161 V until t = 2 ms. At this time the arc arrives in front 162 of the splitter plate. In the case with one splitter, arc 163 voltage increases by squeezing against the splitter, by 164 touching the lower edge of the splitter until arc was 165 totally on the splitter from t = 2 ms to t = 3 ms. The 166 voltage drop contribution dues to the splitter can be 167

2

  limitation by increasing the arc voltage. The increase 3 of the voltage is carried out by various mechanisms: 4 increase of the losses, arc elongation and multiplica-5 tion of the voltage drops in the splitters. To quantify 6 and better describe these mechanisms, studies are con-7 ducted through models and experiments [1][2][3][4].

8

 8

  [START_REF] Sun | Numerical modeling of 221 arc splitting process with ferromagnetic plate[END_REF][8][START_REF] Yang | Numerical simulation 227 of the eddy current effects on the arc splitting process[END_REF][START_REF] Yang | Numerical analysis of 230 the influence of splitter-plate erosion on an air arc in 231 the quenching chamber of a low-voltage circuit breaker[END_REF] 16 studied numerically the splitting process. Considering a nonlinear permeability of ferromagnetic material's 18 they focus their studies on the influence of eddy cur-19 rents[START_REF] Yang | Numerical simulation 227 of the eddy current effects on the arc splitting process[END_REF] and on the influence of metal vapors coming 20 from iron splitter plate erosion[START_REF] Yang | Numerical analysis of 230 the influence of splitter-plate erosion on an air arc in 231 the quenching chamber of a low-voltage circuit breaker[END_REF]. In those studies, 21 the cathode voltage was treated by U-J curves and the 22 anode arc root description was determinedwith the 23 LTE-diffusion model of Lowke and Tanaka[11]. Another group has studied the arc phenomena in LVCB 25 by an imaging system applied to a flexible test ap-26 paratus [12][13][14][15]. The studies were devoted to 27 the influence of vent aperture size on re-ignition and 28 arcing phenomena. They also proposed a modified 29 method to consider the voltage due to the sheath 30 regions. The ability to predict the arc extinction 31

50

  The following assumptions have been used:[START_REF] Quéméneur | Arc motion in low voltage circuit breaker (lvcb) 201 experimental and theoretical approaches[END_REF] The 51 air medium plasma is assumed to be in Local Ther-52 modynamic Equilibrium (LTE). (2) The plasma is a 53 Newtonian fluid and the flow is laminar. (3) Vapors 54 from the walls and electrodes are not taken into ac-55 count. (4) Radiation is treated using the net emission 56 coefficient method.

58Figure 1 .Figure 1 .

 11 Figure 1. These characteristics (a-b-c-d) suggested 74

84

  The geometries for the simulation are presented in 85 the Figure 2. The dimensions of the chambers are 86 40x2.5x11 mm (xyz) and the rails are 40x2.5x1.5 87 mm (xyz). It should be noted that the dimension 88 of the splitters plate are not the same: for one splitter: 89 20x2.5x2 mm (xyz) and for two splitters: 20x2.5x1 90 mm (xyz). D1=8mm represents the distance between 91 the two runners; D2 = 3 mm is the distance between 92 runners and splitters, for the third geometry D3 = 93 2 mm. These dimensions are far from real LVCB 94 but chosen to correspond with Iturregi et al. [17]. A 95 uniform grid is used in the geometry leading to 1.12 96 million cells.

  97

102 3 presentsFigure 2 .

 32 Figure 2. Geometries used for the simulation (From left to right: 0 splitter, 1 splitter and 2 splitters)

  109

Figure 3 .

 3 Figure 3. Voltage drop in the simulation with 1 and 2 splitters

Figure 5 .

 5 Figure 5. Experimental voltage drop and total current Cases with one and without splitter in the chamber

  was developed and the effective electri-182 cal model of conductivity suggested by Lindmayer[START_REF] Lindmayer | The process of arc splitting between metal plates in 205 low voltage arc chutes[END_REF] 183 introduced in the developments to characterize the 184 additional voltage due to the splitter plate sheathes. 185 The feasibility is demonstrated in cases of one and 186 two splitters plates. Interesting results are obtained 187 on the voltage evolutions. In the same time investiga-188 tions are made to characterize the arc behavior in an 189 experimental setup by high speed camera and electri-190 cal measurements. The theoretical and experimental 191 configurations are at this time different nevertheless 192 the same order of magnitude was observed with an ad-193 ditional drop voltage around 20-30V for each splitter 194 plate. The next step of our study will consist to use 195 the same geometry for the model and the experience 196 and to perform one parametric study on the number 197 of splitters plates.


Emilien Bourdy
email: emilien.bourdy@univ-reims.fr

Kandaraj Piamrat
email: kandaraj.piamrat@univ-nantes.fr

Michel Herbin
email: michel.herbin@univ-reims.fr

Hacène Fouchal
email: hacene.fouchal@univ-reims.fr

New Method for Exemplar Selection and Application to VANET Experimentation

Keywords: Exemplars, Sampling, Data mining, Intelligent Transport Systems

Nowadays, huge amount of data are generated and collected in many domains and from various sources. Most of the time, the collected data are processed as common data where simple calculations are applied for the analysis, such as measuring the average, the maximum, the deviation, etc. Exemplar selection has a finer meaning since its aim is to study a few exemplars from common data (the most representative ones). The objective of this paper is to propose a methodology able to extract these representative exemplars from a dataset. The proposed method has been tested against well-known simulated as well as real dataset. It is then experimented on dataset extracted from experimentations of connected vehicle traces.

I. INTRODUCTION

In networking era, the number of connected nodes increases tremendously as well as the number of data to be processed. At the same time, in VANETs (Vehicular Adhoc NETwork), the number of connected vehicles increases exponentially and it is hence impossible to study an individual vehicle one by one due to the huge number of connected cars. In order to analyze their behaviors, we need to explore their data and extract exemplars. To achieve this goal, we propose a new methodology based on an estimation of the local density in a neighborhood of each data. By doing this, we extract representative exemplars that will reflect the different behaviors. We can tune the number of exemplars, as we want to reduce the processing cost and time. If the behavior classes are known, we try to provide at least one exemplar in each class.

The contributions of this paper consist of two aspects. The first one concerns the proposition of the new sampling method and the second one concerns the application of this proposed method to a VANET dataset obtained from the SCOOP@F project [12]. The results achieved with this method can be used in order to identify representative samples for further purposes such as focusing on small samples of cars instead of the whole population or to extract representative behaviors.

The rest of the paper is organized as follows. First, some backgrounds and related works are provided in Section II. Then, the sampling method is presented in Section III along with the assessment on different well-known random and real datasets. The application of methodology to roadway experimentation is presented in Section IV. Finally, we provide conclusion and future works in Section V.

II. RELATED WORKS

For a better comprehension of readers, some backgrounds on sampling techniques are provided in this section.

A set of exemplars is a classical way for storing and representing the cognitive structures [START_REF] Frixione | Prototypes Vs Exemplars in Concept Representation[END_REF]. The exemplars are real data extracted from a large dataset unlike the prototypes that are artificial data obtained from calculating statistics. Thus, the selection of few exemplars that represent the whole dataset is one of the first step when exploring a dataset. For instance, the selection of exemplars is central to several clustering methods [START_REF] Frey | Clustering by Passing Messages Between Data Points[END_REF]. The selection of exemplars is a caseoriented process, which is also called sampling [START_REF] Cochran | Sampling Technique[END_REF]. The goal is to extract a small subset of representative data from the dataset.

The use of sampling technique is necessary when the dataset is too large. However, sampling techniques are also used when the treatment of each individual data needs lots of money, time, effort, etc. Moreover, the selection of exemplars is essential in all the fields where tests or treatments are not possible to implement on the whole population (i.e. the whole dataset). In such trials, there may be risks associated with individual treatment. So ethical reasons involve in testing the treatment on only a small sample of the population. Data are described generally with a large number of variables. Because of the sparsity of high dimensional data space, the selection of exemplars becomes very difficult when data lies within such a space. The phenomenon is known as the curse of dimensionality [START_REF] Houle | Can Shared-Neighbor Distances Defeat the Curse of Dimensionality[END_REF]. The method we propose uses the approach of parallel coordinates [START_REF] Heinrich | State of the Art of Parallel Coordinates[END_REF] to escape the curse of dimensionality when extracting exemplars.

In the literature, many k Nearest Neighbors (kNN) techniques exist [START_REF] Garcia | Prototype selection for nearest neighbor classification: Taxonomy and empirical study[END_REF], but they are used with classified data [START_REF] Pkalska | Prototype selection for dissimilarity-based classifiers[END_REF]- [START_REF] Kim | A fast k nearest neighbor finding algorithm based on the ordered partition[END_REF]. Here we use a kNN methodology with both classified and non-classified data. When used with classified data, we can retrieve them to prove the efficiency of our methodology.

The area of VANET is a very challenging domain nowadays. It attracts many research teams mainly to prepare the future vehicles, which will probably be connected and autonomous. Connected vehicles exchange a lot of messages and the need of analysis on these large amount becomes very urgent. In [START_REF] Nadem | A Comparative Study of Data Dissemination Models for VANETs[END_REF], the authors present a formal model of data dissemination within VANETs and study how VANET characteristics, specifically the bidirectional mobility on well defined paths, have an impact on performance of data dissemination. They investigate the data push model in the context of TrafficView, which have been implemented to disseminate information about the vehicles on the road. In [START_REF] Mekki | Vehicular cloud networks: Challenges, architectures, and future directions[END_REF], [START_REF] Rachedi | BadZak: An hybrid architecture based on virtual backbone and software defined network for Internet of vehicle[END_REF], the recent challenges and futures issues on cloud vehicular domain are presented and the extracted data from such networks is very important to handle. In [START_REF] Castellano | Analysis of urban traffic data sets for VANETs simulations[END_REF], the authors handle two aspects: the derivation of real mobility patterns to be used in a VANET simulator and the simulation of VANET data dissemination achieved with different broadcast protocols in real traffic setting. Most of data analysis are done on one hop sent messages as in [START_REF] Bernard | Slot scheduling for wireless sensor networks[END_REF] but it could be interesting to analyse data over routing issues as presented for different protocols in [START_REF] Ramassamy | Classification of usual protocols over wireless sensor networks Communications (ICC)[END_REF] .

III. THE PROPOSED SAMPLING METHOD

Let Ω be a dataset with n data defined by

Ω = {X 1 , X 2 , X 3 , ...X n }.
The goal of sampling is to select a subset of Ω, which is called the subset Σ where

Σ = {Y 1 , Y 2 , Y 3 , ...Y p } with Y j ∈ Ω.
When sampling, p is much smaller than n (p << n) and Y j (with 1 ≤ j ≤ p) is a representative or exemplar of Ω. This paper describes a new method to select these exemplars.

Our method is based on an estimation of the local density in a neighborhood of each data. The first exemplar we select is the one with the highest local density. Then the nearest neighbors of this exemplar are removed from Ω. We obtain the following exemplars while iterating the process until the dataset is empty.

A. Local Density

In this subsection, we explain how we estimate the local density of each data and how we define the nearest neighbors of an exemplar. Finally, we study the number of exemplars we can propose using this sampling method.

In this paper, we only consider multidimensional quantitative data. Thus, X i with 1 ≤ i ≤ n is a vector defined by:

X i = (v 1 (i), v 2 (i), ...v p (i))
where v 1 , v 2 ,... v p are the p variables that are the features of data. In this context, each data lies a p-dimensional data space.

Usually, the density is defined using a unit hyper-volume. For instance, the hypersphere of radius α can define the unit hyper-volume. In the data space, the local density at X is then equal to the number of data of Ω lying inside the unit hypersphere centred in X. Unfortunately, the definition of density comes up against the curse of dimensionality [START_REF] Houle | Can Shared-Neighbor Distances Defeat the Curse of Dimensionality[END_REF]. When the dimension of the data space increases, the volume of the available data becomes sparse and the classical definition of density has no meaning. For circumventing this drawback, we define the density for each variable using the approach of parallel coordinates [START_REF] Heinrich | State of the Art of Parallel Coordinates[END_REF] (see Fig. 1). Therefore, we have p densities, each defined in a one-dimensional space. The sum of these densities gives us a density-based index that we use in the whole data space.

Let us define the density computed in the one-dimensional space of the variable v j (with 1 ≤ j ≤ p). The dataset Ω is projected in this space and we obtain n values with:

Ω j = {v j (1), v j (2), v j (3), ...v j (n)}.
These values are in the range [min j , max j ] where min j = min 1≤i≤n (v j (i)) and max j = max 1≤i≤n (v j (i)). Let us define the unit interval we use to compute the density at each value x. Let k be an integer between 1 and n. If we expected a local density equal to k, then the length α j we propose for the unit interval is equal to α j = maxj -minj n * k. Thus, the local density at x is equal to the number of elements of Ω j that are in the unit interval [x -α j /2, x + α j /2]. The local density at X i for the variable v j is then defined by:

density j (X i ) = #{ [v j (i) -α j /2, v j (i) + α j /2] ∩ Ω j }.
Finally, the local density at X i for all the variables is defined by:

density(X i ) = ∑ 1≤j≤p density j (X i ).
We select the data, which has the highest local density. This data is the first exemplar of Ω:

Y 1 = arg max Xi∈Ω density(X i ).

B. Nearest Neighbors

The previous procedure enables us to select only one exemplar. We obtain the following exemplars by reducing the dataset and iterating this procedure. The dataset is reduced by removing Y 1 and its nearest neighbors.

Let us describe our definition of the nearest neighbors of a data X in a dataset Ω. The neighbors of X i for the variable v j are the data of Ω that are in the unit interval centered in X i . This neighborhood N j is defined by:

N j (X i ) = {X k ∈ Ω with v j (k) ∈ [v j (i)-α j /2, v j (i)+α j /2]}.
The nearest neighbors of X i for all the variables should be in the neighborhoods for each variable. Thus, the nearest neighbors of X i are in the neighborhood N defined by:

N (X i ) = ∩ 1≤j≤p N j (X i ).
To select the second exemplar Y 2 we exclude the first one Y 1 and its nearest neighbors N (Y 1 ). We apply the procedure defined in the previous section within a reduced dataset Ω \ N (Y 1 ). Then Y 2 the data with the highest local density within the reduced dataset.

We iterate the procedure until the reduced dataset is empty. The exemplars we obtain give us the samples of Ω.

C. Number of Exemplars

We set our method of sampling using the parameter k where k is an expected local density at each data. The value of k lies between 1 and n when the dataset has n data. In this section, we explain how the value of k can change the number of exemplars selected through our sampling method.

Let us consider a toy example with 200 simulated data (n = 200) with 5 variables (p = 5). Fig. 1 displays the profiles of these data with 200 dashed broken lines. The exemplars are selected using the parameter value k = 100. We obtain 7 exemplars (bold broken lines in Fig. 1). The number of selected exemplars decreases when the parameter value k increases. Fig. 2 shows that the number of selected exemplars decreases from 200 to 1 when the density parameter k increases. This property of our method enables us to adapt a strategy to select the number of samples that we extract from the dataset. If we want a specific number of samples selected from the initial dataset, then we can adjust the parameter k to obtain the expected number of exemplars.

D. Assessment of the Method

The exploratory analysis of a dataset is complex for many reasons. The dataset is often divided into classes but the distribution of these classes is unknown. Moreover, the number of these classes is also unknown. To better understand data, the use of a complementary exploratory trial on a smaller dataset is often necessary. The selection of a reduced number of samples should then represent all the classes of the dataset. For this reason, we will evaluate our sampling method under controlled conditions when the distribution of the classes is known. But of course, the method remains designed for applications in exploratory analysis when the classes are unknown. This method is particularly useful when classes have large overlapping and when the classes have very different numbers of data. In such cases, the classical methods of clustering very often fail.

Let us consider a dataset with known distribution of classes for assessing our sampling method. We verify that the distribution of the selected exemplars between classes remains comparable with the distribution of data within the initial dataset. Table I gives the results we obtain with some simulations.

In the first five rows of the table, we use the dataset displayed in Fig. 1. This dataset is simulated using four classes with a large overlapping. The 200 data are randomly distributed between these classes. (42, 51, 65, 42) is the distribution between the four classes. The number of selected exemplars decreases when the parameter k increases. We obtain 25, 18, 13, 9 and 7 exemplars using respectively 50, 60, 70, 80 and 100 as values of k. In these five simulations, the four classes are effectively represented by the exemplars. However, when k increases, the number of selected exemplars becomes too small for representing each class.

In the last five rows of Table I, we simulate five datasets with respectively 4, 5, 6, 7 and 8 classes. The number of data in each class is randomly selected and it could be very different from one class to another one. The datasets have 200 data and the parameter k is equal to 80 when selecting exemplars. When the number of classes increases, the number of exemplars becomes too small for representing each class (see the two last rows of the table). However, these classes are represented if the number of selected exemplars increases (i.e. if we decrease the value of the parameter k).

Let us study the sampling with real datasets. We consider some datasets of UCI repository (see in [START_REF] Bache | UCI Machine learning repository[END_REF]). Table II displays the selection of exemplars using our blind method (i.e. when the classes are unknown) on the classical dataset called "Iris", "Wine", "Glass", "Haberman" and "Ecoli".

These datasets have respectively 3, 3, 5, 2 and 8 classes. Our sampling method gives generally an exemplar in each class. Obviously the method fails if the number of classes is high compared to the number of selected exemplars. Moreover, the method often fails if the number of elements within one class is very low. For instance, in the last line of Table II, two classes have only 2 elements and these classes are not represented by any exemplar. But these classes can be represented by an exemplars if we increase the number of exemplars we select.

For each of these datasets, we have measured the time to execute the algorithm with all possible values of k. It gives us the curves on Fig. 3. In this figure, for all k values, we show the time (in seconds) required to compute the exemplars. For more readability, we only show time values for k < 100 since time execution is less than one second for each database. In this section, the scenario in which VANET data are collected and their description are provided. Then, the obtained results are presented.

A. Scenario and Data Description

In the Scoop@f [12] (Cooperative System at France) project, Intelligent Transport System (ITS) is experimented in the real life. To do so, connected vehicles drive on roadway and communicate with the infrastructure or other vehicles via a specific WiFi called ITS-G5. Messages used in Scoop@f are CAM [13] (Cooperative Awareness Message) and DENM [START_REF][END_REF] (Decentralized Environmental Notification Message). CAM is an application beacon with information about ITS station position, speed if it is a mobile station, etc. DENM is used to warn about events. In this experimentation, the vehicle drives on a roadway and sends DENMs automatically. The event of this experimentation is a slippery road. When vehicle sends DENM, it logs the 30 previous seconds and 30 next seconds. We used these logs with our methodology.

The logs contain 3,201 data of 17 variables (7 for the acceleration control, the steering wheel angle, the strength braking and 8 for the exterior lights). The acceleration control is defined by the brake, gas and emergency brake pedals, collision warning, ACC (Adaptive Cruise Control), cruise control and speed limiter utilization. And the exterior lights are defined by the low and high beam, left and right turn signal (warning is the combination of both), daytime, reverse, fog and parking light.

B. Results

This subsection presents the results obtained with the exemplar selection method proposed previously. The objective is to describe characteristics from the experimentation and trying to model vehicle behavior on this type of road. By using the 3,201 data, we obtain results in Fig. 4 where we can see that there is no big difference among data. This can be explained by the fact that roadway or highway has less variation than urban road. We then used our methodology with different values of k ∈ {40, 80, 100, 140, 180, 200, 500}, which result in Table III. With k = 40, 142 samples are extracted, 102 with k = 80, 92 with k = 100, 58 with k = 140, 48 with k = 180, 46 with k = 200, and 18 with k = 500. The decrease in number of samples in comparison with the number of entries is explained by the fact that there is a lot of data that have equal values. With our methodology, a pre-processing is achieved by dividing by {22, 31, 34, 55, 66, 69, 178} the number of data to be processed. This method can be very interesting for a preprocessing of big data in VANET. In order to illustrate the impact of variation of k to the number of representative samples, Fig. 5 presents the obtained profiles while varying k (for more readability, only the samples are printed). It can be seen that it is very difficult to distinguish exemplars among data. To provide a clearer view, Fig. 6 and Fig. 7 present two extreme cases where one obtained with the smallest values (k=40) and another with the greatest value (k=500). In fact, these figures present the exemplars (central nodes) obtained with the algorithm and their connected neighbors (small nodes), and each number correspond to the number of the observation. It can be seen that the number of exemplars can be limited depending on the need of use case. When the value of k increases, the number of selected exemplars decreases. It has to be noticed here that the most important is to find the compromised tuning between the desired number of exemplars and the desired processing time and costs.

V. CONCLUSION AND FUTURE WORK

In this paper, we have presented a new methodology to select exemplars from a dataset containing multidimensional quantitative data. Our method is based on an estimation of the local density in a neighborhood of each data. With this methodology, it is also possible to select exemplars from classes, and then reduce the number of data in these classes. This methodology was first tested with random data and then with known real data. And, finally, it is applied to VANET data from the Scoop@f [12] project to perform exemplars of the situation described by the experimentation.

In the near future, we will use the methodology with other experimentations and will develop tools to create classes representing each experimentation. Furthermore, this work is

Fig. 1 .

 1 Fig. 1. Profiles of 200 simulated data with 5 variables (dashed lines) election of 7 exemplars with a parameter value k=100 (bold lines)

Fig. 2 .

 2 Fig. 2. Number of selected exemplars decreases from 200 to 1 when the density parameter k increases from 1 to 200

Fig. 3 .

 3 Fig. 3. Execution time of the algorithm with different values of k

Fig. 4 .

 4 Fig. 4. Profiles of 3,201 data with 17 variables from roadway experimentation.

Fig. 5 .

 5 Fig. 5. Profiles of 3,201 data with 17 variables from roadway experimentation with varying values of k.

Fig. 6 .

 6 Fig. 6. Data Network with k=40

Fig. 7 .

 7 Fig. 7. Data Network with k=500

TABLE I DISTRIBUTION

 I BETWEEN CLASSES WITHIN A DATASET (n = 200) AND WITHIN THE SELECTED EXEMPLARS

	Number of	Distribution in dataset	Number of Exemplars distribution
	classes	(n = 200)	selected	between classes
			exemplars
	4	(42, 51, 65, 42)	25	(4, 8, 8, 5)
	4	(42, 51, 65, 42)	18	(5, 6, 4, 3)
	4	(42, 51, 65, 42)

TABLE II DISTRIBUTIONS

 II BETWEEN CLASSES WITH A REAL DATASET AND WITH SELECTED EXEMPLARS (n = NUMBER OF DATA, p =NUMBER OF VARIABLES)

	Name of	n	p	Distribution	Number of	Distribution
	dataset			in dataset	exemplars	of exemplars
	Iris	150	4	(50, 50, 50)	8	(3, 3, 2)
	Wine	178 13	(59, 71, 48)	9	(2, 4, 3)
	Glass	214	9	(70, 76, 17, 13, 9, 29)	19	(1, 6, 1, 5, 1, 5)
	Haberman 306	3	(225, 81)	10	(7, 3)
	Ecoli	336	7	(143, 77, 2, 2, 35, 20, 5, 52)	23	(3,9,0,0,3,3,2,3)

ACKNOWLEDGEMENT This work was made possible by EC Grant No. INEA/CEF/TRAN/A2014/1042281 from the INEA Agency for the SCOOP project. The statements made herein are solely the responsibility of the authors.


Emilien Bourdy
email: emilien.bourdy@univ-reims.fr

Kandaraj Piamrat
email: kandaraj.piamrat@univ-nantes.fr

Michel Herbin
email: michel.herbin@univ-reims.fr

Hacène Fouchal
email: hacene.fouchal@univ-reims.fr

New Method for Selecting Exemplars Application to Roadway Experimentation

Keywords: Exemplars, Sampling, Data mining Intelligent Transport Systems

Nowadays, data are generated and collected in many domains from various sources. In most of the cases, they are handled as common data where some simple calculations are used to analyse them as measuring the average, the maximum, the deviation, etc. For instance, the average number of children in European families is 1.8 children. This kind of assessment is far away from reality: the number of children should be an integer number. For this reason, exemplars have a finer meaning since its aim, in this case, is to look of an exemplar of a common family in Europe which has 2 children (the most representative family). The aim of this paper is to propose a methodology able to extract representative exemplars from a dataset. This methodology has been experimented with dataset extracted from experimentations of connected vehicle traces. This data analysis has shown some interesting features: the vehicle connectivity guarantees that messages are not lost.

Introduction

When we want to study drivers behavior, we have huge amount of data, each corresponds to a behavior. To analyze them, we need to explore these data and we extract exemplars. In order to do this, we propose a new methodology based on an estimation of the local density in a neighborhood of each data. By doing this, we extract representative exemplars that will reflect the different behaviors. We can choose the number of exemplars we want to reduce the processing cost and time. If the classes are known, we try to provide at least one exemplar in each class.

A set of exemplars is a classical way for storing and representing the cognitive structures [START_REF] Frixione | Prototypes vs exemplars in concept representation[END_REF]. The exemplars are real data extracted from a large dataset unlike the prototypes that are artificial data such as the statistics. Thus, the selection of a few exemplars that represent the whole dataset is one of the first step when exploring a dataset. For instance, the selection of exemplars is central to several clustering methods [START_REF] Frey | Clustering by passing messages between data points[END_REF]. The selection of exemplars is a case-oriented process which is also called sampling [START_REF] Cochran | Sampling Technique[END_REF]. The goal is to extract a small subset of representative data from the dataset.

The use of sampling techniques is necessary when the dataset is too large. But sampling techniques are also used when the treatment of each individual data needs for lot of money, time, effort, etc. Moreover the selection of exemplars is essential in all the fields where tests or treatment are impossible to implement on the whole population (i.e. the whole dataset). In such trials it may be risks associated with individual treatment. So ethical reasons involve testing the treatment on only a small sample of the population.

Data are described generally with a large number of variables. Because of the sparsity of high dimensional data space, the selection of exemplars becomes highly difficult when data lies within such a space. The phenomenon is known as the curse of dimensionality [START_REF] Houle | Can sharedneighbor distances defeat the curse of dimensionality?[END_REF]. The method we propose uses the approach of parallel coordinates [START_REF] Heinrich | State of the art of parallel coordinates[END_REF] to escape the curse of dimensionality when extracting exemplars.

The area of VANET is a very challenging domain nowadays. It attracts many research teams mainly to prepare the future vehicles, which will probably be connected and autonomous. Connected vehicles exchange a lot of messages and the need of analysis on these large amount becomes very urgent.

In [START_REF] Nadem | A comparative study of data dissemination models for VANETs[END_REF], the authors present a formal model of data dissemination within VANETs and study how VANET characteristics, specifically the bidirectional mobility on well defined paths, have an impact on performance of data dissemination. They investigate the data push model in the context of TrafficView, which have been implemented to disseminate information about the vehicles on the road.

In [START_REF] Castellano | Analysis of urban traffic data sets for VANETs simulations[END_REF], the authors handle two aspects The derivation of real mobility patterns to be used in a VANET simulator and the simulation of VANET data dissemination achieved with different broadcast protocols in real traffic setting.

Most of data analysis are done on one hop sent messages but it could be interesting to analyse data over routing issues as the in [START_REF] Ayaida | PHRHLS: a movement-prediction-based joint routing and hierarchical location service for VANETs[END_REF]. We will first present the sampling method in Sect. 2, the assessment with different random and known examples in Sect. 3, use the methodology with a roadway experimentation in Sect. 4 and a conclusion in Sect. 5.

Sampling Method

Let Ω be a dataset with n data defined by

Ω = {X 1 , X 2 , X 3 , ...X n }.
The goal of sampling is to select a subset of Ω, which is called the subset Σ where

Σ = {Y 1 , Y 2 , Y 3 , ...Y p } with Y j ∈ Ω.
When sampling, p is much smaller than n (p n) and Y j (with 1 ≤ j ≤ p) is a representative or exemplar of Ω. This paper describes a new method to select these exemplars.

Our method is based on an estimation of the local density in a neighborhood of each data. The first exemplar we select is the one with the highest local density. Then the nearest neighbors of this exemplar are removed from Ω. We obtain the following exemplars while iterating the process until the dataset is empty.

Local Density

In this subsection, we explain how we estimate the local density of each data and how we define the nearest neighbors of an exemplar. Finally, we study the number of exemplars we can propose using this sampling method.

In this paper we only consider multidimensional quantitative data. Thus, X i with 1 ≤ i ≤ n is a vector defined by:

X i = (v 1 (i), v 2 (i), ...v p (i))
where v 1 , v 2 ,... v p are the p variables that are the features of data. In this context each data lies a p-dimensional data space.

Classically, the density is defined using a unit hypervolume. For instance, the hypersphere of radius α can define the unit hypervolume. In the data space, the local density at X is then equal to the number of data of Ω lying inside the unit hypersphere centered in X. Unfortunately the definition of density comes up against the curse of dimensionality [START_REF] Houle | Can sharedneighbor distances defeat the curse of dimensionality?[END_REF]. When the dimension of the data space increases, the volume of the available data becomes sparse and the classical definition of density has no meaning. For circumventing this drawback, we define the density for each variable using the approach of parallel coordinates [START_REF] Heinrich | State of the art of parallel coordinates[END_REF] (see Fig. 1). Therefore, we have p densities, each defined in a one-dimensional space. The sum of these densities gives us a density-based index that we use in the whole data space.

Let us define the density computed in the one-dimensional space of the variable v j (with 1 ≤ j ≤ p). The dataset Ω is projected in this space and we obtain n values with:

Ω j = {v j (1), v j (2), v j (3), ...v j (n)}.
These values are in the range [min j , max j ] where min j = min 1≤i≤n (v j (i)) and

max j = max 1≤i≤n (v j (i)).
Let us define the unit interval we use to compute the density at each value x. Let k be an integer between 1 and n. If we expected a local density equal to k, then the length α j we propose for the unit interval is equal to

α j = maxj -minj n
Finally, the local density at X i for all the variables is defined by:

density(X i ) = 1≤j≤p density j (X i ).
We select the data which has the highest local density. This data is the first exemplar of Ω:

Y 1 = arg max Xi∈Ω density(X i ).

Nearest Neighbors

The previous procedure enables us to select only one exemplar. We obtain the following exemplars by reducing the dataset and iterating this procedure. The dataset is reduced by removing Y 1 and its nearest neighbors. Let us describe our definition of the nearest neighbors of a data X in a dataset Ω. The neighbors of X i for the variable v j are the data of Ω that are in the unit interval centered in X i . This neighborhood N j is defined by:

N j (X i ) = {X k ∈ Ω with v j (k) ∈ [v j (i) -α j /2, v j (i) + α j /2]}.
The nearest neighbors of X i for all the variables should be in the neighborhoods for each variable. Thus the nearest neighbors of X i are in the neighborhood N defined by:

N (X i ) = 1≤j≤p N j (X i ).
To select the second exemplar Y 2 we exclude the first one Y 1 and its nearest neighbors N (Y 1 ). We apply the procedure defined in the previous section within a reduced dataset Ω \ N (Y 1 ). Then Y 2 the data with the highest local density within the reduced dataset.

We iterate the procedure until the reduced dataset is empty. The exemplars we obtain gives us the samples of Ω.

Number of Exemplars

We set our method of sampling using the parameter k where k is an expected local density at each data. The value of k lies between 1 and n when the dataset has n data. In this section, we explain how the value of k can change the number of exemplars selected through our sampling method.

Let us consider a toy example with 200 simulated data (n = 200) with 5 variables (p = 5). Figure 1 displays the profiles of these data with 200 dashed broken lines. The exemplars are selected using the parameter value k = 100. We obtain 7 exemplars (bold broken lines in Fig. 1).

The number of selected exemplars decreases when the parameter value k increases. Figure 2 shows that the number of selected exemplars decreases from 200 to 1 when the density parameter k increases. This property of our method enables us to adapt a strategy to select the number of samples that we extract from the dataset. If we want a specific number of samples selected from the initial dataset, then we can adjust the parameter k to obtain the expected number of exemplars.

Assessment of Sampling

The exploratory analysis of a dataset is complex for many reasons. The dataset is often divided into classes but the distribution of these classes is unknown.

Moreover, the number of these classes is also unknown. To better understand data, the use of an complementary exploratory trial on a smaller dataset is often necessary. The selection of a reduced number of samples should then represent all the classes of the dataset. For this reason, we will evaluate our sampling method under controlled conditions when the distribution of the classes is known. But of course, the method remains designed for applications in exploratory analysis when the classes are unknown. This method is particularly useful when classes have large overlapping and when the classes have very different numbers of data. In such cases, the classical methods of clustering very often fail.

Let us consider a dataset with known distribution of classes for assessing our sampling method. We verify that the distribution of the selected exemplars between classes remains comparable with the distribution of data within the initial dataset. Table 1 gives the results we obtain with some simulations. In the first five rows of the table, we use the dataset displayed in Fig. 1. This dataset is simulated using four classes with a large overlapping. The 200 data are randomly distributed between these classes. (42,51,65,42) is the distribution between the four classes. The number of selected exemplars decreases when the parameter k increases. We obtain 25, 18, 13, 9 and 7 exemplars using respectively 50, 60, 70, 80 and 100 as values of k. In these five simulations, the four classes are effectively represented by the exemplars. However, when k increases, the number of selected exemplars becomes too small for representing each class.

In the last five rows of Table 1, we simulate five datasets with respectively 4, 5, 6, 7 and 8 classes. The number of data in each class is randomly selected and it could be very different from one class to another one. The datasets have 200 data and the parameter k is equal to 80 when selecting exemplars. When the number of classes increases, the number of exemplars becomes too small for representing each class (see the two last rows of the table). However, these classes are represented if the number of selected exemplars increases (i.e. if we decrease the value of the parameter k).

Let us study the sampling with real datasets. We consider some datasets of UCI repository (see in [5]). Table 2 displays the selection of exemplars using our blind method (i.e. when the classes are unknown) on the classical dataset called "Iris", "Wine", "Glass", "Haberman" and "Ecoli".

Ecoli 336 7 (143, 77,

These datasets have respectively 3, 3, 5, 2 and 8 classes. Our sampling method gives generally an exemplar in each classes. Obviously the method fails if the number of classes is high relative to the number of selected exemplars. Moreover, the method often fails if the number of elements within one class is very low. For instance, in the last line of Table 2, two classes have only 2 elements and these classes are not represented by the exemplars. But these classes can be represented by an exemplars if we increase the number of exemplars we select.

Roadway Experimentation

In the Scoop@f [9] (Cooperative System at France) project, Intelligent Transport System (ITS) is experimented in the real life. To do that, connected vehicles drive on roadway and communicate with the infrastructure or other vehicles via a specific WiFi called ITS-G5. Messages used in Scoop@f are CAM [START_REF][END_REF] (Cooperative Awareness Message) and DENM [START_REF][END_REF] (Decentralized Environmental Notification Message). CAM is an application beacon with information about ITS station position, speed if it's a mobile station, etc. DENM is used to warn about events. In this experimentation, the vehicle drives on a roadway and send DENMs automatically. The event of this experimentation is a slippery road. When vehicle send DENM, it logs the 30 previous seconds and 30 next seconds. We used theses logs with our methodology.

Theses logs contain 3 201 data of 17 variables (7 for the acceleration control, the steering wheel angle, the strength braking and 8 for the exterior lights). The acceleration control is defined by the brake, gas and emergency brake pedals, collision warning, ACC (Adaptive Cruise Control), cruise control and speed limiter utilization. And the exterior lights are defined by the low and high beam, left and right turn signal (warning is the combination of both), daytime, reverse, fog and parking light.

Here we want to describe characteristics from the experimentation and trying to modeling vehicle behavior on this type of route. By using the 3 201 data we obtain the Fig. 3, we can see that there no big differences. It's explain by the fact that roadway has less changes than urban road. We then used our methodology with the k ∈ {40, 80, 100, 140, 180, 200} that give us the Table 3. With k = 40, 142 samples are extracted, 102 with k = 80, 92 with k = 100, 58 with k = 140, 46 with k = 180 and 48 with k = 200. The reduction of number of samples in comparison with the number of entries is explained by the fact that there is a lot of data that are equals. With our methodology, a big pre-processing is made, dividing by {22, 31, 34, 55, 69, 66} the number of data to process.

Conclusion and Future Work

In this paper, we have presented a new methodology to select exemplars from a dataset containing multidimensional quantitative data. Our method is based on an estimation of the local density in a neighborhood of each data. With this methodology, it's also possible to select exemplars from classes, and then reduce the number of data in theses classes. This methodology was first used with random data and then with known real data, and with a roadway experimentation from the Scoop@f [9] project to perform exemplars of the situation described by the experimentation.

In the near future, we will use the methodology with other experimentation and developing tools to create classes and representing each experimentation. This work is a contribution for designing tools to analyze data of roadway experimentations in the project Scoop@f.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Profiles of 200 simulated data with 5 variables (dashed lines) election of 7 exemplars with a parameter value k = 100 (bold lines)

Fig. 3 .

 3 Fig. 3. Profiles of 3 201 data with 17 variables from roadway experimentation.

Table 1 .

 1 Distribution between classes within a dataset (n = 200) and within the selected exemplars

	Number of classes Distribution in	Number of	Exemplars
		dataset (n = 200)	selected	distribution between
			exemplars	classes
	4	(42, 51, 65, 42)	25	(4, 8, 8, 5)
	4	(42, 51, 65, 42)	18	(5, 6, 4, 3)
	4	(42, 51, 65, 42)	13	(3, 3, 5, 2)
	4	(42, 51, 65, 42)	9	(2, 3, 3, 1)
	4	(42, 51, 65, 42)	7	(2, 2, 2, 1)
	4	(7, 103, 62, 28)	10	(1, 3, 4, 2)
	5	(40, 47, 55, 9, 49)	10	(2, 3, 2, 2, 1)
	6	(6, 12, 76, 80, 24, 2) 9	(2, 1, 2, 1, 2, 1)
	7	(23, 16, 51, 46, 1,	8	(1, 1, 1, 2, 0, 1, 2)
		36, 27)
	8	(37, 9, 3, 19, 48, 12,	9	(3, 0, 0, 1, 1, 1, 3, 1)
		45, 27)

Table 2 .

 2 Distributions between classes with a real dataset and with selected exemplars (n = number of data, p = number of variables)

	Name of dataset n	p	Distribution in	Number of	Distribution of
				dataset	exemplars	exemplars
	Iris	150 4 (50, 50, 50)	8	(3, 3, 2)
	Wine	178 13 (59, 71, 48)	9	(2, 4, 3)
	Glass	214 9 (70, 76, 17, 13,
				9

Table 3 .

 3 Selection of exemplars with different values of the parameter k from roadway experimentation.

	k	Number of exemplars Division
	40 142	22
	80 102	31
	100	92	34
	140	58	55
	180	46	69
	200	48	66

Acknowledgement. This work was made possible by EC Grant No. INEA/CEF/ TRAN/A2014/1042281 from the INEA Agency for the SCOOP project. The statements made herein are solely the responsibility of the authors.


S Carmona

Y Rouizi

O Quéméner

Frédéric Joly
email: f.joly@iut.univ-evry.fr

Heat flux identification using reduced model and the adjoint method. Application to a brake disk rotating at variable velocity

Keywords: Reduced model, Modal method, Inverse problem, Advection-diffusion equation, Adjoint method Nomenclature .s -2 .K -1 ]

Heat flux identification using reduced model and the adjoint method. Application to a brake disk rotating at variable velocity S Carmona, Yassine Rouizi, O Quéméner, F Joly

Introduction

In the domain of heat conduction, inverse problems are generally ill-posed in the sense of Hadamard and then require complex procedures to obtain satisfactory results. Two techniques are used, the future time step method (Beck) [START_REF] Chantasiriwan | An algorithm for solving multidimensional inverse heat conduction problem[END_REF] which has the particularity of being a sequential method, and the adjoint method [START_REF] Khachfe | Determination of heat sources and heat transfer coefficient for two-a a dimensional heat flow -numerical and experimental study[END_REF] which is an iterative method based on successive computation of descent directions to minimize a criterion taking into account all the data. In these inverse problems, mathematical complexity of the technique limits the size of the characteristic matrices of the thermal problem and the different studied geometries are often reduced to a simple, two-dimensional appearance. This problem is even more blatant when it comes to conduct an online identification, which involves fast calculations [START_REF] Vergnaud | Quasi-online parametric identification of moving heating devices in a 2D geometry[END_REF]. Under these conditions, the use of modal models [START_REF] Quéméner | A specific reduction method for branch modal formulation : Application to highly non linear configuration[END_REF], which allows a significant decrease in the number of unknowns while maintaining a satisfactory accuracy over the entire domain, allows the extension of the inverse techniques to geometries characterized by mesh of large size. Already developed to a diffusion-transport problem, this identification technique using low order models involved the identification by Beck's method of the heat flux dissipated by friction during braking phases of a brake disk [START_REF] Quéméner | On-line heat flux identification from a rotating disk at variable speed[END_REF]. A similar configuration is studied in order to extend the use of reduced models to the adjoint method. A comparison between the two techniques is then presented.

Position of the problem

A brake disk (Fig. 1) rotating at variable speed is considered. It receives during the braking phase a time dependant heat flux on the friction zone with the brake pads (domain Ω 1 ). In the studied case, the surrounding temperature is set to T ext = 0°C and the uniform initial temperature field is T 0 = 0°C. The different time dependant parameters, the radial velocity ω(t), the heat exchange coefficient h(t) and flux dissipated by friction φ(t) are expressed in terms of their maximum values and are therefore dimensionless:

( ) ( ) = u m t t ω ω ω , (1)
( ) ( ) = u m h t h t h , (2)
( ) ( ) ( ) = = ∫ 1 u m u m t t t ( r )d Ω φ φ φ φ ϕ Ω (3)
The heat flux dissipated by friction ϕ m is not uniform on Ω 1 but varies linearly with velocity, so with the radius. The temporal evolution of ω u (t), h u (t), and φ u (t) are shown in Figure 2, and their maximum values are ω m = 2π rad/s, h m = 110W.m -2 .K -1 , and φ m = 600 W.

( ) ( ) ( ) ∂ = - ∇ ⋅∇ - ∇ ∂     - + +     ∫ ∫ ∫ ∫ ∫ ∫ 1 1 2 u m u m m u m

T e c g d e k T g d t e cU . T g d t h t h T g d h T g d t g d

Ω Ω Ω Ω Γ Ω Ω Ω ω Ω Ω Γ φ ϕ Ω     (4) with ( ) 1 g H Ω ∈
a test function, and

1 2 Ω Ω Ω = ∪ .
The discretization of this problem by linear finite element reveals a matrix system of dimension N (number of nodes) which is written in the order of previous terms:

( ) ( ) ( )   = + + +   CT K U H T Π u u u t h t t ω φ  (5)
After a sensitivity analysis, the mesh consists of 9,860 nodes forming 19,362 triangle elements. For a direct problem, the temporal heat flux evolution is known and the evolution of the discrete temperature field T is done by solving Eq. ( 5). Figure 3 represents the evolution of temperature at point A, placed 10mm downstream from the friction area (see Fig. 1). The analysis of the temperature field shows that the local friction on the Ω 1 area leads to the appearance of a sharp temperature front conveyed at the rotational speed. A fixed sensor detects a very rapid temperature variation, which as will be seen later makes the inverse problem difficult to solve.

Modal reduction

The branch problem

The modal decomposition supposes the existence of a base such that the following decomposition is unique:

( ) ( ) ( ) N i i i 1 T M ,t x t V M = = ∑ (6)
where V i (M) are eigenvectors (or modes), and x i (t) are the unknown coefficients named hereafter modal amplitudes. The modes can be seen as elementary thermal fields.

The branch problem associates to the previous physical problem an eigenvalue problem defined by equations ( 7) and (8):

∀ ∈Ω ∪ Ω ∇ = 2 1 2 i i i M , k V z cV (7) ∀ ∈ ∪ ∇ ⋅ = - 1 2 i i i M , k V n z V Γ Γ ζ   (8)
where zi is the eigenvalue associated with the eigenvector Vi.

The boundary condition (Eq. ( 8)) is a non physical condition that involves the eigenvalue of the mode. The number of Steklov ζ ensures dimensional homogeneity of the boundary condition and prevents degeneration of the modal problem, i.e. to balance Eqs. ( 7) and (8).

This special boundary condition reveals two types of modes. The first type is constituted of modes quasi null on the boundary but not on the domain (domain modes), and the second one formed of modes quasi null on the domain but not on the boundary (boundary modes). This second type of modes allows to link the temperature fields on the interface. Examples of such modes are given in Fig. 4. The existence of boundary modes allows one to rebuild temperature and thermal flux density for all convective coefficient. This basis is then adapted to nonstationary and nonlinear thermal problems.

Reduction method

The modal formulation only shifts the problem : instead of being temperature values at the nodes of a mesh, the unknowns are the amplitudes of the modes x i (t). The number of modes needed to approach correctly the solution needs to be reduced. This is done by the amalgam method [START_REF] Quéméner | A specific reduction method for branch modal formulation : Application to highly non linear configuration[END_REF] [START_REF] Quéméner | The generalized amalgam method for modal reduction[END_REF]. In this method, the most influential eigenmodes are kept (they are called major eigenmodes), and the remaining eigenmodes (called minor) are added to them, weighted by a factor α i,p . This results in new amalgamated eigenmodes i V  , which are a linear combination of eigenvectors of the original branch basis.

r n i i,p i,p p 0 V V α = = ∑  (9)
The determination of factors 𝛼𝛼 𝑖𝑖,𝑝𝑝 is performed by minimizing the deviation of energy between a reference model and the reduced model. Note that in our case the reference problem used is constructed independently of the temporal evolution φ u (t) to be identified. With these amalgamated modes, the modal decomposition of temperature is given by : ( ) ( ) ( )

N i i i 1 T M ,t x t V M = ≅ ∑    (10)
The amplitude equation is obtained by replacing the temperature by its modal decomposition (Eq. ( 10)) in the physical problem (Eq. ( 4)), while the test functions are the modes. It replaces

(a) (d) (c) (d)
the problem on temperatures at the nodes of the mesh size by a problem on the temporal amplitudes of the modes. In discrete form, Eq. ( 4) becomes: A reduced base with 50 modes is used. In the case of the direct problem (Eq. ( 11)), the modal model recovers the evolution of the thermal field with an average error compared to the detailed model of 0.046°C and a maximum error of 6.18°C, the temperature range being of 139°C, showing the good agreement between reduced and detailed models. At point A, the error averaged over time is 0.105°C. Figure 5 shows the temperature difference between these two models at the measurement point for the first seconds of the simulation.

( ) ( ) ( ) ( ) ( ) u u u u t h t t t t ω φ φ   = + + +   = + K U H LX M M M X N M X N     (11) where = t L V CV   , = t K M V KV   , t U M = V UV   ,

Inverse problem

The temporal evolution of the heat flux received by friction by the rotating disc is identified from an observable vector Y, consisting here of a single measurement point located at A. Given the size of the discrete problem, modal formulation is used to reduce the size of the inverse problem. The relationship between the output vector Y and the modal amplitude X has to be added to the direct problem defined by equation ( 11) :

= = Y ET E VX   (12
) Two inversion techniques are used, Beck and adjoint method.

Beck's method

Beck's method consists in determining the amplitude of flux at each time step so that the temperature difference between the measurement and the simulation is the smallest possible. An implicit time discretization (at a fixed time-step ∆t = 0.02 s) of Eq. (11) yields the amplitude of each mode:

( )

1 k 1 k k 1 u t t t ∆ ∆ φ - + +     = - +       X L M LX Π   (13)
A least squares minimization between measurement and temperature computed from the estimate at the previous time-step brings the estimation of the searched solicitation :

1 t k 1 u φ - +   =     t Θ Θ Θ Z (14)
with Θ and Z defined by :

[ ] [ ] 1 t t ∆ ∆ - = - Θ E L M Π (15)
[ ]

k 1 k 1 k 1 t ∆ + + -  = - -     Z Y E L M L X  ( 16
)
This technique is first used in an ideal case, wherein the temperature variation at point A used for identification comes directly from the simulation performed by the reduced model. There is then no error in this situation between the measurement and the direct model. The accuracy of the identification carried out is characterized by global error on the flux ( u φ σ ) and the temperature ( T σ ), which are defined by equations ( 17) and ( 18)

( ) i Nt 2 i 1 T Ŷ( i ) Y( i ) Nt σ = = - = ∑ (17) ( ) u i Nt 2 u u i 1 ( i ) ( i ) Nt φ φ φ σ = = - = ∑ (18)
These simulations were performed for a time-step equal to 0.02s. The choice of this reduced time-step is explained by the influence of the transport term that creates sudden temperature changes that need to be taken into account for identification. The choice of the time-step then does not depend on the simple diffusion time between the source and the sensor, but also on the time of transport and the ability of the model to detect sudden changes in temperature. In an ideal case, results are satisfying as u φ σ =0.039 and T σ =0.962°C.

In a real case, the temperature of a probe is simulated by the full thermal model (Eq. ( 5)), to which is added a Gaussian white noise characterized by a quadratic error σ b = 0.3 °C. In this case the identification results are directly unusable: the error on the identified flux is u φ σ =1.59. Indeed, as shown in Fig. 5, the bias to both the use of reduced model in the inverse procedure and measurement error strongly modifies the rapid changes in temperature. The various regularization attempts (increasing the number of measurement points, using a growing number of future time-steps) do not improve significantly the results. This problem had already been shown in previous work [START_REF] Quéméner | On-line heat flux identification from a rotating disk at variable speed[END_REF], and the recommended solution was the use of a low-frequency filter on the identified flux by Fourier transform, assuming that any variation frequency greater than the frequency rotation could only be a numerical distortion. The application of this technique to our configuration is shown in Figure 6. The results are satisfactory since the use of a 0.4 Hz cutoff frequency results in an error on the identified flow equal to u φ σ = 0.038 and an error on the temperature T σ = 0.832 ° C.

Adjoint method

The second inversion technique is the adjoint method. It is a global method in which a quadratic functional built on the differences between the measured temperatures and those computed with the identified heat flux is minimized. This function can also be penalized by a regularization term ε :

( )

  = - +       ∫ Y Y 2 2 u u 0 1 Ĵ ( t ) ( t ) dt ( t ) 2 τ φ ε φ (19)
The identification process consists in finding optimum solicitations u φ such that J is minimum.

u u arg[min J( )] φ φ = ( 20
) This problem is solved using a descent method. These methods require the estimation of the functional gradient with respect to the solicitations. The amplitude equation of the model can be seen as a constraint between the thermal loads and temperatures. It involves the Lagrangian L a associated with the minimization problem under the constraint of the state equation. This term is constructed by summing the functional and the state equation weighted by a Lagrange multiplier (λ):

a u u u 0 d L ( ,T , ) J( ) ( t ) dt dt τ φ λ φ φ   = + - + +     ∫ X λ L MX N   (21)
At the point where the functional is minimal, the derivatives of the Lagrangian with respect to these three variables are null :

a L 0 λ ∂ = ∂ (22) a u L 0 φ ∂ = ∂ (23) a L 0 T ∂ = ∂ (24)
The computation of derivative defined by Eq. ( 22) retrieves the amplitude equation (Eq. ( 11)).

The two last derivatives (Eqs. ( 23) and ( 24)) bring two new relations, called gradient equation and adjoint equation:

ε ∇ = - t J U Π Vλ (25) ( ) ( t ) ( t ) -= + - * t Lλ M λ V E Y Y  (26)
where M* is the adjoint matrix of M.

Thus the interest of this formulation is to compute the gradient J ∇ (Eq. ( 25)) from the resolution of the single equation ( 26). The iterative calculation of the thermal load k u ϕ is done using this gradient J ∇ . Many descent patterns exist. We present here the conjugate gradient method, which combines the flux value at a previous iteration with a descent direction (noted k w here):

+ = + w k 1 k k k u u φ φ ρ
(27) This iterative calculation is finished when one of the following criteria is met. The first is based on the evolution of the functional J (Eq. ( 28)). The second compares the difference between the estimated temperature and the measurements, which should be of the same order of magnitude as the level of uncertainty of the measurement (principle of Morozov Eq. ( 29)).

( ) ( ) ( ) - - < k k 50 u u k u J J 1% J φ φ φ (28) b T σ σ ≈ (29)
The direction of descent k w is a combination between the current and previous descent directions weighted by a coefficient k γ called Fletcher-Reeves conjugation parameter:

k k γ J w w 1 k k ∇ - = - (30) 2 1 k 2 k J J - ∇ ∇ = k γ (31)
and k ρ is the optimal descent step, computed by the secant method (α is a small non null random number)

∇ = - ∇ + -∇ J w J w w J w k k u k k k k k k u u ( ), ( ), ( ), φ ρ α φ α φ (32)
The treated case corresponds to noisy temperatures (σ b = 0.3 °C) issued from the detailed model. In the inverse procedure, the penalization term is null (ε = 0). The above presented algorithm converges to the imposed flux in 379 iterations. As shown in Figure 7, this method does not need additional filtering to recover properly the temporal flux variations. Flux deviation is u φ σ = 0.051, which is very slightly greater than the deviations obtained by Beck's method with low frequency filtering, and in terms of temperatures T σ = 0.389 ° C, which is less than the error obtained by Beck's method with filtering. Obtaining such satisfying results without filtering can be explained by the fact that the estimated flux is issued from a minimization including the entire temperature variation. In contrast, in Beck's method only the next time step is used to estimate the flux at a given time, which makes this technique much more sensitive to sudden changes and noise measurements.

Conclusion

The study first of all showed the interest of using low order models in inverse problems, as the loss of information generated by the reduction remains below the noise measurements.

Regarding the comparison of the two inverse techniques used in this paper, results showed the difficulty in obtaining correct results with Beck's method. In fact, the sequential aspect of this method does not filter the errors directly obtained from the measurement which are amplified significantly during the flux identification process. A solution is possible, however, but at the cost of additional low frequency filtering, which eliminates the sequential aspect of this technique. The effectiveness of the adjoint method was shown, since very satisfactory results were obtained, with no obligation to use any additional filtering or penalty term functional. This method, more comprehensive, naturally filters the noise during the functional minimization process. The price to pay is that the adjoint method requires more computation time (750s) that Beck's method (115s). These are very encouraging results, paving the way for online identification, both by a search for the minimum acceptable reduction of the modal model, and by the development of a more appropriate adjoint technique (order 2 descent method, temporal sliding window).

Figure 1 :Figure 2 :

 12 Figure 1 : Computational domain. Figure 2 :Temporal evolution of thermal loads

Figure 3 :

 3 Figure 3 :Temperature evolution at point A

Figures 4 :

 4 Figures 4 : Examples of branch modes : boundary modes ((a) and (b)) and domain modes ((c) and (d))

Figure 5 :

 5 Figure 5 :Temperature evolution at point A obtained with different models for the first 3 seconds of the simulation

Figure 6 :

 6 Figure 6 :Heat flux identification obtained with Beck's method after filtering

Figure 7 :

 7 Figure 7 : Heat flux identification with adjoint method