P Mouro

G Gary

H Zhao

Dynamic tensile testing of sheet metal

The testing of sheet metal at high strain rates requires the use of the SHPB set-up. This technique provides satisfactory results as far as the compression and shear are concerned. A modified SHPB set-up is presented here, which extends the scope of this technique to the tensile loading of sheet metals. In this case, many measurement problems arise due to the way in which the specimen is connected to the bars, which never completely prevents unexpected slipping and is liable to induce interference waves due to an impedance mismatch. The technique described here was based_ on the "hat specimen" technique proposed by Lindhohn [1]. A two-dimensional "hat shaped" specimen is subjected to compression loading. A tensile state is induced in a symmetrically necked area, where higher strains are concentrated. Connecting supports of an appropriate shape are attached to the oars. They have the same impedance as that of the bars to avoid the occurrence of any interference waves. The "hat" shape prevents sliding, and to ensure perfect initial contact conditions, the complete system is elastically pre-stressed. A special method of wave analysis is then used to determine the forces and displacements occurring at the specimen ends from the strain signals recorded in the bars.

INTRODUCTION

The numerical modelling of crash tests is nowadays widely used in the automotive industry. It requires the use of a reliable constitutive model which can be used with large strains (up to 30%) and large strain rates in the range of 1/s to 1000/s. The corresponding tests at low strain rates are usually performed under tension loading, using a classical hydraulic testing machine. Tests at high strain rates are performed under compression loading with the Split Hopkinson Pressure Bar. The complete range of strain-rates obtained so far suggests that the behaviour of metal sheets has always been implicitly assumed to be symmetric between traction and compression.

To avoid making this assumption, it was necessary to perform accurate tests under both tension and compression at the same average strain rates. For this purpose, we extended the SHPB technique to tension testing, as it yields accurate results in the high strain-rate range. This approach is in line with that used with the study by Quik and a! [START_REF] Quik | Dynamic mechanical properties of automotive thin sheet steel in tension, compression and shear[END_REF], and the present paper deals with a new SHPB device that can be used extensively for the tensile testing of industrial metal sheets.

Dynamic tensile testing using SHPB is a rather old topic in the case of cylindrical specimens that can be connected directly to the bars [START_REF] Harding | Tensile testing of materials at impact rates of strain[END_REF]. Nicholas [START_REF] Nicholas | Tensile testing of materials at high rates of strain[END_REF] loaded the incident bar with a compressive wave that was trapped in a collar surrounding the specimen so that straining was produced when the reflected tensile wave travelled back from the second bar. Albertini and Montagnani [START_REF] Albertini | Testing techniques based on the split Hopkinson Pressure Bar[END_REF], Staab and Gilat [START_REF] Staab | A direct-tension split Hopkinson bar for high strain-rate Testing[END_REF] created a tensile wave directly by inducing the rapid fracture of a clamp at a point on a pre-stressed part of the incident bar.

In the case of metal sheets, the connection between the specimen and the bars is an important point. Clamping the specimen between jaws generally leads to an impedance mismatch, which at best gives only a reliable measurement of the force in the transmitter bar. Recently, Tanimura and Kuriu [START_REF] Tanimura | Mechanical properties of sheet steel at high strain rates[END_REF] presented a set-up based on the compressive SHPB where the axes of the bars were laterally shifted so that the relative displacement of the bar ends induced a traction state in the specimen. This specimen was connected to cylindrical supports which were not aligned with the axes of the bars.

The set-up used here was a modified SHPB, where tension was produced in a very similar way to what occurs with the classical compression technique. Particular attention was paid to the force measurements at both specimen ends to ensure that the stress was accurately determined throughout the test.

EXPERIMENTAL TECHNIQUES

General description

Loading was applied to the incident bar in the form of a compressive wave and tension was induced in the specimen using a hat-shaped geometry (fig 1). This set-up made it possible to obtain a constant impedance while the waves were travelling along the bars on both sides of the specimen. A specific insert (the "pusher") was connected to the tip of the incident bar to load the bottom of the hat. Likewise, a second insert (the "support") was connected to the transmitter bar. Both connections between the inserts and the bars were implemented with aluminium rings screwed onto both the bar and the insert, which were especially designed so that their contribution to the impedance was negligible. As the incident and transmitter inserts could not have the same cross section for geometrical reasons (see fig. 1), the two bars did not have the same impedance.

One important feature of this system is that the region where the impedance change occurred between the bars and the specimen had the same thickness as the metal sheet. Consequently, the impedance change occurring at the bar-specimen connections was very similar to that observed in the case of a classical pure compression test, as illustrated in fig 2. The specimen was connected by means of screws to each external face of the support. To ensure satisfactory contact conditions right from the onset of the dynamic loading, and promote the friction between the metal sheet and the tips of the support, the whole assembly was statically pre-stressed before the test. The value of the corresponding stress in the necked area of the specimen was deduced directly from the compressive strain measured in each bar (at gage stations Gi and Go, fig 2) . This stress is a reasonable and significant fraction of the static yield stress characteristic of the metal sheet tested.

Analysis and processing of gage measurements

A complete data processing method was used cin the waves . This method consisted in shifting the waves from the gage stations where they were first recorded to the bar ends. It included wave dispersion correction and an assisted delay setting based on the elastic transient response of the specimen, as described in [START_REF] Zhao | On the use of SHPB technique to determine the dynamic behavior of the materials in the range of small strains[END_REF]. With a classical SHPB set-up, where the bars have the same diameter and there is no initial pre-stressing of the specimen, the forces and speeds at both specimen ends are given by [START_REF] Zhao | On the use of SHPB technique to determine the dynamic behavior of the materials in the range of small strains[END_REF] :

F;nput(t) = SB E (s;(t) + sr(t)) F;,utpu/t) = SB ESt (t) Vinput(t)=-C 0 (e; (t)-er(t)) Voutput (t)=-Co et (t) (1)
where SB, E and Co are the bar's cross-sectional area, Young's modulus and elastic wave velocity, respectively.

The strains used in these formulas are the virtual incident, reflected and transmitted strains that have been shifted to the bar ends using the data processing method mentioned above.

With the present set up, the appearance of both waves recorded differed from that of the waves observed in a classical SHPB. Figure 3 shows the signals recorded in a test performed with a short striker, to obtain a large delay between the incident and reflected signals. The main difference with a simple compression test was that the three waves no longer had the same vertical origin. In the case of the incident and transmitted waves, this was due to the differences between the diameters and Young's moduli of the two bars. In the case of the reflected wave, it was due to the null stress boundary conditions pertaining at the end of the incident bar after the impact (the input end of the incident bar escapes the static pre-stressing device). The classical formulas (I) have to be modified to take these differences into account. If the origins (on the Y -axis) are taken to be at the constant baseline level of each signal before the onset of each wave (see fig. 3), those formulas become:

1eJISioH (\7.) ' ' ' ' ' 1 ------J-------L------"------1 ' ' ' ----L--

F;nput (f)=S;EJ E; (f)+ Er (f))+ Fps F;,utput (f)=S 1 EA (f)+ Fps Vinput (t)=-C ;(E; (t)-E r (t))

Voutpur (t)=-Crsr(t) [START_REF] Quik | Dynamic mechanical properties of automotive thin sheet steel in tension, compression and shear[END_REF] where Si> E; and Ci are the incident bar's cross-sectional area, Young's modulus and elastic wave velocity, respectively, and St, E, and Ct are the output bar's cross-sectional area, Young's modulus and elastic wave velocity, respectively.

As in the classical case, the strains used in these formulas were the virtual incident, reflected and transmitted strains that were shifted to the bar ends using the data processing method mentioned above. ' ------~-----------~---------- It was checked that the input and output forces were equal before further processing the test data, to make sure that the specimen was in a homogeneous state of stress. An example of the force equilibrium is plotted in fig. 4 in the case of a 270/s test on mild steel.

Specimen geometry

Since the average traction force applied to the specimen was given, it was necessary in order to draw up a stress-strain plot to use a characteristic specimen length to calculate the strain, as the displacement measurements focused in the case of our set-up on the ends of the inserts. Harding and al. [START_REF] Harding | Tensile testing of materials at impact rates of strain[END_REF] , with their cylindrical specimen, carried out numerous static tests to obtain a strain-displacement relation that was non linear in the small strain range.

Using an explicit commercial code (LSDYNA3D) with various constitutive parameters, we investigated whether using the initial length of the necked area sufficed to be able to obtain the constitutive input relation. This simple analysis consisted in neglecting the strain in the wider parts of the specimen, focusing only on the necked area. Typical measured velocities were used as boundary conditions for 2D and 3D calculations. The force calculated was used instead of an experimental measurement to obtain a true stress-true strain plot. As shown in fig. 5, the simple geometry of our specimen appears to give reasonably good results when Ludwik and Johnson-Cook-like hardening laws are used. In addition, a significant improvement can be obtained using a modified geometry where the necked parts are split, with the same equivalent area, to produce a "comb-like" specimen. The effects of this operation were simulated with a specimen split into only two parts, as shown in fig. 5 /""one core specimen The final shape of the initially flat machined specimen can be obtained by means of a folding technique, taking care that no plastification of the necked area can occur. The symmetry of the final specimen can be quite easily obtained provided that the folding device is accurately designed for each sheet thickness.

RESULTS AND DISCUSSION

Tests were performed with a mild-steel sheet that exhibits significant strain-rate effects when subjected to compression loading in the rolling direction. Specimens were cut at an angle of 90 degrees to the rolling direction. Fig. 6 shows• the raw stress-strain curves obtained at four different strain-rates. Each test was performed twice and the results were fairly consistent. The evolution of the strain-rate during the test was naturally similar to that observed in compression tests. This series did not exhibit a rate effect in the range of interest, which covers nearly a decade. Previous compression tests performed at authors laboratory [START_REF] Zhao | A constitutive model for metals over a large range of strain rates. Identification for mildsteel and aluminium sheets[END_REF]10] on the same steel showed the occurrence of a 20% increase in the flow stress between the lowest and highest strain-rates reached here. The results obtained here under tension loading were unexpected, in view of the data available on mild-steel sheeting.

They should be compared with the results of similar tests performed on specimens cut in the rolling direction of the metal sheet.

The initial yield stress was found to be similar to the static value and lower than the yield stress recorded under compression loading, in line with the results obtained by Quik and a! [I]. Since the strain hardening was found to be much greater under tension loading, the difference between the flow stresses disappeared at high strains. The technique described here yields more satisfactory results than the dynamic tensile testing set-ups currently available for testing metal sheets. It can be used extensively on sheets of all thicknesses and specimens with a wide range of geometries, since the mode of connection with the bars does not distort the measuring signals. The results presented here are largely consistent with the known characteristics of mild steel.

CONCLUSION

A modified SHPB device involving the use of a specially designed specimen was developed to evaluate the effects of the strain rate on the mechanical behaviour of metal sheets subjected to dynamic tension loading. The accuracy of the measurements thus obtained was confirmed by checking the balance between the input and output forces during the test and by performing dynamic numerical simulations of the specimen's response. In comparison with results obtained under compression loading, these preliminary results confirm that the loading sign (tension or compression) affects the stress-strain response and the effects of the strain-rate. The widely used rate-dependent isotropic plasticity model is not able to account for the results obtained under both tension and compression loading. FEM simulations, which are commonly used in the automotive industry, where metal sheets are used in car crash simulations, require the implementation of more accurate constitutive models taking these parameters into account.

A challenging issue for the next few years is to complete this model, including finding appropriate criteria that could be used instead of the Von Mises' criteria for simulating metallic shell structures under dynamic loading.

Figure 1 . 3 Figure 2 .

 132 Figure 1. The hat specimen and special supports

Figure 3 .

 3 Figure 3. Basic recorded signals in pre-stressed bars.

Figure 4 .

 4 Figure 4. Equilibrium in the specimen: input and output forces.

Figure 5 .

 5 Figure 5. Influence of the "split" geometry of the specimen

Figure 6 .

 6 Figure 6. Stress-strain curves at various strain-rates.

  .

	siress (MPa)
	input model
	400-
	/'	two cores specimen
	f.
	200
		0.13700	0.1111541
		0.123Bl	0.11JB17
		0.11Il634	0.!941.133
		0.[95929	0.1112979
		0.1112225	O.D71 125
		O.l&li21	0.059271
		0.054817	O.IU7416
		O.GI1113	0.035!li2
		0.1127«11	O.II2YOO
		0.0137().1	0.01 11154
		D.liiXDJ	O.llDID

Acknowledgements

This work is supported by Peugeot-Citroen Automobiles. The authors would also like to thank Dynalis for the LSDYNA3D license.


Han Zhao

A study of specimen thickness effects in the impact tests on polymers by numeric simulations

Experimental data of polymer behaviour at high strain rates in the literature, mostly obtained with a split Hopkinson pressure bar (SHPB), shows a dependence on the thickness of the specimen. This paper investigates this so-called thickness effect and tries to clarify the doubts on the equilibrium assumption in SHPB tests on polymers by numerical simulations. Fictitious specimens of different thickness with rate sensitive polymer-like constitutive model (Eyring's model) are used to simulate basic data of SHPB tests. The comparison between the given behaviours and those derived from the simulated tests provides an objective appreciation of the quality of tests on polymers (with thin or thick specimen). Results of the simulation prove that the equilibrium assumptions can still be safely used in the case of tests on polymers, and indicate that the thickness effect is mainly due to the radial inertia and friction considerations.

INTRODUCTION

The mechanical behaviour of polymers at high strain rates is often involved in safety studies of engineering applications. For example, in the domain of pipelines for the distribution of fuel gas, the catastrophic failure of the polymeric pipes by a very fast crack propagation, even rare, has been observed under accidental impacts 1

• Recent analyses of tests on real scale pipes allow for the establishment of an empirical critical pressure for the initiation of crack propagation. However, to obtain a theoretical criterion or to perform a numerical simulation, the constitutive law of the materials, of which the pipes are made, is indispensable 2 • 3 • A few experimental data of the polymeric behaviour at high strain rates have been reported in the literature, obtained with a split Hopkinson pressure bar (SHPB). Some authors 4 • 5 used a very thin specimen ( < 2 mm) and gave a quasi-linear relation between the yield stress with the logarithm of the strain rate so that Eyring's model 6 (see also Section 2.2) is still valid at high strain rates. By contrast, the others 7 • 8 used a thicker specimen and found that there is a sharp increase in the yield stress at high strain rates. Such a so-called thickness effect is observed for many polymeric materials.

Recently, Dioh and co-workers 9 have systematically investigated this thickness effect and used specimens of different thickness (1.5 mm and 4.4 mm) but of exactly the same materials. For all the materials studied, such as medium density polyethylene (MDPE), high density polyethylene (HDPE), polycarbonate (PC), and poly( ether ether ketone) PEEK, tests on specimens of different thickness do lead to different rate sensitivities. Results are schematically re-drawn in Figure 1 where a sharp increase is found for tests on thick specimens.

Such results suggest not only that there exists some experimental imprecision in the classical SHPB analysis where simplifying assumptions are made, but also that one of experimental data and the corresponding behaviour modelling (why not both two), is not accurate. Dioh and co-workers 10 supposed that the sharp increase observed using thick specimen is not a physical feature but an experimental imprecision which could be probably explained by the wave propagation effects (see Section 2) involved in the SHPB tests.

It is then of interest to examine the SHPB testing process in order to know if the classical SHPB analysis leads to acceptable results in the case of tests on polymers and to find which assumption leads to this thickness effect. For this purpose, this paper presents a numeric simulation of SHPB testing process and the author believes that it is a reliable way to appreciate objectively the two kinds of results obtained from specimens of different thickness. The basic SHPB testing data is simulated for fictitious specimens of different thickness, the rate sensitive behaviour of which is described by Eyring's model. Using the classical equilibrium assumption, the average stress-strain curve of the fictitious specimen can be calculated from those simulated basic data. The quality of the measurement in both cases (thin or thick specimen) can be then estimated by the comparison between the given and the recovered material behaviour. In addition, such a simulation allows also for an appreciation of the equilibrium assumption in SHPB tests on polymers.

SIMULATION OF SHPB TESTS ON POLYMER

Split Hopkinson pressure bar tests and its assumptions

The SHPB 11 , or Kolsky's apparatus 12 is a standard experimental technique in the study of the constitutive law of materials at high strain rates. A typical SHPB set-up is composed of the long input and output bars with a short specimen placed between them. The impact of the projectile Once it arrives at the bar specimen interface, a reflected wave er(t) is developed in the input bar, whereas a transmitted wave e 1 (t) is developed in the output bar (Figure 2). From those basic experimental data (incident, reflected and transmitted waves), the classical analysis assumes homogeneous stress and strain fields in the specimen, and an average strain and an average stress are then given by equation (1).

Ebsb as(t) = --et(t) Ss (1)
where ls and Ss are the length and the cross-sectional area of the specimen, whereas Eb, Sb and C 0 denote Young's modulus, cross-sectional area and wave speed of the bar.

Nevertheless, equation (1) is correct only when the stress and strain fields in the specimen are effectively homogeneous, and this is never true because of wave propagation effects in the specimen. For example, when the input side of the specimen is loaded, the output side remains at rest until the wave goes through the specimen. Such an effect is obviously more important for a thick specimen. However, theoretical 13 , experimental 14 and numerical 15 investigations have shown that the equilibrium assumptions, even approximate, do not introduce significant errors in the case of tests on metals. By contrast, for material such as polymeric foam, concrete, rocks, etc., where homogeneous stress and strain fields cannot be assumed, only an identification technique based on an inverse calculation method is theoretically correct 16 . Furthermore, equation (1) is a one-dimensional analysis and it implies that the three-dimensional effects such as the radial inertia and the friction between the specimen and bars could be neglected. Such an assumption is less convincing when the specimen is a very thin disc. Corrections have been proposed and a smart diameter/thickness ratio d/l, for h . . d 1• . h f"' 17 18 0 t e specimen Is suggeste to Imit t ose e 1ects ' . ne interesting study by Gorham 19 shows that the sharp rise in rate sensitivity of metallic materials would be related to those points. Even those aspects are not yet well studied in the case of polymers, one can imagine that the general trend remains the same. The three-dimensional effect is important for a thin specimen, whereas the wave propagation effect will be strong for a thick one, and the thickness effects can be explained by one of those assumptions, provided that the signal processing is correctly made 20 • 21 .

Transient simulations of a split Hopkinson pressure bar with a polymeric-like constitutive law

A complete simulation of the SHPB test will be a threedimensional one which takes into account those two assumptions mentioned above. However, the onedimensional simulation is chosen here. Indeed, the exact three-dimensional boundary condition at bar-specimen interfaces during the test cannot be known, and a threedimensional simulation (with elasto-plastic model) given by Bertholf and Karnes 15 has shown that the friction and radial inertia hardly interfere with the wave propagation effect. The last can, therefore, be studied separately.

Another practical reason is that there is no possibility in available dynamic FEM codes to introduce a suitable ratesensitive constitutive law for polymers and to deal accurately with the special contact condition of SHPB 22 . This is why a one-dimensional transient numerical simulation is developed here.

The one-dimensional governing equations and the constitutive law are written as follows:

aa(x, t) av(x, t)
---a;-= P _a_r_ ae(x, t) av(x, t) = at ax f(a, e, a, e, ... ) =0

where a, e, v are the stress, the strain and the particle velocity in the specimen. p is the mass density.

The boundary conditions at the two faces of the specimen are given as follows:

EbSb Sb h . .d 3 ) a(x, t)---v(x, t) = 2 -E 8

ei(t) at t e mput si e ( CoSs Ss

a(x, t) + EbSbv(x, t) = 0 at the output side

CoSs

The quality of the conclusion derived from the simulated tests depends on how well the constitutive law used in those simulations describes the reality. On the basis of the thermally activated mechanism, Eyring 6 has proposed a model describing the yield stress a y as a function of strain rate and temperature (equation ( 4)).

a

[ ( e) Q J ( ay Q )

; =A ln eo + RT ore=e0exp TA-RT (4) where A, Q, R are constant coefficients, amd T is the absolute temperature.

Numerical scheme with the method of characteristics

.1!5

An elastic-plastic model with rate sensitivity of yield stress described by Eyring's model can be constructed and e expressed in the form of a Sokolovsky-Malvern type 23 • 24 (equation ( 5)).

-.85

a= E(eel +sine!) if a> ao

where a 0 is the quasi-static yield stress. (5) Using the Eyring model (equation ( 4)) to describe the non-elastic part, we have einel = f:oexp ( ao + B(:-a/E)n -1) (6) The numeric problem defined by the governing equation (equation (2)) and the above constitutive law can be resolved easily by the method of characteristics, because the characteristic network is composed of families of straight lines in this case. The three families of characteristic lines and the characteristic relation that must be satisfied along those lines are defined [START_REF] Cristescu | Dynamic Plasticity[END_REF] in equation (7).

(i): =Co da = -pCodv-pC6eineldt (7) .. )dx c (u -= -o dt (iii)dx = 0 deinel = eineldt
Using a regular discretization grid, the governing equations (equation ( 7)) with the boundary conditions (equation ( 4)) are numerically integrated.

TESTS AND THEIR SIMULATIONS

In order to show the accuracy of the numerical simulation and the describing quality of this model, a set of real SHPB tests on PC at a high strain rate is performed and served as the basis for the comparison test-simulation.

Using the real measured incident wave as input data, the reflected and transmitted waves can be calculated for a fictitious specimen which has the same size as the real one and has the mechanical behaviour described by the model presented above with identified parameters. Figure 3 shows basic measured waves for a test at a strain rate of about 280 s-1 (specimen of PC, diameter 10 mm, length 10 mm), compared with the simulated ones. They are very close, and it proves the confidence of the numerical accuracy. Figure 4 shows a comparison (for a set of tests from 230 s-1 to 1650 s-1 , specimen of PC, diameter 10 mm, length 10 mm) between the stress-strain curves calculated from real measured waves (bold lines) and those from the simulated waves. The describing quality of the model is quite acceptable.

Furthermore, the stress-strain curves calculated from the simulated tests are also very close to the given model at corresponding strain rates. It suggests that the equilibrium

strain (X)

Figure 4 Comparison between stress-strain curves derived from real tests and those from simulated tests assumption is still valid in the case of tests on polymers. It is noted that those real tests (Figure 4) are just the illustrations of the pertinence of the rate sensitive model used for simulations. This constitutive model is not limited in this range of strain rates. Simulations for tests at higher strain rates (up to 10 4 s -I, cf. Dioh et al. 9 ) lead to similar conclusions.

DISCUSSION ON THICKNESS EFFECTS

The confidence on the simulation and on the model allows, therefore, for an objective examination of the thickness effect. Fictitious tests are then simulated for a 15.8 mm aluminium SHPB set-up, as used by Dioh and co-workers 9 , where the thickness effects are experimentally proved. The used incident wave lasts 450 p.s with a rising time of 15 p.s, which corresponds to a real SHPB incident time. Tests on PC with two types of specimen (diameter 12.7 mm, thickness 1.5 and 4.4 mm, as used by Dioh and co-workers 9 ) are simulated at several impact velocities (the strain rates range is from 10 2 s-1 to 10 s-1

). It is noted that the rising time and specimen/bar impedance ratio also have their effects on the equilibrium process in the specimen. The choice of a constant rising time (experimental observation in our tests shows very little variation) and of the same diameter for two kinds of specimen in this simulation is motivated by those considerations.

The stress-strain curves are calculated from simulated basic waves, using classical SHPB analysis. Figure 5 shows the rate sensitivities of the flow stress (at a strain of 10%) derived from those two sets of simulated tests, compared with the given rate sensitivity. There is then no apparent thickness effect for the simulated tests, and it is in contrast to the known simulation works given by Dioh et a/. 10 • 26 . The explanation may be related, in the author's opinion, to the use of the simple elastic-plastic bilinear constitutive law in their simulation. Certainly, the simulation here is not a complete one. The chosen behaviour may not describe all the mechanical feature of polymers and the wave propagation effect could strongly interfere with the three-dimensional effect in the case of polymers. It must be also noted that there will never be a really complete simulation of experiments (because the contact condition is unknown, as is the real behaviour of polymers). The simulation presented here proves at least that the equilibrium assumption does not introduce significant errors and there is no thickness effect when the present rate-sensitive constitutive law, which is probably quite close to reality, is used. This result suggests then that the thickness effect is not due to wave propagation effects in the specimen and is mainly due to the three-dimensional effect, and it gives, hopefully, a reliable indication for further research into the thickness effect.

CONCLUSION

A simulation of SHPB tests on polymers is performed to investigate the specimen thickness effects observed in experimental data reported in the literature. A rate sensitive (Eyring's model) polymer-like constitutive relation is used in this simulation which is performed by the method of characteristics. It is concluded that the equilibrium assumption is still valid for this type of constitutive law. It can be then a priori safely used in the case of tests on many polymers. It is also shown that there are no thickness effects in the simulated tests. It may suggest that the thickness effect is mainly related to the three-dimensional effects in the SHPB test.

Figure 1 Figure 2

 12 Figure 2 Scheme of classical SHPB analysis

-. 18 Figure 3

 183 Figure 3 Simulated basic waves of a test on PC, compared with real measured waves

5 7-9 d po ymer at ow an me mm stram rate ' • , an one of the aims in the study on thickness effects in the literature


Kevin Chapuis
email: kevin.chapuis@gmail.com

Patrick Taillandier

Renaud Misslin

Alexis Drogoul

Chapuis Kevin

Taillandier Patrick

Misslin Renaud

Drogoul Alexis

Gen*: a generic toolkit to generate spatially explicit synthetic populations

Keywords: Synthetic population, Spatialization, Social simulation, Multi-agent model

published or not. The documents may come

Introduction

Agent-based simulations are now widely used by researchers in the social sciences to study complex systems. Whereas early models tended to be very simple and abstractthe KISS (Keep It Simple, Stupid) approach [START_REF] Axelrod | The complexity of cooperation: Agent-based models of competition and collaboration[END_REF]) -the current trend is now towards the development of richer data driven models -the KIDS (Keep It Descriptive, Stupid) approach [START_REF] Edmonds | From KISS to KIDS in Multi-Agent and Multi-Agent Based Simulation[END_REF]. The motivation behind this is principally the use of such models as predictive tools in a decision-support context.

In such models, behaviors of social agents (e.g., individuals, households or institutions) are strongly determined by their attributes, well as by their location in the artificial worlds they populate. In this context, generating synthetic populations of spatially localized agents that conform to the data available on real populations has become a necessity and a concern for most social modelers.

Some recent studies have already proposed methods to generate these types of populations [START_REF] Harland | Creating realistic synthetic populations at varying spatial scales: a comparative critique of population synthesis techniques[END_REF][START_REF] Cornelis | An original synthetic population tool applied to Belgian case: VirtualBelgium[END_REF][START_REF] Swarup | Generating Synthetic Populations for Social Modeling: Tutorial at the Autonomous Agents and Multi-Agents Systems[END_REF]. However, these methods remain underused because of the difficulty to use them with data that is not well-formatted: for example, the simSALUD [START_REF] Kosar | simSALUD: A Web-based Spatial Microsimulation to Model the Health Status for Small Areas Using the Example of Smokers in Austria[END_REF] web-based generator requires input data to fit a template with fixed position columns that describe attribute IDs, weights, etc. Along the same lines, the SPEW [START_REF] Gallagher | SPEW: Synthetic Populations and Ecosystems of the World[END_REF]) R-based generator requires input data to be in form of IPUMS formatted data tables. Therefore, most attempts to generate localized synthetic populations are limited to empirical cases (Arentze et al. 2008, Barthelemy and[START_REF] Barthelemy | Synthetic Population Generation Without a Sample[END_REF] and propose methods with stringent requirements as to the type, availability and format of the data. Furthermore, the classical approach considers spatialization of the population in the same way as the other agent attributes: being a woman or a man is the same as living there or in the next county. Indeed, the spatial information usually comes from the same source of data: either censuses or surveys. Yet, this spatial information is generally not precise enough for many simulation cases. For instance, many residential mobility models work at the building scale [START_REF] Antoni | Simuler les mobilits individuelles -Les enjeux de linformation gographique[END_REF], and therefore require a specific building to be assigned to each household and not just a city or a district. It is thus often necessary to specify or extrapolate a localization from spatial data. In this case, the problem is not simply to infer an inner distribution of demographic attributes but to estimate what actually drives the spatial localization of people. As pointed out by [START_REF] Anderson | Methods for Estimating Population Density in Data-Limited Areas: Evaluating Regression and Tree-Based Models in Peru[END_REF], obtaining a reliable population estimation at a given resolution is a significant challenge.

To our knowledge, there has been only one recent attempt to use generic input data for population generation and localization: the SPEW open source R library [START_REF] Gallagher | SPEW: Synthetic Populations and Ecosystems of the World[END_REF]. However, even with this library, a data sample of a portion of the population from IPUMS is needed in order to generate a reliable synthetic population. Concerning the localization of the population, SPEW proposes some simple methods to uniformly locate entities across a region or along roads, but does not allow to take benefits of multiple layers of information.

To deal with this lack of flexibility, we propose a complete toolkit known as Gen*, which allows modelers to generate a realistic synthetic population even with not-classically formatted data and to localize it at the required scale from one or several layers of geographical data. This toolkit is developed as an open-source Java library. An extension of the GAMA platform [START_REF] Grignard | GAMA 1.6: Advancing the Art of Complex Agent-Based Modeling and Simulation[END_REF]) and a Kepler workflow package that make it possible for non computer scientist to generate a spatially explicit population with Gen* through user friendly interfaces are available as well. In what follows, we focus on the the localization features of Gen*.

The article is structured as follows: Section 2 presents the related studies concerning the spatialization of synthetic populations. Section 3 is dedicated to the presentation of our toolkit. More precisely, we give an overview of the global architecture of the toolkit, and we present the spatialization methods. A case-study of the city of Rouen (France) is described in Section 4. Finally, Section 5 proposes a conclusion and perspectives for this work.

Context

In many agent-based social simulations the geographical environment has to be taken into account with a granularity that depends on the objectives of the model and the geographical data available. Simple models can rely on conceptual environments, i.e. continuous or discrete space. However, applied and descriptive models tend more and more to rely on GIS data, possibly combining geometrical (shapes and set-relationships among shapes) and continuous information (location of objects and agents in the underlying continuous space).

When dealing with GIS data, several methods are classically used in order to integrate census data that describe the population, but they usually do not generate populations of agents. Among the approaches available in this case, the census centroid data [START_REF] Bracken | The generation of spatial population distributions from census centroid data[END_REF] is the most widely used. Dasymetric modeling is also a well-recognized approach for the spatial decomposition of census data in order to increase the spatial resolution of the population distribution [START_REF] Bhaduri | LandScan USA: a high-resolution geospatial and temporal modeling approach for population distribution and dynamics[END_REF]. A combination of the methods described above is often used in an ad hoc manner. This is, for example, the case in the SVERIGE project [START_REF] Holm | The SVERIGE spatial microsimulation model: content, validation, and example applications[END_REF]) that aims at modeling the demographic evolution of Sweden by explicitly modeling (including the localization) each inhabitant of the country.

Two main steps can be used to address the localization problem: the first one referred to as Areal interpolation, consists in translating data from aggregated areas to more precise ones. The second step referred to as Explicit localization, consists in providing individuals with a precise location, that could be a coordinate (x,y) or a geographical object (e.g. a building). In the next sub-sections, we review the available methods that can be used to perform these two steps, and we briefly outline the typical data that are used by modelers to carry out the localization process.

Areal Interpolation Methods (AIM) and the dasymetric method

Areal Interpolation Methods aim at transferring spatial data from one set of spatial units to another [START_REF] Eicher | Dasymetric mapping and areal interpolation: Implementation and evaluation[END_REF]. Redistribution of population data from an administrative unit to a smaller unit (usually a pixel of a grid matching a remote sensing image) is a particular type of areal interpolation known as "dasymetric mapping". This type of method uses ancillary data to refine the population data originally given at a coarser level [START_REF] Li | Fine-scale population estimation: How Landsat ETM+ imagery can improve population distribution mapping[END_REF]. Not only can these methods provide users with a solution to produce more precise population data, but they also make it possible to reduce the errors resulting from the modifiable areal unit problem (MAUP) [START_REF] Su | Multi-layer multi-class dasymetric mapping to estimate population distribution[END_REF].

The redistribution of populations from census units to a finer scale using dasymetric mapping is a two-steps process. The first step involves the fitting of a statistical model like a linear regression model (Li andWeng 2005, Reibel and[START_REF] Reibel | Areal interpolation of population counts using preclassified land cover data[END_REF] or a random forest model [START_REF] Stevens | Disaggregating census data for population mapping using random forests with remotely-sensed and ancillary data[END_REF]. The second step of the dasymetric process consists in applying the statistical relationships resulting from the model fitting at the finer scale. This final step involves some extra normalization processes because of the possible floating point or negative results from the statistical model.

The use of dasymetric mapping methods requires at least two spatial datasets: one containing the aggregated population data and one (or more) containing ancillary data used by the algorithm to disaggregate population data. Original coarse-level population data are usually vector-based census data representing a set of administrative units [START_REF] Yang | Spatial improvement of human population distribution based on multi-sensor remote-sensing data: an input for exposure assessment[END_REF][START_REF] Briggs | Dasymetric modelling of small-area population distribution using land cover and light emissions data[END_REF]. Ancillary data are usually raster-based datasets resulting from the pre-processing of remote sensing images [START_REF] Wu | Population estimation methods in GIS and remote sensing: a review[END_REF]. Most studies use pre-classified land cover data [START_REF] Reibel | Areal interpolation of population counts using preclassified land cover data[END_REF]Agrawal 2007, Su et al. 2010). Various kinds of data can be used, including unprocessed spectral bands [START_REF] Li | Using Landsat ETM+ imagery to measure population density in Indianapolis, Indiana, USA[END_REF], land surface temperature data [START_REF] Li | Fine-scale population estimation: How Landsat ETM+ imagery can improve population distribution mapping[END_REF], elevation data [START_REF] Su | Multi-layer multi-class dasymetric mapping to estimate population distribution[END_REF], night light emission images [START_REF] Yang | Spatial improvement of human population distribution based on multi-sensor remote-sensing data: an input for exposure assessment[END_REF], vegetation indexes [START_REF] Li | Using Landsat ETM+ imagery to measure population density in Indianapolis, Indiana, USA[END_REF], etc. Some studies have used additional vector data as ancillary data such as road networks [START_REF] Su | Multi-layer multi-class dasymetric mapping to estimate population distribution[END_REF].

Explicit localization methods (ELM)

AIM can help to go deeper into population spatial allocation but is not in any way a method to provide a location for individuals. It targets the population as a whole or identifiable subgroups, whereas agent-based simulation requires each individual agent to be localized. To go even further, ELM propose to localize entities across space, based on the most precise population density data which can be built from the AIM process or derived from raw observation data. If no information at all is available, the challenge is to locate agents in the world by following some specific rules defined by the modelers.

There is no methodology to explicitly determine the localization of agents in a synthetic population: one must rely on rule of thumb algorithms. In some cases, the localization of agents is just randomly chosen [START_REF] Wheaton | Synthesized population databases: A US geospatial database for agent-based models[END_REF]. In other cases, roads [START_REF] Gallagher | SPEW: Synthetic Populations and Ecosystems of the World[END_REF] or buildings [START_REF] Zhu | Synthetic Population Generation at Disaggregated Spatial Scales for Land Use and Transportation Microsimulation[END_REF]Ferreira 2014, Fosset et al. 2016) could be used with different spatial allocation techniques.

If most of existing works rely on a ad hoc approach to carry out this localization process, some of them propose a generic process that can be adapted to different case studies and data. Amongst them, SPEW proposes three different localization methods based on the same general approach: use a given probability distribution over a set of geographical objects (regions, roads...) to sample agents according to this distribution. The difference between these three methods comes from the used probability distribution and the type of geographical objects:

• uniformly across a polygon (region)

• uniformly along a polyline (road)

• according to a given spatial distribution of agent locations (weighted polygons or lines).

In addition, as it is sometimes necessary to link agents with spatial objects with which the agents may interact (schools, workplaces...), SPEW proposes a simple method based on a gravity model to assign agents to geographical objects.

The methods proposed by SPEW are adapted when few data are available (just a shapefile of regions or roads), but cannot take benefit of many possible sources of data that can improve the localization process such as density map, expert knowledge or satellite imagery. In the next section we present our generic proposal to localize a synthetic population using the Gen* toolkit.

The Gen* toolkit

General architecture

The Gen* library consists of four modules, each one responsible for a portion of the population synthesis: gospl is responsible for the population generation, spll make it possible to localize the synthetic entities of the generated population and, finally, spin allows us to connect entities to setup a population network. While modules are independent, they all rely on the core module which defines the higher-order abstraction for populations, entities, their attributes and associated values. Hence, a synthetic population is a collection of entities made up of identical attributes but each entity has its own vector of values. Thanks to a common reference to this higher order population definition, the API architecture provides a way to localize or connect, in any order, entities of any synthetic population that comply with the interfaces define in core. In the next subsection, we detail the content of the module that is responsible for population localization.

Spll: Synthetic population localization library

General scope

Spll make it possible to spatialize a synthetic population defined by Gospl (generated with the algorithms provided by Gospl or directly loaded from a csv/xlsx file). The configuration of the population localization is defined through a configuration file that harmonizes the input data. These data should at least contain a geographic file that specifies the spatial objects on which Spll will locate individuals. It can also contain other data such as matching data to provide a link between individuals of the population and geographic objects, for example the number of people per administrative area. The user can provide other data that can help the localization process, like land use raster or road vector file. In addition to the localization of individual, Spll make it possible to link each agent to a set of geographical objects (workplace, schools...). An unlimited number of Linkers can be setup to optionally bound synthetic individual with spatial objects (dashed arrow) Figure 1 describes the conceptual process underlying Spll. As illustrated in the diagram, localized populations is achieved through the nesting procedure (center left). This consists of linking each individual with a nest, i.e. a spatial object in which the individual will be located, and by defining a location in this nest. We chose to locate individuals in parameterized spatial objects in order for each user to define the most suitable location according to the model needs: the nests could be buildings, cells of a raster file or even just x and y coordinates. In order to help the nesting process, Spll can optionally generate a contingency map or a density map from different sources of data, either through raw input data -referred to here as GIS observation at the bottom right -or AIM based generated data at the center right -using for example linear regression. The main nesting phase is take in charge by a linker which encapsulate a spatial distribution and spatial constraints. Those last are constraints based on GIS data: for example, a maximum distance from which to localize individual along roads or within a given distance from a point of interest (POI). Spatial distribution is a function that assigns weights to each nest candidates, and can be setup with GIS data (GIS based observation) and rules to compute weights, e.g. uniformly or based on the area of spatial object. The linkers can also be used to bind individual with spatial objects, for example chose a workplace or a school for an individual agent. Finally, Spll can export the spatialized population in a GIS file.

Spatializing a synthetic population with Spll

The general localization approach is designed in order to be as adaptable as possible to all case-studies. Thus, the only mandatory step is the specification of a geographical data file (vector or raster) to locate the agent inside -i.e. GIS observation. Each individual of the population will be located in a spatial object defined by this input file. For example, individuals can be assigned to buildings taken from a building shapefile. If no spatial object are defined, hence individual will be assigned to the world itself and will have a precise x and y location in it. Spll provides two ways to improve this input observation data: individuals matched with areal spaces and estimated disaggregated spatial distribution. In the next two sections, we show how to build those refined observation. Then, we illustrate how the nesting process works by introducing the spatial distribution function and spatial constraints. Finally, we describe the ability of spll to link individual with any spatial object using linkers.

3.2.2.1. Matching population and space. This optional step can help to distribute each individuals into custom spatial areas. It has two implications: to have a file that describes some spatial areas as input as well as a means to connect individuals with these areas. This allocation should yield a population count per spatial object or a key ID code for the spatial objects to match with individuals. Most of the time, this space will be administrative areas with either a density or an ID. However, it is not limited to this case, and can be a raster density file or any other GIS format file with a population count or pairwise ID. In this latter case, Spll is able to automatically match individuals and space given that both ID key attributes have been bound in the configuration file. As an advantage, this procedure allocates actual synthetic individuals, whereas density allocation only provides a population count. They both end up defining a number of people per spatial object of a space file and this record can then help for the nesting phase or for the interpolation process.

Areal interpolation.

Based on raw observations provided by the previous matching step, Spll make it possible to further specify areal scale density or count using any interpolation technique such as regression. Like in the previous matching phase this interpolation step is optional. The core idea of areal interpolation is to estimate a more precise spatial distribution using two inputs: first, an observation to be explained, which will be the densities or the contingencies per area, and second, a set of explanatory data, which will be spatial objects of finer granularity linked to a numerical or categorical feature. Therefore, Spll provides several estimation techniques to explain observations based on provided spatial variables: the result of the algorithm takes the form of a formula and can then be applied to compute disaggregated areal population density. The current version of Gen* uses linear regression which is a standard estimation technique in dasymetric mapping (see section 2.0.1). It does well when coupled with pretreated land cover or land use data [START_REF] Li | Fine-scale population estimation: How Landsat ETM+ imagery can improve population distribution mapping[END_REF]. On the contrary, when used together with raw satellite imagery, it performs poorly because numerical band values usually do not interact linearly with the population allocation [START_REF] Stevens | Disaggregating census data for population mapping using random forests with remotely-sensed and ancillary data[END_REF]. To overcome this problem, we plan to implement other techniques such as the generalized non-linear regression and other AI-based estimation technique (see section 5).

A major drawback is that the output of areal interpolation techniques is usually in the form of a floating and possibly negative number. The estimated value should therefore be normalized to fit the population count, i.e. a positive or null integer. Some computation procedures have been proposed in the literature to overcome these two issues [START_REF] Yuan | Remodeling census population with spatial information from Landsat TM imagery[END_REF]. First, we chose to upscale values to a given threshold, with a default value of 0 for the population. We then uniformly reduced other values to return to the original overall sum as a compensation process. Second, we used a heuristic algorithm to round off floating values in order to obtain a positive integer. The process iterates over all spatial objects reducing (truncating) or increasing (rounding off) associated floating values, while maintaining the absolute difference between the overall sum and the expected overall population count as close to zero as possible.

At the end, the areal interpolation gives an estimated areal count or density for each explanatory spatial object. Depending on observation provided by the matching step, each cell can refer to any individual or a specific set of individual bound to a more aggregated spatial area. In both cases, the resulting dasymetric map will be used as a spatial constraint in the linker used by the nesting process.

3.2.2.3. Nesting process. Spll provides a set of algorithms -i.e. nesting process -to bind spatial objects -i.e. nests -and individuals of the population. The process is driven by a linker that encapsulate optional constraints and a distribution function. It also must be initialize using any input data file that describes the nests. When an individual has been placed into his/her nest, a "location" inside the assigned spatial object is then computed. In the following paragraphs we detail how Spll link individuals to spatial objects and provides individuals with a particular location in this nest.

As a minimal requirement, the nesting process needs an input file that describes the spatial object in which to locate an individual. If this file is the only constraint given to the algorithm, then each individual of the population will randomly choose a nest among all the possible ones according to the distribution function. In order to obtain a more realistic spatialization of the individuals, the user can define spatial constraints to filter the possible nests. Spll integrates three basic types of spatial constraints: geometric, contingency and density constraints. The first one is used to bind the spatial objectsthe geometries associated with these objects -and the individuals; while the two others rely on matching a predefined number or density of entity per nest. If the constraints do not make it possible to find an appropriate nest, they can be relaxed. More precisely, for each constraint, a relaxation process can be defined as well as a maximal relaxation function that defines whether the constraint can be relaxed yet or not.

Geometric constraint specifies that individuals have to choose a nest inside a given geometry. Typically, this constraint make it possible to link geographic data with one of the individual's attributes which is similar to the previous optional match and binding step. For instance, if we have information about the district where each individual is living (through a district attribute defined by Gospl for example), this constraint can be used to restrict the list of possible nests to those located inside the given district. If there are none inside the reference geometry, the relaxation will allows the constraint to also include the nests that are close to the reference geometry. The notion of closeness depends on a distance variable that will increase as the constraint is more and more relaxed until reaching a given user defined threshold. A geometric constraint could also be used if we want to specify that an individuals location should be close to a roads or near a POI like in the SPEW toolkit [START_REF] Gallagher | SPEW: Synthetic Populations and Ecosystems of the World[END_REF].

Contingency constraint specifies the maximum number of individuals for one or more geometries. Typically, this constraint comes as a demographic count per geographic area. For instance, we can compute a contingency grid from diverse spatial layers that will specify the expected number of individuals per cell. This constraint will filter the possible nests to only keep the ones for which the max contingency constraint is not yet reached. If all nests are full, the relaxation will increase the capacity of the nest whereas the maximal increase is given by a user defined threshold.

Density constraint specifies the maximum density of individuals inside one or more geometries. This constraint works like the contingency one but with density rather than contingency. The relaxation process relies on identical mechanisms and with the same relaxation threshold variable.

Depending on the application, it can be more interesting to first try to relax the geometry constraint before the contingency one and vice-versa. To specify this notion of priority of relaxation, the user can define a priority for each constraint. An ordering process is also performed when searching for a proper nest in which to locate individual, for which each constraint will successively filter the list of possible spatial nests. When several nests satisfy all the constraint, Gen* uses a distribution function to draw one to bind individual with. We provide user with several predefined distribution functions:

Uniform distribution (default distribution function) specifies that all the nests has the same probability to be chosen. [START_REF] Gallagher | SPEW: Synthetic Populations and Ecosystems of the World[END_REF].

Area distribution specifies that the probability of a nest to be chosen is a linear function of its area.

Capacity distribution specifies that the probability of a nest to be chosen is a linear function of its capacity. The capacity of the nest is updated each time it has been drawn to be bound with an individual.

Note that if the list of predefined constraints or distribution functions does not cover the user needs, he/she can define its own new types of constraint by implementing a Java interface, and distribution function by overloading a Java method. For instance, if the user wants to locate the agents in buildings according to the income of the agents and the standard of buildings, he/she will just have to implement a new distribution function that defines the probability for an individual to chose a building according to its standard and the individual's income.

When each agent has been associated with a spatial object, the next procedure is to assign a precise x/y coordinate location inside it. Spll proposes two basic algorithms to do so: centroid and random. First, the centroid mechanism retrieves the centroid of the object. For polygons, it is computed as the weighted sum of the centroids of a decomposition of the area into triangles. For lines, the centroid computes the average of the midpoints of all line segments weighted by the segment length. Finally, for points, the centroid is the average coordinate for all points. Second, the random mechanism retrieves a random point in the object. For a polygon, the algorithms consists in selecting random points in the object envelope until one of the point overlaps the object. For a line, a random value is retrieved between 0 and the perimeter of the line. The point corresponds to the one at this length from the first point following the line. Finally, in the case of points, one of them is chosen at random. At the end of the localization process, each individual from the synthetic population should have a nest -i.e. a spatial object -and a precise x/y coordinates inside it. They can be accessed using attribute accessors in the spatialized population model or exported as a GIS formated file.

3.2.2.4. Geographical object linking. Similarly to SPEW, Gen* makes it possible to link a set of geographical objects to each individual. This linking process is very close to the localization process (in fact, the nesting step is a spatial case of linking): the principle is to choose for each linking function (e.g. workplace, school), a geographical object among a set of possible ones to link individual with. To do that, Spll uses a linker : that is a combination of spatial constraints and a spatial distribution function. However, as it is often necessary to link individuals to close geographical objects, Spll offers the possibility to define a distribution according to the distance between the agent location and the geographical objects. The toolbox provide several predefined distribution function to be readily used:

Distance-based distribution specifies that the probability of a geographical object to be chosen by an individual is a negative linear function of the distance between them.

Gravity model specifies that the probability of a geographical object to be chosen by an individual depends on the mass of the geographical object divided by the distance between them. By default, the distance corresponds to the euclidean distance between the individual and the geographical distance. Two basic functions are proposed for the mass: the sum of distances between the geographical object and all individuals, and the number of individuals within a given buffer around the geographical object. This second function with a buffer of 0.0 corresponds to the one used by SPEW.

All previously detailed spatial distribution functions and constraints can be re-used to drive the linking process. For example, the user can link individual to places based on spatial object area or capacity, while defining a limited geographical zone as a constraints to filter possible spatial objects to bind individual with.

3.2.2.5. Discussion. Spll proposes a flexible tool to localize a population based on the definition of spatial constraints and distribution functions. To compare with SPEW, Spll can do everything SPEW does, but allows to go further. Indeed, the possibility to define spatial constraints allows to integrate new knowledge in the spatialization process. In addition, Spll allows to use Areal interpolation techniques to use information coming from different sources to improve the spatialization. At last, the genericity of the proposed spatialization approach (independent of the type of geographical object concerned) allows to easily extend it through the definition of new spatial constraints and distribution functions. It makes it possible to adapt the process to available input data -the only mandatory piece of data is the space itself -and to adjust the output to specific needs of modelers -spatial precision as well as custom distribution function and spatial constraints.

Case study: the city of Rouen, France

The Gen* library comes with documented examples, that are all freely available on the Web (see closure section that provides useful links). In this section we present one of this example in detail: the population localization of the city of Rouen in France.

Context of the work

Rouen, which is the capital of the Normandy region in France, is a city of 110,755 inhabitants built on both sides of the Seine River. Many chemical industries are located near its city center which makes a potential accident particularly problematic. In order to be able to evaluate the impact of an accident and the possible evacuation scenario that could be implemented, a model was produced [START_REF] Czura | MOSAIIC: City-Level Agent-Based Traffic Simulation Adapted to Emergency Situations[END_REF]. In this model, the properties and the localization of the inhabitants are particularly important. In the following sections, we will review several options that have been proposed for the model developed in [START_REF] Czura | MOSAIIC: City-Level Agent-Based Traffic Simulation Adapted to Emergency Situations[END_REF].

Population spatialization

In this example, we consider that we have generated a population with Gen* from the data available from the French National Institute of Statistics (INSEE) that cover a wide number of demographic and social characteristics. The location is given as an aggregated count of people per IRIS code, the main and most precise statistical areal unit in France. We consider three scenarios corresponding to different types of data available:

• localization inside the IRIS geometries (polygons)

• localization along the roads (polylines)

• localization inside buildings, with information about the IRIS geometries, satellite image and spatial constraints.

Scenario 1: localization inside the IRIS geometries

As a preliminary step, we defined a configuration file that specifies how to spatialize the population. In particular, we specify the geographical data that will be used: the vector files for IRIS geometries. The localization process started using the most precise spatial demographic data: the population of each IRIS in the city, depicted in Figure 2.

Based on this information we can extrapolate a precise location for people inside these areas. To do so, Gen* matches people and IRIS. The matching process is based on the key attribute of home location which is part of the generation process: each individual has an IRIS ID that allows him/her to be located in the city. Once this match is made, we know the list of generated entities living inside each IRIS area. Once the list is defined, we let Gen* give a precise location to each individual of the population inside the corresponding IRIS geometry. In this simple example, we use the random location assignment algorithm to do so. Figure 3 shows an example of result obtained. The overall localization process took less than 3 sec. to be accomplished. This include matching, assigning and finding a location within IRIS for each of the 110 688 individuals of the city, which is totally acceptable. Like for the previous scenario, as a preliminary step, we defined a configuration file that specifies that we want to use a road shapefile for the spatialization. However, as in this scenario individuals have to be located along the roads and not directly on them, we use as a first step a built-in function of Gen* that defines proxy spatial object to locate individual in: it allows to compute for each geographical object of a shapefile a new geometry that corresponds to the area that is between a minimal and a maximal distance to the objects. For this example, the individuals are located at distance between 2.0 and 10.0 meters of the roads. Once this geometry have been defined for all the roads, we just applied the normal localization process of Gen* with the spatial distribution function based on area. This allow us to favor the selection of long roads rather than short one for the nest selection. Figure 4 shows a snapshot of the result we obtain using road based nesting for the city of Rouen. The overall computation time was 99 sec., with 87 sec. dedicated to the computation of the area around the 4700 roads.

Scenario 3: localization inside buildings with spatial interpolation

In this last scenario, we specified in the configuration file that three sources of geographical data will be used: land use pre-processed satellite imagery, vector files for IRIS and building geometries. The first one was performed using raw satellite imagery from Landsat 8, with a raster cell of 30m 2 . We used a well known clustering algorithm known as "isodata" [START_REF] Memarsadeghi | A fast implementation of the ISODATA clustering algorithm[END_REF]) to compute height different categories of land use: very high, high and low density buildings, high and low green space, water, road and bare land space.

As for the first scenario, the localization process started using the IRIS shapefile to determine the list of generated entities living inside each IRIS area. Then, this spatial distribution is used as the objective of an interpolation process and the land use category of the raster file described above as explanatory variables. Spatial interpolation terminates with a function that allows Gen* to determine a desegregated population distribution within each IRIS. More precisely, the algorithm results in a number of people for each 30m 2 raster cell used as an explanatory variable. While aggregated, these computed contingencies fit the known population count at the IRIS level. Map 5 gives an overview of the resulting contingencies for the valid cells (excluding no-data and null values for water), with lighter cells standing for contingency equal to 1 individual (low The next step consists into giving a precise location to each individual of the population: this is done using information on buildings. As we can see in Figure 6, thanks to the input data about buildings, no individual has been placed into water, on bridges or on roads. The density constraint in 30m 2 cells is released when no building is present within the defined spatial entity. In this case, the closest building is chosen to locate the individual. The location process also takes a limited number of people within the buildings: one cell can receive more than it is supposed to hold and, on the same level, one building can exceed the number of individuals expected to be within it if needed. These two constraints are released one after the other, to allow the location algorithm to always find a solution with alternate concessions on density requirements. To sum up, the localization process is composed of three steps: the matching between entities and IRIS, the spatial interpolation to compute density within raster cells and, finally, the localization of each entity within the buildings. The entire process takes less than 25 sec. to be accomplished.

Geographical object linking

As introduced in the previous step, Gen* allows to link geographical objects to individuals. The nesting process is in itself an example of linking, with a specific status assigned to the selected spatial object. The use of linkers make it possible to bind any individual of the synthetic population with an unlimited number of spatial object. As a major difference with the nesting process, the created link between individual and spatial object will not determine the individual location. However, it can drive dynamic localization process during simulation, like to move to linked school or workplace from current location. For this example, we assign a school to each children of the population (here, individuals with an age lower than 15). We used the gravity model for the linking process. Mass of spatial object candidates (i.e. the schools) is defined as the sum of distances between the geographical object and all individuals. Figure 7 shows the results obtained: we clearly see in this map the impact of the distance for the school assignment. The linking process favors to link children with closest schools while also taking into account the number of children within the school. The linking process took less than 0.6 sec. for the 10 501 children.

Conclusion

This article presents the localization component of the Gen* toolkit. It allows to localize a population, i.e. gives to each entity of the population a spatial nest and a location, through the definition of spatial constraints and a distribution function. In addition, Gen* provides tools to increase spatial resolution of the population distribution through estimation technique, as well as methods to link geographical several spatial objects to individuals of the synthetic population. As shown by the case-study, Gen* make it possible to build a credible spatialized population with many options and within a respectable computation time.

In order to validate the genericity of Gen* and its adaptability to all contexts, we plan to carry out other case-studies in the near future: the generation of the population of Bangkok, of Marrakesh and of several cities in France and Vietnam. These case-studies will allow us to evaluate the capacity of Gen* to deal with different population scales (from hundreds of thousands to several millions) and different data sources (both for census data and GIS data).

In terms of development, we plan to add many new features to Gen* and in particular to Spll. A first one concerns the addition of new regression techniques like random forest, which seems very promising for complex localization problem and with raw input satellite imagery [START_REF] Stevens | Disaggregating census data for population mapping using random forests with remotely-sensed and ancillary data[END_REF]. We plan as well to add new types of spatial constraints and distribution functions to improve the flexibility of Spll. In addition, we want to improve the performance of Gen* in terms of computation time by allowing to take benefit of parallel computation. Finally, we plan to continue to work on facilitating the use of Gen* by non-computer scientists by improving the GAMA plugin and the Kepler version of Gen*.

To conclude, Gen* is still an active project. Its goal is not only to develop a library to be used by modelers, but to create a community concerned with the synthetic population generation as well.

Figure 1 .

 1 Figure 1.: The Spll process to localize a synthetic population. It starts at the right with one input data (GIS observation) or optional estimated data (AIM) file to feed the main step: the nesting process. It grants individual entities a unique location based on a spatial distribution function and spatial constraints encapsulated in a dedicated Linker. An unlimited number of Linkers can be setup to optionally bound synthetic individual with spatial objects (dashed arrow)

Figure 2 .

 2 Figure 2.: Geometries of IRIS in the city of Rouen

Figure 4 .

 4 Figure 4.: Scenario 2: precise x and y coordinate population location for a small part of the city of Rouen

Figure 5 .

 5 Figure 5.: Resulting densities of the spatial interpolation process

Figure 6 .

 6 Figure 6.: Scenario 3: precise x and y coordinate population location for a small part of the city of Rouen

Figure 7 .

 7 Figure 7.: Linking between children and schools (similar color): the circle represents the children and the buildings with red border the schools

Acknowledgment

This work is part of the GEN* ("Generation of Synthetic populations") research project funded by the French National Research Agency.

Website references

The Gen* library suite (API, templates and GAMA plugin) could be find on Github 1 while the GAMA platform has its own website. 2 1 https://github.com/ANRGenstar 2 http://gama-platform.org