Keywords: activity-based analysis, time use, travel time

The paper exams the relationships between travel and activity times, in 7 travel surveys from 4 French and 3 Swiss cities, observed at two different periods. First, we test proportional assignment of total daily available time to activities (including transport). Second, proportionality is tested between (1) daily travel time of a given purpose with respect of the daily activity duration and (2) the trip time associated to the duration of the activity at destination. Only daily leisure time and daily travel time are fixed proportion of total daily available time. At disaggregated level, the trip duration do not show proportionality with activity duration. Finally questioning the proportionality and the linear adjustment, we regress the travel time budgets using duration models. This methodology is particularly adapted to the duration analysis and leads to non-linear relation between travel time and activity times. Leisure and shopping activities exhibit increasing and convex travel time intensities.

Introduction

Many studies have analysed travel times and activity times. For example it seems natural to observe a positive correlation between travel time and the time of activity at a given destination. The search for simplified representations of travel times began with the stability hypothesis in the 1970's, known as the "Travel-Time Budgets paradigm" [START_REF] Goodwin | The usefulness of travel budgets[END_REF], the "Brever law" [START_REF] Hupkes | The law of constant travel time and trip-rates[END_REF], or the "Zahavi's hypothesis" [START_REF] Zahavi | The 'UMOT' Project. the U.S. Department of Transportation and the Ministry of Transport of[END_REF]. Less restrictive hypotheses have been proposed on travel times. One of them assumes that a travel time intensity can characterise activity times, therefore making travel times fully determined by activity times. It has been examined at the daily level on time expenditures. [START_REF] Kitamura | A comparative analysis of time use data in the Netherlands and California[END_REF] test the "principle of proportional assignment" [START_REF] Beckmann | A critique of entropy and gravity in travel forecasting[END_REF]. Under this principle, each activity time represents a fixed proportion of the total available time.

"The ratio of the amounts of time assigned to two activities is invariant regardless of the total amount of time available" (Kitamura et al., 1992, p.135). Their results support that total daily travel time is proportional to the amount of daily available time.

More generally, concerning specifically activity, relationships between travel time and activity duration have been analysed and estimated in different estimation frameworks, such as linear model, structural equations model, duration model, etc [START_REF] Van Wissen | A simultaneous dynamic travel and activities time allocation model[END_REF][START_REF] Hamed | Modelling travellers post-work activity involvement : toward a new methodology[END_REF][START_REF] Golob | A model of activity participation and travel interactions between household heads[END_REF][START_REF] Goulias | Perceptions in mode choice using the situational approach: a trip by trip multivariate analysis for public transportation[END_REF][START_REF] Kitamura | The effects of time of day, activity duration and home location on traveller's destination choice behaviour[END_REF]Narayanan 1998, Ma andGoulias 1998;[START_REF] Levinson | Space, money, life-stage, and the allocation of time[END_REF][START_REF] Lu | Socio-demographics, activity participation and travel behaviour[END_REF][START_REF] Joly | Stability of regularity of the daily travel time in Lyon? Application of a duration model[END_REF]. Some propose an equivalent concept to proportionality: travel time intensity. Under this hypothesis, travel time assigned to activity participation is related to activity duration, with the proportional form representing a particular case of the possible relationships.

In this context, [START_REF] Dijst | Travel Time Ratio: the key factor for spatial reach[END_REF] and [START_REF] Schwanen | Travel-time ratios for visits to the workplace: the relationship between commuting time and work duration[END_REF] propose and analyse travel-time ratio for work activities. Estimating a structural equations model for the activities duration [START_REF] Van Wissen | A simultaneous dynamic travel and activities time allocation model[END_REF], [START_REF] Golob | A model of activity participation and travel interactions between household heads[END_REF], [START_REF] Lu | Socio-demographics, activity participation and travel behaviour[END_REF] analysed relationships between travel time and activity duration. Some estimators can be interpreted in terms of travel time intensities for different activity types. For example, Golob and McNally (1997, p. 185) obtained a travel time intensity for maintenance that indicates that one hour of out-of-home maintenance activity requires on the average 7.8 minutes of travel time.

Studies on urban mobility have shown that spatial characteristics have a significant effect on mobility and travel time [START_REF] Timmermans | Time allocation in urban and transport settings: an international, inter-urban perspective[END_REF]. Then, we propose to explore time use deduced from numerous households mobility surveys, in search for distinct relationships between activities duration and the travel time associated. This paper presents results of a comparative study of Swiss and French daily travel times based on mobility surveys data of seven cities (Bern, Geneva, Grenoble, Lyon, Rennes, Strasbourg and Zurich) observed at two different dates. The cities chosen were guided by the disposability of the data and the following criterions:

• The form and spread of the peri-urbanisation process

• The level of equipment in heavy network of public transport

• The transport and urban policy managing the car accessibility Swiss cities are clearly more transit and pedestrian oriented in terms of infrastructure, transport networks and transport policy. In French cities, transport policies, such as restriction of automobile access to the centre and limitation of parking facilities, appeared relatively recently in comparison to Swiss.

Socio-demographic characteristics of individuals and household and indicators of their mobility on a given weekday are collected. Using, starting and stopping times and the types of activities at the origin and destination, the one-day out-of-home activities diary can be deduced from the first to the last trip of the day.

We focus on the relationships between travel and activity times (work, shopping, leisure). Following [START_REF] Kitamura | A comparative analysis of time use data in the Netherlands and California[END_REF] Travel time budgets are calculated as the daily sum of the travel time on all the trips during the day, regardless of the purposes and the transport mode.

Following Goodwin (1982), the term "budget" is used to introduce the rationality, that is supposed in the process of allocation of time to travel. Three measures of travel times can be distinguished:

• the travel time cost of a trip, or the time it takes to reach a localisation given the transport system and the urban structure;

• the travel time expense, corresponding to the time associated to the decision to participate to an activity;

• the travel time budget, or the time rationally allocated by individuals to travel.

The travel time budgets can be viewed as the solution of the competition for the scarce resource of time between activities.

The study of trip time appears to be more complicated, because it is related to the decision to participate in or to renounce an activity. The aggregate measure of time at the daily level permits us to reduce the complexity of the activity behaviour.

Independence of the Travel Time Budgets

Depending on data, previous studies of travel times budgets have used numerous definitions and calculations. The early studies in the 1970's could only calculate daily travel time on car or motorised trips for work (Szalai, 1972;Zahavi, 1973[START_REF] Zahavi | The 'UMOT' Project. the U.S. Department of Transportation and the Ministry of Transport of[END_REF]Hupkes, 1981;van der Hoorn, 1979). With higher quality of mobility data collection, travels time and travel time budgets are now calculated for all modes and all purposes.

The analysis of travel time budgets within the past decades leads to the distinction of two hypotheses. The first suggests stability in time and space of the travel time budget. A travel time budget of one hour per day is supposed to be a relatively good approximation of the mean travel time budget observed in cities from different country and different time frames. Considering the various urban situations observed, this "strong hypothesis of stability" of travel time budget gains sense. Regardless of the economic development, the level of infrastructure, the urban context, etc., the travel time budget remains approximately the same, whether in African villages or European, American, and Japanese cities and agglomerations (see [START_REF] Schafer | The Future mobility of the world population[END_REF], for an actualisation of this observation).

Indeed, this assumption is irrelevant at a disaggregated level. Analyses of travel time budget at an individual level reveal numerous relationships between travel time budget and other variables, such as socio-demographic attributes, mobility characteristics, or urban contexts (for a review of these effects, refer to Chen and [START_REF] Mokhtarian | TTB or not TTB, that is the question: a review and analysis of the empirical literature on travel time (and money) budgets[END_REF][START_REF] Joly | L'allocation du temps au transport -De l'observation internationale des budgets-temps de transport aux modèles de durées[END_REF]. The question of the regularity of these relationships constitutes the "weak hypothesis of regularity" of travel time budgets. Depending on the data and methodologies, variables affect travel time budget on different ways.

Nevertheless, some relationships appears relatively stable between studies, as for example, the negative effect of the presence of children under 5 years, the weekly cycle of travel time budget lower on Monday than Friday, etc.

The dimensions that can be introduced at the daily level limit the search for regularities in travel time budget. For example, the mode of transport or the urban context of activity participation locations is difficult to describe easily at the daily level. Furthermore, some dimensions exceed the daily level, as the planification of activity participation over weeks or months.

Here, we pay special attention to the relationships of travel time budget with activity time budgets. The consideration of the daily process of travel time allocation appears to be an interesting approach to adapt and to test proposition of the microeconomics theory of the allocation of time, which principally considers the competition between activities for the daily available time (or on longer periods) (Jara-Diaz, 2003).

Proportional assignment, travel time ratio, and travel time intensity hypotheses

Analyses of the relationships between travel time and activity time have produced three distinct concepts.

First, the proportional assignment of available time to travel has been proposed by [START_REF] Beckmann | A critique of entropy and gravity in travel forecasting[END_REF]. It has been revisited and empirically studied by [START_REF] Kitamura | A comparative analysis of time use data in the Netherlands and California[END_REF], based on Dutch and Californian time use surveys. The daily travel time, as any other daily time of activities, is supposed to represent a certain proportion of total daily available time. The activity times are calculated excluding associated travel time and the total available time is 24h less work duration.

It leads to the calculation of the following ratio:

( ) β , X f T ATB j =
where ATB j is the time budget of an activity j, T is the daily total available time. This

proportion is supposed to depend on X, variables and β, parameters. Under the hypothesis of proportional assignment, this ratio is a constant, depending on the individual's characteristics and other variables.

Second, the travel time ratio is proposed by [START_REF] Dijst | Travel Time Ratio: the key factor for spatial reach[END_REF]. They suggest that the participation to an activity of a given location is the result of the trade-off between travel and activity time. Individuals participate to activities that satisfy an acceptable ratio. The travel time ratio is supposed to reflect this trade-off and is expected to exhibit regularities.

The travel time ratio is computed for an activity participation, as the ratio of the total associated travel time (round trip) to the sum of the trip time and activity duration at the destination:

a t t a T T T + = τ
Where the travel time ratio for activity a is noted τ a , T t indicates the travel time and Third, the travel time intensity of an activity then represents the supplementary of travel time implied by an increase in activity duration of a given type. It can be found in certain models interpretations, as in [START_REF] Golob | A model of activity participation and travel interactions between household heads[END_REF], [START_REF] Lu | Socio-demographics, activity participation and travel behaviour[END_REF], or in other microeconomics models. To our knowledge, Chen and Mokhtarian (2006) are the first to proposed a "price time of activity" in a microeconomic time allocation model.

A general formulation can be:

j j j j j T TT T TT TTI ∂ ∂ ≅ ∆ ∆ =
Where the travel time intensity of the activity type j (TTI j ) is the relative variation (or the derivative) of the travel time associated to an activity type j with respect of the duration T j of the activity of type j.

Most of the models that estimate the travel time on activity duration lead to interpret estimators in terms of travel time intensity.

The models

Following [START_REF] Kitamura | A comparative analysis of time use data in the Netherlands and California[END_REF], the test of proportionality of time budgets with respect to the total available time (24h -work time) consists of the test of the θ coefficient in the following form: ...

2 2 1 1 β β θ X X T t j =
The ratio of time budget, t j , on available time, T, is a fixed proportion, if θ = 1. This proportion is depending on the X attributes of individuals. The test is performed on the OLS estimates of the logarithmic model: [START_REF] Kitamura | A comparative analysis of time use data in the Netherlands and California[END_REF] The stability of proportionality seems to be preserved by an adjustment mechanism between the travel times for the non-work activities and the commute times. [START_REF] Schwanen | Travel-time ratios for visits to the workplace: the relationship between commuting time and work duration[END_REF] calculate travel time ratio for commute trips by doubling the home-to-work travel time. In Dutch NTS 1998, the mean travel time ratio is 0.105 and the median is 0.085. A majority of individuals then possess a travel time ratio lower than 10% for work activity. This ratio is relatively stable with respect to the socio-demographics attributes. The travel time ratios are higher for the peri-urban residents.

ε β β β θ + + + + + = ... ln ln ln ln ln 3 3 2 2 1 1 X X X T t j
But, the test of the travel time ratio principle is invalid at the daily time allocation level. This concept is valid at the episode activity level. It focuses on the travel time associated with an activity participation and duration. It raises the question of the definition and measure of the travel time for an activity in a non-unique purpose chain of trips.

First, keeping the daily level of the time allocation process, we propose adaptation of the travel time ratio. We calculate the ratio on the basis of the time budgets of an activity and the part of the daily travel time budget associated to this purpose. It is only an approximation of the travel time ratio at the daily level that underestimates or overestimates the daily travel time for a given purpose. Underestimation will appear if the chain of trips is for only one purpose, because our calculation then excludes the trip back home. Overestimation could appear if the travel time associated to an activity in a chain is mostly explained by the other activities in the chain.

The travel time ratio is transform by logarithm and proportionality between travel time and activity time is tested. Then, to test for an activity type, j, the proportionality between the daily sum of travel time associated to the purpose j, TTB j , and the sum of the activity time, TB j , the model is the following:

ε β β β θ + + + + + = ... ln ln ln ln ln 3 3 2 2 1 1 X X X TB TTB j j j
where, θ j is the coefficient to be tested for the activity type j.

Non-linear assignment

The log-logistic model is one of the forms that allow for non-linear relationship. Test of the θ parameter in the model above will indicate the particular case of proportional assignment. Quality of adjustment can be gain by taking into account the non-normal distribution of duration data. In this objective the duration model framework is used to model the travel time budgets. Furthermore, this model estimate a flexible functional form of interaction between modelled time and covariates. Complete presentation of the duration model estimation theory can be found in Kalbfleich and Prentice (1980), [START_REF] Lawless | Statistical models and methods for lifetime data[END_REF], [START_REF] Hosmer | Applied survival analysis[END_REF], [START_REF] Allison | Survival analysis using SAS -A practical guide[END_REF].

The duration model framework assumes a log-linear model form. The logarithm of the duration (T) is estimated by: lnT= g(X,β) + σε where X is the covariable matrix, β is the column vector of parameters associated to covariables and ε denotes the error term which distribution is supposed to be known (for example, usual distribution are normal, logistic or extreme value distribution).

The error distribution choice will imply a particular time distribution.

Usual specification of the covariables transformation function g(.) is:

g(X, µ β) = µ+Xβ
It leads to an easy interpretation of coefficients in terms of variations of the conditional expected value:

E[lnT | X].
Assuming the error distribution, we can deduce the time density and distribution functions (f(t) and F(t)):

( ) ∆ ∆ + < ≤ = + → ∆ t T t P t f 0 lim ) ( and ∫ = ≤ = t du u f t T P t F 0 ) ( ) ( ) (
Interpretation of the duration model estimations is particularly interesting in terms of survival time or survival probability and hazard rate. The survival function is given by the following probability:

) ( 1 ] Pr[ ) ( t F t T t S - = > =
It corresponds to the probability of an activity duration to exceed a given time t, and depends on the assumed distribution function of the error term. The hazard rate is given by the following conditional probability:

( ) ∆ > ∆ + < ≤ = + → ∆ t T t T t P t h 0 lim ) (
The hazard rate corresponds to the conditional probability of interruption of the temporal process at date t+∆, given it has lasted to time t. The hazard can be interpreted as the rate of variation of the survival curve.

The model estimations reveal an interesting interpretation in the median survival time variations with respect to covariables and especially here with the activity duration covariables. Depending on the assumed distribution of error term, median survival time and its derivative with respect of covariables can be deduced. For the classical Weibull model (Weibull distribution for time and extreme value distribution for error term), the median survival time is given by: ( ) ( )

β σ X t - = exp
) 2 ln( 50where σ is the parameter of the Weibull distribution.

The median survival time corresponding to the exponential distribution is obtained by setting σ =1.

For the loglogistic distribution, the median is given by: ( )

β X t - = exp 1 50
The duration model is applied to the travel time budget. Results produces estimates of the impact of covariables (including activity daily time budgets) on travel time and permits to deduced estimated travel time intensity for each activity type.

Section 2: Data The French data originates from the local mobility surveys of each city (Enquête-

Ménage Déplacements).

Restrictions on the availability of the data limit the choice of the cities. The Swiss chosen cities were subject to a local oversampling in the Swiss national survey. The French cities were chosen for their survey periods, which resemble those of Switzerland.

The question of comparability of the different surveys arises regarding survey methodologies and the definition and measure of indicators. Table 1 presents the objectives and the convergence of the methodologies through a series of questions (who is surveyed, where, when, etc).

The French data are constituted of the 8 local mobility surveys, that all follow the unified CERTU methodology. At least 1,500 households are questioned by at-home interview on their composition, motorisation level, mobility and certain local themes.

Swiss information is from the NTS, where 10 regions were oversampled to ensure sufficient information for local analyses. 16,570 households in 1994 and 27,918 in 2000 were questioned using CATI methodology, on their composition, motorisation level, mobility and several local themes.

Divergences appear in the person interviewed. Swiss surveys concentrate only on a reference person older than six years old (2 persons of reference for households with 4 members). Conversely, in French surveys, mobility of all members are recorded.

Swiss interviews are realised during the entire year, while French interviews are only performed between October and May. Swiss surveys include weekends and long distance trips realised at the national level, while French surveys exclusively consider trips on workdays and exclude trips that exit the perimeter of study.

A number of operations were then executed to extract information on the same type of mobility.

Data handling and variable descriptions

In order to ensure comparability, the handling of the data consists in reproducing in Swiss data the conditions of the French CERTU methodology on the following directions.

Zone adaptation

First, Swiss perimeters have been constructed to correspond to the French perimeters.

Second, each perimeter is divided into 3 zone types: centre, suburban and peri-urban, based on the definition given by [START_REF] Jemelin | Pratiques et images des transports: une typologie des comportements modaux à Rennes Métropole[END_REF]. For the French cities, centre zones are composed of communes with more than 5,000 inhabitants per square kilometre and with more than 100,000 inhabitants. Suburban communes possess at least 50% of the residential capacity in collective buildings and are in continuity with a centre commune. The other communes of the perimeter are periurban. For Swiss cities, the classification is an adaptation of the one of the Federal Office of Statistics (Office Fédéral de la Statistique, 2000), which defined centre as an attraction point of the commute trips. The suburban type is defined on the type of buildings and is continuous with centre. The peri-urban zone is composed of the remaining communes (French municipalities).

Purposes adaptation

Swiss and French surveys focus on distinct purposes classifications. To be comparable the purposes are grouped into a simple distinction: work, shopping/services and leisure activities.

Trips selection

Certain trips were systematically excluded from the data sample. Hence, we only preserve mobile persons on workdays that do not exit the perimeter, nor perform trips for professional purposes. Special attention was made to correct or exclude errors on the timing of trips in order to keep only full time diaries. 1994and 2000, Geneva 1994, and Zurich 1994and 2000. The proportional assignment of time to leisure activity is also only valid for the Bern 2000 and Zurich 2000 samples.

The qualities of adjustment are weak, as it is usually the case for linear model of activity duration. By construction of the total time available, the R² is relatively high for the work time budget.

Test of proportionality between travel and activity times

A look on the results of estimations indicates that the daily travel times associated to a purpose is not a fixed proportion of the daily time of the associated activity type (Tables 5 and6). There is no proportionality at the daily expenditure level between the daily time budget of an activity and the daily travel time associated. Individuals, household and trip characteristics are significant variables. We find regular results for most of them: higher work trip time for man, for non-motorised person, for public transport users, centre inhabitant; lower work trip time for households member with children, for driver licence holder; lower shopping trip time for worker, for no child households member, for the young, for driver licence holder and on Monday. Few regularities appear in the trip time for leisure. Only the negative effect of worker and the positive effect of age above 50 are significant.

Non-linear Transport intensity -Duration model

Duration model estimation of the travel time budget produces non-linear travel time intensity associated to activity duration. The derivative of the predicted median value of travel time budget depends on the distribution adopted. Hence, the first step is to perform a non-parametric estimation of the travel time budgets. The lifetable method is known to be a suitable way to explore the distribution of the duration process.

Figures 1 and 2 present the survival function and the hazard rate and its confidence interval estimated in the different cities. The hazard rates appear to be nonmonotonic, given that the confidence limits follow this form in most of the samples.

To complete the graphical conclusions on the hazard form, likelihood ratio test is used to select the distribution of the duration. Only nested distributions can be tested with the LR test. Then, table 9 shows the LR tests for Weibull, log-normal and generalised gamma distributions, for the Lyon 1995 sample. The gamma distribution gives the best adjustment, with estimates of distribution parameters indicating nonmonotone hazard.

Nevertheless to estimate the travel time intensity, we use the log-logistic distribution.

This distribution is non-monotonic too. We can not test this distribution in the LR test, because log-logistic distribution is not nested in the generalised gamma.

However, the likelihood of the loglogistic model is higher than in the other models, it seems to give the best adjustment. Furthermore, the log-logistic distribution permits to deduce easily the derivative expression of the predicted median duration, by opposition with the lognormal or gamma functions. This distribution is nonmonotonic too.

The covariables set are constituted through an adaptation of the set of sociodemographic variables resulting from the Stepwise selection process performed in semi-parametric methodology of the Cox model. This methodology is known to be a suitable way to estimate the covariable impacts on hazard. The Cox models were performed for each city sample, leading to 14 results tables not presented here.

Regularities in covariable effects appear through all the cities: male, worker and motorised persons have lower hazard rate, hence higher travel time budget. Older persons have lower travel time budget. Travel time budget on Friday are lower.

Conversely, some variables have opposite effect between French and Swiss cities:

young have lower hazard in French cities, but higher in Swiss. And the effect of the presence of child is unclear.

Finally, the loglogistic estimations are performed using the non-parametric results relative to the distributions choice and the semi-parametric results on the covariables selection. Estimates for the two national samples, with dummies for cities and dates are given by table 10.

Most of the significant estimates are similar to those obtained in the Cox models.

Gender [START_REF] Dijst | Travel Time Ratio: the key factor for spatial reach[END_REF]. They find that travel time ratio for commute is a fixed proportion of the time devoted to work at destination (work duration plus commute time).

To gain robustness, the eventuality of heterogeneity between individuals needs to be included in this study. Furthermore, the definition of the travel time associated to the activity pursued, is problematic. Future task will be to analyse the time of trips chain in relation with the duration of the activities performed.

  , we test the proportional assignment paradigm, where daily available time is proportionally allocated to different activities with fixed proportions (which may vary across individuals). Furthermore, we test the proportionality of daily travel times for each purpose type with the daily activity times. It clearly indicates that travel times are influenced by duration of activities. Subsequently, the paper observes travel time ratio of work, leisure and shopping activity type. A linear model is estimated for the travel time associated to each type of activity. It supports a discussion on identification and transferability of regularities in the travel time with respect to socio-demographic variables between the seven cities. The final section presents estimations realised in the duration model framework, for the two national samples, including dummies variables for cities fixed effect. Models for the travel time budgets are built and lead to non-linear estimates of the travel time intensities. Gain in quality of adjustment between the linear model and the duration model is finally discussed. Section 1: Representation of the travel time and activity duration relationships The travel times measures considered in the paper are: (a) the individual daily travel time expenditures (or the daily travel time budget), (b) the daily sum of travel times associated to a given purpose (or the daily travel time budget for the activity type i), (c) the trip time.

  T a denotes the activity duration. For the 1998 Dutch National Travel Survey, the mean travel time ratio for a visit to the work place is 0.105. It corresponds to a commuting time of 3.5 minutes an hour spent working (or 28 minutes for an 8 hours workday).

  performed this test using the national time use surveys of Netherlands 1985 and California 1988. Results do not support proportionality for non-travel activities times. Proportionality is tested for daily travel times, commute times and travel time for non-work activities. The tests conclude to a proportional assignment of travel time to available time. Work time is introduced in the regressors set. It leads to a positive correlation between commute and work time. Finally, it suggests that travel time budgets are not stable, because of the variations of the travel time for non-work activity.

2. 1 .

 1 Enquête-Ménage Déplacements and Microrecencement National SuisseThe mobility surveys exploited in this study concern three Swiss cities (Bern, Geneva and Zurich) and four French cities (Grenoble, Lyon, Rennes and Strasbourg) each observed at two time periods. The Swiss data are from the Swiss national mobility survey (Microrecencement National Suisse) conducted in 1994 and 2000.

  The last step in the search for proportionality and regularities is at the trip level. The travel time for the work, shopping and leisure activities are analysed. Estimations of the logarithm of the trip time to an activity participation are realised on the logarithm of the activity duration and the other socio-demographic variables. Stepwise OLS are performed for each activity type on each city sample and on the two national samples, results of which are presented in tables 7 and 8. R² are comparable with other studies, about 0.20. The tests reject proportionality for the three trip purposes. The activity time effect on trip time are positive and weak for work and leisure (respectively, 0.08 and 0.09). For shopping, a 1% increase of shopping time implies a 0.18% of trip time.

Figures 1 (Figure 4 :Figure 5 :

 145 Figures 1 (a to f): Survival curves, hazard curves and hazard's confidence limits for Swiss travel time budgets

Table 2

 2 specifies the sample sizes for each city and date. Separate regressions were performed for each city sample resulting in the same conclusions as in tables 3 and 4. Proportional assignment hypothesis is rejected for work and shopping activities, with high level of significance, in all cities.

	Section 3: Results
	Estimations have been realised for each sample, but to gain space, only results
	concerning estimations based on the two Swiss and French samples are presented
	here.

3.1. Test of the proportional assignment of daily available time

The test of proportional assignment of available time is performed through Stepwise OLS estimations, constrained to include the total daily available time as an explanatory variable. The logarithm of daily time budgets for work, shopping, leisure and transport are regressed on the logarithm of daily total time available and dummy variables for sex, class of age, professional status, residential location, household structure, day of trip, driver licence, motorization, and dummies for cities crossed with periods. assignment hypothesis is rejected, for all activities. Conversely, the tests realised on each city samples indicates that the proportional assignment hypothesis is accepted for the shopping activity for the samples of Bern

  and professional status have positive effect on travel time budget or are insignificant. The days of the week have similar effect between French and Swiss samples. Travel time budget are higher on Friday and lower on Monday. Coefficients of the classes of age have the same sign. Older person have lower travel time budget. Residential location appears to have opposite effects: central location reduces travel time budget in Swiss cities, but increases travel time budget in French cities. The presence of children has significant negative effect in French cities and insignificant effect in Swiss cities. Male singles are characterised by opposite effect on travel time budget between French and Swiss cities. The driver licence dummy has a positive effect only in French cities. Motorization has no significant effect.Using the duration model, to estimate travel time budget, we theoretically gain in quality of adjustment. Duration model take into account the fact that travel time budgets are characterised by a non-normal distribution. The log-logistic distribution appears to give the best adjustment in the two national samples. On the one hand, this non-monotonic hazard form raises the question of the management of high travel time budget.[START_REF] Joly | Stability of regularity of the daily travel time in Lyon? Application of a duration model[END_REF] shows that the decreasing hazard concerns about 20% of the

	Section 4: Conclusion
	We test the three paradigms: proportional assignment of daily total available time;
	proportionality between daily activity time and the travel time associated; and
	proportionality between trip time and the activity duration at destination.
	The proportional assignment of daily available time is only valid for leisure and
	transport activities. The tests of the proportional relationship between daily travel
	time given a purpose and daily activity time indicate that proportionality is not valid,
	regardless the type of activity. Finally, the proportionality hypothesis is rejected
	between trip time and activity time at the destination. The log-log models estimated
	indicate a significant effect of activity duration on travel time, but in a non-linear
	form.
	sample of Lyon in 1995. This temporal dynamic can characterise the representative
	The dummies for cities indicate that travel time budget of Lyon 1995 and Strasbourg process of travel time management. The decreasing hazard part then questions the
	1997 are the highest in the French sample. In the Swiss sample, Zurich and Bern classical vision of travel time as a cost, that individuals want to reduce. On the other
	2000 have the highest travel time budget. hand, this form can result from unobserved heterogeneity. Certain variables not
	included in the model, could explain why these persons with travel time budget
	Figures 3-5 illustrate the predicted median travel time budget and the travel time greater than 100 minutes, have a decreasing conditional probability to stop travelling,
	intensities calculated at the mean of the French sample as function of each activity given their travel time budget is already of 100 minutes. Determinant of the
	time budget. The travel time intensity is increasing and convex with the shopping residential location choice, or the modal choice, or exceptional conditions could lead
	and leisure daily time budgets. Hence, travel time budget increase more than to special travel time management and make particularly high travel time budget
	proportionally with the time spend in leisure and shopping activity. Leisure appears acceptable.
	more intensive in terms of travel time than shopping. Conversely, the travel time Subsequently, we deduce travel time intensities from estimates of the activity time
	intensity for work is constant. Travel time for work increases linearly with work budgets, for French cities. Shopping and leisure times are associated with increasing
	duration. travel time intensities, with daily duration of activity. Conversely the travel time
	intensity associated with the daily work duration is nearly stable. This result echoes
	results of

Table 3 :

 3 Tests of the proportional assignment of total available time (French cities)

		Work TB / T	Shopping TB / T Leisure TB / T Travel TB / T
	R²	0.73	0.076	0.09	0.1063
	N. of significant variables	21	15	17	29
	Test statistics	92284.7	54.53	0.05	1.69
	(P-Value)	(<0.0001)	(<0.0001)	(0.81)	(<0.193)
	TB: Time budget	T: Total time available

Table 4 :

 4 Tests of the proportional assignment of total available time (Swiss cities)

		Work TB / T	Shopping TB / T Leisure TB / T Travel TB / T
	R²	0.70	0.23	0.23	0.06
	N. of significant variables	13	16	20	20
	Test statistics	15622.9	21.44	6.28	51.52
	(P-Value)	(<0.0001)	(<0.0001)	(0.0123)	(<0.0001)
	TB: Time budget	T: Total time available

Table 5 :

 5 Tests of the proportionality of the daily travel times for a purpose with respect to activity time budget (French cities)

		Daily TT for Work/	Daily TT for	Daily TT for
		Work TB	Shopping /	Leisure / Leisure
			Shopping TB	TB
	R²	0.062	0.253	0.142
	N. of significant variables	26	25	26
	Test statistics	6069.09	18021.5	13151.2
	(P-Value)	(<0.0001)	(<0.0001)	(<0.0001)
	TT: Travel time	TB: Time budget

Table 6 :

 6 Tests of the proportionality of the daily travel times for a purpose with respect to activity time budget (Swiss cities)

		Daily TT for	Daily TT for Shopping	Daily TT for Leisure /
		Work/ Work TB	/ Shopping TB	Leisure TB
		TTB / Work	TTB / Shopping	TTB / Leisure
	R²	0.038	0.147	0.144
	N. of significant variables	19	12	11
	Test statistics	1553.28	3989.09	2156.69
	(P-Value)	(<0.0001)	(<0.0001)	(<0.0001)
	TT: Travel time	TB: Time budget

Table 7 :

 7 Regression of the time trip given work, leisure and shopping purposes (Swiss sample)

	Variable	Trip time for Work Trip time for Shopping Trip time for Leisure
	Intercept	2.232 ***	2.013 ***	2.472 ***
	Log Work time	0.036 ***
	Log Shopping time		0.042 ***
	Log Leisure time			0.022 **
	Male
	Worker		-0.107***	-0.299***
	Suburban			0.048 **
	Monday		-0.054**
	Tuesday	0.071 ***
	Friday	0.113 ***		0.122 ***
	Couple with 3 children	-0.376 ***
	Male single no child	0.036 **
	Male single 1 child		0.087
	Female single no child			0.045 *
	Age < 19		-0.234**	-0.123
	18 < Age < 35			-0.059**
	49 < Age <65			0.052 *
	Age > 66	0.108 **		0.122 **
	Driver Licence	-0.039	-0.071***
	0 car	0.039 *
	3 cars	0.161 **	-0.149
	Walk	-0.661 ***		-0.213***
	Bicycle	-0.343 ***		-0.117*
	Motorcycle	-0.109
	Public transport	0.829 ***	1.169 ***	0.849 ***
	Car	0.107 *	0.394 ***	0.129 ***
	Other		0.275 *	0.950 ***
	Bern 94		-0.221***	-0.123***
	Bern 00		-0.057*
	Geneva94			-0.093
	Geneva00	0.083 ***	0.056 *
	Zurich 94		-0.157***	-0.145***
	Zurich 00	0.046 **	-0.053*	0.079 ***
	R²	0.31	0.24	0.17
	Proportionality F Test	15,873.8***	11,961.5***	7,958.31***

Table 8 :

 8 Regression of the time trip given work, leisure and shopping purposes (French sample)

		Trip time for	Trip time for	Trip time for
	Variable	Work	Shopping	Leisure
	Intercept	2.213 ***	1.753 ***	1.997 ***
	Log Work time	0.088 ***
	Log Shopping time		0.187 ***
	Log Leisure time			0.091 ***
	Male	0.129 ***	0.085 ***
	Worker	-	-0.031 ***	-0.096 ***
	Centre	0.055 ***	0.050 ***	0.054 ***
	Peri-urban		-0.018 ***	-0.079 ***
	Monday		-0.027 ***	-0.049 ***
	Tuesday			-0.034 ***
	Thursday	0.028 ***
	Friday	0.023 **	-0.021 ***
	Couple with no child		0.044 ***
	Couple with 1 child			0.028 *
	Couple with 2	-0.026 **	-0.028 ***
	children
	Female single no	0.060 ***	0.063 ***	0.069 ***
	child
	Female single 1 child			0.049 **
	Male single no child			0.078 ***
	Male single 1 child			0.199 ***
	Age < 19	-0.175 ***	-0.104 ***	-0.189 ***
	18 < Age < 35
	34 < Age < 50	-0.021 ***		0.021
	49 < Age <65			0.096 ***
	Age > 66			0.165 ***
	Driver Licence	-0.082 ***	-0.049 ***	-0.037 ***
	High income			0.041 ***
	0 car	0.089 ***	0.044 ***	0.026 *
	1 car	-0.021 ***	0.026 ***	0.023 **
	Walk	-0.650 ***	-0.304 ***	-0.263 ***
	Bicycle	-0.283 ***	-0.220 ***
	Motorcycle	-0.167 ***	-0.168 ***
	Public transport	0.563 ***	0.474 ***	0.664 ***
	Car		-0.081 ***	0.122 ***
	Other	0.374 ***	0.189 ***	0.608 ***
	Grenoble 01		0.01999	-0.089 ***
	Grenoble 92	-0.105 ***
	Lyon 85	0.047 ***	0.036 ***
	Lyon 95	0.108 ***	0.124 ***	0.081 ***
	Rennes 00	0.108 ***		-0.030 ***
	Rennes 91	-0.119 ***	-0.077 ***	-0.048 ***
	Strasbourg 88	-0.136 ***	-0.169 ***
	Strasbourg 97
	R²	0.21	0.21	0.19
	Proportionality F	51,176.7 ***	49,305.5 ***	46,149.4 ***
	Test

Methodology

Unified CERTU methodology At-home interview CATI

Object :

Household equipment, mobility behaviour, and opinions on transport policy and local themes Household equipment, mobility behaviour, and trip with a night out-of-home, and air trip, opinions on Swiss transport policy

Sample size

Which trips ?

All trips realised the day before the survey (workdays)

All trips realised the day before the survey (workdays or weekend)

When ?

A workday of reference chosen in several months (October to May) A day of reference chosen in the whole year

Where ?

The survey perimeter represents the agglomeration (defined by the entrepreneur) The Swiss national country


A L Cruz-Fabiano

R Logé

M Bernacki

Assessment of simplified 2D grain growth models from numerical experiments based on a level set framework

Keywords: Grain growth, Full field model, Level set, Finite Element, Mean field model, Grain size distribution

In this paper, the results of a 2D full field grain growth model are compared with several 2D mean field grain growth models (Burke and Turbull model and Hillert/Abbruzzese model), using simplified assumptions of isotropic grain boundary energy and mobility, and under the absence of precipitates. The full field model is based on a finite element formulation combined with a level set framework, used to describe the granular structure, and model grain boundary motion through a diffusion formulation. The initial digital microstructures are created using a coupled "Voronoï-Laguerre/dense sphere packing" algorithm, which allows to obey different types of initial grain size distributions, in the considered 2D context. The results show that only the Hillert/Abbruzzese model accurately describes grain growth kinetics for all considered grain size distributions. The validity of the Burke and Turnbull model is, on the contrary, restricted to specific distributions.

Introduction

Mechanical and functional properties of metals are strongly related to their microstructures, which are themselves inherited from thermal and mechanical processing. Grain growth phenomena in polycrystalline metals occur during and after full recrystallization, and have the effect of increasing the average grain size at the expense of smaller ones that will tend to disappear. Even if this phenomenon of capillarity is always present, it is generally neglected during primary recrystallization comparatively to the predominant driving force induced by the inhomogeneous spatial distribution of dislocations stored energies. Grain growth becomes however of primary importance when dealing with long annealing treatments, where capillarity effects become predominant. It then largely dictates the final grain size of the material. Over the last decades, several mesoscale numerical models have been developed to simulate the corresponding microstructure evolution [1].

Probabilistic methods associated with a voxel-based grains structure description such as of Monte Carlo (MC) [2][3] and cellular automaton (CA) [4] are widely used. Several workers have preferred to define microstructures in terms of vertices; the interface motion is then imposed by the displacement of a set of points [5]. Another approach found in the literature is the phase-field method, which offers the advantage of avoiding the difficult problem of tracking interfaces [6]. Finally, grain growth can also be modelled using a level set description of interfaces in the context of uniform grids with a finite-difference formulation [7] [8] or in a finite element (FE) context [9] [START_REF] Agnoli | Understanding and modeling of grain boundary pinning in Inconel 718[END_REF], which is the method chosen in this work. First level-set simulations concerning recrystallization in polycrystalline microstructures are described in [START_REF] Bernacki | [END_REF] [12] [13].

The main purpose of the paper is to test, in 2D, two mean field grain growth models existing in the literaturethe one proposed by Burke and Turnbull [14], and the well-known Hillert/Abbruzzese approach [15] [16] [17] [18] under simplified conditions of isotropic grain boundary energy and mobility, constant temperature, and absence of precipitates. The test is performed by comparison with the results obtained with a full field modelling method, starting from different initial grain size distributions. Grain growth models were developed based on theoretical assumptions which are not easily verified experimentally. The idea is therefore to use the full field simulations results in order to verify if and when the mean field model predictions are acceptable. If the latter are sufficiently accurate, it will also mean that full field models are not required in such simplified conditions, unless detailed topological information is needed.

In [19], R.D. Kamachali and I. Steinbach present a study with similar objectives, using a phase field framework combined with a finite-difference modelling technique. Both statistical and topological aspects of ideal grain growth, using the results of 3D simulations, are discussed. It is shown that, despite a few discrepancies with the mean field theories (Burke and Turnbull model, and Hillert model) the parabolic kinetics of grain growth remains valid during the entire process. The simulation reaches a steady state and the grain size distribution in this steady state agrees with the Hillert distribution [15]. It is also shown that the calculated volumetric growth rate compares well with the mean field assumptions. However, the paper analyzes only one initial grain size distribution, putting aside the influence of the distribution on the grain growth kinetics. In this paper, the influence of the initial grain size distribution, for equiaxed microstructures, is discussed. Thanks to a coupled "Voronoï-Laguerre/dense sphere packing" (VLDSP) algorithm [20][21], 2D imposed initial grain size distributions can be introduced accurately in the generated digital microstructures, even for a reasonable number of grains (less than 10000). Microstructures can be generated in 3D as well, leading however to unreasonable numbers of grains (in terms of computing cost) to match accurately a given grain size distribution. This remark explains why discussions addressed in the document are restricted to 2D configurations.

The paper is organized as follows. The following section explains how to generate the initial digital microstructures and immerse them in finite element meshes. It then recalls the level set framework used to describe and evolve the grain boundary network. In section 3, full field simulations are analyzed, and compared in details with the two investigated grain growth models.

Digital microstructures and FE immersion

Digital microstructures are often generated using Voronoï cells [22] [12]. Despite its widespread use, this classical method only allows to impose the mean grain size, and not the grain size distribution. Initial microstructures are therefore generated in this work with a VLDSP algorithm [20] [21]. Seven different grain size distributions (Table 1) are studied in order to evaluate their effect on grain growth kinetics. The first five distributions are "synthetic" log normal, meaning that they are not based on real experimental data. In these cases, the mean grain size is more or less the same, and the standard deviation ranges from 7 to 40 µm. The 304L distribution was generated based on 2D experimental 304L steel data of a fully annealed sample obtained from optical microscope image analysis. It is well approximated by a log-normal law, and can therefore be compared to the first five distributions. Finally, the last distribution in Table 1 corresponds to a "synthetic" bimodal distribution. The initial number of grains is around 10000 for all distributions, which ensures a good statistics during the simulation and a reasonable numerical cost of the FE simulations (few hours in 32 CPU simulations).

Figure 1 (top) describes a comparison between the targeted microstructures and those built numerically whereas Figure 1 (bottom) illustrates the voxelized digital microstructure obtained for the bimodal case. Table 1 gives in parallel the L2 errors based on the Equation 1 below, between the theoretical and the obtained numerical distributions. (

L2 errors increase with the complexity of the distribution and the error is more important for grains families with smaller grain size. In 2D, the Voronoï-Laguerre method consists in using a distribution of non-intersecting circular particles (generated with the dense sphere packing part of our VLDSP algorithm) that serves as a basis for constructing the grains of the polycrystal (the Voronoï-Laguerre cells). Local heterogeneities of the density of the built sphere packing, which are more likely near the smaller circles, involve local decorrelations between the imposed and the obtained cells radii. This explains the concentration of the errors in the families with smaller grain size. However the reported errors are clearly very good in comparison to the state of art [20]. It is also important to clarify that the L2 errors discussed here really correspond to a comparison between the imposed radius distribution and the equivalent radius distribution of the generated Voronoï-Laguerre cells. It is not a comparison with the generated sphere packing radius distribution, otherwise the error would always be null in the context of our LVDSP algorithm.

In order to compute the evolution of the numerical microstructures, these are discretized into a finite element mesh. Different methods can be found in the literature. The most widely used method consists in generating a surface mesh from the sides of the cell, and then generating the volume mesh starting from the surface mesh [START_REF] Rollett | Proc. Conf. Numiform[END_REF]. This method is relevant when modeling polycrystals deformation. However, when modeling recrystallization and grain growth, grain boundaries keep moving, grains disappear, others nucleate, etc. The complexity of the remeshing operations associated with these topological evolutions rules out the above meshing method. Ultimately, finite element methods emphasizing an explicit description of grains boundaries, such as the Vertex methods [5], always meet significant problems of mesh management, especially in 3D. Other approaches therefore favor implicit descriptions of grain boundaries, such as that introduced in the phase field [START_REF] Suwa | [END_REF][25] or level set methods [26][11][13] [9].

In this work, a level set framework is used to implicitly describe the microstructure. A level-set function , defined over a domain Ω, is called distance function of an interface Γ of a sub-domain Ω S if, at any point x of Ω, it corresponds to the signed distance from Γ. In turn, the interface Γ is given by the zero level of the function :

(2) with the characteristic function of Ω S , equal to one in Ω S , and 0 elsewhere. According to Equation 2, inside the domain surrounded by the interface Γ, and outside this domain. Usually, when dealing with a polycrystalline aggregate, a distinct level set function is used for each grain:

, with N G the total number of grains in the aggregate. In [20], the authors explain in details how the level-set function of each grain can be evaluated when the polycrystal is inserted into a finite element mesh. In this framework, the norm of the gradient of each initial distance function is equal to one . The use of a single level set function for each grain may lead to unreasonable computation time, when dealing with statistical numbers of grains as done here. Therefore, a graph coloration technique is used for limiting the number of level set functions needed to [27]. The objective is to use a single level set function to describe several non-neighbouring grains, instead of just one. This brings the total number N G of level set functions to a reduced number N C of "container" level set functions. Figure 2 illustrates a 2D 2000-grains polycrystal described using only five level set functions. In this Figure, each colour represents one container level set function. The above technique however presents the limitation that when two grains belonging to the same container level set function start touching each other, they coalesce. In [7], a method is proposed to avoid this problem: when two different grains belonging to the same container level set function get closer than a critical distance, one of the two grains is removed from the container level set function, and placed into another one. However, if the proposed methodology is applicable in the context of regular grids where the connected components (the individual grains) of each container level set function can easily be extracted, the problem becomes much more complex when dealing with non-uniform FE meshes. Therefore, in this work, it was chosen to delay the onset of grains coalescence by introducing a constraint in the graph coloration, such that only the 4 th nearest neighbour can belong to the same container level set function. The simulation is then stopped as soon as two grains begin to coalesce. With this approach, the 10000 grains microstructure can be fully described using only around 30 container level set functions.

                      . 0 ) ( , ), , ( ) ( ) , ( ) ( ) ( x x x x d x x d x x S S     S   0   0     G i N i   1 ,    1   i  describe the microstructure [20]
As stated earlier, grain boundary mobility and energy are considered isotropic and uniform throughout the domain. Grain boundaries motion can be described by Equation 3 [28][29]:

Δ , (3),
where the subcript denotes now the set of the container level-set functions, M is the grain boundary mobility, the driving force per unit area and the outward unit normal of the boundaries of the grains constituting the container level-set function. Considering only grain growth, the driving force is defined by :

i f  i n  i f  Δ , (4)
where is the grain boundary energy, and the mean curvature of the grains constituting defined by: .

If container level set functions remain distance functions (i.e.

) near the grain interfaces, the mean curvature can be simplified from (5) as the opposite of the Laplacian of the corresponding container level set function [9]. The grain boundary convection problem with the velocity field defined in Equation 3 is then simplified and becomes a simple diffusion problem:

 . ,..., 1 , ) ( ) , 0 ( 0 ) , ( ) , ( 0
G i i i i N i x x t t x M t t x                    (6)
The system ( 6) is solved for the N C container level set functions present in the domain Ω. It describes a pure grain growth problem if and only if the metric property (

) of each container level set function is enforced at all time steps near the zero isovalues. This is done by using a re-initialization treatment [30] at each time step and for each container level set function.

As the diffusion of the container level set function can generate some kinematic incompatibilities at multiple junctions [26], leading to vacuum or overlapping regions, a multiple junction treatment is performed to avoid these phenomena. This multiple junction treatment is detailed in [9].

Using the numerical tools presented above, the following grain growth algorithm is proposed and is solved for each active container level set function: GG1 -Resolution of Equation 6; GG2 -Numerical treatment of the kinematic incompatibilities; GG3 -Re-initialization procedure; GG4 -Deactivation if it is negative over the whole domain, i.e. when all grains of the container level-set have disappeared.

This global strategy was tested and validated for academic test cases in [9].

Results and discussion

For all simulations presented here and referring to the 304L steel properties, the time step is 120 seconds, the temperature is 1050 o C, with M = 1.37 10 -12 m 4 /(J.s) [31], and γ = 0.6 J/m 2 . The finite element mesh is isotropic unstructured with mesh size equal to 0.01 mm over a square domain of 13 mm x 13 mm, leading to around 8,000,000 mesh elements. Figure 3 illustrates the initial and final grain structures of 304L steel considering a 5 hours thermal treatment at 1050°C. All studied distributionsexcept for the Bimodal (which will be discussed later)reach at some point a quasi-steady state i.e. converge toward a quasi-constant distribution until the number of grains becomes insufficient to consider the microstructure as representative. The final grain size distributions calculated from the six initial distributions are compared with the Hillert [15], the Rayleigh and the Weibull distributions [32], [33], [34] in Figure 4. The Rayleigh distribution was originally derived in the 2D grain growth context by Louat [35] and corrected by Mullins in [36]. The Weibull distribution function has two free parameters, α and β. In [33], based on numerical simulation results, the authors found that the quasi-steady state grain size distribution is best represented using the Weibull distribution with α = 1/[Γ(1+1/β)], where Γ is the gamma function, and β = 2.5. Table 2 gives the expressions of these theoretical stationary distributions in 2D. The Hillert distribution has a non-analytic cutoff at = 2, while the others distributions present an infinite tail.   R R Hillert distribution (7) Rayleigh distribution (8) Weibull distribution In Figure 4, the obtained numerical quasi-steady state distributions vary between the Rayleigh and the Weibull distributions. As a conclusion, for the simulations presented in this work, the quasi-steady state grain size distribution is well represented using the Weibull distribution with a β value ranging between 2 and 2.5. On the other hand, the Hillert distribution does not agree with the numerical results. According to Mullins [36], the reason for the discrepancy between the Hillert theoretical distribution and the experimental and/or numerical results comes from topological reasons: in 2D, the Hillert distribution is reached only if the average number of sides relates linearly to the grain size. The relationship between the number of sides and the grain size in the results presented in Figure 4 should therefore be the subject of future work.

                                                     R R R R R R P w exp 1 ( 9
)
                                    R R R R R R e R R P H 2 4 exp 2 2 4 2 3                                   4 exp 2 2 R R R R R R P R  
Burke and Turnbull model [START_REF] Agnoli | Understanding and modeling of grain boundary pinning in Inconel 718[END_REF] This model is based on three main hypotheses:  The driving force for grain growth is proportional to the grain boundary mean curvature approximated as , where R is the equivalent radius. So the grain boundary migrates toward the centre of its curvature, which in turns reduces the interfacial area as well as its associated energy;  The mobility and grain boundary energy are isotropic and uniform. As a consequence the equilibrium angles in the triple junctions are equal to 120 o ;  The heat treatment temperature is constant.

These hypotheses lead to the following grain growth kinetics equation in 2D: , [START_REF] Agnoli | Understanding and modeling of grain boundary pinning in Inconel 718[END_REF] where (resp.

) corresponds to the average grain radius (resp. at t = 0s). With this equation, neither the topological nor the neighbouring effects are taken into account; the grain growth kinetics is characterized only by the average grain size.

To check the consistency between the full field simulation results and this model, the curves

    ) log( log 2 0 2 t f R R  
have been plotted in Figure 5 for each initial distribution described in Table 1, except for the Bimodal distribution, which is treated separately. For these distributions, Equation 9 is generalized according to: [START_REF] Bernacki | [END_REF].

From Equation 11, the validity of the Burke and Turnbull model can be verified if the slope n of

    ) log( log 2 0 2 t f R R  
, is equal to 1, and if the fitted curve leads to an α value around 0.5. Computed curves and their linear fits are given in Figure 5 and lead to the α and n parameters summarized in Table 3. As illustrated in Table 3 and Figure 5, n and α are highly dependent on the initial grain size distribution. Only the Log4 and 304L distributions lead to the values of n and α expected by the Burke and Turnbull model. Both distributions are log-normal with a value of (see Table 1). For the other distributions, it can be noticed in Figure 6 that n values decrease with the increase of , while α values increase with . These evolutions can be approximated by the following simple relationships: (12) Figure 6: Relationship between α and n fitted parameters and σ/μ of the initial grain size distributions (given in Table 1)

  

     R 1 t M R R  2 1 ² 2 0   R 0 R n t M R R     2 0 ²
45 . 0       
Thus for a log-normal distribution, presents a power law dependence on and n a logarithmic dependence. Equations 12 confirm that grain growth kinetics significantly depends on grain structure and neighbourhood effects. Using the numerical grain growth results we propose a correction of the Burke and Turnbull model. Once Equations 11 are experimentally verified, this corrected Burke and Turnbull model can be directly used when modelling grain growth phenomena.

Figure 7 shows that the Bimodal kinetics can be divided into 2 linear parts -'A' and 'B'. Evolution of the grain size distributions is also given at different times 'a' (t = 0s), 'b' (t = 4400s) and 'c' (t = 18000s). The slope change in the kinetics is seen to coincide with the quasi disappearance of the smallest grains population, i.e. with the switching from a bimodal distribution to a single peak distribution. It is therefore concluded that the Burke and Turnbull model is not valid for most of the investigated grain size distributions. The model only behaves well for lognormal grain size distributions, with a value close to 0.45.

Hillert/Abbruzzese model

Hillert [15] and Abbruzzese [16][17] [18] mean field models are derived from the von Neumann-Mullins theory [37][38]. Even though the proposed developments differ, both Hillert and Abbruzzese models finally propose the same grain growth Equation 13. Therefore, no distinction is done here between these two approaches. The central equation is:

          i i R R M R 1 1    , (13)
    
where the grain structure is assumed to be described by a discrete set of grain families (each family being labeled with a subscript i), corresponds to the average grain boundary velocity of the grain family i, corresponds to the average grain size and is a constant generally assumed to be 0.5 in 2D, and 1 in 3D, even though other values can be found in the literature. For example, in [19], Kamachali found a value of 1.25 in 3D, using a phase field framework combined with a finite-difference modelling technique. In [START_REF] Rios | [END_REF], Rios and Glicksman calculated a 3D value of 0.81, using the average N-hedra method (ANH) [40].

To check the consistency between the full field simulations and the Hillert/Abbruzzese model, calculated grain size distributions are compared. The mean field computations are based on Equation 12, implemented within a more general model described in [41] where . Three different times (t = 2 min, t = 76 min and t = 150 min) were analysed, for all initial grain size distributions considered in Table 1. The corresponding L2 errors (see Equation 1) are given in Table 4. They are based on cumulative distribution functions, as illustrated in Figure 8 for the Bimodal and the 304L cases. 8). From Table 4, it can be concluded that the Hillert/Abbruzzese mean field model is overall in good agreement with the full field computations. L2 errors remain below 11% for all considered distributions, including the Bimodal one. A closer look at the distributions, e.g. those illustrated in Figure 8, shows that errors increase in the lowest grain sizes range. An hypothesis which explains this error is related to volume conservation issues discussed in [41]: volume changes of the shrinking grains are computed by redistributing the total volume changes of growing grains. Consequently, the rates of volume change for small grain sizes are slightly modified from the values strictly deriving from Equation 12. Comparing the results obtained using the mean field model discussed in [41] and the direct use of Equation 12, it is observed that the effect of the volume conservation treatment is more important for a small number of representative grain families (around 50). In fact, when using 40 representative grain families (as the case in the comparisons presented in Figure 8), the direct use of Equation 12gives better results than the model presented in [41]. However, for a statistical number of representative grains around 200, both methods lead to the same results, and the volume conservation treatment does not affect the results. A more complete study of the volume conservation issue will be the subject of future work. Full field computations give a β value (Equation 12) around 0.5 which is exactly the expected value.

It is therefore concluded that, in the absence of second phase particles and under simplifying conditions of uniform and isotropic grain boundary energy and mobility, the Hillert/Abbruzzese mean field model describes grain growth with good accuracy, for a wide range of initial grain structures.

Conclusions

The validity of two grain growth models has been discussed based on full field model computations. The full field model is based on a finite element formulation combined with a level set framework. Seven initial grain size distributions have been considered in order to test the investigated grain growth models. Computed quasi-steady state distributions compare well with the Weibull distribution with a β value ranging between 2 and 2.5, and significantly differ from the Hillert distribution.

The simplified Burke and Turnbull model is shown to be valid only for two grain size distributions. Both of them are lognormal and present a standard deviation value equal to 0.45 times the average grain size. It can be noticed that this kind of distribution is classically observed in polycrystalline metals. On the other hand, the Hillert/Abbruzzes model is shown to be accurate for all tested distributions, even the Bimodal one. Consequently, if the development of full field models is justified for the description of grain growth in complex conditions, or for complex microstructures, their use appears disproportionate in the simple configurations investigated here, where the Hillert/Abbruzzese model behaves very well.

It is important to highlight that all conclusions presented in this work are valid under simplified conditions: 2D grain growth, isotropic grain boundary mobility and energy and no dragging forces. In order to verify if these conclusions are still valid under more complex and realistic conditions (3D grain growth, anisotropic grain boundary mobility and energy, etc.), complementary numerical simulations and/or experimental tests must be performed.

Future work will be dedicated to (i) investigating volume conservation issues in grain growth regime, (ii) analyzing the number of sides relationship with the grain size, particularly in the quasi-steady regime, (iii) extending the present study from 2D to 3D, (iv) considering anisotropic grain boundary mobility and energy, and (v) following the same strategy to compare static and dynamic recrystallization predictions of the mean field model presented in [41] with those obtained with the full field approach [12] [13].

Figure 1 :

 1 Figure 1: (top) Numerical initial grain size distributions compared with the targeted ones: (a) Bimodal and 304L distribution; (b) Log1, Log2, Log3, Log4 and Log5 distributions. In the horizontal axis the radius is normalized by the mean grain size <R>. (bottom) Voxelized digital microstructure obtained for the bimodal distribution made of 9933 Voronoï-Laguerre cells.

Figure 2 :

 2 Figure 2: A 2000-grain 2D equiaxed polycrystal described using five level set functions, as shown by the 5 different colors.

Figure 3 :

 3 (a) Initial 304L microstructure generated with the Voronoï-Laguerre method, as shown in Figure1b, (b) zoom of the initial 304L microstructure, (c) simulated grain growth after 5h at 1050°C, (d) zoom of simulated grain growth.

Figure 4 :

 4 Figure 4: Quasi-steady state and comparison with Hillert, Rayleigh and Weibull predictions.

Figure 5 :

 5 Figure 5: Computed evolutions of grain structures starting with different initial grain size distributions, using the full field model. Linear approximations are added for comparison with the Burke and Turnbull model.

Figure 7 :

 7 Figure 7: (Top to botton and left to right) Burke and Turnbull model study for Bimodal distribution; grain size distribution at point a (t = 0 s), at point b (t = 4400 s), and at point c (t = 18000 s).

Figure 8 :

 8 Comparison between full field results and Hillert/Abbruzzese mean field results for (a) Bimodal, and (b) 304L distributions. Results are displayed using cumulative distribution functions.

Table 1 :

 1 Features of the seven considered initial grain size distributions.

		Mean Radius -μ (µm)	Standard Deviation -σ (µm)	σ/ μ	Initial number of grains	Domain Size (mm)	L2 Error (%)
	Log1	61.7	7.2	0.12	9728	11 x 11	2.9
	Log2	63.7	14.1	0.22	10517	12 x 12	7.1
	Log3	67.4	23.5	0.35	10464	13 x 13	2.0
	Log4	71.4	31.6	0.44	9999	14 x 14	1.8
	Log5	75.3	38.4	0.51	8583	14 x 14	4.9
	304L	69.0	31	0.45	9211	13 x 13	5.3
	Bimodal	62.8	24.3	0.37	9933	12 x 12	8.3

Table 2 :

 2 Hillert, Rayleigh and Weibull distribution equations.

Table 3 :

 3 Burke and Turnbull model analysis by comparison with full field simulations results, considering six different initial grain size distributions (described in Table1).

	Distribution Slope (n)	α	Number of grains in the end
	Log1	2.48	1.21 10 -7	4078
	Log2	1.59	1.08 10 -3	3747
	Log3	1.19	0.08	2889
	Log4	1.02	0.38	2782
	Log5	0.89	2.3	2743
	304L	1.04	0.42	3271

Table 4 :

 4 L2 Errors between grain size distributions obtained with the full field model, and with the Hillert/Abbruzzese mean field model. Errors are computed based on the cumulative distribution functions (see Figure

			Error L2 (%)
		t = 2 min	t = 76 min t = 150 min
	Log1	10.7	4.6	3.9
	Log2	4.5	4.6	6.3
	Log3	1.9	4.0	6.0
	Log4	1.6	4.5	7.6
	Log5	1.9	2.4	3.4
	304L	1.9	5.5	5.5
	Bimodal	4.9	6.8	10.3