Alexis Champion

René Mandiau

Stéphane Espié

Christophe Kolski

Multi-Agent Road Traffic Simulation : Towards Coordination by Game Theory Based Mechanism

Road traffic simulation is a flexible tool used for a wide variety of applications. In addition to mathematical based models, the INRETS has conducted researches based on the driver behavior. ARCHISIM is a behavioral and a multi-agent simulation model: the traffic is produced by the interaction of the agents, which are simulated drivers in virtual vehicles. Like for many agent-based applications, the coordination is a crucial issue because managing interactions is essential and proves to be complex. Simulated road traffic is an open system for which resources are limited and whose agents are competitive. Conflicts are frequent and, for some situations, agents are brought to collaborate. In this paper, we first present the behavioral traffic simulation tool ARCHISIM. We then present the coordination issue of the "traffic system" and expose the mechanisms implemented to answer ordinary and some critical traffic situations. Finally, we focus on a game theory based mechanism we are about to use to coordinate traffic at junctions in order to avoid deadlocks.

I. Traffic Simulation with ARCHISIM

Simulation modeling is an increasingly popular and effective tool for analyzing a wide variety of dynamical problems. Road traffic is an example of such problems. Different types of traffic simulation models exist [Lieberman 97]. Macroscopic models represent entities and their activities in some aggregate manner by values of density and speed. Microscopic models describe entities as individuals and the fundamental interactions take the form of mathematical formulas. All these traffic simulation models describe traffic in statistical formats. In addition to these models, the INRETS (French National Institute for Research in Transportation and Safety) has done some research on road traffic simulation based on the driver behavior for more than ten years. The INRETS' ARCHISIM simulation tool makes use of a behavioral sub-model for driver decisions, which results from in-depth studies carried out in driving psychology [Saad 99]. In ARCHISIM, traffic phenomena come from individual actions and interactions of the various actors of the road situation [Espié 99]. ARCHISIM is a behavioral simulation model and its implementation follows the multi-agent principles; a behavioral simulation model is a multi-agent simulation model [Drogoul 93]. At INRETS, our ambition aims at making ARCHISIM an open tool for the study of the "traffic system". Moreover, ARCHISIM has been developed such that the traffic model can host a driving simulator. In this case, the human being in the driving simulator interacts with the traffic within the simulation model. This step appears significant to us because it makes it possible to confront the new concepts with the final users while following an iterative process. ARCHISIM is a flexible tool for which a set of applications has been found: new road design, study of test scenarios (automatic incident detection, ramp metering), etc, in fact any study relating to a modification of the "traffic system" (Figure 1). Driving is a succession of tasks, sequential or parallel, discrete or continuous. The driver perceives its environment, interpret it and tries to act consequently. Studies show that driving is a hierarchically structured activity [Michon 80] for which the concept of risk seems essential. The strategic level deals with route planning. The tactical level deals with the choice of maneuvers (lane changing…). The operational level deals with the achievement of the selected maneuver by acting on the speed and the trajectory. Within ARCHISIM, agents are simulated drivers in virtual vehicles and consist of three subsystems: perception, interpretation and action. We focus on the "interpretation" part. Each agent has a model of its environment and interacts with the other agents (cars, trucks …), the infrastructure (traffic lights) and the road. Each agent has goals and skills (Figure 2). In opposition to works done in robotics [Reece 93], the agents' behavior is not normative. Each agent has its own attitude. The objective is not to build a robot able to drive automatically but to study the driver's behavior and the way in which the traffic phenomena occur. Within ARCHISIM, agents are autonomous and can potentially react to any situations. The system can then show a greater aptitude to organize and to coordinate itself. In this paper, after this short presentation of the behavioral traffic simulation tool ARCHISIM, we present the coordination issue of the "traffic system". We then expose the mechanisms implemented to answer ordinary and some critical traffic situations. Finally, we focus on a game theory based mechanism we are about to use to coordinate traffic at junctions in order to avoid deadlocks.

Strategic level

Tactical level

Operational level

Perception Interpretation

II. Multi-Agent Coordination and Road Traffic

Coordination is the process by which an agent reasons about its actions and the future actions of others in order to ensure the situation develops satisfactorily. The coordination of agents' actions results from their interactions. Many definitions for coordination have been proposed In particular, our study relates to road traffic, which is characterized by the interaction of numerous actors: road users, operators and road designers. Traffic can be considered as a supply and demand problem whose difficulty relies on two opposite postulates. The offer responds to a collective use: the road network is dimensioned to allow a certain flow. The demand is individual: each driver wishes to travel under its conditions. Therefore the "traffic system" characteristics imply strong constraints for the modeling of the agents, their interactions and the multi-agent system (Figure 3). The coordination of simulated road traffic is a distributed resources allocation problem with real-time constraint. Moreover, information perceived by the agents is geographically limited and incomplete. Also, the agents' environment is non-deterministic and the system has an infinite number of possible states. Specific coordination mechanisms have then to be designed and developed to answer the various traffic situations.

Many different road users Variable number of road users Road network of limited capacity

Road traffic

Situated Cognitive

Agent

Heterogeneous Open Competitive Dynamic

M.A. System

Generally non co-operative drivers

Changing objectives Human behavior

Driving task

III. Traffic Coordination Mechanisms for Behavioral Simulation

To coordinate traffic, we have considered two types of traffic situations: car-following situations, that we call ordinary situations, and situations presenting a risk of blocking, that we call critical situations. In every case, the implemented mechanisms are economic mechanisms, which are bound to the notions of profit and cost. We now present the mechanisms that are implemented for ARCHISIM to coordinate the traffic in the common situations.

III. 1. Traffic coordination for ordinary situations

Ordinary traffic situations are situations for which traffic is relatively fluid and does not show a risk of blocking. In ordinary situations, drivers' behavior results from training and is based on routine actions and reflexes [Chaib-draa 97]. In these situations, drivers generally exhibit a non-cooperative behavior and individually manage to coordinate their actions with others. Let us consider an example. Driver A has to slow down because of Driver B and is about to try to overtake it while driver C is coming from the opposite lane. There are dependencies between the drivers' activities. These activities refer to common resources that drivers have to manage with an acceptable risk (Figure 4). At the tactical level, the driver's objective is to determine the lane that enables to drive at a speed close to what psychologists call its "desire" speed. The driver has to consider its current state (speed, position…) and the constraints of its environment (others…) to make choices in order to satisfy its needs. A distributed resource allocation method, based on in-depth studies carried out in driving psychology and using the concepts of profit and cost, has been implemented. Each agent evaluates, for each potential target lane, the profit it can hope for and the cost -bound to the maneuver and the risk-it must support (Figure 5).

Figure 5: At the tactical level, the agent chooses a lane by using an economic model

In our example, driver A is offered two possibilities: to remain on its current lane or to overtake B. Driver A computes the profit and the cost associated with both possible lanes. For the left lane, it estimates that the cost is lower than the profit and decides to change lane. The variables used by each agent to compute its profit and cost functions for the potential target lanes are data related to the agent's state and intentions and data referring to its close or distant environment. For an agent and a potential target lane, the profit mainly relates to: its "desire" speed, its current leader's speed, the distance to its current leader and the stream's speed. The cost is function of the risk taken for the lane changing and relates to: the effort provided for the maneuver and the stability of the traffic on the target lane. This coordination mechanism takes into account the non-cooperative behavior of the agents. Each agent individually manages its goals and its constraints by considering the others' actions but, also, by trying to influence them. Thus, an agent can take a significant risk when overtaking and hope that the agent coming the opposite way decides to slow down.

Compared to mathematical simulation models, the mechanisms implemented for ARCHISIM are relatively costly and limit the road network loading. But, like for any emergent system, the main difficulty is finding generic rules that lead to the production of phenomena identical to the real phenomena. In spite of this, ARCHISIM has been validated for ordinary situations on highway networks [Espié 99, El Hadouaj 00, Champion 01].

III. 2. Traffic Coordination for Critical Situations

A situation is known as critical when it presents a durable risk of blocking for one or more vehicles. In such a situation, drivers, who generally have a non-cooperative behavior, are brought to collaborate. Indeed, the only obedience of the Highway Code does not make it possible to find a satisfactory solution and, while being less individualistic, drivers avoid blocking durably some vehicles. Let us consider the examples below (Figure 6). In the two first examples of critical situation, the drivers understand that some of the vehicles are blocked. Then it occurs what is called the zip effect. Some vehicles on the free lane will let some blocked vehicles go. The third example stands at a junction where the drivers may block each other. At least one of the drivers decides not to enter and awaits the exit of one of the others or, a driver decides to exit although it does not have priority.

A blocking situation appears when: 1) the drivers of vehicles being on a lane that is likely to be blocked strictly follow the Highway Code and 2) the drivers of vehicles being on a lane being likely to block the vehicles on another lane have a non-cooperative behavior.

A blocking situation ends when: 1) the drivers on the blocked lane become proactive and do not follow the Highway Code any more and 2) the drivers on the blocking lane collaborate.

To avoid a blocking situation, drivers adapt their behavior and collaborate.

We differentiate two classes of critical situations: the situations with flows of different priorities and situations with flows of the same priority [Champion 99]. In some situations, a flow of vehicles can be blocked by a priority flow. In some other situations, the flows have the same priority and a deadlock may appear. In critical situations with different priority flows, there is a durable risk of blocking for at least a vehicle of a non-priority flow mainly because the priority flow has a high density. Thus, to prevent this, a priority driver may decide to collaborate and let the blocked vehicle proceed. If not, the non-priority vehicle will have to force to take a gap.

1 1 1 1 1 1 1 1 2 A 1 1 1 1 1 1 1 1 1 C C Conflict
In ARCHISIM, for this type of situations, the priority agents and the non-priority agents coordinate their actions by following the procedures presented below (Figure 7). This method has been tested for accident situations, insertion situations on jammed highways and for simple Tjunction situations. For deadlock situations, the problem is more complex. As drivers must anticipate the consequences of their actions and others' to avoid blocking each other, the interpretation of the whole situation is of primary importance. To try to implement this behavior with a multiagent coordination method, we have decided to work out a game theory based method.

IV. Coordination at the Junctions: Towards a Game Theory Based Mechanism

We now expose the reasons that led us to work out a multi-agent coordination method for the traffic at the junctions. This game theory based method should solve conflicts and avoid deadlocks and livelocks.

IV. 1. The coordination issue at the junctions

What urged us to want to implement a game theory based coordination mechanism is above all that ARCHISIM, although it is validated for the motorway situations, does not allow proposing totally reliable urban network simulations. A major constraint opposes to it: some critical traffic situations at junctions can lead to deadlocks. Therefore, it is not certain that no blocking appears and immobilizes a part of the simulated traffic, what would have for consequence the non-validity of the experiment in progress.

In this context, it has been decided to revise the coordination method of the traffic at junctions. The objective is above all to implement a fast method allowing avoiding the blocking situations at junctions, even if it is made to the detriment of the realism of the local Vehicles on L 1 : -gain for staying on L 1 is decreasing -cost for going to L 2 is increasing ⇒ lane changing more and more interesting Vehicles on L 2 : -gain for staying on L 2 is increasing -cost for going to L 1 is extremely high ⇒ lane changing more and more interesting

Choice

Vehicles on L 1 : slow down, try -more and more-to leave L 1 to go to L 2 Vehicles on L 2 : slow down before merge area (because of vehicles trying to come from L 1 ), accelerate after merge area of the simulated vehicles' behavior. Because it is the traffic as a whole that interests us first and foremost.

Research made in psychology show that any complex junction is perceived and managed by a driver as a succession of elementary crossroads: X-crossroads and T-crossroads [Saad 89]. Therefore it is possible to turn the problem to the development coordination mechanism for X-crossroads and T-crossroads. Besides, a road network in ARCHISIM is modeled by a set of such elementary crossroads.

To answer our problem, we have decided to organize our reflection in two stages. The first stage concerns the realization of a first version of the coordination mechanism, which, as it is done for the most part of the mathematical models of simulation, does not handle the conflicts inside junctions. So at first, we decide to only manage the access to junctions: as soon as a vehicle is inside a crossroad, the other vehicles potentially in conflict with it should not enter.

The second stage deals with the realization of a generic version of the mechanism managing all the conflicts, including those inside junctions, therefore allowing several vehicles to simultaneously reach a junction.

The coordination mechanism has to be easily integrated into ARCHISIM. So, the idea of an economic method based on the notions of profit and cost turns out relevant. After a research made on the various types of multi-agent coordination mechanisms, we have pushed aside the planning and negotiation mechanisms because they usually are slow. We have finally opted for the idea of a game theory based mechanism, which has the advantage to faster than the other methods.

IV. 2. Game theory and multi-agent systems

Game theory is a mathematical theory that is very widespread in the sciences for which the study of conflict situations is relevant: economics, biology, sociology, computer science… A game, according to the common meaning, is a situation for which individuals (the players) have to choose among a certain number of possible actions in a predefined frame (the rules of the game). The result of these choices gives an exit of the game, to which is associated a positive or negative gain for each of the participants. If one disregards the idea of entertainment usually bound to the word, this definition is mathematically satisfactory. One can however add that players' interests are often different or opposite. Since then and over the years, the ever increasing and expanding research on Artificial Intelligence has seen the development and implementation of many commercial and industrial applications that utilize the link between agents and game theory. This ranges from highly sophisticated industrial applications, e.g. defense intelligent systems, to mail filtering and reordering, flight booking, selection of books etc.

Interesting work survey can be found on game theory applied to multi-agent coordination in [Khoo 95] and [Johansson 99].

In the transportation field, relatively few papers can be found about game theory based multiagent coordination. However, some are extremely linked to our work [Kita 99].

Like most of the multi-agent coordination mechanisms, game theory based mechanisms are highly dependent on the application. This is mainly due to the strong hypotheses of the theory.

A game theory based coordination mechanism should allow to be capable of defining the players, the payoff matrix and the players' strategy, which is the process that allows the players to choose the action that seems the most interesting to them.

IV. 3. Towards coordination by game theory based mechanism

Every traffic situation being a priori different, the number of players at the beginning of a game is unknown. So, every agent approaching a junction should first determine who is going to play with it. This first phase of the mechanism can be complex. In the first version of the mechanism every agent considers as player all the agents coming to the same junction as it. This is not very optimal but it has the advantage to consider the problem in its wholeness.

Once the players are known, they should determine the game they are going to play. Once again, every traffic situation being a priori different, the game is rarely the same. A n-player game is modeled by a n-dimensional matrix and, because we are talking about driving, the possible actions are among two: to accelerate or to brake, in other words: to go or to stop. The payoff matrixes for the n-player games are 2 n -dimensional matrixes.

Let us consider the traffic situations for which two vehicles (A and B) are arriving at a junction. We notice that three types of situations can appear. Either both vehicles are not in conflict because none of them has priority (¬Prio (A, B) and ¬Prio (B, A)), or one of the vehicles has priority (Prio (A, B) and ¬Prio (B, A)), or both vehicles have priority (Prio (A, B) and Prio (B, A)) (Figure 8). responsible in the event of an accident. Consequently, it seems logical: to give a more important profit (100) to a priority vehicle than to a vehicle that is not (50); to give a weaker cost (50) to a priority vehicle than to a vehicle that is not (100) when both reach simultaneously a junction; and finally to give an nil payoff to any vehicle staying outside the junction. Naturally, these values are examples; it is only necessary to remember the ratio.

Let us now consider the resolving of the game, that is to say the strategy that every player is going to take. Every player should choose between both possible actions: go and stop. For that purpose, it calculates the average payoff associated to each of these actions by doing an arithmetical mean. One notices that in the situation for which both vehicles have priority, two strategies can be chosen: one or the other vehicle moves (Figure 8). This shows that the players are indecisive: there is a risk of blocking.

So that there is coordination, it is necessary the traffic to be coherent. Therefore the players should choose their strategy so that the resolving of the game leads to a unique ending. In case several endings equal -that is when a deadlock is possible-, it is necessary to set up a method leading to the choice of a single ending. To do so, we have simulated the fact that in such a situation, the traffic rules foresee that the driver who possesses the oldest driving license should go first. Hence, a number is assigned to every player in consideration of its age in the simulation. Then the payoff matrix is computed again: for the action go, a player adds the number of the other and subtracts his; for the action stop, he adds his number and subtracts the other's (Figure 9). A mechanism allowing the coordination of n vehicles is henceforth possible. Let us take as hypothesis of reflection that an n-player game (an n-dimensional matrix) can be represented by (n2-n)÷2 two-player games. The problem is then to do the aggregation of the twodimensional matrixes whose cells are two-value vectors into an n-dimensional unique matrix whose cells are n-value vectors. The aggregation method that has been chosen is the sum the payoffs and the formula for the three-player game is presented below (Formula 1). Formula 1: Calculation of a 3-player game matrix by aggregating the three 2-player game matrixes. We now present the example of a traffic situation for which three vehicles are approaching a junction (Figure 10). It is about a three-player game for which it is necessary to determine three-dimensional matrix from the three two-player game matrixes. The resolving of the game indicates that its solution is (stop, go, stop): only the second player (B) enters the junction. In this example, player A's strategy is indecisive because the sums of the payoffs associated to each of both possible strategies are equal (they are worth 0). In that case, no strategy is better than the other is and the one to be chosen is the most careful: stop. A game theory based multi-agent coordination mechanism, dedicated to the coordination of simulated road traffic at junctions, was proposed. It seems relevant for the situations involving three agents. Tests are in progress to empirically verify its validity whatever the situations are.

A \ B Go Stop Go (-50+2-1 , -50+1-2) (50+2-1 , 0+2-1) Stop (0+1-2 , 50+1-2) (0+1-2 , 0+2 -
The generalization of this first version of the mechanism for the situations involving n agents and the generic version of the mechanism allowing the coordination of the traffic inside the junctions are also in progress.

V. Conclusion

Simulated road traffic is a phenomenon produced from the coordination of a heterogeneous, open, dynamic and competitive system, composed of autonomous cognitive agents. We showed that in behavioral simulation the coordination of the system is a distributed resource allocation issue with a real-time constraint. The mechanisms implemented in ARCHISIM to solve ordinary situations of traffic were exposed. They are mainly based on an economic approach and have allowed the validation of ARCHISIM for ordinary situations on highway network.

For critical traffic situations, such as junctions, the issue is more complex because deadlocks may occur. To coordinate the agents' actions at the junctions, we have implemented a game theory based mechanism. A first version of this mechanism, which solve the conflicts by letting agents reach the junction one by one, was presented. This mechanism works for twoplayer game and tests for n-player games are in progress. Currently, we are also working on a generic version of the mechanism. Like for any mechanism utilizing game theory, some limitations may appear. So, besides the fact that all the agents should have the knowledge of the payoff matrix and consequently the preferences of all the players, they must also be homogeneous, i.e. to have initially fixed identical characteristics and capacities. These limitations are for the greater part inherent to the game theory and any mechanism based on this theory should take them into account. For behavioral road traffic simulation, we should remind that the driver does not have a normative behavior and the notion of rationality has to be defined in the context of the driving task. Indeed, in some situations the driver presents a rather individualistic behavior and generally tries to maximize his route, while in some critical situations he is brought to collaborate. Furthermore, even if a driver model may be defined, every driver is different. For all these reasons, this is why -at the moment-we consider our future generic mechanism more like a game theory based mechanism than a game theoric mechanism. While the generic version of the mechanism is not defined, we consider that game theory is, in our context, more like a tool for reflection than a solution by itself.

Figure 1 :

 1 Figure 1: Examples of traffic situations simulated with ARCHISIM

Figure 2 :

 2 Figure 2: Partial Scheme of a simulated driver model

Figure 3 :

 3 Figure 3: Characteristics of road traffic multi-agent simulation

Figure 4 :

 4 Figure 4: Example of an ordinary traffic situation

Figure 6 :

 6 Figure 6: Examples of critical traffic situations: accident blocking a lane (a); insertion on a jammed highway (b); junction with deadlock risk (c)

Figure 7 :

 7 Figure 7: Coordination mechanism implemented for critical situations with priority flow

  Heavy traffic on adjacent lane L 2 Current flow speed on lane L 1 : S 1 Current flow speed on adjacent lane L 2 : S 2 ≅ S 1 Expected situation Evaluation S 1 = 0 for an infinite time S 2 decreasing ; S 2 > S 1 because lane changing expected from vehicles currently on L 1

Figure 8 :

 8 Figure 8: Payoff matrix for each of the three types of situation implying two vehicles

Figure 9 :

 9 Figure 9: Consideration of the players' number when several ending are possible

∀

  1) Prio(A,B) and Prio(B,A) :A is the oldest : A is player #1 and player B is player #2 P = (player1, player2, player3) : set of the three players of the game ∀ a ∈ P ; b ∈ P -a ; c ∈ P -a -b ∀ S a , S b and S c ∈ (Go, Stop) : possible strategies of the players ∀ M ab , M bc , M bc : matrixes of the 2-player games ∀ M abc : matrix of the 3-player game M abc (S a ,S b ,S c ) a = M ab (S a ,S b ) a + M ac (S a ,S c ) a with M abc (S a ,S b ,S c ) a : payoff for player a and strategy (S a ,S b ,S c ) page 10/12

Figure 10 :

 10 Figure 10: For a three-player game, calculation of the matrix from the three two-player game matrixes

  ,A) and ¬Prio(A,B) ; Prio(A,C) and ¬Prio(C,A) ; Prio(B,C) and Prio(C,B)

  [Holt 88, Malone 90]. In Distributed Artificial Intelligence, one generally agrees that coordination is managing dependencies between agents' activities. An activity is a set of tasks requiring resources. The categorization of the different types of dependencies between tasks and resources leads to the identification and the analysis of the coordination processes. One of the more studied coordination processes is common resource management. In this case, an effective organizational structure manages the conflicts between the agents likely to use the same resource. The coordination mechanisms that manage shared resource dependencies are resource allocation mechanisms[Steeb 81, Sathi 89]. Although the work done to formalize and generalize multi-agent coordination methods is significant[Bond 88, Jennings 96, Lesser 98], coordination mechanisms are often limited to an application. The efficiency of coordination mechanisms mainly depends on the context.

  That allows to consider the games as the simplest models of conflict situations[von Neumann 44, Duncan 85]. A game can be represented by an n-dimension table, n being the number of players. For a two-player game, every line of the table represents one of the actions player P 1 can follow, and every column represents one of player P 2 's. The table's cells correspond to a result of the game, which is a payoff vector. This representation is named strategic form or payoff matrix.

	The use of the game theory to implement a multi-agent coordination mechanism was
	considered, probably for the first time, by Rosenschein [Rosenschein 85]. His work deals with
	coordination by negotiation and a certain generalization of this type of mechanism of
	coordination was evoked [Rosenschein 94].

Acknowledgement

The authors would like to thank Mr Torjemane and Mr Lancelin of SRILOG for their support. This work was partly sponsored by ANRT through a research grant CIFRE.


Matthieu Reymond

Thomas Doualle

Jerome Sercombe

Mathilde Rocca

Mehdi Nasselahsen

Christian Colin

Yves Pontillon

Laurent Gallais

High power laser heating of nuclear ceramics for the generation of controlled spatio-temporal gradients

The use of high-power laser to reach the high-temperatures typical for off-normal conditions in nuclear reactor fuel or to determine its properties at high temperature is a relevant and proven technique in the nuclear research community. Conducting studies at high temperature with full control in space and time of the thermal gradients is however a challenge. In this paper, we present a combination of methodologies, experiments and models to drive nuclear fuel samples in thermal conditions relevant to the study of nuclear fuel materials. Based on the use of high-power (kW) lasers with beam shaping and temporal laser control, our experimental approach allows one to heat depleted UO2 samples and obtain precise control of the temporal gradients, in the ms range, and spatial gradients, in the 100 µm range. The coupling with numerical simulations allows one to determine the temperature distribution in the depth of the sample and to develop a thermo-mechanical interpretation of the results. We will present the methodology allowing one to properly compare experimental and simulation results by taking into account the optical response of the instrumentation, the laser-UO2 interaction and the resulting heat source term. We will show that we can express the heat source term resulting from the laser loading either by an analytical description or statistically based on Monte-Carlo simulation. A fairly good agreement between the recorded temperature and the simulation results is shown. The presented methodology can be extended to other materials with a proper choice of laser wavelength and instrumentation based on the optical properties of the investigated material.

I. Introduction:

Laser heating is a routinely used technique in various fields to weld, cut, vaporize or anneal materials. It is also used to study materials at high temperature [START_REF] Boehler | Portable laser-heating stand for synchrotron applications[END_REF][START_REF] Anzellini | A Practical Review of the Laser-Heated Diamond Anvil Cell for University Laboratories and Synchrotron Applications[END_REF][START_REF] Minissale | The temperature dependence of optical properties of tungsten in the visible and near-infrared domains: an experimental and theoretical study[END_REF][START_REF] Gallais | High power continuous wave laser heating of graphite in a high temperature range up to 3800 K[END_REF]. In the nuclear research community, laser heating is used to study nuclear materials properties at high temperature or to investigate its behavior under off-normal conditions. We can cite for example the work carried out at the Institute for Transuranium Elements (Karlsruhe, Germany) for the determination of the fusion point of various nuclear materials [START_REF] Manara | Solidus and liquidus of UO2+x from high-pressure melting experiments[END_REF][START_REF] Manara | Advances in the Experimental Determination of the Uranium-Oxygen Phase Diagram at High Temperature[END_REF][START_REF] Bruycker | The melting behaviour of plutonium dioxide: A laser-heating study[END_REF][START_REF] Prieur | Melting behaviour of americium-doped uranium dioxide[END_REF] or the determination of thermo-physical properties such as the thermal diffusivity, thermal conductivity or specific heat of UO2 [START_REF] Vlahovic | Thermal diffusivity of UO2 up to the melting point[END_REF][START_REF] Pavlov | Measurement and interpretation of the thermo-physical properties of UO2 at high temperatures: The viral effect of oxygen defects[END_REF][START_REF] Pavlov | A new method for the characterization of temperature dependent thermo-physical properties[END_REF]. Other notables studies include the study of the optical properties of solid and liquid UO2 heated by laser by Bober et al [START_REF] Bober | Spectral Reflectivity and Emissivity Measurements of Solid and Liquid U02 at 458, 514.5 and 647 nm as a Function of Polarization and Angle of Incidence[END_REF][START_REF] Bober | Determination of the Optical Constants of Liquid UO 2 in the Near Infrared at 1064 nm[END_REF] or spectrometric analysis of laser induced vapors of UO2 by M. Joseph et al [START_REF] Joseph | Laser-Induced Vaporization-Mass Spectrometry Studies on Refractory Materials at Ultrahigh Temperatures[END_REF] and R. Pfiegler et al [START_REF] Pflieger | Thermodynamics of Refractory Nuclear Materials Studied by Mass Spectrometry of Laser-Produced Vapors[END_REF].

The work presented hereinafter is part of recent developments, conducted in the framework of nuclear fuel research at CEA (French Atomic Energy Commission), based on a multi-objective laser platform for the study of nuclear fuel in accidental conditions [START_REF] Vidal | Simulation of reactivity initiated accident thermal transients on nuclear fuels with laser remote heating[END_REF][START_REF] Reymond | Thermo-mechanical simulations of laser heating experiments on UO 2[END_REF] or the determination of thermo-physical properties at high-temperatures [START_REF] Doualle | Development of UO 2 thermal diffusivity measurement with laser techniques[END_REF]. These studies are enabled by the fact that laser heating is allowing one to heat any absorbing material of interest to extreme temperature with an unmatched precision regarding the spatial or temporal energy deposition in the sample. The temperature rising rate can also be finely tuned, depending on the pulse duration, the available laser power, absorption of the considered material and heated surface. The laser-heating approach can also be easily coupled with contactless instruments such as pyrometers, infrared cameras, spectrometers, optical microscopy, or any other relevant instrumentation.

However, this approach often needs to be coupled with the precise simulation of the heating sequence if one wishes to interpret experimental results in a meaningful way. In particular, the non-isothermal conditions induced by laser-heating often makes the interpretation of the results difficult without the modeling of the experiment. Above all, the internal temperature gradient in the depth of the sample cannot be observed nor measured. The simulation of the experiment is thus essential to properly interpret the results. This remark is particularly true under non-steady state condition. Finally, the deduction of the mechanical load induced by the thermal gradient also necessitates a thermal and mechanical numerical model.

In this paper, we present laser-heating experiments of various duration (ms range), power, and intensity distributions of the used laser pulse(s) conducted on depleted polycrystalline UO2 disks aiming to induce fast heating (>10 000 K/s) with a precise control of the spatial gradient in order to reach thermal conditions representative of accidental conditions encountered in a nuclear plant such as the Reactivity Initiated Accident (RIA) [START_REF] Vidal | Simulation of reactivity initiated accident thermal transients on nuclear fuels with laser remote heating[END_REF][START_REF] Guénot-Delahaie | Simulation of reactivity-initiated accident transients on UO 2 -M5® fuel rods with ALCYONE V1.4 fuel performance code[END_REF]. The temperature evolution during these parametric studies are obtained through thermographic imaging means and compared to simulations performed with a numerical model based on the finite element method using the thermal solver Cast3m [20] developed by the CEA.

In order to thoroughly analyze the results through numerical means, we first performed the optical characterization of the considered nuclear ceramic. Indeed, the knowledge of the optical properties of the material is important to accurately describe the absorption of the laser radiation by the samples. To do so, we realized reflectance and transmittance measurements from which we deduced the complex refractive index of the ceramic. Then, experimental techniques and methods used to realize the laser heating experiments are presented. In a second time, we present the numerical model used, the associated material parameters and the description of the heat load resulting from the laser beam propagation and absorption in the depth of the sample. We propose two methods to describe the laser induced heat source term, either by an analytical description or statistically by a preliminary step of Monte-Carlo simulation for light propagation in the sample. The later allowed the phenomenological study of the potential contribution in the simulations and resulting temperature fields of the scattering phenomenon. A fairly good agreement between the recorded temperature and the simulation results was obtained, indicating that all the relevant phenomena are taken into account.

To conclude, laser beams of different intensity profiles were combined in order to reach RIA like thermal conditions, thus demonstrating the flexibility of the laser heating technique in terms of reachable temperature transients.

II. Experimental methods:

This work focuses on depleted UO2 disks of thickness 0.5 or 1 mm. These disks are obtained from dense (~96-98% of the theoretical density, measured by hydrostatic weighting in alcohol) and polycrystalline UO2 fuel pellets fabricated at CEA using a wet process followed by isostatic compaction at 450 MPa and sintering at 1650 °C during 2h under flowing H2. These pellets have a diameter of around 8.2 mm and a height of around 13.5 mm. They were manufactured without dishes or chamfers, as opposed to their industrial versions. Their average grain size is about 10 µm. The disks were obtained from the mother pellets with a wire saw and consequently, their faces presented a typical rough-cut surface rugosity.

1) Complex refractive index of polycrystalline UO2:

In order to describe as precisely as possible the laser loading in the simulations, a complete and thorough characterization of the optical properties of the considered material is necessary. In particular, the knowledge of the absorption and of the coefficient of absorption of the material at our laser wavelength (1080 nm) is crucial, the former describing the amount of power coupled in the material and the latter describing the depth of penetration of the incoming laser light in the material. Hence, the value of the absorption and of the coefficient of absorption intervenes at first order on the calculated temperatures at the surface and in the volume of the sample. It is thus important to determine these values.

Very few characterizations of the optical properties of UO2 have been reported in the literature. Moreover, there are some discrepancies in the published values of optical properties of UO2, possibly related to differences in material composition or microstructure. We also note that almost all studies were interested in monocrystalline or amorphous UO2 without consideration for the polycrystalline material [START_REF] Ruello | Thermal variation of the optical absorption of UO2: determination of the small polaron selfenergy[END_REF][START_REF] Ackermann | Visible and Ultraviolet Absorption Properties of Uranium Dioxide Films$\ast[END_REF][START_REF] Meek | Some optical properties of intrinsic and doped UO2 thin films[END_REF][START_REF] Griffiths | Absorption spectrum of single-crystal UO2: Identification of and effect of temperature on the peak positions of essentially all optical transitions in the visible to near infrared regions using derivative spectroscopy[END_REF].

Different approaches exist for the determination of the optical properties of a material, based either on spectro-photometric or ellipsometric measurements. Most studies conducted on UO2 were based on spectro-photometric measurements such as the absorption, transmission or reflection spectra. Briefly, this approach is based on the measurements of one or more spectra (transmission, reflection or absorption) at different angles of incidence or polarization state. By direct inversion of the spectra or through numerical optimization means, one can infer the optical properties of the sample. The spectro-photometric approach is also suitable for scattering samples and is thus more general.

Consequently, the complex refractive index of UO2 was determined in the visible and nearinfrared wavelength range (500 -1100 nm) from spectrophotometric reflectance and transmittance measurements performed of thin samples of thicknesses inferior to 55 µm. Two methods were used to deduce the complex refractive index from the measurements. The first one is a straightforward inverse method using analytical equations describing the light-UO2 interaction and light propagation in the thickness of the sample while the second one is based on a more classical approach using a matrix formalism to describe light propagation in the considered optical system, combined with an appropriate dispersion model coupled with a numerical minimization algorithm. These two methods yielded similar results.

We report for the imaginary part of the refractive index k values around 0.006 below the band-gap energy, in the 650-1100 nm wavelength range. We report decreasing values of n between 2.3 to 2.0 with increasing wavelength. These values are consistent with reported values of the real part of the refractive index in the literature [START_REF] Bober | Spectral Reflectivity and Emissivity Measurements of Solid and Liquid U02 at 458, 514.5 and 647 nm as a Function of Polarization and Angle of Incidence[END_REF][START_REF] Ackermann | Visible and Ultraviolet Absorption Properties of Uranium Dioxide Films$\ast[END_REF]. The absorption coefficient decreases from 200 000 m -1 at 550 nm towards a value of roughly 70 000 m -1 in the nearinfrared part of the spectrum. Its inverse (the so-called depth of penetration) is thus around 15 µm at our laser wavelength (1080 nm).

The samples preparation, measurements, methodology and interpretation of these findings are available in this supplementary file of this study.

2) Experimental configurations:

The experiments were conducted on the CHAUCOLASE platform (CHAUffage COntrôlé par LASEr/laser-controlled heating) located at the Institut Fresnel. This platform notably consists in two continuous wave high-power Ytterbium fiber laser and a dedicated vacuum chamber equipped with multiple viewports allowing for the transmission of the laser beams and optical instrumentation. More details on this platform can be found in [START_REF] Minissale | A high power laser facility to conduct annealing tests at high temperature[END_REF]. To avoid the oxidation of the sample during heating, the experiments were conducted under vacuum (10 -2 mbar). From the aforementioned value of the coefficient of absorption of polycrystalline dense UO2, the use of Ytterbium fiber laser (wavelength 1080 nm) is pertinent to submit uranium dioxide to laser heating. The first laser used can deliver up to 1500 W of maximum power (SPI laser Qube 1500) with a mono-mode Gaussian beam and a rising time of a few microseconds. The beam is non-polarized and is characterized by a central wavelength of 1080 nm with a bandwidth <4 nm. In this study, a beam diameter of 8 mm (Gaussian distribution with a diameter defined at 1/e²) was used. This laser was used to generate an annular Gaussian beam with a Diffractive Optical Element (DOE, Silos Technologies) as previously described in [START_REF] Vidal | Simulation of reactivity initiated accident thermal transients on nuclear fuels with laser remote heating[END_REF][START_REF] Reymond | Thermo-mechanical simulations of laser heating experiments on UO 2[END_REF]. This annular Gaussian beam is characterized by its radius Ran and waist (at 1/e²) ωan. Two dimensions of annular beams were used in this study:  3.9 mm radius and 0.1 mm waist, hereinafter called focused annular beam.  3.660 mm radius and 0.260 mm waist, hereinafter called unfocused annular beam.

Images of these two beams profiles in the sample's plane can be viewed Figure 1. A second laser was used, which can deliver up to 2000 W of maximum power with a multi-mode laser beam (SPI Laser Qube 2000). This beam was used to generate a quasi Top-Hat (TH) beam. To do so the beam is first focused with a lens of focal length 500 mm (Eksma optics, AR coated at 1064 nm), thus obtaining a beam with a diameter (at 1/e²) inferior to 1 mm in the focal plane. This focused beam presents an intensity profile that can be assimilated as a Top-Hat distribution (super Gaussian distribution of power 6). We then reimaged on the sample this quasi TH beam with an air-spaced achromatic doublet (Thorlabs ACA254-100-B) placed in the focal plane of the first lens. This apparatus was placed in such a way that the TH beam has a diameter slightly superior to the heated disk diameter in the image plane where the sample was placed. The obtained diameter was about 9 mm in the sample plane versus 8.2 mm for the sample diameter. An image of the beam profile and normalized profiles plot is shown Figure 2. The standard deviation measured on the normalized profile is 5%. These two beams can then be combined and adjusted in terms of dimensions, temporal profiles or power to tune the generated transient.

In this study, we used a custom sample holder (Sceram Advanced Ceramics) made of ZrO2 with some parts made of stainless steel. This sample holder was specifically designed to ensure known thermal boundary conditions at the surfaces of the sample. It relies on the use of two jaws maintaining the sample vertically by its edge. The top jaw applies a pressure through a spring-piston system. The jaws are made of ZrO2, which has a low thermal conductivity, thus ensuring that the heat losses through the contact points are minimal, especially in transient conditions such as the ones encountered in this study. Two untreated sapphire windows (Edmund Optics) were placed in front of each face of the sample to ensure a containment of the fragments in case of loss of integrity of the sample. This sample-holder was then mounted on a 5-axis alignment stage (Newport, 9081-M) and placed at the center of the vacuum chamber. CAD views of the sample holder with an UO2 disk are shown Figure 3.

Finally, a laser diode of a few tens of watts operating at 808 nm (LIMO, F200-DL808) was used on the non-heated face of the disk to pre-heat it to the desiderated temperature. Its beam is Gaussian with a waist (1/e²) of 8.5 mm.

A schematic of the experimental setup is visible Figure 4. 3) Temperature measurement:

The sample's surface temperature was measured with an high-performance infrared camera (FLIR, x6901sc-MWIR) operating within 3-5 µm wavelength range. The camera's rate of acquisition was set to 1000 Hz. The sample was observed by the camera at a 45° angle and synchronized with the laser pulses through a pulse generator (Berkeley Nucleonics, 577). In this study, the camera was equipped with a 100 mm focal length lens and a 0.75 inches (0.254 cm) extension ring was placed between the camera and the lens. The use of the extension ring improved the spatial resolution, reaching a value of 85 µm/pixel for a field of view of 54x43 mm. The use of the extension ring did not induce any vignetting in the central region of the sensor used to observe the sample.

The optical impulse response (Point Spread Function -PSF) of this thermographic system (camera + lens + extension ring) was characterized with the knife-edge method [START_REF] Estribeau | Fast MTF measurement of CMOS imagers using ISO 12333 slanted-edge methodology[END_REF][START_REF] Lane | Calibration and Measurement Procedures for a High Magnification Thermal Camera[END_REF][START_REF] Mooney | Measurement of the modulation transfer function of infrared cameras[END_REF]. This method is based on a measure of the Edge Spread Function (ESF) and enables the extrapolation of the PSF or the Modulation Transfert Function (MTF) of the imaging system. The ESF measurement was performed by positioning a sharp razor edge in front of a blackbody source (Newport, Oriel) set to a temperature of 323.15 K (50 °C). The camera observed the edge through the same viewport used on the vacuum chamber and same distances as in experimental condition.

The estimated PSF of the camera is visible Figure 5 and can be approximated to a single Gaussian function with a waist (1/e²) of 7 pixels (~600 µm). It is important to note that the dimension of the PSF is thus larger than that of the annular beams, especially in the case of the focused one (0.1 mm waist). We can thus expect the thermographic measurements to be affected by the optical response of the camera. This should be taken into account during the analysis and interpretation of the measurements.

Finally, the emissivity of UO2 is taken as 0.85 according to [START_REF]Thermophysical properties database of materials for light water reactors and heavy water reactors: final[END_REF].

III. Numerical tools:

The thermal simulations were performed on the thermal solver Cast3m of CEA [20] based on the finite elements method. The model considers a 2D axisymmetric representation of a cross-section of a disk. It considers radiative heat losses and laser loading is prescribed as a volumetric heat source. The non-linear heat flow equation is then solved to obtain the temperature distribution in the sample. The thermal properties are taken from [START_REF]Thermophysical properties database of materials for light water reactors and heavy water reactors: final[END_REF][START_REF] Fink | Thermophysical properties of uranium dioxide[END_REF].

A detailed description of this model and of the used properties for UO2 is given in reference [START_REF] Reymond | Thermo-mechanical simulations of laser heating experiments on UO 2[END_REF]. In this previous publication, the laser loading was described as a surface heat flux. We now can propose a refined analytical description of this laser loading as a volumetric heat source considering the optical characterization presented above.

1) Analytical description of the laser loading:

The absorption of the laser radiation in the depth of the sample is described with a Beer-Lambert law:

𝐼(𝑧) = 𝐼 0 𝑒 -𝛼 𝑎 𝑧 ( 1)
With I0 the intensity distribution (W/m²) at the surface of the sample. In 2D axi-symmetrical representation we thus have I0 = f(r). αa is the coefficient of absorption in m -1 expressed as

𝛼 𝑎 = 4𝜋𝑘 𝜆
, with k the coefficient of extinction and λ the wavelength. z is the depth relative to the surface.

The volumetric heat source is easily obtained by derivation:

𝑄(𝑧) = - d dz (𝐼 0 exp(-𝛼 𝑎 𝑧)) = 𝐼 0 𝛼 𝑎 exp(-𝛼 𝑎 𝑧) (W/m 3 ) ( 2)
A. Annular beam:

The intensity distribution of the annular Gaussian beam I0 (in W/m²) is given by the following expression:

𝐼 0 (𝑟, 𝑅 𝑎𝑛 , 𝜔 𝑎𝑛 ) = 𝐼 𝑚𝑎𝑥 (𝑅 𝑎𝑛 , 𝜔 𝑎𝑛 , 𝑃 𝑎𝑛 ) ex p ( -2(𝑟 -𝑅 𝑎𝑛 ) 2 𝜔 𝑎𝑛 2 ) (W/m²) ( 3)
With 𝐼 𝑚𝑎𝑥 the peak intensity (in W/m²) of the Gaussian distribution at 𝑟 = 𝑅 𝑎𝑛 , which is a function of 𝑅 𝑎𝑛 and 𝜔 𝑎𝑛 , the radius and waist of the annular beam, and of beam's power 𝑃 𝑎𝑛 . Hence, we finally have:

𝑄 𝑎𝑛 (𝑟, 𝑧) = 𝐼 𝑚𝑎𝑥 exp ( -2(𝑟-𝑅 𝑎𝑛 ) 2 𝜔 𝑎𝑛 2 ) 𝛼 𝑎 exp(-𝛼 𝑎 𝑧) W/m 3 ( 4)
B. Top-Hat beam:

The intensity distribution of the Top-Hat beam is considered as homogeneous. To take into consideration the fact that the beam diameter is superior to the disk diameter, we consider the ratio the surface of the disk 𝑆 𝑑 and the surface of the beam 𝑆 𝑏 .

Hence, the heat source term is expressed as:

𝑄 𝑇𝐻 (𝑧) = 𝑃 𝑇𝐻 𝑆 𝑏 𝑆 𝑑 𝑆 𝑏 × 𝛼 𝑎 exp(-𝛼 𝑎 𝑧) W/m 3 ( 5)
With 𝑃 𝑇𝐻 the power of the beam. Lasers' power have been measured with a calibrated calorimeter to ensure known values of 𝑃 𝑎𝑛 and of 𝑃 𝑇𝐻 .

These source terms are then multiplied by a loss coefficient taking into account the total reflectance of the sample. The total reflectivity Rtot of the samples that combines specular and diffuse reflected light by the disks was measured with the same setup that the one used for the optical characterization of UO2. This loss coefficient is around 15% at 1080 nm (laser's wavelength). Additionally, the reflection coefficient of the sapphire windows is also took into account. The losses occurring on the laser line were also measured (on mirrors, DOE, viewports etc.) and were found to be negligible (all optics have anti-reflection or high reflective coatings at 1080 nm).

2) Heat source term obtained from Monte-Carlo simulation

The previous expressions for the source terms assume a rectilinear propagation of the laser radiation through the sample without any scattering effects. However, the propagating laser radiation can potentially be scattered by grain boundaries or bulk porosities present in the material. The (bulk) scattering properties of a material is characterized by the angular distribution of the scattered light (i.e., its phase function) and the scattering coefficient 𝛼 𝑠 (m -

1

). With the previously mentioned coefficient of absorption 𝛼 𝑎 (m -1 ), we can thus express the total attenuation coefficient 𝛼 𝑡 = 𝛼 𝑎 + 𝛼 𝑠 . To the best of our knowledge, there are no existing study in the literature regarding light scattering by polycrystalline UO2. The determination of these quantities is not trivial and would require, for example, an interpretation of spectrophotometric measurements trough inverse Monte-Carlo simulation. As a first step, we hereinafter present a Monte-Carlo simulation model that was used to simulate the propagation of light in an UO2 disk to obtain a heat source term that can then be used in the thermal simulation. This model was used to investigate the potential impact of the scattering phenomenon in the results section. It is important to note that if, in the present study, the material considered is close to its theoretical density (~95-98 %), nuclear ceramic can be subject to microstructural changes resulting in an increase in porosity of up to 20 %. As such it is also crucial to develop tools capable of accounting for these microstructural changes that would lead to a modification of the scattering power of the material.

The MCmatlab [START_REF] Marti | MCmatlab: an open-source, userfriendly, MATLAB-integrated three-dimensional Monte Carlo light transport solver with heat diffusion and tissue damage[END_REF] program, a Monte Carlo simulation program coded in C++ and interfaced with MATLAB, was utilized for modeling light propagation in a 3D voxel space .The interested reader may refer to [START_REF] Wilson | A Monte Carlo model for the absorption and flux distributions of light in tissue: Absorption and flux of light in tissue[END_REF][START_REF] Wang | MCML -Monte Carlo modeling of light transport in multi-layered tissues[END_REF] for more information about Monte-Carlo simulations for light propagation in a scattering media.

In this study, we considered a cuboid of dimension 8.4 x 8.4 x (DT+10 -3 ) mm 3 (DT being the Disk Thickness) divided into voxels of dimensions 5 x 5 x 1 µm 3 . Each voxel was assigned a value of αa and αs. The program considers the Henyey-Greenstein [START_REF] Henyey | Diffuse radiation in the galaxy[END_REF][START_REF] Binzoni | The use of the Henyey-Greenstein phase function in Monte Carlo simulations in biomedical optics[END_REF] phase function which reads:

𝑃 𝐻𝐺 (𝛩, 𝑔) = 1 4𝜋 1 -𝑔² (1 + 𝑔 2 -2. 𝑔. cos𝛩) 3 2 ⁄ ( 6)
With 𝛩 the angle of direction of propagation of the scattered light in radian and g the so-called anisotropy factor in the range ]-1;1[. A negative value of g indicates a backward dominant scattering, 0 an isotropic scattering and a positive value a forward dominant scattering. Each voxel of the simulation space was thus also assigned a value of g.

Custom illumination profiles such as annular Gaussian beam can be generated by MCmatlab, making it the suitable tool for this study. It can also simulate refraction and Fresnel reflection at interfaces between media of different refractive index, provided that the interface is parallel to the xy-plane and we thus also assigned a refractive index value (n) for each voxel. The value for UO2 was set to 2.1 based on the measurements and methodology presented in Section II.1) and in the supplementary file. These values were assigned to each voxel in order to obtain the considered disk immersed in air with a thin layer (10 µm) of air on top of it. The disk was then illuminated with the annular or top-hat beam intensity distribution. A laser intensity volumetric distribution (W.m - 2 ) within the disk was obtained at the end of the Monte-Carlo simulation. This intensity distribution was then multiplied by the absorption coefficient of UO2 to obtain a volumetric heat source term (W.m -3 ) that was interpolated on the Cast3m's mesh. The resulting simulation sequence is illustrated in Figure 6. A typical radial temperature profile measured on the vertical axis passing by the center of the disk with a false color image of the thermographic acquisition is visible on Figure 7. In this specific case, it is observed that the highest recorded temperature corresponds to the 19th acquired image. Given the acquisition rate of 1000 Hz and the synchronization of the camera and laser pulses, this observation is consistent with the expected temporal uncertainty of 1 ms. As stated in Section II. 2), it is anticipated that the optical response of the lens+camera system would greatly distort the thermal radiation image of the heated sample in regards of the annular beam dimensions versus the PSF dimensions. As expected, the calculated radial temperature profiles are consistently narrower, with the maximum measured temperature being up to twice as high as the measured value. Consequently, the measurement-calculation error in terms of peak temperature span from 40% to 120%.

To account for the blurring effect caused by camera aberrations, the simulated radial temperature profiles were convolved with the two-dimensional representation of the PSF, known as the Line Spread Function (LSF). This approach was chosen after unsuccessful attempts with available deconvolution algorithms. The convolution operation was implemented numerically using a Python script and the "convolve" function from the Scipy package. The LSF was previously normalized to have a numerical integral equal to 1.

Figure 8 presents the radial temperature profiles measured in two annular heating experiments (AH 11 and 2), along with the raw simulated and simulated-convolved radial temperature profiles. As the AH2 experiment initial temperature was around 573 K, the edge's temperature was also recorded during the transient. This effect is not seen for the AH11 experiment which had an initial temperature set around 303 K. The convolution operation significantly improves the agreement between the calculated and simulated temperature profiles. It is important to note that this implies that the simulated temperature profiles are arguably closer to the actual temperature profiles of the samples compared to the measured profiles, which are heavily distorted by camera aberrations. It is also interesting to examine the normalized temperature profiles. Figure 15 displays the normalized profiles at the end of the heating sequence for the same two annular heating experiments, AH 11 and AH 2. Upon observation, it is evident that the convolution performed with the LSF does not perfectly replicate the shape of the measured profiles. This suggests that scattering occurring within the sample's depth may not be negligible and could result in the broadening of the laser beam within the sample, subsequently affecting the corresponding heat source term. A. Phenomenological study of the impact of laser light scattering:

In this section, the Monte-Carlo solver presented in Section III was used to quantify the potential impact of scattering. To do so, a parametric sweep considering 27 sets of material parameters (𝛼 𝑎 ,𝛼 𝑠 , g) was conducted. The values of the considered absorption and scattering coefficient are given Table 2. For each combination of this table, three source terms were calculated with g taking a value of 0.9, 0.75 or The parameters were chosen in such a way that the coefficient of total attenuation (αt) is equal to 7x10 4 m -1 +/-5x10 4 with different proportion of scattering and absorption. This value of 7x10 4 m -1 roughly corresponds to the value of the coefficient of absorption determined previously where the scattering phenomena (i.e., 𝛼 𝑡 = 𝛼 𝑎 ) was neglected. δa (µm) αa (m -1 ) δa (µm) αs (m -1 ) αt=αa+αs (m -1 ) 20 5x10 4 10 1 x 10 5 1.5 x 10 5 10 1x10 5 20 5 x 10 5 1.5 x 10 Figure 10 illustrates the temperature profiles and normalized temperature profiles obtained with the parametric sweep at the end of the AH12 experiment, along with a comparison to the analytical description of the heat-source term. In Figure 11, the axial temperature profiles at r=Ran for the same experiment at the same time are displayed. The influence of the anisotropy coefficient g was found to be negligible, and therefore, the plotted profiles correspond to a g value of 0.75. Although the Monte-Carlo calculations demonstrates that scattering can indeed alter surface temperatures and the temperature distribution below the surface, we did not observe a significant broadening of the temperature profiles. This can be observed in the normalized profiles presented in Figure 10. We can conclude that while the scattering effect can potentially play an important role, it does not appear to induce a significant broadening of the heat source term within the investigated parameters range. A complete characterization of the scattering behavior of the ceramic at the considered wavelength will however remain of interest if one wishes to simulate as accurately as possible the heat source term resulting from the laser loading.

From this parametric sweep, the set of parameters that led to the best overall improvement in the simulation-measurement comparison was retained. The best fit parameters are as follow:

 αa = 2x10 4 m -1  αs = 2x10 4 m -1  g = 0.75
Figure 12 presents a comparison of the peak surface temperature between the simulations and the measurements for the case of the analytical heat source term, analytical heat source term with the convolution by the LSF and Monte-Carlo heat source term with the convolution by the LSF, considering the aforementioned best fit parameters. As one can see, taking both into accounts the camera's PSF and the scattering phenomena leads to a fairly good result. We however stress again that the scattering parameters were not measured and are issued from a parametric sweep.

Homogeneous laser heating experiments:

Sixteen homogeneous laser heating experiments were performed on 16 different samples, involving variations in parameters such as duration (20 or 100 ms), power (ranging from 375 to 1900 W), and initial temperature (573 or 873 K, equivalent to 300 or 600 °C). The maximum temperatures recorded during these experiments span a range from around 773 to 2773 K, corresponding to 500 and 2500 °C. From the images, it can be observed that the heating is relatively homogeneous across the entire surface. The observed inhomogeneities can be attributed to the modulation of the laser beam, which is not perfectly uniform. Due to the relatively homogeneous temperature profiles, the impact of camera aberrations on the measurements is negligible. The comparison between the simulations using the analytical description of the laser loading and the measurements resulted in an error percentage of less than 15% for most experiments.

N°

Additionally, the use of the Monte Carlo solver, considering the selected set of scattering parameters, did not lead to an improvement in the error percentage compared to the measurements. In fact, the simulated temperatures were consistently lower than the measured temperatures. The comparison between the measurements and simulations using both the analytical and statistical descriptions of the heat source term is summarized in Figure 20. These findings suggest that the parametric sweep conducted with the annular experiments, along with the selected set of parameters, may have compensated for certain error sources specific to the annular experiments. Therefore, a comprehensive analysis of other potential sources of uncertainty is now deemed necessary. However, based on the observation that the error between the simulation and measurements does not vary significantly with increasing temperature during the annular or homogeneous heating experiments, it can be inferred that the emissivity and optical properties of the ceramic are not strongly dependent on temperature. This suggests that considering the evolution of emissivity and complex refractive index with temperature may not have a significant impact on reducing errors in the simulation or measurement.

Finally, in the case of a strongly localized heating such as the one obtained with the thinnest annular beam (waist at 1/e² of 100 µm), it is possible that the thermally affected area was in any case insufficiently resolved with the lens at our disposal. The use of a better performing lens would enhance both the quality of the measurements and the precision of the ESF measurement, and ultimately reduce experimental errors.

Combined laser heating experiments:

In this section, we present two laser heating experiments that combine the annular laser bean and the Top-Hat laser beam. These experiments highlight the versatility of our approach in terms of spatial energy deposition and the resulting temperature profiles. The desired temperature profile, characterized by a uniform temperature increase across the entire sample surface with a peak at the periphery, closely resembles the temperature gradient experienced by nuclear fuel elements during a Reactivity Initiated Accident. In both cases the unfocused annular beam (waist at 1/e² of 260 µm) was used. The experimental parameters for the two combined laser heating (CLH) experiments are summarized Table 4.

We report on Figure 15 the measured temperature profiles at different times for both experiments. These temperature profiles exhibit a nearly flat increase in temperature across the entire radius of the sample with a peak at its periphery, as anticipated. The maximum rate of temperature rise is approximately 75 000 K/s. The profiles display slight dissymmetry due to the characteristics of the Top-Hat beam, which slightly under or overheated one side of the sample. These experiments demonstrate the ability of our technique to induce complex temperature profiles on a sample by combining different beams of different intensity profiles. V. Conclusions:

The aim of this study was to investigate the interaction between laser and UO2 (depleted dense UO2) in order to induce rapid transients and understand the resulting temperature gradient.

Initially, we conducted spectro-photometric measurements on low thickness (< 55 µm) samples of the same material. These measurements allowed us to determine the complex refractive index of polycrystalline UO2. The refractive index values at 1080 nm were found to be approximately 2.1 (real part) and 0.006 (imaginary part).

Next, we presented our experimental approach, which built upon previous developments, along with the associated instruments and their characteristics. This included the impulse response of the thermographic camera used in the experiments. We also presented a 2D numerical model based on the finite element method to analyse the experimental results. Additionally, a 3D Monte-Carlo simulation scheme was introduced to simulate light propagation within the sample, taking into account scattering of the laser beam and its potential impact on the resulting temperature distribution.

For the highly localized annular heating experiments, we demonstrated that it is crucial to consider the impulse response of the camera in order to compare the simulation results with the measurements accurately. We then employed the Monte-Carlo simulation scheme to assess the potential influence of scattering within a specific range of optical parameters. By considering both the camera impulse response and light scattering, the error between the simulations and measurements was reduced from over 80% to less than 20%.

For the Top-Hat beam experiments, the thermographic acquisitions were found to be less affected by optical aberrations of the camera. A percentage error of approximately 15% was obtained without requiring the convolution step used in the annular case. This validates the analytical description of the laser loading, to some extent. However, it was observed that the previously obtained set of parameters for the Monte-Carlo simulation did not improve the comparison between measurements and simulations in this configuration. This indicates the necessity of a comprehensive experimental characterization of the scattering phenomenon. Additionally, determining the temperature dependence of the optical properties would ensure more precise measurements and simulations, even though we anticipate that these properties are not highly temperature-dependent. Furthermore, we presented two heating sequences where the annular and Top-Hat beams were combined to induce complex transients in the sample. The resulting temperature profiles qualitatively resemble the temperature profiles experienced by nuclear fuel elements during Reactivity Initiated Accident (RIA) transients in pressurized water reactors. In a previous publication [START_REF] Reymond | Thermo-mechanical simulations of laser heating experiments on UO 2[END_REF], we demonstrated that the mechanical state in the sample during annular laser heating experiments also resemble, at the periphery of the sample, the state reached in a pellet during a RIA transient in a reactor. We now proposed in this study a complete characterization and interpretation of the thermal behaviour during these experiments and expanded the capabilities of our approach towards more complex thermal transient. This demonstrates the suitability of our technique for studying nuclear fuel under specific accidental conditions.

We believe that our approach can be extended to other applications for studying various materials at high temperatures or under prompt and specific thermal gradients, the induced thermal transient being heavily parametrizable.

Figure 1 :

 1 Figure 1: Left: image of the beam profile of the focused annular beam. Right: image of the beam profile of the unfocused annular beam.

Figure 2 :

 2 Figure 2: Horizontal and vertical normalized intensity profiles. Insert: False color image of the normalized homogeneous beam in the sample's plane.

Figure 3 :

 3 Figure 3: CAD view of the sample holder. The 5-axis alignment stage is visible on the left. All the parts directly in contact of the sample or around it are made of ZrO2, sapphire windows excepted.

Figure 4 :

 4 Figure 4: Schematic of the laser-heating experiment. The grey circle represents the vacuum chamber.

Figure 5 :

 5 Figure 5: Estimated PSF with the knife-edge method. The scale bar has a width of 5 pixels.

Figure 6 :

 6 Figure 6: Schematic of the simulation sequence with the Monte-Carlo for light propagation simulation step

Figure 7 :

 7 Figure 7: Radial temperature profiles measured for the AH11 heating sequence. Insert : false color image of the thermographic acquisition. Scale bar is 4 mm.

Figure 8 :

 8 Figure 8: Measured, simulated and simulated-convolved radial temperatures profiles for the AH11 (left) and AH2 (right) experiments.

Figure 9 :

 9 Figure 9: Measured, simulated and simulated-convolved normalised temperature profiles for the AH11 (left) and AH2 (right) heating experiments.

Figure 10 :

 10 Figure 10: Temperature profiles (left) and normalized temperatures profiles (right) at the end of the AH2 experiment. Each curve corresponds to one set of parameters used in the Monte-Carlo simulation.

Figure 11 :

 11 Figure 11: Temperature as a function of axial position at r=Ran at the end of the AH2 experiment. Each curve corresponds to one set of parameters used in the Monte-Carlo simulation.

Figure 12 :

 12 Figure 12: Annular heating experiments. Left: Measured peak temperatures versus simulated peak temperatures. Right: % Error measurementssimulations. The measurements were done at the end of the heating duration. Peak temperature was measured on 25 points on the circumference of the thermal ring. The error bar corresponds to the standard deviation of these 25 measures.

Figure 13

 13 Figure13displays typical temperature images obtained during the experiments, showcasing the case of the CH10 experiment. From the images, it can be observed that the heating is relatively homogeneous across the entire surface. The observed inhomogeneities can be attributed to the modulation of the laser beam, which is not perfectly uniform. Due to the relatively homogeneous temperature profiles, the impact of camera aberrations on the measurements is negligible.

Figure 13 :

 13 Figure 13: False color image for the CH10 heating experiment. Scale bar is 4 mm. Temperature scale is given in °C.

Figure 14 :

 14 Figure 14 : Homogeneous heating experiments. Left: simulated temperature as a function of the measured temperature. Right: error percentage calculation-measurement. The measurements were done at the end of the heating duration. The error bar corresponds to the standard deviation of the measurement on the whole surface of the sample.

Figure 15 :

 15 Figure 15: Measured temperature profiles for the CLH1 (left) and CLH2 (right) experiment. The lines correspond to the vertical axis passing through the center of the disk while the dotted lines correspond to the horizontal axis passing through the center of the disk.

Table 1 :

 1 IV. Results and discussion:1. Annular laser heating experiments:Twelve annular laser heating experiments were performed on twelve different samples. All experiments had a pulse duration of 20 ms but different beam dimensions (focused or unfocused) and different initial temperature (circa 303 or 573 K, corresponding to 30 or 300 °C). The maximal measured temperature in this series is around 1523 K (1250 °C). The experimental parameters of this series are summarized in Table1.

	N°	Annular Beam	Power (W)	Initial
		Configuration		Temperature (K)
	AH 1	unfocused	120	308
	AH 2	unfocused	120	583
	AH 3	unfocused	320	613
	AH 4	focused	120	573
	AH 5	focused	250	573
	AH 6	unfocused	120	328
	AH 7	unfocused	146	333
	AH 8	unfocused	120	593
	AH 9	unfocused	320	593
	AH 10	unfocused	456	593
	AH 11	focused	120	348
	AH 12	focused	120	603

Test matrix for the Annular laser Heating (AH) experiments

Table 2 :

 2 Values of absorbing and scattering coefficients taken for the parametric sweep in the Monte-Carlo simulations.

Table 3 :

 3 Parameters of the homogenous heating experiments

Table 4 :

 4 Experimental parameters of the combined laser heating experiments.

Acknowledgements:

The authors would like to thank Doris Druan from CEA Cadarache for the development and teaching of the technique used to realize the thin slides samples used for the determination of the complex refractive index of UO2.

M. Reymond and L. Gallais thank Anabela DaSilva from Institut Fresnel for its introduction to MCmatlab and help for the light scattering simulations.

Finally, we also thank EdF and Framatome for partial funding of this research and for the fruitful discussions.

Data available on request from the authors

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Supplementary material:

The methodology employed for the determination of the complex refractive of dense polycrystalline UO2 from spectrophotometric measurements is provided in the supplementary material provided with this study.

CRediT authorship contribution statement: