Pierre Hirel

Sandrine Brochard

Laurent Pizzagalli

Pierre Beauchamp

Effects of temperature and surface step on the incipient plasticity in strained aluminium studied by atomistic simulations

Keywords: computer simulation, aluminium, surfaces & interfaces, dislocations, nucleation

come

Effects of temperature and surface step on the incipient plasticity in strained aluminium studied by atomistic simulations

The study of mechanical properties takes a new and more critical aspect when applied to nanostructured materials. While plasticity in bulk systems is related to dislocations multiplying from pre-existing defects, such as Franck-Read sources [START_REF] Hirth | Theory of dislocations[END_REF], nanostructured materials are too small for such sources to operate, and their plasticity is more likely initiated by dislocations nucleation from surfaces and interfaces [START_REF] Albrecht | Surface ripples, crosshatch pattern, and dislocation formation : cooperating mechanisms in lattice mismatch relaxation[END_REF][START_REF] Xu | Homogeneous nucleation of dislocation loops under stress in perfect crystals[END_REF][START_REF] Brochard | Grilhé Dislocation nucleation from surface steps: atomistic simulation in aluminum[END_REF][START_REF] Godet | Theoretical study of dislocation nucleation from simple surface defects in semiconductors[END_REF]. In particular, nucleation from grain boundaries is of great interest for the understanding of elementary mechanisms occuring in work hardening of nano-grained materials [START_REF] Spearot | Nucleation of dislocations from [001] bicrystal interfaces in aluminum[END_REF][START_REF] Swygenhoven | Atomic mechanism for dislocation emission from nanosized grain boundaries[END_REF][START_REF] Yamakov | Length-scale effects in the nucleation of extended dislocations in nanocrystalline al by molecular dynamics simulation[END_REF][START_REF] Yamakov | Dislocation processes in the deformation of nanocrystalline aluminium by moleculardynamics simulation[END_REF]. The mechanisms involving the nucleation of dislocations from crack tips are also of great importance to account for brittle to ductile transition in semiconductors [START_REF] Cleri | Atomic-scale mechanism of cracktip plasticity : dislocation nucleation and crack-tip shielding[END_REF][START_REF] Zhou | Large-scale molecular dynamics simulations of three-dimensional ductile failure[END_REF][START_REF] Zhu | Atomistic study of dislocation loop emission from a crack tip[END_REF].

In epitaxially-grown thin films, misfit induces a strain and can lead to the formation of dislocations at interfaces [START_REF] Ernst | Interface dislocations forming during epitaxial growth of gesi on (111) si substrates at high temperatures[END_REF][START_REF] Wu | The first stage of stress relaxation in tensile strained in 1-x ga x as 1-y p y films[END_REF][START_REF] Trushin | Surface instability and dislocation nucleation in strained epitaxial layers[END_REF]. The presence of defects in a surface, such as steps, terraces or hillocks, can also initiate plasticity [START_REF] Xu | Analysis of dislocation nucleation from a crystal surface based on the peierls-nabarro dislocation model[END_REF]. In particular, experimental and theoretical investigations have established that stress concentration near surface steps facilitates the nucleation of dislocations from these sites [START_REF] Brochard | Grilhé Stress concentration near a surface step and shear localization[END_REF][START_REF] Zimmerman | Surface step effects on nanoindentation[END_REF]. Dislocations formation in such nanostructures changes their mechanical, electrical, and optical properties, and then may have a dramatic effect on the behaviour of electronic devices [START_REF] Carrasco | Characterizing and controlling surface defects[END_REF]. Hence, the understanding of the mechanisms initiating the formation of dislocations in these nanostructures is of high importance.

Since these mechanisms occur at small spatial and temporal scales, which are difficult to reach experimentally, atomistic simulations are well suited for their study. Face-centered cubic metals are first-choice model materials, because of their ductile behaviour at low temperatures, involving a low thermal activation energies. In addition, the development of semi-empirical potentials for metals has made possible the modelling of large systems, and the accurate reproduction of defects energies and dislocation cores structures. Aluminium is used here as a model material.

In this study we investigate the first stages of plasticity in aluminum f.c.c. slabs by molecular dynamics simulations. Evidence of the role of temperature in the elastic limit reduction and in the nucleation of dislocation half-loops from surface steps is obtained. Steps in real crystals are rarely straight, and it has been proposed that a notch or kinked-step would initiate the nucleation of a dislocation half-loop [START_REF] Pirouz | Partial dislocation formation in semiconductors: structure, properties and their role in strain relaxation[END_REF][START_REF] Edirisinghe | Relaxation mechanisms in single in x ga 1-x as epilayers grown on misoriented gaas( 1 11)b substrates[END_REF]. This is investigated here by comparing the plastic events obtained from either straight and non-straight steps.

Our model consists of a f.c.c. monocrystal, with two {100} free surfaces (Fig. 1). Periodic boundary conditions are applied along the two other directions, X= [0 11] and Z= [011]. On one {100} surface, two opposite, monoatomic steps are built by removing atoms. They lie along Z, which is the intersection between a {111} plane and the surface. Such a geometry is therefore well suited to study glide events occuring in {111} planes. We investigate tensile stress orthogonal to steps. In this case, Schmid analysis reveals that Shockley partials with a Burgers vector orthogonal to the surface step are predicted to be activated in {111} planes in which glide reduces the steps height [START_REF] Brochard | Grilhé Dislocation nucleation from surface steps: atomistic simulation in aluminum[END_REF]. In some calculations, consecutive atoms have been removed in the step edge, forming a notch (Fig. 1), for investigating the effect of step irregularities on the plasticity. Various crystal dimensions have been considered, from 24 × 16 × 10 (3680 atoms), up to 60 × 40 × 60 (142800 atoms). The latter crystal size was shown to be large enough to have no influence on the results.

Interactions between aluminum atoms are described by an embedded atom method (EAM) potential, fitted on experimental values of cohesive energy, elastic moduli, vacancy formation and intrinsic stacking fault energies [START_REF] Aslanides | Atomistic study of dislocation cores in aluminum and copper[END_REF]. It is well suited for our investigations since it correctly reproduces the dislocations core structures. Fig. 1. System used in simulations, with periodic boundary conditions along X and Z, and free {100} surfaces. The {111} glide planes passing through the steps edges are drawn (dashed lines). Here, a notch is built on the right-side step.

Without temperature, the system energy is minimized using a conjugategradient algorithm. The relaxation is stopped when all forces fall below 6.24 × 10 -6 eV. Å-1 . Then the crystal is elongated by 1% of its original length along the X direction, i.e. perpendicular to the step. The corresponding strain is applied along the Z direction, according to the isotropic Poisson's ratio of aluminum (0.35). Use of isotropic elasticity theory is justified here by the very low anisotropy coefficient of this material: A = 2C 44 /(C 11 -C 12 ) = 1.07 (EAM potential used here) ; 1.22 (experiments [START_REF] Zener | Elasticity and Anelasticity of Metals[END_REF][START_REF] Thomas | Third-order elastic constants of aluminum[END_REF]). After deformation, a new energy minimization is performed, and this process is repeated until a plastic event, such as the nucleation of a dislocation, is observed. The occurence of such an event defines the elastic limit of the material at 0K. At finite temperature, molecular dynamics simulations are performed with the xMD code [START_REF] Rifkin | Xmd molecular dynamics program[END_REF], using the same EAM potential. Temperature is introduced by initially assigning an appropriate Maxwell-Boltzmann distribution of atomic velocities, and maintained by smooth rescaling at each dynamics step. The time step is 4 × 10 -15 s, small enough to produce no energy drift during a 300K run. After 5000 steps, ie. 20 ps, the crystal is deformed by 1%, similarly to what is done at 0K, and then the simulation is continued. If a nucleation event occurs, the simulation is restarted from a previously saved state and using a lower 0.1% deformation increment.

To visualize formed defects, atoms are colored as a function of a centrosymmetry criterion [START_REF] Li | Atomeye : An efficient atomistic configuration viewer[END_REF]: atoms not in a perfect f.c.c. environment, ie. atoms on surfaces, in dislocation cores and stacking faults, can then be easily distinguished. In case of dislocation formation, the core position and Burgers vector are determined by computing the relative displacements of atoms in the glide plane. These displacements are then normalized to the edge and screw components of a perfect dislocation. At 0K, the deformation is found to be purely elastic up to an elongation of 10%. Then a significant decrease of the total energy suggests an important atomic reorganisation. Crystal vizualisation reveals the presence of defects located in {111} planes passing through step edges, and a step height reduction by 2/3 (Fig. 2). Atomic displacements analysis in these planes shows that plasticity has occured by the nucleation of dislocations, with Burgers vectors orthogonal to the steps and with a magnitude corresponding to a 90 • partial. This is consistent with the 2/3 reduction of the steps height. The dislocations are straight, the strain being homogeneous all along the steps, and intrinsic stacking faults are left behind. The formation of dislocations from a surface step has already been investigated from 0K simulations, using quasi-bidimensional aluminum crystals [START_REF] Brochard | Grilhé Dislocation nucleation from surface steps: atomistic simulation in aluminum[END_REF]. It has been shown that straight 90 • Shockley partials nucleate from the steps. However, the small dimension of the step line did not allow the bending of dislocations. Here, although this restriction has been removed by considering larger crystals (up to 90 atomic planes along Z), only straight partial dislocations have been obtained.

In order to bring the role of steps to light, calculations are performed with a 30 × 20 × 30 crystal with two free surfaces, but without step. In that case, plasticity occurs for a much larger elongation, 20%, and leads to a complex defects structure. It clearly shows the important role played by steps, by significantly reducing the energy barrier due to dislocation-surface interaction, and initiating the nucleation in specific glide planes.

The effect of temperature has been first investigated at 300K. Plasticity occurs for a 6.6% elongation, showing that thermal activation significantly reduces the elastic limit. Another important difference due to temperature is the geometry of the formed defect. Instead of a straight dislocation, a dislocation half-loop forms and propagates throughout the crystal (Fig. 3). As expected, the nucleation of a half-loop dislocation is thermally activated. Contrary to the 0K simulation, a dislocation has nucleated from only one step: no dislocation is emitted from the other surface step, which remains intact. Atomic dis- placements at different simulation times (Fig. 4) indicates that this half-loop dislocation has a Burgers vector orthogonal to the step, the screw component being almost zero. The formed dislocation is then a Shockley partial, leaving a stacking fault in its path. Atomic displacements have been fitted with an arctan function, according to elasticity theory. This allows to monitor the position of the dislocation core defined as the maximum of the derivative, during the simulation.

Before the complete propagation of a dislocation, several half-loop embryos starting from both steps have been observed, appearing and disappearing (Fig. 3). Only one of them will eventually become large enough and propagate into the crystal (Fig. 3). This is related to the existence of a critical size for the dislocation formation, due to attractive interaction with the free surface. As the dislocation moves through the crystal and reaches the opposite surface, a trailing partial does not nucleate. Though it would significantly reduce the total energy of the system, especially in aluminum which have a high stacking-fault energy, this would require the crossing of a high energy barrier. On the contrary, the successive nucleation of dislocations in adjacent {111} can be achieved with a much lower energy barrier. So, although it relaxes less energy than a trailing partial would, this mechanism is more likely to be activated. This is what we obtained in most simulations, similar to the twinning mechanism proposed by Pirouz [START_REF] Pirouz | Partial dislocation formation in semiconductors: structure, properties and their role in strain relaxation[END_REF]. The remaining smaller step on the top surface, as well as the step created by the emergent dislocation on the bottom surface, become privileged sites for the nucleation of other dislocations in adjacent {111} planes, leading to the formation of a twin. While sufficient stress remains, successive faulted half-loops will be formed in adjacent planes, increasing the thickness of the twin. After 76 ps, the crystal structure does not evolve anymore. The plastic deformation is then characterized by a micro-twin (Fig. 3), located around the previous position of the step, with an extension of eight atomic planes, and delimited by two twin boundaries whose total energy equals the energy of an intrinsic stacking fault.

We have also investigated how the dislocation formation process is modified in the case of irregular steps. We used a crystal with the same geometry, except that 10 consecutive atoms have been removed from one surface step edge (see Fig. 1), creating two step kinks between which lies a notch. The other step remains straight. First, at 0K, no defect is obtained up to 10% elongation, beyond which plasticity occurs. This elastic limit is similar to the one obtained for the system with perfect steps. Moreover, nucleated dislocations are also Shockley partials with a Burgers vector orthogonal to the step, and are emitted from both surface steps, despite the system asymmetry. However, two dislocations have been formed from the irregular step. In fact, a second partial nucleates and propagates in the {111} plane passing through the notch (Fig. 5). Both dislocations remain in their respective glide plane, leaving two stacking faults. This suggests that kinks are strong anchors for dislocations. Nevertheless, at 0K, it seems they have a negligible effect on the elastic limit, or regarding the nature of the nucleation event. At 300K and for the same geometry, the elastic limit is reached for a 6.6% elongation, i.e. similar to the crystal with straight steps. Again, it suggests that irregular steps have no effect on the elastic limit. The dislocation half-loop does not nucleate from a step kink, but about 15 atomic planes away from it (Fig. 6a). It propagates into the crystal, but stays anchored to the kink, which acts like an obstacle to the movement. Then, another dislocation nucleates in the adjacent {111} plane, within the notch (Fig. 6b). Another simulation on a similar system leads to a dislocation nucleation from the straight step, despite the presence of a kinked step. These results show that kinks are not preferential sites for nucleation. It can be explained because step kinks are 0-D defects, contrary to straight steps, what prevents to initiate 1-D defects such as dislocations. After the first nucleation, the twinning mechanism, already described above, is observed. At about 70 ps, the formed twin cannot be distinguished from the one obtained in the crystal with straight steps. Finally, there is no indication left whether the step was initially irregular or not.

Molecular dynamics simulations have been used to investigate the influence of temperature and of step geometry on the first stages of plasticity in f.c.c. aluminum slabs. Surface steps were shown to be privileged sites for the nucleation of dislocations, significantly reducing the elastic limit compared to a perfect surface. Simulations with straight surface steps have revealed that only straight 90 • dislocations could nucleate at 0K. Temperature reduces the elastic limit, and makes possible the nucleation of faulted dislocation half-loops. Due to the system geometry and the strain orientation, only Shockley partials were obtained. Successive nucleation of partials in adjacent {111} planes are observed, similar to the twinning mechanism described by Pirouz in semiconductors. Simulations with an irregular step have shown that a kink is not a systematic site for nucleation. Instead, half-loops have been obtained from a straight portion of the step. The kinks introduced along a step seem to be strong anchor points for dislocations, making their motion more difficult along the step.

During all simulations including temperature, several dislocation half-loop embryos were observed before one eventually becomes large enough and propagates into the crystal. Calculations are in progress to determine the critical size a half-loop must reach to fully propagate. To determine the activation energy of the nucleation from surface steps, two methods may be used. First, the nudged elastic band method [START_REF] Jonsson | Classical and Quantum Dynamics in Condensed Phase Simulations[END_REF][START_REF] Henkelman | Improved tangent estimate in the nudged elastic band method for finding minimum energy paths and saddle points[END_REF][START_REF] Henkelman | A climbing image nudged elastic band method for finding saddle points and minimum energy paths[END_REF], applied to the nucleation and propagation of a half-loop, would provide the minimum energy path for this event.

Second, by performing several simulations at a given strain, one would obtain the average nucleation time as a function of temperature, thus allowing determination of the activation energy from Arrhenius plots. The dislocations speeds, as well as the size and shape of the dislocation half-loops, can be expected to depend on temperature, which will also be investigated through simulations. As a sequel to the nucleation event, several scenarii were observed. The twinning mechanism is supposed to be in competition with the nucleation of a trailing partial, which requires the crossing of a higher energy barrier. However this last mechanism was obtained during a simulation, showing it is still possible. More investigations would allow to determine the exact dependancy on temperature, strain, or other parameters.

P. Hirel's PhD work is supported by the Région Poitou-Charentes. We greatly aknowledge the Agence Nationale de la Recherche for financing the project (number ANR-06-blan-0250).

Fig. 2 .

 2 Fig. 2. Formation of two dislocations at 0K, after a 10% elongation of a 60 × 40 × 60 crystal. Initial positions of the surface steps are shown (arrows). Only atoms which are not in a perfect f.c.c. environment are drawn: surfaces (yellow-green), stacking fault (dark blue), dislocation cores (light blue) (color online).

Fig. 3 .

 3 Fig. 3. Evolution of the aluminum crystal after a 6.6% elongation at 300K. Same color convention as Fig. 2. The origin of time is when the applied strain is increased to 6.6%. (a) At 12 ps, several dislocation embryos appeared on both steps (arrows). (b) At 20 ps, a faulted half-loop dislocation has nucleated on one step. (c) After 76 ps, a stable twin was formed. The other step (on the right) remains intact.

Fig. 4 .

 4 Fig.[START_REF] Brochard | Grilhé Dislocation nucleation from surface steps: atomistic simulation in aluminum[END_REF]. Calculated edge component of the relative displacements of atoms in the activated glide plane and in the Z-layer corresponding to the dislocation front line, at 300K and for different times (triangles). They are fitted with an arctan function (solid lines) according to elasticity theory, for monitoring the dislocation core position during the simulation. The abcissa labels the depth of the atoms from the top surface : 1 corresponds to atoms at the edge of the initial step, and 40 corresponds to the opposite surface, at the bottom of the system.

Fig. 5 .

 5 Fig. 5. Dislocations nucleated in a crystal with one straight step, and one irregular, elongated by 10% at 0K.

Fig. 6 .

 6 Fig. 6. Evolution of the aluminum crystal with an irregular step, under a 6.6% elongation at 300K. Same color and time conventions as in Fig. 3. The position of the notch is highlited in red. (a) After 7.4 ps, a faulted half-loop dislocation nucleates in the original {111} plane. (b) At 10 ps, another dislocation is emitted in the adjacent {111} plane, passing through the notch.


Francois Sanson
email: francois.sanson@inria.fr

Francesco Panerai

Thierry E Magin

Pietro M Congedo

Robust reconstruction of the catalytic properties of thermal protection materials from sparse high-enthalpy facility experimental data

Keywords: Uncertainty Quantification, Bayesian Inference, Catalysis, Thermal Protection Systems

Quantifying the catalytic properties of reusable thermal protection system materials is essential for the design of atmospheric entry vehicles. Their properties quantify the recombination of oxygen and nitrogen atoms into molecules, and allow for accurate computation of the heat flux to the spacecraft. Their

rebuilding from ground test data, however, is not straightforward and subject to uncertainties. We propose a fully Bayesian approach to reconstruct the catalytic properties of ceramic matrix composites from sparse high-enthalpy facility experimental data with uncertainty estimates. The results are compared to those obtained by means of an alternative reconstruction procedure, where the experimental measurements are also treated as random variables but propagated through a deterministic solver. For the testing conditions presented in this work, the contribution to the measured heat flux of the molecular recombination is negligible. Therefore, the material catalytic property

Introduction

In the design of thermal protection systems for atmospheric entry vehicles, the catalytic properties of the heatshield material allow us to quantify the influence of the highly exothermic molecular recombinations occurring at the surface. In order to estimate these properties for a given material, groundbased high-enthalpy facilities are used to simulate flight conditions at the material surface and to provide relevant experimental data [START_REF] Chazot | Hypersonic Nonequilibrium Flows: Fundamentals and Recent Advances[END_REF]. The plasma flow can be achieved using different techniques. In inductively-coupled plasma (ICP) wind tunnels, often referred to as plasmatrons, the plasma is generated by electromagnetic induction. A strong electromagnetic field ionizes the flow confined into a cylindrical torch and the plasma jet exits at subsonic speed into a low pressure test chamber that hosts material probes. The stagnation point conditions corresponding to a given spacecraft entry are reproduced for several minutes and the plasma flow carries sufficient energy to reproduce actual aerothermal loads experienced by a thermal protection system (TPS) in flight. Thanks to a flow of high chemical purity, plasmatron facilities are particularly suited to study gas/surface interaction phenomena for reusable TPS materials [START_REF] Kolesnikov | RTO-EN-AVT-008 -Measurement Techniques for High Enthalpy and Plasma Flows[END_REF][START_REF] Marschall | [END_REF]4,5,6,7,8,9,10,11] or composite ablative material [12] . High-temperature experiments enable characterizing the catalytic properties of the tested TPS sample by combining direct measurements using various diagnostics and a numerical reconstruction based on computational fluid dynamics (CFD) simulations.

Even for well-characterized facilities, the determination of catalytic properties is affected by the noise present in the experimental data. The quantification of uncertainties in high-enthalpy experiments has previously been studied in the literature [13,14,15,16]. In particular, in our previous work [16], we evaluated the uncertainties on catalytic properties by coupling a deterministic catalytic property estimation with a Polynomial Chaos (PC) expansion method. The probabilistic treatment of the uncertainties helped mitigating over-conservative uncertainty estimates found in the literature by computing confidence intervals. The influence of the epistemic uncertainty on the catalytic property of a reference calorimeter used in the reconstruction was also investigated in [16]. However, the method developed has two shortcomings: the number of experiments is limited and statistics about the measurements distribution are not available, even though they are an essential input for the PC approach.

Two important aspects are explored in the present work. First, we develop a robust methodology for quantifying the uncertainties on the catalytic property following a Bayesian approach. The Bayesian framework has already been successfully applied to the study of graphite nitridation [14] and hightemperature kinetics [17], for model parameter estimation as well as for experimental design [18], but it is a novel approach for the case of reusable materials, bringing a new insight on the ceramic matrix composites on which this paper focuses. In a Bayesian approach, one computes the probability distribution of the possible values of a quantity of interest compatible with the experimental results and with prior information about the system. This is fundamentally different from the PC approach proposed in [16]. While both approaches aim at quantifying the uncertainty on the catalytic properties, the experimental data are direct inputs of the deterministic solver combined to the PC method, whereas they are observed outputs of a model for the Bayesian method.

Second, a thorough comparison between the two methods is developed in order to explain the results obtained in view of their conceptual differences. We investigate the case of two experiments necessary for the reconstruction of the flow enthalpy and material catalytic property. The PC approach sequentially considers the experiments, whereas the Bayesian approach merges them into a unique simultaneous reconstruction. Additionally, the Bayesian approach has a major advantage: it allows us to determine the catalytic property of a reference copper calorimeter used in the reconstruction methodology, along with the catalytic property of the sample material. The robustness of the method is also examined for cases where the problem is not well posed, for instance when there are too many parameters to rebuild, and no sufficient information from the experiments.

In this contribution, we propose to revisit measurements performed in a high-enthalpy ICP wind-tunnel (Plasmatron) at the von Karman Institute for Fluid Dynamics (VKI) to characterize the catalytic response of ceramic matrix composites. Based on the robust uncertainty quantification methodology developed, we will assess whether accurate information on the catalytic properties of these thermal protection materials can be extracted from the experimental data. The paper is structured as follows. In section 2, we recall the main features of the combined experimental/numerical methodology developed at VKI to analyze data obtained in the Plasmatron facility, and then, present the sources of experimental uncertainties involved in the process. In section 3, we reformulate the problem of determining the catalytic properties in a Bayesian framework. In Section 4, we apply this approach to experimental data presented in [4] and compare our results to the uncertainty estimate obtained in [16] by means of the PC approach.

Experimental/numerical methodology

The present study uses a set of data measured during an experimental campaign documented in [4]. The first section briefly recalls the quantities measured experimentally for each testing conditions and their associated uncertainties, whereas the next section introduces the numerical simulations performed to rebuild quantities that cannot be directly measured. The last section introduces some uncertainty quantification terminology.

Experimental setup

In order to derive the catalytic property γ of a ceramic matrix composite sample, the reconstruction methodology used in [4] is based on two sequential experiments. The first step consists in rebuilding the free stream enthalpy h e of the plasma flow, using the cold wall heat flux measurement q cw from a copper calorimeter (see Fig. 1) of catalytic property γ ref . The uncertainties on the heat flux measurements were computed to be ±10%. Note that the quantity γ ref is a source of large uncertainties [16]. A commonly adopted assumption is to consider the surface as fully catalytic [START_REF] Kolesnikov | RTO-EN-AVT-008 -Measurement Techniques for High Enthalpy and Plasma Flows[END_REF]19]. While this is a conservative practice, there is compelling evidence that the actual surface of copper calorimeters is not fully catalytic, owing to the rapid oxidation of copper upon exposure to plasma. Numerous studies have been dedicated to characterize the catalytic properties of copper and its surface oxides (CuO and Cu 2 O) [10,13,20,21,[START_REF] Prok | Effect of surface preparation and gas flow on nitrogen atom surface recombination[END_REF][START_REF] Rosner | [END_REF][START_REF] Ammann | Heterogeneous recombination and heat transfer with dissociated nitrogen[END_REF][START_REF] Dickens | [END_REF]26,27,28,29,30,31,32,33,[START_REF] Nawaz | 44th AIAA Thermophysics Conference, AIAA[END_REF][START_REF] Cipullo | [END_REF][START_REF] Viladegut | 45th AIAA Thermophysics Conference[END_REF][START_REF] Driver | 45th AIAA Thermophysics Conference, AIAA 2015-2666[END_REF].

Together with the heat flux, the total pressure is measured during the first experiment. A water-cooled Pitot probe is introduced in the Plasmatron flow in order to measure the dynamic pressure P d (featuring an uncertainty of ±6%). The surface temperature of water-cooled probes T cw is known by measuring the differential of temperature between the inlet and outlet water lines. The static pressure P s of the test chamber is measured with a 2 Pa accuracy.

In a second step, hot wall measurements are performed on the TPS material sample in order to determine its catalytic property γ, for a known test condition determined through the rebuilding of cold-wall measurements.

The emissivity ε of the sample is measured with 10% accuracy. The front It is further assumed that the free stream flow is identical during both experiments and that local thermodynamic equilibrium (LTE) holds at the edge of the boundary layer. At steady state, the surface radiated heat flux is assumed to be equal to the incoming heat flux from the plasma flow.

Numerical computations

The Plasmatron flow conditions in front of the TPS test sample are rebuilt using experimental data and a 1D non-equilibrium Boundary Layer (BL) solver [START_REF] Barbante | Accurate and Efficient Modelling of High Temperature Nonequilibrium Air Flows[END_REF][START_REF] Barbante | [END_REF] that propagates the flow field quantities from the outer edge of the BL to the stagnation point. The rebuilding methodology is sketched in Fig. 2. The BL solver computes the stagnation point heat flux q cw (or q w for the TPS sample) that, mathematically, is a function of the probe geometry, the surface temperature T cw (or T w for the TPS sample), and the wall catalytic property of the reference calorimeter γ ref (or γ for the TPS sample), given the following set of the plasma flow free stream parameters: enthalpy h e , pressure p e , and velocity v e . The PEGASE library [40], embedded with the boundary layer solver, provides the physico-chemical properties of the plasma flow.

The BL solver can be called by a rebuilding code using Newton's method to determine the quantities h e and γ in a two-step strategy involving one rebuilding per experiment. The static pressure p e is assumed to be equal to the static pressure P s measured in the chamber. The enthalpy rebuilding uses the measured dynamic pressure P d to compute the free stream velocity v e using a viscous correction, as well as the heat flux q cw measured at the surface of the reference calorimeter to reconstruct the free stream enthalpy h e . In a second step, the results from the second experiment and the flow field parameters computed during the first step are combined to determine the sample material catalytic property γ.

Despite the fact that a large number of inputs are measured or unknown, the method is fully deterministic and provides no indication about the outputs uncertainty. Our previous work [16] was based on the propagation of uncertainties using this inverse deterministic solver.

Uncertainty characterization in catalytic property reconstruction

The determination of the TPS catalytic property directly depends on experimental data, and intrinsically carries the uncertainty associated with actual measurements. Uncertainty Quantification (UQ) tools model and quantify the error associated to the variables computed using uncertain inputs. Table 1 reviews the measured quantities and their uncertain. The uncertainties can be classified into three categories:

• The measured quantities (MQ) come from the two experimental steps described earlier. The following quantities are measured: T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,meas , namely the calorimeter probe temperature, the calorimeter probe heat flux, the sample temperature, the sample emissivity, the plasma jet dynamic pressure and static pressure. Note that the heat flux from the second experiment (q w,meas )

is not directly measured but derived from quantities T w,meas and ε meas using Stefan-Boltzmann's law: q w,meas = σε meas T 4 w,meas . The MQ are aleatory quantities that are assumed to be noisy versions of their true values denoted T cw , q cw , T w , ε, P d , P s . In this study, they are modeled as realization of a Gaussian distribution. The quantity T cw,meas denotes the measurement of the probe temperature, so we have:

T cw,meas = T cw + ζ, ( 1
)
where ζ is the realization of a zero mean Gaussian random variable.

• The quantities of interest (QoI) are the unknown quantities crucial to engineering applications. In this study, the sample and the probe catalytic properties denoted γ and γ ref , along with the flow enthalpy h e , are the QoIs. The objective is not only to compute the most likely value of the catalytic property or the one that minimizes the square error but to compute the full probability distribution of all admissible values of the QoI given the measurements for a thorough quantification of uncertainties.

• The Nuisance Parameters (NP) are unknown quantities that must be estimated along with the QoI in order to estimate the sample catalytic property. Quantities T cw , T w , P d , P s , ε are NPs as they have to be estimated in order to run the BL solver used to derive the sample catalytic property.

Bayesian-based approach

One objective of this work is to make a joint estimation of the catalytic properties γ ref and γ of the reference calorimeter and sample material, re-spectively, along with the flow enthalpy h e , for a given set of experiments.

In [16], a polynomial chaos expansion was built on top of the inverse deterministic solver described earlier. In this section, we detail the derivation of the probability distribution of these quantities given the experimental results using a Bayesian approach. This probability distribution of these quantities is referred to as the posterior distribution. This distribution carries all the necessary information for the uncertainty quantification analysis. It provides a robust estimate of the uncertainty through confidence intervals and the variance.

In section 3.1, the posterior distribution is decomposed into a ratio of probabilities using Bayes' rule (Eq. 4) that can be numerically evaluated.

Detailed calculations of each terms of the decomposition are then presented in section 5. Finally, the posterior distribution is numerically evaluated using a Markov Chain Monte Carlo (MCMC) algorithm described in appendix.

Figure 4 summarizes the rebuilding methodology from a Bayesian perspective.

Note that, contrary to the deterministic strategy illustrated in Figure 2, the QoI are rebuilt using both experiments simultaneously. The differences in the two approaches are further discussed in section 4.

Bayesian framework

We recall that the heat flux to the sample material wall q w,meas is completely defined by the material emissivity ε meas and temperature T w,meas through Stefan-Boltzmann's law. Introducing the vector of measured quantities m = (T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,meas ), the posterior probability is then indicated as follows: Furthermore, the vector of NP is introduced as ω nuis = (T cw , T w , ε, P d , P s ).

P (γ ref , h e , γ|m) . ( 2
)
The posterior distribution (Eq.

Let us now focus on the non-marginalized posterior from Eq. 3. The flowchart in Fig. 4 shows the relationships between the unknowns γ ref , h e , γ, ω nuis and the MQ (i.e., the vector m) and how they interact with each other.

In order to evaluate Eq. 3, Bayes' rule is applied as follows:

P (γ ref , h e ,

Results

This section illustrates the results derived from the application of the Bayesian framework to the problem of interest. The objective is twofold: i)

to compute an estimate of the QoI (flow enthalpy h e and catalytic properties γ ref and γ of the reference calorimeter and sample material) and ii) to compare the results with the uncertainty estimates obtained in [16] from a more standard PCE approach. In order to demonstrate the potential of the Bayesian approach, two sets of experimental conditions are selected among the experiments presented in [4]. They are denoted as S1 and S8, as detailed in Table 2. For both experiments, we study the following two cases:

a. The calorimeter reference probe catalytic property γ ref is assumed to be constant and equal to 0.1 (Section 4.1). The results for the posterior distribution are presented. Uncertainty estimates are compared with the ones obtained in [16]. Qualitative and quantitative explanations of the differences between the results obtained by the two approaches are given.

b. Secondly, the probe catalytic property is treated as an unknown quantity determined along with the other NPs and QoIs (section 4.3). Again the results are compared against the method developed in [16].

Constant calorimeter reference probe catalytic property

Quantity γ ref is assumed to be constant and equal to 0.1, focusing on the computation of the posterior distribution of the flow enthalpy h e and material catalytic property γ. The statistical moment and the 95% confidence interval are given in Tables 4 and3 for quantities h e and γ, respectively.

Their mean values are in good agreement with the nominal results obtained in [4]. Figure 5 shows their distributions for sample S1. It is observed that the reconstructed quantities h e and γ both have symmetrical distributions.

Theses results can be related to the typical S-shape enthalpy versus catalytic property curve reported in the literature [16,[START_REF] Viladegut | 45th AIAA Thermophysics Conference[END_REF]. In this case, most of the posterior lies within the high gradient zone on the S-shape, meaning that the small changes in catalytic property induce large variations in the computed heat flux at the wall as they are related through a one to one mapping in that region. In other words, if the measured heat flux takes values in that region, it is expected that the catalytic property posterior will have limited variance. The Maximum A Posteriori (MAP) is defined as the maximum of the posterior density probability. It is an alternative point estimator to the mean of a QoI. In the special case of a Gaussian posterior, the MAP and the mean are equals. The analysis of sample S8 yields similar results and conclusions. The relative error computed as the ratio of the mean and the 95% confidence interval (CI) is one order of magnitude large for the catalytic property compared to the rebuilt enthalpy.

Comparison with Polynomial-Chaos approach

This section compares the proposed Bayesian approach to the PC approach presented in [16]. In the deterministic solver, the two steps of the experiments are taken sequentially (cf. Figure 2 in [16]): first the flow field is computed using the results from the first experiment, namely measurements of cold-wall heat flux, as well as the static and dynamic pressures. Then, the sample catalytic property is determined using the quantities rebuilt from the first experiment. In order to propagate uncertainties, a polynomial approximation of the solver was derived and used to generate the statistical moments of the sample catalytic property. More precisely, the MQ are the only inputs to the Sample q cw,meas polynomial approximation in [16], whereas the probe catalytic property γ ref is kept constant. In order to include the uncertainty of the probe catalytic property, several polynomial approximations of the solver are computed in [16] for different values of γ ref .

P s P d T cw,meas h e T w,meas ε meas γ [kW.m -2 ] [Pa] [Pa] [K] [MJ.kg -1 ] [K] [-] [-] S1 195
In the following section, we highlight the differences between the PC and Bayesian methods. Both qualitative (section 4.2.1) and quantitative (section 4.2.2) illustrations are provided. Sample S1 conditions are chosen for this exercise.

Qualitative differences between the PC and Bayesian methods

The PC and Bayesian approaches tackle the problem from different angles leading to different results. The main differences between the two methods can be summarized as follows.

• The experimental data accumulated during the two reconstructions are not exploited in the same way. In the Bayesian formulation, the measurements are treated silmutaneously in order to reconstruct the catalytic property distribution at once (cf. Fig. 4), whereas the PC approach coupled with the deterministic inverse problem use sequential reconstructions of each quantity (see Fig. 2). In particular, the flow enthalpy is estimated in the PC approach only using the first experiment, whereas the Bayesian approach uses information from both experiments to rebuild the flow enthalpy. As mentioned in [41], in [4,16] the link between the two experiments acts like a valve: the information (or uncertainty) only goes one way. The information from the second experiment does not flow back to the determination of the flow enthalpy. Only information from the first experiment goes to the second reconstruction via the boundary layer edge enthalpy h e . This method presents some similarities with the Cut-model used in Bayesian networks [41], but it generally leads to a wrong posterior distribution.

• Input uncertainties are modeled differently. The PC approach makes stronger hypothesis about the input distribution by assuming that its mean is the experimental value. In the Bayesian framework, it is only assumed that the experimental value obtained is sampled from a Gaussian distribution with mean function of the NP and QoI. This is a strong assumption since a single experimental result can be significantly different from the mean value.

• Not only the input measurements are not modeled the same way, but the way they are propagated is also different. The PC approach, and the results presented in [16], depend on the deterministic method used to solve the inverse problem. In fact, the PC approach only provides the variance of the outputs and higher statistical moments. On the other hand, the Bayesian method leads to an unbiased asymptotically efficient estimation of the sample catalytic property [START_REF] Kaipio | Statistical and Computational Inverse Problems[END_REF][START_REF] Gelman | Bayesian Data Analysis, Third Edition, Chapman & Hall/CRC Texts in Statistical Science[END_REF].

• Finally, the Bayesian approach offers more flexibility in order to add uncertainties without major issues in the computational time, whereas the PC approach is limited by the problem of the curse of dimensionality [START_REF] Bellman | Applied dynamic programming[END_REF][START_REF] Foo | [END_REF], i.e., the lack of convergence speed of the numerical method when an increasing number of uncertainties is considered. Moreover, the Bayesian framework is well-suited for modeling epistemic uncertainty, such as the reference probe catalytic property. In the method developed in [16], this property is not modeled as a distribution, since no information is available to characterize it. Therefore, a limited set of values of γ ref on an arbitrary interval are tested to provide an envelope of the uncertainty on the QoI. On the other hand, the Bayesian implementation can use the information collected during the experiments to compute a posterior distribution of the reference probe catalytic property. Using that posterior distribution, the method yields a much more precise estimation of the uncertainty in the QoI along with an estimation of γ ref .

Quantitative differences between the PC and Bayesian methods

In this section, numerical tests are performed with sample S1 (see Table 2). The comparison focuses on the distributions of the material catalytic properties, as well as on the modeling uncertainties coming from the unknown catalytic property γ ref of the reference calorimeter.

The reconstructions of the material catalytic property γ are first compared using a constant value of γ ref equal to 0.1. Although this case may be unrealistic, since the probe catalytic property is rarely well known, it illustrates the differences between the two methods in a basic setting. Figure 6 shows differences in the sample catalytic property distribution obtained with the PC [16] and Bayesian methods. Note that, in Table 5, the first moment of the two distributions are very close, however the standard deviations and the confidence intervals are significantly larger for the distribution obtained with the Bayesian approach. This explains the much larger magnitude of the relative error. Moreover, the MAP estimates are substantially different: for the Bayesian case, the distribution is skewed and the most probable value and the mean values of the sample catalytic property are different. This is not observed when using the Polynomial Chaos, since the catalytic property distribution is Gaussian.

Since γ ref is rarely known, its variability and influence on the QoI are also investigated here. In particular, the approach used in [16] for including the epistemic uncertainty due to γ ref is compared to the Bayesian implementation.

For the PC method, the uncertainty on the unknown. In Figure 7, the cumulative density function (CDF) of the flow enthalpy derived from the PC approach is plotted for the extreme values of γ ref , i.e. 1 and 0.01, as well as derived from the Bayesian approach with an a priori unknown value of γ ref .

For both values of γ ref , the CDF obtained by means of the PC approach exhibits a much steeper increase compared to the state-of-the-art Bayesian approach, leading to a much more precise estimate of the uncertainty on the enthalpy. This is due to the different degree of knowledge of the probe catalytic property for the two methods. Since the Bayesian implementation uses the measurements to estimate the probe catalytic property, the uncertainty due to the epistemic quantity decreases.

Conversely, for the PC implementation, no information about the probe catalytic property is available, leading to an overestimation of the uncertainty in the enthalpy. In summary, the Bayesian method makes a better use of the information available from the experiments and provides an optimal, reliable estimate of the uncertainty. The distributions of the material catalytic property obtained by means of the Bayesian approach with γ ref a priori unknown will be studied in the following section.

Case where the reference probe catalytic property is unknown

In contrast to an approach commonly followed in the literature, we consider here the value of the probe catalytic property to be unknown, instead of arbitrarily set to a constant value. Therefore, γ ref is determined along with the other unknown quantities and the target distribution is the new posterior:

P (γ ref , h e , γ
, ω nuis |T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,meas ). Hence, the influence of the probe catalytic property on the sample catalytic property uncertainty can be rigorously quantified. Due to the increase in the number of unknowns and in order to increase the speed of convergence of the MCMC algorithm, the Markov Chain is adapted using the Adapted Metropolis (AM) algorithm presented in [46] with a modification from [47] (see algorithms 4 and 5). This approach is more precise and more flexible than the approach used in [16] where a robust brute force method is presented to explore the influence of the probe catalytic property. In this work, the Bayesian approach gives finer results thanks to a better knowledge of the probe catalytic property.

Results obtained for the estimation of the material catalytic property γ, flow enthalpy h e , and reference calorimeter catalytic property γ ref for the two samples S1 ans S8 are presented. Figure 8 shows the distribution of γ ref and Table 6 summarizes their statistics. Means and variances results should be used with care as the computed distributions are extremely far from Gaussian. Based on the experimental data of sample S1, the computed value for the reference probe catalytic property is 0.018, as shown in table 6. This result indicates that the assumption of γ ref = 0.1 utilized in [4] is over-conservative. The results obtained for γ ref for the two conditions (S1 and S8) are rather different but not contradictory. The relative error is extremely large. Note that with sample S1, γ ref can be estimated with slightly more accuracy than with sample S8. This observation shows that the precision of the determination of the estimation of γ ref depends on the experimental conditions and not only on the accuracy on the measurements.

The addition of an extra NP increases the uncertainty on the QoI and other NP. Figure 9 shows the distribution of h e for sample S1 that can be compared to earlier results presented in Figure 5 for the case with a constant γ ref .

The distribution support is significantly increased and shifted toward higher values.

This change can be explained by a simple physical reasoning: for the same value of the experimental heat flux measurement, the reference probe catalytic property has been estimated by means of the Bayesian approach to a value of 0.018 much lower than 0.1. Consequently, the contribution to the heat flux due to catalytic recombination is lower than in the γ ref = 0.1 case and the contribution from the convective heat flux therefore becomes larger and the flow enthalpy is estimated as well to a higher value than in the γ ref = 0.1 case.

Figure 10 shows the distribution of the material catalytic property for samples S1 and S8. For both samples, the material catalytic property uncertainty is much more widespread with respect to the previous case where quantity γ ref was assumed to be constant. In particular, the support of the distribution covers eight orders of magnitudes and does not present a clear maximum for a precise a posteriori estimation. In the case of an unknown quantity γ ref , the experiments do not contain sufficient information. Indeed, one can notice that the posterior distribution is similar to the beta prior distribution, meaning that the likelihood is not informative in this case. even though the support of the distribution is extremely large and seems non informative, some remarks can be made about the CDF. In particular even the slight uncertainty on the determination of the flow enthalpy is associated with large uncertainty on the catalytic property of the material. This means that for the range of enthalpy between 4 MJ/kg and 8 MJ/kg (see Fig. 9) it is challenging to precisely estimate the sample catalytic property for those testing conditions. To illustrate the problem, Figure 12 shows the Bayesian reconstruction of the sample catalytic property for a case where the probe catalytic property γ ref is set to a constant value of 0.02. The sample material experiment considered here is S1. The bivariate distribution of the flow enthalpy h e and material catalytic property γ show that, for a given flow enthalpy, the curve of enthalpy versus catalytic property has a very low gradient. Even though the probe catalytic property is known and constant, the uncertainty is comparable to the case where the probe catalytic property has to be computed. Therefore the increase of uncertainty in the sample catalytic property is due to the experimental conditions rather than to the precision of the measurements. This remark shows that while the specific experimental condition had been selected based on a relevant flight environment, it is not optimal for accurately estimating the TPS material catalytic property. A similar conclusion can be made for sample S8.

Conclusion

In this study, a rigorous method for estimating the catalytic property of a material and the associated uncertainties is presented. By comparing a Bayesian approach with an alternative uncertainty quantification method presented in [16], we showed that the two methods do not yield the same results.

By construction, the Bayesian approach is more adapted to cases where a limited number of experiments are available while the approach presented in [16] makes stronger assumptions on the measurement distribution that are only valid when a large number of experiments is available. Moreover, we found that the Bayesian approach is also more flexible as it can naturally include epistemic variables such as the unknown reference calorimeter catalytic property uncertainty.

The uncertainty analysis carried out in the case of the unknown reference calorimeter catalytic property showed that the experimental set up is not adequate to precisely estimate the catalytic property of a given material.

For the testing conditions presented in this work, the contribution to the measured heat flux of the molecular recombination is negligible. Therefore, the material catalytic property cannot be estimated precisely. Conversely, in this study, we were able to have some estimation of the reference calorimeter catalytic property. We have found that the assumption of constant value γ ref = 0.1 is wrong and introduces a bias in the estimation of the material catalytic property.

As future work, we propose to identify experimental conditions that are optimal for accurately estimating the TPS material catalytic properties.

ε meas |γ, h e , ω nuis ) is the likelihood of the measurements obtained during the catalytic property reconstruction. Note that quantity γ ref is solely involved in the first experiment, whereas quantity γ in the second one. However, both experiments are still connected through the free stream conditions (such as the enthalpy h e ) that are assumed to be constant for both probes that are injected sequentially in the plasma jet. The two likelihoods can still be computed in two different steps, as shown in the following sections.

Derivation of the first experiment likelihood

The enthalpy rebuilding step does not involve ε meas and T w,meas . The expression becomes:

P (T cw
since the measurements are considered independent.

Each term from the right hand side of Eq. 1 has to be evaluated individually.

For instance, for the cold wall surface temperature, one has that:

P (T cw,meas |γ ref , h e , ω nuis ) = P (T cw,meas = T cw + ζ|γ ref , h e , ω nuis ) = 1 2πσ 2 Tcw,meas exp - (T cw,meas -T cw ) 2 2σ 2
Tcw,meas .

The last equality comes from the fact that ζ is a zero mean Gaussian random variable. Very similarly, one has P (q cw,meas |γ ref , h e , ω nuis ) = P (q cw,meas = q cw + ζ|γ ref , h e , ω nuis ) = 1 2πσ

Derivation of the second experiment likelihood

For the second set of experiments the material sample is tested in order to measure its catalytic property γ. The catalytic property rebuilding step consists in computing P (T w,meas , ε meas |γ, h e , ω nuis ). In the rebuilding procedure, the heat flux radiated by the TPS is assumed to be equal to the heatflux q w from the flow to the TPS, which is computed by means of the BL solver.

Mathematically we have:

q w (γ, h e , ω nuis ) = σεT 4 w . ( 13
)
Following the same procedure as for the enthalpy rebuilding, the likelihood for the catalytic property rebuilding has the following form:

P (T
Similarly, it follows that: Injecting Eqs. 12 and 17 in Eq. 5 provides an explicit way to numerically evaluate the likelihood. Unfortunately, even though there are analytical solutions for the likelihood and the prior distribution, in order to compute the posterior, it is necessary to compute the normalization factor in Eq. 4. In this study, this is computationally intractable. To bypass that issue, a classical Markov Chain Monte Carlo method is used to directly compute the posterior without having to evaluate the normalization factor. In fact, the Metropolis algorithm enables to sample from the posterior distribution [START_REF] Kaipio | Statistical and Computational Inverse Problems[END_REF][START_REF] Bremaud | Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues[END_REF]49,50].

Details of the implementation are given in appendix B.

1. In our case for an efficient exploration of the distribution of γ it is natural to choose the random walk as h e,n =h e,n-1 + ξ 1

ω n =ω n-1 + ξ 2 log(γ n ) = log(γ n-1 ) + ξ 3 (22)
Therefore the random walk is not symmetrical for γ and the ratio becomes: R = P (T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,meas |γ n , h e,n ω n ) P (T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , p s,meas |γ n , h e,n ω n )

× (γ n -γ max ) 2 (γ n -γ min ) 2 γ n-1 (γ n-1 -γ max ) 2 (γ n-1 -γ min ) 2 γ n ( 23
)
The rest of the algorithm of the implementation follows the MH algorithm described in [START_REF] Kaipio | Statistical and Computational Inverse Problems[END_REF].

Figure 1 :

 1 Figure 1: ESA standard probes (5 cm diameter) used for the measurements performed in the Plasmatron facility: (left to right) stagnation-point probe holding a material sample, copper calorimeter, and water-cooled Pitot probe.

Figure 2 :

 2 Figure 2: Flow chart of the deterministic estimation of material catalytic property

Figure 3 :

 3 Figure 3: Flow chart of the BL solver with its main inputs

Figure 4 :

 4 Figure 4: Flowchart of the Bayesian-based estimation of material catalytic properties.

Figure 5 :

 5 Figure 5: Bayesian reconstruction (γ ref =0.1) of the material catalytic property on a semi-log scale (top) and flow enthalpy on a linear scale (bottom) for material sample S1.

  Figure 11 compares the CDF of γ based on the sample S1 in the two cases where γ ref is either constant (equal to 0.1) or unknown. The constant γ ref case is actually a worst case scenario that overestimates the molecular recombination rate at the surface of the sample. The unknown γ ref case shows that the actual material sample catalytic property is certainly lower. Its distribution is hardly usable as it is, especially for the low values of γ since for those the posterior is very similar to the arbitrary prior chosen for this study. However, the CDF remains useful to estimate probabilities and confidence intervals. Now, we investigate reasons for the large increase in the γ uncertainty for an unknown γ ref quantity compared to the constant case. It is partially due to the addition of γ ref as NP but also to the lower estimation of γ ref =0.018 leading to an increase in the estimated flow enthalpy. The dependence of material catalytic property versus the flow enthalpy is weak. By inspecting the distributions of γ in Figure 10, one notices that these are flat in particular for sample S8. In other words, the sample catalytic property does not influence the measured heat flux for the tested conditions. It follows that scarce information from the measured heat flux can be used to estimate γ.

Figure 8 :

 8 Figure 8: Distribution of the reference probe catalytic property γ ref for samples S1 and S8 on a semi-log scale, obtained by means of the Bayesian approach.

Figure 9 :

 9 Figure 9: Distribution of the flow enthalpy h e for sample S1 obtained by means of the Bayesian approach (γ ref a priori unknown).

Figure 10 :

 10 Figure 10: Distribution of the material catalytic property γ for samples S1 and S8 on a semi-log scale, obtained by means of the Bayesian approach (γ ref a priori unknown).

Figure 11 :

 11 Figure 11: CDF of the material catalytic property γ for sample S1 on semi-log scale, obtained by means of the Bayesian approach (γ ref = 0.1 and γ ref a priori unknown).

Figure 12 :

 12 Figure 12: Bivariate distribution of the flow enthalpy h e and material catalytic property γ for sample S1 on semi-log scale obtained by means of the Bayesian approach (γ ref =0.02).

P 2

 2 (ε meas |γ, h e , ω nuis ) catalytic property likelihood becomes: P (T w,meas , ε meas |γ, h e , ω nuis ) = 1 2πσ Tw,meas exp -(T w,meas -T w )

Table 1 :

 1 Measured quantities used for the flow and sample material characterization

	Symbol Variable	Uncertainty
	P d,meas Dynamic pressure	6%
	P s,meas Static pressure	0.3%
	q cw,meas Heat flux	10%
	T cw,meas Probe temperature	10%
	T w,meas TPS temperature	1%
	ε	Emissivity	5%

  γ, ω nuis |m) = P (m|γ ref , h e , γ, ω nuis ) P (γ ref , h e , γ, ω nuis ) The likelihood quantifies the amount of information carried by the measurements to the QoI and the NP. It is the probability of observing the measured quantities knowing the QoI and the NP. It measures the compatibility between the measurements and the value of unknown parameters, such as the catalytic property of the material sample. When the value of the catalytic property is compatible with the experimental results, the likelihood increases. The amount of this increase is directly related to the amount of

	P (m)	(4)

where P (m|γ ref , h e , γ, ω nuis ) is the likelihood, P (γ ref , h e , γ, ω nuis ) the prior, and P (m) a normalization factor such that the probabilities add up to one. information brought by the measurements. If the measurements are very informative, the increase (or decrease if the catalytic property gets less and less compatible with the experiments) is very steep. The prior accounts for the knowledge of the unknown parameters before any experiment. In our case, as scarce prior information is available for ω nuis and h e , uniform priors are considered. As γ and γ ref are defined on the interval [0;1], a beta distribution with parameters α = 1 and β = 1 is chosen with a support of [10 -8 ; 1]. The next section is devoted to the determination of the likelihood.

Table 2 :

 2 Deterministic conditions for material samples S1 and S8. Here, reported values of h

e and γ are determined using the standard rebuilding procedure detailed in

[4]

.

Table 3 :

 3 Flow enthalpy h e [MJ kg-1 ] statistics obtained by means of the Bayesian approach (γ ref =0.1)

	Sample Mean SD MAP	95% CI	UQ(95% CI) [%]
	S1	6.0	0.43 6.06	[5.06;6.76]	28.3
	S8	9.7	0.43 6.66 [8.80;10.51]	17.6
	Sample Mean	SD	MAP	95% CI	UQ(95% CI) [%]
	S1	7.4e-3 4.1e-03 6.2e-3 [2.4e-3;1.7e-2]	197.2
	S8	3.7e-3 1.8e-03 2.7e-3 [1.4e-3;8.38e-3]	188.6

Table 4 :

 4

Material catalytic property γ statistics obtained by means of the Bayesian approach (γ ref =0.1)

  QoI due to the MQs are computed for discrete values of γ ref , whereas for the Bayesian method, γ ref is a priori

	Method	Mean	SD	MAP	95% CI	UQ(95% CI) [%]
	Polynomial Chaos 0.00747 1.6e-03 0.007 [0.0045 ; 0.0094]	65.6
	Bayesian	0.00747 4.1e-03 0.0059 [0.0024 ; 0.017]	195.4

Table 5 :

 5 Comparison between the statistics of catalytic property γ for material sample S1 obtained by means of the PC and Bayesian approaches (γ ref =0.1).

Table 6 :

 6 Reference probe catalytic property γ ref statistics obtained by means of the Bayesian approach.

	The reason for this

  P (P d,meas |γ ref , h e , ω nuis ) = P (P d,meas = P d + ζ|γ ref , h e , ω nuis ) P (P s,meas |γ ref , h e , ω nuis ) = P (P s,meas = P s + ζ|γ ref , h e , ω nuis ) Note that q cw can be computed using the BL solver as it is a function of γ ref , h e , and ω nuis . Finally Eq. 6 becomes: P (T cw,meas , q cw,meas , P d,meas , P s,meas |γ ref , h e , ω nuis ) = Ps,meas σ P d,meas σ Tcw,meas σ qcw,meas exp -(P s,meas -P s ) 2 2σ 2

						2 qcw,meas	exp -	(q cw,meas -q cw ) 2 qcw,meas 2σ 2	,	(8)
					=	1 2πσ 2 P d,meas	exp	  -	(P d,meas -P d ) 2 P d,meas 2σ 2	  , (9)
				=		1 2πσ 2 Ps,meas	exp -	(P s,meas -P s ) 2 Ps,meas 2σ 2	.	(10)
								(11)
			1
	(2π) 2 σ ps
	× exp	  -	(P d,meas -P d ) 2 2σ 2 P d,meas	-	(T cw,meas -T cw ) 2 2σ 2 Tcw,meas	-	(q cw,meas -q cw ) 2 2σ 2 qcw,meas



 .

(12)

  w,meas , ε meas |γ, h e , ω nuis ) = P (T w,meas |γ, h e , ω nuis ) P (ε meas |γ, h e , ω nuis ) ,

	(14)
	and the following expression can be computed:

P (T w,meas |γ, h e , ω nuis ) = 1 2πσ 2 Tw,meas exp -(T w,meas -T w ) 2 2σ

2

Tw,meas .

Acknowledgment

The authors would like to thanks Andrien Todeschini for the fruitful discussions and remarks on the Bayesian approach.

Appendix A: Determination of the likelihood

The likelihood represents the link between the MQ, the QoI and NP and is directly related to the experiments. The two experiments from the first and second steps are independent, so that the likelihood can be rewritten as: Chain toward the desired distribution [50]. Complete proof of the convergence of the Algorithm can be found in [START_REF] Kaipio | Statistical and Computational Inverse Problems[END_REF][START_REF] Bremaud | Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues[END_REF]. In this section, the basics of the algorithm and the specificity of the implementation are presented.

Consider a random walk Markov Chain X n on state space S. In the case studied, S contains all the admissible values of the NPs and QoIs.

Consider two states x and y ∈ S, the probability to go from x to y is P(x,y) referred as the transition probability. Let π(x) be the distribution of X n , if x∈S π(x)P (x, y) = π(y). Then, the distribution π is said to be invariant or stationary. In the special case of random walks, the invariant distribution is unique and the random walk converges to π asymptotically (see [49] or [START_REF] Bremaud | Markov Chains: Gibbs Fields, Monte Carlo Simulation, and Queues[END_REF]).

In other words, no matter where the Markov Chain started, we have,

The Metropolis algorithm uses the right transition probability P (x, y) such as π is the distribution of interest (the QoI distribution). It uses this results

from Markov Chain theory cf. [START_REF] Kaipio | Statistical and Computational Inverse Problems[END_REF] or [49] for more further details) :

If π(x)P (x, y) = π(y)P (y, x) then π is the limiting distribution for X n π(x)P (x, y) = π(y)P (y, x) is called the detailed balanced equation. In short, the algorithm models a random walk but between each step it adapts the next random step so that the detailed balanced equation is verified. Asymptotically, the MC behaves like the stationary distribution and using Monte Carlo method one can compute the distribution after convergence of the Markov Chain. In our case the state space has 6 or 7 dimensions and the Markov Chain we aim to build is X n = (γ smpl,n , h e,n , ω n ) and since we are interested in the posterior distribution, in our case we choose:

π(γ, h e , ω) = P (T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,meas |γ, ω, h e ) P (T cw,meas , q cw,meas , T w,meas , ε meas , P d , P s ) ×P (γ, ω, h e ), (19) that can be computed up to a normalization factor. The advantage of the Metropolis Hasting (MH) algorithm is that it only uses the ratio

Since the priors for h e,n , ω n are uniform and γ follows a beta distribution, the ratio simplifies into: R = P (T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,meas |γ n , h e,n ω n ) P (T cw,meas , q cw,meas , T w,meas , ε meas , P d,meas , P s,i |γ smpl,n , h e,n ω n )

where P (n -1 → n) is the probability to go from state n -1 to n. If the random walk is symmetrical P (n -1 → n) = P (n → n -1) and the ratio is