Vianney Monnier
email: vianney.monnier@ensma.fr

Vincent Rodriguez

Pierre Vidal

Ratiba Zitoun

An analysis of three-dimensional patterns of experimental detonation cells

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Introduction

The instability of the reaction zones of self-sustained, e.g., Chapman-Jouguet (CJ), or overdriven detonation waves in gaseous mixtures was identified experimentally in the late 1950s by Denisov and Troshin [START_REF] Denisov | Pulsating and spinning detonation of gaseous mixtures in tubes[END_REF], preceded by the observations of marginal detonations by Campbell and Woodhead [START_REF] Campbell | The ignition of gases by an explosion-wave. Part I. Carbon monoxide and hydrogen mixtures[END_REF]. Detonation fronts thus have a cellular structure, which is an impressive example of non-linearities in compressible reactive fluid dynamics such as those investigated by Clavin [START_REF] Clavin | Combustion Waves and Fronts in Flows: Flames, Shocks, Detonations, Ablation Fronts and Explosion of Stars[END_REF] for several varieties of reactive fronts. Higgins [START_REF] Higgins | Steady one-dimensional detonations[END_REF], Ng and Zhang [START_REF] Ng | Detonation instability[END_REF], Vasil'ev [START_REF] Vasil'ev | Dynamic parameters of detonation[END_REF], and Desbordes and Presles [START_REF] Desbordes | Multi-scaled cellular detonation[END_REF] reviewed experimental and numerical progress on detonation instability and cells. Understanding and characterizing this instability is useful for studying chemical kinetics in detonation reaction zones, calibrating numerical simulations and presizing safe industrial devices and advanced propulsion systems [START_REF] Bykovskii | Continuous spin detonations[END_REF][START_REF] Shepherd | Analytical models for the thrust of a rotating detonation engine[END_REF][START_REF] Wolanski | Research on detonative propulsion[END_REF].

To date, the cellular structure of the detonation is characterized based on representative lengths, namely the average width of the cells [START_REF] Vasil'ev | Cell size as the main geometric parameter of a multifront detonation wave[END_REF], and the effective length of the reaction zone, that is, the Soloukhin's hydrodynamic thickness [START_REF] Soloukhin | Nonstationary phenomena in gaseous detonation[END_REF] further investigated by Lee and Radulescu [START_REF] Lee | On the hydrodynamic thickness of cellular detonations[END_REF][START_REF] Radulescu | The hydrodynamic structure of unstable cellular detonations[END_REF]. The increase in computational capabilities makes it possible to consider today predictable numerical simulations and other descriptions than characteristic lengths. This requires first collecting and analysing experimental data specific to the three-dimensional aspects of detonation cells, which is the object of this work.

The reaction zone of detonation in gases is an unsteady three-dimensional flow with unburnt and burnt gas pockets, and preceded by a leading shock made up of cells, that is, Mach waves with longitudinal and transverse fronts. Mach's method of traces on soot-coated plates is the usual technique to record geometrical information about cellular structure, [START_REF] Denisov | Pulsating and spinning detonation of gaseous mixtures in tubes[END_REF][START_REF] Boulal | Experimental investigation on detonation dynamics through a reactivity sink[END_REF]. The typical setup is a straight detonation tube with the coated plates positioned longitudinally against its inner walls. The transverse waves erode the coating, which draws the diamond-shaped patterns that usually identify the detonation cells. Generally, their average widths and regularity [START_REF] Tiggelen | Evolution des variables chimiques et physiques à l'intérieur d'une maille de détonation[END_REF][START_REF] Sharpe | Nonlinear cellular dynamics of the idealized detonation model: Regular cells[END_REF] depend on the equivalence ratio, dilution, initial pressure and temperature of the mixture [START_REF] Stamps | The influence of initial pressure and temperature on hydrogen-air-diluent detonations[END_REF][START_REF] Auffret | Detonation structure of C 2 H 4 -O2 -Ar mixtures at elevated initial temperature[END_REF][START_REF] Auffret | Detonation structure and detonability of C 2 H 2 -O 2 mixtures at elevated initial temperature[END_REF][START_REF] Ciccarelli | The influence of initial temperature on the detonability characteristics of hydrogen-air-steam mixtures[END_REF], and on its confinement, such as a straight or a curved channel. They also depend on the wavefront dynamics, such as steady propagation in a tube, transmission from a tube to a large volume, or spherical expansion after direct ignition from a point source. The reference width is obtained with fronts travelling at the CJ velocity in a straight tube and is the length most often used for characterizing the cellular structure. It is defined as the average of the largest distances between the tracks of two contiguous transverse waves travelling in opposite directions (Subsect. 3.1). Values are available for many reactive mixtures [START_REF] Shepherd | Chemical kinetics of hydrogen-air-diluent detonations[END_REF].

The average width is accepted as a detonability indicator, that is, the capability of detonation to propagate in a system with finite transverse dimensions (another is the distance for the transition of deflagration to detonation, DDT). Thus, the smaller this width (or the DDT distance), the higher the detonability [START_REF] Gavrikov | A model for detonation cell size prediction from chemical kinetics[END_REF]. Shepherd et al. [START_REF] Shepherd | Analyses of the cellular structure of detonations[END_REF] and Zhao et al. [START_REF] Zhao | Quantitative comparison of cellular patterns of stable and unstable mixtures[END_REF] proposed automated methods to quantify cell width and irregularity using spectral and statistical analyses, respectively. More recently, Zhang et al. [START_REF] Zhang | Reconstructing cellular surface of gaseous detonation based on artificial neural network and proper orthogonal decomposition[END_REF] reconstructed the cellular surface using the neural approach.

Cellular patterns on longitudinal soot foils have been categorized as very regular, regular, irregular and very irregular [START_REF] Tiggelen | Evolution des variables chimiques et physiques à l'intérieur d'une maille de détonation[END_REF][START_REF] Manzhalei | Measurement of inhomogeneities of a detonation front in gas mixtures at elevated pressures[END_REF], based essentially on recordings in square-section tubes. The average width of irregular patterns can thus be as large or even smaller than standard deviation, raising questions on its representativeness [START_REF] Sharpe | Nonlinear cellular dynamics of the idealized detonation model: Regular cells[END_REF]. Cell regularity is interpreted based on the chemical kinetics in the detonation reaction zone [START_REF] Strehlow | Transverse wave structure in detonations[END_REF][START_REF] Strehlow | Transverse Waves in Detonations: II. Structure and Spacing in H 2 -O 2 , C 2 H 2 -O 2 , C 2 H 4 -O 2 , and CH 4 -O 2 Systems[END_REF][START_REF] Manzhalei | Stability of an overcompressed gas detonation, Combust[END_REF][START_REF] Manzhalei | Fine structure of the leading front of a gas detonation[END_REF][START_REF] Moen | Influence of Cellular Regularity on the Behavior of Gaseous Detonations[END_REF][START_REF] Paillard | Influence de la cinétique chimique sur la structure des détonations : cas du bioxyde de chlore[END_REF][START_REF] Tiggelen | Evolution des variables chimiques et physiques à l'intérieur d'une maille de détonation[END_REF]. Regular patterns are observed in mixtures with light fuels and oxygen highly diluted by a mono-atomic inert gas, typically argon, and irregular patterns in mixtures with heavier fuels and dilution with nitrogen [START_REF] Desbordes | Multi-scaled cellular detonation[END_REF]. Mixtures with regular or irregular patterns are also termed stable or unstable, respectively [START_REF] Zhao | Quantitative comparison of cellular patterns of stable and unstable mixtures[END_REF][START_REF] Austin | Reaction zones in highly unstable detonations[END_REF][START_REF] Jackson | The influence of the cellular instability on lead shock evolution in weakly unstable detonation[END_REF]. Short [START_REF] Short | Pulsating instability of detonations with a two-step chainbranching reaction model: theory and numerics[END_REF] and Radulescu [START_REF] Radulescu | A universal parameter quantifying explosion hazards, detonability and hot spot formation: χ number[END_REF] carried out a modelling synthesis based on the Zel'dovich-Neuman-Döring (ZND) model of the detonation reaction zone. Irregularity corresponds with ZND induction zones large compared to reaction thicknesses. They proposed a regularity criterion based on the ratio of the former to the latter (the χ number), the larger this ratio, the more unstable (irregular) the mixture. These interpretations thus involves chemical kinetics alone, whereas the shape of the cross-section of the detonation tube also determines how detonation cells behave. For example, regular mixtures show rectangular front-view cellular patterns only in tubes with rectangular cross-section, [START_REF] Hanana | On rectangular and diagonal threedimensional structures of detonation waves[END_REF], and the marginal regimes of detonation propagation in regular mixtures are different depending on the cross-section shape. The usual control parameter for selecting marginal or multi-cellular detonation regimes in a given tube is the initial pressure because the larger it is, the smaller the cells.

There are only a few cells on the front of marginal detonations, and only one for the lower limit of propagation [START_REF] Campbell | The ignition of gases by an explosion-wave. Part I. Carbon monoxide and hydrogen mixtures[END_REF][START_REF] Manson | Sur la structure des ondes explosives dites hélicoidales dans les mélanges gazeux[END_REF][START_REF] Fay | A mechanical theory of spinning detonation[END_REF]. For the round cross-section, the limiting mode shows a spinning wave, that is, a single transverse wave rotating along the tube wall [START_REF] Dove | A photographic investigation of the mechanism of spinning detonation[END_REF][START_REF] Schott | Observations of the structure of spinning detonation[END_REF][START_REF] Macpherson | The three-dimensional wave system of spinning detonation[END_REF][START_REF] Voitsekhovskii | Investigation of the structure of detonation waves in gases[END_REF]. For the square cross-section, the limiting mode shows two transverse waves normal to each other and reflecting on facing walls [START_REF] Dove | A photographic investigation of the mechanism of spinning detonation[END_REF]. For slightly larger initial pressures, there can be two pairs of parallel transverse waves, each pair normal to the other and each wave of the same pair moving in opposite directions. The usual observation is the rectangular mode for which the transverse waves of a same pair are parallel to facing walls [START_REF] Strehlow | Multi-dimensional detonation wave structure[END_REF]. However, Hanana et al. [START_REF] Hanana | On rectangular and diagonal threedimensional structures of detonation waves[END_REF][START_REF] Hanana | Pressure profiles in detonation cells with rectangular and diagonal structures[END_REF] identified a diagonal mode for which the transverse waves are angled by π/4 to the tube walls, by passing the detonation through an obstacle with a square hole angled by π/4 to the walls. Inductively, these experiments indicate that the three-dimensional properties of a cellular structure in a given mixture at given initial pressure and temperature depend on the dimension and the shape of the confinement [START_REF] Strehlow | Gas phase detonations: Recent developments[END_REF] and, for marginal regimes, on the ignition method. Williams et al. [START_REF] Williams | Detailed structure and propagation of threedimensional detonations[END_REF] carried out first 3D numerical simulations of the two one-cell rectangular modes, thus anteceding the experimental observations by Hanana et al. [START_REF] Hanana | On rectangular and diagonal threedimensional structures of detonation waves[END_REF][START_REF] Hanana | Pressure profiles in detonation cells with rectangular and diagonal structures[END_REF]. They described these modes as a pair of superimposed 2D ones, and identified, in particular, the phase shift issue. They also compared 2D and 3D numerical simulations and found the same cell width regardless of the dimensionality. Deledicque et al. [START_REF] Deledicque | Computational study of three-dimensional gaseous detonation structures[END_REF] and Dou et al. [START_REF] Dou | Simulations of detonation wave propagation in rectangular ducts using a three-dimensional WENO scheme[END_REF] reached the same conclusion. Most numerical simulations of rectangular and diagonal modes are based on inviscid (Euler) equations coupled to detailed chemical-kinetics schemes or simple Arrhenius rates combined with the constant-γ equation of state, [START_REF] Tsuboi | Three-dimensional numerical simulation for hydrogen/air detonation: Rectangular and diagonal structures[END_REF][START_REF] Eto | Numerical study on three-dimensional c-j detonation waves: detailed propagating mechanism and existence of oh radical[END_REF][START_REF] Deiterding | Numerical structure analysis of regular hydrogen-oxygen detonations[END_REF][START_REF] Wang | High resolution WENO simulation of 3D detonation waves[END_REF][START_REF] Borisov | Investigation of cellular detonation structure formation via linear stability theory and 2D and 3D numerical simulations[END_REF][START_REF] Taileb | Numerical study of 3d gaseous detonations in a square channel[END_REF].

The converse situation is that of cells small compared to the transverse dimensions of the tube or to the detonation size itself such as the radius of a spherically expanding detonation. Although there are many observations and recordings of multi-cellular regimes of detonation, the geometrical properties of a large set of cells seem to have received less attention, apart from the indication by Takai et al. [START_REF] Takai | Study of detonation wave structure[END_REF] that the volumes drawn by the transverse waves of rectangular modes are octahedrons and tetrahedrons. Therefore, recording implementations aiming at the 3D reconstruction of the cells should also investigate how the structure of multi-cellular fronts depends on the geometrical properties of the confinement, as suggested by Presles et al. [START_REF] Presles | An optical method for the study of the detonation front structure in gaseous explosive mixtures[END_REF].

Our work thus describes and analyses experiments in tubes whose cross-sections have different shapes, namely square, triangular and cylindrical, based on front-view and longitudinal cell recordings on soot-coated foils for the regular mixture 2 H 2 + O 2 + 2 Ar. This study is essentially experimental and descriptive, and intended only to discuss the cellular patterns, not to provide a predicting model for their sizes. Section 2 details the setups and their instrumentation, Section 3 presents the experimental results, Section 4 investigates how elements of graph theory can help interpret these results, and Section 5 discusses and concludes the work.

Experimental methodology and setup

We carried out a series of recordings in tubes with round (R), equilateral triangular (T) and square (Q) cross-sections of the same area 16 cm 2 to within ±4 % . We also carried out some in round and square tubes with larger areas (R' and Q' resp.). Table 1 shows the inner dimensions of the tubes; Figure 1 shows the schematic of a round or square tube. The T tube was 1 m long; it was inserted in the 10 m long R' tube at its end opposite ignition. A spark plug or an exploding wire was used for ignition, Shchelkin spirals then ensured the transition to detonation on a very short distance, typically 1 m.

We selected the stoichiometric reactive mixture 2 H 2 + O 2 + 2 Ar because its regular cellular pattern in square tubes at low-enough initial pressures (Section 1) is a convenient reference for comparison with the patterns in the round and triangular tubes. The mixture was prepared in a separate tank using the partial-pressure method, and then injected at the desired initial pressure p 0 after vacuuming the tubes. We varied the initial pressure p 0 from 15 kPa to 100 kPa and the initial temperature was 294 ± 3 K for all experiments.

Tube Cross-section area (cm We recorded the longitudinal and front views of the cellular structure on soot-coated foils. We obtained the front views using the basic method of the head-on impact of the detonation front with the foil positioned normal to the cross-section at the tube end [START_REF] Denisov | Pulsating and spinning detonation of gaseous mixtures in tubes[END_REF][START_REF] Voitsekhovskii | Trad: The structure of the detonation front in gases[END_REF]. We found it easier to implement than the optical method [START_REF] Presles | An optical method for the study of the detonation front structure in gaseous explosive mixtures[END_REF], originally devised for liquids, of the head-on impact with an illuminated aluminized foil (the cellular-front imprint deflects the light in different directions, and a synchronized camera records the foil deformation at the instant of the impact, hence the cells). We positioned the foils for the longitudinal recordings in the round and square tubes against their inner walls, at their ends opposite ignition. They were shaped into a half-cylinder for the round tubes and a plate for the square tubes, with length and thickness 50 cm and 1 mm, respectively. The three inner faces of the T tube were coated over their whole length (1 m) before being assembled in such a way as to ensure sharp and equal inner angles.

Three Kistler 603B pressure transducers coupled to Kistler 5018A electrostatic charge amplifiers were positioned on the round and square tubes before their soot-foil section. We used the triggering of the transducers to obtain the average velocity of the detonation wave and check its steadiness in the soot-foil section.

We also checked steadiness from the longitudinal recordings with the criterion that cells should have average geometrical properties constant from the beginning to the end of the soot foil. We could not position transducers on the T tube, and we checked steadiness, in this case, from the soot recordings solely. We always observed constant average geometrical properties along the total lengths of the square and round tubes and from about 10 cm from the entry of the 1 m long T tube.

Results

Cell patterns and cross-section shapes and areas

Figures 2 to 5 show typical front-view and longitudinal soot recordings. At lower initial pressures, from p 0 = 15.0 to 20.0 kPa (Figures 2 to 4, respectively), the front-view recordings in the Q tube show the long-time identified regular arrangement of square and rectangular patterns whose distributions depend on the instant of the detonation impact. These distributions are analyzed in subsection 3.2. However, the arrangements in the T and R tubes are stochastic, and repeated experiments show no dependence on the instant of impact. Although counting is difficult, a detonation front in an R tube appears to contain fewer cells than in the T and Q tubes, and in the T tube than in the Q tube, for the same p 0 . The longitudinal soot recordings shows cells more irregular in the T and R tubes than in the Q tube. They also show cells longitudinally slimmer in the Q tube than in the T and R tubes, that is, the cell length (L) to width (λ) aspect ratios in the R and T tubes are closer to unity. In the Q tube, they also show a regular spacing of the reflection impacts of the transverse fronts moving parallel to the longitudinal plate.

At intermediate initial pressures, from 20.0 to 70.0 kPa (Figures 5 and6, respectively), the rectangular patterns on the front-view recordings in the Q tube are locally deformed with complex shapes similar to those on the T and R front-view recordings. The deformation domains have a stochastic distribution and spread with increasing p 0 , although the longitudinal recordings in the Q tube still show a regular spacing of the transverse impacts.

At higher initial pressures, from p 0 ≈ 70.0 kPa (Fig. 6), the deformation domains eventually cover the whole section of the Q tube. There is no transverse waves moving parallel to the walls and no impacts of transverse fronts on the longitudinal plate anymore. The front-view patterns look the same regardless of the cross-section shape. They show the same complexity without evident local and global symmetries. The longitudinal recordings in the Q tube (Fig. 6) also look the same as in the T and R tubes, in particular, they do not show reflection impacts of the transverse fronts anymore, as they did for the lower and intermediate p 0 .

The longitudinal soot recordings point to two other observations. The first is the paradoxical result that the longitudinal patterns are regular even in the cases where the front-view patterns are not, regardless of the tube. The second is that the cells are slimmer in the Q tube than in the T and R tubes. Although perhaps limited to regular mixtures, this opposite evolution participates in raising questions about the representativeness conditions of a single length for characterizing detonation cells, at least about those deduced from longitudinal recordings alone.

Nevertheless, Figure 7 shows the cell average widths λ from longitudinal recordings for each cross-section shape as function of the initial pressure p 0 . The widths were defined classically as the largest distances between two transverse waves projected normal to the average incident shock direction [START_REF] Zhao | Quantitative comparison of cellular patterns of stable and unstable mixtures[END_REF][START_REF] Strehlow | Transverse Waves in Detonations: II. Structure and Spacing in H 2 -O 2 , C 2 H 2 -O 2 , C 2 H 4 -O 2 , and CH 4 -O 2 Systems[END_REF]. Their averaging includes at least 50 measurements along each recording, and the resulting uncertainty is approximately the size of the coordinate markers in the figure. These averages show the usual decreasing trend with increasing p 0 regardless of the cross-section shape and, if p 0 > 25.0 -30 kPa, they meet the well-known empirical relation λ ∝ p -α 0 with 1.1 < α < 1.3 [START_REF] Shchelkin | Gasdynamics of combustion[END_REF]. Their differences from one shape to another increase with decreasing p 0 . The difference from the Q to the R shapes increases progressively with p 0 decreasing from 20.0 kPa whereas that from the T to the R shape is more abrupt at p 0 about 15.0 kPa. The cell widths in the R tube are the larger, those in the Q tube the smaller. Their differences can be as large as ∼ 100% for the lower initial pressure p 0 = 15 kPa. For a given p 0 , the differences are small for the tubes whose cross-sections have the same shape and different areas.

There are two possible interpretations. The first is that no symmetry can result in the long term from the necessarily complex oblique reflections of transverse waves on round and triangular tube walls. The second is a boundary layer at walls. A variation analysis in the limit of large cross-sections appears not to favor the latter. For example, the basic Fay model [START_REF] Fay | Gas pase detonations: Recent developments[END_REF][START_REF] Gao | An experimental investigation of detonation limits in hydrogen-oxygen-argon mixtures[END_REF] reproduces at least qualitatively the increase of the boundary-layer thickness with decreasing p 0 . The lower flow velocity due to viscosity in a boundary layer acts as a mass sink that induces locally a specific mass larger than in the entire cross-section. The consequences are a diverging flow and, therefore, a curvature and a velocity deficit of the average front [START_REF] Wood | Diameter effect in condensed explosives. the relation between velocity and radius of curvature of the detonation wave[END_REF][START_REF] Bdzil | Steady-state two-dimensional detonation[END_REF][START_REF] Chiquete | Characteristic path analysis of confinement influence on steady two-dimensional detonation propagation[END_REF], that should increase with increasing thickness of the boundary layer. This description applies to cellular detonations if the cell widths are small enough compared to the transverse dimensions of the tubes so that a smooth average surface represents well the detonation front. According to Figure 7, this effect should then be more pronounced in the R tube than in the Q tube since detonation cells are larger if the shock temperature, that is, the detonation velocity, is smaller. In the limit of large cross-sections, the boundarylayer thickness, denoted below δ, should be independent of the cross-section shapes and thin compared to the transverse dimensions of the tubes, namely here the radius R of the round tube and the side a of the square tube. Their cross-section areas are A R = πR 2 and A Q = a 2 , those reduced by the cross-section areas of the boundary layers are

A ′ R = πR 2 (1 -δ/R) 2 and A ′ Q = a 2 (1 -2δ/a) 2 , so, since A R = A Q in this work, A ′ Q A ′ R = 1 -2ε 1 - √ πε 2 , (1)
with ε = δ/a. This ratio is smaller than 1 for physically-small δ/a, and decreases with increasing δ/a. The effective cross-section area would thus be larger in a R tube than in a Q tube with the same geometrical cross-section area and would decrease less rapidly in an R tube than in a Q tube. Figure 7 shows the opposite trends, that is, the cell width is larger in the R tube than in the Q tube, and the difference increases with decreasing p 0 . This observation suggest that, in our conditions, possible boundary layers have a characteristic time of development larger than that of chemical-kinetics termination, which tends to favor the hypothesis of asymmetry from complex oblique transverse reflections. Indeed, cells are still small compared to the transverse dimensions of the tubes. For example, 0.05 ≤ λ/a ≤ 0.2 (a = 40 mm, Table 1) for p 0 ∈ [START_REF] Kumar | Detonation cell widths in hydrogen/oxygen/diluent mixtures at low initial pressures[END_REF][START_REF] Auffret | Detonation structure and detonability of C 2 H 2 -O 2 mixtures at elevated initial temperature[END_REF] kPa approximately, a domain where an influence of the cross-section shapes is already observed. Another argument for the complex interplay of the transverse waves and the tube walls is that cells are slimmer in the Q tube than in the R and T tubes at lower p 0 , but all attain the same aspect ratio with increasing p 0 regardless of the cross-section shape. The limiting initial pressures above which the λ values become independent of the cross-section shapes and the front-view patterns in the Q tube irregular appear to be approximately the same. We estimate this limiting p 0 to 70.0 kPa. However, although their cross-section areas are different (Table 1), the Q and Q' square tubes give similar cell widths even for the lower initial pressures. These widths agree well with those reported by Barthel [67] and Strehlow [START_REF] Strehlow | Transverse Waves in Detonations: II. Structure and Spacing in H 2 -O 2 , C 2 H 2 -O 2 , C 2 H 4 -O 2 , and CH 4 -O 2 Systems[END_REF] (also in [START_REF] Kaneshige | Detonation database, California Institute of Technology; Online available[END_REF]). Below p 0 ≤ 22.5 kPa, the cell mean widths for the tubes with greater cross-section area (Q' and R', areas 25 and 70.8 cm 2 , resp.) are smaller than in the T and R tubes (area 16 cm 2 ). It has long been known that the transverse dimension of the tube affects the cellular and propagation dynamics of the detonation, all the more so as there are fewer cells on the front, depending on the initial pressure or temperature. Conversely, the smaller the cells, the more independent of the tube they are and the greater the contribution of chemical kinetics [START_REF] Lee | The Detonation Phenomenon[END_REF], p.194. For example, the increasing divergence of λ with decreasing p 0 was noted by Strehlow and Engel [START_REF] Strehlow | Transverse Waves in Detonations: II. Structure and Spacing in H 2 -O 2 , C 2 H 2 -O 2 , C 2 H 4 -O 2 , and CH 4 -O 2 Systems[END_REF] when they compared their measurements to those of Voitsekhovskii [START_REF] Voitsekhovskii | Investigation of the structure of detonation waves in gases[END_REF], but we could not find the tube dimensions for these experiments. Similarly, Kumar and Dewit [START_REF] Kumar | Detonation cell widths in hydrogen/oxygen/diluent mixtures at low initial pressures[END_REF][START_REF] Kumar | Detonation cell widths in hydrogen/oxygen/diluent mixtures at low initial pressures[END_REF] noted the increasing difference between their measured and theoretical cell widths with decreasing p 0 . They invoked both measurement issues and the influence of the tube size because their modelling involved only the post-shock velocity and specific-heat ratio, the initial temperature, and the induction time.

Essentially, our experiments also point out the increasing influence of the cross-section shape with decreasing initial pressure, that is, with increasing cell widths. They also point out the difficulty to determine which minimum pressures ensure cell properties independent of the tube geometrical properties, that is, dependent on chemical kinetics alone.

Cells patterns and phase shift in square tubes

We carried out a detailed analysis of the detonation propagation in square-section tubes with p 0 low enough so there are transverse waves moving approximately parallel to the tube walls (Subsect. 3.1). We used the grey levels on the front-view recordings to infer the propagation direction of the transverse waves, which is from the lighter to the darker domains because the latter are made up of the soot particles not swept yet by these waves (Figure 8). We thus identified a continuum of rectangular modes of propagation (Section 1). Figure 9 shows a schematic, the coordinates x and y denote the transverse directions, and z the longitudinal one, i.e., that of the detonation propagation. A single longitudinal soot recording identifies the two sets of transverse waves defining a rectangular mode, one by the diamond-shaped patterns, the other by the periodic impacts of the transverse fronts moving about parallel to the soot plate.

It is convenient to introduce a phase shift φ defined as the distance between the longitudinal positions of the transverse-wave impacts in the x and y directions as schematized in Figure 9. Its minimum and maximum values are 0 and half a cell length φ max = L/2, respectively, and the value used in Figure 9 is intermediate. This definition generalizes to multi-cellular regimes that identified numerically by Williams et al. [START_REF] Williams | Detailed structure and propagation of threedimensional detonations[END_REF] and experimentally by Hanana et al. [START_REF] Hanana | On rectangular and diagonal threedimensional structures of detonation waves[END_REF] with one-cell marginal regimes (Section 1). For φ = 0, the two sets of transverse waves are in phase, with impacts at adjacent tube walls at the same position z (Figure 10, impact position z 2 ). The diagonals and the normals to the walls through the centre of the cross-section are symmetry lines of the cellular front regardless of time. Each cell has a fixed center and alternates a square and a rectangular shape, which periodically grows and shrinks with an aspect ratio oscillating between 0 and +∞. The opposite case corresponds to transverse sets in phase opposition. The phase shift φ reaches its maximum value φ max = L/2 (Figure 11). Similarly to the in-phase case, the cell centers have fixed positions and the square and rectangle head-on shapes alternate between each other. There is no symmetry lines any more but the head-on structure looks as though it rotates by π/2 every time interval necessary for a longitudinal propagation be equal to L/2. In Figure 11, this corresponds to the positions z 1 and z 3 , and z 2 and z 4 .

In our conditions, each longitudinal recording showed a continuous variation of the phase shift φ between its minimum 0 and maximum L/2 Fig. 13, so the front-view recordings most often showed out-of-phase modes. Our interpretation is that the detonation propagation is also subjected to long-wavelength and low-amplitude instabilities -compared to the cell dynamics -that distort the average front so its surface is warped, hence the evolutive mode of the rectangular dynamics. This variation might result from an intrinsic instability of the rectangular mode growing with increasing cell number, that is, p 0 , and from the system and ignition imperfections, such as small defects at the tube wall and an asymmetrical shape of the wave front that comes out of the Shchelkin spiral. Thus, the phase and its shift are difficult to identify and measure with a practical accuracy, and appear as concepts restricted to ideal ignition system, tube and cellular detonation front, that is, neither tilted, curved, nor warped.

Another observation is that the blowup of the crossing locus of two transverse waves reveals that each splits into the same third one, although they look about normal to each other (Figure 12, red and blue lines). Each propagates partly behind, partly ahead of the other, that is, on different initial states. Therefore, their front velocities are different, which requires a third wave for their matching (green line). Regular cells in square cross-section tubes are thus bounded by two incident shocks and two Mach stems (Figure 12, states (0) and (2), resp.), so they are octagons and not rectangles. More generally, four-wave crossings on a head-on recording appear to be fortuitous with a lifetime negligible compared to the evolution time of a detonation cell.

Our analysis here points out that long-time scales phenomena affect the regularity and the symmetry of detonation cells. Numerical simulations should thus consider investigating the propagation of cellular detonation based on initial and boundary conditions with stochastic distributions of imperfections representative of actual devices. Also, capturing the details of the transverse wave dynamics necessitates three-dimensional calculations, but whether these details significantly influences the overall detonation dynamics is an open debate.

Equivalent front-view tessellation for many cells

The analysis of the experimental front-view recordings indicate that the influence of the cross-section shape of the tube on the cellular patterns decreases with increasing initial pressure p 0 (Subsect. 3.1). Above a limiting value of p 0 , all front-view patterns show a similar-looking irregularity regardless of this shape, that is, without evident periodicity in the relative positions of the cell edges. Paradoxically, the longitudinal recordings show the classical diamond-shaped regular arrangements typical of the considered mixture, both below and above the limit of influence of the cross-section shape. Therefore, longitudinal recordings alone cannot characterize the cellular structure. Determining average widths and patterns independent of the shape and area of the tube cross-section demands measurements with different cross-section shapes and increasing p 0 (Figure 7). Indeed, the average widths obtained from the longitudinal recordings converge with increasing p 0 .

This subsection thus discusses the definition and the representativeness of an average cell width λ using front-view recordings obtained in conditions presumably independent of the cross-sections, that is, with many irregular cells. By dimensional analysis this width should be of the form j A t /F where A t is the tube cross-section, F the number of cells on the front-view recording and j an O(1) geometrical factor that depends on some representative cell pattern independent of the tube. The premise here is that several front-view recordings obtained with the same composition and the same large enough p 0 show the same statistical distributions of different cell patterns, such as triangle, square, pentagon, hexagon, etc. Equivalently, the front-view distribution, at large F , of the same pattern, is statistically independent of the impact instant of cellular fronts generated in several experiments carried out in the same conditions. Elements from graph theory and tessellation, [START_REF] Aurenhammer | Voronoi diagrams[END_REF] can then be used to obtain an equivalence to this pattern distribution, and hence a representative cell width.

The cells on a front-view recording can thus be viewed as a set of faces that forms a planar graph governed by the Descartes-Euler-Poincaré relation

V -E + F = 2, (2)
which constrains the numbers of faces F , edges E (the transverse waves plus the intersections of the cells with the tube wall(s)), and vertices V (the intersections of transverse waves with themselves plus those with the tube wall(s)). The right-hand side of relation ( 2) is 1 for a polygon. Formally, the number of faces F in (2) thus includes the outer (embedding) face, whose edges physically represent the tube walls. The conditions that each edge is common to exactly 2 faces and each vertex to at least 3 faces then give the assembly constraint

2E -3V = k ≥ 0, (3)
where the integer k is the total number of edges supernumerary to 3 on a vertex, so k = 0 if all vertices are common to exactly 3 edges. Combining (2) and (3) yields the average edge number per face,

2E F = 6 1 - 2 F - 2k F . (4)
In this work, the blowup of the crossing locus of transverse waves suggests that any vertex can be common to only 3 edges (Section 3.2, Figure 8, Figure 12). All intersects are indeed Mach configurations with three transverse waves. Therefore, k = 0 here, so relation (4) for the average number of edges per face and its limit at large F reduce to

2E F = 6 1 - 2 F , lim F →∞ 2E F = 6. (5)
A front-view recording with a large number of irregular cells is thus equivalent to a planar graph with the same number of hexagonal cells, that is, the hexagon is the representative pattern for cells considered as elements of a large set. The corresponding representative cell width can then be defined as the average of the inner and outer diameters of the regular hexagon, that is, of the distances between facing edges and facing vertices, respectively,

λ = 6 π π 6 0 dx cos x × d i = 3 ln(3) π d i ≈ 1.049 d i , (6)
where d i = 2A h / √ 3 and A h = A t /F are the inner diameter and the area of the hexagon, and A t is the cross-section area of the tube; hence,

λ = 3 ln(3) π 2 √ 3 A t F ≈ 1.127 A t F . (7)
If one accepts two-wave transverse crossing, that is, four edges per vertex, such as those of front-view patterns approximated by rectangles and squares in a Q tube at low p 0 (Section 3.2), the number of edges supernumerary to 3 on a vertex, per face, k/F , is 1, not 0. Relation (4) and its limit at large F then reduce to

2E F = 4 - 6 F , lim F →∞ 2E F = 4. (8)
Here, the representation of a large number of irregular cells is a planar graph with the same number of square cells whose average width is

λ = 4 π π 4 0 dx cos x × d i ≈ 1.122 d i = 1.122 A t F , (9)
where d i = A q and A q = A t /F are the distance between facing edges (a) and the area of the square cell, and A t is the cross-section area (a 2 ) of the tube.

There are three observations. The first is that the average cell width for a large number of cells does not essentially depend on the actual representative cell pattern. A rough count from the recording at p 0 = 70 kPa in the 40 × 40-mm 2 Q tube (Fig. 6) gives F ≃ 650, and relation [START_REF] Desbordes | Multi-scaled cellular detonation[END_REF] for F = 600, 650 and 700 then gives λ = 1.84, 1.77 and 1.70 mm, respectively. These averages agree with our measurements (Fig. 7) and those of Barthel [67] and Strehlow [START_REF] Strehlow | Transverse Waves in Detonations: II. Structure and Spacing in H 2 -O 2 , C 2 H 2 -O 2 , C 2 H 4 -O 2 , and CH 4 -O 2 Systems[END_REF] (also in [START_REF] Kaneshige | Detonation database, California Institute of Technology; Online available[END_REF]) although these authors do not indicate the dimensions of their tubes.

The second observation is that the sensitivity ∆F/2F to the front-view cell number is relatively low. The cell count can indeed be difficult, depending on the quality of the frontview recordings. Relation [START_REF] Desbordes | Multi-scaled cellular detonation[END_REF] gives a relevant lower bound for measured cell widths, that is, measured values larger than the average value [START_REF] Desbordes | Multi-scaled cellular detonation[END_REF] indicate the persisting influence of the cross-section shape and area.

The third observation is that relations ( 7) and ( 9) define intrinsic, minimum, uncertainties on the cell width ∆λ/λ as the relative differences of the edge-to-edge and vertex-to-vertex distances, that is, 1 -√ 3/2 ≃ 13.4% for the hexagonal cell and 1 -1/ √ 2 ≃ 29.3% for the square cell. Thus, the uncertainty is about twice smaller if the representative cell is a hexagon. However, both uncertainties are large, which questions whether a single length can characterize front-view detonation cells.

A step forward could result from the comparison in Figure 14) suggesting that a front-view of detonation cells (left) closely resembles a Voronoi tessellation (right).A Voronoi element is the set of the closest points to a single source point in that set. Its boundaries would represent the transverse waves of the detonation front, and its source point the re-ignition locus from which these waves originated. The counting of the non-truncated Voronoi elements in Figure 14) yields F = 73 faces composed of 0 triangles, 12 rectangles, 21 pen-tagons, 25 hexagons, 13 heptagons, and 2 octagons. The majority, 63%, are pentagons or hexagons, i.e., with 5 or 6 edges, which is consistent with the upper bound 6 indicated by the relation [START_REF] Ng | Detonation instability[END_REF]. Here indeed, 2E/F = 5.62, which is close to 6(1 -1/F ) = 5.92. This slight deviation may result from the number of faces F = 73 not being large enough. Statistics from a tenth of other similarly-generated tessellations give about the same distribution and the same deviation, with very few or zero triangles and octagons, and only once a nonagon.

In this example, the boundaries of the tessellation elements are not the contours of the detonation cells on the experimental extract because these are not defined w ell e nough f or a valuable automatic detection. Instead, we generated the Voronoi tessellation from a random distribution of sources, the number of which we chose so that the elements had similar dimensions to the experimental extract. This tessellation technique thus represents a mathematical model for the front-view of a cellular detonation suited to alternate modelling such as a cellular automaton. As its stands, however, a supplementation with a physical criterion is necessary to predict the number of source points per unit front surface.

Discussion and conclusions

The results above aim at providing an experimental basis for revisiting the notions of regularity and characteristic width for detonation cells. We implemented the soot-plate technique to record front-view and longitudinal traces of the cellular structure of the detonation front in the mixture 2 H 2 + O 2 + 2 Ar at the initial temperature T 0 = 294 K and the initial pressure p 0 varying from 15 kPa to 100 kPa (Sect. 2). This mixture is termed stable because its detonation cells in a tube with a square or a rectangular cross-section show regular diamond-shaped patterns on longitudinal recordings and, for low-enough p 0 , alternate square and rectangular patterns on front-view recordings. In this study, we analyzed its cellular dynamics in detonation tubes whose cross-sections have the same surface area but different shapes, namely square, triangular and round (Sect. 3).

This work addressed only the case of the mixture 2 H 2 + O 2 + 2 Ar, for which exists conditions on geometry and chemical kinetics for observing regularity. Therefore, a similar investigation is necessary to characterize the cellular dynamics of irregular mixtures such as heavy fuels diluted with nitrogen, depending on the cross-section shape of the tubes.

The distinction between regular and irregular cells or between stable and unstable mixtures relies essentially on longitudinal recordings in square cross-section tubes. However, we found that longitudinal and front-view recordings do not define the same regularity. Whereas the cell patterns on the longitudinal recordings are regular for all considered p 0 and crosssection shapes, those on the front-view recordings are irregular except for the square shape and low-enough p 0 (Subsect. 3.1). Therefore, the cellular dynamics at the walls of a tube is not representative of that on the whole detonation front, and longitudinal soot traces alone are not sufficient for describing the cellular structure. Our experimental results indicate that a large number of cells is necessary for characterizing cells by a single length. At large enough p 0 , all front-view recordings show irregular cells with the same average width, that is, independent of the cross-section shape and area. The lower limits of p 0 above which cells are irregular and have the same width regardless of the cross-section appear to be the same in our conditions. However, more recordings are necessary to remove uncertainty.

We presented geometrical elements from graph theory suggesting that a tessellation of hexagons can model a large set of irregular cells on a front-view recording (Sect. 4). This defines a cell average width λ and its intrinsic minimum relative error ∆λ/λ ≃ 13.4%. Our cell count on the front-view recording at the estimated limiting p 0 for irregularity, combined with the relation for λ from graph theory, results in the same value as interpolated from measurements on longitudinal recordings. This supports that cells are irregular for this p 0 . The large value O(0.1) of the intrinsic minimum relative error emphasizes the difficulty for numerical simulation to represent a detonation front with many cells, essentially the right balance between numerical accuracy and physical representativeness of the chemical kinetics. For example, considering the simple global Arrhenius rate of chemical progress and denoting E a the activation energy, R the universal gas constant and T H the shock temperature, we have λ ∝ exp(E a /RT H ) in the limit of large reduced activation energy E a /RT H , hence

∆E a E a = RT H E a ∆λ λ . (10)
The typical values T H = 1500 K and E a = 20 kcal/mole, with molecular mass M = 20×10 -3 kg/mole, R = 8.314 J/mole/K, and ∆λ/λ ≈ 0.1 then gives ∆E a /E a ≈ 0.3%. This small number implies that values of E a can always be found for satisfactory restitution of a single characteristic length such as λ, although this Arrhenius rate cannot often represent the actual kinetics. The physical interpretation is that both a detailed scheme of chemical kinetics and more advanced conceptual tools than a single length are necessary for characterizing detonation cells. Our analysis also indicates a continuous variation of the phase shift of the multi-cellular rectangular mode in square tubes at low or moderate initial pressures (Subsect. 3.2). We suggested that detonation propagation in this case is also subjected to long-wavelength and low-amplitude instabilities -compared to the cell dynamics -with characteristic times at least the effective reaction time. These include an intrinsic instability of the rectangular mode growing with increasing cell number, that is, p 0 , small defects at walls and persisting asymmetry of the wave front after ignition.

The simulation of cellular detonations should thus also include initial and boundary conditions with stochastic distributions of defects representative of actual devices. Today, this simulation relies essentially on Euler's equations for compressible reactive fluids. However, the uncertainties in the numerical technique, chemical kinetics, equations of state, multi-physics of interface phenomena, and hence the still very long computation times for three-dimensional waves do not make the physical interpretation of the computations less uneasy and time-consuming than that of the experiments.

Simplified partial differential equations with the same mathematical hyperbolic structure as Euler's, such as Burger's with a non-viscous source term, are modelling substitutes helpful to understand detonation instabilities and structures [START_REF] Fickett | Detonation in miniature[END_REF][START_REF] Kasimov | Model for shock wave chaos[END_REF][START_REF] Kasimov | Detonation analogs revisited[END_REF][START_REF] Lau-Chapdelaine | Detonation model using Burgers equation and a pulsed reaction[END_REF]. Combining simulation and modelling with applied mathematics tools on pattern generation and arrangements, such as information entropy and Voronoi tessellations [START_REF] Aurenhammer | Voronoi diagrams[END_REF], could bring detonation cell analysis closer to the mathematical corpus of dynamical systems and help to better characterize distributions with several cell patterns (Sect. 4).

Figure 1 :

 1 Figure 1: Schematic of a round or square tube. I: ignition, S: Shchelkin spiral, P: pressure transducers, L: longitudinal soot foil, H: end-on soot foil

Figure 2 :

 2 Figure 2: Front-view (top) and longitudinal (bottom) soot recordings in the mixture 2 H 2 + O 2 + 2 Ar at p 0 = 15.0 kPa in round (R), triangular (T) and square (Q) tubes with the same cross-section area (16 cm 2 ).

Figure 3 :

 3 Figure 3: Front-view (top) and longitudinal (bottom) soot recordings in the mixture 2 H 2 + O 2 + 2 Ar at p 0 = 17.5 kPa in round (R), triangular (T) and square (Q) tubes with the same cross-section area (16 cm 2 ).

Figure 4 :

 4 Figure 4: front-view (top) and longitudinal (bottom) soot recordings in the mixture 2 H 2 + O 2 + 2 Ar at p 0 = 20.0 kPa in round (R), triangular (T) and square (Q) tubes with the same cross-section area (16 cm 2 ).

Figure 5 :

 5 Figure 5: Front-view (top) and longitudinal (bottom) soot recordings in the mixture 2 H 2 + O 2 + 2 Ar at p 0 = 22.5 kPa in round (R), triangular (T) and square (Q) tubes with the same cross-section area (16 cm 2 ).

Figure 6 :

 6 Figure 6: Front-view soot recordings in the mixture 2 H 2 + O 2 + 2 Ar in the Q, R and R' tubes and longitudinal soot recording in the Q tube at p 0 = 70.0 kPa. The periphery of the R'-tube cross-section could not be recorded.

  λ (mm) R: Round, 16 cm² T: Triangular, 16 cm² Q: Square, 16 cm² R': Round, 71 cm² Q': Square, 25 cm² Strehlow 1969

Figure 7 :

 7 Figure 7: Cell average widths measured on the longitudinal soot recordings as function of the initial pressure p 0 and the shape and the area of the cross-section of the tubes.Measurement uncertainty is about the same as the marker heights.

Figure 8 :

 8 Figure 8: Left: side-and top-view schematics of the soot layer before (top) and after (bottom) the head-on impact. Right: head-on recording with the transverse-wave directions indicated by white arrow (mixture: H 2 + 1 2 O 2 + Ar, p 0 = 15.0 kPa, tube width a = 40 mm)

Figure 9 :

 9 Figure 9: Schematics of the detonation cell evolution with an arbitrary phase shift 0 < φ < φ max.

Figure 10 :Figure 11 :

 1011 Figure 10: Schematics of the detonation cell evolution with minimum phase shift φ = 0

Figure 12 :

 12 Figure 12: Blowup of crossing transverse waves

Figure 13 :

 13 Figure 13: Example of phase shift φ varying from 0 to φ max on a longitudinal soot recording for the 2 H 2 + O 2 + 2 Ar mixture at p 0 = 15.0 kPa.

Figure 14 :

 14 Figure 14: Front-view recording in the Q tube extracted from Figure 6(left) and Voronoi tessellation with randomly-distributed point sources (right).

Table 1 :

 1 Dimensions of the tubes. R, R': round, T: equilateral triangle, Q, Q': square

	R T Q R' Q'	16.6 16.1 16.0 70.8 25.0	2 ) Diameter or side length (mm) Length (m) 46 6 61 1 40 6 95 10 50 6

Acknowledgements

This work was supported by the Ministry of Higher Education, Research and Innovation (France)


Context & motivations

In engineering, many system design problems come down to finding design parameters for the system to operate under desired conditions. When time-consuming numerical simulations are used to assess the conditions of operation of such systems, it is essential to treat this problem-broadly known as "inversion"-using search methods that are very sparing in the number of simulations. With this constraint in mind, we propose here a sequential strategy for a particular robust inversion problem that we call "reliability-based inversion" (RBI), in which the system has both deterministic and uncertain inputs. In this formulation, the objective is to retrieve the set of deterministic inputs such that the probability of the outputs exceeding a given level is greater than a threshold.

The reliability-based inversion (RBI) problem

We view our system as a black-box function f : X × S -→ R. In this setting, X is the space of deterministic inputs and S the space of uncertain inputs.

• Given a level q ∈ R and a threshold α ∈]0, 1[, our objective is to approximate the set

Γ(f ) = {x ∈ X : P(f (x, S) ≥ q) ≥ α}
where S is a random variable with known distribution P S on S.

• Equivalently, we can formulate this problem as a classification problem, in which we wish to approximate the classifier

c(x) = 1 if x ∈ Γ(f ) 0 otherwise
The Stepwise Uncertainty Reduction (SUR) principle SUR strategies are a subset of the Bayesian strategies in which the points of evaluation are sequentially chosen by minimizing the expected future uncertainty on the object of interest, using a probabilistic surrogate of the unknown function.

• Consider ξ ∼ GP (µ, k) a Gaussian Process prior on f , with µ and k respectively the mean and covariance functions of the process.

• We define the current information I n :

I n = {(X 1 , S 1 , ξ (X 1 , S 1
)) , ... , (X n , S n , ξ (X n , S n ))}

• A SUR strategy choose the point minimizing the expected future uncertainty:

(X n+1 , S n+1 ) ∈ arg min (x,s)∈X×S E n (U n+1 | X n+1 = x, S n+1 = s)
where:

-U n+1 is an uncertainty metric on the object of interest, depending on I n+1 -E n expectation with respect to P n , the distribution of ξ given I n SUR approaches have been successfully used for the resolution of several problems regarding expensive-to-evaluate black box, including optimization [VVW09; MTZ78], function fitting, and more standard inversion problems. [START_REF] Bect | Sequential design of computer experiments for the estimation of a probability of failure[END_REF]

  SEQUENTIAL BAYESIAN INVERSION OF BLACK-BOX FUNCTIONS IN PRESENCE OF UNCERTAINTIES Romain Ait Abdelmalek-Lomenech, Julien Bect, Emmanuel Vazquez Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes SEQUENTIAL BAYESIAN INVERSION OF BLACK-BOX FUNCTIONS IN PRESENCE OF UNCERTAINTIES Romain Ait Abdelmalek-Lomenech, Julien Bect, Emmanuel Vazquez Université Paris-Saclay, CNRS, CentraleSupélec, Laboratoire des Signaux et Systèmes

This work has been funded by the National French Research Agency (ANR) in the context of the project SAMOURAI (ANR-20-CE46-0013).

Construction of a SUR criterion for the RBI problem

• Let us define the integrated process τ and Γ(ξ) the associated random excursion set above the threshold α:

• We consider the uncertainty metric U n defined by

The link between the metric and the uncertainty of the random set Γ(ξ) can be seen more clearly by remarking that:

with V n the conditional variance given I n

• The conditional distributions of τ (ξ, x) are intractable: the SUR criterion does not admit an analytic expression. • Monte Carlo simulations: ξ being Gaussian, we can generate ξ 1 , ..., ξ M sample paths of ξ under

For each candidate point (x, s), the sample paths are reconditioned according to values sampled from the distribution of ξ(x, s) given I n

Numerical results

We compare, iteration by iteration, the proportion of misclassified points obtained with our method to the ones of methods focusing on the excursion set in the joint space X × S of design and uncertainties: the straddle criterion [START_REF] Bryan | Active Learning for Identifying Function Threshold Boundaries[END_REF] and a SUR approach to standard inversion problems. [START_REF] Bect | Sequential design of computer experiments for the estimation of a probability of failure[END_REF] Empirical evidences show that our method generally outperforms state of the art joint space inversion methods. Moreover, in all the test cases, the new SUR strategy never lags behind the others approaches considered.


Manal Hamzaoui

Laetitia Chapel

Minh-Tan Pham

Sébastien Lefèvre

A Hierarchical Prototypical Network for Few-Shot Remote Sensing Scene Classification

Keywords: Few-shot learning, class hierarchy, scene classification, remote sensing

Few-shot learning (FSL) aims at making predictions based on a limited number of labeled samples. It is a hot topic in many fields such as natural language processing, computer vision and more recently, remote sensing. In this work, we focus on few-shot remote sensing scene classification which aims to recognize unseen scene categories at training stage from few or even a single labeled sample at test stage. Although considerable progress has been achieved in this topic, less attention has been paid to leveraging the prior structural knowledge. In this paper, we learn transferable visual features by introducing the class hierarchy which encodes the semantic relationship between the classes. We build on a prototypical network and we define hierarchical prototypes that allow us to encode the different levels of the hierarchy. Experiments conducted on the remote sensing NWPU-RESISC45 dataset demonstrate that the proposed hierarchical prototypical network acts as a regularizer and leads to better performance than the original network in the context of fewshot remote sensing scene classification.

Introduction

Scene classification is an important research topic in remote sensing which aims to automatically assign a specific semantic category to each remote sensing scene image. Deep learning frameworks have been applied to this problem and have achieved outstanding performance on most remote sensing image scene classification (RSISC) datasets. They essentially extract end-to-end features from images using deep neural networks such as Auto-Encoders [START_REF] Esam | Using convolutional features and a sparse autoencoder for land-use scene classification[END_REF] and Convolutional Neural Networks (CNN) [START_REF] Nogueira | Towards better exploiting convolutional neural networks for remote sensing scene classification[END_REF]. However, most of supervised deep remote sensing scene classification algorithms are "data-hungry" as they require a large amount of labeled data for training. When the labeled data are insufficient, there would be an obvious over-fitting and irrelevant extracted features, leading to a degradation of classification performance. However, obtaining labeled samples may be tough as it is labor-intensive, time-consuming and may need strong human expertise.

Inspired by the human ability to learn new abstract concepts from very few, or even one, examples and to generalize quickly to new instances [START_REF] Shi | Relational generalized few-shot learning[END_REF], few-shot learning (FSL) was introduced as one of the alternative ways to deal with the "data-hungry" issue. FSL methods can be divided into three categories [START_REF] Sun | Research progress on few-shot learning for remote sensing image interpretation[END_REF]: metric learning, meta-learning and transfer learning. Metric learning methods learn a distance function that brings samples from the same category as close as possible in the feature space while pushing samples from other categories as far away as possible. As for meta-learning, also known as learning to learn, it is the most common approach in FSL, which efficiently optimizes the model parameters to new tasks. Transfer learning aims at using the knowledge gained from relevant tasks towards new tasks, e.g. fine-tuning the pre-trained models is a powerful transfer method.

Recently, the combination of meta-learning and metric learning has been one of the most studied approaches in FSL for natural image classification [START_REF] Snell | Prototypical networks for few-shot learning[END_REF][START_REF] Vinyals | Matching networks for one shot learning[END_REF] and for remote sensing scene classification [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF]. First, based on meta-learning, these approaches construct tasks with few labeled samples, which enhances the generalization performance of the model for new tasks. Then, the similarity between image features is measured to make predictions. Some of related methods include relation network [START_REF] Sung | Learning to compare: Relation network for few-shot learning[END_REF], classical matching network [START_REF] Vinyals | Matching networks for one shot learning[END_REF] and prototypical network [START_REF] Snell | Prototypical networks for few-shot learning[END_REF].

In addition to the sparsity of labeled data, it may be more challenging to classify remote sensing data than natural images. Indeed, remote sensing scene images may present confusing visual similarity between different classes. The large intra-class variation may even exceed the inter-class variance, thus similar semantic classes may present significant visual dissimilarity [START_REF] Cheng | When deep learning meets metric learning: Remote sensing image scene classification via learning discriminative cnns[END_REF]. Another challenge is that remote sensing images are top-down views and contain inevitably many objects that are not relevant to the semantic class of the scene [START_REF] Li | Few-shot scene classification with attention mechanism in remote sensing[END_REF]. Yet, this characteristic could be very useful in multi-label or hierarchical classification tasks where several levels of semantic granularity are considered.

In the recent years, many approaches were proposed to tackle the problem of few-shot remote sensing scene classification (FSRSSC). In [START_REF] Li | Dla-matchnet for few-shot remote sensing image scene classification[END_REF], the authors adopted the attention mechanism to delve into the inter-channel and inter-spatial relationships to discover discriminative regions in the remote sensing scene images. The authors in [START_REF] Cheng | Spnet: Siameseprototype network for few-shot remote sensing image scene classification[END_REF] used a Siamese-prototype network with prototype selfcalibration and inter-calibration to learn more discriminative prototypes. In [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF], the authors introduced a pre-training step on the base data to provide better initialization of the feature extractor and performed the few-shot remote sensing scene classification using cosine distance metric. However, to the best our knowledge, the majority of these methods have focused only on visual scene information to improve feature representations without considering semantic knowledge that may exist within these classes. Yet this type of semantic knowledge about classes, which can consist of attributes, word embeddings or even a knowledge graph (e.g. WordNet [START_REF] Miller | Wordnet: A lexical database for english[END_REF]), is commonly used in zero-shot learning (ZSL) and increasingly in few-shot natural images classification approaches.

Semantic knowledge is not a novelty in ZSL, since this task can not be accomplished without such knowledge. However, in FSL, this semantic knowledge has hardly been used until recently. [START_REF] Chen | Multi-level semantic feature augmentation for one-shot learning[END_REF] proposed the TriNet to tackle the "1-shot" task by synthesizing the instance features from the semantic space which is given by the label embeddings. In [START_REF] Yang | SEGA: semantic guided attention on visual prototype for few-shot learning[END_REF], the authors proposed a method called Semantic Guided Attention (SEGA) mechanism which leverages semantic knowledge to guide the visual perception in learning the discriminative visual features of each class. Most of these FSL approaches that introduce semantic knowledge involve the text modality. However, few attention has been paid to knowledge transfer based on the class hierarchy which is either built using text modality as in [START_REF] Li | Large-scale few-shot learning: Knowledge transfer with class hierarchy[END_REF] or already predefined as in [START_REF] Liu | Many-class few-shot learning on multi-granularity class hierarchy[END_REF]. In [START_REF] Li | Large-scale few-shot learning: Knowledge transfer with class hierarchy[END_REF], the authors proposed a hierarchical image recognition approach by performing Softmax optimization on all levels of the class hierarchy. This allows learning transferable visual features through this class hierarchy which encodes semantic relationships between seen and unseen classes. In [START_REF] Liu | Many-class few-shot learning on multi-granularity class hierarchy[END_REF], a class hierarchy was introduced to address the multi-class FSL problem. The authors proposed a "memory-augmented hierarchical-classification network (MahiNet)" model which leverages the hierarchy as prior knowledge to train a coarse-to-fine classifier where each coarse class can cover multiple finer classes.

According to [START_REF] Liu | Many-class few-shot learning on multi-granularity class hierarchy[END_REF], FSL with knowledge transfer can be accomplished independently of an additional modality such as text and yields competitive performances, when the class hierarchy is known or easily obtained, which fits well with our research interests. This class hierarchy has been successfully applied to traditional supervised learning tasks. It can be introduced in the learning process according to three approaches [START_REF] Bertinetto | Making better mistakes: Leveraging class hierarchies with deep networks[END_REF]: a label-embeddings approach, a hierarchical loss or hierarchical architectures. In label-embeddings approach, a mapping function is used to encode class relationship information and associate it to class representations such as soft-labels [START_REF] Bertinetto | Making better mistakes: Leveraging class hierarchies with deep networks[END_REF]. Hierarchical loss-based methods adjust the loss to be optimized by assigning a higher penalty to predictions that are distant from the true label in the class hierarchy, such as hierarchical crossentropy loss [START_REF] Bertinetto | Making better mistakes: Leveraging class hierarchies with deep networks[END_REF]. As for hierarchical networks, they introduce the class hierarchy into the classifier architecture without necessarily changing the loss function, allowing them to make super-class predictions at early layers and fine predictions at later layers.

The remote sensing classes can be easily arranged in a hierarchical structure following well-known organizations such as Corine Land Cover (CLC), the European Nature Information System (EUNIS) habitat classification scheme or other structures such as done in [START_REF] Liu | Remote-sensing image retrieval with treetriplet-classification networks[END_REF] where they propose a hierarchical organization of the scene classes of the PatternNet [START_REF] Zhou | Patternnet: A benchmark dataset for performance evaluation of remote sensing image retrieval[END_REF] remote sensing scene dataset.

In this work, we rely on the semantic knowledge associated with scene classes through their hierarchical organization. We build on prototypical networks to define a hierarchical variant: in a nutshell, hierarchical prototypes are attached to each level of the hierarchy, allowing us to first consider high-level aggregated information before making a fine prediction. We show on a remote sensing dataset that it acts as a regularizer, giving better performances not only at the top nodes of the hierarchy, but also at the leaf classes. We also show that it performs better than soft-labels [START_REF] Bertinetto | Making better mistakes: Leveraging class hierarchies with deep networks[END_REF] that we introduce for the first time (to our knowledge) in a remote sensing few-shot learning context. The remainder of our paper is organized as follows. In Section 2, we provide some details about FSL and prototypical networks. Section 3 presents in depth the proposed method. We describe the experimental setup and the obtained results in Section 4. Conclusion and future works are given in Section 5.

2 Few-Shot Classification with Prototype Learning , where x i is an image and y i ∈ C e its corresponding label. Similarly for the episode query set Q, K ′ labeled images are sampled from the base set D base for each class c ∈ C e , resulting in Q = {(x q , y q )} N ×K ′ q=1 . In this training step, the support set S and the query set Q are used to learn the model that projects the input images into the feature space. The testing step is also carried out with the same episodic strategy where we have an unlabeled query set Q (drawn from D novel ) for which we want to predict the class label of each query sample x q ∈ Q using the labeled support set S (also drawn from D novel ). Fig. 1 shows a visualization of the N-way K-shot episodes.

Prototypical Networks

Prototypical networks [START_REF] Snell | Prototypical networks for few-shot learning[END_REF] adopt an episodic strategy to train a meta-learner classifier M. Given an episode with a support set S and a query set Q , we compute the representations of the images in S using the meta-learner feature extractor f Φ (a neural network such as CNN) parameterized by Φ. Thereafter, the representations are averaged to compute the prototypes p c for each class c ∈ C e as follows:

p c = 1 K (xi,yi)∈S c f Φ (x i ) (1)
where S c is the subset of the episode support set S that contains the samples of class c ∈ C e .

To optimize the feature extractor f Φ , we minimize the loss function:

L = - 1 N × K ′ c∈Ce (xq,yq)∈Q c log p Φ (y q = c | x q ) (2)
where Q c is the subset of the episode query set Q that contains the samples of class c, p Φ (y q = c | x q ) is the probability of predicting a query sample (x q , y q ) ∈ Q as class c and is given as:

p Φ (y q = c|x q ) = exp(-d(f Φ (x q ), p c )) c ′ ∈Ce exp(-d(f Φ (x q ), p c ′ )) (3)
where d(.) represents a certain distance measurement, such as the Euclidean distance [START_REF] Snell | Prototypical networks for few-shot learning[END_REF] or the Cosine distance [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF].

3 A Hierarchical Prototypical Network for Few-Shot Image Classification

Overall Framework

We propose a meta-learning framework whose complete pipeline is illustrated in Fig. 2 to solve the few-shot problem when a hierarchy that describes the organization between the classes is available. We train a meta

Hierarchical Prototypical Network

Some works have already attempted to introduce the class hierarchy knowledge into the few-shot classification process. In [START_REF] Li | Large-scale few-shot learning: Knowledge transfer with class hierarchy[END_REF], the authors suggested to perform a Softmax optimization over the different levels of the class hierarchy to enable knowledge transfer from seen to unseen classes. Here, we rather rely on the prototypical networks and introduce the hierarchy knowledge thanks to the definition of hierarchical prototypes. The overall idea is to regularize the latent space by putting closer classes that are in the same branch of the class hierarchy, and pushing apart classes that have common ancestors in higher levels of the class hierarchy To properly formulate our approach, given an episode, we first compute the prototypes per class which are prototypes at the leaf-level of the class hierarchy H (following Eq. 1). We then compute the hierarchical prototypes by aggregating the leaf-level prototypes according to H. The prototypes of the super-classes c ∈ C l e (the hierarchical prototypes) at level l (1 < l < L with l = 1 the root node and L = height(H)) are denoted as P l = {p c l } c∈C l e and computed as the mean of support samples of the super-class sub-tree S c l similarly to Eq. 1:

p c l = 1 |S c l | (xi,yi)∈S c l f Φ (x i ) (4)
Note that when l = L, the prototypes at level l are the prototypes at the lowest level of H (leaf-level prototypes).

The hierarchical prototypical network outputs a distribution over classes for each query sample x q ∈ Q at different levels of H, based on a Softmax over the distances to the prototypes of each level l in H. We then formulate the probability of predicting the query features f Φ (x q ) and the prototype p c l of its super-class c at level l in H as formulated in Eq. 3 as:

p Φ (y l q = c|x q ) = exp(-d(f Φ (x q ), p c l )) c ′ ∈C l e exp(-d(f Φ (x q ), p c ′ l )) , (5)
where y l q is the ancestor of y q at level l, C l e represents the super-classes at level l at the current episode.

We therefore optimize a new loss function given as

L H-proto = L l=2 λ l L l (6)
where

λ l = γ l-1 L l ′ =2 γ l ′ -1
, γ is a hyper-parameter that controls the importance of each level in the hierarchy and L l=2 λ l = 1. L l represents the prototypical network loss at level l of the class hierarchy H.

As such, we can tune the importance of each level of the hierarchy into the learning process: by choosing low values of γ, we put more importance into organizing the higher levels of the hierarchy; a value close to one gives the same importance for all the levels; a high value tends to behave like the flat cross entropy loss formulation.

Few-Shot Learning for Remote Sensing Scene Classification

We evaluate the performance of our hierarchical prototypical approach in a fewshot scene classification task. We first present the remote sensing scene dataset we consider in our study, since its labels are hierarchically organized Then, we describe the parameters and hyper-parameters setting. Finally, in order to assess the interest of introducing semantic knowledge into the few-shot scene classification task, we compare the classification results of the proposed approach to some baseline methods for the 5-way 5-shot and 5-way 1-shot tasks.

a set of episodes randomly sampled from the base set D base , which we set to 1000 episodes per epoch. We optimize the model based on the average loss of 4 episodes, i.e. the batch size is set to 4 episodes. We use SGD optimizer to update the network parameters with a momentum set to 0.9 and a weight decay set to 0.0005. The learning rate is fixed to 0.001. After each training epoch, we test our model on a validation set D val by randomly sampling 1000 episodes, the network weights with the lowest validation loss are retained as the best results.

For the hyper-parameter γ, we assigned different values (γ = 1, γ < 1 andγ > 1) in order to observe its impact on the framework performances.

For the meta-testing stage, we conduct a 5-way 1-shot and 5-way 5-shot classification following the widely used meta-learning protocol. We evaluate the best model on 2000 randomly sampled episodes from the test set D novel . Following the FSL evaluation protocol [START_REF] Snell | Prototypical networks for few-shot learning[END_REF], for 5-way K-shot episode, we randomly sample 5 classes from the unseen classes C novel , K images per class to form the support set S, and 15 images per class to form the query set Q, making a total of 5 × (K + 15) images per episode.

Evaluation Metrics

We use two metrics to evaluate the performance of the different methods:

-The classification accuracy, computed at different levels of the class hierarchy; -The hierarchical precision [START_REF] Silla | A survey of hierarchical classification across different application domains[END_REF] which is defined as the total number of common ancestors between the predicted class and the true class divided by the total number of ancestors of the predicted classes:

hp = i | Ŷi ∩ Y i | | i Ŷi | (7)
where Ŷi = {ŷ i ∪ Ancestor(ŷ i , H)} is the set consisting of the most specific predicted class for test example i and all its ancestor classes in H except the root node and Y i = {y i ∪ Ancestor(y i , H)} is the set consisting of the most specific true class for test example i and all its ancestor classes in H except the root node.

For all evaluation metrics, we report the average of the test episodes with a 95% confidence interval.

Experimental Results

In both 5-way K-shot configurations, K=1 or 5, we compare our method to the original flat prototypical network [START_REF] Snell | Prototypical networks for few-shot learning[END_REF] and to the approach proposed in [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF] that uses the cosine metric as a similarity function, which we denote by ProtoNet and c-ProtoNet respectively. We re-implement both methods with their related parameter setting according to [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF] and use ResNet-12 as a backbone for a fair Few-shot learning has captured the attention of the remote sensing community thanks to the great success it has achieved in other fields. In many cases, when dealing with the problem of scene classification, the organization of the classes is defined in a hierarchical manner, with classes being semantically closer than some others. In this work, we present a novel prototypical network which defines hierarchical prototypes that match the nodes of the label hierarchy1 . We evaluate our method on a benchmarked [START_REF] Cheng | Remote sensing image scene classification: Benchmark and state of the art[END_REF] remote sensing scene dataset in a few-shot learning context and we show that hierarchical prototypes ensure a regularization of the latent space, providing higher performance than flat prototypes but also than a competitive hierarchical loss introduced in another context.

In future work, we plan to investigate the use of the hierarchical prototypes on other tasks such as semantic segmentation, where we consider adding spatial information among regions to output meaningful hierarchical prototypes. We also intend to use graph prototypical networks instead of prototypical networks, which better take into account the class relationships. We further consider relying on other metric spaces than the Euclidean one, e.g. hyperbolic spaces that are known to better encode the distances when the data are hierarchically-organized. Thus, a follow-up of this work could be to build hyperbolic prototypes to enforce the hierarchical information into the learning process.

2. 1

 1 Problem Formulation of the FSL Training (seen classes) Testing (unseen classes)

Fig. 1 .

 1 Fig. 1. Illustration of N-way K-shot classification episodes. The left side shows the M episodes of the training step; each episode consists of N × K support samples and N × K ′ query samples. The testing step is similarly defined on M ′ episodes, as shown on the right.

Fig. 2 .

 2 Fig. 2. Overall framework of the proposed hierarchical prototypical network for fewshot image classification. In this example (one-shot), N = 5, N h = 5, = 1, K ′ > 1 (usually set to 15).

We recently became aware of a paper that proposes a similar approach to classify audio data in the FSL context[START_REF] Garcia | Leveraging hierarchical structures for few-shot musical instrument recognition[END_REF]. The difference lies rather in the experimental part in which we use a deeper network and a pre-training step.

Acknowledgement

This work was supported by the ANR Multiscale project under the reference ANR-18-CE23-0022.

Dataset Description

The NWPU-RESISC45 [START_REF] Cheng | Remote sensing image scene classification: Benchmark and state of the art[END_REF] dataset is a widely used benchmark for remote sensing image scene classification. It consists of 31 500 images of 256 × 256 pixels; the spatial resolution varies from approximately 30 to 0.2 m per pixel. It covers 45 scene categories, each with 700 RGB images, which can be organized hierarchically. Following [START_REF] Liu | Remote-sensing image retrieval with treetriplet-classification networks[END_REF] in which the authors propose a hierarchical organization of the scene classes of the PatternNet [START_REF] Zhou | Patternnet: A benchmark dataset for performance evaluation of remote sensing image retrieval[END_REF] remote sensing scene dataset, we construct a tree-like arrangement of these scene classes which reflects their semantic relationships.We note that the leaf level of the constructed class hierarchy corresponds to the original scene classes of the dataset. A sub-tree of the 3-level label tree is shown in Fig. 3. For a fair comparison, we adopt the same split as done in [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF]. We split the dataset into three disjoint subsets: meta-training D base , meta-validation D val , and meta-test D novel containing 25, 8, and 12 categories, respectively. We note that the meta-validation set is used for hyper-parameter selection in the metatraining step. The meta-training set is further divided into three subsets: training, validation, and test sets. In our experiments, we follow [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF] and resize all the images to 80 × 80 pixels to fit our designed feature extractor.

Implementation Details

Following recent FSRSSC studies [START_REF] Li | Few-shot scene classification with attention mechanism in remote sensing[END_REF][START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF][START_REF] Zhang | Task-adaptive embedding learning with dynamic kernel fusion for few-shot remote sensing scene classification[END_REF][START_REF] Zhang | RS-SSKD: self-supervision equipped with knowledge distillation for few-shot remote sensing scene classification[END_REF], we utilize ResNet-12 as a backbone for feature extraction. We also adopt the pre-training strategy as suggested in [START_REF] Zhang | Few-shot classification of aerial scene images via meta-learning[END_REF] to better initialize the meta-learner feature extractor.

We train our meta-learner for 1000 epochs with an early stopping of 50 epochs. The best model parameters were obtained within the first 300 epochs. In standard deep learning, an epoch implies that the entire train set passes through the deep neural network once. However, in meta-learning, an epoch is comparison. We also compare our prototypes (namely h-ProtoNet) with the results yielded by the soft-label method [START_REF] Bertinetto | Making better mistakes: Leveraging class hierarchies with deep networks[END_REF], which allows taking into account the class hierarchy within the learning process. Results are provided in Table 1 for K = 5 and Table 2 for K = 1. For both 1-shot and 5-shot cases, our proposed h-ProtoNet achieves the highest accuracy and outperforms both flat prototypes and the soft-labels hierarchical loss. We obtain the best performance with γ = 1, that is to say when all the (hierarchical) prototypes have the same weights. When we put more weights on the prototypes that correspond to the higher level of the hierarchy (corresponding to nodes close to the root, γ < 1) or to those that correspond to the leaves (γ > 1), we obtain degraded performances, that are still better than the other methods. Note that this value of γ = 1 would have been selected if we perform a cross-validation on the validation set. We argue that the improvement observed in the case of the hierarchical prototypes is due to an efficient regularization of the latent space, with a loss that encourages leaves within the same branch of the level hierarchy to be closer. As such, the performances at level 2 and 3 are improved, but also the overall accuracy.