Gwendal Jouan
email: gwendal.jouan@ec-nantes.frpanagiotis.kotronis@ec-nantes.fr

Panagiotis Kotronis

Frédéric Collin
email: f.collin@ulg.ac.be

NUMERICAL MODELING OF A 3 POINT BENDING TEST OF A REINFORCED CONCRETE BEAM USING A SECOND GRADIENT THEORY

Keywords: three point bending test, localization, second gradient, reinforced concrete

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Numerical modeling of 3 point bending test of a reinforced concrete beam using a second gradient theory

INTRODUCTION

Strain localization in quasi-brittle materials, or more generally in materials exhibiting strain softening is a well-known problem [START_REF] Hill | Bifurcation phenomena in the plane tension test[END_REF] [2] [START_REF] Rudnicki | Conditions for the localization of deformation in pressuresensitive dilatant materials[END_REF]. It is a phenomenon that can be clearly observed in experimental tests, and yet cannot be modeled with classical continuum mechanics models. Analytically, the solution is a crack with zero energy dissipation [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF]; numerically, it leads to a pathological mesh dependency. These shortcomings are due to the lack of an internal length in the continuum model.

Several ways of introducing an internal length have been proposed. Non local integral models were first formulated by Pijaudier-Cabot et al. [START_REF] Pijaudier-Cabot | Nonlocal damage theory[END_REF] and applied to damage theory. The gradient models of Aifantis [START_REF] Aifantis | On the microstructural origin of certain inelastic models[END_REF] enrich the conventional plasticity and damage theories with gradients of the internal variables. This type of model can be shown to be equivalent to the integral type models. More recently, strain localization due to damage has been treated using the thick level set approach [START_REF] Moës | A level set based model for damage growth: The thick level set approach[END_REF].

A rather natural way of introducing (indirectly) a length parameter in a continuum model is to account for the microstructure of the material. The general class of so called microstructured models or higher order continuum models allows for the description of the kinematics of the microstructure by using an additional tensor in the displacement field. Higher order continuum theories can be traced back to the works of the Cosserat brothers [START_REF] Cosserat | Théorie des corps déformables[END_REF] and have been generalized and properly formulated by Germain [START_REF] Germain | La méthode des puissances virtuelles en mécanique des milieux continues: Première partie : théorie du second gradient[END_REF] [10] using the method of virtual power.

The local second gradient model developed by Chambon et al. can be seen as a particular case of a higher order continua and has been used to regularize problems involving localization in geomaterials [START_REF] Chambon | Plastic continuum with micro-structure, local second gradient theories for geomaterials: localization studies[END_REF][START_REF] Matsushima | Second gradient models as a particular case of microstructured models: a large strain finite elements analysis[END_REF]. It is hereafter used to model a three point bending test of a reinforced concrete beam.

THE SECOND GRADIENT MODEL

Theoretical framework

As detailed in the seminal work of Germain, using the virtual power method one can choose a field of virtual displacements to describe the proper kinematics of the continuum (including its microstructure). The internal stresses, limit conditions and equilibrium equations appear naturally as long as the linear form representing the virtual power is correctly defined and that it respects the principle of material independence.

Following this, if we choose the virtual displacement field as the "field of continuous and continuously differentiable velocities", then the principle of virtual work yields:

* + Σ , * Ω - * Ω - * + * = 0 (1)
Where u i is the displacement field, D ij is the symmetric part of its gradient, σ ij is the macro stress (conjugate of the first gradient of the velocity field) and Σ ijk is a double stress (conjugate of the second gradient of the velocity field). G i is the classical body force, p i is a contact force and P i a double contact force. Du i refers to the normal derivative: Du i = n k u i / x k. . Here and henceforth * denotes a virtual quantity.

The equilibrium equations and boundary conditions are given by:

- Σ + = 0 (2) - Σ - Σ - Σ + Σ - Σ = (3) Σ = (4)
Where Dq/Dx j is the tangential derivative.

Here we suppose that the model uses two constitutive equations, one linking the macro stress to the first gradient of the displacements and the other linking the double stress to the second gradient of the displacements. The two equations are also supposed to be decoupled. The first gradient law can be any classical constitutive law (involving or not damage, plasticity etc…). The constitutive laws can be written in rate form as:

= ! " (5.1) Σ = # $% ,$% (5.2)
If the second gradient law is an isotropic elastic one, the general form of the constitute law involves five material parameters.

& ' ' ' ' ' ' ' ' ( ) *** ) ** ) * * ) * ) ** ) * ) * ) + , , , , , , , , - = &
Where a ijk… are functions of five material parameters a1, a2, a3, a4, and a5 [START_REF] Mindlin | Second gradient of strain and surface tension in linear elasticity[END_REF].

A particular form of this latter constitutive equation has been proposed by Chambon et al. [START_REF] Mindlin | Second gradient of strain and surface tension in linear elasticity[END_REF] with a single modulus B (corresponding to a specific choice of the a i parameters).

The internal length does not appear explicitly in these equations. However, the analytical solution in a one-dimensional localization problem with a bilinear first gradient law of moduli A 1 and A 2 , exhibits a localized strain band with an internal length l s given by [START_REF] Chambon | One-dimensional localization studied with a second grade model[END_REF]:

It is shown that the internal length l s is a function of the ratio B over A 2 . The use of the second gradient model allows for the proper representation of strain localization but it does not restore the uniqueness of the solution. For a one-dimensional tension test or a biaxial compression test, the number of bands and their positions can vary [START_REF] Bésuelle | Switching deformation modes in post-localization solutions with a quasibrittle material[END_REF]. This is in accordance with experimental results which tend to be poorly reproducible when involving strain localization.

Numerical implementation

The second gradient of the displacements in the weak formulation of the problem necessitates the use of C 1 elements in a finite element code. This can be avoided by introducing a new field v ij imposed equal to the gradient of u i through the use of Lagrange multipliers [START_REF] Matsushima | Large strain finite element analysis of a local second gradient model: application to localization[END_REF]. The new weak formulation of the problem becomes: * + Σ @ , * Ω -A B * -@ * C Ω -" = 0 (8.1)

A * B -@ C Ω = 0 (8.2)
The problem is then discretized using a 9 th node second gradient finite element: 8 of the nodes are used for u i , 4 for v ij and the central node for λ ij .

Figure 1: Second gradient finite element [START_REF] Matsushima | Large strain finite element analysis of a local second gradient model: application to localization[END_REF] This element has been implemented in the finite element code Lagamine and the problem is solved using the classical Newton-Raphson method.

THE THREE POINT BENDING TEST

The experimental test was conducted on a reinforced concrete beam according to the specifications of the CEOS.FR benchmark. The beam is of thickness b=200mm, height h=500mm and 5000mm span (see: figure 2). It was subjected to a cyclic controlled load.

MODELISATION OF THE BENDING TEST

The bending test is modeled as a two dimensional problem using the 9 node finite element described above. Two meshes have been used for the simulations. The first mesh consists of 5180 elements, 4148 of which are second gradient elements representing concrete and 1032 truss elements representing the horizontal reinforcement. The average size of the concrete elements for this mesh is of 0.02mx0.035m. The second mesh consists of 13494 elements with an average size of 0.01mx0.017m for the concrete elements. Concrete and steel elements are supposed to be perfectly bonded. The end nodes at the bottom of the beam are blocked vertically; the right node at the bottom is also blocked horizontally. The concrete behavior is reproduced using the classical Mazars damage law [START_REF] Mazars | Application de la mécanique de l'endommagement au comportement non linéaire et à la rupture du béton de structure[END_REF]. The equivalent strain is defined as:

D EF = G〈D〉 + : 〈D〉 + ( 9
)
Where 〈 〉 K denote the positive part and : is the double dot product. The damage is separated in two parts D t and D c , one due to the tension, and one due to compression.

L = 1 - D 0 81 -6 L < D EF -6 L E -7 L ND EF -D 0 O (10.1) 2 = 1 - D 0 81 -6 2 < D EF -6 2 E -7 2 ND EF -D 0 O (10.2)
Where ε d0 , A c , A t , B c and A t are material parameters provided in table 1: On the upper part where the loading is applied and at both supports at the bottom an elastic linear law is introduced to prevent from unwanted damage. An elasto-plastic law with isotropic hardening is used for steel. The second gradient constitutive law for concrete depends on the elastic modulus B taken equal to 1.5.MN.

E (GPa) ε d0 A t B t A c B c β

RESULTS

Overall, the numerical results are in good accordance with the experimental data. Figure 5 shows the numerical global force versus displacement at the center of the beam compared to the experimental results (note: for the experimental test the beam was loaded and unloaded cyclically whereas in the simulation the beam was loaded with a monotonic increasing displacement). The force displacement graph exhibits the classical reinforced concrete behavior in three stages: In the first stage, concrete and steel stay both in the elastic regime; then concrete starts to damage and the slope on the force displacement curve slightly changes. Finally, steel enters in the plastic phase and the second change in the slope appears. The final slope of the numerical simulation doesn't match the experimental one, but results can be improved by doing a parametrical study on the influence of the elastic modulus B.

Figure 6 shows the pattern of the damage variable in concrete for different stages of loading and for the two meshes. The damage pattern develops with sudden "peaks" which experimentally correspond to developing cracks. The crack opening is not modeled directly in this simulation as the displacement field remains continuous, but it can be calculated from the damage model either through energetic equivalence or by simply measuring the jump in displacements between two points located on the opposite sides of the damaged zone. This obviously works only when the damaged bands are clearly separated. The width and separation of the damage bands can be controlled by changing the internal length, which in our case would mean changing the slope of either or both the first gradient and second gradient constitutive laws.

For concrete bending tests, non-local damage models which define an equivalent strain by averaging over a certain distance l c may develop damage on the upper compressed part of the beam even when the local strain is not high enough to trigger damage. This is due to the averaging over an area. There is no such problem with this model as all the variables are local, including the equivalent strain.

CONCLUSIONS

A second gradient model has been used to model the three point bending test of a reinforced concrete beam. The results show that the model is able to reproduce the forcedisplacement curve obtained experimentally. The damage localizes into bands and their width is controlled by the model parameters. The uniqueness of the solution is however not restored as shown by the results obtained with two different meshes. In these two cases the bands are of the same size but their numbers can vary (figure 6). Being a local theory, the second gradient method avoids the limitations caused by the use of a non-local definition of the equivalent strain used in the non-local damage theory.

These results are encouraging and represent the first steps toward a wider use of the local second gradient method for concrete structures.

Figure 2 :

 2 Figure 2: Reinforced concrete beam dimensions [17] Strains were measured with gauges placed as indicated on figure 3.

Figure 3 :

 3 Figure 3: Location of strain gauges[START_REF] Jason | Réponse au benchmark statique monotone du projet national CEOS[END_REF]

Figure 4 :

 4 Figure 4: Finite element meshes[START_REF] Matsushima | Large strain finite element analysis of a local second gradient model: application to localization[END_REF]

Figure 5 :

 5 Figure 5: Numerical results Vs. Experimental data: Force-Displacement at the center of the beam

Figure 6 . 1 .Figure 6 . 3 .Figure 6 . 5 .Figure 6 . 7 .Figure 6 . 9 .Figure 6 . 11 .Figure 6 .

 61636567696116 Figure 6.1. Damage distribution (2mm displacement) Figure 6.2. Damage distribution (3mm displacement)

Table 1 :

 1 Concrete material parameters

	37.2	9.10 -5	0.7	6800	0.42	780	1.1

Table 2 :

 2 Steel material parameters


Blaise Douglas

Bonds in a Selection of Middle English Breton Lays

For the Middle Ages, bond is a key notion, a notion which is fundamental in the proper sense. Indeed medieval society is fashioned according to a series of bonds. In a feudal society everyone is bound. Peasants are bound to the land, which is bound to a lord; lords themselves are then bound to a more powerful lord or a prince or a sovereign (the one who actually gives lands to his/her subjects) who is himself or herself bound to God. This creates a kind of pyramidal chain that binds everyone, from top to bottom, in a society ordered according to God's will.

Jacques Le Goff provides a relevant definition of this system: "La féodalité c'est d'abord l'ensemble des liens personnels qui unissent entre eux dans une hiérarchie les membres des couches dominantes de la société. Ces liens s'appuient sur une base « réelle » : le bénéfice que le seigneur octroie à son vassal en échange d'un certain nombre de services et d'un serment de fidélité." 1 This definition stresses the personal nature of these bonds and the fact they essentially concern the dominant classes of society. The lays 2 , we shall see, give a perfect illustration of this, as they seem to exclusively deal with the noblest individuals and with the bonds that bind them together. Yet they seldom evoke the material benefits that could be at stake, which is significant in itself.

The lays naturally represent bonds in a social perspective. They also explore the various threats to these bonds in their different aspects: bonds of duty, which are socially motivated but also take on a moral dimension, as well as bonds of love, something particularly relevant when dealing with Breton lays. This exploration will disclose an obvious fact: that bonds are most of the time associated with speech. The linguistic dimension is indeed essential. In the world of the lays, one is bound by one's promise, by the word given as well as by one's 1 Jacques Le Goff, La civilisation de l'occident médiéval, Paris, B.

situation in society. But this also implies the poet who is himself or herself bound by a tradition.

It seems proper to begin with considering social bonds, as they prove of paramount importance, in many different ways. The first bond is obviously and logically that of blood.

Its importance appears particularly clear in Lay le Freine and in Sir Degaré in which the eponymous characters can only achieve themselves fully with finding their biological parents.

The mothers who abandon their children want them, in spite of all, to be recognized, if not for who they are, at least for their rank: which means they never completely give up their children. Orphans are therefore provided with valuables that symbolize their status: "Sche toke a riche baudekine That hir lord brought from Costentine And lapped the litel maiden therin, And toke a ring of gold fin, And on hir right arm it knitt, With a lace of silke therin plit; And whoso hir founde schuld have in mende That it were comen of riche kende."

Lay Le Freine (137-44)

A similar token is used in Sir Degaré:

"Four pound she tok of gold, And ten of selver also; Under his fote she laid hit tho…" Sir Degaré These riches are provided together with a letter which states the child's noble nature: 'For hit is comen of gentil blod ' (209). This shows that these mothers are in fact unable, and probably unwilling to abandon their children completely. Indeed blood ties cannot be broken for both moral and spiritual reasons. 'Kin' is here a key word with deep resonance if one considers the polysemy hidden in these three letters. The word comes from the Old English cym which stands for family, race, kind, rank, nature, gender or sex. We can trace it in the word 'kindred' in modern English or in Kind, the child, in German. The Latin equivalent would be genus from which derive the French gène, génétique or genre, which has taken on a literary connotation particularly appropriate for us.

In the lays, the noble nature of a character is expressed with different words. 'Gentil' is one of them, another avatar of the Latin genus. 'Fre' is a frequently used Germanic equivalent that basically means free, i. e. not bound to the land. Yet, to be 'fre' does not imply absolute freedom since nobility comes with duties towards one's people and family: Guroun is bound to comply with his knights and marry a lady of rank, as Le Freine is yet of unknown origin:

"His knightes com and to him speke, […] Sum lordes douhter for to take […] And seyd him were wel more feir In wedlok to geten him an air Than lede his liif with swiche on Of was kin he knewe non."

Lay Le Freine An obvious conclusion is that everyone is bound to some extent. Lord though he be, Guroun must act according to what is expected from a man of his rank.

The great importance of social bonds in the lays is also perfectly demonstrated in Emaré, another poem from the collection by Laskaya and Salisbury. The eponymous heroine twice experiences unfair ostracism. She is twice exiled, first by her own father, then on account of her mother-in-law's hatred, a highly symbolical rejection from society which leads her to change her name into Egaré, which is very telling for a French ear. The formulation used for her second exile is remarkable. Emaré is to be led 'out of that kyth' (594). The polysemy of the word kyth, which means at the same time one's country, one's native land and also one's family, relatives and friends, makes it clear that the young woman is not just sent away physically but socially rejected.

When Emaré becomes aware of her condemnation, which she supposes comes from her lord and husband, she complies with it as she acknowledges her being rankless: "… do my lordes commaundement, God forbede thou spare.

For he weddede so porely On me, a sympull lady, He ys ashamed sore."

Emaré, 629-33 These social bonds are closely related with bonds of love, as love is usually, and quite logically, associated with begetting children. In Sir Degaré, Sir Degaré's father, though he rapes his future wife, announces at once that he loves her and that she will be pregnant.

Having children is also the main argument produced by the knights for the marriage of their lord in Le Freine. The Franklin's Tale and Sir Orfeo are two counter-examples since the bonds of love are clearly dissociated from the need to have children. In Sir Orfeo, what binds Orfeo to his wife is definitely essential. Heurodis' abduction leaves Orfeo incomplete and therefore unable to remain and assume his functions as king, which motivates his exile. Love takes here an ontological dimension: it is part of the essence of the character. This is almost a neo-Platonic vision of love as a union that allows one to become whole again.

In The Franklin's Tale, love is also perceived as a most essential bond: Arveragus agrees to let Dorigen keep her promise out of love for her, which may seem paradoxical, and this proves an act of love, a sacrifice that asserts his noble attitude and compels Aurelius and the clerk, who has helped him, to emulate the knight. This adventure incidentally illustrates the virtue of example. It also demonstrates that a squire can be as noble, as 'fre' as a knight, which is part of the Franklin's aim in telling his tale. He actually wants to honour the squire who has just said, but not finished, his tale. This, in turn, shows how the social bonds work within the world of romance. Aurelius is actually to do the same as Arveragus, to imitate him from the beginning: he loves the same woman as the knight, then proves equally capable of self-denial. A similar emulation occurs in Sir Orfeo, as the steward, who is to eventually succeed his lord, honours harpers the same way his lord used to. Social bonds thus imply that one should take example on those who are higher on the social ladder, that one should look upward. This hierarchical perception is quite clear when the mirage-making clerk of The Franklin's Tale renounces his payment. This is as much as to say that the commercial bond established between him and Aurelius is nothing in comparison with love and honour which frame Arveragus and Dorigen's course of action. "La société féodale était une société d'ordres",3 wrote Gourevitch and this seems particularly true in the world of medieval romance.

Aurelius' final awareness provides a very positive perspective, but this idealistic vision is often confronted with threats at different levels. As any romance, lays rely on chivalric and courtly ideals, yet these ideals are not always respected as they should. A most striking example of this is to be found in Sir Degaré with the 'sterne knight' who wants to ravish the helpless lady. A knight is supposed to protect the weak and show particular respect for women. His failure to act properly means his logical defeat by Sir Degaré according to poetic justice and divine order. This sense of justice might seem surprising considering the fact Sir Degaré's father has behaved in a rather uncourtly way with his future wife. He has indeed done what the 'sterne knight' intends to do with the lonely lady in her castle. This 'sterne knight' could thus be construed as a kind of double of the fairy knight and his defeat by Sir Degaré sounds like the son's atonement for the father's sin. Similarly, when Aurelius frees Dorigen of her promise, one might construe the deed as Aurelius suddenly realizing that he is about to act against chivalric and courtly values. That would mean his failure to achieve knighthood, losing oneself as well as his love.

Another threat to the values of the chivalric world of romance is the repeated use of lying and slandering. The worst example of such a threat is Guenevere's efforts at taking revenge from Sir Launfal. Her action is not just a threat to a single knight but to their social order as a whole. This is true from the very beginning as she is said to have many lovers, which explains why Arthur's knights do not like her: "For the lady bar los of swych word That sche hadde lemmanys under her lord, So fele ther was noon ende."

Sir Launfal This means, if one formulates this differently, that she does not respect the bond of marriage, the seriousness of this breach of faith being much aggravated by the high rank of the lady. She is obviously no example for other ladies, who cannot look upward as Aurelius did with Arveragus.

Guenevere's destructive action becomes even greater when she accuses Sir Launfal falsely. Her slander makes it clear that she respects neither truth nor troth. Sir Launfal's answer to her accusation is quite revealing when he tells her she is not worthy of her rank. He mentions his lover, Dame Tryamour, and describes her thus: "'Hyr lothlokest mayde, wythoute wene, Myghte bet be a Quene Than thou, yn all thy lyve!'" Sir Launfal (697-99) Sir Launfal's judges precisely dismiss the veracity of Guenevere's accusation, only to retain the knight's criticism of his queen's beauty, there being an equation between beauty and rank. Sir Launfal's own reaction is particularly telling: when Guenevere asks him to be her lover, he labels it as treason: "'I nell be traytour day ne nyght…'" (683).

Guenevere's behaviour is all the more condemnable as, according to her position and status, she should set the example for other ladies to follow. She strikingly contrasts with the ideal traditional feminine figure who, like Beaulybon in the Erle of Tolous, embodies a moral standard by maintaining an exemplary conduct at all costs. Le Freine's mother proves as destructive as Arthur's wife, when she slanders her neighbour, suggesting that twins imply two different fathers. The slander is cause for all the disruption. It is perceived as such by everyone and by the poet to begin with who describes the lady as "hokerfulliche misegging" (61), for which the suggested translation is 'maliciously slandering'. Her speech is further styled as "wordes of felonie" (64). Strangely enough, but quite in keeping with the expected poetic justice of the genre, these "wordes of felonie" turn against her at once. No one believes her and her own husband feels ashamed for his wife's unwise claim. The lady even gets cursed for the unreasonable accusation she has voiced: "The messanger was sore aschamed;

The knight himself was sore agramed,

And rebouked his levedy

To speke ani woman vilaynie.

And ich woman therof might here Curssed hir alle yfere,

And bisought God in heven

For His holy name seven That yif hye ever ani child schuld abide A wers aventour hir schuld bitide."

Lay le Freine (73-82)

Cursing occurs here as a linguistic means of action. The curse proves effective when the slandering lady herself gives birth to twin girls. We may feelthis is implicitly suggestedthat God has heard the women's curse and sent a "wers aventour" to the slanderer.

In Emaré, the mother of the king of Galys is yet another woman slanderer. Like Guroun's knights, she does not realize that the heroine's beauty actually marks her as a woman of rank, though nothing is known of her ascendance. On the contrary, this extreme beauty is seen as a means of beguiling men, motivating the unfair mother-in-law's judgement: The old queen's refusal to bless her son's marriage with her unwanted daughter-in-law amounts to curse the union. Here again, poetic justice is perfectly respected as the slanderer is to experience the same punishment she has inflicted upon Emaré. She has unfairly broken the bond of love between the young woman and her son. She has disrupted natural and divine order, just as Emaré's father nearly does when he purposes to marry his daughter, and quite logically is herself cast out of society, suffering exile too.

A similarly proportional punishment is inflicted on the eponymous main character of Sir Gowther. As he acts with complete disregard for any sense of humanity, raping and killing away, his retribution is the very loss of his humanity when the Pope deprives him of speech and only allows him to eat what he may snatch from dogs. Social bonds are utterly broken with those of humanity, and redemption can then only come from God. Sir Gowther, this should be added, is saved thanks to love and the intervention of his beloved, the dumb daughter of the emperor of Germany.

In Sir Launfal, the eponymous hero is not faultless, even if his guilt is far from being as serious as Guenevere's. The generous knight jeopardizes the bond that has been established between him and Dame Tryamour. Sir Launfal is expressly asked not to mention her, but he forgets about it. Once again, the result is a disruption: the harmony between the two worlds, ours and the fairies', is broken. As a consequence the passage from one world to another becomes impossible, at least for a while. In the same way, the bond between the lovers has been broken and is only restored thanks to Dame Tryamour's capacity to forgive her knight, a typical and traditional quality required from a lady.

The bond between Sir Launfal and Dame Tryamour definitely evokes the Irish geis or Welsh tynged, which are taboos, interdictions imposed on a specific individual. The geis has to do with one's coming of age and is often placed by a woman or a goddess. It is often ambivalent as it can be considered as a gift, which seems to be the case for Sir Launfal, or as a curse, which might well be the case for Le Freine's mother.

In the lays, this type of bond is obviously linguistic in nature. In Sir Launfal, Dame Tryamour's own words are given to the audience as the poet resorts to direct speech: "'But of a thyng, Syr Knyght, I warne the, That thou make no bost of me For no kennes mede! And yf thou doost, I warny the before, All my love thou hast forlore!'" Sir Launfal The bond is also the result of a debt in Sir Launfal's case. Dame Tryamour gives him her love and provides her knight with riches, which implies that Sir Launfal is to remain faithful to her and act according to her requirements.

Just as Sir Launfal becomes indebted for the recovery of his rank, since Dame Tryamour's financial help allows him to get back into society, the earl saved by Sir Degaré has a debt with the young knight. As in Sir Launfal, the bond created by the indebtedness of the earl leads to the integration of the character in society: Sir Degaré acquires the title which is due to his rank. Though the indebtedness is reversed, the bond is, in both cases, the means of the restoration of a social status.

In both instances, bonds rely on truth and the promise that one should respect. The Franklin's Tale offers a most precious example of the utmost importance of truth. This is what Arveragus claims when he sacrifices his honour by sending Dorigen to Aurelius so that she can remain true to her word. But the first demonstration of respect for truth is made by Dorigen herself when she discloses her rash promise to Arveragus, so that the whole lay might be construed as a celebration of truth (and of love of course).

Similarly, in Sir Orfeo, the motif of the rash promise shows the importance of one's word once it is given. Even the almighty fairy king must keep his word. When he first refuses Orfeo's claim, the minstrel king points at the indignant nature of such a refusal: "'O sir!' he seyd, 'gentil king, Yete were it a wele fouler thing To here a lesing of thi mouthe! So, sir, as ye seyd nouthe, What ich wold aski, have y schold, And nedes thou most thi word hold.'" Sir Orfeo (463-8) Indeed, words bind even the mightiest, which is true for a king as well as for a poet. The voice of past poets, the ancient speech that has recorded the great deeds and adventures of the knights and ladies of old, these have to be respected, even if this not always the case4 . This should be particularly true for these Middle English Breton lays which are indebted to French lays. Translation is seldom referred to though it could appear as quite a binding relation. The most direct allusion to translation is to be found in Le Freine, when the poet feels the need to explain the meaning of the protagonists' names. Guroun's guests congratulate him for the choice of his wife, which calls for a clarification: "'Fairer maiden nas never seen, Better than Ash is Hazle y ween!' (For in Romaunce Le Frain 'ash' is, And Le Codre 'hazle', y-wis."

Le Freine The poet has to account for his source from a linguistic point of view and this comes together with assessing the fact that we are dealing with the world of Romaunce.

Indeed, the notion of genre, of which we are recurrently reminded in the lays, is a similar bond as that which binds the king of the Fairies in Sir Orfeo and here again a linguistic bond. The prologues to Sir Orfeo and Lay le Freine explicitly remind their audiences of what they should expect from a lay, essentially love and adventures involving fairies. As soon as we are warned that we enter the world of lays, we feel that the poet is bound to produce a specific kind of stories. This bond also involves the poet's duty to preserve men and women from oblivion. The reference to a usually ancient past is recurrent. The lay is thus presented as a bond, a linguistic bond between past and present; between the ideal world of romance, literary in nature, and the real world of the audience; between our world and the world of fairies; between England and Brittany, this kind of fringe that sometimes seems to be confused with Fairyland itself… From a literary point of view too, truth remains a key notion as we are recurrently invited to believe what we are told. This is conveyed by the numerous and conventional formulas used by the poets: "for sothe it is" (Lay le Freine, 26), "now sykyrly" (Sir Degaré, 1090), "I you plyght" (Sir Launfal,33) or "if I sooth seyen shal" (The Franklin's Tale, 770). These recurrent invitations necessarily create a distance and a perspective since we know this is romance, what a modern mind would label as 'fiction'.

Bonds obviously play an essential part in medieval society, but this proves also true from a literary point of view. Bonds provide coherence and order, which means they have both a moral and a spiritual dimension. Bonds are, most of the time, established verbally and even orally. They need to be expressed and uttered, and this linguistic nature is particularly noticeable in the lays. The linguistic bond staged in Sir Orfeo is representative of an ideal that gives shape to the poetical process: a binding relation is established between the performer and his or her audience. Both have duties. The performer has to comply with the genre, to have form and matter match, while the audience is to consider and acknowledge their debt.

"

  The olde qwene spakke wordus unhende And sayde, 'Sone, thys ys a fende, In thys wordy wede! As thou lovest my blessynge, Make thou nevur thys weddynge, Cryst hyt the forbede!'"

Aaron J. Gourévitch, Les catégories de la culture médiévale,Paris, Gallimard, 1983, p. 175.

Cf Agnès Blandeau, "Loyalty and Treason in Some Middle English Breton Lays", in Études Épistémé, 25, 2014. "Gode is the lay, swete is the note" : Résonances dans les lais bretons moyen-anglais -Echoes in the Middle English Breton Lays , http://revue.etudes-episteme.org/?-25-2014-gode-is-the-lay-swete-is-


Alessio Moreschini
email: alessio.moreschini@uniroma1.it

Salvatore Monaco
email: salvatore.monaco@uniroma1.it

Dorothée Normand

Dorothée Normand-Cyrot

Gradient and Hamiltonian dynamics under sampling

Keywords: Sampled-data systems, Discrete gradient methods, Discrete Hamiltonian dynamics

come

INTRODUCTION

Gradient and Hamiltonian dynamics have straight relations with fundamental properties of physical systems such as conservation and/or variational principles; they are widely investigated and are at the basis of ad hoc design approaches (e.g. [START_REF] Wiggins | Introduction to applied nonlinear dynamical systems and chaos[END_REF], [START_REF] Van Der Schaft | Porthamiltonian systems theory: An introductory overview[END_REF]). Digital analysis and design methods, which make use of discrete-time models representing a given plant, possibly under sampling, are faced with the preservation of such properties; properties which are lost under usual sampling techniques (e.g. [START_REF] Stramigioli | Sampled data systems passivity and discrete port-hamiltonian systems[END_REF], [START_REF] Tiefensee | IDA-PBC under sampling for port-controlled hamiltonian systems[END_REF], [START_REF] Monaco | Sampled-data stabilization; a PBC approach[END_REF], [START_REF] Mattioni | Feedforwarding under sampling[END_REF]).

Discrete gradient methods, introduced in [START_REF] Gonzalez | Time integration and discrete hamiltonian systems[END_REF] and [START_REF] Mclachlan | Geometric integration using discrete gradients[END_REF] to solve numerical integration problems, have been employed in the last decade to characterize discrete Hamiltonian structures. Several approaches have been proposed (e.g. [START_REF] Laila | Construction of discretetime models for port-controlled hamiltonian systems with applications[END_REF], [START_REF] Sümer | A direct discrete-time IDA-PBC design method for a class of underactuated hamiltonian systems[END_REF] [START_REF] Yalçin | Discretetime modeling of hamiltonian systems[END_REF], and [START_REF] Aoues | Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability[END_REF]). Such solutions, although preserving dissipative and conservative properties, are based on approximated, essentially Euler type, sampled-data models which do not reproduce the continuous-time behaviours at the sampling instants. A different approach is then proposed by [START_REF] Talasila | Discrete port-hamiltonian systems[END_REF] which directly models the systems in a discrete setting. The aim of this paper is to go further answering the following question: does the exact sampleddata equivalent dynamics exhibit a discrete gradient or discrete Hamiltonian form?

To properly address the problem, a precise characterization of gradient and Hamiltonian dynamics in discretetime must be given; it turns out that the discrete gradi-

Alessio

Moreschini thanks the Université Franco-Italienne/Università Italo-Francese (UFI/UIF) for supporting his mobility between France and Italy.

ent function can be profitably used to this purpose. The problem is addressed in this paper making reference to dynamics associated with quadratic real valued functions in order to explicitly compute the discrete gradient. It is shown that to preserve both the energetic properties and to match the state trajectories of the continuoustime system it is necessary to modify the interconnection and dissipation terms in the state space representation of the discrete dynamics. More precisely, one defines those parts through suitable matrices depending on the sampling period δ. The proof is constructive in the sense that these matrices are described by their asymptotic expansions in powers of δ around the continuous-time solutions. It comes out that the approaches proposed in the literature [START_REF] Aoues | Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability[END_REF], Sümer and[START_REF] Sümer | A direct discrete-time IDA-PBC design method for a class of underactuated hamiltonian systems[END_REF][START_REF] Mclachlan | Geometric integration using discrete gradients[END_REF]) correspond to first-order approximations of the solution here proposed. This preliminary study is performed for gradient dynamics and then extended to conservative and dissipative Hamiltonian dynamics.

The paper is organized as follows. In Section 2 preliminary concepts are introduced. In Section 3 the problem is addressed for gradient dynamics. Section 4 addresses the problem for conservative and dissipative Hamiltonian dynamics. An elementary example is used to stress the different behaviors. The paper ends with some concluding remarks.

NOTATIONS AND PRELIMINARIES

Throughout the paper all the functions and vector fields defining the dynamics are assumed smooth and complete over the respective definition spaces. The sets R and N denote, respectively, the set of real and natural numbers including 0. For any vector v ∈ R n , |v| and v define the norm and transpose of v respectively. For v, w ∈ R n , v, w denotes the inner product, i.e. v, w = v w. I d denotes the identity function or identity matrix while I denotes the identity operator. Given a real-valued function V (•) : R n → R assumed differentiable, ∇V is used to represent the gradient vector, ∇ denoting the differential operator vector. "(•) > 0" and "(•) < 0" denote functions or matrices positive or negative definite (let recall that for a function such definition may be local or global and holds with respect to a point where the function takes the zero value). Given a smooth vector field over R n , e f (indifferently e L f ) denotes the exponential Lie operator e f := I + i≥1

L i f i! in the Lie operator L f = n i=1 f i (x) ∂
∂xi ; for a linear vector field the exponential Lie operator recovers the exponential of the matrix representing the operator. For any smooth function h(•) : R n → R then e f h(x) = h(e f (x)) = e f h| x where | x denotes the evaluation of the function at x. For sampled-data systems, x k := x(kδ) and x k+1 := x((k + 1)δ), ∀k ∈ N and δ ∈]0, T [, a finite time interval; x k := x(k) and x k+1 := x(k+1) in a pure discretetime context. The arguments of the functions are dropped when clear from the context.

Let us recall from the concerned literature (see [START_REF] Gonzalez | Time integration and discrete hamiltonian systems[END_REF], [START_REF] Mclachlan | Geometric integration using discrete gradients[END_REF]) the following definition.

Definition 2.1. (Discrete gradient). Given V (•) : R n → R a differentiable real-valued function, its discrete gradient is a vector-valued function ∇V (v, w) :

R n × R n → R n which satisfies the equality (w -v), ∇V (v, w) = V (w) -V (v), with ∇V (v, v) = ∇V (v) for continuity argument.
According to Definition 2.1 and setting

∇V (v, w) = ∇V (v 1 , w 1 ) • • • ∇V (v n , w n )
for v = (v 1 , . . . , v n ) and w = (w 1 , . . . , w n ) , then the discrete gradient can be computed according to

∇V (v i , w i ) = 1 w i -v i w i v i ∂V (v 1 , ..., v i-1 , ξ, w i+1 , ..., wn) ∂ξ dξ.
Lemma 2.1. Given V (v) = 1 2 v P v, with symmetric P matrix, then the associated discrete gradient verifies

∇V (v, w) = 1 2 P (v + w) = 1 2 (∇V (v) + ∇V (w)). (1)
Proof. This is directly deduced from the equality

V (w) -V (v) = 1 2 w P w - 1 2 v P v = 1 2 (w -v) P (v + w).

GRADIENT DYNAMICS

Gradient dynamics are preliminarily described in both continuous and discrete time. It is shown that sampleddata equivalent models recover discrete gradient forms thanks to the introduction of suitable interconnection matrices which depend on the sampling period. The problem, set in the nonlinear context, is solved when considering gradient dynamics associated with quadratic forms.

Continuous-time and discrete-time gradient dynamics

Given a C r (r ≥ 2) real-valued function V (•) : R n → R, a continuous-time gradient dynamics is defined as

ẋ(t) = f (x(t)) = -∇V (x(t))
(2) so directly concluding that by construction

• any equilibrium x e of (2) (f (x e ) = 0) coincides with a local extremum of V (x) (∇V (x e ) = 0); • V (x) = L f V (x) = -|∇V (x)| 2 ; • x e is an asymptotically stable equilibrium of (2) provided V (x) is positive-definite.
Similarly, a discrete-time gradient dynamics can be defined in terms of the discrete gradient of V (x) as

x k+1 -x k = -∇V (x k , x k+1 )
(3) so verifying by construction that:

• any equilibrium x e of (3) coincides with a local extremum of V (•) ( ∇V (x e , x e ) = ∇V (x e ) = 0); • V (x k+1 ) -V (x k ) = -| ∇V (x k , x k+1 )| 2 ; • x e is an asymptotically stable equilibrium provided V (•) is positive-definite.
When V (•) is a quadratic form then the discrete gradient dynamics (3) can be explicitly computed from (1). Proposition 1. Assume V (x) = 1 2 x P x, with symmetric positive-definite matrix P = P > 0, then (3) rewrites

x k+1 = x k + F x k = I + 1 2 P -1 I - 1 2 P x k
where for any square matrix X over R n so that I + X is invertible, the inverse matrix is formally defined as (I + X) -1 = I + p≥1 (-1) p X p .

Problem statement and motivating example

Let us first recall from [START_REF] Monaco | A combinatorial approach of the nonlinear sampling problem[END_REF]) that the equivalent sampled-data model to the nonlinear dynamics (2) admits for any δ ∈]0, T [ (T small enough) the discrete-time representation

x k+1 = F δ (x k ) (4)
where F δ (x k ) is given by its asymptotic expansion in powers of δ

F δ (x k ) = e δf x k = x k + δL f x k + δ 2 2 L 2 f x k + .... Setting f (x) = -∇V (x)
, we address the question: does the equivalent sampled-data dynamics (4) admit a discrete gradient form? When V (x) is a quadratic function, it is shown in the sequel that an "equivalent" discrete gradient form can be computed; it turns out to be a quadratic δ-dependent function which is specified as a series expansion in powers of δ. The approximation at the first order recovers the discrete gradient dynamics usually adopted in the current literature.

The following elementary example is used to better point out the posed question. Given ẋ(t) = -x(t) = -∇V (x(t))

(5) with V (x) = 1 2 x 2 , does the sampled data equivalent model x k+1 = e -δ x k (6) admit a discrete gradient form?

As said before, it is quite usual (see [START_REF] Aoues | Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability[END_REF], Sümer andYalçın (2011), McLachlan et al. (1999), etc.) to associate to (5) the discrete gradient dynamics below making reference to the same function V (•), i.e.

x k+1 -x k = -δ ∇V (x k , x k+1 ). ( 7) Such a form preserves the stability of the evolutions at the sampling instants t = kδ, since one gets along (7)

V (x k+1 )-V (x k ) = -δ( ∇V (x k , x k+1 )) ∇V (x k , x k+1 ) < 0,
but its equivalent explicit representation does not match the state evolutions of (2) at the sampling instants. As a matter of fact, as ∇V (x k , x k+1 ) = 1 2 (x k + x k+1 ), (7) rewrites as

x k+1 = (1 + δ 2 ) -1 (1 - δ 2 )x k (8)
which is not equivalent to (6).

We will show in the sequel that the exact sampled-data dynamics (6) does satisfy a new discrete gradient form

x k+1 -x k = -δI(δ, -I) ∇V (x k , x k+1 ) (9
) with a suitably defined δ-dependent matrix I(δ, -I). Moreover, ( 7) recovers ( 8) in first approximation in δ.

Gradient dynamics under sampling

Given the continuous-time dynamics (2) with quadratic form V (x) = 1 2 x P x and P = P > 0, then its sampled equivalent dynamics

x k+1 := e -δP x k (10)
satisfies the forward difference inequality

V (x k+1 ) -V (x k ) = - (k+1)δ kδ |∇V (x(τ ))| 2 dτ < 0. ( 11
)
The question now relies on the possibility to rewrite (10) into a discrete gradient form (3) with respect to a suitably defined V (•) function.

The following matrix will be instrumental throughout the rest of the paper. Given X ∈ R n×n and δ ∈]0, T [, we denote by I(δ, X) ∈ R n×n , the matrix which satisfies the algebraic equality below δI(δ, X)X =2(e δX -I)(I + e δX ) -1 (12) where the inverse is again formally defined by the series

(I + e δX ) -1 = I + p≥1 (-1) p e pδX .
Accordingly, one gets by construction the description of I(δ, X) as the series expansion in δ below

I(δ, X) = p≥0 j0≥0,j1,...,jp≥1 (-1) p (δX) p i=0 ji 2 p (j 0 + 1)!j 1 !...j p ! . ( 13
)
For the first terms one computes

I(δ, X) = I - δ 2 3!2 X 2 + δ 4 5! X 4 + O(δ 6 )
so verifying that the coefficients of the odd powers in X ((δX) 2i+1 , i ≥ 0) are equal to zero in the expansion (13).

On these bases the following result can be proved.

Theorem 3.1. Given the gradient dynamics (2) with function V (x) = 1 2 x P x, then for any fixed δ ∈]0, T [, its sampled-data equivalent dynamics (10) admits, (a) the discrete gradient form

x k+1 -x k = -δI(δ, -P ) ∇V (x k , x k+1 ) (14)
with matrix I(δ, -P ) defined as in ( 13); (b) equivalently, the discrete gradient form

x k+1 -x k = -δ ∇V (δ)(x k , x k+1 ) (15)
with respect to the new quadratic function V (δ)(x) = 1 2 x (-P )(δ)x with symmetric positive-definite square matrix (-P )(δ) = I(δ, -P )P ; (c) according to ( 14), one gets

V (x k+1 ) -V (x k ) = -δ( ∇V ) I(δ, -P ) ∇V (16) = - (k+1)δ kδ |∇V (x(τ ))| 2 dτ < 0.
Proof. As far as (a) is concerned, according to the definition of discrete gradient and considering the sampled-data equivalent dynamics (10), the equality ( 14) rewrites as e -δP x = x -δ 2 I(δ, -P )P (I + e -δP )x which holds true by construction of I(δ, -P ) given in (13) when replacing the X matrix with -P , so that the following equality is verified δI(δ, -P )P = 2(I -e -δP )(I + e -δP ) -1 .

Regarding (b), (15) follows from ( 14) when setting V (δ)(x) = 1 2 x (-P )(δ)x with (-P )(δ) = I(δ, -P )P . Moreover (-P )(δ), which is given by the expansion

(-P )(δ) = p≥0 j0≥0,j1,...,jp≥1 (-1) p (-δP ) p i=0 ji 2 p (j 0 + 1)!j 1 !...j p ! P =P - δ 2 3!2 P 3 + δ 4 5! P 5 + O(δ 6 ),
is by construction a symmetric matrix. Its positivity for all δ ∈]0, T [ follows from the equality ( 16) in (c). Such equality is a direct consequence of (11), of the discrete gradient definition, and of the gradient form of the dynamics (14). 2

The results in Theorem 3.1 show that:

• the dynamics (10) exhibits a gradient form ( 14) with respect to the same real-valued function V (•) as in the continuous-time case through a new connection matrix I(δ, -P ) which depends on the function V (•) itself and is described by its series expansion in powers of δ. The negativity of the forward difference V (x k+1 ) -V (x k ) follows as it exactly matches the continuous-time V (x) evolution at the sampling instants according to (16); • the dynamics (15) (equivalent to ( 14)) is defined with respect to a different real-valued function V (δ)(•) parameterized by δ which preserves symmetry and positivity of V (•). The negativity of the forward difference V (δ)(x k+1 )-V (δ)(x k ) follows by definition of the gradient form itself, i.e.

V (δ)(x k+1 ) -V (δ)(x k ) = -δ| ∇V (δ)| 2 < 0 but differs from (16).
For completeness, it is worth mentioning that usual sampled-data gradient structures proposed in [START_REF] Yalçin | Discretetime modeling of hamiltonian systems[END_REF], [START_REF] Aoues | Hamiltonian systems discrete-time approximation: Losslessness, passivity and composability[END_REF], Sümer and[START_REF] Sümer | A direct discrete-time IDA-PBC design method for a class of underactuated hamiltonian systems[END_REF][START_REF] Mclachlan | Geometric integration using discrete gradients[END_REF] correspond to Euler type approximations of the proposed results. More precisely, one can solve in I(δ, -P ) the equation ( 14) when setting x k+1 -x k = -δP x k so restricting the dynamics to the Euler approximation (first-order in δ) of the exact sampled dynamics (10). Equivalently, this corresponds to set in (15) V (δ)(x) = V (x) so getting approximated results at the first-order in δ. To conclude, we note that the discrete gradient form associated to the exact sampled data model of the elementary dynamics ( 5) is given by

x k+1 =x k - δ 2 I(δ, -1)(x k + x k+1 ) =x k -2 1 -e -δ
1 + e -δ ∇V (x k , x k+1 ) which clearly generalizes the usually proposed approximated sampled gradient dynamics (7).

HAMILTONIAN DYNAMICS

The previous results are now extended to Hamiltonian dynamics described over R 2n when the real-valued function H(•) is an "energy-like" function associated with a dynamics expressed in the canonical position and momenta coordinates (q, p). In Section 4.2, the problem of preserving Hamiltonian forms under sampling is discussed for both conservative and dissipative Hamiltonian dynamics with quadratic energy function and constant interconnection and damping matrices.

Continuous and discrete Hamiltonian dynamics

A continuous-time Hamiltonian dynamics is given by ẋ = f (x) = (J(x) -R(x))∇H(x) (17) where H(•) : R 2n → R is assumed a C r (r ≥ 2) function, J(x) ∈ R 2n×2n a skew-symmetric non-degenerate matrix, and R(x)≥ 0 ∈ R 2n×2n a symmetric matrix whose entries are functions of x and characterizing the stored and dissipated energy respectively. The following comments are in order:

• any equilibrium x e of (17) coincides with a local extremum of H(x) (∇H(x e ) = 0);

• assuming H(x) positive-definite, one gets Ḣ(x) = L f H(x) = -(∇H(x)) R(x)∇H(x) < 0 which implies asymptotic stability of x e ; • when R(x) = 0, Ḣ(x) = L f H(x) = 0
corresponding to the conservative property of the Hamiltonian along (17).

Along the same lines, discrete Hamiltonian dynamics can be defined in terms of the discrete gradient of H(•).

Definition 4.1. Given a C r (r ≥ 2) real-valued function H(•) : R 2n → R, a discrete Hamiltonian dynamics is given by

x k+1 = x k + (J(x k ) -R(x k )) ∇H(x k , x k+1 ) (18) where J(x), R(x) are square matrices satisfying J(x) = -J (x), R(x) = R (x) ≥ 0.
Analogously to the continuous-time case, the following holds true:

• any equilibrium of ( 18) coincides with a local extremum of H(•) ( ∇H(x e , x e ) = ∇H(x e ) = 0); • assuming H(x) positive-definite, one gets from ( 18)

H(x k+1 ) -H(x k ) = -( ∇H) R(x k ) ∇H < 0 which implies asymptotic stability of x e ;
• when R(x) = 0, the discrete dynamics is energy conservative

H(x k+1 ) -H(x k ) = -( ∇H) J(x k ) ∇H = 0, with ∇H = ∇H(x k , x k+1 ).
An explicit representation of ( 18) can be computed when H(x) is a quadratic form and the damping and interconnection matrices J and R are constant matrices. Proposition 2. Set V (x) = 1 2 x P x, J = -J , and R = R ≥ 0, then the discrete Hamiltonian dynamics (18) equivalently satisfies the difference equation

x k+1 = x k +F x k = I - 1 2 (J -R)P -1 I + 1 2 (J -R)P x k .

Hamiltonian dynamics under sampling

In what follows it is shown that the exact sampled equivalent dynamics to (17) recovers a discrete Hamiltonian form when assuming a quadratic energy function H(x) = 1 2 x P x with symmetric positive-definite matrix P , and constant interconnection and dissipation matrices

J = -J , R = R ≥ 0. Consider the continuous-time dynamics ẋ(t) = (J -R)P x(t) (19)
with exact sampled equivalent dynamics described by x k+1 := e δ(J-R)P x k .

(20) Since by construction the evolutions of ( 19) and (20) from the initial state x(0) = x 0 coincide at the sampling instants t = kδ, the sampled dynamics (20) satisfies the forward difference

H(x k+1 ) -H(x k ) = - (k+1)δ kδ (∇H(x(τ ))) R∇H(x(τ ))dτ < 0
so matching, at the sampling instants, the energy behavior of the continuous-time dynamics as well.

Remark 4.1. If the matrix JP characterizing the Hamiltonian dynamics (19) with R = 0 is infinitesimally symplectic (Marsden and Ratiu ( 2013)) (i.e. (JP ) Ω = -ΩJP for a skew-symmetric non-degenerate matrix Ω) then its equivalent sampled-data dynamics

x k+1 = e δJP x k
is defined by a symplectic matrix (i.e. (e δJP ) Ωe δJP = Ω).

The result below extends the result of Section 3 to conservative dynamics (19) (R = 0). Theorem 4.1. Given a conservative Hamiltonian dynamics (19) with R = 0, then for any δ ∈]0, T [ its sampled equivalent model (20) admits a conservative discrete Hamiltonian form

x k+1 = x k + δJ (δ) ∇H(x k , x k+1 ) (21)
with skew-symmetric J (δ) = I(δ, JP )J ∈ R 2n×2n satisfying the equality δJ (δ)P = 2(e δJP -I)(I + e δJP ) -1 .

Proof. By definition of I(δ, JP ), the matrix J (δ) = I(δ, JP )J can be computed to satisfy the equality e δJP x = x + δ 2 J (δ)P (I + e δJP )x so recovering (22). Moreover it is a matter of computation to verify that the matrix J (δ) admits the series expansion J (δ) = p≥0 j0≥0,j1,...,jp≥1

(-1) p (δJP )

p i=0 ji 2 p (j 0 + 1)!j 1 !...j p ! J =J - δ 2 3!2 (JP ) 2 J + δ 4 5! (JP ) 4 J + O(δ 6 )
which is characterized by terms in J at odd power indices only so proving that the skew-symmetry of J is preserved for J (δ). 2

By construction, the discrete Hamiltonian dynamics ( 21) is energy preserving, i.e.

H(x k+1 ) -H(x k ) = δ( ∇H) J (δ) ∇H = 0.
Remark 4.2. Defining the matrix P (δ) ∈ R 2n×2n as JP (δ) =I(δ, JP )JP then ( 21) can be rewritten as

x k+1 = x k + δJ ∇H(δ)(x k , x k+1
) so defining a new energy function H(δ) = 1 2 x P (δ)x depending on δ. It can be proven that P (δ) ∈ R 2n×2n is again a symmetric positive-definite matrix for any δ ∈]0, T [ that is described by an infinite sum of matrices of the form P (JP ) i , i = 2, 4, .., which result to be alternatively negative and positive definite, starting from i = 2. For the first terms, one gets

P (δ) = P - δ 2 3!2 P (JP ) 2 + δ 4 5! P (JP ) 4 + O(δ 6 )
with P (JP ) 2 < 0 and P (JP ) 4 > 0.

Let us now address the case of dissipative Hamiltonian dynamics. According to Theorem 4.1, it is easy to show that the sampled equivalent dynamics to (19) can be rewritten in the Hamiltonian form below x k+1 = x k + δQ(δ) ∇H(x k , x k+1 ) with by definition Q(δ) := I(δ, (J -R)P )(J -R) ∈ R 2n×2n satisfying the equality δQ(δ)P = 2(e δ(J-R)P -I)(e δ(J-R)P + I) -1 .

(23) Again, one gets

H(x k+1 ) -H(x k ) = δ( ∇H) Q(δ) ∇H = - (
J(δ) = 1 2 (Q(δ) -Q (δ)), R(δ) = - 1 2 (Q(δ) + Q (δ))
(25) respectively skew-symmetric and symmetric positive semidefinite, and Q(δ) ∈ R 2n×2n defined in (23).

Proof. Setting Q(δ) = I(δ, (J -R)P )(J -R), one immediately deduces (23) by definition of the I(δ, X) matrix which satisfies ( 12). Accordingly, one computes

Q(δ) = p≥0 j 0 ≥0,j 1 ,...,jp≥1 (-1) p (δ(J -R)P ) p i=0 j i 2 p (j 0 + 1)!j 1 !...jp! (J -R) = I - δ 2 3!2 ((J -R)P ) 2 + δ 4 5! ((J -R)P ) 4 (J -R) + O(δ 6 ).
By construction, the matrix J(δ) defined in ( 25) is skew symmetric and R(δ) is symmetric. Moreover, since

H(x k+1 ) -H(x k ) = δ( ∇H) Q(δ) ∇H = -δ( ∇H) R(δ) ∇H = - (
= -δ( ∇H(x k , x k+1 )) R(δ) ∇H(x k , x k+1 ) with ∆H = H(x k+1 ) -H(x k ), (equivalently H(x k+1 ) = H(x k ) when R = 0).

An example.

Let the simple continuous-time second order Hamiltonian dynamics ẋ = (J -R)∇H(x) = q = p ṗ = -q -αp (26)

with quadratic Hamiltonian function H(q, p) = 1 2 q 2 + 1 2 p 2 and positive dissipation coefficient α. According to the current literature the sampled Hamiltonian dynamics associated to ( 26) is assumed of the form

q k+1 p k+1 = q k p k + δ 0 1 -1 -α ∇H(q k , p k , q k+1 , p k+1 ). (27)
When substituting ∇H = 1 2 [q k + q k+1 , p k + p k+1 ] into (27), one gets

q k+1 p k+1 =    4 -δ 2 + 2δα δ 2 + 2δα + 4 4δ δ 2 + 2δα + 4 - 4δ δ 2 + 2δα + 4 4 -δ 2 -2δα δ 2 + 2δα + 4    q k p k
which differs from the equivalent sampled data model

q k+1 p k+1 = e δ 0 1 -1 -α q k p k
which admits a sampled Hamiltonian representation of the form (24).

CONCLUSIONS

The paper describes discrete gradient and Hamiltonian dynamics in terms of the discrete gradient of a certain real-valued "energy-like" function. It is shown that discrete gradient or Hamiltonian representations can be recovered under exact sampling with respect to the same energy function and modified interconnection and dissipation matrices. The proposed method holds both the objectives of satisfying under sampling the energetic properties which characterize Hamiltonian dynamics and that of matching the state evolutions at the sampling instants. In this sense, the proposed method outperforms the discrete-time forms currently set in the related literature. Work is in progress regarding the extension to the general nonlinear context and how to exploit these forms for control purpose as in [START_REF] Ortega | Interconnection and damping assignment passivity-based control: A survey[END_REF] and following the lines of a preliminary study in [START_REF] Tiefensee | IDA-PBC under sampling for port-controlled hamiltonian systems[END_REF]. Finally, it is intriguing to note that in the conservative case, the sampled data Hamiltonian representation can be interpreted as resulting either from the same energy function with δ-dependent interconnection matrix or from a δ-depend energy function and the same interconnection. This suggests that a similar dichotomy should be present in discrete Hamiltonian modeling as in [START_REF] Talasila | Discrete port-hamiltonian systems[END_REF].

  k+1)δ kδ (∇H(x(τ ))) R∇H(x(τ ))dτ < 0 so concluding negativity of Q(δ) for all δ ∈]0, T [. Splitting Q(δ) into a skew-symmetric (conservative) and a symmetric (dissipative) part respectively, the following holds.Theorem 4.2. Given a dissipative Hamiltonian dynamics (19), then for any δ ∈]0, T [ its equivalent sampled model (20) admits a dissipative discrete Hamiltonian formx k+1 = x k + δ(J(δ) -R(δ)) ∇H(x k , x k+1 ) (24) possessing an asymptotically stable equilibrium in x e = 0 with matrices

  k+1)δ kδ (∇H) R∇Hdτ < 0, one concludes positivity of R(δ) from positivity of R. 2 Theorem 4.1 and Theorem 4.2 show that the dynamics (24) (equivalently (21)) preserves the Hamiltonian representation with respect to the same energy function H(•) with modified skew-symmetric J(δ) and symmetric positive-definite R(δ) matrices described by their series expansions in δ so as to verify ∆H

Figures 5 .

 5 Figures 5.1 and 5.2 depict the trajectories and the evolutions of H(•) from the initial state q = p = 2 under


domain: spi phys

text:
J-Y Lestrade
email: jean-yves.lestrade@onera.fr

J Anthoine

A J Musker

A Lecossais

Experimental Demonstration of an End-Burning Swirling Flow Hybrid Rocket Engine

Keywords: Hybrid propulsion, H2020 HYPROGEO project, Innovative combustion chamber, Experimental demonstration, End-burning swirling-flow

Within the framework of the European H2020 HYPROGEO program, an innovative hybrid engine combustion chamber compatible with satellite requirements (constant thrust over very long burn-times) had to be developed. A first test rig was designed and tested in order to better understand the functioning of this innovative combustion chamber, and to help the design of a breadboard to demonstrate the efficiency of this new engine with respect to the mission requirements. Two test campaigns, the first on the test rig with 87.5% hydrogen peroxide, and the second on the hybrid engine breadboard with 98% hydrogen peroxide, were performed under various operating conditions to demonstrate the catalytic ignitability of this new hybrid engine, and the sustainment of a stable combustion over firing durations up to 180 s. The test campaigns also enabled the identification of the main influencing parameter on the fuel regression rate for this innovative combustion chamber.

Introduction

The use of electrical propulsion systems to transfer satellites from GTO to operational GEO orbits, in addition to their current use for station-keeping, seems to be the trend for future European telecommunication satellites. However, the long transfer durations inherent to such low-thrust technologies may not be compatible with all mission timeline requirements. One of the solutions proposed by industry and public agencies is to employ a dedicated high-thrust propulsion system to accelerate the transfer phase. This apogee kick-stage module should also combine environmentally-friendly chemical technology and electrical thrusters, to optimize its efficiency in terms of propulsive performances, mass, costs, etc.

Owing to their advantages over current MMH/NTO bi-propellant engines, such as increased safety, lower environmental impact, better propulsive performances, lower costs, etc., hybrid propulsion systems could be a good candidate for this task. However, the current architecture of such an engine is not suitable for the long burn-time and constant-thrust required for a satellite application, as a result of the variations in oxidizer-to-fuel ratio and fuel grain shape with time. An innovative combustion chamber satisfying the previous requirements is thus to be developed.

In order to have a constant thrust, Rice et al. developed the vortex endburning hybrid engine (VEBH) which provides the advantage of a constant fuel burning surface area [START_REF] Rice | Mars ISRU CO/O2 rocket engine development and testing[END_REF]. The engine mixture ratio can be controlled by varying the oxidizer mass flow rate and/or the combustion chamber diameter (fuel surface area). As presented in Figure 1, this engine, designed for use with HTPB, consists of a ring chamber sandwiched between top and bottom matings, with the gaseous oxidizer (GOX) tangentially injected through ports at the tail-end (nozzle side) of the chamber in order to create a swirl motion of the gaseous phase, and to increase the fuel regression rate. Based on observations of postfiring fuel grains, a counter-swirling pattern of grooves, which appear to indicate flow opposite to that of the swirl GOX injection, exists near the centre of the chamber. According to the authors, a portion of the oxygen may be spiralling along the bottom of the chamber and up to the head end in the central region, where it meets the grain surface and flows back outward. In this case, the flow field might consist of two inter-twined spirals in the central region of the chamber, one spinning upward and the other spinning downward towards the nozzle (Figure 2).

However, this concept is not compatible with the long burn- Figure 2: Flow field in the combustion chamber [START_REF] Chiaverini | Review of Solid-Fuel Regression Rate Behavior in Classical and Nonclassical Hybrid Rocket Motors[END_REF] time required for satellite applications, since the distance between the injection location and the fuel surface evolves with time, which may cause combustion stability and ignition issues. In order to solve this problem, an interesting solution is the end-burning swirling-flow hybrid rocket engine (named SOFT for Swirling Oxidizer Flow Type), using paraffin wax developed by Hayashi and Sakurai for application to a first stage [START_REF] Sakurai | A Fundamental Study of a End-Burning Swirling-Flow Hybrid Rocket Engine using Low Melting Temperature Fuels[END_REF]. This concept is similar to that of Rice et al. in terms of overall dimensions, oxidizer type and injection, but is equipped with an actuator to push the fuel grain axially toward the nozzle side in order to keep the volume of the combustion chamber constant during burning. As in the previous study, the authors observed that the fuel regressed axially with a high regression rate region close to the centre of the fuel grain. According to Volchkov et al., the convection coefficient on the surface is maximum at the axis level because of the phenomena induced by the swirl injection [START_REF] Volchkov | Turbulent heat transfer at the forward face surface of a vortex chamber[END_REF]. Hayashi and Sakurai concluded that the formation of a crater at the centre of the fuel grain was provoked by these same mechanisms, since the energy input was higher.

Although this concept was interesting, the actuator was not useful, as a result of the very short burn duration (only two seconds). Moreover, the fuel employed for this study was not suitable for use on-board a satellite due to its high regression rate: the length needed to make this engine compatible with the requirements in terms of burn duration would make it incompatible with spacecraft sizing. Dedicated studies on pancake hybrid engines were performed in order to satisfy these constraints [START_REF] Caravella | Characterization of Fuel Regression in a Radial Flow Hybrid Rocket[END_REF][START_REF] Haag | Low Cost Propulsion Development for Small Satellites at the Surrey Space Centre[END_REF], but these concepts led to a timevariant change in the oxidizer-to-fuel ratio and the fuel grain shape, which is not coherent with the satellite requirements for constant thrust.

Within the framework of the European H2020 HYPROGEO project, a hybrid engine compatible with a long burn duration (about 5000 s) and with a constant thrust at the required level (250 N) for use on-board a spacecraft had to be developed [START_REF] Odic | HYPROGEO hybrid propulsion: project objectives & coordination[END_REF]. To reach these objectives, the idea was to design an end-burning, swirling-flow hybrid engine combined with a passive actuator to compensate for the fuel regression, operating with high density polyethylene as fuel and 98% hydrogen peroxide as oxidizer; this type of fuel is suitable for use on-board satellites due to its very low regression rate. In addition, catalytic injection combining start and stop capabilities is made possible by the choice of this oxidizer.

The methodology that was followed during the HYPROGEO project led to the development of two innovative hybrid rocket engines. The first one, the MHYCAS facility [START_REF] Lestrade | Develop[END_REF], was designed to better understand the functioning of this new combustion chamber architecture, while the goal of the second one, the Su-perMHYCAS facility, was to demonstrate the efficiency of this new engine with respect to the objectives for constant thrust over long burn durations. This paper will consequently describe the architecture of these new hybrid engines and discuss the test results.

The MHYCAS test campaign

The MHYCAS engine (Figure 3), a French acronym for 'hybrid engine compatible with satellite application', was designed and tested as an intermediate step in order to better define the final breadboard of the HYPROGEO project, to investigate potential difficulties, such as catalytic ignition or oxidizer injection, and to define the operating conditions to have the required oxidizer-to-fuel ratio, chamber pressure and oxidizer mass flow rate. The MHYCAS engine (Figure 4) is composed of a fuel tank containing the high density polyethylene fuel grain, an annular ring providing a swirl injection, with six decomposition chambers filled with the platinum-based PX1 catalyser and supplied with 87.5% hydrogen peroxide mounted to the annular ring (Figure 5), a combustion chamber of 200 mm in-diameter at the fuel location, and a nozzle with an adaptable throat section. The modular design of this hybrid engine enables the operator to easily change the number of active decomposition chambers, the initial position of the fuel grain with respect to the location of the oxidizer slots, the shape of the fuel grain, and the operating conditions in terms of chamber pressure, oxidizer mass flow rate and mass flux (calculated at oxidizer injection). However, contrary to the VEHB and SOFT engines, the oxidizer injection is located very close to the fuel grain to ease the catalytic ignition of the engine. Moreover, since the oxidizer injection and the gaseous flow have a major impact on the fuel regression rate, the ring was designed with the help of idealised and CFD analyses [9]. These analyses led to the selection of six continuous ducts which inject the oxidizer tangentially to the combustion chamber to generate a swirling motion (Figure 5). The engine is equipped with temperature and pressure probes located at the end of one of the six decomposition chambers, and three pressure probes placed around the combustion chamber. The engine is also placed on a thrust bench to measure the propulsive performance. However, it should be noted that the nozzle was not adapted to ambient pressure, since the nozzle throat was only modified to target different values of combustion chamber pressure.

To complete these measurements, thin thermocouples were welded onto the external wall of the engine: 14 were placed on the combustion chamber, one was placed at the end of each decomposition chamber, and two were placed on the fuel tank. By means of a backwards integration method, these temperature measurements enable the evaluation of the inside wall temperature and the thermal flux along the combustion chamber.

Finally, the instrumentation for the test facility includes a Coriolis mass flow meter and temperature and pressure transducers for the liquid oxidizer, The uncertainties related to the measurement techniques implemented on this test bench are specified in Table 1.

Test results

The MHYCAS test campaign was performed through 25 test firings with the same test sequence. As presented in Figure 6, the first phase of the firing tests corresponded to a blipping sequence of the oxidizer injection, to fill the oxidizer lines and to slowly warm-up the decomposition chambers to avoid damaging the catalyser. This sequence was followed by a continuous oxidizer injection phase, which can be split into two parts: a mono-propellant phase, where the combustion process has not started, and a hybrid phase as soon as the engine has ignited. Finally, after the firing test, a drain phase was employed, which consisted of a continuous nitrogen injection to avoid thermal lag and degradation of the engine. This phase was also composed of two distinct parts: the first was the flushing of the hydrogen peroxide contained in the feed lines (associated with an increase in chamber pressure during the flush), whilst the second was the injection of pure nitrogen into the engine. The objectives of the first four tests were to ignite the hybrid engine, and to maintain a stable hybrid mode over several seconds. These tests revealed that the engine was able to ignite, but also that the fuel grain position with respect to the oxidizer injection location had a where the engine re-ignited due to the hydrogen peroxide flush. The first firing tests were conducted with high oxidizer-to-fuel mass ratios compared with the target value of 7.4, which corresponds to the optimal value for a hybrid engine operating with 98% hydrogen peroxide and high-density polyethylene as propellants. Values very close to this target were reached with test MHYCAS 17 (two active decomposition chambers), and MHYCAS 22 (six active decomposition chambers), proving that reaching an optimum value of oxidizer-to-fuel ratio was feasible with such a hybrid engine.

Firing test analysis

The analysis of the firing tests presented in Table 2 will only focus on the behaviour of the fuel regression rate, since this parameter has a major role on the oxidizer-to-fuel ratio, and on the propulsive performances, which are two important criteria to optimize for applications to spacecraft.

The distance between the fuel grain surface and the oxidizer injection has a significant influence on the ignition delay. Consequently, it seems that the heat transfer between the gaseous flow and the fuel grain is increased when this distance decreases. This is confirmed by comparing the MHYCAS 02 and MHYCAS 05 tests, for which the fuel regression rate has been significantly increased by reducing the distance between the fuel surface and the injector. This influence will not be present for the SuperMHYCAS engine, since the distance between the fuel surface and the oxidizer injection will remain constant by compensating for the fuel regression rate.

In a classical hybrid engine, pressure has no significant influence on the fuel regression rate. In order to verify if this assumption is correct with the MHY-CAS engine, tests 07 and 15, as well as tests 09 and 12, can be compared in pairs, since they have similar oxidizer injection conditions with respect to mass flow rates and mass flux, but strong differences in combustion chamber pressures.

It can be seen that the fuel regression rate is not modified when the pressure changes. This conclusion is verified whatever the value of the oxidizer mass flow rate, since it is true for high mass flow rates (geq 100 g/s), or low mass flow rates (leq 45 g/s). Note that these comparisons were performed on firing tests with the same number of active injectors. These results tend to show that the combustion chamber pressure has a negligible effect on the fuel regression rate.

The last factor which could have an important impact on the fuel regression rate is the oxidizer injection, which is well known for classical hybrid engine geometries. This can be separated further into various parameters, such as the mass flow rate, the mass flux, the injection velocity, and the injection geometry (the number of active decomposition chambers). When considering two firing tests with the same mass flow rates and mass fluxes, the only method of obtaining different oxidizer velocities is by changing the oxidizer density, which is achieved by modifying the chamber pressure. However, as discussed previously, the pressure, and thus the injection velocity of the oxidizer, does not have a significant influence on the regression rate. Based on a comparison of tests 16 and 17, and 15 and 22, it is possible to conclude that at a given number of active decomposition chambers and similar oxidizer mass flow rates, an increase in the mass flux improves the fuel regression rate. Finally, the number of active decomposition chambers has a major impact on the fuel regression rate, since even with a similar oxidizer mass flow rate and a different oxidizer mass flux, tests 17 and 22 provided nearly identical oxidizer-to-fuel ratios. It seems that, for an oxidizer mass flux value, an increase in the number of active decomposition chambers increases the fuel regression rate.

The SuperMHYCAS test campaign

The objective of the SuperMHYCAS facility (Figure 7) was to demonstrate the efficiency of this innovative hybrid engine architecture with respect to the satellite requirements, namely providing a constant thrust over a long firing duration. In order to maximize the propulsive performances, this engine operates with 98% hydrogen peroxide, compared to the MHYCAS engine, which is only compatible with the standard grade of peroxide (87.5%). In order to sustain the high temperature expected during operation, most of the parts of the engine were manufactured in Inconel 625.

Description of the test facility

The design of the SuperMHYCAS engine (Figure 7) is very similar to the MHYCAS engine. The engine is composed of a fuel tank containing the high-density polyethylene fuel grain, an annular ring providing a swirl injection, with six decomposition chambers filled with the PX1 catalyser mounted to the annular ring, a combustion chamber with a diameter of 250 mm at the location of the fuel grain, and a graphite nozzle with a throat diameter of 11.5 mm.

This engine is thus slightly bigger than the MHYCAS engine, in order to be compatible with the thrust level required by the mission specifications. However, the design of the annular ring was performed in order to easily change the thickness of the oxidizer slot, since this parameter drives the oxidizer mass flux, which has a major impact on the fuel regression rate (main result of the MHYCAS test campaign). The injector ring (Figure 8) is composed of three parts, one external and two internal; the first internal part (in orange on Figure 9) sets the distance between the fuel grain and the oxidizer slot, while the other internal part (in blue on Figure 9) fixes the thickness of this slot.

The main difference with the design of the previous engine manufactured in the framework of this project lies in the addition of a passive fuel displacement system, in order to compensate for the fuel regression. The displacement of the fuel grain is guaranteed by the pressurisation, at a higher value than the combustion chamber pressure, of the volume located at the end the end-hand of the engine and the fuel grain is stoped at the opposite side (combustion chamber side) by the conical shape of annular injector ring.

The instrumentation of SuperMHYCAS facility is also very similar to that of the MHYCAS facility. The only difference lies in the addition of a linear displacement sensor to give an estimation of the fuel regression rate.

Test results and analysis

Eleven tests were performed during the SuperMHYCAS test campaign. The first four were dedicated to the adjustment of the test sequence, in order to avoid the extrusion of the fuel grain through the conical injection ring at the end of the firing test. The objective of the next three tests was to provide data through different oxidizer mass fluxes, in order to plot the evolution of the average fuel regression rate as a function of the oxidizer mass flux and of the oxidizer-to-fuel 3) but with three different slot thicknesses (0.3, 0.6 and 2.8 mm).

As with the MHYCAS test campaign, all the values provided were averaged over the last five seconds, to avoid significant effects of engine ignition and shut-off on the data averaging. The regression rate values are based on mass loss during the firing test.

Figure 10 presents the time evolutions of the chamber pressure, oxidizer propellant phase then started, and lasted until the engine was ignited and the hybrid phase began. Even if the oxidizer mass flow rate was constant during the hybrid mode, the chamber pressure evolved significantly, which meant that the transient phase for temperature was still not completed for this firing duration.

The drop in the chamber pressure at the end of the firing test was due to the de-pressurization of the fuel grain tank, and the retraction of the fuel grain in order to avoid its extrusion into the combustion chamber when the oxidizer injection stopped. The data presented in Table 3 was collected in the five seconds before the depressurization of the fuel tank.

Based on these three tests, the evolutions of the fuel regression rate as a function of the oxidizer-to-fuel ratio, and of the oxidizer mass flux, are plotted (Figure 11, Figure 12). Based on the interpolation equations of these two plots, an oxidizer-to-fuel ratio of 7.4 (the optimal value with respect to specific impulse for this configuration) Later tests were performed to increase the firing duration up to 300 s, the maximum possible burn time based on transient thermoelastic computations at the optimal combustion efficiency for the combustion chamber. The Su-perMHYCAS 09 and SuperMHYCAS 10 firing tests were performed under the same operating conditions over 90 and 180 s of continuous oxidizer injection, respectively. As presented in Figure 13, the data was very stable over the hybrid mode. The averaged results over the 5 s duration are listed in The reason for this evolution is still unexplained, but it justifies the decrease in the oxidizer-to-fuel ratio when the firing duration increases. Moreover, based on the observation of the fuel grain surfaces after the firing tests (Figure 15), the non-uniformity of the fuel regression rate over the burning surface has an influence on this evolution, but its quantification, if possible, is quite complex.

Conclusion

This paper presented the development of an innovative hybrid engine architecture compatible with satellite requirements (long burning duration associated with a constant thrust). To do this, two facilities were manufactured and tested.

The first, the MHYCAS facility, was designed to better understand the functioning of this new combustion chamber. The test campaign, performed with 87.5% hydrogen peroxide, was successful and achieved all the initial objectives.

The catalytic ignitability of the fuel was demonstrated for various operating conditions: low and high oxidizer mass flow rates (from 30 to 110 g/s), low and medium combustion chamber pressures (from 5 to 20 bar), various numbers of active decomposition chambers, etc. The analysis revealed that the combustion The results of this first test campaign enabled the design of the SuperMHY-CAS hybrid engine, with special attention being given to the oxidizer injection, in order to easily change its mass flux. The test campaign, performed with 98% hydrogen concentration, was also successful, with a combustion efficiency of about 90%. For a first engine of this size, and an engine that was not optimized with respect to thermal losses, this can be considered good. This test campaign included a 55 s firing test at the optimal average oxidizer-to-fuel ratio (7.4) for the required application, but as the firing duration increased, the averaged oxidizer-to-fuel ratio decreased, likely as a result of an escalation of the non-uniformity of the fuel regression rate over the burning surface. This should be addressed in further studies by working on the oxidizer injection..

Figure 1 :

 1 Figure1: Vortex end-burning hybrid engine[START_REF] Rice | Mars ISRU CO/O2 rocket engine development and testing[END_REF]

Figure 3 :

 3 Figure 3: MHYCAS hybrid engine during a test firing

Figure 4 :

 4 Figure 4: Drawing of the MHYCAS engine

Figure 5 :

 5 Figure 5: Drawing of the selected injector ring with the decomposition chambers

Figure 6 :

 6 Figure 6: Example of the firing test phases for the MHYCAS engine

Figure 7 :

 7 Figure 7: SuperMHYCAS hybrid engine

Figure 8 :

 8 Figure 8: Complete assembly of the SuperMHYCAS injector ring

Figure 10 :

 10 Figure 10: SuperMHYCAS 06 test results

Figure 11 :

 11 Figure 11: Evolution of the fuel regression rate as a function of the oxidizer to fuel ratio

Figure 12 :

 12 Figure 12: Evolution of the fuel regression rate as a function of the oxidizer mass flux

Figure 14 :

 14 Figure 14: Evolution of the SuperMHYCAS 10 fuel grain displacement

Figure 15 :

 15 Figure 15: Fuel grain surface after SuperMHYCAS 08 (top) and SuperMHYCAS 10 (bottom) firing tests

Table 1 :

 1 Uncertainties of the measurements techniques implemented on the MHYCAS facility

	Quantity	Uncertainty Unit
	Pressure	1000	Pa
	Oxidizer mass flow rate	0.1	g/s
	Temperature	1	k
	Thrust	2	N

Table 2 :

 2 MHYCAS firing test results

	MHYCAS test number	02	05	07	09	12	15	16	17
	Number of active decomposition chambers	6	6	6	3	3	6	2	2
	Oxidizer slot thickness (mm)	10.5	10.5	10.5	10.5	10.5	10.5	10.5	3.5
	Ignition delay (s)	19.2	35.0	3.7	6.6	4.2	8.6	7.5	9.2
	Firing duration (s)	12.7	18.0	28.8	36.2	29.1	23.8	25.3	22.9
	Chamber pressure (bar)	7.87	8.24 17.89 8.54 18.26 11.03 7.67	7.02
	Oxidizer mass flow rate (g/s)	51.7	52.1 103.1 41.4	44.1 107.9 36.2	30.6
	Oxidizer mass flux (kg/m 2 /s)	16.4	16.5	37.7	26.3	28.0	34.3	34.4	87.5
	Oxidizer-to-fuel mass ratio (-)	34.5	21.3	27.5	13.9	16.2	29.2	11.7	7.4
	Regression rate (mm/s)	0.050 0.082 0.126 0.100 0.124 0.124 0.104 0.142
	Experimental characteristic velocity (m/s) 1049 1071 1187 1364 1276 1212 1386 1436
	Combustion efficiency (%)	89.2	82.9	96.8	96.0	93.0 100.0 93.7	89.9

Table 3 :

 3 SuperMHYCAS test results -tests 05 to 07

	SuperMHYCAS test number	05	06	07
	Slot thickness (mm)	0.6	2.8	0.3
	Ignition delay (s)	9.7	7.2	22.5
	Firing duration (s)	45.3	47.8	32.5
	Chamber pressure (bar)	8.43	8.00	7.27
	Oxidizer mass flow rate (g/s)	51.4	53.5	46.4
	Decomposition temperature (K)	1176 1190 1188
	Oxidizer-to-fuel mass ratio (-)	5.1	9.3	4.1
	Regression rate (mm/s)	0.221 0.125 0.243
	mass flow rate and decomposition chamber temperature observed during the
	sixth test of the SuperMHYCAS engine (with slot thickness of 2.8 mm). The
	evolutions for the two other tests were very similar. The blipping phase al-
	lowed for a gradual increase in decomposition chamber temperature before the
	continuous oxidizer injection, during which it reached about 1200 K. The mono-

Table 4

 4

	, and are

Table 4 :

 4 SuperMHYCAS test results -tests 08 to 10

	SuperMHYCAS test number	08	09	10
	Slot thickness (mm)	1.65	1.65	1.65
	Ignition delay (s)	7.1	5.5	5.9
	Firing duration (s)	47.9	84.5 174.1
	Chamber pressure (bar)	8.35	8.41	7.68

Acknowledgements

This project received funding from the European Union's Horizon 2020 research and innovation programme, under grant agreement No 634534 entitled