E Di Giulio

C Perrot

R Dragonetti

Towards a control of acoustic energy conversion in structured porous materials

Introduction

Manipulating porous material's geometry allows the control of internal viscous and thermal processes. Generally, these processes are interconnected due to the material's microstructure. For instance, simple geometries like uniform cross-section materials exhibit interrelated viscous and thermal reactions. By altering the pore size (i.e. the hydraulic radius), both thermal and viscous effects can be tuned. This applies to conventional acoustic materials such as fibers, open-cell foams, and granular materials. However, there are cases where independent management of thermal or viscous effects is crucial. In thermoacoustics, enhancing heat exchange while minimizing viscous losses is necessary. In acoustics, maximizing both effects is essential for dissipating acoustic energy. Nevertheless, there are situations, like low particle velocity scenarios (e.g., thin materials on rigid surfaces or in low-frequency cavities), where increasing viscous dissipation may not be impactful. For this reason maximizing the thermal power exchanged could improve acoustic absorption. In this work we will show some preliminary studies conducted on a particular geometry of porous material in which it is possible to control these two effects separately

Theoretical formulation

The variation of the sound power Ė in a channel of infinitesimal length dx, can be expressed as [START_REF] Swift | Thermoacoustics: A Unifying Perspective for Some Engines and Refrigerators[END_REF][START_REF] Morse | Theoretical acoustics[END_REF][START_REF] Allard | Propagation of Sound in Porous Media: Modelling Sound Absorbing Materials[END_REF]

d Ė dx = 1 2 ℜ Ũ dp dx + p dU dx , (1)
where p and U represent the acoustic pressure and volume velocity fields. ℜ and ∼ indicate respectively the real part of the complex number and the complex conjugate. By combining the Eq. ( 1) with the solution of momentum and energy equations, one obtains

dE dx = - 1 2 iωA γp m θ κ |p| 2 - 1 2 iωρ m A θ ν |U | 2 + 1 2 ℜ [g pU ] . (2)
Here, ω is the angular frequency, ρ m , p m , γ are respectively the static density, pressure and the specific heat ratio. A is the cross-sectional passage area of the fluid. The term g is function both of the microstructure and the thermal gradient dT m /dx across the porous material. It is fundamental in the formulation of the thermoacoustic phenomena. θ ν and θ κ are respectively some viscous and thermal loss functions. They allow taking into account the dissipated energy occurring from viscous and thermal mechanisms. They strictly depend on the microgeometry and can be expressed as depending on the thermoviscous functions f ν and f κ , as follows

θ ν = ℑ [-f ν ] |1 -f ν | 2 , (3)
θ κ = ℑ [-f κ ] . (4)

Materials and methods

The dynamic frequency dependent responses of a porous material rely on several transport parameters, namely porosity, tortuosity, viscous and thermal characteristic lengths, and static viscous and thermal permeabilities. Nevertheless, all these parameters are correlated to the microgeometry of the material and their values can not be independently tuned. This leads to an interdependent viscous and thermal response in terms of f ν and f κ , as well as θ ν and θ κ . Here, we show a particular unit cell exhibit the property to regulate the viscous and thermal behaviour almost independently. It has been highlighted how the viscous behaviour is closely tied to the throat section area inside the microgeometry, while the thermal behaviour depends on the volume to surface ratio. The identified microstructure is a tetrakaidecahedron refereed as a Kelvin cell-based geometry, optimized to accommodate manufacturing constraints, and characterized by walls with precise opening ratios and thickness (Fig. 1).

Figure 1: Selected unit cell which can be used to control viscous and thermal dissipation mechanisms.

Results and conclusions

Fig. 2

 2 Fig.2displays the results obtained from numerical simulations in terms of θ ν and θ κ versus the frequency axis. It can be highlighted how the variation of a single geometrical parameter, such as the membrane opening ratio, has strong influence of the viscous behaviour with low effect on the thermal part. On the other hand, the variation of the wall thickness influences the thermal behaviour without changing the viscous response. In conclusion, the presented geometry could open a way on the research on this kind of materials. In fact, based on the application, it could be required to maximize the viscous behaviour over the thermal one and vice versa.

Figure 2 :

 2 Figure 2: Illustration of the evolution of the viscous θ ν and thermal θ κ relaxation functions as a function of frequency for varying aperture ratio and wall thickness.


Keywords: Rhetoric, accommodation, flattery, pride

By studying three letters of Erasmus to princes (to Charles, Archduke of Austria in 1516, to the same, became Charles the 5 th in 1523 and to Francis the 1 st in 1524), this present article intends to show that Erasmus constantly tries to resolve a puzzle which is central in Utopia's first book. Erasmus doesn't really choose between the answers which the fictitious More and Hythlodaeus give to the readers. He uses rhetorical accommodation as a way to be at the same time councillor and prophet, at the same time deferential and irreverent. In 1535, his las work, the Ecclesiastes, shows that during all his life he recommended the same solution. Accommodation is the way to be heard and to make words effective for princes' councillors and for preachers too.

Being councillor and prophet? A puzzle for Thomas More and Erasmus

It is well known that the Utopia of Thomas More and the Institutio principis christiani -both published in 1516 -are very similar 1 . In a sense, they seek to answer the same question but by different means. How to speak to the Prince? In the Utopia, Hythlodaeus' answer is quite radical: there's no way to tell the truth to the Prince, that's why the philosopher has to quit the court. The fictitious More thinks that it is possible to be a loyal councillor: for him, there's a rhetorical solution, which is accommodation or decorum in Latin 2 . I would like to show that it is possible to read Erasmus' works like an attempt to diminish or to delete this core contradiction of the Utopia which is, according to Hythlodaeus, irreducible. In other words, Erasmus constantly tries to be at the same time the fictitious More and Hythlodaeus. This article intends to illustrate this idea with three examples of Erasmus' compromise, three letters to

1 Edward Surtz insists on the similarity between the Utopia and the Institutio principis Christiani. According to him, they "are so much alike that the apparent idealism, aloofness, divorce from reality, and lack of practicality in the latter have induced some scholars to classify it as a political utopia rather than a political disquisition. The two works are variations on one theme: the ideal state" (Utopia, ed. J. H. Hexter and Edward Surtz, Yale Edition of the Complete Works of More, (New Haven and London: Yale University press, 1965) [from now on, Utopia], Introduction, p. clxxxi). On contrary, according to James D. Tracy, (The Politics of Erasmus, A Pacifist Intellectual and His Political Milieu, Toronto, Buffalo, London, University of Toronto Press, 1978), the Christian king is not for Erasmus a fictitious character like Utopus or Hythlodaeus. 2 About accommodation or decorum in Erasmus works: see Jacques Chomarat, Grammaire et rhétorique (Paris: Belles Lettres, 1982) and in particular the chapter "Veracité et accommodation" (t. 2, p. 1107-1118) ; about Christ's and teachers accommodation : Manfred Hoffmann, Rhetoric and Theology The Hermeneutics of Erasmus, (Toronto Buffalo London: Toronto University Press, 1994), p. 106-112 ; Kathy Eden, Hermeneutics and the Rhetorical Tradition, (Ithaca and London: Cornell University Press, 1997), Chapter four, p. 64-78 ; Kirk Essary, Erasmus and Calvin on the foolishness of God: Reason and Emotion in the Christian Philosophy, Toronto Buffalo London, Toronto University Press, 2017, p. 79-81 and note 104, p. 220. princes, in which it is possible to observe several rhetorical strategies. The first studied example is a dedicatory letter written in 1516 to the young Charles, Archduke of Austria. The letter seems to be very deferential, but, at the same time, Erasmus wants the young prince to look at himself in a mirror which is not accommodating at all. The second and third analysed examples are dedicatory letters sent to the most powerful princes of his time (to Charles the 5 th in 1523 and to Francis the first in 1524); these letters are both an introduction to the reading of the Paraphrases on the Gospels. In the light of these examples, it appears that there's an obvious similarity between Erasmus and the fictitious More 3 . Both think that the solution of Hythlodaeus' issue hinges on rhetoric. But according to me, Erasmus goes a little bit further than the fictitious More, because he seeks to give a place to the foolishness of god in a rhetorical frame, i. e. to be at the same time councillor and prophet 4 .

-The core dilemma of the Utopia

In More's Utopia, in a famous part of the "Dialogue of counsel", the fictitious More and Raphael Hythlodaeus disagree radically. More thinks that it is possible to be a good and loyal councillor. Raphael is convinced that it means to "accomplish nothing else than to share the madness of others 5 ". At stake is the opposition between otium and negotium, and between philosophy and rhetoric 6 . To give strength to his condemnation of the court and of all councillors, Hythlodaeus uses the paulinian commonplace which is central in the Praise of the folly and which was recently studied by Kirk Essary: "the wisdom of this world is foolishness to God 7 ". Hythlodaeus says:

Truly, if all the things which by the perverse morals of men have come to seem odd are to be dropped as unusual and absurd, we must dissemble almost all the doctrines of Christ. Yet He forbade us to dissemble them to the extent that what He had whispered in the ears of His disciples He commended to be preached openly from the housetops 8 .

Hythlodaeus doesn't want to silence the truth, even if it seems crazy, even if it is hard to hear for princes' ears. He wishes to live in a land where flattery is useless, because there's no place for princes or ministers' pride. He found this land in Utopia. But More (the real More) and 3 Even if Erasmus "chose a life of persistent, humiliating mendicancy and patronage-hunting in preference to the riches and security that would have come to him with a permanent and active place in a prince's court" (Utopia, Introduction, p. xxxiv), even if he chose not to serve a prince like Hythlodaeus, there's a similarity between "Morus' polite, civil and Christian ethos" (Gerard Wegemer, "The Rhetoric of Opposition in Thomas More's "Utopia": Giving Form to Competing Philosophies", Philosophy & Rhetoric, 23, 4 (1990), p. 290) and Erasmus' ethos. Indeed, unlike Erasmus, Raphael Hythlodaeus' style places clarity and candour before charity making little effort to mitigate the harsh effect of his radical proposals" (ibid., p. 293). 4 Which is impossible according to Hythlodaeus, as J. H. Hexter shows it: "[People who entered the service of their princes] were lost to something other than letters, something of greater importance, and Hythlodaeus knew it; they were lost to their own cause and their true calling. For at court in the council of a prince there was no place for a prophet" (Utopia, introduction, p. xci). Erasmus agree: this island doesn't exist, it is nowhere, that's why they called it their "nusquama". Stephen Greenblatt explains that the issue that Hythlodaeus points to is "what C. B. Macpherson calls a 'possessive individualism'9 " and that everything in Utopia tends to diminish "the scope of the ego10 ". Indeed, there's a strong link between private property and pride in the Utopia. Both nurture envy and jealousy: "Pride (superbia), says Hythlodaeus in his peroration, measures prosperity not by her own advantages but by others' disadvantages11 ". In his important treatise about flattery (Libellus Plutarchi de discrimine adulatoris et amici), in order to describe almost the same thing (if it is not the same), Plutarch uses another word, which is philautia12 . Unlike real friends, flatterers, who are bad councillors, protect princes' pride like a treasure, because princes' philautia is the source of their own power. In Utopia, truth doesn't seem absurd or rude, partly because people are without pride and insensitive to flattery. For Hythlodaeus, there is no middle-way between parrhesia and flattery.

-The dedicatory letter of the Institutio principis christiani

Plutarch's treatise De discrimine adulatoris et amici is actually a very effective reading grid to explain the political and rhetorical dilemma which is both at the heart of the Utopia and at the heart of Erasmus' works 13 . How to make a prince listen to the strength and the harshness of the Gospels? In 1516, Erasmus tried already to give a practical answer by writing a dedicatory letter to Charles, Archduke of Austria, future king of Spain. Thereby, he shows his rhetorical way to perceive politics. He inherits this view of politics both from Isocrates and Plutarch. Plutarch's treatise is very present in the Institutio 14 . The second chapter is entirely devoted to the subject of flattery. Further, since 1516, the Institutio is often published with Erasmus' translation of the De discrimine adulatoris et amici 15 . The issue of flattery is underlined at the end of the dedicatory letter. Erasmus admits that he's very complimentary and that he can be suspected of "adulation or impertinence (vel adulationis vel procacitatis)", but he protests that he does not have "any purpose but the common good" and that he would write differently to an older prince 16 . It is worth noting how clever he is in the last words of the letter: Among the countless distinctions (decora) which under God your merit will win for you, it will be no small part of your reputation (haec nonnulla pars laudum tuarum) that Charles was a prince to whom a man need not hesitate to offer the picture of a true and upright Christian prince (integri verique Christiani principis simulachrum) without any flattery, knowing that he would either gladly accept it as an excellent prince already, or wisely imitate it as a young man always in search of self-improvement 17 .

Erasmus seems to know how to speak to the Prince who seeks glory: he shows him that he will win true glory by accepting an unaccommodating and unflattering mirror which is the Institutio. In the words "no small part of your reputation", can be recognised the use of the litotes which Elizabeth McCutcheon has studied in the Utopia 18 . As she writes, it is often a sign of irony. Erasmus plays the role of a good friend more than of a servant or a slave and, contrary to Hythlodaeus he doesn't think that service and servitude are synonyms. He encourages the king to make the difference between good love of himself and philautia, between true and fake glory. The Institutio will help the King to know himself and to discover where he may find real greatness. The letter gives him no choice: either he accepts the mirror and imitates it, or he will lose "no small part of [his] reputation".

-The dedicatory letter to the Paraphrase on the Gospel of Matthew

The dedicatory letter to the Paraphrase on the Gospel of Matthew which Erasmus writes to Charles, when he is Emperor, tends to show that he was sincere in 1516. He's actually harsher with the Emperor than with the young prince. And he uses the same opposition between real and false greatness. The beginning of the letter can seem quite impertinent: Right well I know, invincible Emperor Charles, the great respect and reverence due to all the sacred literature which sainted Fathers put forth for our benefit under the inspiration of the Holy Spirit, and especially to those which give us a faithful account (quae nobis bona fide referunt) bibliography in Érasme, La Formation du prince chrétien, edited and translated by Mario Turchetti, (Paris: Classiques Garnier, 2015), p. 417-420. 16 "Were I writing for an older prince, I might perhaps be suspected by some people of adulation or impertinence As it is, this small book is dedicated to one who, great as are the hopes he inspires, is still very young and recently invested with government, and so has not yet had the opportunity to do very much that in other princes is matter for praise or blame. Consequently, I am free of both suspicions and cannot be thought to have had any purpose but the common good (publicam utilitatem), which should be the sole aim both of kings and of their friends and servants (amicis ac famulis)" (CWE, 27, p. 204). 17 Ibid. 18 "Denying the contrary: More's use of litotes in the Utopia", Essential articles for the study of Thomas More, p. 263-74[first published in Moreana, 31-32, 1971, p. 107-121]. of all that our heavenly Father did or uttered in the person of his Son Jesus Christ for the salvation of the whole world; and I am well aware of my own unworthiness 19 .

The beginning of this letter written to the most powerful man in Europe creates an element of surprise. It is not the "great respect and reverence due" to Charles the 5 th which is at stake, but to "the sacred literature". Even if Erasmus pretends to be unworthy of his task, he's like the sainted Fathers whom he eulogizes, he wants to give a faithful account of all that God did or uttered in the person of his son. The greatness is on his side, not on the Emperor's. Another extract of this letter opposes true to false greatness and at the same time the figure of the good and loyal councillor to the flatterers. Erasmus rewrites one excerpt of Isocrates and claims that the Paraphrase is an appropriate gift for an emperor, much more than "the gifts offered by those who bring precious jewels, high-mettled horses, hounds for the chase, or barbarous tapestries20 ". Thus, he criticises the idea that luxury could be a sign of greatness and he criticises a chivalric ideal too21 . Erasmus adds this warning: "it will be read with the greatest profit, if each person takes it up with the intention of becoming better than he was before, not of adapting the gospel text to fit his own affections (nec Evangelicam scripturam ad suos accommodet affectus) but of correcting his way of life and his desires to match the rule that it lays down". Charles still needs to be educated, and the Gospel is the best mirror he has in which to look at himself. This mirror is not more accommodating than the Institutio and probably even less so. This time, the accommodation doesn't mean that a text is consistent with the usual representation of reality; but it means that the king has to accommodate his behaviour to the Gospel. For Erasmus, this is the best use of decorum: it reveals the madness of the world which stays deaf to the call of the Gospel. The letter finishes with a prayer for peace. This prayer is more urgent when Erasmus writes to Francis the 1 st . The war between Francis the 1 st and Charles the 5 th was formally declared in the summer 1523 22 .

-The dedicatory letter to the Paraphrase on the Gospel of Mark

There's no doubt that the preface to Erasmus' Paraphrase on St Mark, in other words, the dedicatory letter to Francis the 1 st is more complimentary in its beginning, certainly because Erasmus is sincerely grateful -and probably flattered (!) by his invitation to come to France. Nevertheless, the rest of the letter is ruder than Charles' one. These following comparisons are quite clear and humiliating:

It was dishonourable for Nero to contend in the theatre (Quam vero Neroni fuit indecorum in theatro certare) with singers and musicians or in the circus with Charioteers, and it is no less dishonourable for a king to be immersed in low and sordid business which concerns his own private interest […] Again, it would be improper (quam indecorum esset) for a philosopher with his gown and beard to dance some part of ballet […] and it is no less improper for a gospel king to wage war or engage in commerce, to mention nothing more discreditable (tam indecorum est Evangelicum regem belligari aut nundinari, ne quid dicam inhonestius)23 .

There's something ridiculous and absurd when a Gospel King makes war. Again, Erasmus speaks to the king's pride in order to convert him. Further, it is another interesting way to use the commonplace of the world upside-down and the rhetorical idea of decorum. It still means that Francis the 1 st has to play his role not in the artificial and absurd theatre of the social world, but in the light of the Gospel. After that, he gives many examples of both parrhesiastic and christian characters who have dared tell the truth to a tyrant. Like them, Eramus tells the truth and furthermore he is impertinent, as this excerpt shows: "St John was beheaded for speaking his mind; but not all princes are like Herod and not all have a wife like Herodias24 ". Erasmus is not only ironic; he's even threatening when he writes about Babylas who was put to death by an emperor. He warns Francis the 1 st that Babylas "began to be an object of fear not only to a pagan emperor but to the devils who in those days were still worshipped as gods25 ". There's no doubt that Erasmus is completely aware of the harshness of his parrhesia, because he soothes it in the following sentence:

For my part, O best of monarchs, I have such confidence in the emperor Charles and you and the king of England, with your noble natures, as leaves no room for doubt that you would have all followed sound policies long ago had some counsellor arisen in whom a proper respect and freedom of speech could be combined (si quis monitor modeste liber et libere modestus extitisset)26 .

Erasmus is, obviously, not confident in these three European Princes; if he were, he wouldn't use such humiliating comparisons between a Christian king and pagan emperors. Erasmus quits a while his frankness to still be listened to. But he seems pretty sure that his words will be effective, that a good councillor can make a difference. He seems to believe that he manages to be a perfect compromise, a perfect balance between respect and freedom of speech. However, he admits in his letter that his words are a quite violent drug, and they are just an introduction to a more violent drug "the gospel hellebore". The violence of his freedom of speech is necessary, because there's something mad and rotten in the state of France27 . The diagnosis of the "faithful physician", who Erasmus wants to be, is the same as Hythlodaeus': "Wherever ambition and the love of riches and pride (superbia) and anger and revenge and passion to hurt others are supreme, there I infer that the faith of the gospel is not to be found 28 ".

In the light of these three dedicatory letters, I think that the yoking of "flattery and impertinence" present in the Institutio can be better understood. Erasmus speaks to the Princes with the language of philautia, he speaks to them in terms of glory. In a sense, he plays the role of the flatterer. But at the same time, he shows that this language of pride or philautia is absurd. Thereby flattery is turning into impertinence. And actually, when he's saying to Charles the fifth and to Francis the first: I trust you, it can be interpreted like a joke. The rest of the letter shows that he doesn't. He isn't confident in the princes, but he is confident in the power of his words and the Gospel's words. He is a flatterer and he isn't, that's his way to be both Hythlodaeus and the fictitious More of the Utopia.

-Ecclesiastes sive de ratione concionandi: A lifelong questioning:

Erasmus' last work shows that at the end of his life, he's still thinking that in order to make us feel the strength of the Gospel, a writer or an orator needs two things: to tell the truth and accommodate it. Like the Apostle Paul, he wanted to be Christ-like in his accommodation 29 . The idea of accommodation is present in the Paraphrase on the First letter to the Corinthians in which Erasmus writes: "Discourse must be accommodated to each person's capacity 30 ". It is still present in the Ecclesiastes which is published in 1535 just a few months before Erasmus' death. In this rhetorical treatise, Erasmus describes Paul's eloquence in these terms:

[…] euangelicus sermo in dispensatore non fidem modo requirit, verum etiam prudentiam.

Fides ad columbinam simplicitatem pertinet, prudentia ad serpentis cautionem. […]

Prudentiae partes sunt, ex temporum, locorum ac personarum circumstantiis dispicere, quid, quibus, quando, qua moderatione sit adhibendum. Paulus, in quo videmus summam simplicitatem cum pari prudentia fuisse coniunctam, vt sese vertit in omnia, non semper expendens quid liceat, sed quid expediat, […] 31 ! An evangelical speech has to be always accommodated; faith only is not enough. In this excerpt, Prudentia (ex temporum, locorum ac personarum cicumstantiis dispicere) is synonymous with decorum or accommodation. Another idea is worth noting in this quotation: the way to speak is not only dictated by the necessity to fit the words to the circumstances, it is above all dictated by the desire to be effective: non semper expendens quid liceat sed quid expediat. This idea is certainly a key to understand the three dedicatory letters too. And in 1523, Erasmus was confident in the fact that he had found the best way to be listened to; in another famous letter, [which was published under the title Catalogus novus omnium lucubrationum Erasmi Roterodami], he wrote: So in this way [by writing the Institutio principis christiani] I first entered on the duties of a trusty councillor. […] And yet none of the great men took offence at the freedom of this book, small as it was. Good princes tolerate freedom in their advisers; it is factious licence that they cannot stand 32 .

Since he wrote the dedicatory letter to Charles Archduke of Austria, Erasmus tries to maintain a balance: he wants to criticize without causing offence (objurgare citra contumeliam) and eulogize without flattery (laudare citra adulationem), but above all he wants to be effective, he wants to save the peace in Europe. In the short-term, of course, he failed. But the fact that we are still listening to his words offers a small measure of his legacy. Further, as James D. Tracy writes, his situation analysis "that nations were being thrown into war by princes (or by those 29 On this idea, see Kirk Essary, Erasmus and Calvin on the foolishness of God, p. 91. 30 Quoted by Kirk Essary, ibid. 31 Ecclesiastes, ASD, V, IV, p. 64, l. 579-587. 32 CWE 9, Letter 1341, to Johann von Hotzheim, p. 321. who preyed on their power) in quest of personal glory -is at least plausible, and certainly not naïve33 ". In his effort to solve a lifelong dilemma, Erasmus was extremely constant, to such an extent that his rhetorical solution is both theological and political. It is always a way to give voice to the folly of the Gospel. The rhetorical practice which we observed in the political letters is the same which he recommended for the preachers; accommodation is the evangelical way to preach and to speak to princes. For Erasmus, it's just one and the same thing. On contrary, Hythlodaeus refuses accommodation as an unacceptable compromise. But Hythlodaeus and Erasmus agree on one point, the problem is pride.

Renaissance Self-Fashioning from More to Shakespeare [1980] (Chicago and London: The University of Chicago Press, 2005), p. 39.

Ibid.

Utopia, p. 243.

On this notion, see Jean Mesnard, « Sur le terme et la notion de philautie », Mélanges sur la littérature de la Renaissance, à la mémoire de V.-L. Saulnier (Genève: Droz, 1984), p. 197-214 and La « philautie » humaniste héritages et postérité, ed. Anne-Pascale Pouey-Mounou and Charles-Olivier Stiker-Métral, Paris, Classiques Garnier, 2019.

And More certainly read it. As R. J. Schoeck remembers it, More "knew his Plutarch well. In what has been called his list of Great Books (in chapter vi of the Second Book of Utopia, where Hythlodaye itemizes and describes his 'pretye fardell of bookes') Hythlodaye says rather critically, 'They sett greate stoore by Plutarches bookes'" (Essential Articles for the study of Thomas More, ed. R. S. Sylvester and G. P. Marc'hadour,(Hamden, Connecticut: Archon Books, 1977), p. 276 [first published in : Philological Quarterly, 35 (1956), p. 366-375]).

On the importance of Plutarch's treatise in the Institutio, see Blandine Perona, « L'"incommodité de la grandeur" -Lectures de Plutarque d'Érasme à Montaigne », Le Mépris de la cour : la littérature anti-aulique en Europe (XVI e -XVII e siècles), Cahiers V. L.Saulnier, 35 (2018), ed. Nathalie Peyrebonne, Alexandre Tarrête and Marie-Claire Thomine, p. 107-123.

 15 Erasmus published his translation of this treatise for the first time under the title Quomodo pacto possis adulatorem ab amico dignoscere(Plutarchi opuscula, (Basel: Froben, 1514)) and indeed, his first edition of the Institutio principis christiani is published with Plutarch's treatise : Institutio principis christiani saluberrimis referta praeceptis, per Erasmum Roterodamum, cum aliis nonnullis eodem pertinentibus [Isocrates ad Nicoclem regem de institutione principis. Panegyricus gratulatorius de foelici ex Hispania reditu, ad principem Philippum, Maximiliani filium, eodem authore. Libellus Plutarchi de discrimine adulatoris et amici], Basileae, apud Joannem Frobenium, mense Maio 1516). About the editions of the Institutio principis christiani, see Mario Turchetti's

Ep. 1255, lines 1-6 Allen; lines 3-8 CWE.

Ibid., p. 11.

On this point, see: James D. Tracy, The Politics of Erasmus, A Pacifist Intellectual and His Political Milieu, (Toronto, Buffalo, London: University of Toronto Press, 1978), p. 58-59.

About the beginnings of the war between Charles V and Francis I, see : Mia J. Rodriguez-Salgado, « Obeying the Ten Commandments : The First War between Charles V and Francis I, 1520-1529 », The World of Emperor Charles V, ed. Wim Blockmans and Nicolette Mout (Amsterdam: Royal Netherlands Academy ofArts and Sciences, 2004), p. 15-67.

CWE, 10, letter 1400, p. 118.

Ibid., p. 122.

Ibid.,

Ibid., p. 123.

"This pestilence has seized even upon those whose chief duty it was to find a cure for our disastrous state" (ibid., p. 125).

Blandine Perona, "Being Councillor and Prophet? A Puzzle for Thomas More and Erasmus", ErasmusStudies, 40, 2020, p. 65-75.

The Politics of Erasmus, p. 45.


Emmanuel Goutierre
email: emmanuel.goutierre@upsaclay.fr

Christelle Bruni

Johanne Cohen

Hayg Guler

Michèle Sebag

Physics-aware Modeling of an Accelerated Particle Cloud

Particle accelerator simulators, pivotal for acceleration optimization, are computationally heavy; surrogate, machine learning-based models are thus trained to facilitate the accelerator fine-tuning. While these current models are efficient, they do not allow for simulating the beam at the individual particle-level. This paper adapts point cloud deep learning methods, developed for computer vision, to model particle beams.

Introduction

In the field of particle accelerator physics, simulators are crucial for optimizing the acceleration process. These simulators generate (macro-)particles and track their trajectories through the accelerator. They provide data on various parameters such as the beam barycenter, size, charge and emittance of beams. They can provide all information for the simulated particles to support in-depth analysis. Monitoring this information is fundamental for effectively controlling particle beams during acceleration. However, a notable drawback of existing simulators is that they are computationally heavy, often limiting their extensive usage.

Machine learning (ML) surrogate models emerge as a promising solution to address the computational challenges encountered with traditional simulators in particle accelerator physics. These models efficiently generate statistical data about beam properties [START_REF] Edelen | Machine learning for orders of magnitude speedup in multiobjective optimization of particle accelerator systems[END_REF]. However, a significant gap in their functionalities is that they do not provide detailed information about the individual particles within the beams, which is only delivered through tracking particles in traditional simulators. This gap hinders the full grasp of the beam 6D phase space with ML-based models.

This paper is thus devoted to enhancing these surrogate models, allowing them to handle individual particle data. The proposed approach builds upon advancements in 3D computer vision, paving the way to efficient handling of 2 or 3D point clouds. Specifically, PointNet [START_REF] Qi | PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation[END_REF] involves a specific architecture for processing point sets, using a shared network to extract features from individual points and employing a symmetric function to aggregate these features into a global representation of the set. Considering that a particle beam can be represented as an 8D point set comprising three positional dimensions, three momentum dimensions, intrinsic time, and charge, it aligns naturally with the capabilities of PointNet. 1 Overall, this paper presents the adaptation of PointNet for regression tasks targeting particle coordinates within the beam.

Related Works

Surrogate models of particle accelerators. Surrogate models offer a streamlined approach to emulating particle accelerator behaviors without the associated computational burden of direct simulations. Neural network-based surrogates present a viable path to creating rapid and reliable accelerator models. [START_REF] Edelen | Machine learning for orders of magnitude speedup in multiobjective optimization of particle accelerator systems[END_REF][START_REF] Kranjcevic | Multiobjective optimization of the dynamic aperture using surrogate models based on artificial neural networks[END_REF][START_REF] Ivanov | Physics-based deep neural networks for beam dynamics in charged particle accelerators[END_REF] utilize fully connected neural networks to forecast simulator outputs based on the accelerator control parameters.

Polynomial chaos expansion, as highlighted in [START_REF] Adelmann | On Nonintrusive Uncertainty Quantification and Surrogate Model Construction in Particle Accelerator Modeling[END_REF], aids in creating surrogate models for cyclotrons, focusing on evaluating machine sensitivities around specific control settings. Such models have been instrumental in enhancing multi-objective optimization processes [START_REF] Edelen | Machine learning for orders of magnitude speedup in multiobjective optimization of particle accelerator systems[END_REF]. By incorporating deconvolutional layers post feed-forward stages, [START_REF] Edelen | Machine learning models for optimization and control of x-ray free electron lasers[END_REF] projects the Longitudinal Phase Space of beams, while [START_REF] Scheinker | Adaptive machine learning for time-varying systems: low dimensional latent space tuning[END_REF] introduces a convolutional layer for initial laser distribution representation. Further advancements include incorporating deep and invertible neural networks as surrogates, which are efficient alternatives to computationally intensive physics-based models [START_REF] Bellotti | Fast, Efficient and Flexible Particle Accelerator Optimisation Using Densely Connected and Invertible Neural Networks[END_REF].

Point cloud representation. As an electron beam essentially is a set of 8D points, its representation can benefit from advances in computer vision and computer graphic applications. Various methods are developed to capture and analyze the spatial information contained within point clouds.

The first option is based on the voxelization of the space [START_REF] Wu | 3D ShapeNets: A Deep Representation for Volumetric Shapes[END_REF], reporting the particle density and other average specifics in each voxel. This fixed parametric representation suffers from the tradeoff between the precision τ and the number of voxels in τ 8 .

The second option pioneered by [START_REF] Su | Multi-View Convolutional Neural Networks for 3D Shape Recognition[END_REF] is based on multiple 2d-projections of the point cloud, processed using convolutional architectures and concatenated to define an embedding of the distribution.

The third option, introduced by PointNet [START_REF] Qi | PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation[END_REF], relies on direct processing of the points within the point cloud. Given a set of particles {x i }in R d sampled after distribution D, PointNet computes D #f = {f (x i )} with f a mapping from R d to R d , and returns an aggregation of D #f through a symmetric function g. The merit of the approach is that aggregation function g is learned and can adapt to the varying densities and scales of the data. The versatility of this architecture inspires numerous development of applications in computer vision, from object classification to semantic segmentation [START_REF] Ruizhongtai | PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space[END_REF][START_REF] Zhao | Point Transformer[END_REF][START_REF] Wu | Point Transformer V2: Grouped Vector Attention and Partition-based Pooling[END_REF].

The LinacNet approach

The proposed approach emulates the physical structure of the accelerator, involving a sequence of N = 25 segments. Each segment operates on the beam, a set of particles emitted by the former segment, and propagates it according to its control parameters (settings of the various electromagnetic elements attached to the segment), aimed at specific purposes (e.g. controlling the deviation or the diameter of the beam) through diverse physic mechanisms. In the following, D i = {(x i,j , y i,j ) , j = 1 . . . n} denotes the (ground truth) set of n particles forming the input beam of the i-th segment, with x i,j in R 8 describing the position and speed of the j-th particle and y i,j in {0, 1} stating whether the particle is present or has been deviated before reaching segment i. The number n of particles is set to 10,000 in the experiments.

Architectures. According to physics [START_REF] Wiedemann | Particle accelerator physics[END_REF], the propagation of each particle depends on the whole set of particles. For this reason, the propagation effectuated by the i-th segment is modeled using a PointNet architecture. Each layer uses the GELU activation function.

Two PointNet architectures are compared (Fig. 1): in LINACNET.(a), the control settings are part of the input of the PointNet module, whereas in LINACNET.(b), they are concatenated to the output of the PointNet module. The question is whether PointNet better exploits the control settings to form the aggregated representation of the beam or is better used to augment the aggregated representation. The PointNet's max operator is replaced by a weighted average operator to reflect better the overall distribution, where the particle weight is set to the probability of its presence.

Two baseline architectures are also considered to assess the importance of particle interactions. The first baseline architecture is a Siamese network [START_REF] Bromley | Signature Verification using a "Siamese" Time Delay Neural Network[END_REF], involving a (single, shared) MLP that predicts each particle coordinates at a segment's end based on its initial coordinates and the segment control settings.

The second baseline architecture is GNN-like, where the aggregation operation is performed on k local clusters. Each cluster is associated with a (uniformly sampled) particle referred to as a seed, and particles are related to the cluster of their nearest seed. The local information of each particle f (x i ) is concatenated with the aggregation of the information of its cluster before the last regression module. In our experiments, the number k of clusters is set to 100 (1% of the number n of particles). Losses and training strategies. As the whole accelerator is composed of a sequence of segments, the global architecture of LINACNET is made of the composition of neural nets emulating segments 0 to N -1. The loss associated with the i-th neural net includes the mean square error for every particle (MSE: x i,j -x i,j2 ) augmented with the cross entropy for the presence indicator y i,j (CE: y i,j log y i,j + (1 -y i,j ) log (1 -y i,j )). The training strategy comes into three modes. In independent mode, each neural net receives as input the (ground truth) distribution D i and computes the loss associated with the predicted D i+1 , noted L( D i,i+1 , D i+1 ). The weakness of the independent mode is that it does not account for the propagation of the errors along the successive segments. In end-to-end mode (E2E), each neural net receives as input the output of the previous segment, and the associated loss is noted L( D 0,i+1 , D i+1 ). In hybrid mode, the overall loss is a weighted combination of the independent and E2E losses:

L = N -1 i=0 λL( D i,i+1 , D i+1 ) + (1 -λ)L( D 0,i+1 , D i+1 ) (1)
Experimental setting. The goal of the experiments is to comparatively investigate the merits of the different architectures and losses, and the impact of the trade-off weight λ on the overall performance. The overall dataset includes 14000 simulations, computed by Astra [START_REF] Flöttmann | ASTRA: A space charge tracking algorithm[END_REF] and uniformly divided into a training set (80%), a validation set (10%) and a test set (10%). The overall dimension of the control settings is 24 [START_REF] Purwar | Random error propagation on electron beam dynamics for a 50 MeV S-band linac[END_REF].

The LINACNET model is trained on an Nvidia A100 SXM4 80 Go GPU. 2 The impact of the E2E vs independent loss is investigated by varying the trade-off parameter λ in [0, 1]. The inference time per batch is 0.8 seconds, significantly faster than the 192 minutes required for a 1-core CPU to simulate the same 16 instances in the batch with Astra.

Results

Hybrid loss, Overall results. For a hybrid loss with λ = .5, using architecture LINACNET.(a), the particle predictions at the output of the accelerator are quite good, as measured from the R2 indicator above .99 (Table 1) and as visually evidenced in Fig. 2. Most surprisingly, however, it turns out that the end predictions are slightly (but statistically significantly) better than the predictions at the end of the first segment (bottom line in Table 1). A tentative interpretation of this fact is that the first segment appears to be the most complex one, particularly from a numerical viewpoint, as it increases the particle speed from 0 to a fraction of the speed of light in this only segment. Table 1: LINACNET.(a), hybrid loss (λ = .5): R2 score of the model prediction over the 8 particle coordinates.

x y z p x p y p z t q R 2 D 0,N , D N 0.9918 0.9934 0.9999 0.9926 0.9926 0.9989 0.9999 0.9978 R 2 D 0,1 , D 1 0.9847 0.9780 0.9924 0.9754 0.9754 0.9871 0.9933 0.9956

Impact of the architecture. The histogram of the independent losses, aggregated over all segments on the test samples and displayed in Fig. 3, shows that LINACNET.(a) dominates LINACNET.(b), with a difference that is significant at level 10 -4 after paired t-test: the impact of the control settings associated with each segment is slightly better taken into account when provided to the PointNet module. The significant overlap of the two histograms however suggests that the difference is not critical for the overall result.

Impact of the architecture. The histogram of the independent losses aggregated over all segments on the test samples and displayed in Fig. 3, shows that LINACNET.(a) dominates LINACNET.(b), with a difference that is significant at level 10 -4 after paired t-test: the impact of the control settings associated with each segment is slightly better taken into account when provided to the PointNet module. The significant overlap of the two histograms suggests that the difference is not critical for the overall result.

The Siamese architecture performs significantly worse than LINACNET on the first segment (Table 2). This result confirms the importance of particle interactions regarding the modeling of an accelerated beam. The GNN-like model performance, though better than the Siamese, is lower than the LINAC-NETs; this suggests that particle interactions extend beyond local neighborhoods, emphasizing the advantage of modeling particle interactions at the global, PointNet-like level. Impact of the training strategy. The impact of the trade-off parameter λ is depicted in Fig. 4, with two lessons. Firstly and as expected, the independent loss's magnitude increases when the associated weight λ decreases. Secondly, and more surprisingly, the magnitude of the E2E loss cannot be decreased to 0, whatever the value of λ. Complementary experiments, taking inspiration from [START_REF] Sener | Multi-Task Learning as Multi-Objective Optimization[END_REF] and considering the simulator modeling as the multi-objective optimization problem aimed at minimizing both the independent and the E2E losses, have been conducted with a sophisticated adaptive schedule of the trade-off weight λ, to no avail. Furthermore, complementary experiments with λ = 1 (independent loss only) show that the E2E loss increases up to 10 5 because the errors done in each segment cumulate and exponentially increase.

Discussion and Perspectives

The main contribution of this paper is to demonstrate the feasibility of predicting the whole beam behavior in a particle accelerator. This result, original to our best knowledge, relies on adapting computer vision approaches to the particle physics domain.

The presented experiments and the discussion shed some light on the difficulty of the compound optimization problem associated with modelling the different segments, opening several perspectives for further research. The first perspective is to design a curriculum learning schedule, gradually optimizing the i-th neural net using E2E loss while freezing the i < i neural modules. Another perspective, building upon the fact that a pretty small weight on the E2E loss prevents the explosion of the errors, is to revisit the projection of the gradients proposed in [START_REF] Sener | Multi-Task Learning as Multi-Objective Optimization[END_REF]: the goal is to avoid any progress made on the independent losses from deteriorating the E2E loss.

Figure 1 :

 1 Figure 1: LINACNET Architecture. In LINACNET.(a), the local control settings are provided as input of the particles processed by PointNet. In LINACNET.(b), the control settings is concatenated to the output of the PointNet. Each layer is followed by a GELU [7] activation function.

Figure 2 :

 2 Figure 2: Predicted vs. actual values of the particle coordinates at the end of the accelerator for LINACNET.(a) with hybrid loss (λ = 0.5). The spread of data points along the diagonal line underscores the low disparity of errors for the particle beam coordinates.

Figure 3 :

 3 Figure 3: Impact of the LINACNET architecture (LINACNET.(a) in blue, LINACNET.(b) in orange): histograms of independent losses over all segments (λ = 1).

Figure 4 :

 4 Figure 4: Independent and E2E losses for different values of λ. The markers' size indicates the error's standard deviation across the test set (1400 samples).

Table 2 :

 2 Comparative validation of LINACNET and baseline architectures on the first segment.

	Architecture	Loss
	LINACNET.(a) 2.43 • 10 -5
	LINACNET.(b) 2.29 • 10 -5
	Siamese	2.62 • 10 -4
	GNN-like	1.08 • 10 -4

Graph Neural Networks (GNN) constitute another way of representing interacting particles. After preliminary experiments, however, full GNN shows that beam simulator modelling is too computationally expensive. A partial GNN architecture is considered in Section 3.

Using the RAdam optimizer[START_REF] Liu | On the Variance of the Adaptive Learning Rate and Beyond[END_REF], with dynamic learning rate adjustments (halving every ten epochs upon stagnation of validation loss), with batch size 16, aggregated gradients across 4 batches and gradient clipping at 10 for stability. Training is stopped after 80 hours or after 20 epochs without a loss improvement.


Emmanuel Goutierre

Christelle Bruni

Johanne Cohen

Hayg Guler

Michèle Sebag

Surrogate Model for Linear Accelerator: A fast Neural Network approximation of ThomX's simulator

Surrogate Model for Linear

2 PERSPECTIVES-

 2 -Saclay, CNRS, IJCLab, 91405, Orsay, France. ² Université Paris-Saclay, CNRS, Laboratoire Interdisciplinaire des Sciences du Numérique, 91190 Gif-sur-Yvette, France * emmanuel.goutierre@universite-paris-saclay.fr (3rd year PhD Student) WEIGHTING STRATEGY (𝛼 𝑖, 𝑗 ) -Equal weights : Each combination of the modules is deemed equally important -End-to-End : Only the 𝛼 0, 𝑗 are considered -Independent : Each module 𝑚 𝑖 is learnt independently -MGDA² (Multiple Gradient Descent Analysis) : Dynamic weighting of the module that moderates conflicting loss between modules LinacNet accurately reproduces the beam distribution at the end of the Linac, despite very deteriorated conditions GOAL LINACNET Modular Neural Network architecture base on physical description of the machine The beam is handled with a PointNet¹-like architecture (particle-based representation) Design a new fast-executing, physics-aware surrogate model of ThomX, expected to: 1. Quickly estimate the behavior of the beam, depending on the setting; 2. Efficiently support the surrogate optimization 3. Accommodate the wide variability of settings and beams corresponding to the diversity observed on an in-commissioning machine Applied to the ThomX's Linac, a Compton X-ray source in commissioning at Orsay, France • Each module 𝑚 𝑖 predicts the distribution at the next diagnostic given: • 𝑑 𝑖 -1 : previous distribution • 𝑎 𝑖 : value of the control setting • 𝜃 𝑖 : eight of the neural network • Modules can be combined to represents a sub-part of the machine: 𝑓 𝑖, 𝑗 𝑑 𝑖 , 𝑎 ; 𝜃 = 𝑚 𝑗 𝑚 𝑗 -1 ⋯ 𝑚 𝑖 + 1 𝑑 𝑖 , 𝑎 𝑖 + 1 ; 𝜃 𝑖 + 1 , 𝑎 𝑖 + 2 ; 𝜃 𝑖 + 2 ⋯ , 𝑎 𝑗 ; 𝜃 𝑗 • The final loss is the weighted combination of these predictions: ℒ 𝛼 𝑑, 𝑎; 𝜃 = 𝑁 𝑖 = 1 𝑁 𝑗 = 𝑖 + 1 𝛼 𝑖, 𝑗 𝑓 𝑖, 𝑗 𝑑 𝑖 , 𝑎 𝑖 ; 𝜃 -𝑑 𝑗 2 SURROGATE OPTIMIZATION: Validate the performance of the surrogate model with the optimization task -MODULARITY: An evolution of the machine only need an evolution of the corresponding modules -REALITY GAP: The large sampling domain is well-suited to handle the gap between simulation and real-world machines The simulated point cloud is the result of the simulation up to the end of the Linac of a misaligned machine with suboptimal control parameters. The resulting beam is a bi-modal, truncated beam in the transverse phase space The surrogate model successfully recovers the properties of the beam. The projections of the predicted beam (red histogram) match those of the simulated beam (green histogram)