"A Crash Is Apt to Grate Agreeably on our Ears": Thoreau and Dissonance

Danielle Follett

Université Sorbonne Nouvelle For Thoreau, music refers not only to material acoustic vibrations but also carries a metaphysical and moral significance. Thoreau's conception of sound is inextricable from his metaphysical views, and in his writings sonorous harmony resonates with overtones of spiritual harmony. Simultaneously, he expands his definition of musical harmony to include sounds conventionally considered to be dissonant, and states that all sounds are musical. This essay investigates the philosophical implications of Thoreau's aesthetics of all-musicality and dissonant harmony, especially in the context of his spiritual and moral interpretation of sound. The discussion will necessarily take us by way of his views on violence.

In his earliest journal writings, under the rubrics of "harmony" and "sphere music," Thoreau sketches a theory of sound in which spiritual and acoustic harmony are intertwined.

For example, in 1840 he writes: "[Music] is God's voice -the divine breath audible." [START_REF]Elizabeth Hall Witherell[END_REF] Such thoughts are not unique to Thoreau's youth but are found throughout his writings, although the word "God" becomes less common. In 1853 he writes that "The strains of the aeolian harp & of the wood thrush are the truest & loftiest preachers that I know now left on this earth." 2 And in "Higher Laws," Thoreau imagines John Farmer's experience of the sound of a flute which brought a moral message from a higher sphere. [START_REF] Walden | [END_REF] For Thoreau, just as for Emerson, the spiritual is inseparable from the sensory; Thoreau was no materialist, and although his idealist metaphysical views often remain implicit, they inform his writings from the first to the last. [START_REF]universe contains nothing but matter (which may be dynamic), and "idealism" as the ontological perspective that spirit objectively exists in or above matter. The fact that Thoreau focused on particularity and did not often use[END_REF] Harmony, for Thoreau, seems to refer to the holistic interrelatedness of existence, which is acoustically embodied in the physiological experience of resonance; it is physically manifested in natural law and spiritually in the moral principles he terms "higher laws." In A Week, Thoreau writes, "Music is the sound of the universal laws promulgated." [START_REF]A Week on the Concord and Merrimack Rivers[END_REF] Historically, the term also includes a political aspect, an idea of solidarity and justice, in the progressive uses of the term in 18 th and 19 th century America, such as in Robert Owen's New Harmony experiment and in Fourierist discourse. Encompassing these various dimensions -natural, spiritual, moral, aesthetic, political -the term was widely used in Thoreau's time, and Thoreau's own uses of it are very numerous.

Thoreau's association of the musical and the metaphysical participates in a longer philosophical history, to which it cannot be reduced, but nor can it be fully understood if seen outside of this context. He offers a radical rewriting and modernization of the Neoplatonic idea of the music of the spheres by bringing cosmic music down to earth, hearing metaphysical harmony in all sounds, from the lowing of a cow to the squeaking of a pump.

The idea of sphere music, associated with Pythagoras, was passed through the immanentist, idealist tradition of Plotinus, Iamblichus (quoted in A Week, in the context of sphere music), Ralph Cudworth (whom Thoreau read in the spring of 1840), and Emerson, among many others. 6 However, while at times Thoreau maintains this metaphysics of quasi-monistic immanence, at others he expresses a strong dualism between a higher and a lower sphere. On one hand, the sonorous and spiritual harmony he perceives is immanent and ubiquitous; for example, in 1851 he writes: "My profession is to be always on the alert to find God in nature -to know his lurking places. To attend all the oratorios -the operas in nature." 7 On the other, sometimes Thoreau says that literal sounds are hints of a higher harmony, thus introducing a form of dualism between the earthly and the heavenly. In his early poem, "Rumors from an AEolian Harp," sounds are "rumors" of an ideal place, source of virtue. Similarly, in 1853, he writes that the ringing telegraph wire "is my redeemer -It always brings a special & a general message to me from the highest." 8 Sound is thus for Thoreau simultaneously the immediate and concrete embodiment of divine immanence, and the hint or reminder of a higher spiritual realm. As Christopher A. Dustin concisely explains, "Thoreau's vision of God as immanent in nature is incomplete without the moment of transcendence that relates the natural and the human to a source lying beyond both." 9 The coexistence of these two spiritual views, immanence and dualism, will be important in the following.

The metaphysical value of sound is a constant throughout Thoreau's life; however, his writings on harmony evolve from the abstract, more overtly religious vocabulary of his youth toward the more sensorial, phenomenal descriptions of real sounds with spiritual overtones of the 1850s. That these sounds retain a metaphysical charge for Thoreau is clear, for example, in the following description of the song of a bobolink: He is just touching the strings of his theorbo -his glassicord -his water organ -& one or two notes globe themselves & fall in liquid bubbles from his teeming throat. It is as if he touched his harp within a vase of liquid melody -& when he lifted it out the notes fell like bubbles from the trembling strings -Methinks they are the most liquidly sweet & melodious sounds I ever heard. … The very divinest part of his strain droppingfrom overflowing his breast -in globes of melody It is the foretaste of such strains as never fell on mortal ears -to hear which we should rush to our doors & contribute all that we possess & are -Or it seemed as if in that vase full of melody some notes sphered themselves & from time to time bubbled up to the surface & were with difficulty repressed. [START_REF]Bradford Torrey and Francis Allen[END_REF] Musical birdsong analogically slides into the language of liquidity and spherical shapes like bubbles that drop and burst. Thoreau's language mimes the singularity of the song by inventing new reflexive verbs (the notes "globe themselves," "sphere themselves"). The sound is vividly actual and yet it evokes the divine and the afterlife. We can hear the echoes of sphere music: "one or two notes globe themselves & fall in liquid teeming bubbles from his throat…. The very divinest part of his strain dropping from overflowing his full breastin globes of melody… it seemed as if in that vase full of melody some notes sphered themselves & from time to time bubbled up to the surface & were with difficulty repressed." Spheres, bubbles, globes: this is Thoreau's immanent and actual terrestrialization of the music of the spheres.

Thoreau heard metaphysical harmony not only in nature but in the everyday sounds of human life. He desires to "hear a music in the rattling of the tool we work with," writes that "the squeaking of the pump sounds as necessary as the music of the spheres," and evokes "the hum of the shaft or other machinery of a steamboat which at length might become music in a divine hand." 11 He states several times that all varieties of sounds are musical and harmonious. In 1839, he writes: "All sounds, and more than all silence, do fife and drum for us"; in 1851: "Every sound is music now"; and in 1857: "Music is perpetual, and only hearing is intermittent." 12 Based on Thoreau's view of thoroughgoing divine and harmonic immanence, we might name this idea of all-musicality his panharmonium. [START_REF]Many commentators have studied the ways in which Thoreau's panharmonium prefigures the aesthetics of John Cage; perhaps the most comprehensive study is Jannika Bock[END_REF] This includes sounds often thought of as discordant, like the sound of a storm: under the rubric "Harmony," he writes in 1837: "Nature makes no noise. The howling storm -the rustling leaf -the pattering rain -are no disturbance, There is an essential and unexplored harmony in them." 14 The cosmic "sphere music" is called a "din": "The earth goes gyrating ahead amid such a din of sphere music." 15 Thoreau's panharmonium integrates cacophony into the category of harmony or aesthetically agreable sound. [START_REF]As a young man, Thoreau may have read William Gardiner's popular book The Music of Nature; Or, An Attempt to Prove That What is Passionate and Pleasing in the Art of Singing, Speaking and Performing upon Musical Instruments, Is Derived from the Sounds of the Animated World[END_REF] Indeed, even "a crash is apt to grate agreeably on our ears." 17 In 1840, in the first version of the different drummer trope, Thoreau writes: "A man's life should be a stately march to a sweet but unheard music, and when to his fellows it shall seem irregular and inharmonious, he will only be stepping to a livelier measure; or his nicer ear hurry him into a thousand symphonies and concordant variations." 18 Discord is only perceived as such by those who are not attuned to the universe's higher harmonies. In Walden, Thoreau describes the "harsh and tremendous voice" of a cat-owl: "Boo-hoo, boohoo, boo-hoo! It was one of the most thrilling discords I ever heard. And yet, if you had a discriminating ear, there were in it the elements of a concord such as these plains never saw nor heard." [START_REF][END_REF] In nature there is no dissonance; all acoustic vibrations are ultimately harmonious as they are the sound of the promulgation of universal laws. Harmony, for Thoreau, does not exclude what is commonly considered discord.

However, hearing sphere music in the ruckus of our earth may entail certain complications when we consider dissonance from a spiritual and moral point of view. Are the sounds of all earthly actions truly harmonious? As much as we may be tempted to divorce the sonorous from the moral when he starts talking about crashes and dins, that would be abandoning half of the radicality of Thoreau's thought. What does it mean to say that "violent cacophony is the sound of the universal laws promulgated"? Does a crash preach, like the thrush? What would John Farmer think, if he were visited not by gentle flute music, but by a wild clatter? These questions reflect a larger one, concerning violence. The sonorous expression of violence is crashing and cacophony. Can violence, like dissonance, be subsumed within the larger notions of metaphysical harmony and higher law? We may begin to seek an answer in Thoreau's recurrent, somewhat disturbing expressions in which violence is assimilated into what might be called an effective promulgation of higher laws. In a famous passage in "Spring," he writes: I love to see that Nature is so rife with life that myriads can be afforded to be sacrificed and suffered to prey on one another; that tender organizations can be so serenely squashed out of existence like pulp, -tadpoles which herons gobble up, and tortoises and toads run over in the road; and that sometimes it has rained flesh and blood! With the liability to accident, we must see how little account is to be made of it. The impression made on a wise man is that of universal innocence. Poison is not poisonous after all, nor are any wounds fatal. 20 Thoreau knows that death, including violent death, is part of life: accident and predation are natural and ultimately innocent, just as crashes are harmonious and "there never was yet such a storm but it was AEolian music to the innocent ear." 21 The death of individuals does not adversely affect the life of the species and the health of nature in general. Commenting on this passage, David M. Robinson writes: "What, from a limited standpoint, may appear to be cruelty or ruthlessness Thoreau tries to understand instead as the process or mechanism that serves as a guarantor of nature's power and richness. … Enormous loss, measured on a human scale, becomes less significant when seen from a higher and more comprehensive perspective." [START_REF] Robinson | Thoreau, Modernity and Nature's Seasons[END_REF] Although discord exists among the parts, harmony reigns still on the level of the whole. The subsumption of natural violence into higher innocence is the metaphysical equivalent of the incorporation of dissonance into harmony; discord is not discordant after all, nor are any sounds unmusical.

Thoreau held this view from an early age. After his beloved brother John died tragically in 1842, Thoreau wrote to Emerson the following words:

The wind still roars in the wood, as if nothing had happened out of the course of nature. … Nature is not ruffled by the rudest blast -the hurricane only snaps a few twigs in some nook of the forest. … old laws prevail in spite of pestilence and famine. … How plain that death is only the phenomenon of the individual or class. Nature does not recognize it, she finds her own again under new forms without loss. Yet death is beautiful when seen to be a law, and not an accident … One might as well go into mourning for every sere leaf -but the more innocent and wiser soul will snuff a fragrance in the gales of autumn, and congratulate Nature upon her health. [START_REF]Letter to Emerson[END_REF] The crashes and roaring of the storm do not interrupt nature's harmony. Death is one of the natural laws constantly promulgated, and is beautiful even when tragic. This idea returns in Cape Cod, in a Stoic description of the aftermath of a shipwreck: "I sympathized rather with the winds and waves, as if to toss and mangle these poor human bodies was the order of the day. If this was the law of Nature, why waste any time in awe or pity?" [START_REF] Cod | [END_REF] This is the crash, whose sound "grates agreably" on his ears: the shipwreck. "I saw that the beauty of the shore itself was wrecked for many a lonely walker there, until he could perceive, at last, how its beauty was enhanced by wrecks like this, and it acquired thus a rarer and sublimer beauty still." [START_REF] Cod | See also Thoreau's reflections on the "strengthening sight" of a dead sucker[END_REF] Paradoxically, the presence of death, or of dissonance, enhances aesthetic beauty, as it reveals the continuing vitality of nature despite the loss. According to higher laws, natural violence is an inescapable part of life, and no matter how grating the discord, it is still always subsumed into nature's harmony. The emphasis on natural law in these passages ("death is beautiful when seen to be a law," "If this was the law of Nature") shows that even as Thoreau embraces discord, such dissonance does not interrupt or displace his fundamental metaphysics of harmony, but expands it.

In his understanding of natural discord as harmonious, Thoreau is anticipating Darwin. At the same moment Thoreau was developing these ideas, Darwin was incubating his Origin of Species, in which he would argue that the death or non-reproduction of the weak is for the greater good of the species. In an epigraph to his work, he quoted the natural theologian William Whewell. In the last paragraph he evokes the laws governing evolution including that of natural selection, and expresses a view somewhat similar to Thoreau's philosophy of panharmonium: "Thus, from the war of nature, from famine and death, the most exalted object which we are capable of conceiving, namely, the production of the higher animals, directly follows. There is grandeur in this view of life…." 26 Underlying the strife, a certain harmony is preserved.

One may reasonably wonder at this point what Thoreau means by a metaphysical

harmony that is not always peaceful and sometimes involves violent crashes. It certainly does not mean static symmetry or perfect order, unless that last term also be redefined. Beyond the importance of natural law underpinning his idea of harmony, Thoreau seems to mean the unity of the dynamic universe-system and the relatedness of all of its parts with each other, which he designated with terms like sympathy, resonance and relation, and today we might call holistic interconnectedness and interdependence. [START_REF] Dassow | Walls introduces the useful concept of "empirical holism" to describe Thoreau's worldview, which coincides rather well with the vision I am suggesting: Seeing New Worlds[END_REF] This holistic relatedness is not broken when there is conflict between the parts. [START_REF] Arsić | absence of a self to idealize it into prosodic coherence[END_REF] It includes particularity and difference, and also means that there are no individuals who are absolutely other or foreign. It is characterized neither by pure homogeneity nor by radically heterogeneous particularity, but rather by interdependent particularity. Discord and disorder for Thoreau seem to be related to the dynamic conflict between entities and not to an ontology of objective chance. [START_REF] David Robinson Writes | Thoreau consistently tries to see a particular fact or event not as a random or unique occurrence but as indicative of a more comprehensive idea or law[END_REF] Thoreau's conception of a dynamic and dissonant whole is based upon the idea that spirit and natural law are immanent throughout all of nature, including its violent parts, and is related to Emerson's Neoplatonic and immanentist view of "Unity in Variety." [START_REF] Emerson | It is theoretically possible to conceive of a form of unity or holism that is materialist and not idealist, but there is[END_REF] Although Thoreau places extra emphasis on variety, he does not neglect unity. One might also cite Alexander von Humboldt, who wrote in his Cosmos, "Nature … is a unity in diversity of phenomena, a harmony." 31 Thoreau's extension of the definition of harmony to include dissonance, difference and discord occurs within the tradition of idealist metaphysics, rather than breaking with it. The incorporation of discord into a larger harmony might be seen as a Thoreauvian form of dialectics, comparable to Emerson's dialectics of "circles." But we have not yet fully answered our question: are all acoustic phenomena really harmonious? What about the sound of a gun? Does it too sound "as necessary as the music of the spheres"? In "Higher Laws," Thoreau is somewhat ambiguous about human predation: he says clearly that eating other animals is a lower stage of spiritual development, but he also admits that he "reverences" both the spiritual and the "savage" life, and at times has been tempted to devour a woodchuck raw. 32 He tends away from animal food because he finds it unclean and less spiritual, and "no human being, past the thoughtless age of boyhood, will wantonly murder any creature." 33 And yet if humanity is just another animal, as the nonanthropocentric worldview maintains, it should have no more qualms than other predators when it kills and eats other animals -as long as it takes only what it needs, as they do, and kills for food and not for wanton murder. [START_REF] Cafaro | states (though in the context of farming) that this is the "responsible answer[END_REF] Nature is so "rife with life" that this local violence will not upset the universal balance. Thoreau seems to waver between these two spiritual perspectives: on one hand the dualism of a lower, meat-eating stage vs. a higher spiritual stage in which humans forego eating other animals, and on the other the immanentism of the universal innocence of all of nature's predation, including that of humans. In 1859, he writes: I hear these guns going today -& I must confess that they are to me a spring-like & exhilarating sound -like the cock-crowing -though each one may report the death of a musquash. This, methinks, or the like of this, with whatever mixture of dross -is the real morning or evening hymn that goes up from these vales today -& which the stars echo. This is the best sort of gloryfying [sic] of God & enjoying him -that at all prevails here today -without any clarified butter or sacred ladles.

As a mother loves to see her child imbibe nourishment & expand -so God loves to see his children thrive on the nutriment he has furnished them. In the musquash hunters I see the Almouchicois still pushing swiftly over the dark stream in their canoes -These aboriginal men cannot be repressed -but under some guise or other they survive & reappear continually -Just as simply as the crow picks up the worms which all over the fields have been washed out by the thaw -these men pick up the musquash that have been washed out the banks. 35 Although there is clearly some irony in the statement that there is no better hymn to God happening on that day than gunshots, it is surely not only ironic, as he goes on to say that this "springlike and exhilarating sound" is indeed a way of glorifying God, since God loves the natural processes of life including predation. In this passage, which recalls those cited above concerning natural violence, humanity is an animal among others, governed by the same laws of nature which require eating to maintain health, and therefore hunting is innocent. Strictly speaking, meat-eating is compatible with the immanentist, nonanthropocentric view, whereas vegetarianism imposes a dualism between the ways of humans and those of other carnivores.

Thoreau's approach is thus ambiguous: at times he maintains a dualistic view (higher/lower, clean/unclean, humans/other animals) where humans are held to a higher standard than other animals, and then at others he displays an immanentism where all predation, including that of human animals, accords with natural harmony. This tension between two spiritual views, immanence and dualism, seems to remain unresolved. [START_REF] Cornell Dolan | notes a similar ambiguity in a journal entry which first condemns eating animals and then describes Eskimos hunting[END_REF] The sound of the hunter's gun is thus somewhat ambiguous. What of the soldier's?

The tension we have just seen reappears in this context in a slightly different form. In many early journal entries, and in his 1840 essay "The Service," Thoreau associates music with a soldier's bravery and heroism. He essentially argues that a soldier fighting for a just cause is acting in harmony with the universe. The essay seems rather to be written to inspire a spirit of "service" and a soldier's courage in its readers' everyday lives, than to advocate any concrete use of guns, though it also criticizes the "insincerity and sloth of peace." [START_REF]The Service[END_REF] Rejected for publication in The Dial by Margaret Fuller, it may indeed seem rough terrain to readers because it juxtaposes two quite different registers, the highly abstract praise for spiritual courage and the evocation of the concrete practice of war, without offering the needed intermediary term -the particular moral justification for any historically situated use of violence, without which the violence is incomprehensible. Thoreau takes it for granted that the battle cry he praises is one in defense of justice and higher laws, such as John Brown's, and not, for example, that of the US Army against the Seminoles. [START_REF] Reynolds | A number of commentators have observed that Thoreau advocated not only non-violent resistance but also, when necessary, violent action against injustice; see for example Lance Newman, "Thoreau and Violence[END_REF] He states in "Slavery in Massachussetts," "Show me a free State, and a court truly of justice, and I will fight for them, if need be." 39 In this context, Thoreau believes that the universe has an immanent rhythm and music, and acting in sync with it inspires bravery and overcomes isolation. This helps explain the following passage:

Especially the soldier insists on agreement and harmony always. … War is but the compelling of peace. If the soldier marches to the sack of a town, he must be preceded by drum and trumpet, which shall identify his cause with the accordant universe. All things thus echo back his own spirit, and thus the hostile territory is preoccupied for him. He is no longer insulated, but infinitely related and familiar. 40 It is difficult to see how the sack of a town, which is very different from two armed forces meeting on a battlefield, could ever be considered to accord with the higher law of justice and make the sacker infinitely related to the rest of creation. This passage may perhaps be best understood as the overexcited froth of a young man who had not yet outgrown his guns, the process he recounts in "Higher Laws." At any rate, Thoreau's view here and elsewhere seems to be that through bold actions theoretically in accordance with higher principles, the soldier is not only related to his companions but to the universe. [START_REF] Arsić | What Music?" 206). She relates harmony to forced harmonization and colonialization (207). However, Thoreau's soldier does not want to subjugate discord or difference, but to conquer the foe of "Falsehood" or injustice, and in so doing he necessarily brings with him not only band music but the cacophony of war[END_REF] The sound of a just war is for Thoreau harmonious because it resonates with the rhythm of the universe's immanent laws; but simultaneously, he states that the sound of all war is terribly dissonant: in 1840 he writes that "war is the sympathy of concussion." 42 It seems strange that sympathy, a romantic term related to harmony, could be found in the collision of battling bodies and the crash of concussion. But this idea reminds us again of the harmony he finds in storms and shipwrecks: the sympathy of concussion would be the accordance of discord, or the harmony of dissonance. He continues:

Men do not peep into heaven but they see embattled hosts there. Miltons' [sic] heaven was a camp. … The soldier is the practical idealist -he has no sympathy with matter, he revels in the annihilation of it. So do we all at times. When a freshet destroys the works of man -or a fire consumes them -or a Lisbon earthquake shakes them down -our sympathy with persons is swallowed up in a wider sympathy with the universe.

A crash is apt to grate agreeably on our ears. 43 Even in heaven, where ultimate harmony should theoretically reign, the discord of war continues. This passage may be relatively incomprehensible if not seen in the context of the idea of a just war. The harmony of a just war's discord is thus related to the harmony of nature's violence, prefiguring the arguments cited above in the letter to Emerson and in "Spring": our sympathy with an individual is sacrificed to a sympathy with the whole. In the interest of higher ideals or the laws of nature, even "matter" may be destroyed: here we see a glimpse of Thoreau's dualism. Thoreau is not a simple sword-rattler; he prefers staying at home and braving the everyday experiences of life. 44 However, when necessary for a just cause, he supports the use of violence, for example in rising up against slavery: "I do not wish to kill nor to be killed, but I can foresee circumstances in which both these things would be by me unavoidable. … The question is not about the weapon, but the spirit in which you use it." [START_REF]A Plea for Captain John Brown[END_REF] The discordant sound of a gun used in the spirit of justice thus "grates agreeably" on Thoreau's ears, and is incorporated into the immanence of nature's harmony. But guns are not always used in this spirit, and this example shows that there are indeed some sounds that are not musical: if it is necessary to go to war for justice, then injustice exists, whose sound will always for Thoreau remain unbearable. Speaking of the execution of John Brown, he writes: "While these things are being done, beauty stands veiled and music is a screeching lie." 46 And in "Slavery in Massachussetts": "Justice is sweet and musical; but injustice is harsh and discordant. The judge still sits grinding at his organ, but it yields no music, and we hear only the sound of the handle. He believes that all the music resides in the handle." 47 The noise of this organ handle is evidently not as musical as the "rattling of the tool we work with" or "the squeaking of the pump." 48 The image is quite evocative, as it vividly depicts the absence of harmony though not the absence of sound, just as true justice is not embodied in the "justice" of the court who sent Anthony Burns back to slavery. Dissonance exists not only in metaphorical sounds: the rousing drum-beat that inspires the soldier in "The Service" is quite different from "this rub-a-dub" that Thoreau hears at night as soldiers, like those who executed the judge's decision, train in the street. 49 The immanence of divine harmony clearly does not extend so far as to include all forms of violence, in particular human acts of injustice. Here we see Thoreau's idealist dualism between higher principles and human society, well analysed by Daniel S. Malachuk in his Two Cities in terms of the city of God and the city of man. Thoreau's first writings on music sometimes express this view as well: "Some sounds seem to reverberate along the plain, and then settle to earth again like dust; such are Noise, Discord, Jargon. But such only as spring heavenward … are the true sphere music." 50 Thoreau's dualism thus enters into tension with the immanentism of his panharmonium: not all sounds are musical after all. The dualist view does not undermine the radicality of Thoreau's panharmonium, but nuances it. From the perspective of immanence, discord does not exist; from that of dualism, some sounds are irretrievably dissonant.

Thoreau holds both views.

It was not uncommon for romantics to believe simultaneously in a quasi-monistic immanence of spirit in nature and in a dualist split between higher and lower. The overlapping perspectives do not necessarily create an overtly problematic tension; we must not demand perfect consistency. However, moments of dissonance, if not direct clashes, remain. For example, beyond the question mentioned above of whether humans eating other animals is a form of harmony or of injustice (especially in light of nonanthropocentric views of equality between species), we can wonder whether Darwin's law can still be seen as harmonious when applied to human society. Would Thoreau say that he loves to see that "Nature is so rife with life" that the humans of London in 1854 can be afforded to be sacrificed and "serenely squashed out of existence" by malnutrition, disease and unemployment? Or would he hold humans to a higher standard than other animals? We can wonder if his quasi-monist, dissonant panharmonium extends to such social violence, which can thus be condoned, or if this violence should rather be understood in terms of the dualism of higher law vs. lower lawlessness, and thus be denounced. Thoreau lived long enough to have assimilated Darwin's theory of evolution, but not long enough to have witnessed or participated in the social darwinism debates that his work sparked. We may thus hear a certain dissonance in the latent clash between these two views.


domain: spi phys

text:
F X Irisarri

T Macquart
email: terence.macquart@bristol.ac.uk

C Julien

D Espinassou
email: denis.espinassou@cetim.fr

A novel design method for the fast and cost-effective manufacture of composite parts employing the Quilted Stratum Process

Keywords: A. Layered structures, A. Thermoplastic resin, A. Discontinuous reinforcement, B. Buckling, Variable-stiffness optimization

Composite materials, and carbon fiber reinforced plastics (CFRP) in particular, are nowadays widely used in the design of high performance aeronautical and aerospace lightweight structures. Applications to other sectors have, however, been limited due to their high costs and long processing times. To address these issues, the CETIM has developed the Quilted Stratum Process (QSP); a fully automated high-volume production line of thermoplastic composite parts. The QSP manufactures variable-thickness and variable-stiffness composite laminates by assembling composite layers of various orientations and shapes. The focus of this work is to bring together the great versatility of composite design offered by the QSP under a single coherent framework, automating and simplifying the design of QSP-manufactured structures. To this end, we develop the Quilted Stratum Design (QSD). The QSD is a three-step laminate optimization procedure during which the structure is first ideally optimized, then simplified and finally detailed layer by layer to be interpreted for QSP-manufacturing. The method is presented in this paper and applied to both an academic and an industrial case study for which the trade-off between the manufacturing complexity and mechanical performances is studied. In both our applications, partitioning the structure into small numbers of constant stiffness areas was found to be sufficient to significantly over-perform conventional constant stiffness designs and well-compatible with the QSP.

Introduction

Composite materials, and carbon fiber reinforced plastics (CFRP) in particular, are nowadays widely used in the design of high performance aeronautical and aerospatial lightweight structures. Besides these flagship applications, other industries such as offshore oil, wind energy, and automotive have also found numerous applications availing to the high specific stiffness and strength-to-weight ratios of composite materials.

Compared to light metal alloy, CFRPs also offer additional structural degrees of freedom, related to the fiber orientations. The resulting increase in design freedom has been and still is responsible for much of the enthusiasm given to composite laminate optimization in the literature; a review of which can be found in [START_REF] Ghiasi | Optimum stacking sequence design of composite materials Part I: Constant stiffness design[END_REF][START_REF] Ghiasi | Optimum stacking sequence design of composite materials Part II: Variable stiffness design[END_REF].

The mechanical properties of composite laminates are characterized by their constituents, e.g. fiber and matrix materials, but also by their manufacturing process which influences fiber-matrix adhesion, interlaminar strength and residual stresses. From a design point of view, the manufacturing process defines and constrains the design space, generally trading-off complexity for reduced manufacturing costs. Consequently, many design and optimization methods for composite structures are specific to manufacturing processes.

In the aeronautic industry, the design of classical laminates manufactured by hand lay-up or automated tape placement processes is subjected to numerous design guidelines motivated either by the manufacturing process itself or by past experiences about the mechanical behavior of composite laminates [START_REF]Polymer Matrix Composites Guidelines for Characterization of Structural Materials[END_REF][START_REF] Bailie | A summary and review of composite laminate design guidelines[END_REF]. Designing a single laminate is known in the literature as a constant-stiffness design problem. Laminate design guidelines such as symmetry and balance of the laminate have been considered since the very first developments about constant-stiffness optimization [START_REF] Le Riche | Optimization of laminate stacking sequence for buckling load maximization by genetic algorithm[END_REF][START_REF] Nagendra | Design of a blade stiffened composite panel by a genetic algorithm[END_REF][START_REF] Liu | Permutation genetic algorithm for stacking sequence design of composite laminates[END_REF].

Contrary to constant-stiffness structures, variable-stiffness structures consist of multiple stacking sequences.

Large composite panels with thickness variations obtained by adding or terminating plies form a first class of variable-stiffness structures. The design of such panels is commonly referred to as the laminate blending optimization problem [START_REF] Birna | Optimal design of large composite panels with varying loads[END_REF][START_REF] Liu | Composite wing structural design optimization with continuity constraints[END_REF][START_REF] Soremekun | Stacking sequence blending of multiple composite laminates using genetic algorithms[END_REF][START_REF] Adams | Genetic algorithm optimization and blending of composite laminates by locally reducing laminate thickness[END_REF]. Specific developments about handling ply-drop design guidelines can be found in [START_REF] Irisarri | Optimal design of laminated composite structures with ply drops using stacking sequence tables[END_REF][START_REF] Zein | A constraint satisfaction programming approach for computing manufacturable stacking sequences[END_REF].

The development of Automated Fiber Placement (AFP) machines has driven a significant research effort to devise specialized optimization methods. Contrary to conventional laminates, AFP processes enable continuous variation of the fiber orientation within each layer. This results in an enlarged design space for variable-stiffness design whose exploration requires specialized optimization methods. Numerous direct parameterizations of the tow paths have been proposed, for instance in [START_REF] Hyer | Use of curvilinear fiber format in composite structure design[END_REF][START_REF] Gürdal | Tow-Placement Technology and Fabrication Issues for Laminated Composite Structures[END_REF][START_REF] Gürdal | Variable stiffness composite panels: Effects of stiffness variation on the in-plane and buckling response[END_REF][START_REF] Lopes | Tailoring for strength of composite steered-fibre panels with cutouts[END_REF][START_REF] Wu | Buckling analysis and optimisation of variable angle tow composite plates[END_REF][START_REF] Byung | Continuous tow shearing for manufacturing variable angle tow composites[END_REF][START_REF] Montemurro | On the effective integration of manufacturability constraints within the multi-scale methodology for designing variable angle-tow laminates[END_REF].

The most advanced methods for variable-stiffness design combine, within multi-step approaches, gradient based optimization and evolutionary algorithms for their complementary efficiency in solving continuous and combinatorial problems respectively. Lamination parameters are used to find the optimal stiffness distributions in combination with Finite Element (FE) analyses [START_REF] Miki | Optimum Design of Laminated Composite Plates Using Lamination Parameters[END_REF][START_REF] Grenestedt | Layup Optimization of Composite Material Structures[END_REF][START_REF] Ijsselmuiden | Optimization of Variable-Stiffness Panels for Maximum Buckling Load Using Lamination Parameters[END_REF][START_REF] Liu | A lamination parameter-based strategy for solving an integercontinuous problem arising in composite optimization[END_REF][START_REF] Wu | Framework for the Buckling Optimization of Variable-Angle Tow Composite Plates[END_REF]. Alternatively, the polar parameterization can also been used by [START_REF] Vannucci | Stiffness design of laminates using the polar method[END_REF][START_REF] Jibawy | Hierarchical structural optimization of laminated plates using polar representation[END_REF][START_REF] Catapano | Stiffness and Strength Optimization of the Anisotropy Distribution for Laminated Structures[END_REF] for this purpose. The subsequent optimization steps aim at retrieving the corresponding fiber angle distributions [START_REF] Irisarri | Optimisation of ply drop order in variable stiffness laminates[END_REF] and tow path [START_REF] Blom | Optimization of course locations in fiberplaced panels for general fiber angle distributions[END_REF] while taking into account manufacturing constraints [START_REF] Daniël | Stacking sequence optimisation of variable stiffness laminates with manufacturing constraints[END_REF].

The present work is a first step towards devising a composite structure design method specialized for the Quilted Stratum Process (QSP) developed by the CETIM [START_REF] Wedhorn | Quilted Stratum Process for High-performance CFRP Production[END_REF]. The QSP, in short, is a multi-step process transforming raw materials (e.g. fibres and resin) into sophisticated composite parts, providing an economically viable solution for high volume production. QSP structures are built-up from an assembly of composite patches. A patch is a composite layer whose material, size, shape, orientation, position and stacking order in the assembly are design variables. Assembling patches enables piecewise linear fiber orientation variations as well as thickness variations. As such, QSP structures constitute a specific class of variable-stiffness structures.

The optimization of laminated structures made of patches has been studied by Zehnder and Ermanni in [START_REF] Zehnder | A methodology for the global optimization of laminated composite structures[END_REF] and further developed by Giger et al. [START_REF] Giger | A graph-based parameterization concept for global laminate optimization[END_REF]. Both studies focus on devising an appropriate parametrization to solve the optimization problem using a genetic algorithm. Such an approach, however, entails computational costs incompatible with time and cost constraints of most design offices. In this study, in order to better control the computational costs, a multi-step design procedure is proposed, derived from variablestiffness methods.

From the general presentation of the QSP in Section 2, degrees of freedom and constraints imposed by the manufacturing process on the design of composite structures are determined. Section 3 presents the proposed three-step Quilted Stratum Design (QSD) method. The first step starts with a variable thickness and stiffness structural optimization employing lamination parameters. In the subsequent step, the design is simplified into a reduced number of constant-stiffness zones. The last step aims at retrieving the fiber orientations per zone in order to sketch out a design that can be manufactured using the QSP. Results of two design case studies are presented and discussed in Section 4 whilst concluding remarks are given in Section 5.

The Quilted Stratum Process design problem

The use of CFRP composites in mass-production vehicles remains scarce, mostly due to their high costs and long processing times. Reducing material and manufacturing costs as well as increasing production rates are key issues that must be addressed before the automotive industry can fully integrate CFRPs within mass-produced vehicles. Aeronautic manufacturers face similar challenges for parts such as seat structures, window frames, or FAN blades. The CETIM, in an effort to address these challenges, has developed the Quilted Stratum Process (QSP).

The QSP, presented in Figure 1, is a fully automated four-step process transforming raw materials (i.e. fibers and resin) into complex composite parts in a short turnaround time. The production line starts with the continuous pultrusion of thermoplastic composite tapes that feeds into a cutting module. Alternately, any thermoplastic composite tape avaible from material suppliers can be used. Ultrasonic blades are used to cut the tape into patches of variable shapes. Patches are then picked from carriers and placed and welded in precise position on a conveyor system. Patches can be joined end to end to form composite layers with piecewise linear fiber orientation variations. The layers are laid in successive assembly stations to build up a variable-thickness preform on a flat surface tool. The preform is then heated using a combination of infrared and conduction heating devices in order to provide sufficient energy to the thickest areas without degrading the thinnest areas. Finally the preform is thermo-formed in a hot press and demolded. The QSP minimizes material waste and produces net-shape parts with short processing time, thus providing an economically viable solution for high volume production. The cycle time is driven by the press cycle which varies form one to ten minutes depending on the maximal thickness of the part (up to 6 mm thick) and the materials.

Net shape parts can be obtained by simultaneous thermoforming and overmolding of the part edges. A large variety of inserts can also be added in the same press-forming cycle. From a design viewpoint, the key issue is the patch assembly. The assembly is defined by multiples patches, positioned, rotated, and stacked on a flat surface, to obtain a preform. Additionally, the QSP is a versatile process which enables numerous variations in manufacturing patches which must also be considered, thus a patch is defined by a shape, a size, a position, a material and its orientation. The degrees of freedom listed above form an extremely large and complex design space due to their number and diverse nature.

Direct sampling and search in such a design space could be very costly in terms of computational cost. This problem is thoroughly discussed in [START_REF] Zehnder | A methodology for the global optimization of laminated composite structures[END_REF][START_REF] Giger | A graph-based parameterization concept for global laminate optimization[END_REF]. Thus, in the following, a multi-step approach is proposed. As this is the first attempt at developing a method for the design of QSP-manufacturable composite structures, several simplifications are made to restrict the process versatility to a manageable level of complexity.

1. It is assumed that all patches are made from the same base ply material in order to reduce the design space.

2. Minimizing the amount of material waste during the cutting step is out of the scope of this work.

3. Devising the shape and position of the patches in the patch assembly is a complex task that is not fully completed in the present work. The proposed method returns an optimized partition of the structure into a reduced number of constant-stiffness areas for which feasible laminates are retrieved. The work focuses on the search for efficient trade-offs between the complexity of the partition of the structure and the mechanical performances of the optimized part.

4. The influence of process specific features such as cuts and thickness transitions on the strength of the composite structures are not considered. Instead, the present work focuses on stiffness-related design criteria.

5. Patch sliding occurring due to the flow of the matrix during thermoforming is neglected. Nonetheless, material orientation variations induced by the forming have to be taken into account in the design process for complex part shapes.

The Quilted Stratum Design (QSD) method

The design method is based on a three-step decomposition of the problem. The first step provides an idealized target design, relying on a gradient-based structural optimization and a homogenized description of the laminates. In the second step, the idealized continuous design obtained after the first step is simplified using a clustering algorithm to subdivide the structure into a small number of constant stiffness zones.

Finally, the last step aims at retrieving the fiber orientations per layer and per zone.

During the first and second optimization steps, structural analyses are performed on the final thermoformed shape of the structure. By contrast, the third step is based on the flat perform. The forming of a 0 • -ply is simulated using Altair Drape Estimator to estimate both the flat shape of the patch assembly and the spatial evolution of the material longitudinal direction on the final thermoformed shape. Structural optimizations are performed using Altair OptiStruct whilst the overall design approach is implemented in MATLAB.

Step 1: structural optimization using lamination parameters

In the first optimization step, thickness and stiffness distributions are used as design variables. Any mechanical response accessible through linear elastic FE analysis can be used either as a design objective or a constraint. The stiffness distribution is parameterized using lamination parameters [START_REF] Gurdal | Design and optimization of laminated composite materials[END_REF]. Lamination parameters provide a compact and continuous parameterization of the stiffness behavior of composite laminates.

The number of lamination parameters, twelve for the general case, is independent of the number of plies.

The number of lamination parameters falls to eight in the case of uncoupled laminates (with the coupled stiffness matrix B = 0) and to four in the case of uncoupled laminates with orthotropic membrane A and bending D properties. Furthermore, the design domain for lamination parameters is continuous and convex, making it suitable for gradient-based optimization. Individually, lamination parameters can take any value in the interval [-1, 1]. They are, however, not independent since they are computed from a given stacking sequence. Deriving compact sets of feasibility constraints for the lamination parameters space has attracted the attention of many researchers since the first studies using lamination parameters as design variables for laminate optimization and different sets of equations are available depending on the number of non-zero parameters [START_REF] Miki | Optimum Design of Laminated Composite Plates Using Lamination Parameters[END_REF][START_REF] Grenestedt | Layup Optimization of Composite Material Structures[END_REF][START_REF] Wu | Framework for the Buckling Optimization of Variable-Angle Tow Composite Plates[END_REF][START_REF] Gabriel Diaconu | Feasible Region in General Design Space of Lamination Parameters for Laminated Composites[END_REF][START_REF] Bloomfield | On feasible regions of lamination parameters for lay-up optimization of laminated composites[END_REF][START_REF] Macquart | Optimisation of composite structures Enforcing the feasibility of lamination parameter constraints with computationally-efficient maps[END_REF].

The membrane stiffness matrix A and the bending stiffness matrix D are defined as linear functions of the lamination parameters:

A = h Γ 0 + Γ 1 V A 1 + Γ 2 V A 2 + Γ 3 V A 3 + Γ 4 V A 4 , D = h 3 12 Γ 0 + Γ 1 V D 1 + Γ 2 V D 2 + Γ 3 V D 3 + Γ 4 V D 4 .
(

) 1
where h is the thickness of the laminate and Γ i=1,...,4 are matrices dependent on the base ply in-plane stiffness properties. The membrane lamination parameters V A i=1,...,4 and bending lamination parameters V D i=1,...,4 are defined as follows:

V A 1 ,V A 2 ,V A 3 ,V A 4 = 1/2 -1/2 (cos 2θ (z), sin 2θ (z), cos 4θ (z), sin 4θ (z)) dz, V D 1 ,V D 2 ,V D 3 ,V D 4 = 1/2 -1/2 z2 (cos 2θ (z), sin 2θ (z), cos 4θ (z), sin 4θ (z)) dz, ( 2
)
where z is the normalized through the thickness dimension and θ (z) is the laminate stacking sequence.

Step 2: zone clustering

The total number of design variables at step 1 depends on the assumptions made on the stiffness behavior of the laminate (which are closely related to laminate design guidelines such as symmetry and balance), the use of constant-thickness or variable-thickness laminates, and the desired type of thickness variations, which is closely related to the chosen manufacturing process. In the case of steered laminates, continuous stiffness variations over the structure are possible which can be described using a set of design variables per element of the model. In the case of more conventional variable-thickness straight-fiber laminates, the ply-drop locations are usually predefined and design variables are assigned per constant-stiffness areas.

The design of QSP laminates constitutes an intermediate class of problems. The process does not enable continuous variations of the laminate stiffness and the subdivision of the structure into constant stiffness areas is not known a priori.

From a manufacturing viewpoint, transitions between constant stiffness areas are obtained by terminating patches or joining patches with different orientations. The smoother the stiffness variations over the structure, the smaller and more numerous are the patches and the more complicated is the patch assembly.

In Step 1, a first structural optimization is performed using a fine initial subdivision of the structure into constant-stiffness zones. For small FE models, finite elements can directly be used as zones. For larger models, however, elements are grouped into clusters to limit the number of design variables and feasibility constraints. The optimization result can be interpreted as the best achievable solution using the smallest possible patch size. The zoning of the structure is simplified in Step 2 to search for an efficient trade-off between the simplicity of the patch assembly and the performance of the structure. Clusters of zones are formed based on the structural similarities between each zone employing a k-means clustering method [START_REF] Arthur | k-means++: the advantages of careful seeding[END_REF].

Each zone is represented as a vector containing the optimal values of the corresponding design variables (e.g. thickness and lamination parameters) and the coordinates of the center of the zone in order to promote the formation of clusters of adjacent zones.

The k-means clustering finds a partition in which elements within a cluster are closer to the cluster centroid than to any other cluster centroid. A heuristic is used to find the centroids that maximize the separation between the clusters. The optimal number of clusters k can be determined by experimenting with a limited range of values for k. In this work, the average silhouette value is used as a quantitative indicator of how well-separated the clusters are [START_REF] Kaufman | Finding groups in data: an introduction to cluster analysis[END_REF]. The best values of k tend to maximize the average silhouette value.

The corresponding clusters form a meaningful partition of the structure with respect to the optimal stiffness distribution. However, in most practical cases, perfect separation between neighboring clusters cannot be achieved. User intervention is often required to check and eventually adjust the clusters. Once the number of clusters chosen and the clusters validated, a simplified zoning of the structure is defined. Each cluster is attributed a single set of design variables and a second structural optimization is run to determine the optimal thickness and lamination parameters distributions and ensure that the performance loss with respect to the first optimal design is acceptable. The feasible lamination parameters space being convex, the average values of the first optimization design variables over each cluster can be used as a starting point for the second optimization run.

Step 3: patch assembly retrieval

Step 2 results in a simplified zoning of the part, with optimal membrane and bending stiffness matrices per zone. In Step 3 the laminates that best matches the target stiffness distribution are retrieved. The QSP allows to join patches of different material orientation end to end. Thus, within a single layer, no fiber continuity constraint applies with the QSP. Therefore, a first solution can be obtained by searching for the best laminate in each zone considered separately. In the present work, independent stiffness matching optimizations are performed for each zone. Proximity between a laminate, with membrane stiffness matrix A and bending stiffness matrix D, and the target stiffness matrices A * and D * is assessed using the A-distance d A and D-distance d D values [START_REF] Irisarri | Improved Shepard's Method for the Optimization of Composite Structures[END_REF]:

d A (A, A * ) = Tr A -1 A * + Tr AA * -1 -6, d D (D 1 , D * ) = Tr D -1 D * + Tr DD * -1 -6.
(

) 3
The weighting between d A and d D depends on the application, nevertheless giving the same weight to both criteria is usually good practice.

Laminate stiffness matching optimization can be performed using specialized algorithms. However, with the QSP, the number of layers in the patch assembly is limited. On the one hand, the laminate total thickness is limited due to process-related constraints for the final thermoforming phase. Laminates more than 10 mm thick would be very difficult to heat correctly in a limited time. On the second hand, each layer is laid by a dedicated assembly station. Thus the number of layers of the patch assembly drives the design and cost of the scalable assembly system. Last but not least, thick plies are generally less expensive than thin plies and will be preferred to minimize the cost of the parts. Material cost is indeed one of the main cost driver for mass production. All the above considerations tend to reduce the number of layers, usually bellow 30 layers. With limited sets of allowed ply orientations, such as 0 • , ±45 • , 90 • , the problem becomes tractable for database enumeration. Using a database of laminates also simplifies the enforcement of laminate design guidelines such as symmetry, balance, continuity of the surface plies or 10% design guideline.

Applications

Two applications of the QSD method are presented and discussed in this Section. The first application serves as a demonstrative academic benchmark case study for the proposed design method, showcasing its potential and efficiency. The second application, industrially-focused, illustrates the optimization of a more complex automotive part, targeting a reduction in mass.

Buckling optimization of a square plate

The first application deals with the buckling optimization of a square plate under uniaxial compression, a problem previously studied by IJsselmuiden et al. [START_REF] Ijsselmuiden | Optimization of Variable-Stiffness Panels for Maximum Buckling Load Using Lamination Parameters[END_REF]. All four edges are simply supported and remain straight. Furthermore, the orthotropic material properties are E 11 = 181 GPa, E 22 = 10.3 GPa, G 12 = 7.17

GPa and ν 12 = 0.28. The laminate is assumed to be symmetric and balanced and the total number of plies is set to 16 with a total thickness of 1.524 mm. The plate is modeled in OptiStruct, using a square mesh of 20 × 20 quadrangular shell elements.

Step 1 of the optimization is performed using an independent set of design variables per element: the lamination parameters V A 1 and V A 3 for the membrane behavior and V D 1 and V D 3 for bending. The optimized lamination parameters distributions are presented in Figure 2, which shows obvious similarities with the results previously reported in the literature [START_REF] Ijsselmuiden | Optimization of Variable-Stiffness Panels for Maximum Buckling Load Using Lamination Parameters[END_REF]. Despite some differences, especially in V A 3 and V D 3 distributions, the optimized buckling load, normalized with respect to the "black aluminum" solution (i.e. all lamination parameters equal to zero), matches the one reported in [START_REF] Ijsselmuiden | Optimization of Variable-Stiffness Panels for Maximum Buckling Load Using Lamination Parameters[END_REF]. That is, the buckling load of the variable stiffness (VS) solution is about 2.53 times higher than the quasi-isotropic solution. In Step 2, the plate is partitioned into constant stiffness (CS) areas, base on the results of Step 1. Several clustering attempts are performed using the k-means algorithm for different numbers of clusters. In the VS plate the buckling improvement with respect to CS design is related to load redistribution induced by the spatial variation of the membrane stiffness properties. Thus, in the present case, the clustering is performed based on the position of the center of the elements and on the optimal values of the membrane lamination parameters V A 1 and V A 3 obtained from Step 1. The evolution of the mean silhouette value (MSV) as a function of the number of clusters is plotted in Figure 3. The maximal MSV value is reached for 7 clusters, but the curve shows that partition of similar quality can be obtained for 3 or 5 clusters. The corresponding partitions of the plate are shown in Figure 4. The 3-cluster solution gives a simple and seemingly relevant partitioning of the structure. It is interesting to note that clusters can cover disconnected regions of the part. Hence, from a process-related point of view, a cluster can correspond to several patches. The same comment applies to the 5-cluster solution. However, this solution is much less relevant since cluster 1 is far too dispersed. Such a solution would require a manual correction. The solution with 7 clusters presents transition zones between a few main clusters that are very similar to the ones of the 3-cluster solution.

In order to assess the influence of clustering on the mechanical behavior, a lamination parameter based whose buckling load λ /λ 0 = 1.69 is consistent with the plateau observed in Figure 5. This plateau is due to the low number of possible laminates evaluated at step 3. Both the angular discretization and the thickness discretization due to the use of a single base ply prevent accurate matching of the target stiffness distributions. These results show that a simple partitioning of the structure is enough to obtain a significant improvement over the CS designs (about 73 % improvement with respect to the "black aluminum solution"

and 10 % with respect to the CS optimum.

The four design layers of the 3-cluster solution are detailed in Figure 6. For each layer, the elements are grouped per material orientation. The groups of elements sketch the shape of the patches. For example, layer 1 can be covered using a single patch with material orientation 45 • . Layers 3 and 5 could be covered using 2 patches at 0 • , 2 patches at 45 • and 3 patches at 90 • . At this stage of the design process, however, further work is required to obtain a complete definition of the patches. The patch edges at Step 3 are defined by the exterior edges of the element sets and should be simplified without being constrained by the FE mesh. Additionally, in the proposed design method, patch edges correspond to the boundaries of the constant stiffness areas obtained after simplification (i.e.

Step 2). Therefore the final design will display clear thickness discontinuities at the interface between patches, resulting in weak areas subject to failure.

However, as previously mentioned in Section 2, the present work focuses on stiffness-based criteria and the development of a post-processing algorithm for the design of smooth transition between patches is beyond the scope of this work.

Note that the calculation time at

Step 1 is about 3 minutes on a regular laptop (4 Intel Core i7-37440QM

CPUs and 8 GB of RAM). The overall calculation time for Step 2 and 3, including application of the clustering algorithm for k ranging from 1 to 15, continuous optimization of a simplified solution for a chosen number of clusters k and finally laminate retrieval for this design is less than 5 minutes.

Layer 1 (𝜃 1 ) Layer 3 (𝜃 2 ) Layer 5 (𝜃 3 ) Layer 7 (𝜃 4 )

Mass minimization of an automotive composite suspension arm

The second application deals with the mass minimization of a car suspension arm under nodal displacements constraints. The FE model of the suspension arm is shown in Figure 7. The wishbone is made of two symmetrical shells welded together along the plane (O, x, y). Points O and A correspond to the junction of the part with the car chassis. Point B corresponds to the junction with the wheel. Ball joints are used at all three points, thus no bending moments are transferred to the part. The double composite shell geometry provides very high bending stiffness to the part so that global buckling is not a concern. Thus only half of the part is modeled and a symmetry condition is enforced on the (O, x, y) plane. The load F corresponds to a critical design case with respect to the wheel displacement. At point B, the translation along the x-direction is constrained by an upper bound U x (B) UB and the translation along the y-direction is constrained by a lower bound U y (B) LB . The part is made from a T700S/PA66 thermoplastic material with elastic properties E 11 = 104.2 GPa, E 22 = 5450 MPa, G 12 = 2022 GPa and ν 12 = 0.34 (see [START_REF] Todeschini | Analysis of damage and failure mechanisms of Quilted Stratum Process composite parts[END_REF]). The tape thickness is 0.316 mm.

The reference solution is a 12-ply constant stiffness laminate with quasi-isotropic membrane properties.

The mass of the part is about m 0 = 500 g. The aim of the optimization is to design a feasible solution with a mass reduced by 20 %. The first step optimization problem is formulated as follows. The original mass minimization problem with constrained displacements at point B is converted into the minimization of an elliptic function of the displacements at point B under constrained mass, better suited to a stiffness-matching approach.

minimize U x (B)

U UB x (B) 2 + U y (B) U LB y (B) 2 , submitted to m ≤ 0.8 × m 0 . (4)
The design variables are the laminate thickness T and the lamination parameters V A 1 and V A 3 for the membrane behavior and V D 1 and V D 3 for the bending behavior. The laminate thickness ranges from 2.528 mm (8 plies) to 5.056 mm (16 plies). The FE model of the part has about 10,000 shell elements. Using a set of design variables per element at Step 1 would result in a very large number of design variables and feasibility constraints. This seems quite irrelevant with respect to Step 2 which aims at reducing as much as possible the complexity of the optimal thickness and stiffness distributions from Step 1. In order to reduce the number of design variables and the calculation cost, elements are grouped into zones of uniform stiffness properties using the k-means algorithm applied to the 3D coordinates of the centroid of the elements. Here k is determined by specifying a characteristic diameter for the zones, supposing they are circular. In the present case, a 20 mm diameter results in 292 zones. The resulting zones are somewhat similar to Voronoï cells. The optimal thickness and lamination parameters distributions after Step 1 are shown in Figure 8. A simplified partition of the part is built at Step 2, based on the optimum solution achieved at Step 1. The maximal average silhouette value is obtained using 6 clusters. The partition is manually corrected to form compact clusters of zones. A seventh cluster is added in order to enforce constant stiffness around the hole centered on point O. The simplified partition is shown in Figure 9. The figure also presents the optimal design variables distributions after continuous optimization on the simplified partition of the part. Using the simplified partition enables to filter the irregularities observed on the thickness distribution in Figure 8. Zones 3 and 4 reproduce features that are clearly observable in the distributions of the lamination parameters values in Figure 8. Zone 3 corresponds to a laminate mostly oriented in the 90 • direction (V A 1 < -0.8 and V A 3 > 0.6) while Zone 4 is mostly oriented in the 0 • direction (V A 1 > 0.9 and V A 3 > 0.9) . Zone 6 is directly inherited from the thickness distribution shown in Figure 8

Concluding Remarks

The work presented in this paper is a first step towards a composite structure design method specialized for the Quilted Stratum Process. The main challenge was to bring together the great versatility of composite designs offered by the QSP under a single coherent framework, automating and simplifying the design of QSP-manufactured structures for future applications. To this end, the Quilted Stratum Design (QSD) method was proposed. The QSD is a three-step optimization procedure during which the structure is first ideally optimized, then simplified and finally converted into a layered design that can be interpreted for QSP-manufacturing.

The proposed QSD method was successfully applied to both an academic and an industrial application case study. The first level optimization provides ideally optimized variable-thickness and variable-stiffness structural designs used both as starting points and references for the subsequent optimization steps. Considering the QSP manufacturing constraints and the inherent manufacturing complexity of continuously varying thickness and stiffness distributions, partitioning the structure into constant stiffness zones was found to provide good trade-off designs, significantly reducing the structure complexity at the cost of a small reduction in performance. In both our applications, a small number of zones was found to be sufficient to significantly over-perform conventional constant-stiffness designs and well-compatible with the QSP.

With the current state of the method, the solution is described by independent laminates for each constant stiffness areas. Future work should investigate how to retrieve a detailed description of the patch assembly at

Step 3. Two main points have to be taken into account. First, the shape of the patches have to be constrained in order to minimize material wastes during the cutting phase, since for high volume production material waste drives the overall cost of the composite part. Second, discontinuities at the interfaces between patches create weak areas subject to failure. Strength related design guidelines have to be introduced in Step 3 in order to design efficient patch overlapping schemes with respect to failure and avoid matrix-dominated failure modes driven by the coalescence of patch delamination and matrix cracks at the junction between patches.

Figure 1 :

 1 Figure 1: The Quilted Stratum Process

Figure 2 :

 2 Figure 2: Step1. Optimal lamination parameters distributions.

Figure 3 :

 3 Figure 3: Step2. Evolution of the Mean Silhouette Value as a function of the number of clusters.

Figure 4 :

 4 Figure 4: Step2. Clusterization examples. The clusters are identified by the colorbars.

Figure 5 :

 5 Figure 5: Evolution of the critical buckling factor as a function of the number of clusters. Results are presented for each step of the proposed approach. All buckling factors are normalized with respect to λ 0 the buckling load the "black aluminum" solution. Step 1 and Step 2 solutions correspond to stiffness distributions. Laminates are known for Step 3 solutions.

Figure 6 :

 6 Figure 6: Step3. patch assembly for the solution with 3 clusters. Layers 1 is on the surface of the laminate. Layer 7 is close to the midplane.

Figure 7 :

 7 Figure 7: Mesh and loading conditions for the FE model of the upper half of the automotive wishbone. The plane (O, x, y) is a plane of symmetry. Points A and B belong to that plane as well as the load vector F. All translations are blocked at point O. Translations along the y-direction and z-direction are blocked at point A. Translation along the z-direction is blocked at point B.

Thickness

Figure 8 :

 8 Figure 8: Step1. Optimal design variables distributions.

Figure 9 :

 9 Figure 9: Step2. Optimal design variables distributions.

Figure 11 :

 11 Figure 11: Step3. Solution 2. Stiffness matching with balanced laminates. Prescribed (0 • /90 • ) pair of plies on the bottom surface of the part and a (±45 • ) pair of plies on the top surface. Allowed ply orientations are 0 • , ±45 • , 90 • .

Table 1 :

 1 Table1summarizes the results for all three optimization steps. Figure10shows that solution 1 is a close match of Step 2 continuous optimum. Due to the prescribed ply orientations on the surfaces of the part, solution 2 is farther from the target, especially in terms of bending stiffness. Table1shows that the displacements at point B are very sensitive to small stiffness variations. However, both layered solutions obtained at Step 3 significantly reduce the magnitude of the displacement at point B with a 20 % mass saving. Step 3 solutions are detailed in Table2. Overview of the designs obtained after each optimization step, compared with the reference CS design [(90/45/0/-45) 3 ] s .

		Solution	U x (B) U UB x (B)	U y (B) U LB y (B)	Mass [g]
		Reference CS design	0.87	0.88	500
		Step 1	0.67	0.15	392
		Step 2	0.76	0.26	400
		Step 3 (solution 1)	0.83	0.34	400
		Step 3 (solution 2)	0.89	0.53	400
	Zone Solution 1
	1	[90 45 -45 0 0 45 -45 90]

Table 2 :

 2 Step 3. Laminate per zone for solution 1 and solution 2. Figure 10: Step3. Solution 1. Stiffness matching with balanced laminates. Allowed ply orientations are 0 • , ±45 • , 90 • .

	Zone 1	Zone 2	Zone 3	Zone 4	Zone 5	Zone 6	Zone 7
				A xx [MPa]
				D xx [MPa]
	Zone 1	Zone 2	Zone 3	Zone 4	Zone 5	Zone 6	Zone 7
				A xx [MPa]
				D xx [MPa]