A Little Fire of No Consequence Gisèle Pineau Sonia put on one of her white mini-shorts and her pink tank top that had LOVE written in black block letters on it. She wasn't wearing a bra and her small, round, tight breasts and nipples protruded under the taut cotton, similar to unripe mangoes that gave the promise of maturing again and again, until they produced a smooth juice that you would never get tired of having in your mouth. She had squeezed herself into those shorts andshe was only too aware of ither buttocks, which her tight shorts held up high under the denim, aspired to taunt people and tempt the devil. If her old aunt had something to say about her clothes, or her navel that looked scornfully at the world like an arrogant eye rooted in the midst of her belly, she would shorten her visit, free in no time from the burden that had been imposed on her.

Corinne had insisted on the phone: "Please Sonia! I'm asking you to go. You'll be there for an hour or so, tops. Do this for your mommy, please. Go visit Tata Raymonde at the hospital. It'll do her good. She will be so pleased to see you so grown-up after all this time. I'm begging you. God will reward you… Come on! Do this for me, don't forget that she brought me up. I owe her so much! God will reward you a hundred times over, mark my words! If I could've come, I would have taken the plane to stay by her bedside. But you know the situation…" On her way up the morne to the hospital in Pointe-à-Pitre, Sonia was dragging her feet under the lofty midday sun, cursing her mother who always adopted a childlike voice and used sickly pleading to make her cave in, before calling upon the Good Lord as a last resort.

Sonia had arrived at her father's parents' about two weeks before and had no time to lose, as her stay in Guadeloupe would end in less than a month. Her grandma, who was unbelievably permissive and jolly let her come and go as she pleased. Every morning at breakfast she would encourage Sonia to make the most of her holidays, just as she was making the most of her retirement, filling her days with the time-consuming activities of her Senior Citizens Clubaquafit, bridge, English, ikebana, quadrille, macramé, and scrabble… In Sainte-Anne, Sonia had gotten back in touch with her family, especially her cousins. She had grown up under her mother's supervision, far away from them, in Paris, the land of four seasons. Her relatives had welcomed her as if she had always been part of their world. They were delighted to act as her tour guide and took her out every day to discover the island with its unusual rites and its whimsical nature, its deceivingly quiet landscapes, its spicy dishes and its creole dialect teeming with a thousand metaphors.

At the end of the first week, imitating her cousins and their friends, Sonia had muzzled the too reasonable teenager who, day and night, kept her thoughts in check. In less than no time, freed from the moral straightjacket in which her mother had confined her in Paris, she had metamorphosed into a carefree, feverish little creature, eager to taste life's delights and in a hurry to succumb to the temptations of her flesh, which violently twitched under the ravenous gaze of men. She felt both light and tormented, determined and vulnerable. Yet, with a ferocious desire to go through the mirror, to give in to her body's desires and dive at last into the deep end of carnal pleasures that made up most of girl talk.

Aged between fifteen and eighteen, they formed a fascinating little clique of rowdy, thoughtless girls who seemingly enjoyed a life of freedom. They spent their time filling the void of days and nights with their boyfriends, throwing beach parties, setting up improvised barbecues and private parties, driving to nightclubs on the Riviera, skinny dipping in the calm waters of nearby beaches guarded by coconut trees that looked like dishevelled factotums and where furtive shadows would pass through. Such an abundance of activities only served as a pretext for just one aim: the exhilarating game of seduction. A single purpose: sex. At sixteen, Sonia was the only virgin of the group and therefore highly preyed upon; yet she was granted a little respite that certainly would not last. Sometimes she felt as though she had some disability and wondered if, behind her back, she was not the laughing stock of the group, deemed useless, someone who watched the pleasure train pass by while ruminating over the fantasies of a retarded high-school girl.

Would the world be more beautiful once you'd taken the Would life be more joyful right after you'd spread your legs? Sonia was particularly jealous of Candy, a friend of her cousin Jada's. That girl was the most exuberant, the most daring one. She took her time adorning herself to arouse boys and lent her body unconditionally. Her favorite pastime was abandoning her flesh to the ferocious appetite of those young, rabid dogs swollen with pride and testosterone. Submit her odorous crack to the hard-wood sticks between their legs. Candy was passed around, without any qualms. They penetrated her, free of charge. They rammed into her for their own pleasure and for the insane orgasm that surged up at the end of their male exertions that were impetuous, violent and so invigorating, yet always strangely unexpected, like the bite of a scolopendra, a streak of bad luck, a little death.

The girls walked around half nakedbecause of the sun and the heat, they said. Low-cut tank tops, see-through blouses, booty shorts, teeny-weeny mini-skirts and slinky dresses. The body parts that they exposed for everyone to see were both insidious invitations and odes to their raw beauty. Full thighs, slender legs, breasts of exotic Venuses and goddess-like bellies adorned with fake diamond piercings. They devoted hours to their hair, straightening, cutting and dyeing it. Some of them had acquired real expertise in this. They would dexterously sew in hair extensions and plait their hair in braids of all shapes and sizes. The day before, Sonia had observed how Mathilde had planted on her head a load of dreadlocks made in china, threaded in wooden pearls and plastic cowries that were vaguely reminiscent of the girls' African roots. You had to seem them. They strutted around with their cheap lure, enjoying the sensation of the synthetic hair extensions caressing their shoulders and cascading down their backs. Of course, they were all wearing outrageous make-up. Black-penciled eyes, eyelids streaked with green, blue or purple shades. They certainly made no spare on foundation, pressed powder and trendy reds. But their lips, painted over with lipsticks that shouted murder and ever so tender hues of pink, confessed that in the end, behind misleading colors, they were only little girls adorned to be sacrificed in the frenetic dance of their bodies to the first boy who came alongjust so they could call themselves women. And then, the sidelong glances would tell others that they had done it.

Beyond the admiration she had for them, Sonia was confused. Coraline had brought up her first time in a hushed voice. She felt mild pride out of the soft, burning sensation that persisted between her legs. She had confessed a feeling of having been sullied remained nestled inside of her, stifled, stifling. But she had done it again, as if there had been no other road than this tortuous, demeaning one. As if her own pleasure had only been a bonus, an option, a measly tip, that her boyfriend had consented to give her. And, nice as she was, she kept on abandoning her body to him, a body that he always straddled with haste and fury. To keep him on the hook -or so she thought -Coraline acted like a beauty queen, tottered on high heels that forced her to place one foot in front of the other, strolling like a drunken woman. She wore make-up in the fashion of Beyoncé's clones. Swayed her hips before him, just to convince herself that he was doing her good, to assure him that he was her god and that she liked it.

Several boys were hovering around Sonia, but she didn't fancy any of them.

One evening, Mike tried to kiss her furiously in the oppressing twilight of her grandparents' garden. He reeked of cold tobacco and beer. His intrusive tongue twisted around probing the deepest recesses of Sonia's mouth, to the point that she felt queasy. Then, as if her mind had detached itself from her body, she positioned herself as an observer above the stage, waiting for a moment of pleasure -even a fleeting one -to arise from this thorough exploration. Alas, she could have kept on like this for a long time, letting her mouth be open to Mike's saliva. Luckily, her grandmother had saved her by calling out her name from the kitchen window. It was dinner time.

Rosan had taken advantage of a zouk-love dance to hold her close on the dancefloor of a nightclub in Saint-François. While Patrick Saint-Eloi was singing at the top of his magnificent voice… An ka rantré an somey aw Paskè lannuit an ka véyé asi'w Pas an pa nip on dot jwa anko Ké dé viv de vou é mwen pou nou… Sonia had suddenly felt her dance partner's penis harden and rub against her thigh. As she realized that a frenetic urge was growing inside her and, along with it, the perilous desire to give in to this attraction, to surrender, to let herself get carried away, she pushed him back right away. With his big wolfish mouth and his square teeth, Yann had gently sucked and bitten her right breast, away from prying eyes, one afternoon on the beach of the Contrebandiers. At the same time, he had unhesitatingly slipped his hand into Sonia's bikini. His fingers had been as impatient as the legs of a crab that has just seized a dead toad, about to relish it and not wanting to share it with any of his kind.

Lying in her hospital bed, her old aunt looked poorly. She was nearly eighty-five and must have weighed no more than forty kilos. Reluctantly, Sonia bent over her to lay a kiss on her wrinkled cheek. Her eyes were glazed over and her complexion sepia. The big veins running along her arms formed an intricate network of rivers under her withered skin. Her grey hair was thinning and tied up in four pathetic buns. She smelled of urine and cheap perfume. Death seemed to be embracing her already.

The old woman lightened up a little when she saw her appear.

"You look so much like your Manman… I remember when she was your age. You've become quite a young woman…" And then, haltingly, they began chatting about life in France, the harsh winters there and about Guadeloupe caught in the tangle of modern times, the rain seasons that gushed out its load of cyclones… Sonia, with a fixed smile, her hands tightly gripping her cell phone, had not even been there for ten minutes but was already thinking of leaving. All of a sudden, Raymonde went quiet, shut her eyes. Sonia tried to find something cheerful to talk about. But the only words that came to her were about sickness, pain and old age… She looked at her watch, anxious with boredom, assuming that the old lady was falling asleep. At such an old age, every little effort made one feel tired. That's it! Ten more minutes of being patient and she'd be out of that room. She had done her good deed of the day and her mother wouldn't be able to say anything.

"Do you have a fiancé?" Tata Raymonde suddenly whispered. Her voice was slower, quavering. It felt as if she had come back from the Outer World.

Sonia emerged from the boredom that was crushing her.

"Do you have a fiancé yet?" the old woman repeated. Her eyes glistening now.

Sonia lifted her eyebrows.

A fiancé! That was an old-fashioned concept… All that she knew was that it was time for her to have sex. Whatever it would cost her. She was tired of satisfying herself, of trying to reach orgasm on her own. Alone in her bed, with the joy of her body hidden under the sheets, her mind guilty and her senses acute. Alone with her dexterous fingers, her legs open, her short breathing. Alone, caressing herself slowly and imagining faces of men grimacing in the act of possession and pleasure. Images of truncated bodies: muscular male chests, outstretched arms and legs, disproportionate phalluses. Negroes looking like statues of Apollo chiselled in bronze, ebony or ebonite. Powerful young black men who, by means of constraining her, immobilised her under the weight of their bodies, their implacable brutality and their spiteful, pressing need to penetrate her.

Sonia had sacrificed a lunch on the beach to keep the promise made to her mother. She tried to look friendly and cast another glance at her watch on the sly. The old aunt was not delirious… She really seemed to be expecting a reply, and by the looks of it, the conversation had only just begun.

"No, I don't have a fiancé," Sonia blurted out.

Raymonde smiled and heaved a little sigh that was difficult to interpret, because it might mean a thousand and one things or their exact opposite. She seemed to be scanning through her past, recollecting whole chapters of her life, recovering long-forgotten sensations and joys. Then, she turned on her side with great effort. Sonia glimpsed her saggy breasts through the neckline of her nightdress. They had once been full, heavy and probably voluptuous. Now they looked like two old whales stranded on the beach, at the end of life.

"Do you want to hear a story? A story that deserves to be told, at least once. I don't want to take it with me to the grave…" "Nonsense, Tata! You're not going to die any time soon!" Sonia protested.

"You know, my dear girl, a long time ago, I was young, fresh and beautiful like you… Imagine that! I don't know why, but I wanted to get married. I had this stuck in my head, a fixation, a disease. Having a ring on my finger and changing my name, that was all I was asking for in all my prayers to God Almighty. To have myself be called Mrs. So-and-So! To me, it was some sort of crowned achievement, you know, an accomplishment, a blessing. I was a complete idiot and not even sixteen …" "Help!" a little voice shouted in Sonia's mind. "You're going to have to stay here and listen to Tata Raymonde rambling on and on, whether you like it or not." "Well, my prayers were answered on the eve of my seventeenth birthday. Douglas Benoît proposed without even taking time to court me. I did not hesitate, you know. I told my mother 'yes!', I had thought it through, he was the one I wanted and it was God himself who had sent him to me. In double-quick time, I found myself at the Registry Office and before the priest… And I had won: I had become a 'Madame', Madame Douglas Benoît." At these words, the old woman started laughing wickedly at herself. And her bursts of laughter that sounded like jarring jingle-bells came up from her belly to fill her throat and her mouth before they rolled out in the small hospital room, reaching Sonia's ears as a crackle.

"Tell her that you have to go! Make something up! An appointment at the dentist's… Tell her you have a toothache…, the little voice inside her whined, while Sonia sank in her chair, suddenly taking the bait on the story of that rushed wedding. She wanted to know how it ended." "Lord! Do you want to hear about the wedding night?" Sonia nodded.

"We went to bed. Next to each other. I had got myself ready. I'd covered myself with perfume from head to toes and had put on my white brushed cotton nightdress, all new, well ironed, all embroidered down the front. Monsieur was wearing blue and green striped pyjamas. He turned off the light and I waited in the darkness. I was not without knowing that something big was about to happen. Maman had warned me… Something huge was about to pop out from my young husband's crotch. Something stupendous was about to enter my body and turn me first into a woman, and then into a mother…" At that point, a nursing auxiliary pushed the door and put a tray on the small wheeled table. "It's time for your meal, Madame Benoît! Good to see you have a visitor!" the woman cheered. "And a young one at that! This will lift your spirits! You have to eat today, okay?" She turned to Sonia: "How nice of you to visit her! You'll give her a hand, won't you? Are you family? The poor woman doesn't see anybody apart from us." "She comes from France. She is my niece, Corinne's daughter, the one who lives in Paris…" "Ooh, lucky you, Madame Benoît! Alright then, enjoy your meal!" Sonia uncovered the dishes. Grated carrots out of a tin. Steak hâché with giraumon squash. A vanilla flan. "Let's eat, Tata." "Wait! Let me tell you the end of my story. Just give me a little drop of water to freshen my memories, will you? ..." So, we are lying there like two lumps of wood. Time goes by. Nothing happens. I am starting to fall asleep as the waiting persists. Suddenly, he takes my hand and lays it on his underbelly. Aha! I search for something to hold on to. Alas! I come across some sort of slug that slithers out of my fingers. Nothing! No vigour. A helpless thing, as if cursed! Will you believe me if I tell you that I had to get used to it? All my life, trying to make it become erect, encouraging it to become hard, to straighten up. Hours I spent sweet-talking it, coaxing it, taming it.

Exhausting and despairing nights flattering it with my hands, my mouth, my thighs, hoping for a jolt, a bounce, a miracle.

Alas!

But what a nice man he was, Monsieur Benoît! He was a handsome, tall negro and had one of those big smiles. A lot of women were jealous of me, not knowing the truth of the matter. They said that I was lucky to have found a faithful husband. Poor man! Nature had endowed him with a little, flabby plastic toy gun. He could not even fire it once, can you imagine? And he never gave me the least bit of pleasure. No… all he had were his smile, his arms and his courage. He was no lazy man, certainly not. He worked all his life. Built us a beautiful concrete house, with his own two hands and some strong friends.

Yes, he had some good friends… "Friends who wanted to sleep with you? Did they make a move on you?" Sonia asked as if chatting with Jada or Coraline.

"Yes, they did… You've got it. One day, one of them cornered me in the courtyard. Monsieur Benoît had gone to Basse-Terre. A bit of the terrace still had to be finished. That one friend offered his help. He was a man of his word… I had been feeling him staring at me for weeks. He had the eyes of a very soft, innocent sheep that you wanted to stroke and who promised to offer tenderness in return, simply by the look in his eyes. Every time he was around, I felt a little fire starting between my legs. A harmless little fire, yet that could not be extinguished, that burned on, persisted and fought against the headwinds until even after the said friend left. A little fire that kept burning, low and gently in the bed as I lay awake at night next to Monsieur Benoît, sound asleep, certain as he was of having a woman with no desire by his side. You see, what was bound to happen happened." "You cheated on him?" Sonia interrupted, wide-eyed.

"Never in my life! No, I could not do that…" "Oh, you didn't?" Sonia's voice showed her disappointment.

"Wait! The friend cornered me in the courtyard. He tried to get to my mouth and stick his tongue in it. His concrete mixer was making a hellish din. With the little fire between my legs, I told him to follow me in the kitchen. And he started to -" "What?!" "He kneeled before me. I remained standing by the side of the sink where I could keep an eye through the window on the people passing by in the street… He took my panties down, his face disappeared between my legs. It was as if he'd known that that little fire that kept burning there had to be attended to. And he started playing with his tongue, sticking it inside like a child in a pot of jam, licking all the recesses and hidden folds. Moving in and out of the quivering crack with alternating rhythm, speeding up and slowing down. At first, I stayed there with my arms dangling in the air, when I suddenly came out of my inertia. I let the little fire set me ablaze. I then grabbed his head, as if I wanted the whole of it to come inside my body, up that narrow path uncharted by Monsieur Benoît's little, lifeless gun. And I felt that it was coming, all of a sudden, like a big wave that comes towards you full speed, about to break on the beach, that nobody will be able to stop… Even better! A hurricane able to rip up the sky and wreak havoc on the earth in an instant. I felt that it was coming with fury from the innermost parts of my being. And I didn't care if Monsieur Benoît came out unexpected from the end of the street, at the kitchen door. Nothing could prevent me from fulfilling myself, from living this big explosion that poured out this over-brimming sense of myself into a joy that you cannot control and that shows you that you are alive, that you are on this earth to experience this very joy. And that it is within your reach… That it is not reserved to a privileged few… That it is not a sin to take this joy fully in and get pleasure from your body… That it isn't cheating on your husband to offer that crack of yours to the mouth of a Good Samaritan who asks nothing in return, and certainly doesn't want to stir up trouble in your household."

Raymonde heaved a long sigh of contentment.

"I stayed married forty-eight years to my husband. The day he died, he thanked me for having been a good wife. I was the one who closed his eyes. He left in peace, I believe… and unaware of my little fire of no consequence…"

A drizzle started to tap on the window pane.

"Are you hungry, Tata? It's getting cold…" "I had my pleasure, that way… All my life, I had men at my feet. They stuffed their head between my legs and took care of my little fire. They gently sucked me, with such kindness, such patience and such ardour! Up until I could no longer contain my joy and let it flow into their mouths. Not one of them forced me to do anything I did not want to do…" "Do you want to eat a few carrots?" Sonia whispered. Now the old woman's crude confessions were almost unbearable to hear.

"You have to eat! You have to recharge your batteries, Tata…" "What for? I've had my time, you know. I don't have anyone anymore. Don't you worry, I am fine with my memories. I have no regrets. Please, give a kiss to your mother for me. Don't you forget!" When Sonia went down the morne of the hospital that afternoon, she felt that her old Tata's confessions had made her grow up all at once. The rain had stopped. Along the pot-holed pavements, pools of water had formed like shapeless mirrors that reflected shattered suns and, at times, her fragmented shape. She knew she would take her share of joy. She knew she would not be a toy, an obedient doll in the hands of her boyfriends. She knew that deep down inside of her there too burned a little fire of no consequence.


Ludovic Maire

Charbel Moussa

Nathalie Bozzolo

Daniel Pino Muñoz

Marc Bernacki

Marc Bernacki Full Field

Full field simulation of dynamic and post-dynamic recrystallization in 304L steel

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

During hot deformation, plasticity increases the dislocation density while dynamic recovery decreases the dislocation density in the microstructure. Globally, the dislocation density increases. When a critical dislocation density is locally reached, nucleation which is the appearance of new grains free of defects can occur. In parallel, grain boundary migration occurs due to stored energy gradients across interfaces and due to capillary effects. These mechanisms lead to the so-called dynamic recrystallization (DRX) mechanism. When plastic deformation stops, the material is still at high temperature, thus grain boundary migration is still active under the same thermodynamic driving forces. Furthermore, nucleation can also occur in the most hardened grains. These mechanisms appearing right after hot deformation lead to the so-called post-dynamic recrystallization process (PDRX). In the present work, a 3D finite element (FE) model based on the level-set (LS) method in the context of unstructured tetrahedral mesh is proposed to model the DRX and PDRX phenomena in austenitic stainless steel 304L submitted to large deformation and with relatively low computational costs. The LS approach coupled to a remesher provides an accurate tracking of interfaces (i.e. grain boundaries) all along the simulation while phenomenological laws are used for controlling nucleation, work hardening and recovery.

Constitutive laws of the model

Representation of the digital microstructure

The model is based on a LS description of the interfaces within a FE framework [START_REF] Bernacki | Level Set Framework for the Numerical Modelling of Primary Recrystallization in Polycrystalline Materials[END_REF][START_REF] Bernacki | Finite Element Model of Primary Recrystallization in Polycrystalline Aggregates Using a Level Set Framework[END_REF]). First, grain interfaces are virtually generated by using the Voronoï [START_REF] Weaire | On the Distribution of Cell Areas in a Voronoi Network[END_REF] or Laguerre-Voronoï [START_REF] Fan | Simulation of Polycrystalline Structure with Voronoi Diagram in Laguerre Geometry Based on Random Closed Packing of Spheres[END_REF]) method. The virtual interfaces are then described into an unstructured FE mesh thanks to LS functions. Furthermore at the initial stage of the simulation, an energy field considered constant per grain is defined in the microstructure.

Grain boundary migration

At high temperatures, grain interfaces migrate driven by stored energy gradients across interfaces and by capillarity forces induced by grain boundary mean curvature. To simulate grain boundary migration, the strategy presented in [START_REF] Scholtes | 3D Level Set Modeling of Static Recrystallization Considering Stored Energy Fields[END_REF]) is adopted.

Strain hardening and recovery

The strain hardening and recovery mechanisms appearing during hot deformation can be modeled at different scales: at a local scale with crystal plasticity algorithms or at a higher scale with phenomenological laws. In this model, phenomenological laws are used in order to limit the computational cost of the 3D simulations. Considering N G grains in the microstructure, the averaged dislocation density field in each grain j noted 〈𝜌 𝑗 〉 is assumed to evolve according to the Yoshie-Laasraoui-Jonas law [START_REF] Yoshie | Formulation of Static Recrystallization of Austenite Hot Rolling Process of Steel Plate[END_REF]) :

where 𝜀̇𝑝 𝑒𝑒𝑒 denotes the rate of effective plastic strain, K 1 and K 2 are two constants which represent respectively the strain hardening and dynamic recovery term. A superposed dot denotes differentiation with respect to time.

Nucleation mechanism

In the considered model, a nucleus can appear where the averaged dislocation density reaches a critical value noted 𝜌 𝑐𝑐 . Concerning the computing of 𝜌 𝑐𝑐 , a first approximation is made and an iterative calculation is done according to the following equation [START_REF] Beltran | A Mean Field Model of Dynamic and Post-Dynamic Recrystallization Predicting Kinetics, Grain Size and Flow Stress[END_REF]:

where τ is the dislocation line energy, δ(𝜀) ̇ is a strain rate dependent parameter, M b is the grain boundary mobility and ɣ b is the grain boundary energy.

When a new nucleus appears in the microstructure, its critical radius must be high enough so that the growth driving force resulting from the difference in stored energy with the neighborhood compensates for the shrinkage driving force induced by capillarity forces. This condition is approximated by the so-called Bailey-Hirsch criterion [START_REF] Bailey | The Recrystallization Process in Some Polycrystalline Metals[END_REF]:

where ω > 1 is a safety factor ensuring that the created nucleus has the required driving force for growth.

The nucleation rate 𝑉 ̇, representing the volume of created nuclei per unit of time, is calculated according to a variant of the proportional nucleation model of [START_REF] Peczak | The Effect of Nucleation Models on Dynamic Recrystallization I. Homogeneous Stored Energy Distribution[END_REF]:

where K g is a probability coefficient related to the thermomechanical conditions (i.e. temperature and effective plastic strain rate) and φ represents the total boundary area (for necklace-type nucleation) or total volume (for bulktype nucleation) of grains verifying 𝜌 𝑖 >𝜌 𝑐𝑐 .

Simulation of a hot compression

A simulation has been launched on 4*24 CPU processors using the above-described model. Results are presented on Fig. 1. The simulated process is a channel-die compression at 1000°C up to 1.90 strain, at a strain rate of 0.01s -1 . The number of initial grains is 64 while the final number of grains is around 50.000. The final recrystallized grain radius predicted by the model is 6.1μm while experimental investigations have predicted 5μm at the same instant and in similar thermomechanical conditions [START_REF] Beltran | A Mean Field Model of Dynamic and Post-Dynamic Recrystallization Predicting Kinetics, Grain Size and Flow Stress[END_REF]. The microstructure is recrystallized at 96% after 190s of process. The same framework is able to account for post-dynamic evolutions as well.

Figure 1 .

 1 Figure 1. Simulation of a channel-die compression of the 304L steel at 1000°C and at a strain rate of 0.01s-1. Four different instants are represented: (a) ε = 0, (b) ε = 0.65, (c) ε = 1.3, (d) ε = 1.9. The initial number of grains is 64 and the final number of grains is around 50.000.

Acknowledgements

The authors thank the AREVA NP, ARCELORMITTAL, ASCOMETAL, AUBERT & DUVAL, CEA, SAFRAN, TIMET and TRANSVALOR companies and the ANR for their financial support through the DIGIMU consortium and ANR industrial Chair.


Benoit Geller
email: benoit.geller@ensta-paristech.fr

ADVANCED SYNCHRONIZATION TECHNIQUES FOR THE INTERNET OF THINGS

Keywords: Internet of Things (IoT), Single carrier systems, Phase locked loops (PLL), Code Aided (CA) synchronization, Bayesian synchronization

In this paper, we review advanced synchronization techniques for the internet of things. Our study can be directly applied to the single-carrier IEEE 802.15.4 and IEEE 802.15.6 standards. In particular we display the advantages of using a Code-Aided approach and a Bayesian packet-oriented approach. The performances of several phase-locked loop techniques are compared to different Cramer Rao Bounds so that one can see the advantage of respectively the Bayesian approach compared to the on-line approach and of the Code Aided technique compared to the Non Data Aided/ Data Aided approach.

INTRODUCTION

The Internet of Things (IoT) brings a radical change to the telecommunications paradigm. Up to now, the digital world stopped in our digital terminals and most communications involved human beings. With the IoT, all these frontiers will disappear. Any physical process can be measured with embedded sensors and the IoT aims at creating an "Ambient Intelligence" where many wireless nodes gather and process information to control physical processes. The applications are numerous: industrial surveillance and preventive maintenance (factories 2.0), intelligent buildings, environment control and biodiversity mapping, precision agriculture, heath care and medicine, intelligent roads, smart cities, wearables… Wireless sensors are definitely different from other networks on many points :

-If up to now, a few billions of human beings could communicate, it is expected that in a few years, tens of billions of wireless sensors could be deployed. With the IEEE 802.15.4 standard, the number of connected nodes to an internet gateway can theoretically reach up to 65000 items, which is far more than one is used to manipulate in an ordinary wireless network cell.

-The nodes have generally limited calculus capacities and the targeted prices for the electronic chips are several orders less than the one of traditional chips such as Wifi, and even more 4 or 5G.

-The main problem of a sensor network is its life-time. Often a node is abandoned during its deployment and it has to rely on its own battery. The main source of energy consumption of a node is due to telecommunications [START_REF] Pottie | Wireless integrated sensor networks[END_REF] ; for instance, a 2200 mA.h battery node working with a 20mW listening/transmiting consumption will hardly survive more than a couple of weeks ; then the sensor network will have to autonomously insure its self-configurability to bypass the dead relay nodes. Nevertheless, sometimes the sensor network should be able to work reliably during more than 10 years. The only way to reach such a duration is to turn off the radio most of the time with duty cycles inferior to 1%.

In this context, synchronization is of utmost importance. A node should wake up at the right moment to communicate with its environment. In addition, as synchonization is achieved at the front-end before any further processing, it should be reliable enough, not to block any of the following tasks processed by the node. Finally, localization is necessary for more and more applications and localization mainly rely on an accurate synchronization. This is why synchronization has to be accurate even with difficult transmission channels.

There are several kinds of synchronization in each node : packet synchronization, time synchronization, phase synchronization. In this paper, we describe phase synchronization techniques but similar results are available for time synchronization [START_REF] Nasr | Time synchronization in IoT Networks: Case of a WBAN[END_REF]- [START_REF] Nasr | A Soft Maximum Likelihood Technique for Time Delay Recovery[END_REF]. In addition, as for simplicity reasons, most presently deployed sensor networks rely on single-carrier technology, we describe several techniques which can improve the traditional phase locked loop (PLL) techniques, in particular in difficult channels (low signal-to-noise ratio, fast fading,…)

This paper is organized as follows. Traditional phase synchronization is reviewed in the next section. In section 3 we describe the advantage of using a code-aided synchronization. Section 4 describes a very low-complexity block Bayesian synchronization technique. Results are compared to Cramer-Rao bounds. Finally a conclusion is drawn at the end of this paper.

WELL-KNOWN SYNCHRONIZATION PHASE LOOP TECHNIQUES

We assume that a received modulated signal suffering from AWGN noise (with variance 2 n  ) and from an unknown phase offset  is sampled at time k, respecting the Nyquist criterion :

y k = a k e jq k + n k , (1)
                                           y (2)
The Maximum-Likelihood (ML) estimator is a practical popular estimator because if an efficient estimator exists the maximum likelihood procedure will produce it [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]; with model (1), the maximization of the log-likelihood gives:

                  K k j k k n e a y ArgMax p ArgMax 1 2 Re 2 ) ( ln      y         , 0 . Im 2 . Im 2 Im 2 0 ) ( ln ) . ( 2
2 1 2                           j Arg j n j n K k j k k n e e e

ˆ) (

Arg ML   , which is zero when there is no noise. In practice, one does not need to receive the whole observation vector and proceeds sequentially using a stochastic gradient:

               j k k k k k k e a y p p k         Re ) y ( ln ) y ( ln ˆwith 1   k j k k k k e a y     1 Im ˆ     . (4)
However, in telecommunications networks, one often receives unknown information data (Non Data Aided -NDA approach). When the unknown data are assumed to be equally probable and the alphabet constellation is binary A={-1,+1}, the likelihood corresponding to model (1) becomes:

      ) 6 ( ), Re 2 ( cosh 2 1 2 1 ) ( (5) 1 ) , ( ) , ( 2 1 _ 1 2 Re 2 _ Re 2 1 _ 1 2 1 2 1 2 2 2 2
                                                      y a y
so that the maximization of the log-likelihood produces the following sequential procedure :

                 k k j k n j k k k e y e y       2 1
Re 2 tanh Im ˆ. [START_REF] Kay | Fundamentals of statistical signal processing[END_REF] (7) looks like (4) for binary symbols, where the tanh() weight of the updating term belongs to ]-1,+1[ and depends on the signal to noise ratio. When the signal to noise ratio is high, [START_REF] Kay | Fundamentals of statistical signal processing[END_REF] becomes equivalent to the decision feedback loop, whereas when the signal to noise is low, ( 7) is equivalent to the Costas-loop:

  k j k k k e y     2 2 1 Im ' ˆ   
. [START_REF] Brossier | Self Adaptive PLL for General QAM Constellations[END_REF] The sequential updating procedures corresponding to equations ( 4), ( 7) and ( 8) are known as digital Phase Locked Loops (PLL). Not only the PLLs are low complexity algorithms, but they also have an inherent adaptativity feature to time-varying estimation due to the sequential estimation. However, the PLLs also suffer from several drawbacks : -The choice of the step-size might be problematic. If it is chosen too small, the PLL hardly converges and is unable to follow non stationnarities. If it is chosen too large, the asymptotic MSE remains too large. The choice of an adaptative step-size [START_REF] Brossier | Self Adaptive PLL for General QAM Constellations[END_REF]- [START_REF] Geller | Equalizer for Video Rate Transmission in Multipath Underwater Communications[END_REF] can prevent those phenomena but is beyond the scope of this paper.

-Noise can make the PLL diverge from a fixed stable point (Cycle slip problem).

-An arbitrary initialization can make a slow PLL converge (Hang up problem).

-The algorithm can get stuck in a local minimum. To prevent as much as possible, the latest situations the previous PLLs can be improved as seen in the next sections.

CODE-AIDED PHASE LOCKED LOOPS

Clearly, the main goal of a receiver is not to only locally solve the problem of synchronization, but rather to recover reliably some useful data. To achieve this reliability and have a near-Shannon performance, the transmitter is helped by the redundancy of a channel coder. If the synchronization is not good, the reliability of the decoded data will be low and this can be highlighted by the soft information at the output of the decoder [START_REF] Lin | Error Control Coding[END_REF]- [START_REF] Vanstraceele | A Low Complexity Block Turbo Decoder Architecture[END_REF]. Some ad-hoc methods take advantage of the inherent link between the quality of the synchronization and the decoder's output by maximizing the average power of the soft information [START_REF] Oh | Joint decoding and carrier phase recovery algorithm for turbo codes[END_REF]- [START_REF] Vanstraceele | A Block Turbo Phase Synchronization Scheme[END_REF]. Other methods such as [START_REF] Colavolpe | Algorithms for iterative decoding in the presence of strong phase noise[END_REF]- [START_REF] Lottici | Embedding carrier phase recovery into iterative decoding of turbo-coded linear modulations[END_REF] have been devised but suffer from their complexity.

In fact, the PLLs introduced in the previous section can easily be modified to benefit from the decoder's soft output with a moderate complexity increase [START_REF] Geller | System for Compensating Turbodecoder Phase Shift[END_REF]- [START_REF] Zhang | Iterative carrier phase recovery suited for turbo-coded systems[END_REF].

We assume that the observation model is still given by equation (1) but that a soft decoder is able to provide some LLR values k  . If the constellation alphabet A is first supposed to be binary, we have :

) 2 cosh( 2 ) 1 ( 2 k k k e a p       (9)
Introducing [START_REF] Geller | Equalizer for High Rate Transmission[END_REF] in the likelihood, we then obtain : [START_REF] Geller | Equalizer for Video Rate Transmission in Multipath Underwater Communications[END_REF] As in the previous section, we obtain a sequential algorithm by maximizing at each instant, the following quantity: [START_REF] Lin | Error Control Coding[END_REF] so that the derivation gives :

       K k e a y n K k k k n j k k e a y p p 1 _ 2 1 2 2 1 ) , ( ) , (      a y                                        2 2 2 2 Re 2 _ 2 Re 2 _ 2 1 _ 1 2 ) 2 cosh( 2 ) 2 cosh(
  ) Re 2 2 cosh( ) 2 cosh( 2 2 1 2 1 1 _ 2 2 2 n j k k K k k y K n e y e n k                    
                                     j k n k k e y Max y p Max Re 2 2 cosh ln ) ( ln 2
                           j k n k j k n k e y e y y p Re 2 2 tanh Im 2 ) ( ln 2 2 (12)
We then obtain the Code-Aided PLL updating :

    ) 13 ( Re 2 2 tanh Im ˆ2 1                      j k n k j k k k
e y e y k Equation ( 13) is very similar to equation [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]. The only difference lies in the argument of the tanh() weight. In the case of equation ( 13), the weight does not only take into account the SNR, but also the reliability given by the decoder's output.

Fig.1. Convergence of the CA synchronizer in 2 iterations

In practice and as illustrated by figure 1, a CA PLL needs (at least) two iterations. When processing the received data at the first iteration, the synchronization device has no information from the decoder's output so that with LLR values equal to zero, equation ( 13) is equivalent to equation [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]. Then the receiver is able to demodulate its data and provide LLR values which are fed back at the front-end synchronizer working with equation ( 13). This then gives a more reliable phase estimate vector; it can be seen that sometimes two iterations are sufficient to reach the Cramer-Rao bound [START_REF] Kay | Fundamentals of statistical signal processing[END_REF]. In general, some Exit chart analysis [START_REF] Lin | Error Control Coding[END_REF] should be made to forecast the convergence conditions but in many situations two or three iterations (between the decoder's output and the PLL's input) are sufficient to obtain a good convergence of the CA phase synchronizer. Clearly with CA synchronization, the synchronizer can take advantage of the decoder soft information as long as the decoder works correctly; this allows some improvement, when compared to NDA mode, from medium to (near Shannon) low SNRs. The previous algorithm described by equation ( 13) can readily be extended to a QAM constellation alphabet A of size M. By proceeding analogously, we obtain:

  k k j k k y QAM k k e y a E     / 1 ) ( Im ˆ    , (14)
where the mean soft symbol is obtained by : 14) is equivalent to equation ( 4) for the DA case; also, we can simplify equation ( 14) with a decision directed loop at high SNR. In short, the CA mode allows to achieve a better spectrum efficiency, when compared to the DA mode, and a better performance at lower SNR than the NDA mode.

) ( ) ( / v QAM a v k y QAM a p a a E v k    (15)      QAM a k k k v k k k v v v y a w y a w a p ) , , , ( ) , , , ( ) ( with

LOW COMPLEXITY BAYESIAN SYNCHRONIZATION

The Bayesian approach is a natural choice when a parameter is time-varying because this feeds some a priori to the estimation process. We add the following Brownian phase model to the observation model (1) : 1) and ( 16) is neither linear, nor stationary. The MAP approach which weights the likelihood can then take advantage of the natural a priori introduced by equation ( 16):

 k =  k-1 +  + w k , (16)
        ) , ( ln max arg ) , ( max arg ) 17 ( ) ( ) ( max arg ) ( / ) ( ) ( max arg ) ( max MAP p p p p p p p p     
Bayesian (resp. Hybrid) Cramer-Rao bounds were created to measure the performance of a random (resp. random and deterministic) parameters [START_REF] Kay | Fundamentals of statistical signal processing[END_REF] [START_REF] Bay | On the Hybrid Cramér-Rao bound and its Application to Dynamical Phase Estimation[END_REF]. These bounds indicate that there is room for a good estimation improvement brought by the Bayesian approach [START_REF] Yang | Bayesian and Hybrid Cramer-Rao Bounds for QAM Dynamical Phase Estimation[END_REF]- [START_REF] Yang | Bayesian and Hybrid Cramer-Rao Bounds for QAM Dynamical Phase Estimation[END_REF], [START_REF] Yang | Bayesian and Hybrid Cramer-Rao Bounds for the Carrier Recovery under Dynamic Phase Uncertain Channels[END_REF]. Different estimators were devised to approach those bounds but they usually suffer from form a complexity brought by the Bayesian canvas [START_REF] Colavolpe | Algorithms for iterative decoding in the presence of strong phase noise[END_REF], [START_REF] Dauwels | Phase Estimation by Message Passing[END_REF]- [START_REF] Wymeersch | Iterative Receiver Design[END_REF]. We now see that the previous PLLs can be easily adapted to the Bayesian context at a very moderate complexity price [START_REF] Yang | Bayesian and Hybrid Cramér-Rao Bounds for the Carrier Recovery under Dynamic Phase Uncertain Channels[END_REF]. We start with a binary constellation set A. With the model given by ( 1) and ( 16), the joint probability density can be written as:

) / ( ) / ( ) / ( ) ( ) ( ) / ( ) , ( 1 2         k k K k k k p y p y p p p p p      θ θ y θ y (18)
where:

 

               
In order to find the maximum of equation ( 18), we set

  0 ) , ( ln ) (     MAP k k k p    θ y
Assuming no initial a priori on  1 , we then find: 21) is just like equation ( 7), whereas equation ( 22) is similar to equation ( 21) with the role of the indexes k+1 and k being inversed. In other words, the approximate MAP estimator of equation ( 20) is just the mean of a forward PLL given by equation ( 21) and of a backward PLL (working from the end of the block towards its beginning). In practice, we can first run an ordinary PLL, then, at the end of the block, we run a backward PLL towards the beginning of the block, we repeat this procedure until convergence (in practice 2 or 3 forward-backward recursions), and in the end, we average at each time index, the last forward and backward estimates to obtain the approximate MAP estimator as described by equation [START_REF] Geller | System for Compensating Turbodecoder Phase Shift[END_REF]. Just like in [START_REF] Cochran | Carrier Phase Synchronization by Reverse Playback[END_REF], the initialization has a lower impact because of the various forward and backward iterations, but differently from [START_REF] Cochran | Carrier Phase Synchronization by Reverse Playback[END_REF], the MAP performance improvement is brought by the averaging between the two trajectories. 20)-( 22) can be generalized to any coded QAM systems [START_REF] Yang | Smoothing PLLs for QAM Dynamical Phase Estimation[END_REF]. Proceeding analogously, we obtain: [START_REF] Vanstraceele | A Low Complexity Block Turbo Decoder Architecture[END_REF]. The performance of such an algorithm after N S-PLL =3 iterations is displayed in Figure 3 for a (possibly turbo-coded) 16 QAM signal. The advantage of the MAP (Off-line) estimation over the ML (On-line) is very clear with a MSE gain of several dBs for any SNR. In addition, the MAP estimator performance is very near from the HCRB and one recovers that the CA mode stays near the bound for lower SNRs than the NDA mode.

                                                       K k e y e y K k k e y e y F K F K B B j K n j K n w F K B k F k j n j n w B MAP k                  Îm Re 2 tanh 2 ˆ) 20 ( 1 2 2 1 1 Îm Re 2 tanh 2 ˆ) ( 1 ) ( 1 ) ( 2
) ( 2 1 2 1 2 2 ) ( 1
) ( ) ( 2 2 2 2 2 ) ( 2
) ( with:         (22) Im Re 2 tanh ând (21) Im Re 2 tanh ˆ) ( 1 ) ( 1 ) ( 1 ) ( 1 2 2 2 ) ( 1 ) ( 2 2 2 ) ( 1 )
                                  Equation (
                                K k e a E y K k k e a E y F K K B j QAM K n w F K B k F k j QAM n w B MAP k              ) ( Im 2 ˆ) 23 ( 1 2 2 1 1 ) ( Im 2 ˆ) ( 1 1 ) ( 2 1 2 2 ) ( 1
) ( ) ( 2 2 2 2 ) ( 2 ) ( with :     ) ( 1 ) ( 1 1 1 2 2 ) ( 1
) ( 1 1 2 2 ) ( 1 ) ( ) ( Im ˆ) ( Im ˆB k F k j k QAM k n w B k B k j k QAM k n w F k F k

CONCLUSION

In this paper, we have presented some single-carrier synchronization algorithms that are able to reach the Cramer-Rao bounds in difficult conditions (phase noise) with a very low complexity. These algorithms are well suited to limited devices as sensor nodes of the IoT. Some future work aims at including the synchronization in a crosslayer context [START_REF] Zhou | Multi-user video streaming over multiple heterogeneous wireless networks: a distributed, cross-layer design paradigm[END_REF]- [START_REF] Akyildiz | Can Vuran "Wireless Sensor Networks[END_REF], so as to ease up some localization functionality, often useful with the IoT.

  where the phase at time k is modified by a linear drift  and some Gaussian noise w k of variance 2 w  . This noise takes into account the poor oscillator's quality as encountered in sensor networks. The phases are stacked in . Note that the model described by equations (

Fig. 2 .Fig. 3

 23 Fig.2. Comparison of the Forward and of the Forward-Backward MAP estimation performance

  .0002rad 2 ,  = 0.003rad, N S-PLL = 3 ) NDA Sim (On-Line) CA Sim (On-Line) DA Sim (On-Line) NDA Sim (Off-Line) CA Sim (Off-Line) DA Sim (Off-Line) DA HCRB (Off-Line)