"Be Careful What You Wish For: Geoffrey Chaucer"s Object of Desire" Jonathan Fruoco Université Grenoble Alpes, ILCEA4 "Objects of Desire" -International Conference at Lille Catholic University, 24-26 May 2018

If there is one concept that we might consider central to medieval poetry, it is desire; whether it is celestial bliss for theologians, love for courtly love poets or fame for many aspiring artists. In this lecture, I propose to focus on England"s most famous courtly love poet, namely Geoffrey Chaucer. But it is not love that will interest me today, but a different object.

If trying to get is indeed, as Elizabeth Anscombe remarked, the primitive sign of wanting, then Chaucer was, despite his use of self-mockery and his modesty, as guilty as anyone of looking for his fifteen minutes of fame. He was no Petrarch, who turned self-promotion into an art, but we can still find in Chaucer"s life and poetry the signs of a wish to leave the shadows of Aldgate, where he worked for so many years as a controller of customs, and to be heard. Yet, Chaucer was also profoundly critical of the desirability of fame. I will not be psychoanalyzing a poet who has been dead for 600 years, but one may wonder if the ambiguity of his desire provided the poet with a motive for action or if the subsequent realization of his desire was merely an accident that would have struck Chaucer as deeply illusory. For if desire was indeed at the heart of medieval poetry, it is, in fact, its unattainability that made it philosophically relevant and artistically inspiring.

The poem that most clearly illustrates Chaucer"s most intense reaction to the object of his desire is The House of Fame, which he composed sometime after his return from Florence in 1373. It is during his time in Italy that Chaucer discovered Dante"s poetry, and most notably the Divine Comedy, but as important as this particular poem might have been for Chaucer and the history of English literature, one cannot help but notice that both poets had a completely different understanding and vision of the world. More importantly, they apparently did not share the same approach to fame-seeking. Dante had indeed greatly developed and written his own legend with the Comedy, while Chaucer seemed to be profoundly critical of this need of recognition.

The House of Fame invites us to follow a narrator unsurprisingly named Geffrey within a dream vision: the poet"s persona wakes up in the temple of Venus and finds himself face to face with paintings illustrating various scenes from the Eneas. He goes out of the temple, realizes he is lost and is then snatched by an eagle that takes him to the House of Fame, where the poet will strongly criticize the vanity of other artists and underline the very limits of the human mind and its implication in the creative process. The eagle is undoubtedly the most significant motif borrowed by Chaucer to Dante, who meets this golden-feathered eagle in the ninth canto of the Purgatorio. Yet, although the eagle is in appearance quite similar in both poems, it becomes quite clear here as soon as it starts talking that Chaucer had no intention of following Dante any further. The flight of the eagle had a deep theological and philosophical meaning in the Comedy for it helped the narrator get physically and spiritually closer to God and Heaven. But in Fame, it has a radically different role since it will systematically be used by Chaucer to present an ironic counterpart to Dante"s attitude. Faced with elevation, Chaucer chooses a lowering process, an artistic principle at the very heart of grotesque realism and of carnivalesque laughter. When the eagle takes off with Geffrey in its claws, for instance, it gently reminds the narrator how fat he actually is: "Seynte Marye, / Thou art noyous for to carye!" (II. l.572-574). And whereas Dante wondered if he was worthy to follow Virgil, noting that he is neither Eneas, nor Saint Paul (Inf. II, l. 32), Geffrey asks himself: The eagle obviously has its own opinion of the question and answers "Thow demest of thyself amys" (II. l. 596). Jupiter has no intention whatsoever of turning poor Geffrey into a star. As we can see, Chaucer adopts an openly ironic attitude and does not hesitate to mock Dante. He does not share the Tuscan"s interest for fame and even seems to show true contempt for such a need for attention.

When Geffrey finally arrives on the steep pick upon which is built the house of Fame, he notices that the House"s foundations are made out of ice. Describing the frailty of those foundations, Geffrey remarks that the builder has very little reasons to boast (III. l. 1128(III. l. -1135)). Worst of all, the ice, covered with the name of famous people is starting to melt, thus erasing the inscriptions that cover its surface. When Geffrey finally moves closer to the House, his vision illustrates the very state of literature during the XIV th century. Poetry had indeed reached a key moment in its development, half-way between oral and written transmission. Chaucer was more than anyone else conscious of this duality and expresses it in his description of the house of Fame. The exterior is thus dedicated to oral culture: not only does Geffrey see Orpheus, Orion and other legendary harpists playing music, along with musicians of lesser rank, but he also meets magicians, illusionists, and soothsayers (III. l. 1201-1281). Besides, even though the inside of the temple is dedicated to written literature, Chaucer keeps reminding us that the two forms of expression are complementary. Geffrey hears, for example, the poets who immortalized the Trojan War argue about the Iliad, with some spirits defending Homer, while Trojan supporters accuse him of having favored the Greeks (III. l. 1477-1480). Despite their glory and fame, these artists are condemned, as souls wandering in the limbos, to pass the eternity in conflict with one another. As a result, when a spirit asks Geffrey what is his desire, he answers that he has not entered the house looking for Fame, on the contrary, he had rather "[t]hat no wight have my name in honde" (l.1877) after his death. He is aware of his own value and does not need to be admired. He seems thus far to have Dante"s pretension and makes sure that we know it. Whereas Dante had prayed Apollo to be worthy of his laurels (Par. I, l. 13-18), Geffrey explainsnot without self-mockerythat he would be happy to just kiss Apollo"s tree (III. l. 1091-1108). He then guides us out of the House of Fame and straight to the House of Rumor, which is described by the narrator as a whirling wicker cage where gossip is filtered. The air going through this maze produces a whistling reminding Geffrey of the sound of a stone being launched by a catapult (l. 1916-1955).

In the end, the news, or "tydings" going from the House of Rumor to the House of Fame are merely verbal creations. The eagle had explained Geffrey that sound "ys noght but eyr ybroken" (II. l. 765): the words uttered in our world are, in other words, compressed air reaching the Houses of Rumor and Fame and there taking a physical shape. But Chaucer"s description of sound as "broken" air, that is to say of form of flatulence, gives us once more a pretty good idea of what he thinks about fame. He does not hide his aversion for other artists" fame-seeking and tries to represent himself as a discreet man, writing merely for his own pleasure.

But although part of this representation of Chaucer is probably true, we must ask ourselves what does that tell us about the artist? Did Chaucer really despise fame so much in his early career, or should we perceive this mockery as a sign of envy? If we look at the archives, we notice that the poet was a relatively minor figure in his own time. Amongst the 493 official documents gathered in the Chaucer Life-Records, none mention his career as a poet and it seems that his most powerful supporters at the court were none other than his wife Philippa, one the Queen"s domicella, and his sister-in-law, Katherine, mistress and future wife of John of Gant. Chaucer was, in other words, evolving on the fringes of the court. His artistic career evolved in the shadow of his official missions: as a true Juno, Chaucer presented himself to the world with two faces: one secret and artistic; the other public. But this duality must have been terribly frustrating for such a talented poet. His only audience was probably a small group of friends, yet his poems surely deserved the attention of the court. The Book of the Duchess was, in this regard, clearly written with this aim in mind: shortly after the death of Blanche of Lancaster in 1368, Chaucer composed this dream visionhis first major composition in Englishin which he symbolically attempts to comfort Blanche"s husband, John of Gant. Why would Chaucer then try to attract the attention of one of the most powerful men in England, if fame was so inherently despicable to him? Was he only trying to be rewarded by the prince? If so, being paid for his poem would mean being officially recognized as a poet, and not merely as a civil servant. The fact that he remained anonymous after the composition of Duchess might have profoundly disappointed him. His subsequent visit to Italy, and more precisely to Florence, could, therefore, be considered as an aggravating factor. Indeed, Chaucer discovered in Florence a city-state in which artists and poets were celebrated and respected, where art was loved. One can easily imagine the shock felt by a man coming from a country that was still to make its first steps towards the Renaissance. England was in perpetual war with France and its court sadly more easily impressed by feats of arms and military exploits than by beautiful poetry. As Judith Butler then asks, "what sort of journey is desire if it leads to such an impasse? And what sort of vehicle is desire? Does it stop elsewhere before reaching its mortal destination?" (Butler, 2011, 17) Chaucer"s reaction in The House of Fame becomes then easily understandable, for the desiring subject follows a journey made of illusions and failures, but also of occasional moments of acknowledgment perceived as a source of temporary redemption. Chaucer, however, was yet to experience such moments. Having failed so far to fulfill his desire, he had no other choice but to try to domesticate it. He had to desire to do something with desire, whether it is controlling it or silencing it, if he were to find some sort of inner harmony (Butler, 2011, 22).

A moment of temporary redemption eventually presented itself to the poet with the composition of Troilus and Criseyde in 1385-86, some twenty years after The Book of the Duchess. Despite his attempt to silence his desire for fame, Chaucer found himself in a situation that would have profoundly surprized the Geffrey wondering in the House of Fame.

When he started translating Boccaccio"s Filostrato, Chaucer probably did not anticipate that he was about to produce one of English literature"s most important work. As Paul Strohm remarks, Chaucer responds in an agitated way to an idea that appears to have just occurred to him, or that is striking him with unprecedented force: He has written a poem deserving of broad admiration, a poem that may circulate to present and future audiences in manuscript form, and thus be read without his superintending presence and beyond his personal control (2014,209). This sudden realization of the value of his poetical production was bound to unravel the control he had over his desire. His attitude towards the circulation of his poetry slowly began to change as he realized that to reach the audience he deserved, he had to embrace a brand new method of diffusion. Up until that moment, he only ever dealt with a small audience, which strongly limited the fulfillment of his desire but granted him control over the reception of his work. Yet, the larger diffusion of his poetry in manuscript form implied losing the control he had during public readings. Although he strongly desired recognition, he did not want his fame to be conditioned by a falsification of the real produced by a bad diffusion of the transmitted poetical information. He managed, however, in the conclusion of Troilus and Criseyde to let his "litel bok" go and seek its own fame. If Chaucer had previously thought himself unworthy of Apollo"s laurels, he now started to reflect upon his place in a literary tradition going back to the classics of the Antiquity: But litel book, no making thou n"envye, But subject be to alle poesye; And kis the steppes wher as thow seest pace Virgile, Ovide, Omer, Lucan, and Stace. (V. l. 1789-1792) He does not take the title of auctor but presents himself in these lines as a possible heir to those great names. He remains worried by a bad diffusion of his manuscripts (l. 1793-1798) and preys that his poetry might be understood properly and not perverted during its transmission.

Looking at Chaucer"s career, we can easily define him as a Hegelian subject on a journey of desire based on experience. Chaucer evolved as he grew older and experienced failures and successes: if we put it in philosophical terms, the subject that encounters an object or experiences an external and ontologically disparate element of the world cannot be considered identical to the subject that eventually sees himself reflected in those external phenomena (Butler, 2011, 29). The subject needs, in other words, some form of auto-reflexion or he will keep on considering himself as more limited than he potentially is. Chaucer"s frustration after the composition of The Book of the Duchess and the subsequent silencing of his desire in The House of Fame could thus be seen as the result of a lack of auto-reflexion. The change of perception seen in Troilus and Criseyde shows, on the other hand, an evolution one might consider as the result of the poet"s reflexion on his relation with the world. Experience allowed him to reach a broadened conception of his poetry and of his place in the literary world. But it is important to understand that the Hegelian subject is not a fixed subject traveling from one ontological position to another; on the contrary, he embodies his own journey and is each position in which he finds himself. Desire is in this regard the very principle of the reflexivity of consciousness, and Chaucer"s own journey through dissatisfaction shows how the Hegelian subject internalizes the world he desires and deploys himself to integrate external elements perceived as Other. The final satisfaction of desire is, to quote Butler, the "discovery of substance as subject, the experience of the world as a confirmation of the feeling the subject has about the immanence of his metaphysical position" (2011, 29). Chaucer"s ontological position at the end of his career shows the progress of his journey. His ambitions and his desire were freed from any form of domestication and he managed to adapt his creation to the dramatic changes he faced in 1386, when he suddenly lost his work as controller of customs, his lodging at Aldgate, and was forced to leave London, effectively cutting him from his audience. But instead of succumbing once more to frustration and dissatisfaction, he integrated those changes and these new experiences; he internalized his new position and used it as an inspiration in the creation of his magnum opus, The Canterbury Tales. If he could not reach his audience, he would create one in the form of a group of pilgrims, and travel alongside them. When Hegel wrote that "self-consciousness is desire" ( §167, p.164), he underlined the importance of the reflexivity of consciousness in desire. Consciousness has to become Other, it has to become self-consciousness. It provides the Hegelian subject with a mediation allowing him to understand his own structure. The subject must become Other and be outside of himself, and this reflexive movement is necessary for the subject to know himself. Chaucer had previously used avatars in his other poems, but in The Canterbury Tales he appeared outside of himself inside the narrative as the Chaucer pilgrim, whose poetry is both admired and criticized by the other characters. The exotopic perception of himself as an artist, this self-consciousness of his own successes and failures as a poet led him to the end of his journey of desire, and to its mortal destination.

Chaucer"s death in 1400 prevented him from witnessing the larger diffusion of his work.

Though unfinished, The Canterbury Tales became his most famous creation, the one that would ensure his reputation. It is difficult to imagine what would have been the poet"s reaction to the realization of his desire: it might have struck him as deeply illusory, as he so eloquently remarked in the poetry composed after his return from Florence. Or, he might have found some sort of peace, for when we desire, we question the metaphysical situation of human identity; the satisfaction of this desire gives us an answer to that question (Butler, 2011, 30).

Chaucer was discreetly buried in Westminster Abbey. No matter what he might have thought of the end of his journey of desire, he probably never anticipated that his grave would become the foundation of the Poet"s Corner, where he would later be joined by Edmund Spenser, Alfred Lord Tennyson, John Dryden, and many others. After having fought his desire, he became the founding member of his very own House of Fame, showing once more that you really should be careful what you wish for.


Simon Meunier

Loic Queval
email: loic.queval@centralesupelec.fr

Arouna Darga

Philippe Dessante

Claude Marchand

Matthias Heinrich

Judith Cherni

Vido Lionel

Bernard Multon

Lionel Vido

Bernard Multon Satie

Influence of the temporal resolution of the water consumption profile on photovoltaic water pumping systems modelling and sizing

Keywords: Photovoltaic, Pumping, Temporal resolution, Water consumption

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

I. INTRODUCTION

In Sub-Saharan Africa, more than 300 million people use unimproved water sources and most of them live in rural areas [START_REF]The Millennium Development Goals Report[END_REF]. This unimproved water access is associated with significant ill-health and time losses [START_REF] Howard | Domestic water quantity, service level, and health[END_REF] [START_REF] Bartram | Hygiene, Sanitation, and Water: Forgotten Foundations of Health[END_REF]. Amongst the different alternatives for improving water access, photovoltaic (PV) pumping constitutes a suitable solution especially for offgrid rural areas. Indeed, photovoltaic water pumping systems (PVWPS) are economically competitive in rural areas [START_REF] Singh | Solar Powered Water Pumping System Employing Switched Reluctance Motor Drive[END_REF], reliable [START_REF] Singh | Solar Powered Water Pumping System Employing Switched Reluctance Motor Drive[END_REF] [START_REF] Chergui | Strategy photovoltaic pumping system in scattered area[END_REF] and maintenance needs are low [START_REF] Mapurunga Caracas | Implementation of a High-Efficiency, High-Lifetime, and Low-Cost Converter for an Autonomous Photovoltaic Water Pumping System[END_REF] [START_REF] Jha | Application of solar photovoltaic system in Oman-Overview of technology, opportunities and challenges[END_REF].

The modelling, design, control and application of PVWPS have been extensively discussed in the literature. Components which are well adapted to photovoltaic water pumping have been designed, such as the motor drive using PV transistors detailed in [START_REF] Quéval | A Switched Reluctance Motor Drive using Photovoltaic Transistors: Principle, Prototype, Experimental and Numerical Results[END_REF]. Several control strategies of PVWPS have been investigated such as the parallel distributed compensation control [START_REF] Ouachani | Optimization and control for a photovoltaic pumping system[END_REF] and the fuzzy logic control [START_REF] Bahloul | Effective Fuzzy Logic Control of a Stand-alone Photovoltaic Pumping System[END_REF]. Krim et al. also investigated the use of the Field-Programmable Gate Array (FPGA) to control the converters of PVWPS [START_REF] Krim | FPGA Contribution in Photovoltaic Pumping Systems: Models of MPPT and DTC-SVM Algorithms[END_REF], Koreboina et al. studied the switched reluctance motor Pulse Width Modulation (PWM) control for PV water pumping [START_REF] Koreboina | Performance Evaluation of Switched Reluctance Motor PWM Control in PV-fed Water Pump System[END_REF] and Akeyo et al. proposed a strategy to control an array of pumps [START_REF] Akeyo | Multi-MW solar PV pumping system with capacity modulation and battery voltage support[END_REF]. As to cases of applications, Yahyaoui et al. studied the use of PVWPS for tomato irrigation [START_REF] Yahyaoui | An Algorithm for Sizing Photovoltaic Pumping Systems for Tomatoes Irrigation[END_REF] and Sodiki investigated the application of PVWPS for agriculture and domestic water access in Nigeria [START_REF] Sodiki | Solar-powered groundwater pumping systems for Nigerian water sheds[END_REF].

Most of the PVWPS models considered in previous studies compute the pumped water volume using climatic data (irradiance, temperature) as only input and then compare this pumped volume to the water volume required. However, to our best knowledge, none of available data validated models has taken the water consumption profile as an input. Such overlook represents a problem for PVWPS which include a tank and a controller that stops and restarts the pump depending on the water level in the tank. Indeed, for these systems the water consumption profile has an impact on the water level in the tank and therefore on the evolution of the pumped flow rate. We have developed and validated a PVWPS model which takes into account for the first time, to our knowledge, water consumption as an input. This model allows to simulate the operation of PVWPS with the architecture mentioned above, which are very common in rural areas, and it permits to optimize the sizing of these systems.

As water consumption is included as model input, it is important to determine the temporal resolution required for this input. Indeed, this resolution may influence the model accuracy and therefore the results of sizing studies that are based on this model. Currently, the greatest majority of studies on communal PV water pumping in rural areas consider the total volume of water consumed per day. In addition, no study regarding the influence of the water consumption profile temporal resolution on PVWPS model accuracy has been encountered. However, similar studies have been found regarding the influence of the electrical consumption profile temporal resolution on the accuracy of PV-battery systems models and on the optimal sizing of these systems [START_REF] Notton | Autonomous photovoltaic systems: Influences of some parameters on the sizing: Simulation timestep, input and output power profile[END_REF] [START_REF] Hoevenaars | Implications of temporal resolution for modeling renewables-based power systems[END_REF]. These latter systems are close to PVWPS with tank as the battery plays the role of the tank and the load profile can be assimilated to the water consumption profile. For PV-battery systems, Notton et al. advice to avoid time steps larger than 1 hour for the electrical consumption profile [START_REF] Notton | Autonomous photovoltaic systems: Influences of some parameters on the sizing: Simulation timestep, input and output power profile[END_REF]. Hoevenaars and Crawford showed that using simulation time steps of 1 hour instead of 1 minute had a small effect on the PV-battery systems optimal sizing [START_REF] Hoevenaars | Implications of temporal resolution for modeling renewables-based power systems[END_REF]. However, the simultaneous variation of the temporal resolution of the irradiance and of the electrical consumption is performed in this study, which prevents from isolating the effect of the electrical consumption profile temporal resolution.

In this article, we study the impact of the temporal resolution of the water consumption profile on the PVWPS model accuracy and on the optimal sizing. This provides information for selecting time steps for data acquisition, modelling and optimization that save data storage capacity and computational time without greatly impacting the results.

The PVWPS model and the test system to which it is applied are described in section 2; the different temporal resolutions of the water consumption profile are presented in section 3; the effects of the change of the temporal resolution on PVWPS modelling and sizing are detailed in sections 4 and 5 respectively.

II. PHOTOVOLTAIC WATER PUMPING SYSTEM MODEL

The PVWPS model developed is presented in detail in [START_REF] Meunier | Modélisation et validation expérimentale d'un système de pompage photovoltaïque dans une communauté rurale isolée du Burkina Faso[END_REF]. This model applies to systems with the architecture presented in Fig. 1. In this architecture, the controller regulates the energy provided by the PV modules to the motor-pump according to two set points of the water level in the tank, which is obtained by a float switch. Pipe assemblies PA1 and PA2 are two piping systems that link the motor-pump to the tank and the tank to the fountain respectively. The inhabitants collect water at the fountain. The model is used to simulate the operation of a real PVWPS installed in the rural village of Gogma in the centereastern Burkina Faso (GPS coordinates: 11.724586; -0.572290). The system was set up in December 2017 by the company DargaTech, based in Ouagadougou, in collaboration with our research groups. In Gogma the inhabitants mostly work on agriculture for less than 1 $ per day and houses neither have access to electricity nor to piped water. The PVWPS provides water to approximately 250 people each day for 4 types of domestic uses: drinking, cooking, body hygiene and laundry. Fig. 3 shows this PVWPS. There are 750 Wp of PV modules, which corresponds to a total surface of 3.9 m 2 , the motor-pump is immersed at 30 m underground, the volume of the tank is 9.1 m 3 and there are three taps at the fountain. The average daily volume collected by the users is 8.7 m 3 .

In Gogma, the inputs and the output of the model have been measured from January 2018 onward by using a data logger that we developed. Here we use measurements separated by a 1 minute time interval. The comparison between the measured pumped flow rate and the one simulated by the model for several periods of the year allowed to validate the model experimentally. Fig. 4 shows this comparison for the 22 nd of February 2018. On this figure, the interruptions in the pumped flow rate profile correspond to the moments at which the tank is full and show that Gogma's PVWPS is oversized. In order to observe the impact of a decreasing temporal resolution of the water consumption profile, we build several profiles from the measured dataset. To do so, we replace all the values of water consumption flow rate in a time interval T by the average of those values. This averaging corresponds to acquiring data from a water meter at a given time interval T. Fig. 5 presents the water consumption profile measured by the data logger and the profiles obtained with some selected time intervals, for the 22 nd of February 2018. In total 15 time intervals, ranging from 1 min to 1 day have been considered for this study and for all of these intervals the total volume consumed during the day is the same.

IV. EFFECT OF THE TEMPORAL RESOLUTION ON MODEL

ACCURACY

The impact of the temporal resolution on the model accuracy is evaluated by computing the pumped flow rate from the PVWPS model for different temporal resolutions of the water consumption profile. Fig. 6 compares, for the 22 nd of February, the pumped flow rate measured by the data logger to the pumped flow rate computed by the model for a temporal resolution of the water consumption profile of 1 hour. The initial water level in the tank is equal to the one at the beginning of the 22 nd of February, 2.0 m. Fig. 7 performs the same comparison as Fig. 6 but a temporal resolution of 1 day is considered. We observe that the larger the time interval used for the water consumption profile, the higher the mismatch between the measured and the modelled pumped flow rate.

To quantify the loss of accuracy, the root mean square error (RMSE) between the measured and simulated pumped flow rate is computed for each temporal resolution. This RMSE is normalized by the maximum measured pumped flow rate, which is equal to 1.2×10 -3 m 3 /s on the 22 nd of February. The results on the normalized RMSE for the 22 nd of February are presented on Fig. 8. We observe that, for the 22 nd of February, the normalized RMSE is multiplied by 3 as the temporal resolution goes from 1 minute to 4 hours. Moreover, the first sharp increase of the RMSE take place for time intervals strictly larger than 20 minutes. The same study was performed for other days of the data set and trends similar to the one for the 22 nd of February were observed.

We also notice that, despite the fact that the temporal resolution strongly influences the simulated instantaneous pumped flow rate, it has a smaller impact on the simulated total volume pumped during the day. For instance, for the data of the 22 nd of February, the simulated total volume pumped during the day is comprised between 10 m 3 and 13 m 3 for all of the temporal resolutions.

V. EFFECT OF THE TEMPORAL RESOLUTION ON SYSTEM SIZING

We now study the effect of the temporal resolution of the water consumption profile on the PVWPS optimal conception. In this optimization problem, we search with the help of a differential evolution algorithm [START_REF] Price | Differential Evolution: A Practical Approach to Global Optimization (Natural Computing Series)[END_REF] for the values of the PV modules area APV and of the tank volume VT that minimize the part of the capital cost that depends on the size of the PV modules and of the tank, CAPEX. The fixed part of the capital cost is not included in the objective function as it does not influence the result of the optimization. The CAPEX, expressed in k$, is given by:

CAPEX = 0.16 APV + 0.61 VT (1)
The coefficients have been obtained by using prices provided by companies located in Burkina Faso [START_REF] De | A financial and technical assessment of solar versus hand water pumping for off-grid areathe case of Burkina Faso[END_REF]. We have set the constraints that the water level in the tank must remain higher than zero and that the instantaneous pumped flow rate must remain lower than 1.3×10 -3 m 3 /s. Indeed, we have noticed during pumping tests that were performed at Gogma's PVPWS that too high pumped flow rates produce a significant decrease of the water level in the borehole and may therefore cause the pump to run dry. It is important to notice that the PVWPS model is required to verify these two constraints. This is therefore at this stage that the temporal resolution of the water consumption profile has an influence. Indeed the temporal resolution of the water consumption profile impacts the evolution of the pumped flow rate and of the water level in the tank. During the optimization process, the model is run over three periods of 24 h and the model input data are the same for these three periods, equal to the data of the 22 nd of February 2018. Indeed, the results of the model for the following periods are identical to the results for the third one, which means that the system has reached a steady state at the third period. Fig. 9 presents, for the data of the 22 nd of February, the values taken by the optimization's variables and the optimum CAPEX obtained for different temporal resolutions of the water consumption profile.

It appears on Fig. 9 that the first significant changes in the optimal PV panels area take place for time intervals strictly larger than 45 minutes. However, the first significant changes in the tank volume and in the cost function take place for time intervals strictly larger than 4 hours. This difference may be due to the fact that the weight of the tank volume in the cost function is higher than the weight of the PV panels area. When we performed the same study for other days of the dataset, it appeared that the results follow the same trend as the ones of the 22 nd of February. The main contribution of this article is to provide a first estimation of the water consumption profile temporal resolutions required for accurately modelling and sizing PVWPS for domestic consumption in remote villages. It appears that, for the day presented in this article, time intervals strictly larger than 20 minutes significantly change the model accuracy and that time intervals strictly larger than 4 hours have a strong impact on the optimal system sizing. The results presented here and simulations performed for other days of our dataset highlight that the temporal resolution strongly influences the model accuracy and the optimal system sizing. This suggests that current sizing methods, which are often based on the forecasted daily water consumption and therefore use a time interval of 1 day, may lead to badly sized PVWPS. This highlights the importance of developing methods that allow to forecast the water consumption profile with a time step smaller than one day. The methodology presented in this article could also be used to study the effects of the water consumption profile temporal resolution for other types of pumping systems with tanks, such as wind powered pumping systems or piped water pumping systems. In the future, we will perform the simulations by using larger datasets, such as periods of one week instead of periods of one day. We will also study the influence of the temporal resolution of the irradiance profile and of the ambient temperature profile on PVWPS modelling and optimal conception.

Fig. 1

 1 Fig. 1 PVWPS architecture considered PA: pipe assembly The model allows to simulate the pumped flow rate from climatic and water consumption data. The inputs of the model are the irradiance on the plane of the PV modules IrrPV, the ambient temperature Ta and the water consumption at the fountain Qc. The model output is the pumped flow rate Qp. The block diagram of the model is given on Fig. 2.

  [START_REF] Howard | Domestic water quantity, service level, and health[END_REF]

  .

Fig. 2

 2 Fig. 2 Block diagram of the PVWPS model t: time, PPV: Input power to the motor-pump, TDH: total dynamic head, HT: water level in the tank, b: triggering signal from the controller

Fig. 4

 4 Fig. 4 Comparison between the measured and modelled pumped flow rate III. WATER CONSUMPTION PROFILE TEMPORAL RESOLUTION

Fig. 5

 5 Fig. 5 Water consumption profiles obtained for various temporal resolutions

Fig. 9

 9 Fig. 9 Effect of the temporal resolution on PVWPS optimal sizing Results on PV modules area (a), tank volume (b) and CAPEX (c) Results for the data of the 22 nd of February VI. CONCLUSION

This work is supported by a public grant overseen by the French National research Agency (ANR) as part of the « Investissement d'Avenir » program, through the "IDI 2015" project funded by the IDEX Paris-Saclay, ANR-11-IDEX0003-02.


Ali Beydoun

Hamze Alaeddine
email: hamze.alaeddine@ul.edu.lb

Haidar Elmokdad
email: hmokdad@ul.edu.lb

New fast time synchronization method for MIMO-OFDM systems

Keywords: MIMO-OFDM system, Timing synchronization, CAZAC sequence, Maximum ratio combiner

This paper presents a fast and low complexity timing synchronization method for MIMO-OFDM system. The proposed method uses simple CAZAC sequence as synchronization preamble and the combination of a set of correlators at the receiver side in order to increase the probability of correct timing acquisition. Simulation results in AWGN-Rayleigh fading channel show that an acquisition probability of 1 could be achieved for a Signal to Noise Ratio (SNR) higher than -20dB with a preamble length of 1024.

Introduction

The current trend in mobile communication systems goes towards the integration of novel data intensive applications such that high-definition video streaming, smart health monitoring, Internet of Things (IoT) and other multimedia services which requires a large increase in the data rate [1] [2]. To achieve this target,the combination of Multiple Input Multiple-Output (MIMO) system with Orthogonal Frequency Division Multiplexing (OFDM) presents a promising solution due to its capability of supporting high data rates [START_REF] Cho | MIMO-OFDM wireless communications with MATLAB[END_REF]. In fact, OFDM has attracted a lot of attention to achieve higher data rate over wireless channel thanks to its higher spectral efficiency, its simplicity and its robustness against frequency selective channels by dividing the entire channel into many narrow flat fading orthogonal sub-channels which reduces the Inter-Carrier Interference and the Inter-Symbol Interference (ISI) [START_REF] Jha | OFDM towards fixed and mobile broadband wireless access[END_REF]. Moreover, the MIMO system [START_REF] Tsoulos | MIMO system technology for wireless communications[END_REF], using multiple antennas at the both side of transceiver, allows to increase the link capacity by sending different data stream over different transmit antenna or to improve the link reliability by sending the same data stream over different antenna using Space Time Block Code (STBC) [START_REF] Tarokh | Space-time codes for high data rate wireless communication: performance criteria in the presence of channel estimation errors, mobility, and multiple paths[END_REF].

One of the main problems of MIMO-OFDM system is the timing synchronization between the transmitter and the receiver. Indeed, incorrect timing synchronization introduces inter-carrier interference and leads to apply the FFT at the receiver side at the undesired position which prevents the OFDM signal to be correctly decoded and decreases significantly the expected performances. Therefore, accurate timing synchronization is crucial for reliable reception of the transmitted data using MIMO-OFDM [START_REF] Tsoulos | MIMO system technology for wireless communications[END_REF].

Several timing synchronization approaches have been proposed for MIMO-OFDM system in the literature [START_REF] Schmidl | Robust frequency and timing synchronization for OFDM[END_REF], [START_REF] Jieping | An improved training sequence based timing synchronization algorithm for MIMO-OFDM system[END_REF], [START_REF] Sreedhar | An efficient preamble design for timing synchronization in MIMO-OFDM systems[END_REF], [START_REF] Nasraoui | Robust Synchronization Approach for MIMO-OFDM Systems with Space-Time Diversity[END_REF]. The main idea of Schmidl and Cox algorithm is to transmit a synchronization preamble, in front of the signal to be transmitted, composed of two identical halves of an orthogonal sequence. At the receiver side, the synchronization is performed in two steps:

1) Coarse synchronization: This steps tries to roughly localize the synchronization preamble. A running window correlation of every N samples of the received signal with the next consecutive N samples is performed in order to determine the Region Of Interest (ROI) where the synchronization preamble is present. 2) Fine synchronization: In order to find the exact position of the first sample, the received signal is correlated with the known preamble sequence in the region of interest to detect a correlation peak that precisely locates the first sample of the synchronization preamble and deduce consequently the start of the data OFDM symbols.

This method produces broad plateau in the autocorrelation metric due to the presence of two identical sequences in the preamble which increases the region of interest and therefore the computational complexity and the execution time will be increased [START_REF] Sreedhar | An efficient preamble design for timing synchronization in MIMO-OFDM systems[END_REF]. In [START_REF] Nasraoui | Robust Synchronization Approach for MIMO-OFDM Systems with Space-Time Diversity[END_REF], the plateau effect was removed using sliding window differentiator after the coarse synchronization. However the execution time remains high as the synchronization is always done in two steps. In order to reduce the execution time, the solution proposed in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF] avoids the plateau problem and achieves the synchronization in one step using different preamble structures based on CAZAC Sequence. These preambles are inserted in the frequency domain before the Inverse Fourier Transform (IFFT) and the cyclic prefix insertion which increases the computational resources. For this purpose, this paper proposes a new synchronization scheme aiming to insert preambles in the time domain in order to not involve the FFT in the synchronization process and to reduce consequently the execution time. The remaining of the paper is organized as follows. Section 2 introduces briefly the MIMO-OFDM system model. The proposed timing synchronization method is described in section 3. Simulations results and comparison are discussed in section 4 . Finally, the conclusion is drawn in section 5.

MIMO-OFDM system model

A simple 2 × 2 MIMO-OFDM system with two transmit antennas and two receive antennas using the STBC-Alamouti encoder is shown in Figure 1. The binary data to be transmitted is first mapped to the complex modulation symbols by using M-QAM signal constellation. The QAM symbols are then introduced into a STBC (Space-Time Block Coding) encoder that allows to exploit the spatial diversity and increases the reliability of transmission [START_REF] Larsson | Space-time block coding for wireless communications[END_REF]. Thereafter, Inverse Fast Fourier Transform (IFFT) is applied to generate the OFDM symbols. A cyclic prefix (CP), the replica of last part of any given OFDM symbol, is attached at the beginning of each time domain OFDM symbol to counteract the distortion caused by ISI in the channel. The length of CP should exceed the maximum excess delay of the multipath propagation channel [START_REF] Jha | OFDM towards fixed and mobile broadband wireless access[END_REF]. After the insertion of CP, the resultant OFDM symbols are converted to serial form and are transmitted over the 2×2 AWGN Rayleigh channel. The OFDM symbol transmitted from the i th transmitting antenna is given by [START_REF] Jha | OFDM towards fixed and mobile broadband wireless access[END_REF]:

s i (n) = 1 √ N N k=0 S i (k)e j 2πnk N (1)
where S i (k) is the data in the frequency domain before the IFFT and N is the number of sub-carriers for one OFDM symbol. The Rayleigh channel between i th transmit antenna and j th receive antenna is given by:

h ij (t) = L l=0 α l ij δ t -τ l ij ( 2
)
where L is the number of multi-path channels between i th transmitter and j th receiver. α l ij denotes the channel gain and τ l ij denotes the delay of the path. The received signal r j on each receive antenna R j is given by:

r j (t) = 2 i=1 L l=0 α l ij δ t -τ l ij * s i (t) + n j (t), j = 1, 2 (3)
where s i is the transmitted signal on the transmit antenna T i , n j is the additive white Gaussian noise (AWGN).

At the receiver side, the first block after the analog to digital converter (ADC) is the timing synchronization block. After a good timing synchronization, the cyclic prefix of each OFDM symbol is removed. The signal returns back into frequency domain thanks to the FFT block. The OFDM symbols are then decoded and combined by the STBC decoder. Finally, a QAM demodulator is used to demodulate and recover the binary data.

Preamble timing synchronization algorithm for mimo-ofdm system

Review of timing synchronization method

In order to reduce the execution time, the timing synchronization method proposed in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF] performs the synchronization in single step. It is based on sending a synchronization preamble at the beginning of each OFDM frame that has the same length as an OFDM symbol. The main characteristic of the synchronization preamble is to have good autocorrelation and cross-correlation properties in order to detect a correlation peak as closed as possible to a Dirac pulse. On the other hand, the synchronization preambles should be orthogonal to eliminate the interferences between preambles at simultaneous transmissions in MIMO systems. Several types of preambles have been proposed in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF] using Constant Amplitude Zero Auto-Correlation (CAZAC) sequences [START_REF] Fan | Sequence design for communications applications[END_REF]. These sequences are characterized by their constant amplitude and good correlation properties. They are expressed by:

C(k) = e j πM k(k+1) Lc if L c is odd e j πM k 2 Lc if L c is even (4)
where L C is the length of CAZAC sequence, M is a natural number relatively prime to

L C and 0 ≤ k ≤ L c -1.
The two types of preambles proposed in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF] are shown in Figure 2. For type 1, the CAZAC sequence of length L c = Lp 2 is combined with its opposite conjugate to build the preamble of length L P . The values of the CAZAC sequence are mapped to odd sub-carriers while the opposite conjugate values are mapped to even sub-carriers. For type 2, the CAZAC sequence and its opposite conjugate are placed side by side to form the synchronization preamble. For both types, the proposed preambles show good correlation properties and do not destroy the orthogonality between sub-carriers while maintaining the orthogonality between different preambles over transmit antennas.

2 PC LL = Preamble length Ant2 Ant1 2* 1 C L C - - 2 1 C L C - 2* 1 C - 2 1 C 2* 0 C - 2 0 C 1* 1 C L C - - 1 1 C L C - 1* 1 C - 1 1 C 1* 0 C - 1 0 C Type 1 Ant2 Ant1 1* 0 C - 1* 1 C L C - - 1 1 C L C - 1 1 C 1 0 C Type 2 2* 0 C - 2* 1 C L C - - 2 1 C L C - 2 1 C 2 0 C 1 P 2 P 1 P 2 P Figure 2
. Schematic diagram of proposed preambles in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF].

It is important to recall that the preamble is inserted in the frequency domain which requires that the size of the preamble must be the same as the IFFT size. Therefore, the IFFT and the Add Cyclic Prefix blocks will be involved in the synchronization process which increases the complexity and the execution time of the method.

Proposed timing synchronization method

In order to reduce the complexity, the proposed method inserts the synchronization preamble in the time domain after the OFDM modulator (see Figure 1). In this case, the preamble structure is simply the CAZAC sequence without any restriction on the length L p as shown in Figure 3. However, the prime number M in the equation of the CAZAC sequence (4) must be carefully chosen from the set of prime numbers M. Indeed, not all prime numbers give the same auto-correlation function. Figure 4 shows the normalized auto-correlation function with different prime numbers for a preamble length L P = 128.

It can be noticed that the difference between the first and the second peak is not the same for all prime numbers. This difference is considered as the main criteria in our method to select the best two prime numbers, from the set M, that correspond to the CAZAC sequences with the highest difference. Therefore, the auto-correlation of the synchronization preambles are as close as possible to a Dirac function.

At the receiver side, in order to detect the timing synchronization peak, the proposed synchronization algorithm correlates simultaneously the received signal on each antenna R j with the known sequences P 1 and P 2 as shown is chosen as the argument that maximizes λ(d) 2 and is denoted by τ :

τ = arg max d λ(d) 2 (5)
Three expressions could be considered for the timing metric λ(d):

1)

λ(d) = Λ j i (d) = r j ⊗ P i = Lp-1 m=0 r j (d + m)P * i (m) (6
) where i and j ∈ [START_REF] Zhuang | Future Internet bandwidth trends: An investigation on current and future disruptive technologies[END_REF][START_REF] Din | Smart health monitoring and management system: Toward autonomous wearable sensing for internet of things using big data analytics[END_REF] and ⊗ denotes the correlation operation.

In this case, one correlator is needed and the timing metric λ(d) is simply the correlation between any received signal r j with any preamble sequence P i .

2)

λ(d) = Λ i (d) = 2 j=1 λ j i (d) i = 1, 2 (7)
The received signals at both antennas are correlated with the same sequence P i and added together in order to increase the acquisition probability. 3)

λ(d) = Λ t (d) = 2 i=1 2 j=1 λ j i (d) (8)
To further increase the acquisition probability, the received signals at different antennas are correlated with both preambles and combined together in a way similar to Maximum Ratio Combining (MRC).

With purpose of validating this concept, the correlation of the received signal at antenna R 1 following the transmission of the two synchronization preambles with the known preamble P 1 is given by:

Λ 1 1 = (P 1 * h 11 + P 2 * h 21 + n 1 ) ⊗ P 1 (9)
where ⊗ and * denote the correlation and the convolution operations respectively. n 1 is the white noise added to the signal. Without loss of generality, we can interchange the correlation and convolution operations and rearrange equation 9 in the following way:

Λ 1 1 = (P 1 ⊗ P 1 ) δ * h 11 + (P 2 ⊗ P 1 ) =0 * h 12 + n ′ 1 (10)
Based on the orthogonality between preambles and the autocorrelation function of each preamble, the expression of Λ 1 1 is simplified to be :

Λ 1 1 = h 11 + n ′ 1 (11)
This results can be easily extended for other correlators and we can conclude that the correlation of the received signal at the receive antenna R j with the local sequence P i produces the impulse response between the i th transmitter and the j th receiver plus some correlated noise.

Theoretically, the first tap of the channel impulse response, that corresponds to the direct path, has the highest amplitude and the start of the preamble can be efficiently detected. However, due to the presence of the noise in the channel (n ′ 1 in (11)), it is difficult to identify the first tap in the channel as the first tap may not be necessarily the strongest peak especially at low Signal to Noise Ratio (SNR) values. For this purpose, the proposed method suggests the combination of the outputs of two or four correlators to increase the probability of correct detection at very low SNR values.

It is important to mention that this method could be easily extended without any loss of generality to higher order MIMO systems.

simulation results

The performance of the proposed method is evaluated by computer simulation using Matlab. In these simulations, a 2 × 2 MIMO-OFDM system is considered with the parameters listed in Table 1. [START_REF] Ohara | The IEEE 802.11 Handbook: A Designers Companion[END_REF] Propagation delay

[0 Ts 2Ts 3Ts 4Ts 5Ts] between multi path [START_REF] Ohara | The IEEE 802.11 Handbook: A Designers Companion[END_REF] The power of each [0.8111 0.1532 0.0289 0.0055 multi-path [START_REF] Ohara | The IEEE 802.11 Handbook: A Designers Companion[END_REF] 0.001 0.0002] Figure 6 and Figure 7 show the probability of correct timing synchronization over an SNR range of [-26 . . . 30] dB with the proposed method for different timing metrics (equations: 6, 7, 8). Each probability value is estimated using an average of 100 independent Monte Carlo trials. It can be noticed that for short preamble length (L P = 64) the performances with only one correlator (Λ j i , Eq. ( 6)) are almost the same. Increasing the number of correlators to 2 (Λ i , Eq.( 7)) or 4 (Λ t , Eq. ( 8)) improves the algorithm's ability to detect the correct timing.

Figure 8 shows the effect of the preamble length L P using the timing metric Λ t and Figure 9 shows the evolution of the SN R threshold from which the acquisition probability is 1 in terms of the preamble length. It can be noticed that the higher the preamble length, the better the expected performance and the lower the SNR value above which 100% of correct detection is reached. Figure 10 compares the acquisition probability in terms of SN R using the timing metric Λ t and a preamble length L P = 1024 between the proposed method and the one presented in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF]. It can be shown that both methods give the same performance if the CAZAC sequence is well chosen with good auto-correlation function. However, the main advantage of the proposed method lies in the significant reduction of complexity and execution time.

conclusion

A robust synchronization method for MIMO-OFDM system was proposed in this paper. The proposed method aims to detect the timing start of the OFDM frame by correlating the received signal at both antenna with local orthogonal preambles and combining them in order to increase Figure 10. Comparison of the proposed method with the method proposed in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF].

the correct timing acquisition probability. The considered orthogonal preamble is simply a CAZAC sequence where the prime number in the sequence equation is carefully chosen in order to obtain an auto-correlation function as close as possible to a Dirac function. The proposed method shows satisfactory performance with an acquisition probability of 1 for SN R > -20dB with a preamble length of 1024 while reducing considerably the computational complexity and the execution time compared to the solution presented in [START_REF] Rachini | A Novel Timing Synchronization Method for MIMO-OFDM Systems[END_REF].

11

 11

Figure 1 .

 1 Figure 1. Simplified block diagram of MIMO-OFDM system.

Figure 3 .

 3 Figure 3. Proposed preamble structure using only CAZAC sequence.

Figure 4 . 1 LFigure 5 .

 415 Figure 4. Normalized auto-correlation of CAZAC sequence with length L P = 128 for the prime number (a) M = 2, (b) M = 53,(c) M = 127.

Figure 6 .

 6 Figure 6. Probability of correct timing synchronization with the proposed method with preamble length L P = 64 for different timing metrics.

Figure 7 .Figure 8 .

 78 Figure 7. Probability of correct timing synchronization with the proposed method with preamble length L P = 1024 for different timing metric.

Figure 9 .

 9 Figure 9. Evolution of the SN R threshold.

TABLE 1 .

 1 SIMULATION PARAMETERS

	Parameter	Value
	FFT/IFFT length	N = 512
	Cyclic Prefix length	48
	Channel type	Multi-path Rayleigh and AWGN
	Number of synchronization	1
	symbol
	Modulation tecnique	4 QAM
	Number of channel multi-path	L = 6
	between different antenna


Pauline Schlosser
email: pauline.schlosser@hmg.inpg.fr

Denis Favier
email: bdenis.favier@hmg.inpg.fr

Hervé Louche
email: cherve.louche@univ-savoie.fr

Laurent Orgéas
email: laurent.orgeas@hmg.inpg.fr

Experimental characterization of NiTi SMAs thermomechanical behaviour using

Keywords: NiTi, shape memory alloy, superelasticity, localisation, strain and temperature full field measurement, heat sources estimation

Introduction

Shape Memory Alloys (SMAs) are now employed in a large number of applications in the fields of aeronautical, biomedical and structural engineering. The applications based on the peculiar properties of these SMAs are more and more designed using numerical simulation and finite element softwares in which three dimensional constitutive equations are implemented to model the thermomechanical behaviour of these SMAs. These constitutive equations are based on uniaxial and multiaxial experiments within a wide temperature range assuming homogeneous deformation of the specimens. The aim of the present paper is to illustrate, on two examples, information that can be deduced from temperature and strain full field measurements performed during these tests. This information is fundamental for the establishment of more efficient model.

The present paper is focused on the superelastic behaviour of NiTi polycrystalline SMAs. Owing to their outstanding superelastic behaviour at human body temperature and to their biocompatibility, polycrystalline Ti-50.8%at Ni SMAs are being increasingly used for biomedical applications (human implants and surgery instruments). Design of these applications uses constitutive equations mostly based on the conventional understanding of the superelastic tensile engineering strain-stress curve shown in Fig. 1. Initially, the material is in austenite phase (A) since the testing temperature is above A f . A f is the temperature at which the martensite phase (M) stable at low temperature is completely transformed by heating to the austenite phase A stable at high temperature. The loading curve of Fig. 1 is usually divided in a first linear part (1) associated with the elastic deformation of A, a second one (2) associated with the transformation A-M occurring along an upper stress plateau and then a third one (3) associated with the elastic deformation of oriented martensite variants. When the load is progressively removed, the reverse transformation M-A occurs along a lower stress plateau [START_REF] Orgéas | [END_REF] after an initial linear stage (4) associated with the elastic unloading of the oriented martensite M. The complete reverse transformation is followed by the elastic unloading of A during the stage (6). Modelling of the superelastic behaviour of NiTi SMAs based on this above simple analysis is a preliminary step. However, a better understanding of the complex phenomena occurring during all these stages is necessary to improve this modelling. In that respect, within the past decades, intensive experimental investigations have been carried out to characterize and understand deformation mechanisms associated with the superelasticity of NiTi SMAs. Most of these studies have been achieved in tension using wires, strips or bone-shaped samples. It is now well-established that uniaxial tensile superelastic deformation of polycrystalline NiTi SMAs often exhibits localised Lüders-like deformation modes [1][2][3][START_REF] Favier | Solid Mechanics and Its Applications[END_REF][START_REF] Orgéas | [END_REF][6][7][8][9] which occur during the upper (2) and lower (5) plateaus sketched in Fig. 1. This heterogeneous deformation may disappear for other mechanical loadings such as compression or shear [START_REF] Orgéas | [END_REF] or tension-torsion combined tests [6], and can also be observed during the ferroelastic tensile deformation of polycrystalline NiTi SMAs [10]. The localisation has been studied using qualitative optical observations [1,3,[6][7][8], multiple extensometers [2,[START_REF] Favier | Solid Mechanics and Its Applications[END_REF][START_REF] Orgéas | [END_REF] and full-field temperature measurements [3,9]. Comparatively, the other stages (1), ( 3), ( 4) and ( 6) have been less studied. However, Brinson et al [7] present some results showing clearly the occurrence of several deformation mechanisms other than elastic distortion of the crystalline structure during each of these three stages.

Studies in the references cited earlier and other numerous works have brought up precious issues in the fundamental understanding and modelling of stress-induced phase transformation in polycrystalline NiTi SMAs. For optimal implementation of NiTi SMAs in engineering applications, a thorough understanding of the material behavior is necessary. In that order, the present contribution reports and analyses results concerning tensile tests on NiTi polycrystalline thin walled tubes during which thermal as well as kinematical full-field measurements have been simultaneously carried out.

Experimental

Experimental set-up. A special designed gripping system (Fig. 2a) was created using a pair of standardized collets and collet holders (2) with a pin inside to avoid crushing the tube (1) [11]. Both collet holders are linked to an universal mechanical tensile testing machine (Instron 5569, 50kN) via two driving shafts (3) in order to avoid any bending of the tensile specimen. During the tests, the usual mechanical data (axial force F, crosshead displacement U) as well as the thermal and kinematical fields were acquired simultaneously with two digital cameras, respectively an infrared and a visible one (Fig. 2.a).

The temperature field was obtained with a fast multidetector infrared camera (Cedip Jade III MW, 145 Hz). The infrared camera has a resolution of 320 × 240 pixels. The spatial resolution (pixel size), that depends on the adjustment of the focal distance, was estimated to be close to 0.4 mm for the carried tests. Prior to the tests, the surface of the tube was coated by a highly emissive paint in order to obtain blackbody properties compatible with the calibration law of the camera, this leaded to an accuracy on temperature variations less than 0.1 K. The displacement field of any particle of the outer surface of the tube has been obtained using a digital visible camera (Hamamatsu, 1280 × 1024 pixels, 9 Hz, ambient lighting) and a Digital Image Correlation (DIC) processing ("7D" software [START_REF] Vacher | Proc. of Instn Mech Engrs[END_REF]). To obtain blackbody properties a black paint was spread on the tube. Moreover white paint speckles were superimposed to enhance DIC. The elementary cell of the correlation grid was 10 × 10 pixels, so the spatial resolution was estimated to be of the order of 0.5 × 0.5 mm 2 , as shown in Fig. 2.c. With such parameters, the accuracy on the strain measurement was <0.1%.

The nominal stress σ 0 is obtained by dividing the measured force by the initial cross section of the tube. The nominal strain is obtained by dividing the difference of displacements Δl of two points A and B initially distant of the gauge length L 0 by this length (Fig. . Before being clamped, each tube was heated up to 373K and cooled down to the ambient temperature T a , plotted in dashed lines in Fig. 3 . The initial tube temperature T 0 is homogeneous and equal to ambient temperature T a so that the material was in the austenite state with a small amount of initial R-phase for sample 1 and almost totally austenitic for sample 2.

Results

Macroscopic results. The curves plotted in Fig. 4.a and 4.b show the nominal axial stress σ 0 = F/S 0 (S 0 being the initial cross-section of the tube) as a function of the nominal axial strain ε respectively on the first and the second sample. The nominal axial strain ε is deduced from the displacements of the two extremities A and B obtained with DIC. Markers in the two curves show typical times.

Discussion

Specimen 1 is fully superelastic at the testing temperature but the maximum strain during loading is 5.8% so that the unloading proceeds before the end of the loading stress plateau (stage 2 of Fig. 2). Since specimen 2 is initially austenite, A-R and R-M forward transformations are important deformation mechanisms during loading which is performed beyond the stress plateau. However, the testing temperature was not high enough to observe the reverse stress plateau during the unloading (stage 5 of Fig. 2).

The stages 1, 2 and 4 will be thus analysed for the two specimens, stage 3 only for the specimen 2 and stage 5 and 6 for the sample 1. 4b): During this stage, the deformation of the two samples proceeds homogeneously in the observed region, as demonstrated in the snapshot (a) of Fig. 5 and along the axial profiles of Fig. 6(a) and 7(a). An almost homogeneous temperature increase is observed for both specimens, starting as soon as the tensile deformation proceeds. It is also observed that both stress σ 0 and strain ε but also temperature variation θ increase almost linearly with time. Such temperature increase shows that deformation mechanisms do not involve pure elasticity, which alone would induce a temperature decrease.

Stage 2 (b-e of Fig. 4a andc-f of Fig. 4b): As soon as the stress peak appears, localisation phenomena are observed both in strain and temperature fields. In both tensile tests, the localizations take the shape of helical bands. The helical bands turn around the tube with a constant angle of approximately 59° toward the vertical axis. Only one band is observed for specimen 1 whereas, on the second specimen, a second band starts at the end of the stress plateau (profile e in Stage 3 (g-h of Fig. 4b): The θ and E xx profiles of Fig. 7.c and 7.d show that the temperature is not homogeneous at time (g) in spite of a homogeneous deformation of the tube. This is obviously due to the temperature history during the loading. It is worth noting that the almost homogeneous strain increase between times (g) and (h) is accompanied at any point of the tube by temperature decreases.

Stage 4 (e-f of Fig. 4a and h-j of Fig. 4b): The strain decreases homogeneously, even for the sample 1 which is deformed heterogeneously at time (e); this homogeneous strain decrease is accompanied by a strong temperature decrease, with smooth bell-shaped temperature profiles.

Stage 5 (g-i of Fig. 4a): The strain decreases heterogeneously (times (f), (g) and (h) in Fig. 6.c). The temperature decreases below room temperature, as shown in Fig. 6.d.

Stage 6 (i-j of Fig. 4a): The strain field is homogeneous again and the temperatures θ are heterogeneous and below ambient temperature.

The analysis of both strain and temperature variation fields reveal homogeneous and localised phenomena during tensile tests. The kinematic and morphology of localisation can be investigated. Since the measured temperature is only the consequence of internal heat sources and conduction phenomena with the gripping system and through the sample, the analysis of temperature remain qualitative and difficult. An estimation of the associated heat sources would provide precious quantitative information. Moreover it will give interesting data for future theoretical development concerning modelling of the thermomechanical behaviour of NiTi shape memory alloy.

So far, the nature of the cooling during stage 4 is unclear for both specimens 1 and 2. The next section will focus this stage 4 showing homogeneous strain decrease and try to answer this question by estimating homogeneous heat sources.

During the plateau (2 and 5) the morphology and kinematic of the localisation is obtained using the strain fields. As soon as the localisation starts, temperature localisations can be observed but are difficult to analysis after few seconds especially during the unloading. At the beginning of the unloading, the temperature field is strongly heterogeneous and the mean temperature is above room temperature. The last section will present a method to estimate 2D heat sources necessary in stage involving heterogeneous heat sources and temperature fields such as stage 2 and 5.

From full-field measurements to the analysis of deformation mechanisms

Heat conduction equations. Neglecting any heat source of external origin, heat transfers are ruled by the following local heat conduction equation [14][15][16]:

q T k T C & & = ρ -lap , (1)
where T and T & denote the temperature and its rate at any point of the body, respectively. The term "lapT" stands for the laplacian operator applied to the temperature field. It is assumed that the heat conduction is governed by the standard Fourier's law, with an isotropic, uniform and constant thermal conductivity k. The terms ρ and C denote respectively the mass density and the specific heat assumed both uniform and constant, is the specific heat source rate (in W/kg). The experimental values for NiTi alloy of C range from 450 to 600 J kg q & -1 K -1 , and those for k from 8 to 28 W m -1 K -1 [17]. In this study, C and k were taken constant and equal to 490 J kg -1 K -1 and 15 W m - 1 K -1 , respectively.

If the temperature field is homogeneous, the equation ( 1) can be written in a simpler form when heat sources are homogeneously distributed in a specimen surrounded by ambient medium at temperature T q & 0 and when specimen temperatures are not too far from the thermal equilibrium [15]. At any point of the outer surface of the tube, this simplified model reads :

( ) q C θ t θ & = + τ ∂ ∂ eq , ( 2
)
where θ =T-T 0 is the temperature variation. The parameter τ eq represents a characteristic time reflecting heat losses both by convection through inner and outer surfaces of the tube and by conduction towards the grip's zones. More detail on the estimation of τ eq and the hypothesis are presented in [18].

Hypotheses for the deformation mechanisms. It is well known that deformation mechanisms for NiTi shape memory alloys include elastic distortion of the atomic lattice and additional mechanisms associated with martensitic and R phase(s) transformation(s) and variants (de)twinning [19]. Local plastic accommodation of the transformation(s) may also be involved but is effective only for large local strain [7]. The heat source rate involved in ( 2) is thus divided into a first rate due to phase(s) transformation(s), a second one due to the usual thermoelastic coupling and a last one due to intrinsic dissipation induced by irreversible phenomena like plasticity and frictional barriers opposing interfacial motion :

q & tr q & thel q & diss q & 00 with tr thel diss thel T qq q q q α σ ρ =+ + = -, (3)
where α is the thermal expansion coefficient. The classical additive decomposition of the strain rate is assumed :

ε & 0 with E el in el σ εε ε ε =+ = & && & & , (4)
where and are the elastic and inelastic strain rates, respectively. The inelastic strain rate includes the rate due to transformation(s)

el ε & in ε & tr ε &
and the rate due to variants (de)twinning . In the present analysis, the rate due to plasticity is neglected, together with =0.

de ε & diss q &
Lastly, for any phase transformation, both strain rate tr ε & and heat source rate will be assumed proportional to the transformation fraction rate [16,20] :

tr q & f & tr tr tr tr H f q and f Δ = ε = ε , (5)
where ε tr and ΔH tr stand for the transformation strain and specific enthalpy for a complete transformation (f=1).

Deformation mechanisms during homogeneous unloading stage. Equations developed above are now used in order to further analyse the beginning of the unloading of the sample 2 (stage 4 conventionally attributed to elastic deformation and detwinning of the oriented M variants formed during loading). In this stage, temperature variation θ, stress σ 0 and strain ε decrease almost linearly with time (Fig 8 .a). Due to the decrease of temperature (endothermic heat source), three main mechanisms are involved during the unloading, i.e. elasticity, detwinning and reverse transformation, plasticity being neglected. Taking into account the equations (4), the additive decomposition of the strain rate is thus expressed as : As evident from fig. 8.b, deformation mechanisms other than pure elastic distortion of crystalline lattice occur as soon as the stress starts to decrease. From the above rough analysis, it is suggested that reverse phase transformation has a predominant effect during the whole unloading. Its contribution to the overall strain rate is almost constant and equal to half the total strain rate.

Decreasing the strain ε leads to reduce the elasticity contribution and to increase the detwinning phenomena. It is worth concluding that these results deny the usual assumption of elastic unloading which is commonly made in thermomechanical modelling for stage 4 of Fig. 1.

Heat source estimation during heterogenous stages. It is very difficult to analyse quantitatively the observed heterogeneous deformation mechanisms from temperature maps such as that ones shown in Fig. 3. Instead, the determination of local heat source fields are much more appropriated, to detect and analyse the phase transformations.

The local heat sources can be estimated from the heterogeneous temperature field by solving the heat source equation (1). Unfortunately, the IR camera allows measuring only temperatures at the outer surface of the considered tubes, so that further assumptions are required to circumvent this problem. The following sets of hypotheses are stated, as previously achieved in the case of thin flat samples [15,21]: the initial temperature of the tube T 0 is supposed to be homogeneous and equal to the ambient air temperature T a assumed constant during the whole tensile test ; the thickness of the tube e is small compared to its mean radius R and the temperature and heat sources are assumed to be homogeneous within the thickness.

By integrating equation (1) along the thickness of the tube, it is now possible to obtain an estimation of the averaged heat sources , approximating the surface temperature given by the IR camera to the averaged temperature variation (7) where d = k/(ρC) is the thermal diffusivity and τ th = ρC/h eq represents the characteristic time of The process to estimate the heat sources fields is based on the left hand side estimation of the equation ( 7) by using the experimental temperature variation fields 〈θ〉 ( [15]). These fields are discrete and noisy data, obtained at the outer surface (r = R+e/2) of the tube. From the noisy and discrete temperature field, time derivative and laplacien are then estimated in cylindrical coordinates using image processing filter and discrete derivation operators, in time and space ( [14]). The image filtering, necessary to estimate operators on the noisy data, is a delicate operation and is obtained by low pass filters applied by Fast Fourier Transform.

〉 〈 i s /2 0 /2 1 () Re Re TT d r e θ - - 〈〉 = - ∫ . : , s lap d C i th t , 〉 〈 = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ 〉 〈 - 〉 〈 + 〉 〈 θ τ θ θ ρ 2

Example of local heat sources estimation at two characteristics times on sample 1.

The processing method was applied on the temperature fields of specimen 1. The comparison of temperature field and heat sources field at time (d) of Fig. 4.a are plotted in Figure 9.a and b respectively. At this time (middle of the stress plateau), the temperature variation field was about 7 K above room temperature and strongly heterogeneous (Profile (d) in Fig. 6.b). This temperature variation field was hard to interpret mostly due to strong conduction phenomena. The heat source field in Fig. 9.b shows several peaks that are with a similar angle than the strain field discontinuities (Figure 5). Three strong peaks were close to the grips with a magnitude of 4 to 6 10 6 W/m 3 . In the middle of the sample two sets of parallel fronts close to each other with a smaller amplitude (1 to 2 10 6 W/m 3 ) are observed. Those fronts were moving away from each other. Outside of theses peaks, the heat sources were strongly reduced. The heat source profiles along a generating line of the sample are compared to the axial strain profiles in Figure 9.d and c at times (a) and (d) respectively. In the homogeneous stage 1 (time a), both heat source and strain profiles were homogeneous. The heat sources were exothermic estimated of the order of 10 6 W/m 3 . During the stress plateau (stage 2, time d), the strain field was strongly localised. The helical band initiated on the top grip has reached the bottom one; four peaks were present along the sample. Although, the temperature variation are hard to analyse (cf Figure 6.b, time (d)), the heat source profile shows clearly peaks and discontinuities at the same position as the strain discontinuities. The amplitudes of the discontinuities give information on the kinetics of the phase transformations. The fronts propagate faster where the heat sources are higher. When no strain variation was observed, the heat sources were weak. Inside the highly deformed area, the heat sources decreased strongly, revealing a reduction of the phase transformation kinetic.

Conclusions

Although experimental observations of tensile tests on NiTi SMA and of localisation phenomena occurring during these tests can be found in the references cited earlier [1][2][3][START_REF] Favier | Solid Mechanics and Its Applications[END_REF][START_REF] Orgéas | [END_REF][6][7][8][9]22], simultaneous measurements of temperature and strain fields are unique. They provide new results enabling to revisit the conventional understanding of homogeneous and heterogeneous phenomena occurring during tensile tests on NiTi SMA.

These simultaneous measurements, have allowed us to observe both homogeneous and localized stages during the tensile tests. The strain field is especially useful to analyze the kinetic of localized stage. Temperature field only gives qualitative information. An estimation of local heat source can be obtained from the temperature field. Depending of the observed phenomena, two models have been presented: A 1D thermal estimation suitable for homogeneous stage has been proposed, and a more complex, 2D model resolution when the temperature field are localised. Such heat source estimation enables to better analyse the deformation mechanism.

Figure 1 :

 1 Figure 1: Oversimplified superelastic strain-stress curve. Conventional understanding: (1) Elastic deformation of austenite A. (2) Austenite to oriented martensite transformation A-M. (3) and (4) Elastic deformation of oriented martensite. (5) Oriented martensite to austenite reverse transformation M-A. (6) Elastic deformation of austenite A.

  2.b).

Figure 2 :

 2 Figure 2 : a) Experimental setup with the two infrared and visible digital cameras, b) view of the tube (1) in the centre, c) Close-up showing the virtual grid and the surface of the tube coated with a random pattern of white painted speckles superimposed on black paint.

Figure 3 :

 3 DSC measurements of a) the first sample, b) the second sample. The dash lines represent the tensile test room temperature T a . Tensile tests. The tensile tests were conducted in air at constant cross head velocity for specimen 1 and for specimen 2. The nominal axial strain rates are

Figure 4 :

 4 Figure 4 : Nominal stress-strain curve obtained during tensile tests performed at room temperature: a) at 1mm/min on the first superelastic sample. b) at 5 mm/min on the second sample initially austenitic, but not totally superelastic at the testing temperature.

Figure 5 :

 5 Figure 5: Tensile test on sample 1. (1) sample with in yellow the limit of the image correlation area. (2 a to h) Coloured maps showing the local Green-Lagrange strain E xx (M,t) (left) and the temperature variation θ(M,t) (right) fields on the surface of the deformed tube at the five selected times (a,c,e,f,h) marked in Fig.4.a.

Fig. 6

 6 Fig.6represents, for specimen 1, the evolution of the axial Green-Lagrange strain E xx (x,y=0,t) and the temperature variation θ(x,y=0,t) along one tube generating line (y=0) at the selected times marked in Figure4.a.

Figure 6 :

 6 Figure 6 : Profiles evolution along one tube generating line during the tensile test performed on specimen 1. a) Axial strain profile E xx (x,y=0,t) during loading. b) Temperature variation θ (x,y=0,t) during loading. c) E xx (x,y=0,t) during unloading. d) θ(x,y=0,t) during unloading.

Fig. 7

 7 Fig.7 represents, for specimen 2, the evolution of the Green-Lagrange axial strain profiles E xx (x,y=0,t) and the temperature variation θ(x,y=0,t) along one tube generating line at the selected times marked in Figure 4.b.

Figure 7 :

 7 Figure 7: Profiles evolution along one tube generating line during the tensile test performed on specimen 2. a) Axial strain profile E xx (x,y=0,t) during loading. b) Temperature variation θ (x,y=0,t) during loading. c) E xx (x,y=0,t) during unloading. d) θ(x,y=0,t) during unloading.

Fig. 5

 5 to 7 conjure up the following comments: Stage 1 (O-b of Fig.4a and O-b of Fig.

  Figs 7.a). The end of the loading on specimen 1(time (e) Fig 4.a) is before the end of the stress plateau. At time (e) on sample 1, the strain profile is still strongly localized (Fig 6.a). Due to thermal conduction with the gripping system, the temperature field analysis is difficult.

  From these measurements and the equation (6), the absolute values of the elastic , reverse transformation and detwinning strain rates are plotted in Fig.8.b as functions of the strain ε

Figure 8 :

 8 Figure 8: (a) Stress σ 0 (thick curve), strains ε (dashed curve) and temperature variations (thin curve) as functions of time during stage 4 of sample 2. (b) Estimation of the evolution of the elastic, transformation and detwinning strain rates.

  the radial heat loss by convection and radiation, represent the time derivative of the averaged temperature variation field,

Figure 9 :

 9 Figure 9 : Tensile test on sample 1. a) Temperature variation field θ(M,t) at time (d) of Fig.4.a. b) Heat sources <s i (M,t)> estimated at the same time. c) Axial strain profiles along a generating line of the tube E xx (x,y=0,t) at times (a) and (d) of Fig.4.a. b) Heat sources <s i (x,y=0,t)> profiles along the same generating line at times (a) and (d).

1mm/min U = & 5mm/min U = &

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