Philippe Jacquet
email: philippe.jacquet@inria.fr

Liubov Tupikina
email: liubov.tupikina@nokia-bell-labs.com

Geometry of Outdoor Virus Avoidance in Cities

Keywords: Virus Exposure, Random Walks, City Maps

We present low complexity and efficient path selection algorithms for limiting the exposure to pandemic virus during and between lock-downs. The average outdoor exposure rate reduction is around 3.

Introduction

The Covid-19 pandemic has caused repetitive lock-downs for most of mankind in order to limit the virus circulation. In many countries outdoor excursion were permitted during lock-downs under strict restrictions; in France, no more than 1 hour within 1km from home. All efforts done in order to reduce the daily exposure time to virus, the lock-down's aim is to "flatten the curve" in order to limit the congestion in hospitals and must be repeated every time the curve becomes critical and before vaccines and herd-immunity take over. But social distancing needs to be applied during inter lock-down periods.

This paper is divided in two parts. In the first part we introduce an application, called Ariadne Covid. It selects outdoor excursion path with limited exposure to virus during lock-downs based on its geometrical properties. The second part is dedicated to inter lock-downs periods, and we introduce a class of algorithms to select commuting paths.

The fundamental tool in Ariadne Covid is the use of random walks which naturally leads to to a better load balance between streets while avoiding crowded areas. The application is shown to give a benefit to the first user, although to have an impact on the pandemic, it must be used by a majority of users.

During inter lock-downs periods people are allowed to commute from home to work. The path selection presented in the second part is based on drifted random walks in order to reach the destinations without too much elongations while offering balanced loads on city streets. Both applications take benefit from the geometry of city maps which leads to naturally homogeneous street networks.

Covid exposure during outdoor excursion in cities

The virus of Covid-19 is supposed to have some probability to be transferred when an infected person is within 1 m of another person during a certain time (10 m when running or biking). The larger is the exposure time the higher is the contamination probability (the average contamination delay is around 30 min). We call exposure time the cumulated time a given person is within 1m of any other person during an outdoor excursions.

There are 2,900 km of cumulative pedestrian path length in Paris for a population of 2.2 million inhabitants [1]. Therefore there is ample room for all of the inhabitants to walk 24/24h within safe distance. Since only a fraction of the inhabitant are expected to be walking at the same time, it should be easy to get the safe spacing.

The problem is that people show a social trend to gather in the same streets, even during lock-downs, rising the exposure to virus. Data sampled via Google Street View show how negative is the impact of the social trends. In the paper we will call social walkers (or social joggers) the walkers who follow the social trends during outdoor excursions in the city.

We have sampled 55 km of Paris street via Google Street View. We have made 11 independent random walks starting in different arrondissements. A segment (30 m) of pedestrian path is sampled every 100 m. Figure 1 displays the histogram of segment densities sampled in Paris. The Street View pictures date before the lockdown, and mix several periods. The histogram shows a profile close to a Zipf distribution of coefficient close to 0.75. We have no yet records for the period during the lockdown, but we expect a more concentrated distribution with a larger Zipf coefficient, since the most of the commercial areas are closed and therefore attraction streets are reduced. We represent the city by an abstract graph, with the street intersection (vertices) linked by street segments (edge). We denote V the set of intersections and E the set of street segments.

3 Path selection during lock-downs

Ariadne String protocol description

The application consists into delivering on request, a random path for the excursion and a time slot in the day. This path will reduce the exposure to the virus by avoiding the dangerous areas with an abnormal density of social walkers. In its most basic setting the path will be a pure random walk, since it shows to actually reduce the exposure rate even if the user faces only social walkers. Indeed it is shown that the outdoor exposure to virus is even more reduced compared to the situation where all inhabitants use the application.

During the excursion the user will have his/her own position displayed on her/his mobile phone. At half of the excursion time the walker will receive a notification to reverse the path in order to come back home by keeping path uniformization. The path may contain loops, because they are necessary in order to get the uniform density. If it is the case the walker will proceed with the loops. The walker may skip the loop in the way back but this will shorten the delay of the excursion. The application terminates (at least its embedded mobile part) when the user is back home. There are several options in the randomized path selection.

Time slot determination

The time slot is selected uniformly on the time schedule proposed by the user in his/her request. In France, the slot has a maximum duration of one hour.

Path Initialisation The first step is to randomly select the initial direction: left or right with probability 1/2, 1/2 on the street/edge of the home address.

Non-backtracking algorithm: at each intersection J the algorithm randomly selects the exit road excluding the entry street. If the intersection has four roads, thus three eligible exits, the weight of each exit is 1/3, 1/3, 1/3. In case of deadend, the algorithm selects the U turn. This guarantees the uniformisation of the densities. If the cartesian distance to home address exceeds the authorized distance (1km in France), the path should make a U-turn.

Non-backtracking algorithm with reduced path diversions: if the intersection has an even number of streets (thus an odd number of exits) more weight will be given to the exit street facing the entry street, i.e. the median exit. For example, the median exit has 1/2 probability to be selected, and the other exits are uniformly selected. This minor trick still maintain the uniformization of the densities.

Estimation of exposure reduction during Covid-19 lock-downs

Let E be the street set and for s ∈ E, let (s) its length and λ(s, t) its linear density at time t. The total length of the pedestrian network is L = s∈E (s). The total pedestrian population at time t is N (t) = s∈E λ(s, t) (s). The probability that a pedestrian walk on segment s at time t is ρ(s, t) = (s)λ(s,t) N (t) . If we assume that the distribution on every segment is uniform (and ignoring side effects) the probability that a given pedestrian on segment s is within distance r to another given pedestrian on the same segment is 2r (s) which could be reduced to r (s) if the dangerous zone stands forward to the pedestrian. Thus if a total population of N pedestrians are present on the streets, each independently following the density pattern of the λ(s, t), the probability that a pedestrian on segment s is not on situation of contact at time t is

1 -ρ(s, t) 2r (s) N or equivalently 1 -2r λ(s,t) N (t) N .
Let N T the total population of the city. If the proportion of inhabitants walking outdoor is ν, the average exposure rate E G (ν) satisfies

E G (ν) = 1 - s∈E ρ(s) 1 -ρ(s) 2r (s) νN T . ( 1
)
assuming that ρ(s) := ρ(s, t) does not vary (too much) during the permitted hours. If ρ(s, t) varies during the day and ρ(s) is its average on segment s, then by concavity the above expression is an upper bound of the exposure rate.

Theorem 1. The stationary distribution of the non-backtracking random walk is the uniform distribution density.

By uniform distribution we mean that the average densities in every street are identical.

Proof. This is mainly a state of the art result.

But the random walk may be long to converge to its steady state. In our case it is faster because the density of inhabitant in Paris is almost uniform along each street, thanks to Hausmannian construction rules dating from the XIXth century. From now we take the simplified assumption that walkers using the random walk are indeed in uniform density in the streets. Let E U (ν) be the average exposure rate when all walkers are random walkers. It satisfies:

E U (ν) = 1 -1 - 2r L νN T . (2)
Indeed it suffices to consider the whole pedestrian network as a single segment. More interestingly we have the case when a single random walker proceeds among νN T walkers following their social trends. Let's call E U G (ν) her/his exposure rate:

E U G (ν) = 1 - s∈E (s) L 1 -ρ(s) 2r (s) νN T . (3)
We assume that Parisians have a daily time span (10 hour) when excursions are permitted.

The figure 2 shows the cumulative time spent within unsafe distance to any another individual. Notice that with one hour of excursion time with all social walkers, each walker cumulates nearly half of this time at less than 1m of another individual. This duration drops to 8 minutes with random walking. If the excursions were extended to 5 hours, the social walkers would cumulate 4h30mn within unsafe distance, the random walkers around 2h30mn. (Left) red: all walkers are social but randomly dispatched on 10 hours (solid) on 5 hours (dashed); blue: when walkers are all random. (Right) red: for a single social walker, when all other walkers are social walkers during 1 hour, green: the single walker is random, dashed: social walkers select slot on an interval of 5 hours instead of 20 hours.

Path selection during lock-downs periods

Contrary to lock-down outdoor excursions which are mostly aimless (walking, jogging), the excursions during inter lock-downs periods are now aimed (commuting between home and work place). The public transportation will be mostly discarded because of poor social distancing. The city of Paris has increased the number of biking lanes. However the biking lanes will be very busy because people commute around the same hours. Our work is to extend Ariadne Covid application in order to help the user to find biking routes from home to work, which limit the virus exposure.

Path Selection Algorithm

An initial position of a user is on a segment (I U , I D ) ∈ E. The path will take the intersection among I U and I D which lies ahead of the destination z

f , i.e if z f -z 0 |I U -I D ≥ 0 then I U is selected: z 1 = I U , otherwise z 1 = I D .
At step number k of the algorithm, let z k be the intersection, where the path is currently ending. Let d be the degree of the intersection z k . If d = 1, then the path backtracks. If d > 1 then with probability the path proceeds as a nonbacktracking random walk, i.e. the path takes any other neighbor edge distinct of the entrance street with equal probability. Otherwise, with probability 1 -, the path takes one of the two best sectored streets, defined as follows. Assuming the heading of the segment (z k , z k-1 ) is θ 0 , and we enumerate θ 1 , θ 2 , . . . θ d-1 the headings in increasing order of the other exit segments of the intersection z k . Let θ be the heading of the vector (z k , z f ) toward the destination.

The isotropic walk algorithm Let j be such that j 2π d < θ ≤ (j + 1) 2π d . Note here that j + 1 must be considered by modulo of d. (4)

The selection of the next intersection z k+1 is done as follows: with probability α d (θ) it selects the edge corresponding to angle θ j , and with probability β d (θ) the edge corresponding to the angle θ j+1 . The algorithm is much less complex than the Dijkstra shortest path algorithm and contrary to the later provides path diversity. Indeed the shortest path algorithm leads to a complexity larger than the number of edges, while the randomized algorithm is at most linear, in the number of vertices, in fact proportional to the diameter of the graph.

Definition 1 (Isotropic walk condition). A walk is isotropic, when reaching any given intersection I, the difference of an angle θ toward a destination with an angle θ 0 to an arriving edge, satisfies a certain distribution P I (θ -θ 0 ). Theorem 2. Under the isotropic walk condition, and assuming the same constant speed v of all travellers in each street, the aggregation of paths leads to uniform densities of travellers on streets.

Proof. We show that uniform rate situation is stationary. Let's consider that a walker enters an intersection I of degree d. In the epsilon mode the walker uniformly exits on the other streets. In the sectored mode, enumerating the edges in counter clockwise way from the entrance street, the isotropic walk conditions leads to the probability p d (j) that the edge j is selected: (

p d (j) =
Let us assume that before time t all streets have same exit rate ρ. Therefore the entrance flow on any of the streets is ρ + (1 -) j=d j=0 p d (j)ρ = ρ and consequently the exit rates are uniform beyond t. Assuming a uniform speed v on each street, the density of travellers on each street is ρ/v per unit length.

The Geo-routing Algorithm

We introduce the geo-routing algorithm inspired from [START_REF] Jacquet | Opportunistic in wireless ad hoc networks: Upper bounds for the packet propagation speed[END_REF], which is similar to the isotropic walk with the difference that we replace the 2jπ/d by the actual angles of the streets. The consequences are twofold: (i) the average exit path is heading toward the destination, (ii) we lose the pure isotropic property, the densities in the exit streets will now vary with the angles between the streets.

Simulation of the algorithms

We have simulated the algorithms on the map of the city of Kaliningrad, famous for its network of bridges, river and marshes (figure 3). Instead of the classic grid maps of modern cities, the natural obstacles, blocking direct headings, hamper the performances of the algorithms. Indeed we get some blocking loops. In order to cope with looping problems we used the concept of the so-called accessibility graph, see [START_REF] Lentz | Unfolding accessibility provides a macroscopic approach to temporal networks[END_REF]. Let G = G(V, E) be the street graph. The second degree accessibility graph of the graph G * G or G * 2 is the graph whose vertex set is V and the edge set is E 2 such that for (x, y) ∈ V 2 , (xy) ∈ E 2 if there exists z ∈ V such that (xz) and (zy) belong to E. We have computed G * 2 , G * 4 and G * 8 . If the algorithm on a path determination fails on G because of a permanent loop, we run the algorithm on G * 2 . Then, if it again fails on G * 2 k we run the algorithm on the accessibility graph of the next degree G * 2 k+1 until the accessibility graph becomes fully connected graph (we never get to this). When the path is determined one unfolds the path on the original graph G. We have simulated the application of the isotropic walk and the geo-routing algorithm over 1,000 requests between the same pair initial point and destination point. We set = 0. The figure 4 shows the various path provided by the algorithms. Let s ∈ E, (s) is the segment length, and let λ(s) be the traffic load after N 0 initial-destination random pairs. The average path length is L G = 1 N0 s∈E λ(s) (s). The average travel time is L G /v. If the simulated time is T , the segment s sees an average entrance rate λ(s)

T and the density on the segment length is λ(s) vT , assumed to be Poisson. If we consider a total number of travellers N , then the density on a segment is λ(s) vT N N0 , where N 0 is a number of travellers needed to simulate the estimate of λ(s). Given the Poisson density, a probability that a random traveller at a random time on the segment is not within distance R 0 of another traveller is exp -λ(s) 2R0 vT N N0 , R 0 is the safe distance against the virus (for biking 10m). We write the expression of an average cumulative exposure time E(N ) for a random walk (traveller):

E(N ) = s∈E λ(s) (s) vN 0 1 -exp -λ(s) 2R 0 vT N N 0 . ( 6
)
Figure 5 shows the simulation of the path length and exposure times with both algorithms after N 0 = 5, 000 random initial-destination pairs. The geo-routing algorithm shows a slightly better path length and cumulated exposure.

Conclusions

We have presented the path selection algorithms and efficient path selection, which limits outdoor virus exposure during or between lock-down periods by path diversity and exploiting the geometry of street networks in cities. The proposed solution is based on low complexity algorithms. For practical applications one needs to take into account that success rate of the algorithm implementation strongly depends on users acceptance of those rules, which is left as an outlook.

Fig. 1 .

 1 Fig. 1. Histogram of Pedestrian densities in Paris

Fig. 2 .

 2 Fig. 2. Virus exposure time (min) as function of outdoor excursion duration (in hours).(Left) red: all walkers are social but randomly dispatched on 10 hours (solid) on 5 hours (dashed); blue: when walkers are all random. (Right) red: for a single social walker, when all other walkers are social walkers during 1 hour, green: the single walker is random, dashed: social walkers select slot on an interval of 5 hours instead of 20 hours.

  Let variables α d (θ) and β d (θ) such that α d (θ) + β d (θ) = 1 and α d (θ)e 2ijπ/d + β d (θ)e 2i(j+1)π/d be proportional to e iθ . Numerically we have α d (θ) = sin(2(j+1)π/d-θ) sin(θ-2jπ/d)+sin(2(j+1)π/d-θ) β d (θ) = sin(θ-2jπ/d) sin(θ-2jπ/d)+sin(2(j+1)π/d-θ) .

P

  I (θ)α d (θ) + 2jπ/d 2(j-1)π/d P I (θ)β d (θ).

Fig. 3 .

 3 Fig. 3. Left: Illustration of steps of the randomized geo-routing algorithm. The traveller enters the intersection via the large blue arrow. Right: the map of Kaliningrad.

Fig. 4 .

 4 Fig. 4. Path diversity in central Königsberg. Left: geo-routing algorithm. Right: isotropic walks. In green the streets belonging to less than 25 paths, in blue, between 25 and 50, in red, between 50 and 200. In black, above 200.

Fig. 5 .

 5 Fig. 5. In blue the isotropic routing, in brown the geo-routing in Kaliningrad (Königsberg). (Left) average path length in km. (Right) the exposure time in minute as function of travellers number N , peak time duration: 2 hours.


David Gérard

Françoise Méchin

René Saint-Loup

Etienne Fleury

Jean-Pierre Pascault

Study of the carbamate / aldehyde reaction, a new pathway towards NIPU materials

Keywords: aldehyde, primary carbamate, non-isocyanate polyurethane (NIPU), geminated urethanes, thermosetting coating, biobased, vanillin

An innovative alternative to the synthesis of non-isocyanate polyurethanes (NIPUs) thermoset materials was thoroughly investigated. This pathway, involving the reaction between primary carbamate functions and aldehyde functions, was studied using model reactions between ethyl carbamate and different aldehydes that lead to the highlights of the key parameters as well as the first reported proposed mechanism for this reaction. In a second time, polycarbamates synthesized from two different polyols were synthesized via transcarbamoylation before participating in a polycondensation reaction with different monoaldehydes. First a linear NIPU oligomer was synthesized using a dicarbamate derived from ethylene glycol and benzaldehyde. This product was analyzed by NMR spectroscopy and MALDI-TOF mass spectrometry. This oligomer exhibited low molar masses attributed to many secondary termination reactions, mainly due to the presence of acetic anhydride used as solvent in the reaction. Cross-linked NIPU materials were then synthesized without solvent, from the reaction between a tricarbamate derived from trimethylol propane and vanillin, a biobased monoaldehyde, for the first time. The process parameters for the synthesis of these materials were optimized to obtain biobased NIPU thermosetting networks with gel fraction (GF) in dimethylsulfoxide (DMSO) up to 82%, with Tg values ranging between 39°C and 65°C depending on the GF value. This study brings a much needed highlight on the little-known carbamate / aldehyde reaction and demonstrates its potential for the synthesis of novel biobased NIPU thermosetting coatings using a simple monoaldehyde such as vanillin.

INTRODUCTION

The vast majority of the crosslinking systems currently involved in the synthesis of thermosetting polymers require the use of hazardous components for the environment as well as for the operators. Polyurethanes (PU), which are among the most important polymer materials thanks to their versatile properties, are not an exception to this issue. Polyurethanes are indeed classically obtained by polyaddition between a polyol and a polyisocyanate, the synthesis of the latter involving the use of highly toxic derivatives such as phosgene [START_REF] Bayer | Das Di-Isocyanat-Polyadditionsverfahren (Polyurethane)[END_REF]. The development of safer and eco-friendlier alternatives to those already existing crosslinking systems are thus of particular interest for both academic and industrial research teams [START_REF] Maisonneuve | Isocyanate-free routes to polyurethanes and poly(hydroxy urethane)s[END_REF][START_REF] Argyropoulos | Ambient Temperature Curable Isocyanate-Free Compositions for Preparing Crosslinked Polyurethanes[END_REF]. Two synthesis pathways attracted most of the attention these last few years in order to obtain non-isocyanate polyurethane (NIPU): the transurethanization of biscarbamates and polyols [START_REF] Guan | Progress in study of nonisocyanate polyurethane[END_REF][START_REF] Delebecq | On the versatility of urethane/urea bonds: reversibility, blocked isocyanate, and non-isocyanate polyurethane[END_REF][START_REF] Hahn | Hydroxyl-functional polyurethanes and polyesters: Synthesis, properties and potential biomedical application[END_REF][START_REF] Unverferth | Renewable non-isocyanate based thermoplastic polyurethanes via polycondensation of dimethyl carbamate monomers with diols[END_REF], and the polyaddition of polyamines and bis-or poly-cyclic carbonates [START_REF] Kathalewar | Non-isocyanate polyurethanes: from chemistry to applications[END_REF][START_REF] Tomita | Model reaction for the synthesis of polyhydroxyurethanes from cyclic carbonates with amines: substituent effect on the reactivity and selectivity of ring-opening direction in the reaction of five-membered cyclic carbonates with amine[END_REF][START_REF] Nohra | From petrochemical polyurethanes to biobased polyhydroxyurethanes[END_REF][START_REF] Carré | From the synthesis of biobased cyclic carbonate to polyhydroxyurethanes: a promising route towards renewable non-isocyanate polyurethanes[END_REF][START_REF] Pyo | Cyclic carbonates as monomers for phosgene-and isocyanate-free polyurethanes and polycarbonates[END_REF]. If both ways lead to the formation of PU without the direct use of any isocyanate or phosgene derivatives, they still suffer from many weaknesses. In the first case, processing at high temperatures is required. In the second case, a slow kinetics (particularly for 5-membered ring carbonates) as well as side reactions leading to low molar masses are often encountered [START_REF] Besse | How to explain low molar masses in polyhydroxyurethanes (PHUs)[END_REF]. Although recently some of those pitfalls were circumvented by the use of multifunctional precursors lowering the gel conversion [START_REF] Capar | Fast curing of polyhydroxyurethanes via ring opening polyaddition of low viscosity cyclic carbonates and amines[END_REF], or avoided by that of activated carbonates such as a-alkylidene 5-membered carbonates [START_REF] Gennen | CO2-sourced α-alkylidene cyclic carbonates: a step forward in the quest for functional regioregular poly(urethane)s and poly(carbonate)s[END_REF][START_REF] Habets | Advancing the synthesis of isocyanate-free poly(oxazolidones)s: scope and limitations[END_REF] leading to thermally stable unsaturated polyoxazolidones, the problem remains when considering the synthesis of standard β-hydroxy urethane bonds. This explains why these synthetic routes are still limited to academic researches despite an outstanding applicative potential [START_REF] Khatoon | A review on the production, properties and applications of non-isocyanate polyurethane: a greener perspective[END_REF].

The present work aims to study an alternative pathway to making novel NIPU materials using the carbamate / aldehyde (CA) reaction. This reaction is operated in acidic conditions and implies two carbamate functions for each aldehyde group, thus forming a double urethane function on a single carbon that will be called a geminated diurethane function in this paper (FIGURE 1). It is important to note that only primary carbamates can be involved in this reaction. Moreover, this reaction would be particularly well suited to the preparation of innovative and potentially biobased organic coatings, or even foams, since it generates water.

FIGURE 1 Geminated diurethane synthesis from the reaction between primary carbamates and an aldehyde under acidic conditions.

Although being known since the 19 th century with the works of D. Roth in 1870 [START_REF] Roth | Über einige Verbindungen von Benzoëaldehyd mit primären Monamiden[END_REF], this reaction has not been extensively studied even to this day, providing very little information about the reaction's key parameters or mechanism. However, quite recently, it received more attention as polycarbamate / polyaldehyde cross-linking has been reported to make NIPU coatings [START_REF] Argyropoulos | Ambient Temperature Curable Isocyanate-Free Compositions for Preparing Crosslinked Polyurethanes[END_REF][START_REF] Anderson | Crosslinkable Composition, a Method of Making the Same and a Crosslinked Composition Produced Therefrom[END_REF][START_REF] Greyson | Crosslinkable Composition and Method of Producing the Same[END_REF][START_REF] Silbert | Biobased, Nonisocyanate, 2K polyurethane coatings produced from polycarbamate and dialdehyde cross-linking[END_REF]. Indeed, it possesses many advantages that make it suitable and interesting to obtain thermosetting PU resins, such as versatility and possibility to cure at ambient temperature with a simple acid catalyst. However, a significant limitation lies in the lack of readily available polyaldehydes. In fact, publications on this cross-linking systems are often limited to a mixture of 1,3-and 1,4-cyclohexanedicarboxaldehyde (CHDA), a petro-derived dialdehyde [START_REF] Argyropoulos | Ambient Temperature Curable Isocyanate-Free Compositions for Preparing Crosslinked Polyurethanes[END_REF][START_REF] Anderson | Crosslinkable Composition, a Method of Making the Same and a Crosslinked Composition Produced Therefrom[END_REF][START_REF] Greyson | Crosslinkable Composition and Method of Producing the Same[END_REF]. Silbert et al. proposed the diformylfuran (DFF) as a bioderived alternative to CHDA, but until recently the market availability and price of this product was still an issue [START_REF] Silbert | Biobased, Nonisocyanate, 2K polyurethane coatings produced from polycarbamate and dialdehyde cross-linking[END_REF]. In fact, most of the authors using DFF as the precursor of new polymer materials synthesized this reactant by themselves [START_REF] Silbert | Biobased, Nonisocyanate, 2K polyurethane coatings produced from polycarbamate and dialdehyde cross-linking[END_REF][START_REF] Hajj | Network Design to Control Polyimine Vitrimer Properties: Physical Versus Chemical Approach[END_REF].

In this paper, we propose an in-depth study of the CA reaction using model compounds that allowed us to establish a mechanism highlighting the key parameters of this pathway. Thereafter, the synthesis of new NIPU linear oligomers and networks was explored using monoaldehydes, much more available than their difunctional homologues. Moreover, in addition to petro-based monoaldehydes, bio-based monoaldehydes were also studied in this system, and especially vanillin considered as a promising biobased building-block in polymer chemistry [START_REF] Fache | Vanillin, a promising biobased building-block for monomer synthesis[END_REF]. Two main processes were explored: the first is operated at low temperature (<80°C) in presence of acetic anhydride that, as shown below does not only act as a solvent (process A) but also takes part in the reaction by scavenging water, also leading to undesired side products. The second one is operated in bulk at higher temperature (>100°C)

(process B). The obtained NIPU oligomers and networks were then characterized and the results were discussed in the lights of the observations highlighted in the mechanistic study.

EXPERIMENTAL SECTION

Chemicals

All chemicals were used as received. Products for model reactions are: ethyl carbamate, benzaldehyde, vanillin, 5-methyl furfural (5-MF), cyclohexane carboxaldehyde (CC), n-heptanaldehyde, trifluoroacetic acid (TFA), ethylphosphonic acid (EPA), acetic acid, and acetic anhydride, all purchased from Sigma-Aldrich. For the synthesis of NIPU linear oligomers and networks, polycarbamates were synthesised directly using ethylene glycol, trimethylolpropane (TMP) methyl carbamate and dibutyltin oxide (DBTO), all obtained from Sigma-Aldrich. Dimethyl sulfoxide (DMSO) from Sigma-Aldrich was used in synthesis as well as in analysis, and especially for NMR spectroscopy using its deuterated form (DMSO-d6) purchased from Euriso-Top. This list, completed with molecular structures and CAS numbers, is presented in the supporting information (SI).

Analytical Methods

NMR spectra were recorded at room temperature on a Bruker Avance spectrometer at 400.13 MHz for 1 H and 100.61 MHz for 13 C with a 5 mm BBFO1 probe at 300K and processed with Bruker Topspin 3.2 software. 2D analyses were also recorded using this spectrometer, especially 1 H-1 H homonuclear correlation spectroscopy (COSY) and 1 H- 13 C heteronuclear spectroscopy (HSQC and HMBC).

Thermal properties were obtained with a DSC Q22 (TA Instruments). Aluminum hermetic capsules were used, and samples of about 5 mg were weighed. Samples were heated from -80°C to 180°C, at a rate of 10°C/min.

Thermogravimetric analyses were performed with a Q500 equipment (TA Instruments). Samples were weighed in an aluminum pan, then placed in an oven heated at a fixed isotherm (100°C, 110°C or 120°C). A nitrogen or air flow (90 mL/min) was introduced into the oven to study the role of oxidation on the samples.

The MALDI-TOF mass spectrometry analysis was performed with a Voyager-DE PRO (Applied Biosystems, Framingham, MA) equipped with a nitrogen laser emitting at 337 nm with a 3-ns pulse duration. The instrument was operated in linear and reflectron modes. Samples were prepared by mixing them in a CHCA (α-Cyano-4-Hydroxycinnamic Acid) matrix with a sodium salt to enhance cationization, using DMSO as co-solvent. The resulting mixtures were spotted on the MALDI sample plate and air-dried before analysis.

Gel fractions (GF) of NIPU networks were assessed by inserting a known mass mi of the recovered material in a known volume of DMSO, a good solvent of the monomers and PU oligomers. After 2h at 25°C, the sample was filtered, washed with DMSO and acetone before drying and weighing (mf). The gel fraction was then given by GF = mf/mi. The aim is to obtain a fully insoluble material (GF = 100%) in order to prove the complete reticulation of the network.

Typical procedure for the model reaction synthesis

The procedure used to study the CA reaction was inspired by the literature [START_REF] Dmitriev | New opinions on the amidoalkylation of hydrophosphorylic compounds[END_REF]. A typical synthesis was as follows: benzaldehyde (2.4 g, 1 eq) and ethyl carbamate (4 g, 2 eq) were introduced in a roundbottom flask with 8 mL (3.8 eq) of acetic anhydride and stirred at room temperature until complete homogenisation. Trifluoroacetic acid, TFA was then added (0.27 g, 0.1 eq), marking the start of the reaction. In each case, a white precipitate was observed after only few minutes of reaction and therefore the reaction was stopped after 1 hour, more than enough to complete the reaction. The precipitate was then filtered and washed by acetic anhydride, then dried in an oven at 40°C before characterization. Finally, 1 H and 13 C NMR analyses were performed to confirm the geminated diurethane structure of the product.

Typical procedure for the synthesis of polycarbamate precursors from polyol by transurethanization reaction

In all cases for the subsequent synthesis, polycarbamates were synthesized as a first step from polyols which underwent a transcarbamoylation (or transurethanization) reaction with methyl carbamate in presence of DBTO as described in the literature [START_REF] Argyropoulos | Ambient Temperature Curable Isocyanate-Free Compositions for Preparing Crosslinked Polyurethanes[END_REF][START_REF] Anderson | Crosslinkable Composition, a Method of Making the Same and a Crosslinked Composition Produced Therefrom[END_REF][START_REF] Greyson | Crosslinkable Composition and Method of Producing the Same[END_REF]. Ethylene glycol was used in the case of the difunctional carbamate leading to linear NIPU oligomers while TMP was used as a precursor of the trifunctional carbamate to obtain NIPU networks.

The polyol, being either ethylene glycol or TMP, (40 mmol) and methyl carbamate (320 -400 mmol)

were introduced in a two-neck round-bottom flask and heated at 120°C under magnetic stirring. Upon complete fusion of the methyl carbamate, the catalyst (DBTO, 2 mmol) was then added, and the formed mixture was stirred at 120°C under a constant nitrogen flow until completion of the reaction (18-24h), as indicated by the NMR spectra of the reaction mixture (SI, Figures S1-S3). Finally, the crude product was washed by acetone and dried before being analysed and used in further synthesis.

Methyl carbamate was used in excess because of its tendency to sublime during the reaction, which can impact the stoichiometry if not taken into account. In order to lessen the impact of this phenomenon, the flask was equipped with a water column, with the temperature of water flowing through it fixed at 60°C. This device allowed most of the methyl carbamate vapours to condense in the column and thus fall back into the reaction medium.

Typical procedure for the synthesis of NIPU linear oligomers and networks

The previously obtained polycarbamates were further used in two different processes.

Synthesis of linear oligomers

The first one (process A) is a direct application of the aforementioned model reaction synthesis process and was only used for the synthesis of linear NIPU oligomers. A typical synthesis for A is as follows: benzaldehyde (0.17 g, 1 eq) and ethylene dicarbamate (0.25 g, 1 eq), obtained from ethylene glycol, were introduced in a round-bottom flask with 0.6 mL (3.8 eq) of acetic anhydride and stirred at 80°C until complete homogenisation. TFA was then added (0.02 g, 0.1 eq), marking the start of the reaction.

The reaction was then continued for 15 h before washing the product with the reaction solvent acetic anhydride (to remove unreacted monomers) and filtration. The product underwent NMR analysis in order to validate its chemical structure.

Synthesis of NIPU networks

A second process (process B) was carried out in bulk and allowed to obtain NIPU networks. In a typical procedure, the tricarbamate (0.5 g, 1 eq), obtained from TMP, and vanillin (0.43 g, 1.5 eq) were mixed in a ceramic cup. This step can be helped by heating the premix above the melting temperature of the aldehyde (81.5°C). EPA (0.02-0.08 g, 0.1-0.4 eq) was then added and mixed with the blend to start the reaction. The obtained premix was then placed in an oven where the temperature and atmosphere of the curing step can be adjusted. After 24 h of curing, the resulting material was cooled at room temperature before characterization, and especially determination of the GF.

RESULTS AND DISCUSSION

Model reactions using monofunctional compounds

In order to understand the fundamental characteristics and key parameters of the reaction, model reactions between an aldehyde and ethyl carbamate were first studied. Inspired by a simple experimental procedure of the literature that seemed to lead to good yields with rather similar precursors, acetic anhydride was chosen as solvent [START_REF] Dmitriev | New opinions on the amidoalkylation of hydrophosphorylic compounds[END_REF]. Investigations were more particularly focused on three main parameters: the nature of the aldehyde's substituent used in the reaction, the nature and influence of the acid catalyst, and the role of acetic anhydride in the synthesis. All model reactions were performed in the same conditions: 1 h at room temperature, and a yield was calculated by weighing the washed product and comparing it to the theoretical mass, as presented in Table 1. The reaction between benzaldehyde and ethyl carbamate was first operated in order to validate this reaction (Table 1, entry 1) and a yield of 69 % was obtained in accordance with the literature [START_REF] Dmitriev | New opinions on the amidoalkylation of hydrophosphorylic compounds[END_REF].

Subsequent trials made clear that this reaction can be extended to various aldehydes, aromatic as well as linear aliphatic (Table 1, entries 2-5) with the same efficiency. Moreover, three of these aldehydes are potentially biobased (5-methyl furfural, heptanaldehyde and vanillin), showing that at least partially biobased PU can be obtained through this pathway. During the reaction, and as shown in Figure 1, water is formed and can subsequently react with the acetic anhydride used as a solvent but that obviously also plays the role of desiccant. Acetic acid is then formed in situ, both from ambient humidity and from the reaction with water.

The question was then: is it possible to use only this created acetic acid as the required acid catalyst?

After 1 h when no acid catalyst was used (Table 1, entry 6), no reaction was observed, showing that the in situ formation of acetic acid from acetic anhydride is not enough to catalyse the reaction.

Moreover, it was observed that the catalytic activity of acetic acid was weaker than that of TFA in the same conditions (Table 1, entry 7). This presumably results from the difference of pKa of the two acids (pKa(acetic acid) = 4.7 whereas pKa(TFA) = 0.3). In order to replace TFA with a safer acid catalyst, different other organic acids were tested that would offer better alternatives than metal catalysts both for health and environment. Indeed, metallic catalysts are often hazardous and difficult to eliminate in the final product, which is why organic catalysts are seen more and more as the preferable alternative [START_REF] Bossion | Opportunities for organocatalysis in polymer synthesis via step-growth methods[END_REF]. In this case, we chose to use phosphonic acids, since they have already shown a good activity for PU synthesis [START_REF] Belmokaddem | Novel nucleophilic/basic and acidic organocatalysts for reaction between poorly reactive diisocyanate and diols[END_REF]. And with our system, ethylphosphonic acid, EPA was a good candidate : it gives similar results as TFA under the same conditions (Table 1, entry 8) but, although displaying a rather comparable toxicity, it has a significantly lower impact on the environment (no hazard statement relative to environment whereas TFA is classified H412 "Harmful to aquatic life with long lasting effects"). Finally, the reaction was operated in bulk (Table 1, entry 9), and a negative impact on the reaction from the absence of acetic anhydride could be noted, which shows that this compound does not only act as a solvent. We hypothesized that acetic anhydride also acts as a hygroscopic agent, consuming the water formed during the reaction and thus shifting the equilibrium of the reaction towards the product.

Mechanistic study of the CA reaction

From the previous experimental data, a tentative mechanism of reaction was proposed and further validated, as presented in FIGURE 2.

This mechanism is divided into 6 steps that can be detailed as follows:

① Activation of the carbonyl by the acidic proton from the catalyst.

② Nucleophilic attack of the carbonyl by the nitrogen atom of a first carbamate group.

③ Structural rearrangement in order to obtain a hemiaminal structure ④ Release of a water molecule and formation of an intermediate in the form of an iminium ion that can stabilize itself directly in its imine form. However, even when fostering the monosubstitution by an excess of aldehyde, the imine form was never observed in our model monofunctional reactions. Indeed, this imine might not be stable if the released water is not properly eliminated from the medium.

⑤ Stabilization of the iminium intermediate by the nucleophilic attack of a second carbamate group.

⑥ Release of an acidic proton in order to obtain the geminated diurethane structure and thus closing the catalytic cycle of the reaction.

FIGURE 2 Proposed mechanism for the carbamate / aldehyde reaction

Moreover, in order to improve the conversion of the reaction the necessity to eliminate water from the reaction medium as soon as it is formed was already pointed out. This translates to the existence of equilibria in the steps ① through ④. Likewise, the last two steps can be considered as equilibria;

this would be consistent with the acidic hydrolysis observed in the past under appropriate conditions (i.e. by boiling for a short time with normal aqueous hydrochloric acid) for the geminated diurethane compounds and leading back to the original reactants [START_REF] Martell | Condensation of amides with carbonyl compounds: benzyl carbamate with aldehydes and alpha keto acids[END_REF].

Evidences of the iminium intermediate's existence

In order to validate this mechanism, the existence of the proposed iminium intermediate needs to be proved. It was first attempted to observe directly the intermediate in situ using FTIR-ATR monitoring of the CA reaction at 100°C for 2 hours (SI, Figure S4). However, this analysis was not convincing enough to prove the existence of this iminium intermediate, therefore a second method was used to observe it, indirectly this time. Indeed, when looking at the mechanism of the reductive amination of aldehydes in FIGURE 3, it appears that the four first steps of this reaction are identical to those of the proposed mechanism (with R = COOR 1 ). The idea was then to add a reducing agent such as STAB (Sodium Triacetoxyborohydride) in the model CA reaction between ethylcarbamate and benzaldehyde (Table 1, entry 1). The raw product of this reaction was then analysed by complete NMR spectroscopy ( 1 H, 13 C, COSY, HSQC and HMBC) in order to identify the different structures obtained thereof. The obtained spectra are displayed in FIGURE 4 and in SI (Figures S5-S9).

FIGURE 3 Reaction mechanism of the reductive amination of aldehydes [28]

FIGURE 4 1

H NMR spectrum in DMSO-d6 at room temperature of the raw products of the CA reaction between ethylcarbamate and benzaldehyde in presence of STAB. Crossed signals are due to the presence of acetic anhydride and acetic acid in the raw sample.

The 1 H NMR spectrum displayed in FIGURE 4 shows that 3 different structures can be identified: a geminated diurethane structure (1) from the CA reaction between ethylcarbamate and benzaldehyde (whose presence is confirmed by the small "c" signal at 6.05 ppm); a monourethane structure (2) from the reductive amination of benzaldehyde by ethylcarbamate and STAB; and finally, benzyl acetate as the majority structure (3). The 1 H NMR spectra of structures 2 and 3 (respectively ethyl N-benzyl carbamate [2621-78-5] for product 2, and benzyl acetate [140-11-4] for compound 3) can be found in several literature databases and confirm the presence of those 2 compounds. Structure 3, although unwanted, can be explained by a reaction between benzyl alcohol and acetic acid in acidic conditions, which explains its high proportion since acetic anhydride, giving acetic acid directly through hydrolysis, is the solvent of the reaction. Benzyl alcohol comes directly from the reduction of benzaldehyde by a reducing agent such as STAB (FIGURE 5).

FIGURE 5 Origin of the observed benzyl acetate from benzaldehyde in presence of a reducing agent (STAB)

However, an important result is the concomitant presence of both the structures 1 and 2 that means that there was a competition between the CA reaction and the reductive amination, hence our conclusion that both reactions imply the same iminium intermediate.

As a conclusion to this part, we first tried to observe directly the hypothesized iminium intermediate for the CA reaction by FTIR-ATR analysis. However, the results were not conclusive enough to validate the existence of such an intermediate. We then showed that the CA reaction indeed seems to pass through an iminium intermediate by stabilizing it in another form using the reductive amination reaction.

Polycarbamate synthesis from the transcarbamoylation reaction of polyols

The first step towards the synthesis of PU using CA reaction is the synthesis of polycarbamate precursors. These precursors are obtained from the transcarbamoylation reaction of polyols with methyl carbamate (FIGURE 6).

FIGURE 6 Synthesis of a polycarbamate from a polyol and methyl carbamate through transcarbamoylation reaction

Two polyols were used depending on the functionality wanted on the final polycarbamate precursor:

ethylene glycol for the synthesis of a dicarbamate, and TMP for the synthesis of a tricarbamate. After purification, the yield was 35-45% for the dicarbamate, and 60-70% for the tricarbamate, with the best yields corresponding to larger excesses of methyl carbamate. In both cases, the washed product was analyzed by NMR spectroscopy showing a satisfactory purity for subsequent CA reactions. However, the 1 H NMR spectra displayed in the SI (Figures S1,S3) show that traces of excess methyl carbamate can remain in the final polycarbamate. Likewise, some alcohol functions from the original polyol can remain on the polycarbamate chain, even if these traces are not observed on the NMR spectra. These residual traces can be the cause of unwanted secondary reactions, as observed later in this paper.

Linear polymerization through CA reaction

Using process A, a NIPU oligomer was synthesized from ethylene dicarbamate and benzaldehyde, using acetic anhydride as solvent and TFA as acid catalyst (FIGURE 7 i)). The obtained oligomer was then analysed by NMR spectroscopy as well as mass spectroscopy (MALDI-TOF). Using this last technique, molar masses up to 3760 g.mol -1 (DP=15) with a very broad distribution were detected (SI, Figure S10). SEC analysis was unable to confirm these results due to insolubility issues of the oligomer in all tested possible eluents. However, mass spectroscopy analysis was especially useful to identify the different structures in our sample, explaining the broad distribution and low molar masses (FIGURES 7 and 8). The first structure (a) is in majority, and has been identified as the expected product of the CA reaction between ethylene dicarbamate and benzaldehyde (FIGURE 7 i)). Structure b presents one imine group as chain end, which can be due to the direct stabilization of the iminium intermediate during the CA reaction (FIGURE 7 i)). Although this imine group was not observed in the monofunctional model reactions, our hypothesis is that this structure becomes more stable as the chains grow. Structure c presents an alcohol function that terminated the reaction (FIGURE 7 ii)). This alcohol function may originate from residual alcohol functions in the ethylene dicarbamate used for the synthesis, as noted previously. Finally, both structures d and e come from a secondary reaction involving acetic anhydride and terminating the reaction (FIGURE 7 iii) and iv)). One can assume that the low molar masses obtained for the studied oligomers are directly linked to the many different secondary end-capping reactions brought into play, especially due to the presence of acetic anhydride that behaves as a chain stopper. This justifies why it was decided to rule out acetic anhydride in the next step.

FIGURE 7 Synthesis of a NIPU linear oligomer from ethylene dicarbamate and benzaldehyde, showing secondary reactions leading to alternative structures identified in the sample

Several attempts of linear polymerization were at first made in the presence of TFA, using DMSO as solvent and replacing acetic anhydride with other water scavengers such as anhydrous magnesium sulfate, ethyl orthoformate or even molecular sieves (3 Å); however for all those trials no reaction was observed, neither at 25 nor at 80°C.

Another attempt was then made in DMSO at 100°C, without acetic anhydride nor any other desiccant, and after 24 h led to a yield ~ 52%, without the side products ascribed to acetic anhydride (see SI, Figure S11). The full chemical characterization of this product was impeded by its poor solubility, except for the shortest oligomers, but this process could be a suitable way of preparing satisfactory thin coatings by CA reaction.

In order to prepare thick, solid materials, it was finally decided to use a different process (process B), which also allowed to successfully eliminate most secondary reactions.

NIPU network synthesis through CA reaction FIGURE 9 General synthesis of a NIPU network from a tricarbamate and a monoaldehyde

To the best of our knowledge, only two other teams developed NIPU networks using CA reaction [START_REF] Argyropoulos | Ambient Temperature Curable Isocyanate-Free Compositions for Preparing Crosslinked Polyurethanes[END_REF][START_REF] Silbert | Biobased, Nonisocyanate, 2K polyurethane coatings produced from polycarbamate and dialdehyde cross-linking[END_REF],

and in each case, dialdehydes were used in order to obtain the materials. However, since each aldehyde function has a functionality of 2 in the CA reaction, reacting a monoaldehyde with a polycarbamate with a functionality strictly higher than 2 in stoichiometric proportions is enough to obtain a cross-linked material (FIGURE 9) [START_REF] Pascault | Thermosetting Polymers[END_REF]. In this paper, we report the formation of a NIPU network by reacting a tricarbamate obtained from TMP with a biobased monoaldehyde: vanillin.

In those preliminary trials only aiming at a first overview of the material aspect, we chose not to use a solvent (e.g. DMSO) that would probably allow the preparation of bubble-free samples with welldefined geometries. Therefore a bulk process (process B) was used to avoid using acetic anhydride that causes secondary reactions, and the used curing temperatures were above 100°C in order to eliminate most of the water formed during the reaction. However, rather thick samples (several millimetres) with numerous remaining bubbles were obtained. EPA has been chosen as the acid catalyst since it was the most adapted to the process conditions (mp = 61°C and low volatility). The temperature and atmosphere of the oven used for the 24h curing step were both adjustable parameters, as presented in the Table 2. The obtained NIPUs were found insoluble in all tested organic polar solvents (tetrahydrofuran, N,Ndimethyl formamide, N,N-dimethylacetamide and dimethyl sulfoxide, DMSO that was subsequently used to determine the gel fractions). The aim of the trials in Table 2 was to identify the important parameters in order to obtain the NIPU material with the most important gel fraction (GF) as possible.

Indeed, the first trial (Table 2, entry 1) led to a mere 2 % GF, far from the 100 % needed to validate the completion of the reaction. First, we investigated the impact of the proportion of acid catalyst used on the GF, but even when multiplying it by 4, the GF never exceeded 9 % (Table 2, entries 2 and 3).

The next investigated parameter was the temperature of the curing step and a significant raise of the GF could be observed, from 9 % at 100°C to 63 % at 120°C (Table 2, entry 3 to 5). However, raising the temperature of the curing step does not only affect the reaction rate, but also promotes the thermal degradation of primary carbamate functions. Indeed, a thermal degradation of polycarbamate molecules can be observed that accelerates with the temperature, with a mass loss of ~ 1 wt% per hour at 100°C, up to 6 wt% per hours at 120°C, both in air (SI, Figure S12). This explains the rather low GF obtained even at 120°C because this thermal degradation of primary carbamate functions competing with the CA reaction has a direct impact on its stoichiometry, hence limiting the maximum theoretical conversion.

However, it was observed that this thermal degradation could be reduced by half at 120°C by simply changing the atmosphere for an inert gas such as argon (SI, Figure S13). This can be explained by the fact that primary carbamate functions are particularly sensitive to thermooxidation, as illustrated by the release of CO2 and derivatives of cyanic acid or short isocyanate molecules suggested by ATG/FTIR coupling analysis (SI, Figure S14) in agreement with literature [START_REF] Adams | Esters of carbamic acid[END_REF]. Indeed, using an argon atmosphere during the curing step of the CA reaction allowed to lessen the degradation's impact on the stoichiometry, thus significantly increasing the GF, regardless of the temperature, up to 82 % at 120°C (Table 2, entries 6 to 8). Moreover, the increase in the Tg value of the material with the GF, from 39°C at 9 % GF up to 65°C at 82 % GF confirms the increase in the cross-linking degree of the system. Both DSC thermograms are given in the SI (Figures S15,S16). It can be noted that in both cases, the signal deteriorates, showing the degradation of the sample at high temperature. Moreover, those curves are difficult to exploit as they show many erratic fluctuations.

We could not obtain materials with a GF of 100 %, however such materials should presumably be obtained by finding the appropriate trade-off between reactivity and thermal degradation. For example other polycarbamates with a different chemical structure could be easier to mix with the used aldehydes, and might also display a higher thermal stability that would allow a broader temperature range for processing.

CONCLUSION

The comprehensive study of the carbamate / aldehyde reaction developed in this paper allowed to propose a mechanism for the first time, in accordance with all experimental observations. The suspected intervention of an iminium intermediate could be indirectly confirmed by the use of a reducting agent. Moreover, extensive NMR analysis also supports the reaction intermediate proposed in our mechanism.

Following the literature, in the preliminary studies involving low-functionality precursors (synthesis of model geminated diurethane and linear polymers), acetic anhydride was successfully used, both as solvent and water scavenger for the CA reaction; however, it turned out that it could also play the role of chain stopper and therefore limit the molar masses. It was then ruled out for the preparation of new materials.

As a follow up, NIPU oligomers and networks were synthesized, and we report for the first time the preparation of NIPU materials using vanillin through CA reaction. These materials present a gel fraction up to 82 % because of the trade-off between reactivity and thermal degradation during the curing step that disturbs the reaction stoichiometry. Although a few improvements are still needed, this work can be used as a bedrock for future investigations on the synthesis of new NIPU coatings based on CA reaction, and especially on using different polycarbamates in order to adjust the material's final properties. The use of a suitable solvent that could allow the preparation of thin, bubble-free coatings with optimized curing would also provide valuable additional characterizations of those new materials.

Finally, this reaction implies the formation of water molecules, sometimes in the form of bubbles that should be totally evacuated for coatings; but one could also take advantage of their presence to generate innovative foams, i.e. by adding appropriate surfactants to those formulations.

Supporting Information:

5.

MALDI-TOF analysis of the linear NIPU oligomer obtained from the CA reaction between benzaldehyde and ethylene dicarbamate 6.

Comparison between compounds synthesized with or without acetic anhydride 7.

Thermal degradation of primary carbamate functions TGA thermogram of methyl dicarbamate at different isotherms in air TGA thermogram of methyl dicarbamate at a 120°C isotherm in air and inert atmosphere TGA / FTIR coupling analysis of methyl dicarbamate in air, showing the TGA thermogram as well as its derivated curve

2206-27-1

Euriso-Top

Synthesis of polycarbamates from polyols

Ethylene Dicarbamate was synthesized using ethylene glycol as the polyol.

Comparison between compounds synthesized with or without acetic anhydride

FIGURE 8

 8 FIGURE 8 MALDI-TOF mass spectrum of the NIPU oligomers obtained from ethylene dicarbamate and benzaldehyde. All identified structures are highlighted.

8 .

 8 Thermal properties of NIPU networks DSC thermogram of NIPU network 1 (IP = 9 %) DSC thermogram of NIPU network 8 (IP = 82 %)

Figure S1 : 1 H

 1 Figure S1 : 1 H NMR spectrum in DMSO-d6 of the purified product

Figure S3 : 1 H

 1 Figure S3 : 1 H NMR spectrum in DMSO-d6 of the purified product

Figure S11 -

 S11 Figure S11 -Comparison of the 1 H NMR spectra (DMSO-d6, 25°C) of compound PL-2 (synthesized in solution in acetic anhydride, 80°C) and of the soluble fraction of compound PL-3 (synthesized in solution in DMSO, 100°C, without acetic anhydride) showing the clear difference in the δ area below 2,5 ppm

TABLE 1

 1 Model reactions between ethyl carbamate (2 eq) and different aldehydes after 1 h at 25°C

	Entry	Aldehyde (1 eq)	Acid catalyst (0.1 eq)	Acetic anhydride	Yield
				(eq)	(%)
	1	Benzaldehyde	TFA	3.8	69
	2	5-Methyl furfural	TFA	3.8	41
	3	Cyclohexane carboxaldehyde	TFA	3.8	66
	4	Heptanaldehyde	TFA	3.8	64
	5	Vanillin	TFA	3.8	82
	6	Benzaldehyde	none	3.8	0
	7	Benzaldehyde	Acetic acid	3.8	31
	8	5-Methyl furfural	EPA	3.8	43
	9	5-Methyl furfural	TFA	0	26

TABLE 2

 2 NIPU networks synthesis between vanillin (1.5 eq) and the tricarbamate from TMP (1 eq)

	Entry	Acid catalyst	Temperature	Curing	Gel fraction (%) Tg (°C)
		(EPA)	(°C)	atmosphere
	1	0.1 eq	100°C	Air	2	-
	2	0.2 eq	100°C	Air	8	-
	3	0.4 eq	100°C	Air	9	39°C
	4	0.4 eq	110°C	Air	33	-
	5	0.4 eq	120°C	Air	63	-
	6	0.4 eq	100°C	Argon	57	-
	7	0.4 eq	110°C	Argon	61	-
	8	0.4 eq	120°C	Argon	82	65°C

Study of the carbamate / aldehyde reaction, a new pathway towards NIPU materials David GÉRARD, 1 Françoise MÉCHIN, 1 René SAINT-LOUP, 2 Etienne FLEURY 1 , Jean-Pierre PASCAULT 1

	1 Univ Lyon, INSA Lyon, CNRS, IMP UMR 5223, F-69621, Villeurbanne, France
	2 Roquette Frères SA, Lestrem 62136, France
	1.	List of all used chemicals
	2.	Synthesis of polycarbamates from polyols
		1 H NMR spectrum in DMSO-d6 of the purified product
		13 C NMR spectrum in DMSO-d6 of the purified product
		1 H NMR spectrum in DMSO-d6 of the purified product
	3.	FTIR-ATR

kinetic analysis for the direct observation of the suspected iminium intermediate 4. Evidences of the presence of iminium intermediate's existence: complete NMR spectra of the reaction's raw product

	1 H spectrum
	13 C spectrum
	COSY spectrum
	HSQC spectrum
	HMBC spectrum

ACKNOWLEDGEMENTS

We thank the NMR Polymer Center of Institut de Chimie de Lyon (CNRS FR3023), for assistance and access to the NMR facilities. Authors also express their grateful thanks to Dr Catherine Ladavière (IMP -University Lyon1) for the MALDI-TOF analyses. This work was funded by the French Research Agency (ANR), as well as the society Roquette Frères. This work is dedicated to Jean-Pierre PASCAULT, whose knowledge and wisdom made this work possible.

FTIR-ATR kinetic analysis for the direct observation of the suspected iminium intermediate

The experiment was run in Attenuated Total Reflectance (ATR) mode using the UATR 2 device from Perkin-Elmer equipped with a heated top plate. As expected, the decrease in the signal at 1698 cm -1 corresponding to the carbonyl vibration of the aldehyde can be observed. This signal, together with the carbonyl from ethyl carbamate initially around 1710 cm -1 , never completely disappears because of the contribution of the formed urethane groups in the same carbonyl area during the reaction.

However, most of the characteristic bands of benzaldehyde tend to strongly decrease with time, suggesting a possible partial evaporation of this compound, and therefore making this analysis particularly difficult to interpret. Still, the formation of urethanes is also confirmed by the appearance of an NH band at 3310 cm -1 that is clearly visible after 1 or 2 h.

Small band at 1495 and 1511 cm -1 also caught our interest. Indeed, they appear during the first moments of the reaction and then fade away as the reaction approaches to completion. This behaviour is typical of a reaction intermediate. Moreover, such wavenumbers are theoretically coherent with the vibration of an iminium-like structure.

c/ COSY spectrum

MALDI-TOF analysis of the linear NIPU oligomer obtained from the CA reaction between benzaldehyde and ethylene dicarbamate.


Yoann Le Lamer
email: yoann.le-lamer@isae-supaero.fr

Giovanni Quaglia

Emmanuel Benard
email: emmanuel.benard@isae-supaero.fr@isae-supaero.fr

Joseph Morlier
email: joseph.morlier@isae-supaero.fr

MULTIFIDELITY AEROELASTIC OPTIMIZATION APPLIED TO HAR WINGS

Keywords: Multifidelity, High Aspect Ratio wings, Aeroelasticity, Multidisciplinary Design Optimization

As part of a global effort toward the reduction of CO2 emissions, the study of High Aspect Ratio wings is a particularly promising avenue of research. Considering this type of configuration at preliminary design stage is thus critical for optimization purposes. The use of multifidelity can help reduce computational costs by mainly running low-fidelity computations and only resorting to high-fidelity computations when necessary. The goal of this study is to develop a sufficiently accurate aero-structural model in order to fully exploit the drag reduction potential of HAR wings. The first step is to develop and apply a multifidelity approach on the CRM wing by creating a low-fidelity model. It is made sure that there is an adequate correlation between the structural modes of both high-fidelity and low-fidelity models. The so obtained low-fidelity structural model is then implemented within a in-house aeroelastic analysis and optimization framework. Afterwards, a modified version of the CRM wing with a higher aspect ratio (AR=13.5) is considered in order to apply the same methodology with the goal to reduce significantly computational costs.

INTRODUCTION

Increasing the aspect ratio of wings is one promising way to reduce CO2 emissions by limiting induced drag. In order to benefit from the full potential of these aircraft configurations, it is necessary to resort to aerostructural optimization [START_REF] Kenway | Multipoint High-Fidelity Aerostructural Optimization of a Transport Aircraft Configuration[END_REF]. In particular, considering computational costs is critical to achieve this goal. Indeed, in the last century, complex mathematical models have been developed in order to simulate real systems in many areas of scientific research, but the main obstacle to an intensive exploitation of such models (visualization and analysis) is the computational cost which can easily become prohibitive even for simple cases [START_REF] Kennedy | Predicting the output from a complex computer code when fast approximations are available[END_REF]. This problem is crucial in the field of aerostructural design and optimization since many simulations are required and one single CFD run may require hours of CPU time. The challenge has been tackled by creating surrogate models, sometimes also referred to as meta-models, which are an approximation of the actual system based on a number of collected samples [START_REF] Jones | A taxonomy of global optimization methods based on response surfaces[END_REF]. In recent years, multifidelity (MF) has been proposed as a valuable strategy to reduce computational time. With such an approach, the results employed to train and enrich surrogate models derive from sources of different precision and are balanced in order to reduce the overall cost. These sources can be seen as black-box functions [START_REF] Jones | Efficient global optimization of expensive black-box functions[END_REF] and are connected to each other in an optimization environment. In the last decade, ONERA, in a joint effort with the university ISAE-Supaero, has developed the SEGOMOE (Super Efficient Global Optimization with Mixture Of Experts) framework [START_REF] Bartoli | Improvement of efficient global optimization with mixture of experts: methodology developments and preliminary results in aircraft wing design[END_REF]. It is a single-fidelity, surrogate-model based, Bayesian optimization tool, able to deal with high-dimensional problems with a large number of constraints. As a further progress, [START_REF] Bartoli | Multifidelity efficient global optimization: Methodology and application to airfoil shape design[END_REF] exploited this technique as a benchmark to demonstrate the improvements offered by the use of a multifidelity approach in terms of computational time. In this research project, a subsonic airfoil with 15 variables has been designed using MFEGO (MultiFidelity Efficient Global Optimization). This work only considers unconstrained multifidelity optimization for now. Finally, [START_REF] Mas-Colomer | Aeroelastic Similarity of a Flight Demonstrator via Multidisciplinary Optimization[END_REF] produced an aeroelastic black-box function, the aerostructure package [START_REF] Colomer | Aerostructure package[END_REF], also used in [START_REF] Jiménez | Multifidelity aeroelastic optimization with application to a bwb[END_REF], implemented in the OpenMDAO framework [START_REF] Openmdao | Openmdao source docs[END_REF], which optimizes the NASA CRM (Common Research Model) wing using computationally cheap aerodynamic and structural solvers.

On the one hand, regarding aerodynamics, two levels of accuracy are currently represented by a CFD and a panels code. On the other hand, the subject of this article is the development of a low-fidelity version of the structural model. It features a reduced mesh resolution, a shorter analysis time but keeps a substantial correlation with respect to the original high-fidelity model. In particular, a MATLAB tool has been initially designed in order to produce a FEM mesh with a level of complexity fixed by the user. Later the model properties have been assessed through different tests. In section 2, the different theoretical arguments are discussed, along with their mathematical formulation, to support the general framework. Section 3 presents the simplifications made on the original CRM model and how the low-fidelity structural model is created. In section 4, static and dynamic behaviours are validated and the model is tested on a simple unconstrained multifidelity optimization. Then, in section 5 an aeroelastic model for the HAR uCRM-13.5 wing is presented and first aeroelastic analysis results are provided. Lastly, section 6 reports the conclusions and lists the future objectives.

THEORETICAL FRAMEWORK

The aerostructural problem

During the design phase of an aircraft the focus is the enhancement of multiple objective functions, namely the QoI (Quantities of Interest), such as drag or fuel consumption under mass and maximum stress constraints. In the case of steady cruise flight, the QoI are computed as result of a static aeroelastic problem. In literature, two approaches to solve it can be found, the monolithic and the partitioned. The latter is the most applied nowadays and it consists in the coupling of two different solvers for the structural and fluid problems, such as a FEM and a CFD software. The theory is reported with the notation used in [START_REF] Scholcz | Multi-fidelity methods for fluid-structure interaction and uncertainty quantification[END_REF].

y = F(x), x = S(y) (1)
Here, x and y are the vectors containing the structural and aerodynamic variables, namely the nodes displacements and the fluid pressure along the airfoil. S and F are the structural and fluid problems. The coupling can be expressed as:

x * = arg min x∈IR N s || R(x) || (2) with R(x) = S • F(x) -x (3)
Once the two software required by the disciplines are chosen, they are wrapped in a framework able to process the outputs in order to be provided as inputs one to the other. Given a set of design variable, the solver iterates until a predefined convergence criterion is reached, and finally the QoI and the constraints values are extracted. Hence, the ensemble can be seen as a black-box function as shown in figure 1.

Multifidelity

The concept of multifidelity has already been introduced in the previous sections. When it is not possible to have more than few high-fidelity samples, the use of a cheaper solver allows to explore the variable design space. The large number of LF simulations permits to enrich the surrogate model and to address the optimum search. Moreover, the space of possible sources is huge. Samples used to train the model could come from experiments, software of different levels of modelling, such as CFD or panels codes in aerodynamics, a coarse and a fine mesh on the same problem or even a change in the convergence criteria for an iterative algorithm [START_REF] Scholcz | Multi-fidelity methods for fluid-structure interaction and uncertainty quantification[END_REF]. Multifidelity is efficient if the accuracy of the optimum found is maintained while saving computational time performing less HF simulations. The main property required is the correlation between the results. The LF model must evolve in accordance with the HF one to correctly predict the behaviour of the objective function. Furthermore, the cheap code should be one or more orders of magnitude faster, allowing consistent scaling and time saving. The theoretical background regarding MF methods is still ambiguous. While the results in terms of computational time are generally positive, it has been proven that the use of LF data can worsen the surrogate precision [START_REF] Guo | Analysis of dataset selection for multi-fidelity surrogates for a turbine problem[END_REF]. [START_REF] Fernandez | Issues in deciding whether to use multifidelity surrogates[END_REF] delve into over 100 papers which make use of a MF approach. The research compared the studies in terms of levels of fidelity, disciplines, types of solvers, and data of correlation and time savings when made available. The conclusion is the absence of a clear pattern, able to determine whether or not it is convenient an MF implementation, and the lack of research in this direction.

DESIGN AND ANALYSIS OF A LOW FIDELITY STRUCTURAL MODEL

General framework

The focus in the current research is optimizing the NASA CRM wing in the SEGO framework enhanced by the use of multifidelity. First of all, the programming language of the SEGO optimization algorithm is python and, in this sense, the adoption of the OpenMDAO library results extremely useful. OpenMDAO is an open-source computing platform for multidisciplinary analysis and optimization. Its structure makes it well suited for wrapping the two disciplines solvers, creating an aeroelastic black-box function. In general OpenMDAO codes are written connecting components, which can be explicit or implicit, each of them representing a part of the problem. Every component has inputs and outputs which must all be connected in the main function. Once run, the solver iterates until also the implicit parts reach convergence.

One key part of this next step in the research is the OpenMDAO package aerostructures [START_REF] Mas-Colomer | Aeroelastic Similarity of a Flight Demonstrator via Multidisciplinary Optimization[END_REF]. The framework is used for solving static and dynamic aeroelastic analysis and optimizations. The XDSM diagram 2 presents its functioning. Each block represents a component, one can find the two main disciplines which run at their interior Panair [START_REF] Boeing | A502I User's Guide-PAN AIR Technology Program for Solving Potential Flow about Arbitrary Configurations[END_REF], an open-source aerodynamic panels code and Nastran95 [START_REF]The Nastran User's Manual[END_REF], used for solving the structural problem. All the other boxes have linking scopes being the interpolation and the transfer of loads and displacements between the fluid and structural mesh. Taking advantage of this package, setting up the new optimization framework has been straightforward. The MF-EGO/MF-SEGO environment can easily encapsulate aerostructures and creates two versions of the main function. The first type is directly the same as before, while in the other case Panair has been substituted by a component calling the execution of the CFD software ADFlow. In this way two levels of fidelity are available. While this first version is currently being tested, one more step was needed to improve the multifidelity approach. In fact, both functions had as an input the original CRM FEM model, which is given with a high resolution mesh, weakening the possible advantages of MF. Finally, the scope of this work has been twofold. Firstly, the accuracy of the structural model has been reduced to allow faster simulations, secondly, the preserved correlation with the initial system has been analyzed. Figure 3 illustrates how the framework should resemble once this tasks have been achieved.

The NASA CRM model

The Common Research Model is an attempt made by NASA to provide an experimental database for the validation of application of Computational Fluid Dynamics [START_REF]Nasa common research model[END_REF]. The related website is a repository of results from numerical and experimental analysis. In particular, some versions of different FEM models for the aircraft wing are released in Nastran language, one is drawn in figure 4. The one considered here has the following characteristics. The material is an aluminum alloy with ρ = 2795.67 and E = 68.9 GPa. The model consists of over 10500 nodes connected in 26600 elements. The structure is modelled as wing boxes delimited by 58 ribs closed by the upper and lower skins and with two spars at the leading and trailing edge. All this parts are explicitly represented by CQUAD4 elements depending on 12 PSHELL cards of different thickness. Moreover, ribs and skins are reinforced by stringers created with the CBAR class and with 6 different sections and principals of inertia.

Design phase

The first task of the project consisted in coding a routine able to receive as input the nastran .bdf file related to the original model and producing as outputs the reduced list of nodes and the new mesh description. As an additional objective it was planned to Figure 4: Illustration of the NASA CRM wing FEM model [START_REF]Nasa common research model[END_REF] parametrise the code in order to allow choosing the desired complexity of the model. The environment used for coding the tool has been MATLAB. An early difficulty has been faced when dealing with the NASA input file. The list containing the nodes was given with no apparent order in the IDs so that it was impossible to find a mathematical law able to define their succession. As a consequence the problem as been split in writing two separate functions to order the nodes and to write the new mesh. MainCreatePlanes.m is the first routine. One initial assumption has been made neglecting the intermediate nodes between two ribs with the outcome of having the elements representing skins and spars as long as each section. The removal of the additional rows of nodes in the cord direction diminishes the overall model flexibility with respect to the rotation around the body axis. As a result of such hypothesis the function operates with the following procedure. First, it reads a sub-list of the stringers elements inside the ribs and takes the two extreme points defining the first one. After that, it searches for a third node as the closest among the remaining and computes the 4 geometrical parameters of a plane passing through them. Finally, the node list is scanned, selecting and removing the entries whose distance from the plane is less than a fixed tolerance. The picked nodes are saved in a structure referring to a particular rib, and the process is iterated on the reduced file. As a final step, the 58 sections are sorted root to tip and the nodes belonging to each of them are locally enumerated. The result is presented in figure 5 where the initial chaotic cloud of points has been ordered. The second routine MainCreateMesh.m loads the structure containing the sorted nodes and writes a new coarser model. At the beginning of the function two main parameters, L and V, are defined, as they set the number of points in the reduced mesh on the vertical and cord directions. One first sub-routine creates the list with the new nodes of the model. In order to accomplish this goal for each section the coordinates are transformed in a temporary reference frame so that the first two axes are in the rib plane and the third one is orthogonal to them. Then a spline interpolation is created for each row of nodes in the section, and it is evaluated in the number of points requested by L and V. The 57th rib has less nodes than the others and it is joint with the last one. Since the precise location of the intersection cannot be respected considering a coarser mesh, the function shifts all the points in the rib in order to place it as close as possible to the original. Once created, the list with the new set of nodes is taken as database by a second subroutine which has the scope of redefining all the elements as in the starting model. The last hypothesis regards the definition of the spars. In the original CRM model they are modelled explicitly by CQUAD4 elements in a I-beam. In the spirit of reducing the complexity of the system, the flanges were substituted with simple CROD elements as they are loaded only axially in first approximation. All the elements preserve the original properties, and in particular the V vectors are reassigned to the CBAR entries looking for the closest and identically oriented stringer. One realization of the LF structural model is presented in figure 6 for the choices of L = 4 and V = 3. The last task during the design phase consisted in adapting the python aerostructures package to receive the same model with different mesh definitions. In fact the structure created by the MATLAB tool would have a decreased stiffness, due to the lower number of stringers, if the geometrical and inertial properties are kept as in the HF model. The problem was overcome providing to the main function the parameters L and V, the summation on all the stringer areas and principals of inertia as well as the size of the spar flanges. A new component is then added to the OpenMDAO group and it reassigns the properties according to the actual number of bars.

MODEL VALIDATION

Static analysis

In a preliminary phase of the validation process, all the simulations performed were static aeroelastic problems representing the aircraft in cruise condition. After computing the converged solution in the OpenMDAO framework, Gmsh has been used to visualize the deformation of the wing subject to the aerodynamic loads along with the Von Mises stress distribution. As an example, figure 7 compares the solution of the original model with the one obtained with the L = 4 low fidelity structure. The standard geometrical values for wing surface, wing span and sweep angle were taken from the original NASA model [START_REF]Nasa common research model[END_REF], while the case parameters are reported in table 1. The QoI selected to be compared in the analysis have been the lift coefficient (Cl) and the induced drag (CDi) computed by Panair, and the tip vertical displacement (δ tip ) and the maximum Von Mises stress (VM max ) that come from the postprocessing of the Nastran95 output file.

Mesh resolution influence

As a first step, the influence of the mesh resolution has been studied. The MATLAB tool designed was used to produce different FEM models for the parameter L varying in the interval . The lower bound allows to have at least 2 stringers running along the wing span, while the upper bound reaches the same value of nodes in the cord direction as for the original model. The computational cost is the major interest of MF, to this purpose, two timers, measuring the setup and run times of the OpenMDAO routine, were added to the main function . Figure 8 shows the results normalized by the values obtained running the full model. The run time includes all the executions of Nastran95, hence it presents the greatest reduction. As expected, the line grows with a power trend due to the increasing number of elements for each node added. The key outcome is the ratio between the reference and the fastest simulation time (4), computed with L = 4. This is the maximum gain obtained and it can be called Fidelity Cost Ratio (FCR) and employed as an indicator, for example in the criterion for the next point fidelity.

Figure 9 shows the computed values of Cl depending on the L parameter along with the reference produced by the original model. The LF results have a general convergent trend, as the structure representation becomes more and more refined with the mesh, proving the coherent design of the model. The offset between the final values at L = 28 and the HF solution is the trade-off of a MF approach. It is caused by the simplifications assumed during the design phase, namely the removal of the rib intermediate nodes, the redefinition of spars and the shifting of the next to last rib.

Figure 9: coefficient as a function of the L parameter with reference HF value.

Multifidelity correlation analysis

Once the MATLAB tool has been validated and the FCR computed, it was possible to focus on the correlation analysis between the LF model with L = 4 and the HF original one. It is hopeful that the LF maintains the same offset found in the previous section even for other design variables inputs. For this reason maximum and minimum relative variances must be always checked during the design space exploration. Moreover, the most important indicator is the correlation between the two fidelity, since it states if the LF can provide true information on the HF behaviour. This property has been tested in two cases, one with a random approach and one with predefined parameter steps. In a first experiment the influence of a physical property of the structure has been analyzed. The vector containing the thickness of the 12 different PSHELL cards has been randomly modified changing its components independently inside the ±40% range. 10 different realizations of the properties have been considered, the 10th being the original one. The two fidelity have been tested on each configuration. The first outcome has been the representation of the four QoI values as a function of the scalar product between the modified and original thickness vectors normalized by themselves. This index is equal to 1 when the parameters are unchanged and it decreases expressing an overall distance in the variable design space. Figure 10 presents the two fidelity results for δ tip and as well as the other graphs, it shows a perfect accordance in the results. Moreover, the computation of the actual Pearson's correlation coefficient (PCC) between the HF and LF QoI validates the trend. Table 2 reports its values along with the relative percent variances with respect to the HF results. The second experiment has focused on the effect of two geometrical properties, the wing span (b) and the sweep angle (ϕ). Their standard values were b = 58.7629 [m] and ϕ = 37.16, they were updated two times in both direction by a ∆b = 5 [m] and ∆ϕ = 2.5 step, resulting in 25 configuration cases. The main function was modified by the use of the aerostructures components able to deform the mesh according to a variation in those parameters. Figure 11 shows three realizations of the wing in the software Gmsh, the original and the two extreme cases. In this second test the QoI has been represented on the Z axis of a 3D graph with the design variables on the XY plane. Figure 12 shows the surfaces for the VM max quantity. Once more the LF is able to correctly predict the behaviour of the HF. All the figures present a partial crossing of the surfaces for high values of the b parameter. Such long-span configuration could present some nonlinearity badly predicted by the aeroelastic function and should be object of further analysis. Finally the comparative indices, obtained from the matrices containing the QoI values, are reported in table 3.

Dynamic analysis

The second phase of the validation has concerned the dynamic behaviour of the LF model. The objective has been the assessment of the correlation between the frequency response of the simplified structure and the original one. Nastran95 provides a solution for normal mode extraction even if it does not offer the same advanced options as in commercial releases. For example the standard Lanczos method is not available and thus a basic inverse algorithm has been chosen for all the simulations. One common performance indicator for a dynamic comparison is the Modal Assurance Criterion (MAC) [START_REF] Pástor | Modal assurance criterion[END_REF]. The Modal Assurance Criterion is a statistical indicator that is most sensitive to large differences and relatively insensitive to small differences in the mode shapes. When comparing two sets of eigenvectors in the same space, MAC indices are usually stacked in a matrix where a maximum value of 1 states full correlation, 0 being no correlation. Thus, the MAC matrix is calculated as the normalized scalar products of the two sets of vectors Φ HF and Φ LF as in [START_REF] Bartoli | Improvement of efficient global optimization with mixture of experts: methodology developments and preliminary results in aircraft wing design[END_REF].

M AC(i, j) = Φ HF T i Φ LF j 2 (Φ HF T i Φ HF i )(Φ LF T j Φ LF j ) (5)
Mode shapes must belong to the same space, therefore the HF ones are projected, through an interpolation, onto the LF subspace. Moreover, a good practice during the comparison is to apply a mode tracking algorithm able to find the most correlated modes regardless of their extraction order. Finally, this correlation criterion takes into account only eigenvectors while frequencies must be separately compared on a relative error basis.

The first result presented in table 4 is the FCR between a 10 modes extraction for the two fidelity. The LF model gives a sharp decrease in the computational time while the FCR value can be use as a cost index in the multifidelity optimization. Following the static validation, the fidelity correlation has been tested on the b and ϕ design variables with the same original values and variations. Since mode shapes might shift in order from one model to the other, the code extracts 10 eigenvectors from the HF and 30 from the LF, using mode tracking to find the best matches. From each analysis it is possible to compute the MAC matrix and a table of relative errors on the first 10 modes. Figure 13 and table 5 show the results for the CRM original values of the variables. Models are proven to have great dynamic correlation, in particular no mode tracking is needed since LF frequencies are ordered and the relative error is almost zero for the first three modes.

Multifidelity test case application

The final part of the validation process has consisted in a simple unconstrained optimization using the MFEGO algorithm. Both the static and dynamic HF behaviours are well represented by the LF model but the latter has been chosen as analysis type. The objective function to be minimize is the inverse of the distance between the first two frequencies, b and ϕ being taken as design variables in accordance with the tests made. Thus the problem is defined as in [START_REF] Bartoli | Multifidelity efficient global optimization: Methodology and application to airfoil shape design[END_REF].

1 f 2 -f 1 for b ∈ [40m, 70m] ϕ ∈ [0 • , 30 • ] (6)
The MFEGO algorithm requires two black box functions, namely the LF and HF codes for normal modes extraction, the number of points used to initially train the Co-Kriging surrogate, 7 for each fidelity, and a cost evaluation. Based on the previous section the last vector is set to [0.01,1] and a budget of 15 is chosen. Each call of a function evaluation counts for a cost against the budget. Figure 14 shows the initial LHS sampling of the design space. The same number of LF and HF points is chosen to train the first surrogate model.

Once the initial meta-model is obtained, the EGO algorithm successfully balances the exploration and exploitation of the design space choosing for each enrichment point the level of fidelity to be used. The final landscape is presented in figure 15 along with the location of the last point considered as optimum. Table 6 summarises the analysis results.

[3] The simple test performed verifies the possibility to use the new LF in a multifidelity environment. The imposed cost ratio leads the algorithm to exploit three regions using many calls of the cheap code while the HF points are fewer and better spread. Nevertheless, the number of calls to the HF model is still consistent and a further reduction of the computational cost ratio would be an interesting development. The dynamic analysis is the first step towards flutter computing in the preliminary design loop.

APPLICATION TO HAR WING

From now on, we will consider the case of an HAR wing that will be refered to uCRM-13.5, that has been developed in [START_REF] Brooks | Undeflected Common Research Model (uCRM): An Aerostructural Model for the Study of High Aspect Ratio Transport Aircraft Wings[END_REF] based on on the NASA CRM by mostly increasing overall span, also refered as uCRM-9.0 in [START_REF] Brooks | Undeflected Common Research Model (uCRM): An Aerostructural Model for the Study of High Aspect Ratio Transport Aircraft Wings[END_REF].

Design phase of HAR wing structural model

A Python script has been written to automatically creates the different sections of the wingbox and the connections between them in a BDF file format to allow analysis using MSC Nastran solver. That is done by asking the user to specify the coordinates of leading edge and trailing edge of each section. This script is parametric and thus enables one to create different wing geometries.

This script has first been tuned/developed to be able to reproduce the properties of the original NASA CRM model Therefore, thanks to the script being parametric, it has been used to create a new model with the geometric properties of uCRM-13.5, basically with an aspect ratio increased by 50%.

As for the design of the low-fidelity model, the same procedure than the one presented in subsection 3.3 is applied to the high-fidelity HAR wing model developed previously.

Aeroelastic static analysis

In this subsection HF and LF aeroelastic analysis were carried for uCRM-9.0 and uCRM-13.5 models in order to determine wingtip displacement for each case. Cruise parameters used in subsection 4.3 remain unchanged. As an example, figure 16 compares wing deformations of HF and LF models for uCRM-9.0 and uCRM-13.5. Also, table 7 compares wingtip displacements for the different cases. As expected, higher deflections are observed for the high aspect ratio wing with regard to regular aspect ratio configuration.

The deviation of LF with regard to HF computation is satisfactory (below 5%) for uCRM-9.0 model. This deviation is higher for the high aspect ratio uCRM-13.5 model but remains reasonable (below 8%). That could be explained by the higher displacements, and the increased flexibility of HAR wing.

CONCLUSIONS

The first objectives of this research project have been achieved. First of all, the MATLAB tool developed in this paper is able to create a low-fidelity version of a high-fidelity wing model, whether it is the original NASA CRM model or the uCRM-13.5 model. Static and dynamic validation results of the LF model are satisfactory in term of accuracy and correlation with regard to the HF model. Then, we were able to develop a HF model of HAR wing uCRM-13.5, and the same simplification procedure was applied successfully. First results, show greater wingtip displacements than the original CRM as expected with reasonable deviation between HF and LF solutions.

Finally, the next step of this research project will be to accurately assess HAR wing performance while keeping reasonable computational costs for analysis and optimization. To be more specific, as higher aspect ratio wings show high rotations and displacements, structural geometrical non-linearities should be taken into account [START_REF] Gray | Geometrically Nonlinear High-fidelity Aerostructural Optimization for Highly Flexible Wings[END_REF]. This will be done by using MSC Nastran SOL 400 non-linear solver. Another goal will be the implementation of uCRM-13.5 wing structural model within the optimization framework in order to apply multifidelity to unconstrained, and then unconstrained optimization problems.

Figure 1 :

 1 Figure 1: Black-box aeroelastic function diagram.

Figure 2 :

 2 Figure 2: XDSM diagram of the static aeroelastic OpenMDAO LF solver [7].

Figure 3 :

 3 Figure 3: Diagram representing the current research framework.

Figure 5 :

 5 Figure 5: Matlab representation of the ordered nodes colored by rib section.

Figure 6 :

 6 Figure 6: Low Fidelity structural model visualized in Patran for the choices of L = 4 and V = 3.

Figure 7 :

 7 Figure 7: Gmsh visualizations of the converged solutions of the CRM wing for the HF case (up) and the L = 4 LF case (down).

Figure 8 :

 8 Figure 8: Setup and execution normalized timers as a function of the L parameter.

Figure 10 :

 10 Figure 10: Tip vertical displacement as a function of the normalized projection of the shell elements thickness vectors onto the standard CRM property.

Figure 11 :

 11 Figure 11: Gmsh visualization of the standard wing and of the two extreme configurations tested.

Figure 12 :

 12 Figure 12: Maximum Von Mises stress as a function of the wing span and the sweep angle.

Figure 13 :

 13 Figure 13: MAC matrix for the original values of b and ϕ.

Figure 14 :

 14 Figure 14: Initial LHS sampling in the design space.

Figure 15 :

 15 Figure 15: Design space after the optimization.

  (a) Comparison of deformations of uCRM-9.0 wing for HF case (up) and LF case (down). (b) Comparison of deformations of uCRM-13.5 wing for HF case (up) and LF case (down).

Figure 16 :

 16 Figure 16: Comparison of deformations for uCRM-9.0 and uCRM-13.5 wings for HF and LF cases.

Table 1 :

 1 Aircraft cruise input parameters.

	Name	Value
	Flight speed	252.16168 [m/s]
	Mach number	0.85
	Air density	0.380580 [kg/m 3 ]
	Wing incidence angle	1.34 [deg]

Table 2 :

 2 Comparative indices between HF and LF for different QoI, panels thickness case.

	Indices:	Cl	CDi	δ tip	VM max
	PCC	0.9995 0.9987 0.9903 0.9849
	∆ max	1.365% 1.693% 0.704% 3.477%
	∆ min	1.126% 0.790% 0.404% 0.219%

Table 3 :

 3 Comparative indices between HF and LF for different QoI, 2 variables case.

	Indices	Cl	CDi	δ tip	VM max
	PCC	0.9998 0.9982 0.9997 0.9945
	∆ max 1.694% 1.547% 0.884% 2.841%
	∆ min 0.354% 0.119% 0.015% 0.118%

Table 4 :

 4 Computational cost of modal analysis.

	HF extraction LF extraction FCR
	840.53 s	21.44 s	2.55 %

Table 5 :

 5 Frequencies [Hz] and relative error.

	HF	LF	Relative error
	1.9320 1.9398	-0.400
	5.7476 5.7409	0.117
	7.9694 7.9275	0.525
	11.650 11.402	2.126
	17.672 16.227	8.172
	19.002 18.413	3.102
	22.289 22.067	0.992
	26.817 26.038	2.906
	31.016 28.620	7.725
	35.742 34.826	2.561
	min b,ϕ

Table 6 :

 6 MFEGO test results.

	LF calls	127
	HF call	21
	Total iterations	134
	b solution	41.53 m
	ϕ solution	20.69 •
	Objective function 3.393786858646402e-05

Table 7 :

 7 Comparison of wingtip displacements between HF and LF models for uCRM-9.0 and uCRM-13.5 wings.

	Model	HF model (m) LF model (m) Deviation (LF vs. HF in %)
	uCRM-9.0	7.58	7.30	-3.69
	uCRM-13.5	10.21	9.55	-6.46

ACKNOWLEDGEMENTS

The authors would like to thank FONISEN and European project U-HARWARD for funding this research project and ONERA strong partnership in the development of MF-EGO/MF-SEGO (N. Bartoli, T. Lefevbre, R. Lafage, in particular). We also want to thanks Joan Mas Colomer for his constant help.