Ben Chiheb

Emmanuel Ameur

Bernard Mory

Eugen Cousin

Dedu

Tcphas

Chiheb Ben Ameur
email: chiheb.benameur@yahoo.fr

Emmanuel Mory
email: emmanuel.mory@orange.com

Bernard Cousin
email: bernard.cousin@irisa.fr

Eugen Dedu
email: eugen.dedu@univ-fcomte.fr

TcpHas: TCP for HTTP Adaptive Streaming

published or not. The documents may come   L'archive ouverte pluridisciplinaire

I. INTRODUCTION

Video streaming is a widely used service. According to 2016 Sandvine report [START_REF] Sandvine | Global internet phenomena report[END_REF], in North America, video and audio streaming in fixed access networks accounts for over 70% of the downstream bandwidth in evening hours. Given this high usage, it is of extreme importance to optimize its use. This is usually done by adapting the video to the available bandwidth. Numerous adaptation methods have been proposed in the literature and by major companies, and their differences mainly rely on the entity that does the adaptation (client or server), the parameter used for adaptation (the network or sender or client buffers), and the protocols used; the major companies having finally opted for HTTP [START_REF] Dedu | A taxonomy of the parameters used by decision methods for adaptive video transmission[END_REF].

HTTP Adaptive Streaming (HAS) is a streaming technology where video contents are encoded and stored at different qualities at the server and where players (clients) can choose periodically the quality according to the available resources. Popular implementations are Microsoft Smooth Streaming, Apple HTTP Live Streaming, and MPEG DASH. Still, this technology is not optimal for video streaming, mainly because its HTTP data is transported using the TCP protocol. Indeed, video data is encoded at distinct bitrates, and TCP does not increase the throughput sufficiently quickly when the bitrate changes. TCP variants (such as Cubic, Illinois, and West-wood+) specific to high bandwidth-delay product networks achieve high bandwidth more quickly and seem to give better performance for HAS service than classical TCP variants such as NewReno and Vegas [START_REF] Ben Ameur | Combining traffic shaping methods with congestion control variants for HTTP adaptive streaming[END_REF], but the improvement is limited.

Another reason is the highly periodic ON-OFF activity pattern specific to HAS. Server-based shaping methods at application layer have been proposed to reduce its occurrence.

They are cross-layer because they interact with the TCP layer and its parameters such as the congestion window, cwnd, and the round-trip time, RTT. Hence, implementing HAS traffic shaping at the TCP level is naturally more practical and easier to manage; in addition, this should offer better bandwidth share among HAS streams, reduce congestion events and improve the Quality of Experience (QoE) of HAS users.

Despite the advantages of using the transport layer for HAS, and in contrast with other types of streaming, where methods at the transport layer have already been proposed (RTP and TFRC [START_REF] Floyd | TCP Friendly Rate Control (TFRC): Protocol specification[END_REF]), to the best of our knowledge, there is no proposition at the transport level specifically designed for HAS. For commercial video providers YouTube, Dailymotion, Vimeo and Netflix, according to [START_REF] Hoquea | Mobile multimedia streaming techniques: QoE and energy saving perspective[END_REF], "The quality switching algorithms are implemented in the client players. A player estimates the bandwidth continuously and transitions to a lower or to a higher quality stream if the bandwidth permits." The streaming depends on many parameters, such as player, video quality, device and video service provider etc., and uses various techniques such as several TCP connections, variable chunk sizes, different processing for audio and video flows, different throttling factors etc. To conclude, all these providers use numerous techniques, all of them based on client.

Therefore, in this paper, we propose a HAS-based TCP congestion control, TcpHas, that aims to minimize the aforementioned issues and to unify all these techniques. It uses four sub-modules: bandwidth estimator, optimal quality level estimator, slow-start threshold ssthresh updating, and cwnd updating to the shaping rate. Simulation results show that TcpHas considerably improves both QoS (queuing delay, packet drop rate) and QoE (stability) and performs well even with several concurrent clients.

The remainder of this paper is organized as follows: Section II presents server-based shaping methods and describes possible optimizations at TCP level. Then, Section III describes TcpHas congestion control and Section IV evaluates it. Section V concludes the article.

II. BACKGROUND AND RELATED WORKS

Many server-based shaping methods have been proposed in the literature to improve QoE and QoS of HAS. Their functioning is usually separated into two modules:

1) Estimation of the optimal quality level, based on network conditions, such as bandwidth, delay, and/or history of selected quality levels, and available encoding bitrates.

2) Shaping function of the sending rate, which should be suitable to the encoding bitrate of the estimated optimal quality level.

The next two subsections describe constraints and proposed solutions for each module. The last subsection presents some possible ways of optimization, which provides the basis for the TcpHas design.

A. Optimal Quality Level Estimation

A major constraint of optimal quality level estimation is that the server has no visibility on the flows that share the home network.

Ramadan et al. [START_REF] Ramadan | Avoiding quality oscillations during adaptive streaming of video[END_REF] propose an algorithm to reduce the oscillations of quality during video adaptation. During streaming, it marks each quality as unsuccessful or successful, depending on whether it has led to lost packets or not. We note that, to discover the available bandwidth, this method increases throughput and pushes to packet drop, which is different from our proposed method, where the available bandwidth is computed using an algorithm.

Akhshabi et al. [START_REF] Akhshabi | Serverbased traffic shaping for stabilizing oscillating adaptive streaming players[END_REF] propose a server-based shaping method that aims to stabilize the quality level sent by the server by detecting oscillation events. The shaping function is activated only when oscillations are detected. The optimal quality level is based on the history of quality level oscillations. Then, the server shapes its sending rate based on the encoding bitrate of the estimated optimal quality level. However, when the endto-end available bandwidth increases, the HAS player cannot increase its quality level when the shaper is activated. Also, to verify whether the optimal quality level has been increased or not, the server is obliged to deactivate the shaper to let the TCP congestion control algorithm occupy the remaining capacity available for the HAS stream. What is missing is a good estimation of the available bandwidth for the HAS flow.

B. Traffic Shaping Methods

Ghobadi et al. propose a shaping method on the server side called Trickle [START_REF] Ghobadi | Trickle: Rate limiting youtube video streaming[END_REF]. This method was proposed for YouTube in 2011, when it adopted progressive download technology. The key idea of Trickle is to place a dynamic upper bound on the congestion window (cwnd) such that TCP itself limits the overall data rate. The server application periodically computes the cwnd bound from the product between the round-trip time (RTT) and the target streaming bitrate.

The selection of the shaping rate by a server-based shaping method does not mean that the player will automatically start requesting that next higher quality level [START_REF] Akhshabi | Serverbased traffic shaping for stabilizing oscillating adaptive streaming players[END_REF]. Furthermore, it was reported [START_REF] Ben Ameur | Combining traffic shaping methods with congestion control variants for HTTP adaptive streaming[END_REF] that ssthresh has a predominant effect on accelerating the selection of the desired optimal quality level by the HAS client. Indeed, when ssthresh is set higher than the product of shaping rate and round-trip time, the server becomes aggressive and causes congestions and a reduction of quality level selection on the player side. In contrast, when ssthresh is set lower than this product, the congestion window cwnd takes several round-trip times to reach the value of this product. Hence, to improve the performance of the shaper, ssthresh needs to be modified too.

C. Optimization of Current Solutions

What can be noted from the different proposed methods for estimating the optimal quality level is that an efficient end-to-end estimator of available bandwidth can improve their performance, as shown in Subsection II-A. In addition, the only parameter from the application layer needed for shaping the HAS traffic is the encoding bitrate of each available quality level of the corresponding HAS stream. As explained in Subsection II-B, the remaining parameters are found in the TCP layer: the congestion window cwnd, the slow-start threshold ssthresh, and the round-trip time RTT. We are particularly interested in adjusting ssthresh to accelerate the reactivity of the HAS player. Naturally, what is missing here is an efficient TCP-based method for end-to-end bandwidth estimation. We also need a mechanism that adjusts ssthresh based on the output of the bandwidth estimator scheme.

Both this mechanism and estimation schemes used by various TCP variants are introduced in the following.

1) Adaptive Decrease Mechanism: In the literature, we found a specific category of TCP variants that set ssthresh using bandwidth estimation. Even if the estimation is updated over time, TCP uses it only when a congestion event is detected. The usefulness of this mechanism, known as adaptive decrease mechanism, is described in [START_REF] Mascolo | Testing TCP Westwood+ over transatlantic links at 10 gigabit/second rate[END_REF] as follows: "it provides a congestion window that is decreased more in the presence of heavy congestion and less in the presence of light congestion or losses that are not due to congestion". The adaptive decrease mechanism is described in Algorithm 1 [START_REF] Mascolo | Testing TCP Westwood+ over transatlantic links at 10 gigabit/second rate[END_REF] when detecting a congestion, i.e., when receiving three duplicated ACKs or when the retransmission timeout expires. The algorithm uses the estimated bandwidth, Bwe, multiplied by RT T min to update the ssthresh value.

Algorithm 1 TCP adaptive decrease mechanism.

ssthresh = Bwe × RT Tmin 9: cwnd = initial cwnd i.e. 2 × M SS 10: end if
where Bwe is the estimated bandwidth and RT Tmin is the lowest RT T measurement 2) Bandwidth Estimation Schemes: The most common TCP variant that uses bandwidth estimation to set ssthresh is Westwood. Other newer variants have been proposed, such as Westwood+ and TIBET. The only difference between them is the bandwidth estimation scheme used. Because of space limitation, we describe only TIBET in the following.

The basic idea of TIBET [START_REF] Capone | Bandwidth estimates in the TCP congestion control scheme[END_REF] is to perform a run-time sender-side estimate of the average packet length and the average inter-arrival separately. The bandwidth estimation scheme is applied to the stream of the received ACKs and is described in Algorithm 2 [START_REF] Capone | Bandwidth estimation schemes for TCP over wireless networks[END_REF], where acked is the number of segments acknowledged by the last ACK, packet size is the average segment size in bytes, now is the current time and last ack time is the time of the previous ACK reception. Average packet length and Average interval are the lowpass filtered measures of the packet length and the interval between sending times. Algorithm 2 Bandwidth estimation scheme. Alpha (0 ≤ alpha ≤ 1) is the pole of the two low-pass filters. The value of alpha is critical to TIBET performance: If alpha is set to a low value, TIBET is highly responsive to changes in the available bandwidth, but the oscillations of Bwe are quite large. In contrast, if alpha approaches 1, TIBET produces more stable estimates, but is less responsive to network changes. If alpha is set to zero Algorithm 2 will be equivalent to Westwood, where the sample Bwe varies between 0 and the bottleneck bandwidth.

TIBET uses a second low-pass filtering, with parameter γ, on the estimated available bandwidth Bwe to give a better smoothed estimation Bwe. γ is a variable parameter, equal to e -T k , where T k = t k -t k-1 is the time interval between the two last received ACKs. This means that bandwidth estimation samples Bwe with high T k values are given more importance than those with low T k values.

Simulations [START_REF] Capone | Bandwidth estimation schemes for TCP over wireless networks[END_REF] indicate that TIBET gives bandwidth estimations very close to the correct values, even in the presence of other UDP or TCP flows. It is also robust against packet clustering and ACK compression in contrast to Westwood. Moreover, TIBET is less dependent to cwnd than Westwood+. Indeed, Westwood+ estimates bandwidth only once per RTT, hence its estimation is always bounded by min(cwnd,rwnd) RT T .

III. TCPHAS DESCRIPTION

As shown in the previous section, a protocol specific to HAS needs to modify several TCP parameters and consists of several algorithms. Our HAS-based TCP congestion control, TcpHas, is based on the two modules of server-based shaping solution: optimal quality level estimation and sending traffic shaping itself, both with two submodules. The first module uses a bandwidth estimator submodule inspired by the TIBET scheme and adapted to HAS context, and an optimal quality level estimator submodule to define the quality level, QLevel, based on the estimated bandwidth. The second module uses QLevel in two submodules that update the values of ssthresh and cwnd over time. This section progressively presents TcpHas by describing these four submodules.

A. Bandwidth Estimator of TcpHas

As described in Section II, TIBET performs better than other proposed schemes. It reduces the effect of ACK compression and packet clustering and is less dependent on the congestion window, cwnd, than Westwood+.

As explained in Section II-C2, the parameter γ used by TIBET to smooth Bwe estimations is variable (γ = e -T k ). However, this variability is not suited to HAS. Indeed, when the HAS stream has a large OFF period, the HTTP Get request packet sent from client to server to ask for a new chunk is considered by the server as a new ACK. As a consequence, the new bandwidth estimation sample, Bwe, will have an underestimated value and γ will be wrongly reduced. We therefore make parameter γ constant.

B. Optimal Quality Level Estimator of TcpHas

TcpHas' optimal quality level estimator is based on the estimated bandwidth, Bwe, as described in Subsection III-A. This estimator is a function that adapts HAS features to TCP congestion control and replaces Bwe value by the encoding bitrate of the estimated optimal quality level QLevel. One piece of information from the application layer is needed: the video encoding bitrates of the HAS stream that are available in the server. These encoding bitrates are given in the index file of the HAS stream. The EncodingRate vector of TcpHas contains the available encoding bitrates in ascending order. Our estimator is defined by the function QLevelEstimator, described in Algorithm 3, which selects the highest quality level whose encoding bitrate is equal to or lower than the estimated bandwidth, Bwe.

Algorithm 3 QLevelEstimator function.

1: i = length(EncodingRate) -1 2: while (EncodingRate[i] > Bwe AND i > 0) do 3: i -- 4: QLevel = i
QLevel parameter is updated only by this function. However, the time and frequency of its updating is a delicate issue:

• We need to use the adaptive decrease mechanism because when a congestion occurs, QLevel needs to be updated to the new network conditions. Hence, this function is called after each congestion detection. • Given that TcpHas performs a shaping rate that reduces OF F occupancy, when TcpHas detects an OF F period, it may mean that some network conditions have changed (e.g. an incorrect increase of the shaping rate). Accordingly, to better estimate the optimal quality level, this function is called after each OF F period. The EncodingRate vector is also used by TcpHas during application initialization to differentiate between a HAS application and a normal one: when the application returns an empty vector, it is a normal application, and TcpHas just makes this application be processed by classical TCP, without being involved at all.

C. Ssthresh Modification of TcpHas

The TCP variants that use the TCP decrease mechanism use RT T min multiplied by the estimated bandwidth, Bwe, to update ssthresh. However, given that the value of ssthresh affects the reactivity of the player, it should correspond to the desired shaping rate instead of Bwe. Also, the shaping rate is defined in Trickle [START_REF] Ghobadi | Trickle: Rate limiting youtube video streaming[END_REF] to be 20% higher than the encoding bitrate, which allows the server to deal better with transient network congestion.

Hence, for TcpHas we decided to replace Bwe by Enco-dingRate[ QLevel] × 1.2, which represents its shaping rate:

ssthresh = EncodingRate[ QLevel] × RT T min × 1.2 (1)
The timing of ssthresh updating is the same as that of QLevel: when detecting a congestion event and just after an idle OF F period. Moreover, the initial value of ssthresh should be modified to correspond to the context of HAS. These three points are presented in the following.

1) Congestion Events: Inspired by Algorithm 1, the TcpHas algorithm when detecting a congestion event is described in Algorithm 4. It includes the two cases of congestion events: three duplicated ACKs, and retransmission timeout. In both cases, Qlevel is updated from Bwe using the QLevelEstimator function. Then, ssthresh is updated according to [START_REF] Sandvine | Global internet phenomena report[END_REF]. The update of cwnd is as in Algorithm 1.

Algorithm 4 TcpHas algorithm when congestion occurs. 2) Idle Periods: As explained in [START_REF] Allman | TCP congestion control[END_REF], [START_REF] Ben Ameur | Evaluation of gatewaybased shaping methods for HTTP adaptive streaming[END_REF], [START_REF] Ben Ameur | Combining traffic shaping methods with congestion control variants for HTTP adaptive streaming[END_REF], the congestion window, cwnd, is reduced when the idle period exceeds the retransmission timeout RTO, and ssthresh is updated to max(ssthresh, 3/4 × cwnd). In HAS context, the idle period coincides with the OFF period. In addition, we denote by OF F the OFF period whose duration exceeds RTO. Accordingly, reducing cwnd after an OF F period will force cwnd to switch to slow-start phase although the server is asked to deliver the video content with the optimal shaping rate.

To avoid this, we propose to remove the cwnd reduction after the OF F period. Instead, as presented in Algorithm 5, TcpHas updates Qlevel and ssthresh, then sets cwnd to ssthresh. This modification is very useful in the context of HAS. On the one hand, it eliminates the sending rate reduction after each OF F period, which adds additional delay to deliver the next chunk and may cause a reduction of quality level selection on the player side. On the other hand, the update of ssthresh each OF F period allows the server to adjust its sending rate more correctly, especially when the client generates a high OF F period between two consecutive chunks.

Algorithm 5 TcpHas algorithm after an OF F period. 3) Initialization: By default, TCP congestion control uses an initial value of ssthresh, initial ssthresh, of 65535 bytes. However, in HAS context, initial ssthresh is better to match an encoding bitrate. We decided to set it to the highest quality level at the beginning of streaming for two reasons: 1) give a similar initial aggressiveness as classical TCP and 2) avoid to be too higher than highest encoding bitrate to maintain the HAS traffic shaping concept.

This initialization should be done in conformity with (1), hence the computation of RTT is needed. Consequently, TcpHas just updates the ssthresh when the first RTT is computed. In this case, our updated ssthresh serves the same purpose as initial ssthresh. TcpHas initialization is presented in Algorithm 6. if RT T = 0 then i.e. when the first RTT is computed 7:

ssthresh = EncodingRate[ QLevel] × RT T × 1.2 8:
end if 9: end if

D. Cwnd Modification of TcpHas for Traffic Shaping

As shown in Section II-B, Trickle does traffic shaping on the server-side by setting a maximum threshold for cwnd, equal to the shaping rate multiplied by the current RTT. However, during congestion avoidance phase (i.e., when cwnd > ssthresh), cwnd is increased very slowly by one MSS each RTT. Consequently, when cwnd is lower than this threshold, it takes several RTTs to reach it, i.e. a slow reactivity.

To increase its reactivity, we modify TCP congestion avoidance algorithm by directly tuning cwnd to match the shaping rate. For this, we use smoothed RTT computations through a low-pass filter. The smoothed RTT that is updated at each ACK reception is:

RT T k = ψ × RT T k-1 + (1 -ψ) × RT T k ( 2
)
where 0 ≤ ψ ≤ 1. TcpHas algorithm during the congestion avoidance phase is described in Algorithm 7, where EncodingRate[ QLevel] × 1.2 is the shaping rate.

Algorithm 7 TcpHas algorithm in congestion avoidance.

1: if new ACK is received and cwnd ≥ ssthresh then

2: cwnd = EncodingRate[ QLevel] × RT T × 1.2 3: end if
Finally, TcpHas coexists gracefully with TCP on server, the transport layer checking whether the EncodingRate vector returned by application is empty or not, as explained before.

IV. TCPHAS EVALUATION

The final goal of our work is to implement our idea in real software. However, at this stage of the work, we preferred instead to use a simulated player because of the classical advantages of simulation over experimentation, such as reproducibility of results, and measurement of individual parameters for better tuning of the parameters of our idea.

In this section, we evaluate TcpHas using ns-3 simulator, version 3.17. In our scenario, HAS players share the same bottleneck link and compete for bandwidth in a home network. We first describe the network setup used in the simulations. Then, we describe the parameter settings used during the simulations for both evaluation and configuration of TcpHas. Then, we show the behavior of TcpHas. Finally, we analyse results for 1 to 9 competing HAS flows in the home network.

A. Network Architecture in ns-3

Fig. 1 presents the architecture we used, which is compliant with the fixed broadband access network architecture used by Cisco to present its products [START_REF]Broadband network gateway overview[END_REF]. The HAS clients are located inside the home network, a local network with 100 Mbps bandwidth. The Home Network (HG) is connected to the DSLAM. The bottleneck link is located between HG and DSLAM and has 8 Mbps. The queue of the DSLAM uses Drop Tail discipline with a length that corresponds to the bandwidthdelay product. Nodes BNG (Broadband Network Gateway) and IR (Internet Router), and links AggLine (that simulates the aggregate line), ISPLink (that simulates the Internet Service Provider core network) and NetLink (that simulates the route between the IR and the HAS server) are configured so that their queues are large enough (1000 packets) to support a large bandwidth of 100 Mbps and high delay of 100 ms without causing significant packet losses.

We generate Internet traffic that crosses ISPLink and Ag-gLine, because the two simulated links are supposed to support a heavy traffic from ISP networks. For Internet traffic, we use the Poisson Pareto Burst Process (PPBP) model [START_REF] Zukerman | Internet traffic modeling and future technology implications[END_REF], considered as a simple and accurate traffic model that matches statistical properties of real-life IP networks (such as their bursty behavior). PPBP is a process based on the overlapping of multiple bursts with heavy-tailed distributed lengths. Events in this process represent points of time at which one of an infinite population of users begins or stops transmitting a traffic burst. PPBP is closely related to the M/G/∞ queue model [START_REF] Zukerman | Internet traffic modeling and future technology implications[END_REF]. We use the PPBP implementation in ns-3 [START_REF] Ammar | PPBP in ns-3[END_REF], [START_REF] Ammar | A new tool for generating realistic internet traffic in ns-3[END_REF]. In our configuration, the overall rate of PPBP traffic is 40 Mbps, which corresponds to 40% of ISPLink capacity.

We use an ns-3 simulated HAS player similar to the emulated player described in [START_REF] Ben Ameur | Combining traffic shaping methods with congestion control variants for HTTP adaptive streaming[END_REF], with a chunk duration of 2 seconds and a playback buffer of 30 seconds (maximum video size the buffer can hold). Note that [START_REF] Ben Ameur | Combining traffic shaping methods with congestion control variants for HTTP adaptive streaming[END_REF] compares four TCP variants and two router-based traffic shaping methods, whereas the current article proposes a new congestion control executed on server. The player is classified as Rate and Buffer based (RBB) player, following classification proposed in [START_REF] Yin | A control-theoretic approach for dynamic adaptive video streaming over http[END_REF], [START_REF] Yin | Toward a principled framework to design dynamic adaptive streaming algorithms over http[END_REF]. Using buffer occupancy information is increasingly proposed and used due to its advantages for reducing stalling events. In addition, the bandwidth estimator we use consists in dividing the size of received chunk by its download duration. The buffer occupancy information is used only to define an aggressiveness level of the player, which allows the player to ask a quality level higher than the estimated bandwidth.

All tests use five video quality levels with constant encoding rates presented in Table I, and correspond to the quality levels usually used by many video service providers.

We also use the HTTP traffic generator module given in [START_REF] Cheng | HTTP traffic generator[END_REF], [START_REF] Cheng | Transactional traffic generator implementation in ns-3[END_REF]. This module allows communication between two nodes using HTTP protocol, and includes all features that generate and control HTTP Get Request and HTTP response messages. We wrote additional code into this HTTP module by integrating the simulated HAS player. We call this implementation the HAS module, as presented in Fig. 1. Streaming is done from S f to C f , where 0 ≤ f < N . The round-trip propagation delay between S f and C f is 100 ms.

For all evaluations, we use competing players that are playing simultaneously during K seconds. We set K = 180 seconds, which allows the HAS players to reach stationary behavior when they are competing for bandwidth [START_REF] Houdaille | Shaping HTTP adaptive streams for a better user experience[END_REF].

B. TcpHas Parameter Settings

We set γ = 0.99 (see Subsection III-A) to reduce the oscillation of bandwidth estimations, Bwe, over time. We set the initial bandwidth estimation value of Bwe to the highest encoding bitrate, as explained in Subsection III-C.

The parameter alpha of the TIBET estimation scheme (see Algorithm 2) is set to 0.8. A higher value produces more stable estimations but is less responsive to network changes, and a lower value makes TcpHas too aggressive.

The parameter ψ used for low-pass filtering the RTT measurements in Subsection III-D is set to 0.99, which leads to better performance in our tests, especially when network conditions degrade.

C. TcpHas Behavior

We present results for a scenario with 8 competing clients. The optimal quality level that should be selected by the competing players is n • 2.

We also show the behavior of Westwood+, TIBET and Trickle for the same scenario. We do not show the behavior of Westwood because Westwood+ is supposed to replace Westwood since it performs better in case of ACK compression and clustering. Concerning Trickle, it is a traffic shaping method that was proposed in the context of progressive download, as described in II-B. In order to adapt it to HAS, we added to it the estimator of optimal quality level of TcpHas, the adaptive decrease mechanism of Westwood+ (the same as TIBET), and applied the Trickle traffic shaping based on the estimated optimal quality level. This HAS adaptation of Trickle is simply denoted by "Trickle" in the reminder of this paper.

Fig. 2 shows the quality level selection over time of one of the competing players for the above methods. During the buffering phase, all players select the lowest quality level, as allowed by slow start phase. However, during the steady-phase the results diverge: Westwood+ player frequently changes the quality level between n • 0 and n • 3, which means that the player yields an unstable HAS stream and risks to generate stalling events. TIBET and Trickle players have an improved performance but still with frequent quality changes. In contrast, TcpHas player is stable at the optimal quality level, hence it performs the best among all the methods.

D. Performance Evaluation

Here, we vary the number of competing players from 1 to 9. We select a maximum of 9 competing HAS clients because in practice the number of users inside a home network does not exceed 9. We use one objective QoE metric: the instability of video quality level as defined in [START_REF] Jiang | Improving fairness, efficiency, and stability in HTTP-based adaptive video streaming with festive[END_REF], [START_REF] Ben Ameur | Evaluation of gatewaybased shaping methods for HTTP adaptive streaming[END_REF], [START_REF] Ben Ameur | Combining traffic shaping methods with congestion control variants for HTTP adaptive streaming[END_REF] (0% means the same quality, 100% means the quality changes each second). We also use two QoS metrics: the average queuing delay and the average packet drop rate at the bottleneck. This packet drop rate gives an idea of the congestion severity of the bottleneck link and is defined during a K-second test duration as follows:

DropP kt(K) = number of dropped packets during K seconds K × 100 The mean performance measurements of the QoE and QoS metrics among competing players and for 16 runs are shown in Fig. 3. We chose 16 runs because the relative difference between mean measurements of instability of 16 and 64 runs is less than 4%. We present the performance unfairness measurements among HAS clients with vertical error bars.

Fig. 3a shows that TcpHas yields the lowest instability rate (less than 4%), with a negligible instability unfairness between players. Trickle shows similar instability for 4 and 7 players, but for the other cases it has a high instability rate, the reason being that Trickle still initializes cwnd after each OF F period which causes a low sending rate. In contrast, the instability of Westwood+ and TIBET is much greater and increases with the number of competing players.

As presented in Fig. 3b, although the queuing delay of the four methods increases with the number of competing clients, TcpHas and Trickle show a lower queuing delay than West-wood+ and TIBET. The reason is that both TcpHas and Trickle shape the HAS flows by reducing the sending rate of the server which reduces queue overflow in the bottleneck. Additionally, we notice that TcpHas reduces more the queuing delay than Trickle; TcpHas has roughly half of the queuing delay of Westwood+ and TIBET. Besides, TcpHas does not increase its queuing delay more than 25 ms even for 9 competing players, while Trickle increases it to about 50 ms. This result is mainly due to the high stability of the HAS quality level generated by TcpHas which offers better fluidity of HAS flows inside the bottleneck. The same reason applies for the very low packet drop rate at the bottleneck of TcpHas, given in Fig. 3c. In addition, due to its corresponding traffic shaping method that reduces the sending rate of HAS server, the packet drop rate of Trickle is quite similar to that of TcpHas, as shown in Fig. 3c.

V. CONCLUSION

This paper first presents and analyses server-based shaping methods that aim to stabilize the video quality level and improve the QoE of HAS users.

Based on this analysis, we propose and describe TcpHas, a HAS-based TCP congestion control that acts like a server- based HAS traffic shaping method. It is inspired by the TCP adaptive decrease mechanism and uses the end-to-end bandwidth estimation of TIBET to estimate the optimal quality level. Then, it shapes the sending rate to match the encoding bitrate of the estimated optimal quality level. The traffic shaping process is based on updating ssthresh when detecting a congestion event or after an idle period, and on modifying cwnd during congestion avoidance phase. We evaluate TcpHas when several HAS clients share the bottleneck link and compete for the same home network. Simulation results indicate that TcpHas considerably improves both HAS QoE and network QoS. Concerning QoE, it offers a high stability of quality level and has tendency to select the optimal quality level. Concerning QoS, it reduces queuing delay, and reduces considerably the packet drop rate in the shared bottleneck queue. Moreover, TcpHas performs well when increasing the number of competing HAS clients.

As future work, we intend to optimize TcpHas so that it adapt its performance to Live streaming service and unstable network conditions, and also to implement and evaluate TcpHas in real software and networks.

1: if ACK is received then 2 : 3 : 4 : 5 : 6 :

 23456 sample length = acked × packet size × 8 sample interval = now -last ack time Average packet length = alpha × Average packet length + (1 -alpha) × sample length Average interval = alpha × Average interval + (1alpha) × sample interval Bwe = Average packet length/Average interval 7: end if

1: if 3 duplicate ACKs are received then 2 : 8 : 9 :

 289 QLevel = QLevelEstimator( Bwe) 3: ssthresh = EncodingRate[ QLevel] × RT Tmin × 1if retransmission timeout expires then QLevel = QLevelEstimator( Bwe) 10: ssthresh = EncodingRate[ QLevel] × RT Tmin × 1.2 11: cwnd = initial cwnd i.e. 2 × M SS 12: end if

1 :period detected 2 : 3 : 4 :

 1234 if idle > RT O then OF F QLevel = QLevelEstimator( Bwe) ssthresh = EncodingRate[ QLevel] × RT Tmin × 1.2 cwnd = ssthresh 5: end if

Algorithm 6 1 :

 61 TcpHas initialization. QLevel = length(EncodingRate) -1 highest quality level 2: cwnd = initial cwnd i.e. 2 × M SS 3: ssthresh = initial ssthresh i.e. 65535 bytes 4: RT T = 0 5: if new ACK is received then 6:

Fig. 1 :

 1 Fig. 1: Network architecture used in ns-3.

Fig. 2 :

 2 Fig. 2: Quality level selection over time.

  Packet drop rate.

Fig. 3 :

 3 Fig. 3: QoE and QoS measurements when increasing the number of HAS competing clients.

TABLE I :

 I Video encoding bitrates.


Mehdi Zaoui

Alexandre Bourceret

Yves Gaillard

Sylvain Giljean
email: sylvain.giljean@uha.fr

Christophe Rousselot

Marie Pac

Fabrice Richard

Zaoui Mehdi

Bourceret Alexandre

Giljean Sylvain

Rousselot Christophe

Richard Fabrice

Relation between hardness of multilayered (Ti,Al)N based coatings and periods of their stacking

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

ABSTRACT. This study aims to model, by using a finite element method, the relationship between the hardness and the period Λ of metal/nitride multilayer coatings (Ti0.54Al0.46/Ti0.54Al0.46N)n in order to understand the increase of the hardness at the low periods [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF] and then optimise the multilayer coating architecture to obtain the best mechanical properties.

A 2D axisymmetric finite element model of the Berkovich nanoindentation test was developed. The coating was designed as a stacking of Ti0.54Al0.46 and Ti0.54Al0.46N nanolayers with, in the first hypothesis, equal thickness and perfect interface. The elastoplastic behaviours of the metal and nitride layers were identified by Berkovich nanoindentation experiments and inverse analysis on thick monolayer samples. The indentation curves (P-h) obtained by this model depend on the period Λ of the stacking.

Simulated (P-h) curves were compared with experimental data on 2 µm thick films with different periods Λ ranging from 10 to 50 nm deposited by RF magnetron sputtering using reactive gas pulsing process (RGPP). The model forecasts are very consistent with the experience for the largest period but the model does not reproduce the hardness increase at the lowest periods.

The Λ = 10 nm coating was analysed by electron energy loss spectroscopy (EELS) on a transmission electron microscope. Results show intermixion of the layers with the presence of nitrogen atoms in the metal layer over a few nanometers [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF]. It was concluded that the metal/ceramic interface plays an important role at low periods. The addition in the model of a transition layer in the metal/nitride stacking, with an elastoplastic metal/ceramic medium behaviour, allows to reproduce the nanoindentation experimental curves. The thickness of this transition layer deduced from model updating method is in very good agreement with EELS observations. KEYWORDS: nanoindentation, finite element method, TiAl/TiAlN multilayer films, hardness, indentation modulus.

INTRODUCTION

Titanium and aluminium nitrides (TiAl)N coatings are extensively used in the industry since many years for their hardness, toughness and wear and oxidation resistances [START_REF] Münz | Titanium aluminum nitride films: A new alternative to TiN coatings[END_REF][START_REF] Paldey | Single layer and multilayer wear resistant coatings of (Ti,Al)N: a review[END_REF]. It has been shown that the indentation modulus and the hardness of Ti1-xAlxN coatings depend on the aluminium content and that the optimal properties are obtained when the maximum Al content is substituted for Ti in face centred cubic structure close to x = 0.46 [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF][START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF]. One way to improve the hardness of that kind of coatings consists in depositing multilayer films.

In this study, multilayer coatings are studied and consist in metal/ceramic stacking with alternative deposition of Ti0,54Al0,46 and Ti0,54Al0,46N nanolayers with three different periods of the bilayer metal/ceramic Λ = 50, 16, 10 nm. In order to facilitate the reading of the paper, Ti0,54Al0,46 and Ti0,54Al0,46N are respectively written as TiAl and TiAlN. The main objective of the paper is to study the relationship between hardness of as-design multilayer films and periods of their stacking to understand the elastoplastic phenomenons that are involved in the hardness increasing experimentally observed at the lowest periods [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF].

A 2D axisymmetric model of the Berkovich nanoindentation is developed. Then a method to identify the elastoplastic mechanical properties of the monolayer TiAl and TiAlN is proposed. The results are introduced in a multilayer stacking model that allows to obtain the indentation curves (Ph) as a function of the period Λ. As a first hypothesis the interface between TiAl and TiAlN layers is considered as perfect without thickness. Then a transition layer is added and allows to reproduce the experimental indentation curves.

MATERIALS AND METHODS

TiAl and TiAlN monolayer thin films were deposited onto silicon (100) wafer by RF magnetron sputtering using one sintered titanium/aluminium 66/33 at.% alloy target as described elsewhere [START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF]. (TiAl/TiAlN)n multilayer films with different periods were deposited using Reactive Gas Pulsing Process (RGPP) [START_REF] Martin | Procédé de pulvérisation réactive à signal de commande cyclique et dispositif correspondant[END_REF] with a controlled pulsing flow rate of the nitrogen reactive gas as described elsewhere [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF]. Finally, the samples considered in this paper are summarised in Table 1. Nano-indentation tests were performed using an Ultra Nano-indentation Hardness Tester provided by Anton Paar. Experiments were conducted using a Berkovich tip and the multicycle procedure in load control up to 8 mN [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF][START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF].

TABLE 1. Sample characteristics Sample

A 2D axisymmetric finite element model of the Berkovich nanoindentation test was developed with the ANSYS finite element software as described on Figure 1. An implicit method with non-linear geometric formulation has been used with higher order 2D elements. A deformable indenter, with isotropic linear elastic law, was modelled with a half contact angle of 70,3° and a tip radius to fit the experimental tip defects. A friction coefficient of 0.1 was imposed at the contact between the indenter and the film. The mesh density has been refined in order to the smallest elements are localised in the contact region. The film can be either monolayer TiAl or TiAlN or multilayer (TiAl/TiAlN)n. For the monolayer TiAl and TiAlN, isotropic elastoplastic behaviour laws were considered with linear hardening. Elastic properties were obtained from experimental nanoindentation and plastic properties were estimated using finite element model updating.

The approach used to extract the constitutive parameters of plastic law of TiAl and TiAlN consists in an optimisation problem; the objective of which is to minimise a cost function ω by using an algorithm based on the Levenberg-Marquardt [START_REF] Levenberg | A méthod for the solution of certain non-linear problems in least squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF] method using the MIC2M software [START_REF] Richard | MIC2M Software: Modeling and Identification of the Mechanical Behavior of Materials[END_REF]: [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF] where is the vector of the unknown parameters and ω the cost function i.e., the gap between the experimental data and the numerical results. Then the cost function can be defined as:

(2

)
where T is the number of acquisition times of the load and the maximum value of the experimental indentation load.

RESULTS AND DISCUSSION

3.1

Mechanical properties of TiAl and TiAlN monolayers All the experimental data of the indentation tests performed on the TiAl and TiAlN monolayer films have already been published elsewhere [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF][START_REF] Pac | Microstructural and elastoplastic material parameters identification by inverse finite elements method of Ti(1-x)AlxN (0[END_REF]. Hardness and indentation modulus values of the metal TiAl are significantly lower than the ones of nitride TiAlN. By postulating the Poisson's ratio as known, the elastic properties of the monolayer films can be deduced from the experimental data. Then, plastic parameters were numerically deduced from the optimisation approach described in method section. Results are presented in Table 2. The comparison of the experimental and model indentation load curves are plotted in Figure 2 and show a good agreement.

3.2

Simulation of (TiAl/TiAlN)n multilayer films with perfect interfaces All the experimental data of the indentation tests performed on the multilayer (TiAl/TiAlN)n films have already been published elsewhere [START_REF] Pac | Investigation of Ti0.54Al0.46/Ti0.54Al0.46N multilayer films deposited by reactive gas pulsing process by nano-indentation and electron energy-loss spectroscopy[END_REF]. Interesting observation is that both the hardness and the elastic modulus of the multilayer films follow a Hall and Petch law as the values linearly increase as a function of . In a first approach, the multilayer film was modelled as a stacking of nanolayers of TiAl and TiAlN with a perfect interface without thickness as showed in figure 3.a. The model was set for each stacking period Λ = 10, 16 and 50 nm. A convergence study was carried out with 1, 2 or 3 elements in the thickness of the nanolayers. Since the relative variation of the load is less than 0.5%, only one element, with a size, was finally used in the thickness (Figure 3.a), in order to minimise the calculation time. Results of the as simulated (P-h) curves are plotted on figure 4. For the largest period Λ = 50 nm, the model is in good agreement with the experience. Furthermore, the gap between the experimental curves and the simulated ones increases when the period decreases to become important for the smallest period. Therefore, the phenomenon that generates an increase in hardness when period decreases is not taken into account in the coating model without interfaces.

3.3

Simulation of (TiAl/TiAlN)n multilayer films with a transition layer In the stacking, between each nanolayer of TiAl and TiAlN, a transition layer was added (figure 3.b.). For this transition layer, an elastoplastic metal/ceramic medium behaviour was considered. Each elastoplastic parameter of the transition layer was assumed to be an average (mixture law 50/50) of the parameters of the TiAl and TiAlN nanolayers. The thickness of this transition layer was optimised by minimising the gap between experimental and simulated data as previously described. Result of the simulate (P-h) curve with a transition layer is plotted on figure 5.a and show a good agreement between experimental and simulated curves for the lowest period. For Λ = 10 nm, the simulated thickness of the transition layer, epTr = 2.12 nm, is close to the one of TiAl and TiAlN layers. This result is in very good agreement with the thickness of rough interfaces observed between TiAl and TiAlN layers specified by Electron Energy-Loss Spectroscopy (Figure 5.b).

CONCLUSION

The elastoplastic properties of Ti0.54Al0.46 metal and Ti0.54Al0.46N ceramic monolayers were identified by comparison of Berkovich nanoindentation experimental data and 2D axisymmetric finite element simulations. The mechanical properties of the monolayers was input in a multilayer model to forecast the mechanical properties of (Ti0.54Al0.46/Ti0.54Al0.46N)n multilayer coatings, with various periods, designed by RF magnetron sputtering using RGPP technique. The multilayer model was considered, in first hypothesis, as a stacking of TiAlN and TiAl nanolayers with equal thickness and perfect interface. That model allows to reproduce the indentation curves of the highest 50 nm period films. To fit the experimental indentation curve at the lowest periods, a transition layer was introduced in the model between each metal/ceramic interface. Using an optimisation process of the model, this transition layer was found to be 2.12 nm thick for the lowest period Λ = 10 nm, which is in good agreement with the results of Electron Energy-Loss Spectroscopy experiments.

FIGURE 1 .

 1 FIGURE 1. Description of the 2D axisymmetric finite element indentation model.

FIGURE 2 .

 2 FIGURE 2. Loading indentation curves (P-h) of (a) TiAl and (b) TiAlN monolayers.

FIGURE 3 .

 3 FIGURE 3. Modelisation of multilayer films, (a) with perfect interfaces (b) with a transition layer.

FIGURE 4 .

 4 FIGURE 4. Comparison of the (P-h) experimental and simulated curves for Λ = 10, 16 and 50 nm with a perfect interface between TiAl and TiAlN nanolayers.

FIGURE 5 .

 5 FIGURE 5. For Λ = 10 nm, (a) Comparison of the (P-h) experimental and simulated curves with a transition layer, (b) EELS results

TABLE 2 .

 2 Elastoplastic parameters of the metal TiAl and the nitride TiAlN.

		Young Modulus	Poisson ratio Yield stress	Linear hardening
		(GPA)		(GPa)	(GPa)
	TiAl	199	0.25	0.88	56.6
	TiAlN	417	0.21	5.70	405.0

ACKNOWLEDGEMENT

This work was performed in the frame of the research project RESEM 2019, managed by the Institut de Recherche Technologique Matériaux Métallurgie Procédés (IRT M2P) and financially supported by the French program Plan d'Investissement d'Avenir (PIA).


Gael Chevallier
email: gael.chevallier@univ-fcomte.fr

Emmanuel Ramasso

P Butaud

Detection and analysis of loosening in jointed structures using acoustic emission sensors and smart bolts

Keywords:

come

Detection and analysis of loosening in jointed structures using acoustic emission sensors and smart bolts

In aeronautic, automotive or civil engineering structures, the parts are assembled together by means of rivets welding points or bolted joints. The bolted joints are likely to self-loosen when the structure is subjected to vibrations [START_REF]Aerospace series -Dynamic testing of the locking characteristics of fasteners under transverse loading conditions (vibration test)[END_REF]. As loosening can lead to serious accidents, our project focuses on the early detection of self-loosening. To achieve this goal, it is possible to integrate a sensor in a screw to monitor it, but it is impossible to instrument the thousands of screws that contains a structure. Our project therefore aims to try to overload all links with a network of some Acoustic Emission sensors well chosen and well positioned. The setup ORION has been firstly developed to perform research activity about damping and nonlinear vibrations [START_REF] Festjens | A numerical tool for the design of assembled structures under dynamic loads[END_REF]. It is made with two plates linked with three bolted joints. To facilitate the understanding of the physical phenomena, the contact between the plates is made by patches machined on the surfaces. A piezo-electric actuator (CEDRAT) has been especially developed in order to cause the loosening. A sensor (TEXYS) has been embedded into a M4 bolt head to measure the tightening loads. Acoustic Emission sensors (MISTRAS) are used to measure acoustic activity, see figure 1. AE techniques are able to detect friction and contact induced waves. The « Hit start » seems to be a good descriptor for loosening detection. The « Amplitude » is a good descriptor to measure the joint loading amplitude. Future work will focus on stick slip transition and contact separation simulation and self loosening simulation. Better understanding the physics involved in order to perform smart description of the AE datas

Fig. 1 :Futek

 1 Fig. 1: Studied structure

Fig. 4 :

 4 Fig. 4: AE according to tightening


A Del Masto

F Trivaudey

V Guicheret-Retel

V Placet

A multiscale model-based investigation of the nonlinear tensile behaviour of plant fibre composites

Keywords: Plant fibre composites, FEM, RVE, Multiscale-stochastic approach, Sensitivity analysis

U

nidirectional plant fibres reinforced composites (UD PFRCs) are complex, hetorogeneous and hierarchical materials and represent the most interesting family of biobased camposites for structural high-performance applications. Experimental studies pointed out that UD PFRCs show a nonlinear tensile behaviour when loaded in the reinforcement direction, along with a slight variability in their mechanical properties. The origins of this particular mechanical response are still open to discussion. In this study we propose to numerically assess the impact of material and morphological parameters belonging to the nano, micro and mesoscales of the UD composite on the apparent stiffness of the single ply through sensitivity analysis. A multiscale-stochastic approach is developped through the definition of a parametric Representative Volume Element (RVE) of the UD ply, comprised of a unidirectional roving of about 100 individualised fibres embedded in an epoxy matrix. Results of polynomial chaos expansion based sensitivity analysis allow the identification of the most influential parameters on the stiffness of the ply, which are the longitudinal modulus of the crystalline cellulose, the fraction of cellulose and the initial microfibrillar angle.

Introduction

Ecological and economical issues prompt an increasing number of industrial and academic researchers to focus on high-performance materials that comply with all the new environmental preservation interests. Natural fibres are good candidates for replacing synthetic man-made fibres in reinforcing organic matrices in high-performance composite applications [START_REF] Bourmaud | Towards the design of high-performance plant fibre composites[END_REF]. Two major diffculties lie with the use of plant fibres composites (PFCs): (i) their nonlinear mechanical behaviour and (ii) the variability of their mechanical properties and behaviour [START_REF] Shah | Damage in biocomposites: Stiffness evolution of aligned plant fibre composites during monotonic and cyclic fatigue loading[END_REF]. The mechanisms responsible of this particular response are not yet well understood. Nonlinearity and variability are also generally observed at the lower scale of the elementary fibres [START_REF] Placet | Nonlinear tensile behaviour of elementary hemp fibres. Part I: Investigation of the possible origins using repeated progressive loading with in situ microscopic observations[END_REF]. These results seem to support the hypothesis that the nonlinear behaviour of PFRCs may result, at least in part, from the nonlinear behaviour of the single fibres themselves. And since the tensile behaviour of fibres depends on their morphology, ultrastructure and material properties of biopolymer constituents of the fibre wall at the nanoscale, these features may affect the mechanical response at the higher scales of the composite too. This study is dedicated to the assessment of the impact of several morphological and material parameters belonging to different scales on the tensile stiffness of the UD biobased composite using a multiscale stochastic approach. The mechanical behaviour is investigated at the scale of an elementary volume, assumed to be representative of the UD ply. This Representative Volume Element (RVE) is made of a non-twisted roving of about 100 individualised fibres embedded in an epoxy resin, modelled considering a perfectly elastic behaviour. Within the RVE, the fibre morphology is simplified (circular and elliptical shape of the transverse section) and the fibre wall is considered to be an anisotropic viscoelastic material of which behaviour model is derived from work previously developed in our team [START_REF] Trivaudey | Nonlinear tensile behaviour of elementary hemp fibres. Part II: Modelling using an anisotropic viscoelastic constitutive law in a material rotating frame[END_REF]. The stochastic aspect of the proposed approach is expressed through a succession of deterministic FE simulations made at the scale of RVE with randomly generated microstructures, each one representing an outcome of the probability distribution functions (PDF) of material and morphological parameters experimentally observed at nano, micro and mesoscale within the RVE itself. The behaviour of all generated RVEs is then evaluated through a FEM simulation of the tensile test. The developed approach is used to assess the impact of the mean values of some morphological and material parameters' PDFs within the RVE on its tensile properties through a polynomial chaos expansion based sensitivity analysis [START_REF] Sudret | Global sensitivity analysis using polynomial chaos expansions[END_REF].

Methods and materials

Multiscale-stochastic approach

In order to implement the proposed approach, a software tool is developed. For each simulation, five steps succed one other:

1) The analysis is initialised and the PDFs of each marphological and material parameter considered are setted from means and standard deviations randomly drawn from the experimental ones proposed in the literature (flax fibres); 2) The RVE two-dimensional microstructure is generated adding one fibre at the time. The geometrical and material properties of the constituents at the nanoscale of the wall affected to each fibre are randomly drawn from the PDFs determined at the first step. The fibres are embedded in a elastic epoxy matrix, modelled as a isotropic perfectly elastic material (E = 4GP a and ν = 0.3). The interphases between fibres and matrix are modelled as a thin layer expressing an isotropic elastic behaviour (E = 1GP a and ν = 0.3); 3) The generated microstructure is meshed in 2D using a combination of Matlab ® code and free mesh generator Gmsh. The 2D mesh is and then extruded in the third dimension to obtain the 3D model of the impregnated yarn; 4) A simulation of the tensile test is carried out on the generated 3D model using the FE code Abaqus ® ; 5) The results of the simulation are post-processed using Python and Matlab ® codes in order to obtain the global tensile behaviour of the simulated RVE.

Polynomial chaos expansion-based sensitivity analysis

The developed multiscale-stochastic approach is used to represent the RVE's apparent tangent modulus at the beginning of the loading on a polynomial chaos basis. The observed output Y can be expressed as Y = α∈N k y α Ψ α (X), where y α represent the coefficients of the polynomial chaos expansion, Ψ(X) are multidimensional polynomes of degree |α| = α i and k is the number of input parameters. The y α coefficients are computed using a regression method and then used to compute the sensitivity indices, i.e. to estimate the impact on the observed output, of the considered morphological and material parameters. Two families of indices are computed: first order, which assess the linear effect of the i th parameter on the observed output, and total index, which estimate the global impact of the i th parameter, including the effect of interactions with other variables:

S i , S T i = α∈Ai y 2 α D for A i = {α ∈ N k : α i > 0, α j =i = 0} → S i A i = {α ∈ N k : α i > 0} → S T i (1)
where D represents the variance of the observed output. Five parameters are considered, both material and morphological, which have been previously identified using a grouped screening analysis as the most influential on the apparent stiffness of the RVE: longitudinal modulus of crystalline cellulose E Lcc , fraction of cellulose m c , initial microfibrillar angle ψ, fibre D and lumen d diameter. It's worth notice that among the previously identified most influential variables there are, quite surprisingly, two morphological parameters (D and d) alongside the expected material ones (E Lcc , m c , ψ).

Results

The results in terms of first order and total sensitivity indices are shown in figure below. The computed sensitivity indices shows that three parameters have the greatest impact on the initial apparent stiffness of the UD ply: the longitudinal modulus of crystalline cellulose E Lcc , the fraction of cellulose m c and the initial microfibrillar angle ψ.

Sensitivity indices

Conclusions

A stochastic-multiscale numerical approach is developped in order to explore the origins of the nonlinear behaviour of UD PFRCs. The composite ply is represented by an elementary volume consisting of a roving of flax fibres embedded in an epoxy matrix. The microstructures of the RVEs are generated automatically using random draws from the probability distribution functions of morphological an material parameters belonging to different material scales.

The proposed approach is applied to a polynomial chaos expansion-based sensitivity analysis, in order to assess the impact the considered parameters on the apparent initial stiffness of the UD ply. Results show that the most influential parameters on the stiffness of the UD ply are the longitudinal modulus of crystalline cellulose E Lcc , the fraction of cellulose m c and the initial microfibrillar angle ψ. This study allows a first evaluation of the propagation of the mechanical behaviour through the characteristic material scales of UD biobased composites. The next step will be to model more complex morphologies of fibres and to include in the analysis irreversible phenomena, like plasticity and damage, in order to evaluate the impact of morphological and material parameters on the rupture behaviour of the ply.

Figure 1 :

 1 Figure 1: Multiscale-stochastic approach

Figure 2 :

 2 Figure 2: Sensitivity indices computed through polynomial chaos expansion