Sébastien Bindel

Serge Chaumette
email: serge.chaumette@labri.fr

Benoît Hilt

a predictive link state estimator for mobile networks

Keywords: MANET, link quality, forecasting, routing

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

Introduction

Due to their inherent characteristics, including selforganization, scalability and their potential for managing load balancing, ad-hoc networks are the best choice to deploy on-the-fly a wireless network. They rely on the cooperation of stations to provide an affordable solution to share data or video streams.

Nodes are free to move with a potential high speed making the topology of Mobile Ad-hoc Networks (MANETs) dynamic. They also have to deal with specific constraints related to their environment and the desired application. This includes vehicular networks, so called VANETs, characterized by potential high speed nodes which could drive in an urban, suburban or freeway environment. Besides, unmanned systems are emerging and include aerial, terrestrial and maritime vehicles. For example, terrestrial vehicles also called Unmanned Ground Vehicle (UGV) represent an actual challenge since they accomplish missions in various environments.

Since distributed applications require cooperation of nodes, they are bounded by connectivity and reliability issues. They are partially solved by routing protocols ensuring an end-to-end communication with a multihop relaying technique. However, a relevant problem for routing protocols is the assessment technique of the link state in order to rank and select the best one. Indeed routing and applications performances in terms of delay and reception rate depend on the reliability of the link state assessment. The traditional hop count metric relates the cost of a path to the number of hops required to reach the destination. However, this approach is insufficient in wireless networks, as demonstrated by De Couto et al. [START_REF] De Couto | Performance of multihop wireless networks: Shortest path is not enough[END_REF].

To overcome the limitations of the hop count metric, a current approach relates the link state to the link quality. This approach takes into account the lossy and the dynamic of a link to determine its quality. Two kinds of Link Quality Estimator (LQE) have been proposed in the literature. The first one is based on a measurement performed at the physical layer during the packet reception. However, the correlation between the Packet Reception Ratio (PRR) and their assessments is not directly deductible at all, which limits their efficiency [START_REF] Baccour | Radio link quality estimation in wireless sensor networks: A survey[END_REF] [3] [START_REF] Srinivasan | An empirical study of low-power wireless[END_REF]. The second one includes estimators which perform measurements at upper layers and may assess the link according to application requirements. Related work has mainly designed estimators for Wireless Sensor Networks (WSNs), which limits their effectiveness in dynamic networks. For example, most of them use either a fixed historic or a data filter, which limits the reactivity of the estimators. There is a need for a reliable and reactive estimator for mobile networks.

In order to overcome the limitations of current LQEs in a highly mobile environment, we suggest a novel link quality estimator which performs at the MAC layer. Our approach is based on a dynamic history to deal with the unpredictable link lifetime. In opposition to other traditional estimators, ours adapts its assessment according to the estimated link lifetime. Indeed, we assess the link lifetime also referred to the link stability in order to adapt the link quality evaluation. In practice, the history of the packet reception is maintained through a window, whose length determines the volume of acquired data. The handling of a predefined static window size limits both the reactivity and the accuracy (the number of values) of the assessment. This issue can be solved with the use of a dynamic window size. Its length is re-sized according to the measured link stability in order to adjust the reactivity and the accuracy of the assessment.

We have suggested [START_REF] Bindel | F-etx: An enhancement of etx metric for wireless mobile networks[END_REF] a LQE to address the problem of the link quality assessment in mobile networks. It has been shown as more reactive and accurate than the Expected Transmission Count (ETX) metric [START_REF] De Couto | Performance of multihop wireless networks: Shortest path is not enough[END_REF]. From experiments, we observed that an estimation expressed by a single value provides only a short-term vision and remains insufficient when several links need to be compared. Figure 1 shows the fluctuation of link quality, while a short-term estimation (represented by the couple of points on the plot) leads to selecting continually different link, once its quality becomes worse than another. At the opposite, a long term estimation (represented by the couple of arrows on the plot) can highlight a tendency of the link quality meanwhile one grows (Link N o 1) and another decreases (Link N o 2). In the second related work [START_REF] Bindel | A novel predictive link quality metric for mobile adhoc networks in urban contexts[END_REF] we improved our LQE with the development of additional estimators assessing multiple features of a link in order to provide a reliable vision of the link state. This paper is an extended version of this work which provides additional assessment on the estimator performances and describes a framework in order to integrate our final metric called F-ETX (LQE + its companion estimators) with routing protocols. We increase the level of description of our solution and submit an innovative integration process of F-ETX in a routing protocol and show its benefits compared to other solutions. We have simulated realistic scenarios carefully and shown novel results confirming the performance of our metric.

The main contributions of this paper are : • A discussion concerning the challenges of LQEs in MANETs. We focus our attention on requirements needed for evaluating link quality estimators, especially to compare their performances. We survey the taxonomy and give a short analysis on estimators. Then, we give our challenges met during the development of our estimators (section 2).

• A detailed description of the design of our solution, including the link quality estimator, its companion estimators and also a framework to integrate the final metric into routing applications (section 3).

• A realistic evaluation of the efficiency of our solution within realistic simulations and provide detailed results to prove its usefulness (section 5).

The remainder of this paper is organised as follows. Section 2 presents the background including the link quality estimator requirements and an overview of current estimators. We detail in section 3 the design of our LQE, its companion estimators and we outline the framework to integrate the metric into a routing protocol. Section 4 describes realistic scenarios defined for our simulations. Their results are depicted in Section 5. At the end, we review the related work in Section 6.

Background

Evaluating the link quality in MANETs and its subsets is a challenging endeavour, especially with regards to the current practices, which develop most of link quality estimators for static wireless sensor networks. In this section, we detail some challenges to highlight problems of finding an efficient solution to assess the link quality and determine the link states in MANETs.

Link Quality Estimators Requirements

To get a reliable assessment of link quality, estimators have to meet requirements as described in [START_REF] Baccour | Radio link quality estimation in wireless sensor networks: A survey[END_REF].

Energy efficiency: energy is an essential concern, but is not required if nodes do not own a limited battery capacity, as in VANETs. It is nevertheless advisable for LQEs to minimise their computations and communication overhead. Complex estimation techniques and data structure such as learning might be inappropriate. About communication, even if an active link monitoring technique tends to consume much more power than passive link monitoring, a high beaconing rate should be avoided as it consumes too much energy.

Accuracy: the accuracy of a LQE is related to its ability to characterize the behaviour of a link. It has a significant impact on the protocol efficiency since the selection of the best link is relied on a comparison of available links.

Reactivity: it refers to the ability of an estimator to react quickly to a persistent change of the link quality. In wireless mobile networks, it is important for a LQE to be highly reactive in order to ensure an up-to-date estimation and also to detect quickly a link disruption. The reactivity depends on two factors : the monitoring window and the link monitoring scheme. The monitoring window is used to count received and lost packets in order to build an history of the packet reception. The monitoring scheme provides a tracking system based on three kinds of monitoring: active, passive and hybrid. In case of active monitoring, a node monitors a link by sending periodically a probe packet. In case of passive monitoring, a node exploits current traffic by overhearing the media without generating additional communication overhead. In a hybrid monitoring, a node uses both active and passive methods, thereby enabling an efficient balance between up-to-date measurement and overhead efficiency. Finally, the use of a reduced window and an active monitoring with an high beaconing rate leads to making a reactive LQE.

Stability: unlike the reactivity property, it refers to the ability to tolerate transient variations of the link quality. Indeed, reactivity and stability are at odds. For instance, if a transient link quality change occurs, the routing protocol does not have to find another route, since routing computation involves an useless energy consumption. In [START_REF] Lin | Towards stable network performance in wireless sensor networks[END_REF], the authors argue that stability is met with a long term estimation. For example, EWMA filters with a large smoothing factor can deliver a long term estimation.

As a matter of fact, the design of a LQE meeting all quoted requirements is a challenging task. Indeed, reactivity and stability are at odds, meaning that a LQE cannot be both reactive and stable. Even if in MANETs LQEs have to be reactive, stability is also advocated to avoid for the protocol too often re-computation, e.g., routes change for routing protocols. The reliability of a LQE is relied on the accuracy, since the best link selection is based on a link quality comparison. Last but not the least, energy consumption deals with the lifetime of a node and must be considered, to have a trade-off between energy efficiency and estimation accuracy.

Physical vs Logical information

In order to assess the link quality estimation, an LQE retrieves information from either the physical layer or upper layers such as data link or network. Indeed, LQE can be classified in two categories: hardware-based and software-based.

Classical hardware-based estimators are based on the received signal properties such as the Received Signal Strength (RSSI), the Signal-to-Noise Ratio (SNR) and the Link Quality Indicator (LQI). Besides, these estimators read directly information from the radio transceiver 1 to provide an estimation correlated with the Packet Reception Ratio (PRR) in order to observe the estimation accuracy. However, several research activities have shown their limitations to evaluate the link quality if it is situated in the transitional region. To overcome this drawback a novel kind of estimator retrieves information from the radio chip decoding process. The first one deals with the DSSS (Direct Sequence Spread Spectrum) decoding error analysis [START_REF] Spuhler | Blitz: Wireless link quality estimation in the dark[END_REF] [9] meanwhile a second one [START_REF] Gabteni | A novel predictive link state indicator for ad-hoc networks[END_REF] extracts information from the OFDM (Orthogonal frequencydivision multiplexing) decoding process to assess the link state and predict future link disruptions. These estimators profit from information reported by the radio chip for making an accurate estimation of the link quality and even more the link state determination. Unfortunately, they impose by their computation to be set into specific radio chips, since they exploit data returned by the decoding process.

At the opposite, logical-based estimators assess the link quality according to the application point of view, i.e., the successful packet reception ratio or packet transmitted. Unlike hardware-based estimators, their assessments are directly correlated with the packet reception. They are able to evaluate the link quality not only through the PRR but also through the number of packets needed to ensure a successful transmission, called Required Number of Packet (RNP). Indeed, the estimation involves the computation of the up and downlink packet reception ratio by the sender which counts the number of packets transmitted and transmitted back. Another strategy adopted by the Expected Transmission Count (ETX) estimator is to maintain by the receiver-side the probability of the forward ratio and the backward delivery ratio via an active monitoring. Performance analysis shows that RNP/ETX based estimators are able to estimate the link quality with a better accuracy than PRR based, for the reason that RNP estimation takes into account the underlying distribution of packets loss. Thus, these estimators appear also more reactive by catching packet losses of the both sides of a link.

Challenges

The requirements detailed in section 2.1 are not sufficient to consider LQEs as suitable in mobile networks. In order to address the challenges of the link quality estimation in mobile networks, we detail the new challenges encountered.

• Support mobility and disturbances. The first challenge that we met is the support of the mobility of the nodes. Traditional estimators have been designed for static WSNs and do not support quick network topology changes. Thus, our estimators must be reactive enough to deal with the volatility of nodes. A mobile node may change its speed, location and environment. This leads to an important change in the signal disturbance via a different shadowing, fast-fading and Doppler effects. Our approach should take into account the volatility of a channel that encounters the nodes to ensure an up-to-date estimation but also the signal disturbance to deliver an accurate estimation of the link quality.

• Link selection. Most of LQEs assesses the quality through a single value that leads to catching only a snapshot. This expressiveness of a such estimation leads to comparing only short-term estimations without the possibility to take into account the dynamic of the link. There is a need to take into account this dynamic into the link quality assessment in order to ensure a reliable and robust link selection made by upper protocols.

• Integration. Protocols and estimators interact together through a single value to select the best link. Even if the multi-estimator approach assesses different features of the link with a multiple estimation, the results are fitted into a single one. This ensures a standard interaction but does not profit from a multi-estimation. Indeed, the protocol cannot adopt a strategy according to each estimation, that limits its efficiency. Our approach will ensure to take into account each estimation making the protocol more reactive to link state changes.

Estimators Design

In this section we describe our estimators and the framework designed to integrate them into a routing protocol. We detail in first the design of our LQE. Then, we present the companion estimators which complete the link quality estimation in order to provide a reliable link state assessment that is able to forecast the link behaviour. Current metrics which use a multi-estimation technique merge their different assessments into a single value. This would guarantee to be compatible with current routing protocols, because they use directly the results returned by estimators to adapt their routing policy. We argue this strategy is not the best way for the reason of multi-estimators assess different features of the link not directly related. To this end, we suggest a framework to integrate the link quality estimator and its companions with a routing protocol. The aim for the protocol is to profit from all estimators to trigger events as soon as they detect a link state change.

Overview

In recent years, a number of research efforts have been done to develop link quality estimators for WSNs. However, their performances are limited in a mobile environment. To overcome the identified issues described in section 2, we designed a novel link quality indicator based on a link quality estimation enhanced by additional estimators in order to predict the link state. Indeed, our estimators give both an instant assessment and a time based trend. The proposed estimator is based on a dynamic window size used to build an history of the packets reception. Our approach makes our estimators suitable for dynamic and mobile environments. Even if our LQE is based on the ETX, it overcomes the main problem of the accuracy and the reactivity of the estimator. Indeed, ETX is based on a static window size, implying a tradeoff a between the reactivity and the granularity of the estimation. This problem is inherent to the use of a static window size as we shown in [START_REF] Bindel | F-etx: An enhancement of etx metric for wireless mobile networks[END_REF]. To solve this issue, the estimator uses a dynamic window size to adjust the link quality estimation to the link lifetime. Indeed, our estimator can estimate both the quality and the link lifetime also referred as the stability. We also developed additional estimators to make our estimator more robust and able to make predictions about the link state.

In order to fulfill our goals, we address the issue of the integration of our estimators with routing protocols. Current solutions such as F-LQE [START_REF] Baccour | Flqe: A fuzzy link quality estimator for wireless sensor networks[END_REF] and HoPS [START_REF] Renner | Prediction accuracy of link-quality estimators[END_REF] merge the multi-estimations into a single metric. This approach tends to hide the property of each estimator. We propose to trigger specific routing mechanisms according to each assessment given by estimators.

Window management

Using a window allows to maintain an history of the packets reception state (received or lost) and its size sets the number of observed packets. Like ETX, our estimator is based on an active monitoring in which probe packets are periodically sent, that leads to filling or emptying the window at the same rate. Thus, the window size impacts both the reactivity and the accuracy of the assessment. The reactivity relies on the convergence time to declare a link disrupted or a link with a maximum quality (time to fulfill the window). Beside, the accuracy depends on the number of observed packet. Therefore, the use of a static window size implies a trade-off between accuracy and reactivity. We suggest the use of a window with a dynamic size in order to manage the history according to the link lifetime. Indeed, a node with an high velocity has a reduced link lifetime but can have an important link quality. Our main goal is to re-size the window as soon as a disruption occurs. To this end, two algorithms are required, one to reduce the window size and the other to grow the window.

Window Size Reduction Algorithm. As reactivity is a major concern for link quality metrics, F-ETX uses an algorithm which ensures both a fresh and an accurate estimation. Indeed, the algorithm maintains an history of packets, called window and tracks their reception state. Let a packet p ∈ P a finite whole of observed packets such as, P = {p 0 , p 1 , p 2 , • • • , p n-1 } with n the number of observations. Each observed packet, p is labelled according to its reception state such as, a label L ∈ [0, 1] in which, L ← 0 indicates a loss and L ← 1 a reception. The number of packet considered as received, a, and lost ā are computed as follows :

n = a + ā, a = n i=1 L i , ā = n - n i=1 L i • (1)
Thus, n is reduced according to the number of packet considered as lost :

n = n 2 ā • (2)
As shown in [START_REF] Bindel | F-etx: An enhancement of etx metric for wireless mobile networks[END_REF], in case of successive packet losses, the algorithm enables an exponential (2 ā) decrease of n.

In praxis, maintaining a window let us create a history of packets identified with a unique ID such as a sequence number. A packet is declared received or lost according to a timer, also used to send periodically a probe packet. The figure 2 depicts an example of a window size reduction. The timer is scheduled to trigger a transmission of a probe packet every seconds. To avoid a synchronization among nodes, transmissions are delayed according to a random variable fixed at [0, 210] ms. We also consider the signal propagation delay as insignificant, for the reason that the celerity of an electromagnetic wave in the air is at 3 × 10 8 m.s -1 . Therefore, a delayed reception cannot exceed one second.

Window Size Growing Algorithm. To ensure a suitable link quality estimation, it is required to adjust the estimation to the link stability. The suggested algorithm attempts to adjust the window by keeping or increasing its size. Indeed, once a packet is received, the algorithm determines if the window is increased or slid. To this end, a threshold T h is set as the window size, once a packet is declared as lost. Until the window size is lower than T h , the widow size is incremented by one for each new packet received. Indeed, the algorithm tries to recover the last window size when a disruption occurs. This phase is called the recovery phase. If the window size is equals or higher than T h , each new received packet increments a dedicated counter, C. Then, the window (W ) is increased or shifted (left) as described in the algorithm 1.

This approach limits the speed of the growth of the window size according to the loss occurrence and adjusts the accuracy of the estimation according to

Algorithm 1 Window Size growing algorithm if W n ≥ T h ∧ C W n 2 then W ← increased by one C ← 0 else W ← Slid C ← C + 1 end if
the link stability. In this phase, the algorithm senses the link stability and is called stability sensing. The figure 3 depicts the behaviour of the window with our algorithms.

Link quality estimation

The link quality assessment aims to find the highest throughput link. As for ETX, the estimation is based on a forward and a delivery ratio. The forward ratio (d f ) measures the probability that a probe packet sent by a direct neighbour is received. The reverse delivery ratio (d r ) measures the probability that a probe packet is successfully forwarded by a direct neighbour. Indeed, the previous sender hears its own probe packet and takes it as an acknowledgement. The data flow depicted in the figure 4 shows how the both ratios are computed.

The expected probability that a message is successfully received and acquitted is d f × d r . If we consider a packet transmission as a Bernoulli trial (success or fail), the link quality estimation is determined by the probability to loss a packet. This probability is computed as follows:

χ LQ = 1 (1 -d f ) × (1 -d r ) • (3) Node 0 Compute d r Compute d f Node 1 Compute d f Compute d r
From : 0, To: all, Pre v : 0

From : 0, To: all, Pre v : 1

From : 1, To: all, Pre v : 1

From : 1, To: all, Pre v : 0

Figure 4. Case of probe transmissions

We draw the attention to the fact that a couple of windows must be maintained, one dedicated to compute d f and another for the d r computation. Besides, an only timer is needed to schedule the sending of probe packets and update windows. To conclude, we give in the table 1 the convergence time of the classical ETX and our LQE to detect a link disruption, once the full windows.

Companion estimators

Since a short term estimation is not sufficient, we developed three additional estimators, each evaluating a specific feature of the link to give a comprehensive assessment of the link state. These estimators relies on the Exponentially Weighted Moving Average (EWMA) to track the trend of each feature of the link.

Link quality trend. This indicator tracks the course of the link quality by computing the variation between the current χ LQ t

and the previous estimation χ LQ t-1 . To provide a long term estimation the result is averaged with an EWMA.

∆ LQ t = χ LQ t -χ LQ t-1 , χ T rend t = β × ∆ LQ t + (1 -β) × χ T rend t-1 , (4)
the coefficient β influences the sensitivity of the estimator. Choosing a small β value is advisable to achieve a long term estimation. Note that two successive nulls χ LQ indicate a disruption and reset the link quality trend estimator.

Link stability estimation.

We observed that a fine analysis of the windows content provides a link stability information. Let a binary state [0, 1] that represents the reception state of an excepted packet in a window. We note W max the maximum window size, W n the current window size and the W i the i th element in the window. The windows maintained to compute the d f and d r probabilities are respectively noted W d f and W d r . The link stability indicator is computed with an EWMA filter tacking into account the absolute Ξ and the relative stability ξ:

Ξ = W d f n i=1 W d f i + W d r n i=1 W d r i 2W max , ξ = W d f n i=1 W d f i + W d r n i=1 W d r i W d f n + W d r n , χ Stab t = Ξ t × γ + (1 -γ) × ξ t • (5)
The absolute estimation (Ξ) computed from the maximum window size (fixed value) represents the absolute level of stability of the link. The relative estimation (ξ) computed from the current window size (dynamic value) represents the relative stability. This third estimation gives the level of the link stability according to the current window size. This information is useful, since, for a same absolute value, the relative link estimation gives an additional assessment taking into account losses which took place recently. Both absolute and relative information are suitable to assess the link stability. They must be taken into account in the same way. Hence, we advise a γ value fixed at 0.5.

Unidirectional link level.

This last estimator deals with the detection of bidirectional links becoming unidirectional. Current approaches like F-LQE with the ASL estimator, track the difference between the uplink and downlink reception rates. Such a method becomes inefficient if the link has a short life time or experiment an high level of packet losses. In this case, windows are not sufficient trained to give a trustworthy estimation.

Our method overcomes this limitation by measuring the variation of the up and downlink reception ratios. This makes it independent of the window size and does not require any training periods. Let W be a window and W t n its size at time t. The variation of the reception ratio provided by the window W at time t is noted ∆ W in t . The indicator is given by:

∆ W in t = W t n i=1 W i - W t-1 n i=1 W i , χ U LL t = χ U LL t-1 × λ + (1 -λ) × ϕ(∆ d f t , ∆ d r t ), with ϕ(x, y) =          -1 x < 0 ∧ y > 0 1 x > 0 ∧ y < 0 0 else • (6)
To give a tendency we advise a λ value fixed at an high value. A link may become unidirectional (e.g. nodes with different transmit power level) if the assessment becomes negative.

Routing Integration Framework

In this section, we describe the framework designed to integrate all estimators in a routing protocol. Each estimator assesses a specific property of a link in order to provide a different vision of a link state. It is a key concept for addressing issues of routing protocols. Thus, the routing protocol must have holds all the cards to select the best link in order to maximize the packet delivery ratio and minimize the delay. Current metrics using multi-estimators such as F-LQE [START_REF] Baccour | Flqe: A fuzzy link quality estimator for wireless sensor networks[END_REF] and HoPS [START_REF] Renner | Prediction accuracy of link-quality estimators[END_REF] compute a scored quality link estimation in order to provide an unique assessment. Even if Baccour et al. [START_REF] Baccour | Reliable link quality estimation in low-power wireless networks and its impact on tree-routing[END_REF] have implemented F-LQE in the Collection Tree Protocol (CTP), there are no silver bullets to compute an ultimate single estimation including all values returned by estimators.

To solve this issue, we propose a framework that is able to integrate each estimation into the routing process. Indeed, each estimator would be related to the routing table in order to indicate specific link state if a event occurs. Based on an active monitoring, each estimation is computed after the reception of a probe packet. Then, they assign their assessment to the associated entry into the routing table. The proposed framework is illustrated in the figure 5 including the routing protocol and our metric (including all estimators).

We consider the routing table as the intermediary part to integrate the metric into a routing protocol. Indeed, the metric assesses both the link quality and indicates the link state and store this information into the routing table. Then, the routing algorithm selects the best link according its quality and its state. Thus, we need to define how the quality is assessed and which state is stored in the routing table. In addition, we detail how a routing algorithm may interpret information stored in the routing table.

Local Link Evaluation. Our metric includes four estimators including a short term, long term link quality, stability estimator and unidirectional link indicator.

F-ETX

Routing protocol

Link quality

Link quality trend

Link stability

Undirectionnal link

Routing table

Routing algorithm

Link selection

Assesses the link quality

Declares the link available

Long term assessment

Detects unidirectionnal links

Figure 5. Routing framework

The unidirectional link indicator detects transient looses which turn a bidirectional link into an unidirectional link. Besides, persistent unidirectional links can be detected with a direct observation on the d r ratio indicating the number of packet retransmitted by a neighbour. Thus, our metric indicates two possible states indicating the unidirectional property of the link:

• U p : persistent unidirectional link • U t : transient unidirectional link
The stability indicator provides information about the stability of the link. It enables to determine if a link can be relevant for routing data. This estimator is used to determine if a routing entry is valid or not. Indeed, a null estimation brands a routing entry as disable even if the quality is not null. Thus, our estimator indicates also two possible states on the link stability:

• χ Stab = 0: disable routing entry • χ Stab > 0: available routing entry As we shown in the figure 1, the major problem happens when the quality of two links are closed but have an opposite trend. Hence, we suggest an estimation merging both the short and the long term to assess the link quality, computed as follows :

Link quality = χ LQ + χ T rend [START_REF] Lin | Towards stable network performance in wireless sensor networks[END_REF] Taking into account both the short and the long term estimation, enables to penalize the short term estimation according to its current trend. If a couple of links has a close link quality, the metric is able to select the best one according to its quality trend.

Routing in Praxis.

When a packet is received, the routing algorithm is responsible to route the packet to the destination by selecting the best link. This selection process is performed by selecting the corresponding entry in the routing table. That why, the protocol ranks for each destination node the potential best neighbour according to the link quality (χ LQ + χ T rend ). Then, the protocol inspects the stability indicator to determine if a route is declared available or disable. In the disable case, the algorithm selects the next route and restarts the same approach. At the end, the protocol checks if the link is unidirectional. In the case of a transient state, the route is selected, else the routing algorithm looks for another route.

Simulation Settings

To assess our approach, we use the ns-3 simulator [START_REF] Riley | The ns-3 network simulator[END_REF].

We define two realistic scenarios including, a realistic mobility pattern (generated with BonnMotion [START_REF] Aschenbruck | Bonnmotion: A mobility scenario generation and analysis tool[END_REF]) and a signal propagation environment. The first scenario describes vehicular nodes moving in an urban area and communicate in an highly disturbed environment. In the second scenario, nodes evolve in a disaster area and go through predefined Point of Interests (POIs) and communicate in a free space environment.

Urban Scenario

In this scenario, 40 vehicles drive in an urban area of 200m × 200m with a Manhattan mobility model 4 × 4. We set the mean speed of vehicles at 30m.s -1 in order to simulate an high speed traffic. From [START_REF] Benin | Vehicular network simulation propagation loss model parameter standardization in ns-3 and beyond[END_REF], we fix the channel propagation parameters, with a ThreeLogDistanceLossModel as a shadowing model and Rayleigh model as a fast-fading model, to recreate a realistic urban channel propagation environment. Table 2a details simulation parameters carefully.

Disaster Scenario

In this scenario, 40 vehicles are deployed on an freespace area of 275m × 250m and move according to the Disaster mobility model, within nodes must inspect some Point of Interests (POI)s. From [START_REF] Benin | Vehicular network simulation propagation loss model parameter standardization in ns-3 and beyond[END_REF], we fix the channel propagation parameters, with a Friis loss model as a shadowing model and Nakagami model as a fast-fading model, to recreate a realistic free space propagation environment. Table 2b details simulation parameters carefully.

Simulation results

We now describe the results given by our experiments and assess the performance of our approach. As mentioned in the description of our scenarios, we test our estimators in an urban and free space environment. We explore both the performance of (i) the quality estimator, (ii) the multi-estimators (iii) and the routing process using our framework.

Performance of the link quality estimator

We proceed to investigate the performance of the ETX and the LQE of our F-ETX metric. We focus on three to assess the link quality. We also observe the correlation between the number of nodes in the network and the energy consumed. The figure 6 shows the average amount of energy consumed by nodes, each measurement is marked with its standard-deviation. The graph shows that the energy consumption increases when the number of node expends. It results from the increase number of retransmission of probe packets required to assess the link quality. In the Disaster scenario, the energy consumed is more important than in the Manhattan scenario. Indeed, in the disaster Scenario the propagation environment disturbs less signals that increases the reception rate and by consequence the number of messages exchanged. A slight decrease on the plot of the Disaster scenario can be observed from 40 nodes in the network. It can be explained by the increase of the amount of transmitted data that leads to increasing the number of collisions and contribute to reduce the number of retransmitted packets. We conclude with ETX and our F-ETX have the same energy consumption profile, since they broadcast probe packets at the same frequency. to resist to transient variations (short-term). We assess the stability by computing the coefficient of variation of the link quality estimation. It is defined as the ratio of the standard deviation to the mean. We computed this coefficient on each link presents in the network. In this experiment, we have defined a maximum window size of 15 Sequence Numbers. The figure 7 depicts the coefficient of variation of the couple of metrics. ETX appears more unstable in Manhattan than in the Disaster scenario, since urban environment disturbs more the signal than in a free space (Disaster). However, F-ETX has the same stability in the both cases and is more stable than ETX.

Stability. We proceed to investigate the stability of ETX and F-ETX that represents the ability of an estimator

Reliability. The reliability refers to the ability of the estimator to correctly assess the link quality. In order to evaluate this feature, we plot the empirical cumulative distribution function (CDF) and observe the distribution of estimations given by ETX and F-ETX. Our main goal is to determine if estimators overestimate and underestimate the link quality. The figure 8 depicts the empirical CDF. As the link quality is coded on 8 bits, the estimation is bounded between [0 : 255]. We observe that F-ETX has a better distribution value than ETX in both case. ETX tends to underestimate the link quality with a large number of link estimated with a low quality. Beside, F-ETX declares less links with a low quality and tends not overestimate or underestimate the remainder of links. To this end, F-ETX appears as a more reliable estimator than ETX.

Performance of the multi-estimators

We now investigate the performances and the robustness of all estimators of F-ETX. To this end, we compare the performance of our estimators and current metrics namely F-LQE [START_REF] Baccour | Flqe: A fuzzy link quality estimator for wireless sensor networks[END_REF] and HoPS [START_REF] Renner | Prediction accuracy of link-quality estimators[END_REF]. We set the parameters of each estimator according to [START_REF] Baccour | Flqe: A fuzzy link quality estimator for wireless sensor networks[END_REF] for F-LQE and [START_REF] Renner | Prediction accuracy of link-quality estimators[END_REF] for HoPS.

We fix the parameters of F-LQE as follows. We set the coefficient of the WMEWMA filter used to compute the packet delivery ratio at 0.6. The assessment of the link quality and the link stability require an history of PRR that we set at 30. A minimal history is maintained at 5 PRR until to reach 30. About HoPS we set the parameters as follows. Coefficients are respectively set to 0.9 and 0.997 and its initializes short-and longterm estimations at 50% for new links. At the end, we determine for F-ETX the parameters of EWMA filters used by companion estimators called λ, β and γ to 0.9, 0.1 and 0.5. Our main goal is to assess both the agility of estimators by observing their ability to track fluctuation and their accuracy/robustness. We achieve both temporal and statistical evaluations. Through the temporal experiments, we observe the behaviour of estimators in order to have an overview on their ability to assess the link property. We made also a statistical evaluation of estimators to measure their forecasting properties.

Link Quality Tracking. The reliability of estimator were tested under two scenarios, involving two nodes. The first one takes place in an urban environment and defines a fast speed scenario wherein nodes are able to communicate during few time (4s). The second one takes place in a free space environment (Disaster) where nodes communicate longer during 19s.

Fast speed crossing: the figure 9 depicts the result of the first scenario. Regarding the distribution of the d r and d f in the figure 9(a) nodes are able to communicate during few time (from 6s to 10s while stochastic losses can be observed resulting from a significant fading (Rayleigh) effect). The PRR computed over an history of 5 packets declares the link disrupted at 15s, but the The SPRR follows the corresponding PRR (fig 9) trace with a smoothing trend, but the estimator is clearly not enough reactive and detect the disruption too late. This result from the EWMA filter that gives more stability than reactivity to the estimator. The SF estimator detects that the link quality is changing at 11s, because the link is disrupted. However, the variation indicated by the estimator does not reflect a disruption case but only a slight variation of the link quality. The disruption can be clearly detected at the end with a more important value of the indicator. About the ASL indicator, the variation of d r and d f distribution introduces light fluctuations indicating a low probability to have an asymmetric link.

Regarding estimators from HoPS (fig 9(d) and (e)) we observed the slow convergence time of the short-term estimation, while a link is disrupted, the estimator declares the link quality as not disrupted. But the longterm estimator indicates a correct decreasing trend. Consequently, the EWMA filter is well used for longterm estimation but is not suitable for a short-term estimation which smooth the estimation too. In the same manner, the link quality trend and the variation indicators are affected by the long reactivity of the short and long-term estimations and react too slowly when a disruption occurs. The estimators of F-ETX are depicted in figure 9(f) and (g). In contrast with other LQEs, F-ETX is more reactive than the others and declares the link disrupted earlier (at 13s). The trend estimation indicates a degradation of the link quality via consecutive negative values. This is confirmed by the link stability estimator indicating a low level of stability and a decrease. We also observe that the stability estimator declares the link as disrupted earlier than the link quality estimator (12s). About the unidirectional indicator, its gives a positive value (at 10s) indicating that the link can be unidirectional.

Low speed crossing: the figure 10 Statistical analysis. While previous evaluations give a detail about the strength and weakness of LQEs. We extend our assessment with a statistical analysis of all the links from both the Manhattan (fig. 11a) and the Disaster (fig. 11b) scenarios. We have observed that the link quality estimator of F-ETX is more reactive than F-LQE and HoPs. In this statistical study we are focus on how this estimator can anticipate disruptions compared to the PRR solution based on an history of 5 packets. Figures 11a (a) and 11b (a) show that F-ETX is clearly the best solution by anticipated disruptions before the PRR solution. Since it is based on the dynamic window size, the metric is more reactive and track link states changes very well. In addition, F-ETX assesses both link direction unlike a PRR solution that only evaluates the downlink. The rest of the statistical analysis is made with the Mean Absolute Error (MAE) that measures the magnitude of the predicted estimation and the current outcome. A low score indicates a good prediction while a bigger value indicates a greater error between the prediction and the current value.

The link quality trend is an additional information that determines the current course of the link quality. Fig 11a (b) and 11b (b) show the link quality trend of HoPs and F-ETX. We observe the better ability of our estimator to give the tendency of the link quality compared to HoPS. Even if each of them is based on the link quality estimator and both are computed with an EWMA filter. Their abilities to track the link quality course depends on the ability of short-term link quality estimator. HoPS-ST suffers from lag with the use EWMA filter impacting the HoPS-LT. On the other hand, the link quality estimator from F-ETX is react but unstable. That is why with the use of EWMA the estimation is stabilized given a better longterm estimation than the HoPS-LT overestimating the tendency. Figures 11a (c) and 11b (c) show the unidirectional link estimator of the F-LQE and F-ETX. During the simulation, any effective unidirectional links are present. The ASL estimation makes a single reading on the reception ratio of the up and downlink often different when highly propagation disturbances are present. Our indicator adopts another strategy based on the variation between the up and downlink, as a result our estimation is more robust to disturbances and gives more suitable information about the potential of a bidirectional link to become unidirectional. Tracking link stability is an essential feature to detect and differentiate transient and persistent links. We have compared in the figure 11a (d) and 11b (d) the variation of the value given by these estimators to the current variation observed from the delivery and forward ratios. F-ETX estimator gives the lowest MAE compared to the other. Because HoPS indicator only tracks the variation between the HoPs ST and LT estimations, that not really related with the link stability. About F-LQE, the estimation is based on a PRR history generating consecutive error predictions.

Influence on Routing Performances

In the last experiment we are looking the influence of our metric on the routing performance. We enable both the AODV routing protocol and a developed protocol based on the BATMAN [START_REF]Collaborative authors[END_REF] routing protocol, using our framework. Five simultaneous communications run in the Manhattan and Disaster scenarios and we observe the packet delivery rate obtained with the AODV and with our protocol. The figure 12 depicts the obtained results and shows has the best performance of our solution in the both scenarios. Even if the propagation channel is highly disturbed in the Manhattan scenario, our approach is able to find a reliable path. Beside, in the disaster scenario, the free-space environment tends to disturb less the signal, explaining the better performance of protocols.

Related works

Previous work suggested several logical LQEs. Although, proposed approaches are mainly designed for WSNs, our approach is still inspired by these estimators. In this section we describe these LQEs and highlight some observations from experiment.

Traditional metrics in wired networks, (i.e. hop count) fail to give an estimation on the paths reliability in wireless networks. De Couto et al. suggested the Expected Transmission Count (ETX) metric [START_REF] De Couto | A high-throughput path metric for multihop wireless routing[END_REF], which combines forward and backward packet reception ratios to compute a packet delivery success probability. Draves et al [START_REF] Draves | Comparison of routing metrics for static multi-hop wireless networks[END_REF] experimented four type of metrics (ETX, hop count, Per-hop Round Trip Time and Perhop Packet Pair Delay) through a test bed placed in an indoor environment. They observed that ETX performs better than the others but in a mobility case, the performance of the hop count metric exceeds the ETX metric.

The Required Number of Packet (RNP) metric suggested by Cerpa et al [START_REF] Cerpa | Temporal properties of low power wireless links: Modeling and implications on multi-hop routing[END_REF] performs at the MAC layer and counts the average number of packet retransmissions required to deliver it successfully. They observed the temporal properties of low power wireless links and described the usefulness of such a metric to measure the link quality compared to a reception rate (RR). Unlike the RR information, RNP metric takes into account the underlying distribution of losses because the link quality is characterized in both directions. However, this metric requires an ARQ (Automatic Repeat-Request) mechanism to retransmit lost packets in order to determine the number of retransmission required to achieve a successful reception.

Woo et al [START_REF] Woo | Taming the underlying challenges of reliable multihop routing in sensor networks[END_REF] designed a routing protocol based on the channel snopping and introduced an agile filter called Window Mean with EWMA (WMEWMA) as an empirical estimator technique [START_REF] Woo | Evaluation of Efficient Link Reliability Estimators for Low-Power Wireless Networks[END_REF]. Authors showed its performance, over the others filters based on the Exponential Weighted Moving Average (EWMA).

An interesting novel approach supports that the link assessment can be made with multi-estimators to have a multi-faced vision of the link in order to compute a scored quality link estimation. Baccour et al. [START_REF] Baccour | Flqe: A fuzzy link quality estimator for wireless sensor networks[END_REF] designed an hybrid metric based on logical and physical information to provide multi-estimators that assess the packet delivery ratio, the link asymmetry level, the link stability and the channel quality. These estimators are aggregated into a single metric following a fuzzy logic method. In addition, they implemented F-LQE into the CTP routing protocol and proved its effectiveness in wireless sensor networks [START_REF] Baccour | Reliable link quality estimation in low-power wireless networks and its impact on tree-routing[END_REF].

The Holistic Packet Statistic (HoPS) metric suggested by Renner et al. [START_REF] Renner | Prediction accuracy of link-quality estimators[END_REF] incorporates four estimators; namely short-term, long-term, absolute deviation and trend estimation; computed with the EWMA filter.

However an intrinsic problem of the use of this filter limits the agility of estimators. It also has the disadvantage to require a large amount of traffic to train the estimators and consequently increases the detection time of link state changes.

Conclusion

MANET networks are self-organizing mobile networks resulting from the cooperation of stations could be cars or unmanned vehicles (including aerial and terrestrial engines). The flexibility of these networks allow them to be deployed in several environments. However, these features tend to disturb the routing performance such as the packet delivery rate. To resolve this issue, protocols use link quality estimators but most of them were designed for static WSNs.

In order to deal with the inherent dynamic of MANETs, we proposed a link quality estimator based on a dynamic window size [START_REF] Bindel | F-etx: An enhancement of etx metric for wireless mobile networks[END_REF]. Then, we have developed additional estimators to reinforce the link quality estimation. Each of them evaluates a specific link feature: (i) the trend of the link quality, (ii) the link stability and (iii) the detection of unidirectional links. Indeed, these estimators complete the link quality assessment by evaluating different features of a link. The benefit of such approach is to provide a multifaced estimation on the link. Thus, the multi-estimator is able to determine and forecast the link state. Unlike current multi-estimators such as F-LQE [START_REF] Baccour | Flqe: A fuzzy link quality estimator for wireless sensor networks[END_REF] and HoPS [START_REF] Renner | Prediction accuracy of link-quality estimators[END_REF] which mix each estimation into a single one, we propose a framework that enables the use of each estimator in the routing process. This approach makes the routing process efficient. Simulation results show that the link quality estimation given by F-ETX is more accurate and reliable than given by ETX. In addition, we have compared our multi-estimator with current approaches, by tracking their abilities to assess the link features. Besides, we highlighted prediction capabilities of our estimators through a statistical analysis that shows how our estimator overcomes other LQEs to predict link state changing. At the end of this paper, we showed the benefit our framework where the packet reception rate remains around 95% even if the signal is highly disturbed.

For future work, we plan to test the F-ETX and companion-metrics in other contexts such as building environments to confirm the suitability of our approach. In addition, tests in real environments will give us an important feedback on the estimator behaviour.

Figure 1 .

 1 Figure 1. Link selection : short vs long term

Figure 2 .

 2 Figure 2. The window size reduction

Figure 3 .

 3 Figure 3. Behaviour of the windows

Figure 6 .

 6 Figure 6. ETX vs F-ETX : Energy consumption

Figure 7 .

 7 Figure 7. ETX vs F-ETX : Stability

Figure 8 .

 8 Figure 8. ETX vs F-ETX : Reliability

Figure 9 .

 9 Figure 9. Fast speed crossing

Figure 10 .

 10 Figure 10. Low speed crossing

  depicts the result of the second scenario.The behaviours of d r and d f and the PRR indicate a progressive increase of the link quality until 13s then the quality decreases progressively, fig 10(a). Concerning F-LQE, the SPRR (fig 10(b)) increases progressively then decreases and confirms this tendency. The stability is correctly estimated by the SF estimator, indicating a decrease from 13s. However, the indicator indicates a false variation at 9s. About the ASL, stochastic losses disturb the estimator (fig 10(c)), that indicates the possibility of the link to be unidirectional. About HoPs the short-term link quality estimator also indicates a correct tendency (fig 10(d)). We note that the HoPS ST increases and decreases faster than HoPS LT. But the HoPS trend and variation (fig 10(e)) take the same course at the beginning indicating an increase of the link stability then, the variation and the quality trend indicate a decrease of the link quality. Regarding F-ETX (fig 10(f) and (g)) the link quality is estimated stable, since few packet are lost. This is confirmed by the trend estimator also indicating the same tendency then a decrease of the link quality. The same observation can be pointed out on the link stability indicator. The link unidirectionality level estimator of F-ETX indicates at 8s and 12s transients unidirectional property of link.

Figure 12 .

 12 Figure 11. Statistical evaluation

Table 1 .

 1 Detection time

	Maximum window's size	ETX	Our LQE
	50	50s	5s
	30	30s	4s
	10	10s	3s

Table 2 .

 2 Simulation parameters

	(a) Urban Scenario		(b) Disaster Scenario
	PHY parameters		PHY parameters
	Tx/Rx power (dbm)	0	Tx/Rx power (dbm)	0
	Gain of antenna (dB)	0	Gain of antenna (dB)	0
	Power Detection Threshold (dbm)	-96	Power Detection Threshold (dbm)	-96
	MAC parameters		MAC parameters
	Standard	802.11g	Standard	802.11g
	Mode	OFDM 6 Mpbs	Mode	OFDM 6 Mpbs
	Rate adaptation	ARF	Rate adaptation	ARF
	Propagation Loss Parameters		Propagation Loss Parameters
	ThreeLogDistance		Friss Propagation Model
	Exponent 0	2.5	Frequency (Hz)	2.5 × 10 9
	Exponent 1	5	SystemLoss	1
	Exponent 2	10	MinLoss	0
	Distance0 Distance 1 Distance 2	1 75 114	Nakagami-m m0 m1	1.5 0.75
	Nakagami-m		m2	0
	Rayleight	m=1	d1	80
	Simulation Parameters		d2	320
	Seed	5155236	Simulation Parameters
	Run	9287851	Seed	5155236
			Run	9287851
	important aspects to asses the reliability of a estimator:
	the energy consumption, the stability and the reliability
	of the estimators.
	Energy Consumption. In order to asses the energy
	consumption, we use an energy model retracing a
	Lithium Ion battery 2 and a model retracing a typical
	energy consumption done by a wireless network
	interface 3 . The power consumptions of the model are
	summarized in the table 3.

Table 3 .

 3 Power consumption in Watts

	Actions	Power consumption
	Transmission	1.14
	Reception	0.94
	Idle	0.82
	Sleep	0.10
	We are interested on how ETX and F-ETX stress the
	wireless interface, in term of transmissions required
	2 see: ns-3 LiIonEnergySource model
	3 see: ns-3 WifiRadioEnergyModel

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Valentin Meyer
email: valentin.meyer@insa-lyon.fr

Laurent Maxit

Jean-Louis Guyader

Thomas Leissing

Christian Audoly

A Condensed Transfer Function Method as a Tool for Solving Vibroacoustic Problems

Keywords: Vibroacoustic, Sub-structuring, Admittance, Numerical Analysis

Substructuring approaches are nowadays widely used to predict numerically the vibroacoustic behavior of complex mechanical systems. Some of these methods are based on admittance or mobility frequency transfer functions at the coupling interfaces. They have already been used intensively to couple subsystems linked by point contacts and enable to solve problems at higher frequency while saving computation costs. In the case of subsystems coupled along lines, a Condensed Transfer Function (CTF) method is developed in the present paper. The admittances on the coupling line are condensed in order to reduce the number of coupling forces evaluated. Three variants are presented, where the transfer functions are condensed using three different functions. After describing the principle of the CTF method, simple structures will be given as test cases for validation.

Introduction

Sub-structuring methods in dynamics or vibroacoustics have been intensively studied over the last decades in order to push always further the computational limitations of numerical models. The most widespread techniques today are certainly the modal synthesis methods. They are based on the Rayleigh-Ritz method of modal estimation and consist in splitting the system and calculating modal basis with different boundary conditions. Hurty [START_REF] Hurty | Dynamic analysis of structural systems using component modes[END_REF] was one of the first to introduce the Component Mode Synthesis (CMS) and was then followed by many developments. Craig and Bampton 2 formulated a variant of the method using the constraint modes and the shape modes with fixed interfaces. Gladwell [START_REF] Gladwell | Branch mode analysis of vibrating systems[END_REF] developed the branch mode analysis, which is particularly adapted to systems defined by a main component on which secondary components are mounted. The basis for the main component is characterized by its modes with free interfaces while the secondary components are characterized by the constraint modes and the shape modes with fixed interface (similarly to Craig and Bampton). MacNeal 4 presented a hybrid method, mixing free and fixed interfaces with several corrections, to minimize the effects of modal truncature and improve the convergence. More recently, Tournour et al. [START_REF] Tournour | Validation, performance, convergence and application of free interface component mode synthesis[END_REF] gave convergence criteria and benchmark data for the free CMS, which main advantage lies in the fact that the degrees of freedom at the interface are eliminated from the final system of equations during the assembly process.

Modal basis can however not always be calculated easily and methods based on impedance or mobility concepts are then a common alternative to the CMS techniques. [START_REF] O'hara | Mechanical impedance and mobility concepts[END_REF][START_REF] Hixson | Mechanical impedance. Shock and vibration handbook[END_REF] They offer the possibility to couple subsystems which characteristics are determined by different means (numerical, analytical, experimental). Petersson and Plunt [START_REF] Petersson | On effective mobilities in the prediction of structure-borne sound transmission between a source structure and a receiving structure, part i: Theoretical background and basic experimental studies[END_REF][START_REF] Petersson | On effective mobilities in the prediction of structure-borne sound transmission between a source structure and a receiving structure, part ii: Procedures for the estimation of mobilities[END_REF] characterize the vibratory power input in a structure by defining effective mobilities to account for multi-point coupled structure-borne sound sources. The concepts have been experimentally investigated and show a great practical interest. For linear problems, Ouisse et al. [START_REF] Ouisse | Patch transfer functions as a tool to couple linear acoustic problems[END_REF] developed the Patch Transfer Functions (PTF) method based on the mobility method to couple subsystems along surfaces. These surfaces, that can be either the junction between a structure and a fluid domain or between two fluid domains, are divided into patches to calculate transfer functions. The method is first validated on an acoustic cavity divided in two domains, before being applied to the automotive industry. Chazot and Guyader 11 later reused the method in order to evaluate the transmission loss through double panels. Aucejo et al. [START_REF] Aucejo | Convergence acceleration using the residual shape technique when solving structure-acoustic coupling with the patch transfer functions method[END_REF] focused on the heavy fluid loading and improved the convergence of the method by introducing residual mode shapes. Maxit et al. [START_REF] Maxit | Improving the patch transfer function approach for fluid-structure modelling in heavy fluid[END_REF] have shown that the size of the patches can be increased by choosing coupling surfaces outside the acoustic near-field. An example of application is given on the sound transmission in the ballasts of a submarine vehicle. The results of these studies point out a convergence criterion on the size of the patches, namely that they should remain smaller than half a wavelength. More recently, the PTF approach has been applied to solve a vibroacoustic problem involving porous materials, [START_REF] Veronesi | Patch transfer function approach for analysis of coupled vibro-acoustic poblems involving porous materials[END_REF] or has been used to allow coupling between geometrical acoustics and another domain. [START_REF] Boucher | Coupled (vibro)-acoustic problems using geometrical acoustics and patch transfer functions[END_REF] In this paper, a sub-structuring approach called the Condensed Transfer Function (CTF) method is proposed as an extension to couple structures along a line. This development is motivated by the fact that line coupling can often be met in industrial applications.

In the aeronautical and naval industry for instance, some internal frames that stiffen the structure and support the machinery are connected to the shell along lines. A set of orthonormal functions called condensation functions, that depend on the curvilinear abscissa along the coupling line, is considered. This set is used as a basis for approximating and decomposing the displacements and the applied forces at the line junctions. Thanks to the definition and calculation of condensed transfer functions for each uncoupled subsystem and by using the superposition principle for passive linear system, the behavior of the coupled subsystems can be deduced. Three examples of condensation sets are given and the method is applied to a plate in order to study the convergence with respect to the number of transfer functions considered. Part of this work has been presented previously at NOVEM2015. Introduced first by Firestone, 17 the admittance method has been used to calculate the harmonic response of subsystems coupled by contact points (see Figures 1 and2), and under an external load F ext . The contacts are assumed to be perfect, without friction loss, and bilateral (meaning that the forces transmitted have the same amplitude but are of opposite sign).

For general 3-D problems,

U α n = U α n,p p
is the vector of the 3 translations and 3

rotations of space at the point n and F α m = F α m,q q the vector of the 3 forces and 3 moments applied on m. The admittance Y α nm of a subsystem α is defined as the ratio of the displacement at the point n over the force applied at the point m:

Y α nm = U α n,p F α m,q 1≤p≤6,1≤q≤6
.

(

) 1
For two coupling points (n, m), the admittance Y α nm is thus a 6 × 6 matrix to account for all the degrees of freedom.

Moreover, the displacement vector at point n of the uncoupled subsystems when only external loading is applied, is called free displacement and is written Ũα n . Based on the superposition principle, the displacements of an uncoupled subsystem α at the contact point n can be written as

U α n = Ũα n + Y α nn F α n + Y α nm F α m , α ∈ {1, 2}. (2)
The displacements at the contact point m are simply derived from the previous equation by interchanging n and m.

The displacement continuity and force equilibrium between subsystem 1 and 2 are written for all connection points as

U 1 n = U 2 n F 1 n = -F 2 n = F c n , ∀n. (3)
Equations 2 and 3 yield to Equation 4 and the coupling forces F c n at the connection points can thus be deduced by inverting the admittance term in the following equation:

Y 1 + Y 2 F c = Ũ2 -Ũ1 . ( 4
)
where the notation without subscripts denotes vector and matrices assembled for all connection points. In this case, as no external load is applied on subsystem 2, Ũ2 = 0. This method can easily be extended to more than two coupling points by increasing the size of the admittance matrices and free displacement vectors. [START_REF] Rubin | Mechanical immittance-and transmission-matrix concepts[END_REF]

Extension to line coupled subsystems

In industrial applications, it is likely that subsystems are coupled along lines rather than on points, as shown on Figures 3 and4. That is why the admittance method needs to be extended to allow line coupling. A set of N functions called condensation functions is considered: {ϕ n } 1≤n≤N . They are functions of the curvilinear abscissa s along the coupling line Γ. The number of transfer functions N plays a key role in the convergence of the method and its value will be discussed later in this paper. The set of condensation functions is supposed to be orthonormal. For an uncoupled subsystem α, a condensed transfer function is defined as follows:

Y α nm = Ū α m,q,p , ϕ n F α m,q , ϕ m 1≤p≤6,1≤q≤6 = Ū α m,q,p , ϕ n p,q , ( 5
)
where •, • is a scalar product. F α m is the vector of the efforts on Γ, where the components in each direction of space is equal to ϕ m . The indexes p and q denote directions of space. Ūα m,q is the displacements vector of the junction Γ when the subsystem is excited in the direction q by a force of magnitude F α m,q .

F s It is assumed that the displacements vector U α at the junction can be written for each subsystem α ∈ {1, 2} as a linear combination of the condensation functions:

U 1 (s) = N n=1 u 1 n ϕ n (s) U 2 (s) = N n=1 u 2 n ϕ n (s) , ( 6
)
where s is the coordinate along the arc Γ, and u α n the condensed displacements vector of subsystem α associated to the condensation function ϕ n .

Let us write F α (s) = N n=1 f α n ϕ n (s) the efforts applied on the junction of the subsystem α. In response to these linear excitation forces, the superposition principle for passive linear systems enables to write the condensed displacements u α n as

u 1 n = ũ1 n + N m=1 Y 1 nm f 1 m u 2 n = ũ2 n + N m=1 Y 2 nm f 2 m , ∀s ∈ Γ, ∀n ∈ [[1; N ]], (7)
where ũα n = Ũ 1 p , ϕ n p are the free displacements of the uncoupled subsystems. As on this example the subsystem 2 has no external load: ũ2 n = 0, ∀n. Besides, the displacement continuity and force equilibrium lead to

U 1 (s) = U 2 (s) F 1 (s) + F 2 (s) = 0 , ∀s ∈ Γ (8)
The set of condensation functions being orthonormal, Equation 8 yields to

u 1 n = u 2 n f 1 n = -f 2 n = f c n , ∀n ∈ [[1; N ]] (9)
Assembling the vectors and matrices for all the condensation functions and injecting Equation 7 in the previous equation results in a similar formula than the classical admittance method to deduce the coupling forces between the subsystems:

Y 1 + Y 2 F c = -Ũ1 (10)
To clarify this point, the general expression for the admittance matrix and the free displacement vectors are given in the following equations:

Y α =           . . . . . . . . . . . . Y α nn . . . Y α nm . . . . . . . . . . . . . . . Y α mn . . . Y α mm . . . . . . . . . . . .           (11)
and

Ũα =           . . . ũα n . . . ũα m . . .           ( 12
)
where it is reminded that one element of the matrix Y α is a square matrix which size depends on the number of degrees of freedom (generally 6 in 3-D problems), to take into account all the possible directions of excitations and displacements. Similarly, one element of Ũα is a 6 elements vector (still considering having a general 3-D problem) to calculate the response of the junction in all directions of space to the external load. The size of this system is therefore equal to the number of transfer functions multiplied by the number of degrees of freedom.

In practice, the admittance matrices are calculated by exciting each uncoupled subsystems by all the condensation functions one after the other. Indeed, as the set is orthonormal, the denominator of the admittance term is in this case equal to 1. The admittance is deduced by projecting the displacements on the condensation functions.

Finally the behavior of subsystem 1 is obtained by applying the external loading and the coupling forces thus calculated. Similarly, the behavior of subsystem 2 is known by applying the coupling forces at the junction, and the whole system has thus been described.

The method has been presented in the case of the coupling between only two subsystems but can easily be extended to more complex systems, with more than one joint. Examples of condensation functions with their associated scalar product are given in the next section, before being applied to a plane plate test case.

Numerical study on a plate

Test case parameters

To illustrate the method presented above, let us consider in this section two rectangular plates made of the same material and thickness and which characteristics and dimensions are given in Table 1. The plates lie in the z = 0 plane and the boundary conditions are free on all edges. The aim is to couple the two plates together along their longest edge, as shown on Figure 6.

A reference calculation is made by the Finite Element Method (FEM) on a 1.5 × 2.5 m 2 plate. Harmonic responses are calculated for frequencies lying between 10 Hz and 1500 Hz, with almost 400 values logarithmically spread over the domain. This choice ensures enough values to describe properly the resonances of the system regarding the value of the structural damping coefficient, [START_REF] Fahy | Sound and structural vibration: radiation, transmission and response[END_REF] which is accounted for as a complex factor in the [START_REF] Mindlin | Influence of rotary inertia and shear on flexural motions of isotropic elastic plates[END_REF] is considered to take into account the effects of shear and rotational inertia. A virtual stiffness, which has no influence on the flexural waves, is added on the normal rotation in order to avoid numerical problems. The mesh is fine enough in order to respect the criterion of 6 elements per bending wavelength which is commonly used for such problems.

The two subsystems are modeled by FEM and the condensed admittances and free displacements need to be calculated for each of the two uncoupled subsystems. Thanks to its good convergence and calculation time, the frequency responses of the subsystems under different excitations are estimated by modal superposition including quasi-static modes. [START_REF] Rubin | Improved component-mode representation for structural dynamic analysis[END_REF] This method consists in adding very low frequency mode shapes to the modal basis in order to compensate for the basis truncation. Results of the CTF method are presented in the next subsection. The basis truncation is made so that the frequency of the highest mode is at least 1.5 times higher than the highest frequency. In the case of the two plates described previously, the basis for plate 1 counts 116 modes (highest eigenfrequency at f = 2848.6 Hz) and 125 modes for plate 2 (highest eigenfrequency at f = 2839.3 Hz). Three examples of condensation functions are given in the next subsection and the results for the present test case are discussed.

Three examples of condensation functions

Gate functions:

The gate functions are defined depending on their length, as follows:

ϕ n (s) = 1 √ Ls if (n -1)L s ≤ s < nL s 0 elsewhere (13)
The scalar product associated to the gate function is defined in Equation 14. Considering the set of piecewise continuous functions on segments [a, b[, it can be easily verified that the gate functions form an orthonormal set for this scalar product.

f, g = Γ f (s)g * (s)ds ( 14
)
where * means the complex conjugate.

s φ n (s) (L s ) -1/2 (n-1)L s nL s L s Figure 5: Gate function of length L s .
The shape of one of these functions can be seen on the plot on Figure 5. Building the admittance matrix of each subsystem in this case can be seen as dividing the coupling line into N segments, as shown on Figure 6. The segments are excited one after the other, and the observation of the displacements allows to obtain a 6N × 6N matrix, considering a 6 degrees of freedom problem. Each term of the free displacement vector is the result of the integral of the displacement on one segment due to the external load (point force at the point A in this case).

Exponential functions:

Instead of cutting the edge into segments, the idea of projecting the displacements on exponential functions is considered:

ϕ n (s) = 1 √ L exp( nπs L ) ( 15
)
where L is the length of the junction Γ and  = √ -1. This choice is motivated by the fact that the flexural motion of a plate can be described by trigonometric functions. The associated scalar product is defined as follows, ensuring the set to be orthonormal:

f, g = Γ f (s)g * (s)ds (16)
In practice, the whole line is excited by a force that varies as a complex exponential function along its curvilinear coordinate. A term of the admittance matrix is calculated by projecting the resulting displacement over an exponential function. Setting a certain i max , this calculation has to be done for all the combinations (n, m) ∈ [[-i max , i max ]] 2 . The free displacements are also projected on the exponential functions.

Chebyshev polynomials:

The Chebyshev polynomials are widely used to interpolate functions and are thus an interesting alternative for the condensation functions. [START_REF] Fornberg | A practical guide to pseudospectral methods[END_REF] They can be defined recursively, and the 5 first functions are plotted on Figure 7 with the substitution X = 2s L -1:

     T 0 (X) = 1 T 1 (X) = X T n+2 (X) = 2XT n+1 (X) -T n (X) (17)
The scalar product associated to the Chebyshev polynomials takes into account the weight function

1 √ s(L-s) : f, g = L π Γ f (s)g * (s) 1 s(L -s) ds ( 18
)
The weight function ensures the set T 0 √ 2 ∪ (T n ) n∈N * to be orthonormal and reduces numerical errors by quadratically clustering the nodes at the end of the segment.

Results with N = 3 transfer functions

The mean quadratic transverse velocity on the surface of the two plates coupled together is calculated thanks to the CTF method with the 3 different condensation functions presented previously and is compared to a reference calculation. Considering the coordinates origin at the bottom left of plate 1, the system is excited by a transverse harmonic excitation at the point of coordinates (0.9; 0.5). On Figures 8 and9, the responses and the relative errors are plotted as a function of the frequency in narrow band, with 3 transfer functions for the CTF methods (N = 3). This criterion means:

• dividing the arc in 3 segments when the condensation functions are gates;

• taking i max = 1 for the exponential functions, i.e.: ϕ(s

) ∈ { 1 √ L exp( -πs L ); 1 √ L ; 1 √ L exp( πs L )};
• using the 3 Chebyshev polynomials T 0 , T 1 and T 2 ; Based on the observations on Figures 8 and9, it seems that the results fit well until around f lim ≃ 150 Hz. For this test case, some differences up to 1 dB can be seen at the resonance frequencies in the case of the gate functions. These differences are due to a slight frequency shift in the resonance and are acceptable for industrial applications. Above this frequency, the transfer functions are not able to describe properly the phenomena. In other terms, the forces and displacements cannot be correctly interpolated by the condensation functions at higher frequencies.

Convergence criteria

Assuming that at least 2 points per wavelength are necessary to sample a signal (cf. Nyquist-Shannon sampling theorem), one can link the number of transfer functions N with the frequency limit f lim ensuring convergence of the results. Considering that λ f is the flexural wavelength of the plate at f lim , and based on the observations of the previous subsection, the convergence criteria can be written as follow:

• for gate functions, two segments are needed to describe a wavelength:

L S ≤ λ f 2 (19)
• for Chebyshev polynomials, the degree of the polynomial must be at least equal to the number of points that the function needs to interpolate:

N max ≥ 2L y λ f -1 (20)
• the same reasoning can be made for exponential functions, noting that in this case i can take negative values:

i max ≥ L y λ f - 1 2 (21)
These criteria are equivalent, meaning that a frequency results in the same number of transfer functions for each condensation function. In the case of a 17 mm thick plate, N = 3 leads to a frequency limit of f lim = 168 Hz, which fits the observations on figure 8. Applying those criteria, N = 10 transfer functions are needed to ensure the convergence of the results above 1500 Hz. The convergence is moreover checked on Figure 10 for the transversal displacements of points A and B. The coordinates of the point B are (1.2; 0.9) and the point is supposed to be located on the subsystem 2 for the CTF calculation. The plots on Figure 10 have only been presented with the CTF method using the gate functions, but the complex exponentials and the Chebyshev polynomials show good convergence too.

Position of the excitation point

To check whether the method is still valid when the vibration shape at the line junction becomes more intricate, the excitation is moved to the point of coordinates (1.2; 0.5) and lies thus on the edge of plate 1, at the junction with plate 2. The amplitude of the transversal displacement is plotted as a function of the abscissa on the coupling line on Figure 11, for the cases where the excitation is on or further away from the junction. The frequency of 1414 Hz has been chosen, because important discrepancies have been seen at this frequency for the CTF method with Chebyshev polynomials. The mesh for the finite element calculation is 5 times finer in order to have smoother curves, but the CTF calculations were still performed with the coarser mesh used previsouly in this paper. In the case where the excitation point is on the junction, the vibration shape is less regular along the line than in the case where the excitation point is further from the junction. In particular, a singularity can be seen on the red solid curve at y = 0.5 m, corresponding to the location of the excitation point. The relative errors of the mean quadratic velocity compared to FEM reference calculations are evaluated for the two positions of the excitation point, for the three different sets of condensation functions. The number of condensation functions are chosen accordingly to the convergence criteria. It can be seen that the position of the excitation point has no influence on the convergence of the CTF method in the case of the gate and exponential functions (not plotted here for the sake of clarity). Nevertheless, in the case of the Chebyshev polynomials, the set is not suited to describe properly the singularity on the junction and leads to more important errors, as seen on the plot on Figure 12. As a general comment, it is thus advisable to avoid having singularities on the coupling line, to be able to approximate more easily the physical values at the junction. In other words, in the case of the Chebyshev polynomials at least, the coupling line should remain far enough from the excitation to get rid of the near-field effects.

Influence of the cross admittance

One can wonder if the cross admittance terms (i.e. the extra-diagonal blocks of the admittance matrix: n = m) play an important role and if there is a coupling between the different condensed transfer functions. The results are plotted on Figure 13 with N = 10 transfer functions, to ensure good convergence on the whole frequency domain. They are given for the case of full admittance matrices, and the case where the extra-diagonal blocks are set to zero. It is clear that it poorly converges in the latter case. Thus it can be said that the cross admittance is not insignificant and cannot be neglected. The results have been shown with the gate functions only but similar results are observed with other condensation functions.

Numerical sensitivity

Like most of the numerical methods, it is important to check the stability of the method with regard to the input data. As seen on Equation 10, the CTF method involves the inversion of the sum of the admittance matrices. The condition number with respect to inversion is used to check if a problem is ill-conditionned and is usually defined as the ratio of the largest singular value of the matrix to the smallest. [START_REF] Nelson | Estimation of acoustic source strength by inverse methods: Part I, conditioning of the inverse problem[END_REF] If this number is particularly high, it indicates that the problem is sensitive to numerical errors.

In the case of the CTF methods with 10 transfer functions, the condition number is plotted as a function of the frequency on Figure 14. Although small fluctuations on the condition number at 734.6 and 1019 Hz can be seen, it can be considered that it remains around 10 8 for the gate functions and the Chebyshev polynomials on the whole frequency domain . The condition number for the exponential functions is much higher (around 10 14 ) and it can thus be said that the method is more sensitive to numerical errors in this case, even if the results show good agreement with the reference calculation.

Validation on a curved shell

To show that the convergence criteria defined earlier in this paper remains valid in the case of subsystems coupled along non-straight line connection, a curved shell as shown on Figure 15 is considered. It has the same dimensions and material parameters as the plate presented in the beginning of the section, but a 2.5 m curvature is added widthwise. The excitation is normal to the shell at the point of coordinates (0.9; 0.5), the coupling line is selected at x = 1.2 m and the normal mean quadratic velocity on the surface of the whole plate is calculated by FEM (reference) and by the CTF method. For the sake of simplicity and because it has proven to give the best results in the previous sections, only the set of gate functions is considered for this case.

On Figure 16, the response of the plate is recalled (black dotted line) and is compared to the response of the curved shell (red solid line). The increase in the natural frequencies of the system shows that the curvature increases the stiffness of the structure. Thus, at a given frequency, the flexural wavelength is longer in the curved shell than in the planar structure, and could result in a less restrictive criteria (i.e. wider gate functions for instance). Nevertheless, as the flexural wavelength in a curved shell cannot be calculated easily, the convergence criterion is calculated from the flexural wavelength in the planar structure of same material properties and thickness. L s = 0.15 m wide gate functions are accordingly considered to ensure the convergence of the method. Through this example, the plot on Figure 16 shows that the CTF method is still valid on more complex cases.

Conclusion and further work

A sub-structuring approach based on the mobility method has been developed to couple mechanical systems along a line junction. Efforts and displacements are condensed thanks to a set of orthonormal functions and the coupling forces between the subsystems are deduced. The method has shown good results on simple structures and convergence criteria have been highlighted. In particular, the so-called gate functions seem to be more efficient in terms of numerical sensitivity and at the same time more robust to singularities. These condensation functions can be compared to the patches used in the PTF method 10 in the case of the coupling with acoustic domains. Unlike classical reduction methods, such as Component Mode Synthesis 2 or branch mode analysis, [START_REF] Gladwell | Branch mode analysis of vibrating systems[END_REF] the subsystems in the CTF method do not necessarily need to be described by Finite Element Method (FEM) or by projections on a modal basis. Indeed, it is sufficient to be able to evaluate the admittance at the interfaces by any mean to apply the method.

The CTF method can now be further applied to the case of non-straight line connections. More particularly, it presents a great interest to add non-axisymmetric internal frames to submerged cylindrical shells: structures such as longitudinal floors or curved plates linking adjacent stiffeners can be found in industrial applications. Many studies have focused on the vibroacoustic behavior of shells immersed in water so far, but only a very few of them have considered non-axisymmetry. To tackle this issue, the idea is to couple together through CTF:

• an axisymmetric stiffened submerged shell described by the dedicated Circumferential Admittance Approach (CAA). [START_REF] Maxit | Prediction of the vibro-acoustic behavior of a submerged shell non periodically stiffened by internal frames[END_REF] • non-axisymmetric internal frames modeled by FEM to offer a great flexibility on their design.

Results showing good agreement for a test case will soon be released and the influence of the non-axisymmetric internal frames will be discussed.

1 2Figure 2 :

 12 Figure 2: Coupling forces between the two subsystems.

Figure 3 :

 3 Figure 3: Subsystems coupled along a line Γ.

Figure 4 :

 4 Figure 4: Uncoupled systems, interface forces and displacements at the junction.

Figure 6 :

 6 Figure 6: Two rectangular plates that are to couple along a line. Partition of the coupling line into 10 segments.

4 Figure 7 :

 47 Figure 7: Chebyshev polynomials of first kind for n ∈ {0, 1, 2, 3, 4}.

Figure 8 :

 8 Figure 8: Mean quadratic transversal velocity of the system made of the two coupled plates and under a point harmonic load. Comparison between the reference and the CTF methods with N = 3.

Figure 9 :

 9 Figure 9: Relative error (dB) of the mean quadratic transversal velocity in comparison to the FEM reference for the three CTF methods with N = 3.

Figure 10 :

 10 Figure 10: Transversal displacement of point A (top) and point B (bottom). Comparison between the reference and the CTF method with 10 gate functions (L s = 0.15 m).

Figure 11 :

 11 Figure 11: Amplitude of the transversal displacement along the coupling line for two different positions of the excitation point at f = 1414 Hz.

Figure 12 :

 12 Figure 12: Relative error (dB) of the mean quadratic transversal velocity in comparison to the FEM references for the CTF method with N = 10 Chebyshev polynomials. Comparison for two positions of the excitation point.

  functions -full CTF 10 gate functions -only diag

Figure 13 :

 13 Figure13: Mean quadratic transversal velocity on the system made of the two plates coupled and under a punctual harmonic load.

Figure 14 :

 14 Figure 14: Condition number with respect to inversion of the sum of the admittance matrices, as a function of the frequency.

Figure 15 :

 15 Figure 15: Mesh of a curved shell.

Figure 16 :

 16 Figure 16: Mean normal quadratic transversal velocity on the system made of two curved shells and under a point harmonic load. Comparison between the reference and the CTF method with N = 10 gate functions.

2 Principle of the Condensed Transfer Function (CTF) method 2.1 Classical admittance approach for point coupled subsystems

1

Figure 1: Subsystems linked by two point contacts.

Table 1 :

 1 Material characteristics and plates dimensions.

	Parameter	Notation	Value	Unit
	Young modulus Poisson coefficient	E ν	2.1 × 10 11 0.3	Pa -
	Density	ρ p	7800	kg/m 3
	Structural damping coeff.	η	0.02	-
	Length of plate 1 & 2	L	1.5	m
	Width of plate 1	l 1	1.2	m
	Width of plate 2	l 2	1.3	m
	Thickness	h	0.017	m

stiffness matrix. One layer of quadrilateral isotropic shell elements of thickness h is chosen with a linear interpolation function. The Mindlin-Timoshenko formulation

Acknowledgment

This work was funded by DCNS and performed within the framework of the LabEx CeLyA of Université de Lyon, operated by the French National Research Agency (ANR-10-LABX-0060/ANR-11-IDEX-0007).