Lilian Besson

Centralesupelec Fr

A Note on the Ei Function and a Useful Sum-Inequality

Keywords: Analysis, Inequalities, Primitive

This short note defines formally the Ei function, and gives some interesting inequalities and integration using it. I illustrate the inequalities and detail what is still to be proved.

Introduction and Motivation

Take a > 1, b > 1 and γ > 0, and for an integer L > 0 consider the sum L i=1 (a b i ) γ . We want to bound it, and the goal is to show that it is bounded by a constant times its last term. A first naive bound is L i=1 (a b i ) γ ≤ (L + 1)(a b L ) γ which is too brutal as soon as L → ∞. I first remind and prove two useful elementary results, and then we define and study the Ei function, to finally prove the desired inequality.

Lemma and Proof

Lemma 1 For any n ∈ N * , a > 1, b > 1 and γ > 0, we have

n i=0 (a b i ) γ ≤ a γ + 1 + 1 (log(a))(log(b γ )) (a b n ) γ = O (a b n ) γ . (1)
Proof We first isolate both the first and last term in the sum and focus on the from i = 1 sum up to i = n -1. As the function t → (a b t ) γ is increasing for t ≥ 1, we use a sum-integral inequality, and then the change of variable u := γb t , of Jacobian dt = 1 log b du u , gives

n-1 i=1 (a b i ) γ ≤ n 1 a γb t dt ≤ 1 log(b γ ) γb n γb a u u du
Now for u ≥ 1, observe that a u u ≤ a u , and as γb > 1, we have

≤ 1 log(b γ ) γb n γb a u du ≤ 1 log(b γ ) 1 log(a) a γb n = 1 (log(a))(log(b γ )) (a b n ) γ .
Finally, we obtain as desired,

n i=0 (a b i ) γ ≤ a γ + (a b n ) γ + 1 (log(a))(log(b γ )) (a b n ) γ .
c 2018 L. Besson.

Elementary Results

Integration by Part

The Integration by Part is a basic but useful result to establish inequalities, e.g., for Lemma 10 using Lemma 3, and to prove the existence of finite integrals, e.g., for Lemma 5 using two chained IP.

Lemma 2 (Integration by Part (IP)) Let x, y ∈ R, x ≤ y, and u, v two functions of class1 C 1 , and with this notation uv

y x := u(y)v(y) -u(x)v(x), then y x u(t)v ′ (t) dt = uv y x - y x u ′ (t)v(t) dt. ( 2
)
Proof The two integrals and the two evaluations are well defined by the C 1 hypothesis on both u and v (u and v are continuous at x and y and u ′ v is continuous so integrable on the interval [x, y]).

The product function uv is differentiable, and (uv

) ′ = u ′ v + uv ′ , so uv y x = y x (uv) ′ (t) dt = y x u(t)v ′ (t) dt + y x u ′ (t)v(t)
dt as wanted, by the linearity of the integral.

Lemma 3 (IP Inequality) If both u, v are non-negative, and non-decreasing, then

y x u(t)v ′ (t) dt ≤ u(y)v(y). (3)
Proof The non-negativeness gives that -u(x)v(x) ≤ 0 and the monotony hypothesis gives that u(t)v ′ (t) is non-negative on the interval [x, y], and so -y x u(t)v ′ (t) dt ≤ 0, so an Integration by Part gives the desired inequality.

Sum-Integral Inequality

A well known result is the following, which bound a discrete sum y i=x f (i) by two integrals for non-decreasing functions, and it is used for Lemma 10.

Lemma 4 For any x, y ∈ N * , x ≤ y, and f a non-decreasing function on [0, +∞), then

y x-1 f (t) dt ≤ y i=x f (i) ≤ y+1 x f (t) dt, (4)
and

f (x) + y x f (t) dt ≤ y i=x f (i) ≤ f (y) + y x f (t) dt. ( 5
)
Proof For the first inequality, both parts comes from the monotony of f and monotony and additivity of the integral. On any interval

[i, i + 1], f (i) = min t∈[i,i+1] f (t) ≤ i+1 i f (t) dt, and f (i) = max t∈[i-1,i] f (t) ≥ i i-1 f (t) dt,
And so, if we sum these terms from i = x to y, we get

y i=x f (i) ≤ y i=x i+1 i f (t) dt = y+1 x f (t) dt.
as well as

y i=x f (i) ≥ y i=x i i-1 f (t) dt = y x-1 f (t) dt.
The two sides of second inequality are immediate by isolating the first (or last) term of the sum f (y) (or f (x)), and applying the first inequality to x -1 instead of x (or y -1 instead of y).

The Exponential Integral Ei Function

This last Subsection is rather long, and actually not required to obtain the Lemma 1. But I find this Ei function to be quite interesting, so I wanted to write down these proofs. We define the Ei function [START_REF] Weisstein | Exponential Integral[END_REF]Collective, 2017), by carefully justifying its existence, and then we give two results using it, to obtain the non-trivial Lemma 10.

Lemma 5 For any ε > 0, I(ε

) := ε -ε e u
u du exists and is finite, it satisfies this identity

I(ε) = (e ε -e -ε ) log ε -(e ε + e -ε )(ε log ε -ε) + ε 0 (e u -e -u )(u log u -u) du. (6)
Additionally, it stays finite when ε → 0, and lim ε→0

I(ε) = lim ε→0 ε -ε e u u du = 0.
Proof Fix ε > 0, and let

I(ε) := ε -ε e u
u du. Roughly, one just needs to observe 2 that for u close to 0, e u u ∼ 1 u as e u ∼ 1, and 1 u can be integrated on [-ε, ε], even if it is not defined at 0, because it is odd:

ε -ε 1 u du = lim t→0 ( t -ε 1 u du +
(u) = log u, that is a ′ (u) = e u -e -u and b(u) = u log u -u (C 1 on (0, ε]), gives I 2 (ε) = ε 0 (e u + e -u ) log u du = (e u + e -u )(u log u -u) ε 0 - ε 0 (e u -e -u )(u log u -u) du.
2. The notation f (u) ∼ g(u) for u → u0 means that g(u) = 0 and f (u)/g(u) → 1 for u → u0.

Indeed, b(u) = u log u -u is well defined for u → 0, as u log u → 0, so we can define b(0) = 0 to have b of class C 1 on [0, ε]. Therefore, I 2 (ε) exists, and we have, as wanted, the following identity

I(ε) = (e ε -e -ε ) log ε -(e ε + e -ε )(ε log ε -ε) + ε 0 (e u -e -u )(u log u -u) du.
The last integral is well defined and finite, as the integrated function is continuous and finite for all u, even at 0. So this proves that I(ε) is finite for any ε > 0.

Now, taking ε → 0 gives, for each of the three terms in I(ε),

     (e ε -e -ε ) log ε ∼ ((1 + ε) -(1 -ε)) log ε = 2ε log ε → 0 (e ε + e -ε )(ε log ε -ε) ∼ 2b(ε) → 0 ε 0 (e u -e -u )(u log u -u) du → 0, so I(ε) → 0, as wanted.
Lemma 6 For any 0 < ε ≤ 1, I(ε) satisfies I(ε) ≤ e εe -ε . In particular, I(1) ≤ ee -1 .

Proof For 0 < ε ≤ 1, b(1) ≥ -1, and (e ε + e -ε ) log(ε) ≤ 0, and so the identity (6) gives

I(ε) ≤ (e ε -e -ε ) + ε 0 (e u -e -u )(u log u -u) du, but (e u -e -u )(u log u -u) ≤ 0 for all u ∈ [0, 1], so I(ε) ≤ (e ε -e -ε
) as wanted. In particular, I(1) ≤ (ee -1 ).

Definition 7

The Exponential Integral Ei function is defined for

x ∈ R * = R \ {0} by Ei(x) := x -∞ e u u du, (7)
where the Cauchy's principal value of the integral is taken.

Proof This integral exists and is finite for x < 0, as the function u → e u u is of class C 1 on (-∞, 0). For x → 0 (from above or from below), Ei(x) → -∞. And for x > 0, let ε > 0, and observe that we can write the integral from -∞ to x as three terms,

Ei(x) = Ei(-ε) + ε -ε e u u du +
x ε e u u du. Ei(-ε) and the last integral both exists and are finite, thanks to the first case of x < 0 and as the function u → e u u is of class C 1 on (ε, +∞). And thanks to Lemma 5,

ε -ε e u
u du is finite. So all the three terms in the decomposition of Ei(x) exist and are finite, therefore Ei(x) is well defined.

A few properties of Ei worth noting include the following: it has a unique zero (located at x 0 ≃ 0.327), it is negative for x < x 0 and in particular for x < 0, it is positive for x > x 0 , and it is decreasing on (-∞, 0) and increasing on (0, +∞). Ei is also concave on (-∞, 0) and (0, 1), and convex on (1, +∞).

Illustration We can plot this function3 :

Using Ei to compute primitives

(a b t ) γ dt = 1 log b b L 1 1 u (a u ) γ du = 1 log b b L 1 1 u (a γ ) u du
And a second change of variable with v := log(a γ )u = γ log(a)u gives 1 u du = 1 v dv (and no change in the order of the integral's bounds, as log a > 0 as a > 1), and so

= 1 log b γ log(a)b L γ log(a) e v v dv = 1 log b Ei(v) γ log a b L γ log(a) = 1 log b Ei γ log a b L -Ei (γ log(a)) .

Inequalities for Ei

Lemma 9 (First Inequalities Using Ei) For any

x ∈ R * = R \ {0}, Ei(x) ≤ e x .
Moreover, for x ≥ 1, we also have Ei(x) ≥ Ei(1) + e x -e x ≥ -1 + e x x . A useful consequence is that for any y ≥ 1 and 0 ≤ γ ≤ 1, Ei(log(y γ )) = Ei(γ log(y)) ≤ y γ .

(9)

Proof Let x ∈ R. First, if x < 0, then clearly Ei(x) ≤ 0 < e x . If 0 < x < 1, we can split the integral defining Ei(x) in two terms, and as

I(x) = x -x e u
u du ≤ e xe -x (see Lemma 6),

Ei(x) = -x -∞ e u u du =Ei(-x)≤0 + x -x e u u du =I(x) ≤ I(x) ≤ e x -e -x ≤ e x .
If x > 1, we do the same with three terms, and by using Ei(-1) = Now for the lower bound, let x ≥ 1, and we use the same splitting. For I(1), we use conversely that I(1) ≥ 0 (see Lemma 6), and for the integral we have

x 1 e u u du ≥ 1

x (e xe). So Ei(x) ≥ Ei(1) + e x -e

x . We also have -e

x ≥ -e and numerically, Ei(1)e ≥ -1 (as Ei(1) ≃ 1.895), so Ei(x) ≥ -1 + e x

x . Finally, if x = log(y γ ) and y ≥ 0, then e x = y γ , so Ei(x) = Ei (γ log(y)) ≤ e x = y γ .

Illustration We can check this inequality Ei(x) ≤ e x graphically, as well as a tighter inequality Ei(x) ≤ Ei(-1) -1 e + e x . This last sum-inequality is the result we were looking for.

Lemma 10 (Sum Inequality Using Ei) For any a, b, γ ∈ R and L ∈ N such that a, b > 1, γ > 0 and L > 0, and if Ei (γ log(a)) ≥ 0, then

L-1 i=0 (a b i ) γ ≤ 1 log b (a b L ) γ . ( 10
)
And by isolating the last term, we also have

L i=0 (a b i ) γ ≤ 1 + 1 log b (a b L ) γ (11)
Proof Using the sum-integral inequality (Lemma 4) and then Lemma 8, we have directly that

L-1 i=0 (a b i ) γ ≤ L 0 (a b t ) γ dt ≤ 1 log b Ei γ log(a b L ) ≤ 1 log b (a b L ) γ .
In particular, this inequality (11) holds as soon as a ≥ e 0.373/γ , as γ log(a) ≥ 0.373 > x 0 =⇒ Ei (γ log(a)) ≥ Ei(x 0 ) > 0 and x 0 ≃ 0.372507 ≥ 0.373. For instance, γ = 1/2 gives a ≥ e 0.373/γ = e 0.746 ≃ 2.107, close to the simplest value a = 2.

And if γ = 0, then Ei(γ log(a)) cannot be ≥ 0, but the sum in ( 10) is constant and equals to L.

Conclusion

This small note defines and studies a useful non-canonical function called the "exponential integral" function, Ei, and we use it to find a bound on any sum of the form L i=0 (a b i ) γ .

Note: the simulation code used for the experiments is using Python 3, (Foundation, 2017), and Matplotlib [START_REF] Hunter | Matplotlib: A 2D graphics environment[END_REF] for plotting, as well as SciPy [START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF]. It is open-sourced at github.com/Naereen/notebooks/blob/master/Exponential_Integral_Python.ipynb. This document is also distributed under the open-source MIT License, and is available online at perso.crans.org/besson/publis/A_note_on_the_Ei_function.pdf.

v

  dv with the change of variable v = -u. So ε -ε 1 u du = 0 for any ε ≥ 0. But we have to justify more properly that I(ε) exists for any ε > 0 and that I(ε) → 0 for ε → 0. A first Integration by Part (Lemma 2) with a(u) = e u and b ′ (u) = 1 u , that is a ′ (u) = e u and by choosing b(u) = log |u|, gives I|u| du = (e εe -ε ) log ε -ε 0 (e u + e -u ) log |u| du. Let I 2 (ε) := ε 0 (e u + e -u ) log u du. A second Integration by Part (Lemma 2) with a(u) = e u + e -u and b ′

Figure 1 :

 1 Figure 1: The Ei function on [-1, 1].

Lemma 8

 8 For any a, b, γ ∈ R and L ∈ N such that a, b > 1, γ > 0 and L > 0, L 0 (a b t ) γ dt = 1 log b Ei γ log(a b L ) -Ei (γ log(a)) . (8) Proof A first change of variable with u := b t gives dt = 1 log b 1 u du (log b > 0 as b > 1), and so L 0

  + e xe ≤ ee -1 + e xe = e xe -1 ≤ e x .

Figure 2 :

 2 Figure 2: The Ei function and two upper-bounds valid respectively on R and [1, +∞).

A function of class C 1 is continuous, differentiable and of continuous derivative on its interval of definition.

See for instance, the scipy.special.expi function, on https://docs.scipy.org/doc/scipy/ reference/generated/scipy.special.expi.html, if you usePython and SciPy (Foundation, 2017;[START_REF] Jones | SciPy: Open source scientific tools for Python[END_REF].


Bogdan Uscumlic

Dominique Chiaroni

Brice Leclerc

Thierry Zami
email: thierry.zami@asn.com

Annie Gravey

Philippe Gravey

Michel Morvan

Dominique Barth
email: dominique.barth@uvsq.fr

Djamel Amar

Scalable Deterministic Scheduling for WDM Slot Switching Xhaul with Zero-Jitter

Keywords: 5G, Xhaul, zero-jitter, deterministic scheduling, scalability, WDM slot switching

A low-cost WDM slot switching "N-GREEN" network is studied for the Xhaul application. We assess the impact of inter-slot intervals on the jitter in N-GREEN and propose a deterministic scheduler ensuring a zero-jitter performance as needed by CPRI traffic. We also propose an integer linear program and a corresponding scalable heuristic and use these tools to evaluate the efficiency of the new scheduler within the N-GREEN technology. The results show important savings and improvements in cost, energy consumption, latency and jitter using N-GREEN w.r.t. state-of-the-art Ethernet Xhaul.

I. INTRODUCTION

The Ethernet-based fronthaul scheduling problem has recently become a topic of many research groups. Although Ethernet is a mature technology, Ethernet-based fronthaul exploiting statistical multiplexing has difficulties to support synchronization constraints, low jitter (<65ns) and latency (<100 µs) for the CPRI (Common Public Radio Interface) traffic [START_REF] Chitimalla | 5G fronthaul-latency and jitter studies of CPRI over ethernet[END_REF] in this network segment.

Recently, a new WDM slot switching technology called WDM Slotted Add/Drop Multiplexer (WSADM), investigated in the ANR N-GREEN project, has been proposed for the 5G fronthaul/Xhaul networks [START_REF] Chiaroni | Potential of WDM Packets[END_REF], [START_REF] Triki | Long-Term CAPEX Evolution for Slotted Optical Packet Switching in a Metropolitan Network[END_REF]. The WSADM technology of the N-GREEN project exploits the WDM transparency for the transit traffic to lower the cost of optical components and adopt off-the-shelf devices. The WDM slot technology (in which the data is carried simultaneously over 10 wavelengths [START_REF] Chiaroni | Potential of WDM Packets[END_REF]) has the potential to provide a low-cost [START_REF] Triki | Long-Term CAPEX Evolution for Slotted Optical Packet Switching in a Metropolitan Network[END_REF] and performant fronthaul/Xhaul. However, the problem of the deterministic scheduling of isochronous (CPRI) traffic over a time slotted ring (such as adopted in N-GREEN), where each time slot starts after a fixed size inter-slot interval ("guard time", 𝑇 𝐺 ), has not yet been solved. Furthermore, the scalable scheduling method has not yet been proposed. Indeed, the scalability of the scheduling mechanism is needed to reduce the network reconfiguration time, since the scheduler will be implemented at the SDN (software defined networking) controller, enabling the logically centralized network control.

Our main contributions are as follows. For a first time, the impact of guard time on the jitter performance is considered in the scope of deterministic scheduling. Then, a deterministic scheduler with zero-jitter is proposed for N-GREEN. For this scheduler, a solution is proposed in the form of Integer Linear Program (ILP), enabling to achieve the scheduling at optimal network cost. Next, a heuristic algorithm based on the greedy approach is designed as an alternative scalable solution for the same scheduling. Finally, by using the previously developed tools, we evaluate the cost, jitter and latency advantages of the N-GREEN technology when compared to a state-of-the-art Ethernet Xhaul.

The remainder of the paper is organized as follows. In Section II we present the N-GREEN network and node architecture and define the properties of the WDM transponders (WDM-TRX) used for the network operation. Section III defines the scheduling solution that we propose for the WDM slot switching Xhaul network. In Section IV, we detail the mathematical model (based on ILP) implementing the previous scheduling algorithm. Section V introduces the greedy algorithm for the scalable scheduling, while Section VI provides detailed numerical results, evaluating the cost and the power consumption of the N-GREEN network. Finally, concluding remarks are provided in Section VII.

II. N-GREEN NETWORK ARCHITECTURE

An example of the N-GREEN ring in the fronthaul and N-GREEN's node architecture are illustrated in Fig. 1. Network is composed of 𝑛 nodes, one of which is the master node, having a role of hub and interconnecting the ring with the edge cloud. The other nodes are called the "bridge nodes".

The N-GREEN node is composed of: a) splitters and couplers for the traffic reception/extraction via a set of WDM-TRX transponders operating at 100 Gbit/s (a single WDM-TRX operates physically with 10 single-wavelength receivers, RX, and 10 single-wavelength transmitters, TX; each RX and TX operate at 10 Gbit/s); b) variable optical attenuator (VOA) and semiconductor optical amplifier (SOA) for WDM slot blocking/bypassing and c) Fiber Delay Line (FDL) selection box, for adjusting the propagation time of the transit traffic to enable the processing of the control channel [START_REF] Chiaroni | Potential of WDM Packets[END_REF].

N-GREEN ring operates on a time slot basis (with time slot duration of 𝑇 𝑆 =1 µs). The time slots carry the "WDM slots" which encapsulate and transport the client data simultaneously over 10 wavelengths. Note that the privileged solution for N-GREEN consists in using the WDM-TRX transponders (case called: "CONF. 1"), but to validate the techno-economic interest of N-GREEN, we also consider the case "CONF. 2" where the TRX transponders at 10 Gbit/s are used (1 TRX is composed of 1 single-wavelength RX and of 1 singlewavelength TX). The inserted WDM slots are separated by a guard time of 𝑇 𝐺 =50 ns. Regarding the control plane, the network uses a dedicated wavelength channel for the node and network synchronization and for the transport of the control information, that is processed at each network node. The master node is connected to the SDN controller.

For statistical scheduling, WDM slot switching has been already demonstrated as highly cost efficient when compared to an electronic Ethernet technology or to a single channel approach [START_REF] Chiaroni | Potential of WDM Packets[END_REF]. Indeed, WDM slots allow to reduce the costs of the active components, either immediately for the optical gates that can serve for the entire ring bandwidth, either by benefiting from the savings resulting from the integrations of WDM-TRX transponders. Furthermore, N-GREEN network offers the same functionalities as Ethernet (through a processing of all the bus), while being in line with the recent Datacom technology evolution (towards WDM-TRX).

III. ZERO-JITTER AND DETERMINISTIC LATENCY SCHEDULING

FOR N-GREEN XHAUL The scheduling of CPRI (or isochronous) traffic in N-GREEN consists in: a) aggregating different flows over the same transmission resource (wavelength or waveband) so that the resource use is maximized; b) properly positioning the CPRI packets within subsequent time slots to avoid any jitter and latency due to the packet insertion process and c) including the perturbation caused by the guard time in the jitter guarantees.

To address the scheduling problem, we first note that the positions of the scheduled subsequent CPRI packets in the N-GREEN time slots will vary from slot to slot, since the period of the CPRI flow is not necessarily the same as the time slot duration 𝑇 𝑆 . This means that the transmission resource allocated to the transport of CPRI flows needs to be "continuous". The continuity can be assured either by allocating a physical channel to this transport (as seen so far) or by allocating a virtual channel to it (in CONF. 1 only). For instance, to ensure a 10 Gbit/s transport, we could either allocate a wavelength of 10 Gbit/s to it, or use a periodic "temporal slice" of sufficient capacity, e.g. we could use a 100 Gbit/s WDM TRX during 1/10 of the time. For a physical channel, the jitter introduced by the guard time 𝑇 𝐺 (and limited to the value of 𝑇 𝐺 ), in a general case, is experienced when inserting the CPRI traffic into the optical medium. For a virtual channel, the guard time does not impact the scheduling (as it is not perceived during the emission by the source), and scheduling with zero-jitter is then achieved.

The scheduling that we propose is based on the optimal solution in form of ILP and on the greedy algorithm, defined over a scheduling cycle of size 𝑆, and minimizing the network cost (equal to the cost of network's transponders and channels, transponders being the most expensive components). To precisely account for the CPRI packet position within a time slot, we introduce a notion of "subslot", i.e. we suppose that each time slot is composed of exactly 𝐿 equally sized "subslots" (𝐿 is a given parameter). Let us suppose that 𝑆=𝑚 • 𝐿, where 𝑚 is the number of time slots after which the scheduling is repeated cyclically by all network nodes. Next, for CPRI flow 𝑑, exchanged between nodes 𝑖 and 𝑗, we define the parameters:

𝛼 (𝑖,𝑗)
𝑑 -the number of subsequent subslots corresponding to a transmission duration of a single CPRI packet and 𝛽 (𝑖,𝑗)

𝑑

-the number of subsequent subslots between the consecutive CPRI packets (both calculated from 𝐿), and 𝐹 (𝑖,𝑗) 𝑑 -the number of times the CPRI packets have to be scheduled in a single scheduler cycle 𝑆 (calculated from 𝑆). To get 𝑆, we define it as the least common multiple of numbers 𝛼 (𝑖 1 ,𝑗 1 )

𝑑 1 + 𝛽 (𝑖 1 ,𝑗 1 ) 𝑑 1 , 𝛼 (𝑖 2 ,𝑗 2 ) 𝑑 2 + 𝛽 (𝑖 2 ,𝑗 2 ) 𝑑 2
, …, 𝐿. Finally, the value of 𝑆 allows us to find 𝑚. As a theoretical contribution, we formulate a theorem on the scheduling of isochronous flows, that we have confirmed by the simulations. The theorem can be easily proven by contradiction and shows that the isochronous flows with periods that are not the multiples of the same base period cannot be scheduled on the same channel.

Theorem. Let 𝑃 (𝑖,𝑗) 𝑑 = 𝛼 (𝑖,𝑗) 𝑑 + 𝛽 (𝑖,𝑗) 𝑑 , for all isochronous flows defined in the scheduling problem. Then, if numbers

𝑃 (𝑖 1 ,𝑗 1 ) 𝑑 1 , 𝑃 (𝑖 2 ,𝑗 2 ) 𝑑 2
, …, are mutually prime, the isochronous flows cannot be scheduled on the same channel.

Note that the previous scheduling can be used also for CoE (CPRI over Ethernet) traffic. In the following, the scheduling is realized by two mathematical models: 1) as an optimal ILP program, and 2) as a scalable greedy algorithm.

IV. ILP MODEL FOR THE COST OPTIMIZED SCHEDULING

This section introduces the ILP model for calculating the slot allocation according to the previous scheduling mechanism. The resulting scheduling is cost optimized, and has the minimum cost of channel interfaces and channels, needed to support all the traffic flows in the network. We define channel interface either as a transponder (for physical channels) or as a duration of a single temporal slice of the transponder (for virtual channel). For instance, in CONF. 1 for virtual channels, the cost of channel interface is the same as the cost of a single virtual channel, and is lower than the cost of WDM-TRX. In CONF. 1 for waveband channels and in CONF. 2, the cost of channel interface is the same as WDM-TRX and TRX cost, respectively. The input parameter to the ILP model is also the traffic matrix, defining the number of slots needed for each traffic flow. The output of the ILP model is the scheduling algorithm that shall be applied by each network node, and the network configuration, expressed in the number of the required channel interfaces and channels at each node and in the network. The full list of the input parameters for the ILP model is provided in Tab. I. For instance, the scheduling cycle size and the cost parameters are defined in this table.

Next, in Tab. II we can see all the output variables that are used for the integer linear program. Some variables are not included in Tab. II, since they are auxiliary.

Tab. III provides the list of the constraints that build the ILP model. Eq. ( 1) is the objective function. This function minimizes the overall cost of the N-GREEN ring, i.e. the cost of the channel interfaces and channels required in the ring. Equation ( 2) is the traffic-load constraint, ensuring that right amount of ring capacity is allocated to each isochronous traffic flow. Constraint (3) ensures the periodic slot allocation for the isochronous traffic demands. The constraint (4) implements the logical "IF-THEN-ELSE" condition (connecting the variables 𝑝 𝑞 (𝑖,𝑗),𝑑,𝑠 and 𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 ), so it uses the auxiliary variables and constants. Each slot can be allocated only once, per each link and channel, which is ensured by constraint [START_REF]Cost of 10G DWDM SFP+ Transceivers[END_REF]. Constraint (6) ensures that sufficient number of transmitters and receivers are allocated at each network node. Set of the links in the ring belonging to the routing path between the nodes 𝑖 and 𝑗; 𝐷 (𝑖,𝑗)

Set of CPRI (isochronous) traffic demands 𝑑 between nodes 𝑖 and 𝑗 (𝐷 𝑚𝑎𝑥 -the maximum number of demands between any source-destination pair (𝑖, 𝑗)); 𝛼 (𝑖,𝑗)

𝑑

The number of subsequent slots corresponding to a transmission duration of a single CPRI packet for the demand 𝑑 between source-destination pair (𝑖, 𝑗); 𝛽 (𝑖,𝑗)

𝑑

The number of subsequent slots between the consecutive CPRI packets for the demand 𝑑 between source-destination pair (𝑖, 𝑗); 𝐹 (𝑖,𝑗)

𝑑

The number of times the CPRI packets (for the demand 𝑑 between source-destination pair (𝑖, 𝑗)) are to be scheduled in a single scheduler cycle 𝑆 ; Finally, the constraint [START_REF]Cisco Catalyst 2960-S Series Switches Data Sheet[END_REF] ensures the allocation of the correct number of channel interfaces at each node and channels in the network.

Input

V. HEURISTIC SOLUTION FOR THE SCALABLE SCHEDULING IN N-GREEN XHAUL

The cost-optimized scheduling described in the previous section consumes important computing time when the ILP program is implemented, but no formal proof about its computational complexity is available yet. For the real network implementation of the scheduling, it would be excellent to have a faster algorithm, that is adapted to a fast execution at the SDN controller. With goal of addressing this problem, in the current section we propose a greedy algorithm, that addresses the same scheduling problem that is already introduced, and that has the same input parameters and output variables as the ILP (with exception of the constants 𝑀 1 , 𝑀 2 ).

The name of the proposed algorithm is "Greedy Source Cost Minimization" (GSCM), and its pseudocode is provided in Tab. IV. The algorithm name comes from its intention to ensure the minimization of the number of channel interfaces (transponders) and consequently the cost of each source in the network. The "group of flows" in the algorithm is defined as a set of flows generated by the same source. In steps 2-6 of the algorithm, the group of flows are taken in the decreasing order and routed according to the first-fit channel assignment method. Next, in steps 7-13, the algorithm applies the same procedure, but this time for each network node separately, which has shown to have cost saving advantages for more centralized traffic matrices (like "hub and spoke" scheme, as introduced in the following section). Finally, in the steps 14-16, the algorithm GSCM chooses its best solution. First-fit channel assignment of the group of flows, by always assigning first the resources (channel interfaces and channels) to the highest flow in the group of flows 5: end for 6: Compute the overall network cost 𝑪 𝑻𝑶𝑻 7: for each network node A 8:

Sort the groups of flows exchanged between nodes A and its destinations in order of decreasing traffic rate 9: for each group of flows taken in this order do 10:

First-fit channel assignment of the group of flows, by always assigning first the resources to the highest flow in the group of flows 11: end for 12: end for 13: Compute the new overall network cost 𝑪 𝑻𝑶𝑻 ′ 14: if (𝑪 𝑻𝑶𝑻 ′ < 𝑪 𝑻𝑶𝑻 ) then 15: Accept the new design 16: end

VI. NUMERICAL RESULTS

In this section, we report the simulation results obtained by using both the ILP and GSCM tools, to estimate the cost and energy of the N-GREEN network.

Regarding the traffic matrix, CPRI or isochronous traffic flows, for the packet size of 1250 bytes and 10 Gbit/s channel capacity, are chosen from the sets 𝐴 1 = {2.5, 5, } Gbit/s [START_REF] Chitimalla | 5G fronthaul-latency and jitter studies of CPRI over ethernet[END_REF] and 𝐴 2 = { .67, 3.33, 6.67} Gbit/s, that have different basic periods. Under these assumptions, for different simulations, we consider one of the following four traffic scenarios: a) Scenario 1: The traffic profile is "hub-and-spoke" (more precisely we suppose that each bridge node communicates with the master node, and vice versa, and that the amount of traffic exchanged in both directions in this communication is the same). Next, 𝑚 = 4, 𝐿 = , 𝑆 = 4. Each bridge node sends the traffic from set 𝐴 1 (to the master node), resulting in a uniform and symmetric traffic. b) Scenario 2: The traffic profile is huband-spoke, with 𝑚 = 4, 𝐿 = 3, 𝑆 = 2. Each bridge nodes sends three randomly chosen flows from the set 𝐴 1 ∪ 𝐴 2 . c) Scenario 3: The source, destination and flow type are all randomly distributed. Each bridge nodes sends three randomly chosen flows from the set 𝐴 1 ∪ 𝐴 2 . We suppose that 𝑚 = 4, 𝐿 = 3, 𝑆 = 2. d) Scenario 4: The traffic profile is hub-andspoke, with 𝑚 = 4, 𝐿 = 3, 𝑆 = 2. Each bridge nodes sends three randomly chosen flows from the set 𝐴 1 ∪ 𝐴 2 , for different ring size 𝑛.

A. The results obtained by using the ILP model

We implement the ILP model in IBM CPLEX software (for 𝑛 = 6, 𝐶 𝑡 = [a.u.] for 10 Gbit/s channel interfaces (CONF. 1 or 2), 𝐶 𝑞 << 𝐶 𝑡 , i.e. 𝐶 𝑞 = . [a.u.]) and report the optimal simulation results in Figs. 2345, for Scenarios 1 and 2. In all the simulations, the cost 𝐶 𝑞 corresponds either to the cost of a wavelength or of a virtual channel. Indeed, for such channels, the value of cost 𝐶 𝑞 is the same, since the equivalent capacity of a wavelength or of a virtual channel is the same and equal to 10 Gbit/s. Finally, in all scenarios in this work, the simulations are run for the gradually increasing traffic, by adding to the traffic matrix in each step the traffic contribution (the sent and received traffic) for the next bridge node in the network, until all the bridge nodes have been accounted for.

Fig. 2 shows the number of channel interfaces and channels in the network, for Scenarios 1 and 2. The number of channel interfaces is higher than number of channels for both scenarios, while Scenario 2 is more expensive. For Scenario 2, the traffic is random, resulting in a non-linear increase of the number of channel interfaces and channels. The channel occupancy (of physical channels, i.e. wavelengths or virtual channels) in N-GREEN is illustrated in Fig. 3. The channel occupancy reaches high values (≈90%) meaning that the N-GREEN network with the proposed scheduler is highly efficient in the resource use. The number of transponders in Scenario 2 for an equivalent Ethernet ring (with TRXs at 10 Gbit/s) and for different variants of N-GREEN are compared in Fig. 4. For N-GREEN we consider two cases: 1) CONF. 2, i.e. nodes equipped with several TRX at 10 Gbit/s (with physical channels, i.e. with single wavelength per TRX) and 2) CONF. 1, i.e. nodes equipped with several WDM-TRX at 100 Gbit/s (supporting 10 virtual channels at 10 Gbit/s). Fig. 4 shows the significant savings when the WDM TRX are used (up to 9 times).

Next, the transponder cost is compared in Fig. 5, for the following assumptions:

1) Since burst mode TRX operation at 10 Gbit/s results in a small additional logic [START_REF]Burst-Mode Clock Data Recovery with GTH and GTY Transceivers[END_REF], the cost of TRX in Ethernet and N-GREEN (CONF. 2, physical channels at 10 Gbit/s) is approximately the same and equal to 𝐶 𝑡 = [a.u.]; 2) The cost of WDM-TRX (CONF. 1, virtual channels) is supposed to be 𝛿 • 𝐶 𝑡 (where parameter 𝛿 ∈ [START_REF] Chitimalla | 5G fronthaul-latency and jitter studies of CPRI over ethernet[END_REF]10]).

Fig. 5 shows important savings of N-GREEN in transponder cost w.r.t. Ethernet for physical channels. For virtual channels, the savings up to 33% are achieved for 𝛿 ≤ 6, which seems achievable since the cost of a single 10 Gbit/s TRX is ≈250 $ [START_REF]Cost of 10G DWDM SFP+ Transceivers[END_REF] and the cost of WDM-TRX is expected to be ≤6•250 =1500$ thanks to the laser integration design of these devices.

Finally, Tab. V compares the sources of jitter and latency in N-GREEN and Ethernet. While variable in Ethernet [START_REF] Chitimalla | 5G fronthaul-latency and jitter studies of CPRI over ethernet[END_REF], latency is fixed in N-GREEN in transit and at the insertion, and the jitter is different than zero (and limited to 𝑇 𝐺 ) only for the physical channel scheduling in N-GREEN. When channel is virtual the jitter is equal to zero.

B. The validation of the GSCM algorithm

To validate the GSCM algorithm, we compare the optimal solution (OPT) calculated by the ILP model, and the solution found by GSCM algorithm, in a randomly generated Scenario 3 (to test the GSCM performance in the most general case). The results are presented in Fig. 6. The simulations are performed for different value of the cost ratio between the cost of channel interface and the cost of channel. The following cost assumptions are considered: 1) 𝐶 𝑡 = , 𝐶 𝑞 = . , 2) 𝐶 𝑡 = , 𝐶 𝑞 = , and 3) 𝐶 𝑡 = . , 𝐶 𝑞 = . The results suggest that GSCM obtains excellent performances on the selected random traffic scenario, and its results are very close to the optimal solution. The highest discrepancy from the optimal solution was less than 5%, and this performance was not affected by the change of the cost ratio of channel interface and channels.

We have also compared the solution found by the GSCM and ILP model on the hub-and-spoke Scenario 2, that is centralized, and 0% error of the GSCM (w.r.t. the optimal solution) has been observed. Such good results of GSCM algorithm can be explained by the fact that the channels for transport of the isochronous traffic need to be allocated per source, to preserve the "continuity" of the channel, as previously discussed, which allows to eliminate or to limit the jitter. Because of this, different sources are not allowed to use the same channel over the same links, which allows good results to the heuristics based on the greedy approach in optimizing the source cost.

C. The results obtained by using the GSCM algorithm

The proposed GSCM is scalable. Indeed, its complexity is limited to 𝑂(𝑛 3 𝐷 𝑚𝑎𝑥 ), i.e. this is a polynomial time algorithm. In the current section, we exploit the algorithm GSCM to compare the cost and energy consumption of N-GREEN and Ethernet for a greater and a realistic size of the N-GREEN network of up to 10 nodes.

The simulations are performed for Scenario 4, and for different rings sizes. The transponder cost comparison between N-GREEN and Ethernet is presented in Fig. 7. From the figure, we can see that the savings of N-GREEN w.r.t. Ethernet increase with the increase of ring size. When the network is sufficiently loaded, even if the ring size is not large (𝑛 = 6), N-GREEN network costs much less than Ethernet, measured in transponder cost. If the integration technology used for production of the WDM-TRX transponders enables the cost N-GREEN Ethernet Jitter at insertion 0 or ≤ 𝑇 𝐺 (for physical channels)

~ µ𝑠 [START_REF] Chitimalla | 5G fronthaul-latency and jitter studies of CPRI over ethernet[END_REF] Jitter in transit 0 Depends on traffic load and network size

Latency at insertion & transit

Fixed reduction of these devices of 𝛿 ≤ 3, the potential savings of costs go to more than 6 times, obtained for 𝑛 = . For 𝛿 ≤ 6, the savings are more than 3 times (result not shown in the figure), etc. Fig. 8 shows the power consumption performance of N-GREEN and Ethernet, in Scenario 4. For Ethernet, we suppose the consumption of each SFP+ 10 Gbit/s TRX to be 1.5 W [6] and the switching fabric consumption to be ≈5W/port (calculated from [START_REF]Cisco Catalyst 2960-S Series Switches Data Sheet[END_REF]). For N-GREEN, the consumption of SOA is 1.5W, while 5W is the estimated consumption of the electronic part of the node. The consumption of WDM-TRX is 5W [START_REF]Cisco CPAK 100GBASE Modules Data Sheet[END_REF]. From Fig. 8 we can see that N-GREEN achieves significant savings (up to 10 times) in the power consumption, when compared with Ethernet.

VII. CONCLUSION

A cost optimized and deterministic scheduler for the N-GREEN WSADM technology, supporting the time-sensitive CPRI traffic with zero jitter has been proposed. Then, an Integer Linear Program and a highly performant heuristic have been provided as tools enabling to calculate this scheduling. The benchmarking studies have been performed and have shown that significant cost and energy consumption reductions could be obtained when compared to a same capacity state-of-the-art Ethernet technology. The WDM technology proposed in the N-GREEN project is positioning as a highly competitive solution for a future generation of Xhaul networks.

Fig. 1 .

 1 Fig. 1. N-GREEN network architecture (example of the application in the fronthaul)

Fig. 2 .Fig. 3 .

 23 Fig. 2. Number of channel interfaces and channels (for Scenarios 1 and 2)

Fig. 4 .

 4 Fig. 4. Number of transponders for Scenario 2. Results show potential savings of N-GREEN vs Ethernet (up to 9 times) in number of transponders.

Fig. 5 .

 5 Fig. 5. Transponder cost for Scenario 2. Results show potential savings of N-GREEN vs Ethernet in the cost of transponders (savings up to 33% when WDM-TRX are used).

Fig. 7 .Fig. 8 .

 78 Fig. 7. Transponder cost for different ring sizes in Scenario 4 (results of GSCM algorithm). Results show high potential savings of N-GREEN vs Ethernet in the cost of transponders (more than 3 times when WDM-TRX are used)

TABLE I

 I

		.	INPUT PARAMETERS
	Input Parameters		Definition
	𝐺(𝑉, 𝐸)	A directed graph representing the unidirectional N-
		GREEN ring, where 𝑉 is the set of nodes, 𝐸 is the
		set of (unidirectional) links; (|𝑉| = 𝑛 in Fig. 1);
	𝑄	Set of channels (wavelengths/wavebands or virtual
			channels);
	𝑆	Size of the scheduling cycle (in number of subslots);
	𝜋 (𝑖,𝑗)

  𝑞 is used at node 𝑖 for the transport of traffic, and eq. to otherwise; 𝑦 𝑞 Binary, equal to if channel 𝑞 is used in the ring for the transport of traffic, and equal to otherwise.

	Parameters		Definition
	𝐶 𝑡		Cost of channel interfaces;
	𝐶 𝑞	Cost of a wavelength (CONF. 2), of a waveband or
			of a virtual channel (CONF. 1);
	𝑀 1 , 𝑀 2		Large constants.
		TABLE II.	OUTPUT VARIABLES
	Output Variables		Definition
	𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 , (𝑖,𝑗),𝑑,𝑠 𝑝 𝑞	Binaries, equal to if the demand 𝑑 between (𝑖, 𝑗) is routed over channel 𝑞 starting from/by using (respectively) the subslot 𝑠, and equal to otherwise;
	𝑡 𝑞 𝑖 , 𝑟 𝑞 𝑖 , 𝑢 𝑞 𝑖	Binaries, equal to if transmitter, receiver, channel
		interface (respectively) at channel

TABLE III

 III

				.	ILP FORMULATION
	No.				Constraint Definition
	( )	𝑀𝑖𝑛(∑ ∑ 𝐶 𝑡 𝑢 𝑞 𝑖 𝑄 𝑞=1 𝑖∈𝑉	+ ∑ 𝐶 𝑞 𝑦 𝑞 𝑄 𝑞=1	)
	(2)	∑ ∑ 𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 = 𝐹 (𝑖,𝑗) 𝑑 𝑆 𝑠=1 𝑄 𝑞=1	, ∀(𝑖, 𝑗)	∈ 𝑉 2 , ∀𝑑 ∈ 𝐷 (𝑖,𝑗)
	(3)	𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 1 =𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 2 , ∀𝑞 ∈ 𝑄, ∀(𝑖, 𝑗) ∈ 𝑉 2 , ∀𝑑 ∈
		𝐷 (𝑖,𝑗) , (∀𝑠 1 , 𝑠 2 ) ( 𝑠 2 ≡ (𝑠 1 + 𝑘 • (𝛼 (𝑖,𝑗) ≤ 𝑠 1 , 𝑠 2 ≤ 𝑆 ∧ 𝑑 + 𝛽 (𝑖,𝑗) 𝑑 ))𝑚𝑜𝑑 𝑆	)
					(∀𝑘, 𝐹 (𝑖,𝑗) 𝑑	> 𝑘 ≥ )
	(4)	≤ 𝑝 𝑞 (𝑖,𝑗),𝑑,𝑠 2 -𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 1 + 𝑀 1 • 𝑧 𝑞 (𝑖,𝑗),𝑑,𝑠 1 , 𝐵 𝑞 (𝑖,𝑗),𝑑,𝑠 1 ≤ +
	(5)	(𝑖,𝑗),𝑑,𝑠 1 ), ∀𝑞 ∈ 𝑄, ∀(𝑖, 𝑗) ∈ 𝑉 2 , ∀𝑑 ∈ 𝑀 2 • ( -𝑧 𝑞 𝐷 (𝑖,𝑗) , (∀𝑠 1 , 𝑠 2 ) ( ≤ 𝑠 1 , 𝑠 2 ≤ 𝑆 ∧ ) (∀𝑘, 𝛼 (𝑖,𝑗) 𝑑 > 𝑘 ≥ ) 𝑠 2 ≡ (𝑠 1 + 𝑘)𝑚𝑜𝑑 𝑆 ∑ ∑ 𝑝 𝑞 (𝑖,𝑗),𝑑,𝑠 ≤ (𝑖,𝑗):𝑙∈𝜋 (𝑖,𝑗) , ∀𝑞 ∈ 𝑄 𝑑∈𝐷 (𝑖,𝑗) ,
					∀𝑠 ∈ 𝑆 , ∀𝑙 ∈ 𝐸
	(6)	∑ 𝑗∈𝑉:𝑙∈𝜋 (𝑖,𝑗)	∑ 𝑑: 𝑖=𝑠𝑜𝑢𝑟𝑐𝑒(𝑑), 𝑑∈𝐷 (𝑖,𝑗)	∑ 𝑝 𝑞 (𝑖,𝑗),𝑑,𝑠 𝑆 𝑠=1	≤ 𝑆 • 𝑡 𝑞 𝑖,𝑙 ,
		∀𝑖 ∈ 𝑉, ∀𝑞 ∈ 𝑄, ∀𝑙 ∈ 𝐸; ∑ 𝑡 𝑞 𝑖,𝑙 ≤ |𝐸|𝑡 𝑞 𝑖 , 𝑙∈𝐸
		∑ ∑ 𝑑: 𝑖=𝑑𝑒𝑠𝑡(𝑑), 𝑗∈𝑉	∑ 𝑝 𝑞 (𝑗,𝑖),𝑑,𝑠 𝑆 𝑠=1	≤ 𝑆 • 𝑟 𝑞 𝑖 ,
					𝑑∈𝐷 (𝑖,𝑗)
					∀𝑖 ∈ 𝑉, ∀𝑞 ∈ 𝑄
	(7)	𝑡 𝑞 𝑖 + 𝑟 𝑞 𝑖 ≤ 2 • 𝑢 𝑞 𝑖 , ∀𝑖 ∈ 𝑉, ∀𝑞 ∈ 𝑄;
		∑ 𝑡 𝑞 𝑖,𝑙 ≤ , 𝑖∈𝑉	∀𝑞 ∈ 𝑄, ∀𝑙 ∈ 𝐸; ∑ 𝑢 𝑞 𝑖 ≤ |𝑉| • 𝑦 𝑞 ,

𝑖∈𝑉 ∀𝑞 ∈ 𝑄

TABLE IV

 IV

	.	PSEUDOCODE OF THE ALGORITHM: GREEDY SOURCE COST
		MINIMIZATION (GSCM)

1: Input: the same as for the ILP model (with exception of the constants 𝑴 𝟏 , 𝑴 𝟐 which are not needed) 2: Sort the groups of flows in order of decreasing traffic rate 3: for each group of flows taken in this order do 4:

TABLE V .

 V COMPARISON BETWEEN N-GREEN AND ETHERNET FRONTHAUL

ACKNOWLEDGMENT

We thank François Taburet for the insightful exchanges about the topic. We thank the French National Agency of Research (ANR) for partial funding of the N-GREEN project.


L Duigou

F Boumediene
email: boumedienefaiza@yahoo.fr

J M Cadou

E M Daya
email: daya@lpmm.sciences.univ-metz.fr

REDUCTION METHOD APPLIED TO VIBRATIONS OF VISCOELASTIC SANDWICH SHELLS

Keywords: Frequency, damping, asymptotic numerical method, reduction

This work deals with the vibrations of sandwich visco-elastic structures. The central visco-elastic layer included between two metallic layers leads to solve a problem of complex and non-linear eigenvalues. Indeed the core material depends on the frequency. Several methods exist to solve this type of problem. But the dimension of matrices to be manipulated can become very large and generate substantial computational times. A reduction technique is proposed herein. It is applied and compared to the high order Newton method.

L . ∆U+G . U 0 ∆p+ 2p 0 ∆pM∆U+ε ( ∆E ( p ) E' ( p 0 ) ∆p )K υ .U 0 +∆E ( p ) K υ . ∆U+( ∆p ) 2 M . U 0 +( ∆p ) 2 M . ∆U+εR 0 =0 (3)
with L=K ( 0 )+E ( p 0 ) K υ +p 0 2 M linear operator and R 0 =( K (0 )+E ( p 0 )K υ +p 0 2 M ) .U 0 , initial residue.

3-Finally, the perturbation technique is applied, that is to say, the increment (∆U ,∆p ) is sought in the truncated series form depending on the perturbation parameter ε :

(4)

4-After inserting these series into the modified problem(3), a series of linear problems for each order is obtained and solved.

As the number of unknowns is greater than the number of equations, an equation to normalize the mode is introduced :

∆U t G U 0 = 0 (5)
Then, the linear problems can be written at the i th truncature order, in the following matrix form : [START_REF] De Lima | Component mode synthesis combining robust enriched Ritz approach for viscoelastically damped structures[END_REF] where Lt refers to the complex tangent operator and F nl i is a second member vector changing at each 'i' order but depending only on the variables calculated in previous orders and ( ) t indicates the transposed operator.

The characteristic parameters of this method are the truncature order of (N) series and a small tolerance parameter enabling to check if the method has converged towards the solution. This resolution technique, named high order Newton method gives good results compared to experimental results. The disadvantage of this method is computational time. Indeed, the problems to solve need triangulating large-size matrices. However, one must take into consideration that the high order Newton method needs one matrix triangulation only and the resolution of "N" linear systems (6) at each iteration for an initial solution (U 0 , p 0 ). In order to reduce computational times, the authors propose -in the reference [START_REF] Duigou | Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells[END_REF] -to use real matrices instead of consistent complex matrices, corresponding to the Lt Operator. This method is faster but it diverges in some cases.

The objective of this work is to apply a reduction model to the high order Newton algorithm in order to reduce computational times while preserving the convergence properties of the algorithm.

REDUCTION METHOD

The reduction model is applied to the equation (1) : the displacement vector is projected on a small-sized base :

U= ℜ u (7) ∆U= ∑ i=1 N ε i U i et ∆p= ∑ i=1 N ε i p i [ L t ] [ U i p i ] = [ L GU 0 GU 0 t 0 ][ U i p i ] = [ F i nl 0 ] avec L=K ( 0 )+E ( p 0 ) K υ -p 0 2 M
one is discretized with 424 D.o.F., like in reference [START_REF] Duigou | Iterative algorithms for non-linear eigenvalue problems. Application to vibrations of viscoelastic shells[END_REF]. The used parameters by the high order Newton method are : a truncature order N=20, a tolerance equal to 1.10 -7 . In order to apply the reduction method, 20 vectors are used. Then, the basis ( 8) is composed of 10 vectors Φ (from a linear vibration computation) and of 10 vectors Ψ solutions of the linear problem [START_REF] Büchter | Three dimensional extension of non-linear shell formulation based on the enhanced assumed strain concept[END_REF]. The results obtained by the high order Newton method with -referred to as R-HONA) and without reduction -referred to as HONA -are given in Table 2 for the first two bending modes of the beam defined in Table 1. Table 2 shows that the reduction technique proposed enables to obtain the same values as the high order Newton method.

Mechanical properties

Example with a Young modulus depending on the temperature

A second cantilever sandwich beam with dimension (178*10 mm2) is considered. But this time, the Young modulus of the central visco-elastic layer is not constant. It is represented by the generalized Maxwell model :

E ( ω )=k 0 +η 0 iω+ ∑ j=1 N max iω ( iω k j + 1 η j ) (4)
The Maxwell number N max is equal to 129. The other characteristics of the layer are shown in Table 3.

Here, for the finite element simulation, the temperature used is 20°C and the number of nodes used for discretization is equal to 53, that is to say, 424 D.o.F.. The parameters used by the high-order Newton algorithm with and without reduction method are the followings : a truncature order N=20, a tolerance parameter equal to 1.10 -7 and 20 vectors in the projection basis. The first three bending modes are studied.

In the Table 4, we can note that the computational results carried out with the high order Newton with or without reduction diverge when the truncature order is equal to 20 in the series. That is why, in Table5, the computation of frequencies and dampings is carried out using a truncature order equal to 5. In this case, the two algorithms give good results.

Mechanical properties

Elastic layer -steel Visco-elastic layer

Young modulus These works deal with the vibrations of sandwich visco-elastic structures. The central layer leads to solve a non-linear eigenvalue problem. To do so, two techniques are compared herein : the high order Newton algorithms with and without reduction. Two examples of cantilever sandwich visco-elastic beams are studied. The first one has a constant complex Young modulus (E= E 0 (1+iηc)) and the second one a Young modulus depending on frequency (Model of Maxwell for 20°C). In the two cases, the proposed methods give good results. Numerical tests of these works show that the use of a truncature order close to 5 leads to better convergence than a high order. Currently, numerical tests on more complex geometrical examples needing more final spatial discretization are being carried out in order to quantify computational time gains.

E a = 2,

  1 10 11 N/m 2 E p = 27,216 10 6 N/m 2 Poisson ratio υ a = 0,3 υ p = 0,44 Density ρ a = 7800 kg/m 3 ρ p = 1200 kg/m 3 Thickness h a = 0,6 mm h p = 0,045 mm

Table 1 .

 1 Mechanical and geometrical properties of Beam 1.

		Elastic layer -Aluminum	Visco-elastic layer
	Young modulus	Ea = 6,9 1010 N/m2		E0 = 1794 103 N/m2
	Poisson ratio	υa = 0,3			υp = 0,3
	Density	ρa = 2766 kg/m3		ρp = 968,1 kg/m3
	Thickness	ha = 1,524 mm		hp = 0,172 mm
	Mode		HONA [8]	R-HONA
	64.32	Iteration	2	3
		Damped frequency	70.19647	70.19646
		Damping coefficient	0.228979	0.228978
		η m / η c	0.152653	0.152652
	297.85	Iteration	2	3
		Damped frequency	310.4359	310.436
		Damping coefficient	0.295438	0.295435
		η m / η c	0.196959	0.196957

Table 2 .

 2 First two bending modes of Beam 1. E=E0(1+i ηc)= cst, ηc=1.5 and 20 vectors in the reduction base.

Table 3 .

 3 Properties of the beam 2.

	Modes

Table 4 .

 4 First three bending modes of Beam 2. N=20.

	Mode		HONA	R-HONA
	29.99	Iteration	2	4
		Damped frequency	32.7842474	32.785868
		Damping coefficient	0.0335581189	0.0335753581
		η m / η c	42.9769478	42.9990273
	143.75	Iteration	2	3
		Damped frequency	198.384908	198.635266
		Damping coefficient	0.0928710797	0.0931472569
		η m / η c	118.937401	119.291092
	349.78	Iteration	3	4
		Damped frequency	548.090895	548.325729
		Damping coefficient	0.0922039246	0.117323568
		η m / η c	118.082993	150.253021

Table 5 .

 5 First benfing modes of the beam 2. N=5.