Vladislav A Yastrebov
email: vladislav.yastrebov@mines-paristech.fr

Sliding without slipping under Coulomb friction: opening waves and inversion of frictional force

Keywords: Friction, Slip pulses, Opening waves, Supersonic pulses, Negative friction

Sliding without slipping under Coulomb friction :

opening waves and inversion of frictional force

Introduction

Frictional slip on interfaces, which separate solids with different elastic properties (referred as bi-material interfaces) controls propagation of interfacial mixed mode cracks in composite materials [START_REF] Coker | Dynamic crack growth along a polymer composite-homalite interface[END_REF], energy dissipation in brake systems [Anderson andKnapp, 1990, Moirot et al., 2003] and fault slip [Scholz, 2002, Rice, 1983, Kawamura et al., 2012]. Regardless the huge difference in scales between these examples, frictional sliding demonstrates scale invariant properties, which enable to mimic Earthscale phenomena, like earthquakes, in the laboratory [START_REF] Xia | Laboratory earthquakes: the sub-rayleigh-to-supershear rupture transition[END_REF], Ben-David et al., 2010].

However, the scaling laws and the coupling of different mechanisms involved in frictional slip are not yet fully understood and present an active research area in mechanical, physical and geophysical communities [M üser et al., 2001, Carpinteri and Paggi, 2005, Urbakh et al., 2004, Kawamura et al., 2012].

Difficulties in studying the elastodynamic frictional sliding on bi-material interfaces originates in unknown exact interface laws, complex interaction of the surface and the body waves as well as non-trivial interplay of the involved time and spatial scales. The fact that the slip occurs on the bi-material interface implies that the frictional slip locally changes the contact pressure, which is not the case for similar materials. For the Coulomb's friction law, this strong coupling between normal and tangential tractions results in a mathematically ill-posed problem or flutter instability [Renardy, 1992, Martins et al., 1995, Ranjith and Rice, 2001] for a wide range of combinations of materials and friction coefficients. For example, for material pairs, for which a generalized Rayleigh interface wave does not exist, the frictional slip is well-posed only if the coefficient of friction is below a certain limit: in the case of contact between an elastic half-space and a rigid flat, this limit is a unity [Renardy, 1992]. Above this limit the well-posedness depends on the Poisson's ratio ν and the coefficient of friction f [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF].

Figure 1: Problem set-up: infinite elastic layer of thickness H with imposed period λ slides on a frictional interface with a rigid flat being sheared at constant rate V 0 .

In this Letter we demonstrate that the mathematical ill-posedness loses its destructive effect for a finite system, whose frictional dynamics appears to be governed by the structural dynamics. We report accurate numerical results showing that the frictional sliding in the "ill-posed" regime propagates as a series of stick-slip pulses. These pulses, after had run a certain distance, transform into stick-slip-opening waves; both modes of slip propagation imply very low frictional dissipation. The mechanism is described in detail and a parametric study is presented.

Methods

We consider (Fig. 1) an infinitely long elastic layer of height H (Young's modulus E, Poisson's ratio ν, mass density ρ) pressed against a rigid flat by a constant displacement U y < 0 applied on the top and resulting in uniform interface pressure p. The contact interface is governed by Coulomb's friction law with constant coefficient of friction f independent of velocity and slip distance. From the physical point of view, this implies that at small scale the interface and adjacent materials are assumed to be time independent and not affected by generated frictional heat. The solid is sheared at a constant velocity V 0 c s applied on the top surface, where c s is the shear wave celerity. We use the finite element method with implicit time integration to solve the elastodynamic equation with contact and frictional constraints.

We solve this problem for an elastic layer of length λ. To imitate an infinite elastic layer, periodic boundary conditions were used on lateral sides (displacement degrees of freedom of corresponding nodes are coupled). The simulations were carried out in in-house displacement-based finite element software Z-set [Zset, Besson and Foerch, 1997] 1 . To avoid spurious internal wave reflections, we used a regular structured finite element mesh of squareshaped linear elements with reduced integration (one Gauss point per element). To verify the mesh convergence, we used finite elements with the side h = H/32, H/64, H/128. Implicit Hilber-Hughes-Taylor [START_REF] Hilber | Improved numerical dissipation for time integration algorithms in structural dynamics[END_REF] time integration scheme was used with a moderate dissipation of high frequency modes α = 0.1. The method establishes the following general time-discrete equations of motion:

M üt+∆t + (1 -α)Ku t+∆t + αKu t = (1 -α)F t+∆t ext + αF t ext ,
where M is the mass matrix, K is the stiffness matrix, F ext is the vector of external forces, u, u, ü are vectors of displacement, velocity and acceleration, respectively; the upper index refers to time, and ∆t is the time step. Plane strain formulation with infinitesimal deformations is used. We solved elastodynamic equation of motion

∇•σ = ρ ü,
where σ is the Cauchy stress tensor and ü is the acceleration vector. The stress-strain relationship is given by Hooke's law σ = Eν (1+ν)(1-2ν) tr(ε)I + E (1+ν) ε, where tr(ε) is the trace of the deformation tensor, and I is the second rank identity tensor. Since we consider small strain elasticity, the strain tensor is given by ε = 1/2(∇u + u∇). The boundary conditions are the following: on the top of the rectangle y = H, vertical displacement u y = -0.001 H is prescribed, which results in contact pressure p = U y E(1 -ν)/[H(1 + ν)(1 -2ν)], on lateral sides periodic boundary conditions are imposed for every couple of corresponding nodes u x (x = 0, y) = u x (x = λ, y) and u y (x = 0, y) = u y (x = λ, y), these conditions are kept constant during the simulation. On the contact interface the classic Hertz-Signorini-Moreau contact conditions are used [Wriggers, 2006, Yastrebov, 2013]: p ≤ 0, u y ≥ 0, pu y = 0 and frictional conditions for Coulomb friction [Wriggers, 2006, Yastrebov, 2013]

|σ xy | -f |p| ≤ 0, | ux | ≥ 0, (|σ xy | -f |p|) ux = 0,
where σ xy is the tangential traction on the contact surface defined as σ xy = t•σ•n, where n is the unit outward normal and t is the unit tangent vector. The frictional contact constraints are taken into account via a direct method, which uses local stiffness matrices to resolve the contact constraints within a nested loop separated from the global convergence loop, in which contact and frictional forces enter as external forces, the method was developed in [Francavilla andZienkiewicz, 1975, Jean, 1995].

The elastic layer was first loaded by vertical displacement u y and next the top surface was sheared by u x until |T/N| = 0.95 f , where

T = λ 0 σ xy dx, N = λ 0 pdx
are the tangent and the normal forces, respectively; both initial steps are solved quasi-statically (without dynamic effects). Next, a dynamic simulation started (initial velocity field is zero), the top surface is sheared at a constant rate ux = V 0 , this rate is chosen to be significantly smaller than the shear wave celerity; for most simulations presented in the Letter we used V 0 = 10 -5 c s .

In almost all simulations we used Young's modulus E = 1000 Pa, mass density ρ = 1000 kg/m3 , H = 25 m, Poisson's ratio was varied in the range ν ∈ [0, 0.49] and the coefficient of friction was varied in the range f ∈ [0.5, 2.]. The time step was chosen as ∆t ≈ 0.5m/c s To verify the validity of results, various mesh periods were tested λ = 2H, 4H, 6H, mass density was varied as ρ = 0.1, 10, 1000 kg/m 3 , the time step was adjusted proportionally to ρ 1/2 . In all cases the slip dynamics retains all its properties. Different shear rates were also tested V 0 /c s = 10 -6 , 5•10 -6 , 10 -5 , 5•10 -5 , 10 -4 , they can change the slip dynamics at longer time scales, but for moderate shear rates the initial frictional drop and the formation of slip pulses and opening waves were observed for the most of studied parameters. Higher shear rate V 0 /c s = 10 -4 for some parameters may suppress the inversion of frictional force. Note that the influence of the shear rate was not the focus of the present study and requires a detailed analysis.

Remark on the choice of the time step and the mesh density

The time step in implicit time integration can be arbitrary large as the method is unconditionally stable, however, too large time steps smooth out rapid wave dynamics leading to slow structural dynamic simulation. But if the time step is chosen sufficiently small, of the order of stable explicit time step (Courant-Friedrichs-Lewy condition), the wave dynamics is finely resolved and for time steps ∆t < 1m/c s , the choice of the time step does not affect the solution for quasi-stationary regime, where only major slip pulses survive and secondary ones die.

The mesh size is a more critical parameter, but it affects only the short transitional period from the start of sliding up to the localization of slip pulses. According to [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF], for high enough friction coefficients, the homogeneous slip is unstable and the amplitude of interface displacements grows with an exponent proportional to the wavenumber or, in our case, to the inverse of mesh size. This destabilization may result in localization of stick-slip pulses spaced by a single finite element. However, this instability is only observed within the stabilization cycle before the finite size of the system manifests itself. When the system starts to act as a whole, only major slip pulses localize (their number is determined by the ratio of height to period and the slip front celerity), at this stage the mesh size does not change solution characteristics, if a sufficient number of elements is used enabling to capture all relevant fields' variations. We tested meshes with 32, 64 and 128 elements in height, and for the stabilized quasi-periodic solution we could capture quantitatively equivalent results. To summarize, the dependence on mesh and time step exists, it is slightly more subtle than in classic dynamic simulations, but this dependence is eliminated in the stable regime when the finite size of the system manifests itself. Note, that to get rid of any manifestation of the ill-posedness, one may use Prakash-Clifton regularization of frictional law [START_REF] Prakash | Time resolved dynamic friction measurements in pressure-shear[END_REF], Ranjith and Rice, 2001, Cochard and Rice, 2000, Kammer et al., 2014]. However, we believe that introducing the regularized friction law at the interface will not affect the qualitative results within the stable regime. Pa, ρ = 1000 kg/m 3 , H = 25 m, λ = 50 m, V 0 /c s = 10 -5 ). Material velocity, slip velocity and opening profiles at distinct time moments, which correspond to times marked with white dashed lines in Fig. 2, are depicted in Fig. 3 (an animation of the slip onset and formation of slip and opening pulses is provided in supplemental material [Sup]). Evolution of frictional force for a longer time interval (Fig. 2,d) is similar to the classical stick-slip behavior. Note, however, that here this global stick-slip dynamics emerges from propagation of rapid slip pulses within time intervals corresponding to frictional drops, in other time intervals the entire interface remains in stick state accumulating elastic energy. Note also that the frictional drops, except the very first one, occur when T/N ≈ 1 regardless the fact that the local coefficient of friction f = 1.5.

The observed local propagation mode is referred as train of slip pulses [Adams, 2000, Coker et al., 2003, Bui and Oueslati, 2010]; solitary pulses in finite systems were also observed [START_REF] Baumberger | Self-healing slip pulses along a gel/glass interface[END_REF]. For moderate Poisson's ratios, slip pulses propagate at intersonic celerity c p (c s < c p < c l ), where c l is the longitudinal wave celerity. Life period of many of these initially formed pulses is short: being filtered by wave dynamics and the resonance of the finite size system, their propagation speed and slip intensity decrease before they arrest in a manner rather similar to what was observed in [START_REF] Kammer | The existence of a critical length scale in regularised friction[END_REF]. However, the major pulses keep going through the interface at the constant speed. After a certain propagation distance an opening occurs within the slip pulse. This opening implies that the interface can slide almost without local slipping and thus without any energy dissipation. One may think of a carpet fold analogy: a carpet can be moved along a surface by simply propagating a fold, which requires much less energy than sliding the entire carpet [START_REF] Comninou | Can two solids slide without slipping?[END_REF]. However, here the opening starts at location, where the slip velocity reaches its maximum (Fig. 3,c-e), thus a marginal frictional dissipation still takes place. The opening pulses demonstrate a self-sustained behavior and have a certain "inertia", so that their propagation often results in bringing the contact surface ahead of the top one. This unconventional mechanism results in inversion of the frictional force direction and thus in instantaneous negative apparent friction (Fig. 2,c). It is partly similar to the snap-through instability with accompanied negative dynamic stiffness [START_REF] Lakes | Extreme damping in composite materials with negative-stiffness inclusions[END_REF]. The frictional dissipation is marginal in this slip mode, thus the total energy is approximately conserved. The elastic energy liberated within the first frictional drop via stress waves, is partly restored in the solid being sheared in the opposite direction (Fig. 3,f). However, in the particular case of the softly applied shear load via a "pusher" [START_REF] Ben-David | The dynamics of the onset of frictional slip[END_REF], Radiguet et al., 2013], this stored energy will be also liberated. Such a huge drop in frictional force down to negative values was observed in molecular dynamic simulations [START_REF] Sørensen | Simulations of atomic-scale sliding friction[END_REF], Urbakh et al., 2004]; also, within a simple stick-slip model (a rigid block pulling by a spring over a frictional surface under velocity weakening friction with the static and kinetic coefficients of friction f s > f k ) the spring force may change the sign if f k < f s /2. Evidently, for systems governed by Coulomb fricton with the coefficient of friction f independent of velocity and slip distance, frictional drop is impossible in the quasi-static case when inertial effects are neglected. For elastodynamic case, it was argued that the apparent kinetic friction (the ratio of the tangential to the normal force) at bi-material interfaces should be smaller than the local one [Adams, 2000] or even vanish completely [Adams, 1998, Ranjith andRice, 2001]. This is possible due to the drop of contact pressure within the slip pulse theorized by Weertman [Weertman, 1980]. However, for the best of our knowledge, the changing of the frictional force direction was not observed at macroscopic scale.

Bibliographical remarks

The opening waves, also known as Schallamach waves, is a familiar phenomenon in rubber friction [Schallamach, 1971, Gent, 1974, Barquins, 1985]: when a rigid slab slides over a rubber substrate at moderately high velocity, the relative motion between two solids occurs in the form of detachment waves, the energy is dissipated mainly in the bulk of the rubber. This phenomenon is qualitatively similar to our opening waves, however, the physics of the Schallamach waves, which propagate at smaller speeds than what is observed here, includes visco-elastic material behavior and adhesion on the contact surface, both are not present in our model. The possibility of detachment waves was also speculated in [Comninou andDundurs, 1977, Comninou andDundurs, 1978] either for similar materials or for frictionless contacts, however, some inconsistency in energy balance in these results was pointed out in [Freund, 1978]. Possibility of detachment was discussed in numerous works [Weertman, 1980, Adams, 1995], but obtaining rigorous theoretical results is associated with great mathematical difficulties. Nevertheless, the detachment waves on frictional interfaces, different from Schallamach waves, were induced in atomistic simulations [START_REF] Gerde | Friction and fracture[END_REF]] and, for the first time, were obtained in finite element simulations for a set-up with rotational symmetry in [START_REF] Moirot | An example of stick-slip and stick-slip-separation waves[END_REF].

Stick-slip and stick-slip-opening pulses

Here, we consider in detail the formation of an intersonic slip pulse and the mechanism of its self-sustained propagation [Weertman, 1980, Heaton, 1990, Adams, 1995]. The strong coupling between normal and shear tractions on bi-material interface creates favorable conditions to reach the frictional limit within a local zone and make slip occur and propagate in the direction of sliding of the softer material [START_REF] Andrews | Wrinklelike slip pulse on a fault between different materials[END_REF], Cochard and Rice, 2000, Ben-Zion and Huang, 2002]. Since the study of a solitary pulse in a periodic system is problematic and the number of stable pulses depends on the ratio λ/H and on the pulse speed c p , we consider a train of stick-slip pulses propagating along the interface at constant speed c p . Let l denote the spacing between pulses and s the width of the pulse. Such type of motion was conjectured theoretically [Adams, 1998, Adams, 2000, Bui and Oueslati, 2010], observed in experiments [START_REF] Coker | Dynamic crack growth along a polymer composite-homalite interface[END_REF] and in simulations [Cochard andRice, 2000, Coker et al., 2003]. However, in theoretical findings too restrictive conditions were imposed: the mean slip velocity on the interface was assumed to be equal to the shear velocity V 0 . This condition is natural if the motion is assumed stationary, it is equivalent to the following relation ux = V 0 l/s, where ux is the mean slip velocity in the pulse. Such a stationary solution is observed for low coefficients of friction. However, such a stationary train of pulses cannot result in a huge frictional drop bringing the system in a global pinned state. Moreover, we observe that the contact surface may be brought ahead of the top one, thus the link between the local and the global speeds should be stated as an inequality ux ≥ V 0 l/s. Hence, to characterize the train of stick-slip waves, we have four unknowns c p , s, l, ux and one inequality.

Pulse celerity

As observed in simulations, in most cases the slip front propagates in the sliding direction (direction of the imposed shear) at intersonic velocity c p (see Fig. 2 for an example, and Fig. 4,b and Table 1 for the summary). Thanks to the similarity between propagation of the mode-II crack and the slip front [START_REF] Palmer | The growth of slip surfaces in the progressive failure of over-consolidated clay[END_REF], Andrews, 1976, Svetlizky and Fineberg, 2014, Kammer et al., 2015], to find the slip pulse celerity, one may resort to the help of results obtained for intersonic cracks at the interface between an elastic solid and a rigid substrate [START_REF] Liu | Shear dominated transonic interfacial crack growth in a bimaterial i-ii. asymptotic fields and favorable velocity regimes[END_REF]. It was shown that the sign of the product between vertical displacement at a certain distance behind the crack tip and the contact pressure ahead of the crack tip is determined by the sign of (c p /c s -√

2). Since the pressure ahead of the slip pulse is always compressive (negative), the vertical displacement is positive if c p < √ 2c s , thus propagation at this speed is favorable for opening. In almost all simulations, in which slip and, in subsequence, opening fronts propagate in the direction of sliding, their celerities were found to be confined in the interval c s < c p < √ 2c s . The slip front celerity weakly depends on the coefficient of friction, and, in general, the most energetic pulses are obtained for friction coefficients, which result in c p ≈ √ 2c s . In Table 1 slip-pulse celerities c p are provided for several combinations of parameters, which result in opening waves. Celerity of slip fronts is positive if they propagate in the direction of sliding, and negative if they propagate in the opposite one. The data are obtained for E = 1000 Pa, ρ = 1000 kg/m 3 and λ = 2H; the average celerity is measured over a time period of 200 s using a threshold in the slip velocity ux , and was also verified geometrically. The slip celerity remains almost constant when a major slip pulse is localized and does not change when the opening occurs, at least within the precision of simulations and measurements. For ν = 0 and f = 2, in contrast to all other studied combinations of parameters, the slip front propagates in the sliding direction at supersonic speed, however, this result does not contradict the theoretical basis [Adams, 2000]. More generally, it is usual in non-linear dynamic systems that for certain combinations of parameters several stable (unstable) solutions coexist, whose realization depends on initial conditions. Probably, here we face a similar situation.

Pulse period

A relation can be established between c p and the period length l. Since the slip pulse propagates at intersonic celerity, it radiates a shear Mach wave which propagates in the bulk inclined at angle θ s = arccos(c s /c p ) with respect to the rigid flat. Being reflected from the top surface the Mach wave returns to the frictional interface and might create favorable stress state to initiate or sustain another pulse. Thus l = 2H tan(θ s ) = 2H c 2 p /c 2 s -1, and if c p ≈ √ 2c s then l ≈ 2H. This hypothesis was numerically verified for f = 1.5, ν = 0.2 by varying the ratio λ/H = 2, 4, 6 resulting in 1, 2, 3 major slip-opening pulses. The last unknown, slip length s, remains undetermined.

Summary

Opening waves were observed for certain combinations of friction coefficients and Poisson's ratios (Fig. 4). Such waves are accompanied with frictional drop (but it does not necessarily leads to inversion of the frictional force). This result holds for all shear velocities in the considered interval V 0 /c s ∈ [10 -6 ; 10 -4 ] (see Fig. 2,c). For high Poisson's ratios, within a certain interval of coefficients of friction, slip and opening pulses propagate at supersonic celerity c p > c l in the direction opposite to the sliding; their supersonic celerity is compatible with theoretical results [Adams, 2000]. For higher friction coefficients, slip pulses propagate again in the direction of sliding at intersonic celerity. Seemingly unrealistic, the supersonic slip propagation does not contradict the causality in the system with maximal signal speed equal to the longitudinal wave speed c l . It means simply that the slip advancement is not a result of the local deformation due to the slip, but due to elastic waves traveling in the elastic layer, some experimental and numerical results on supersonic slip can be found in [START_REF] Coker | Dynamic crack growth along a polymer composite-homalite interface[END_REF], Coker et al., 2005, Kammer et al., 2012]. The time interval of the frictional drop and its value is not affected by the applied shear velocity V 0 . The time step and mesh density affect the slip dynamics only within the transitional phase as discussed in Section 2.1, but do not alter the formation of major slip pulse and its transformation into opening wave if the time step is small enough to resolve accurately the wave dynamics.

Conclusion

Experimental verification of the observed phenomena may be carried out on a set-up with rotational symmetry [START_REF] Moirot | An example of stick-slip and stick-slip-separation waves[END_REF], which would ensure an initial uniform stress state; non-uniform stress distribution, usual for finite size contacts, results in triggering frictional slip before the local frictional limit f is reached on the global scale. Thus, in finite systems the apparent or global coefficient of static friction f s gl should be smaller than the local friction f , which is compatible with experimental observations [START_REF] Ben-David | The dynamics of the onset of frictional slip[END_REF]Fineberg, 2011]. In this light, the behavior of frictional system with a uniform stress distribution resembles to the process of supercooling: the slip may not start before the entire interface reaches the frictional limit, but when it does, the process is rapid and energetic. On contrary, in systems with stress heterogeneities, easily triggered slip front may propagate along the entire interface resulting in global sliding [START_REF] Ben-David | The dynamics of the onset of frictional slip[END_REF], Kammer et al., 2015].

The sliding mechanism discussed in this Letter adds a new piece to the puzzle of elastodynamic sliding under Coulomb's friction law in finite-size systems, but the full picture is still missing especially in the region of high coefficients of friction and Poisson's ratios. The prospective study is planned to focus on the dynamics at longer time intervals.

Acknowledgment

The author is grateful to David S. Kammer and Jean-Pierre Vilotte for valuable discussions.

Figure 4: (a) Intersonic opening waves propagating in the direction of sliding, which are accompanied with high frictional drop, were observed for Poisson's ratios and friction coefficients marked with a square (double square for cases with inversion of frictional force), diamonds mark supersonic opening fronts propagating in the opposite direction, open circles correspond to parameters, for which opening was not observed; parameters corresponding to theoretical flutter instability for slip without opening [START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF] is located in the wedge-zone bounded by a dash-dotted line. (b) shows celerities of opening waves for different combinations of Poisson's ratios and friction coefficients; c l , c s as well as the critical value [START_REF] Liu | Shear dominated transonic interfacial crack growth in a bimaterial i-ii. asymptotic fields and favorable velocity regimes[END_REF] √ 2c s are also plotted. Examples of slip localization are shown in spatio-temporal maps (black zones correspond to stick state, colored zones correspond to slipping) (c-f) for (ν = 0, f = 1.5), (ν = 0.2, f = 1.1), (ν = 0.3, f = 1.1) and (ν = 0.45, f = 1), respectively; the latter combination of parameters results in supersonic slip and opening pulses propagating in the direction opposite to sliding.

Figure 2 :

 2 Figure 2: Spatio-temporal maps showing formation and propagation of (a) slip pulse ( ux > 0) and (b) opening wave (u y > 0), which result (c) in a drop of frictional reaction (ratio of tangential T to normal force N on the interface) down to negative values; longitudinal, shear and slip wave celerities are denoted c l , c s , c p , respectively; maps (a,b) and the frictional force (c, red line) are shown for V 0 /c s = 10 -5 , frictional forces plotted in gray in (c) are for V 0 /c s ∈ [10 -6 ; 10 -4 ]. In (d), the ratio T/N for the same simulation is plotted for a longer time interval, t = 0 corresponds to T/N ≈ 0.95 f .

Figure 3 :

 3 Figure 3: Instantaneous maps of material velocity in the elastic layer, slip velocity and opening on the interface are presented for f = 1.5, ν = 0.2, V 0 /c s = 10 -5 at (a) t ≈ 787 s, (b) t ≈ 840 s, (c) t ≈ 890 s, (d) t ≈ 960 s, (e) t ≈ 1010 s, (f) t ≈ 1077 s; deformed state is scaled by a factor 150; dashed line shows the configuration just before the slip starts.

Table 1 :

 1 Slip front celerity in (m/s) for various combinations of Poisson's ratios and coefficients of Coulomb's friction. In the three last lines, c s ,

				Poisson's ratio, ν
	f	0.	0.1	0.2	0.3	0.4	0.45	0.49
	0.50							-5.250
	0.80							-5.253
	0.90						-2.506
	0.95						-2.523
	1.00						-2.523
	1.05				0.851 0.825 -2.511 -5.248
	1.10 0.861 0.829		0.857
	1.15 0.843 0.824 0.880 0.858
	1.20 0.861 0.921 0.875			-2.504 -5.085
	1.30						0.708
	1.40						0.708
	1.50 0.982		0.882 0.774 0.679 0.656 -5.399
	1.80		0.945 0.831 0.764 0.717
	2.00 1.213			0.723 0.666 0.659	0.607
	c s √	0.707 0.674 0.645 0.620 0.598 0.587	0.579
	2c s 1.000 0.953 0.913 0.877 0.845 0.830	0.819
	c l	1.000 1.011 1.054 1.160 1.464 1.948	4.137
							√
								2c s , c l are
	provided.

For a single case, the simulations were run also in commercial finite element software ABAQUS, which properly reproduced the results obtained with our software.

Mechanism of frictional slidingAt reaching the critical shear stress τ = f |p|, the frictional interface, initially in stick (pinned) state, starts to slip uniformly. But since this uniform slip is unstable for high friction coefficients[START_REF] Martins | Dynamic surface solutions in linear elasticity and viscoelasticity with frictional boundary conditions[END_REF], after a short transitional period, slip localizes within several slip pulses propagating along the interface (see an example in Fig.2obtained for ν = 0.2, f = 1.5, E = 1000


Chérif Amrouche

Macaire Batchi

Jean Batina

Navier-Stokes equations with periodic boundary conditions and pressure loss

Keywords: Navier-Stokes equations, incompressible fluid, bidimensional channel, periodic boundary conditions, pressure loss

may come

Navier-Stokes equations with periodic boundary conditions and pressure loss

Chérif Amrouche, Macaire Batchi, Jean

Introduction

The problem which one proposes to study here is that modelling a laminar flow inside a two-dimensional plane channel with periodic section. Let Ω be an open bounded connected lipschtzian of R 2 (see figure hereafter), where

Γ 0 = {0} × ]-1, 1[ and Γ 1 = {1} × ]-1, 1[ .

One defines the space

V = v ∈ H 1 (Ω) ; div v = 0, v = 0 on Γ 2 , v | Γ 0 = v | Γ 1
and for λ ∈ R given, one considers the problem (S) Initially one proposes to study the problem (P) . One has it Theorem 2.1 Problem (S) has an unique solution u ∈ V. Moreover, there is a constant C (Ω) > 0 such that:

               Find u ∈ V, such that ∀v ∈ V,
u H 1 (Ω) ≤ λC (Ω) . (1)
Proof: Let us note initially that space V provided the norm H 1 (Ω) 2 being a closed subspace of H 1 (Ω) 2 is thus an Hilbert space. Let us set

a (u,v) = Ω ∇u.∇v dx, l (v) = λ +1 -1 v 1 (1, y) dy.
It is clear, thanks to the Poincaré inequality, that the bilinear continuous form is V -coercive. It is easy to also see that l ∈ V ′ . One deduces from Lax-Milgram Theorem the existence and uniqueness of u solution of (S) . Moreover,

Ω |∇u| 2 dx ≤ λ √ 2 +1 -1 |u 1 (1, y)| 2 dy 1/2 , i.e. ∇u 2 L 2 (Ω) ≤ λ √ 2 u L 2 (Γ) ≤ λ √ 2 u H 1/2 (Γ) ≤ λC 1 (Ω) u H 1 (Ω)
Thus there is the estimate [START_REF] Amrouche | Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension[END_REF].

We now will give an interpretation of the problem (S) . One introduces the space

V = v ∈ D (Ω) 2 ; div v = 0 .
Let u be the solution of (S) . Then, for all v ∈ V, one has :

-∆u, v D ′ (Ω)×D(Ω) = 0.
So that thanks to De Rham Theorem, there exists p ∈ D ′ (Ω) such that

-∆u + ∇p = 0 in Ω. (2)
Moreover, since ∇p ∈ H -1 (Ω) 2 , it is known that there exists q ∈ L 2 (Ω) such that (see [START_REF] Amrouche | Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension[END_REF])

∇q = ∇p in Ω. (3)
The open Ω being connected, there exists C ∈ R such that p = q + C, what means that p ∈ L 2 (Ω). Let us recall that (see [START_REF] Amrouche | Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension[END_REF])

inf K ∈ R p + K L 2 (Ω) ≤ C ∇p H -1 (Ω) 2 .
One deduces from the estimate (1) and from (2) that inf

K ∈ R p + K L 2 (Ω) ≤ C ∆u H -1 (Ω) 2 ≤ C u H 1 (Ω) 2 ≤ λC (Ω) . Since u ∈ H 1 (Ω) 2 and 0 = -∆u + ∇p ∈ L 2 (Ω) 2 , it is shown that - ∂u ∂n + pn ∈ H -1/2 (Γ) 2 and one has the Green formula: for all v ∈ V Ω (-△u + ∇p) .v dx = Ω ∇u.∇v dx + - ∂u ∂n + pn, v , (4)
where the bracket represents the duality product H -1/2 (Γ) × H 1/2 (Γ) . Moreover, as p ∈ L 2 (Ω) and △p = 0 in Ω, one has p ∈ H -1/2 (Γ) . Consequently, one has therefore ∂u ∂n ∈ H -1/2 (Γ) 2 . The function u being solution of (S) , for all v ∈ V one has:

∂u ∂n -pn,v = λ +1 -1 v 1 (1, y) dy, (5)
i.e.

∂u ∂x -pe 1 ,v Γ 1 + - ∂u ∂x + pe 1 ,v Γ 0 = λe 1, v Γ 1 . (6)
where

e 1 = (1, 0). i) Let µ ∈ H 1/2
00 (Γ 1 ) and let us set

µ 2 =                µ on Γ 0 ∪ Γ 1 0 on Γ 2 and µ =         0 µ 2        
where (see [START_REF] Dautray | Analyse mathématique et calcul numérique pour les sciences et les techniques[END_REF])

H 1/2 00 (Γ 1 ) = ϕ ∈ L 2 (Γ 1 ); ∃ v ∈ H 1 (Ω), with v| Γ 2 = 0, v| Γ 0 ∪Γ 1 = ϕ .
It is checked easily that

µ ∈ H 1/2 (Γ) 2 and Γ µ.n dσ = 0.
So that there exists v ∈ H 1 (Ω) 2 satisfying (see [START_REF] Girault | Finite Element Methods for Navier-Stokes Equations[END_REF])

div v = 0 in Ω and v = µ on Γ.
In particular v ∈ V and according to [START_REF] Temam | Navier-Stokes Equations.Theory and Analysis[END_REF], one has

∂u 2 ∂x , µ Γ 1 = ∂u 2 ∂x , µ Γ 0 , which means that ∂u 2 ∂x | Γ 1 = ∂u 2 ∂x | Γ 0 . (7)
One deduces now from ( 6) that for all v ∈ V,

∂u 1 ∂x -p, v 1 Γ 1 + - ∂u 1 ∂x + p, v 1 Γ 0 = λ, v 1 Γ 1 . (8)
But, div u = 0 and

u 2 | Γ 1 = u 2 | Γ 0 , one thus has ∂u 2 ∂y | Γ 1 = ∂u 2 ∂y | Γ 0 and ∂u 1 ∂x | Γ 1 = ∂u 1 ∂x | Γ 0 . (9)
Consequently, thanks to (8) one has:

-p, v 1 Γ 1 + p, v 1 Γ 0 = λ, v 1 Γ 1 (10) ii) Let ν ∈ H 1/2
00 (Γ 1 ) and let us set

ν 1 =                ν on Γ 0 ∪ Γ 1 0 on Γ 2 and ν =         ν 1 0        
.

One easily checks that

ν ∈ H 1/2 (Γ) 2 and Γ ν.n dσ = 0. So that there exists v ∈ H 1 (Ω) 2 satisfying div v = 0 in Ω and v = ν on Γ.
In particular v ∈ V and according to ( 14), one has

-p, ν Γ 1 + p, ν Γ 0 = λ, ν Γ 1 i.e. p |Γ 1 = p |Γ 0 -λ (11)
where the equality takes place with the H 1/2 sense. In short, if u ∈ H 1 (Ω) 2 is solution of (S) , then there exists p ∈ L 2 (Ω) , unique with an additive constant such that: 12)-( 16), then u is solution of (S). Thus it Theorem 2.2 The problem ( 12)-( 16) has an unique solution (u, p) ∈ H 1 (Ω) 2 × L 2 (Ω) up to an additive constant for p. Moreover, u verifies (S) and

-∆u + ∇p = 0 in Ω, ( 12
) div u = 0 in Ω, (13)
u = 0 on Γ 2, u| Γ 1 = u | Γ 0 , ( 14
)
∂u ∂x | Γ 1 = ∂u ∂x | Γ 0 , (15)
p |Γ 1 = p |Γ 0 -λ. ( 16
) It is clear that if (u, p) ∈ H 1 (Ω) 2 × L 2 (Ω) checks (
u H 1 (Ω) + p L 2 (Ω)/ R ≤ λC (Ω) .
Remark 1 The pressure verifies the relation ( 16), which means that p satisfies the relation of Patankar et al. [START_REF] Patankar | Fully developed flow and heat transfer in ducts having streamwise-periodic variations of cross sectional area[END_REF] .

Navier-Stokes Equations

One takes again the assumptions of the Stokes problem given above. For λ ∈ R

given, the one considers the following problem

(N S)                Find u ∈ V, such that ∀v ∈ V, Ω ∇u.∇v dx + b (u, u, v) = λ +1 -1 v 1 (1, y) dy with b (u, v, w) = Ω (u.∇) v.w dx
With an aim of establishing the existence of the solutions of the problem (N S) , one uses the Brouwer fixed point theorem (see [START_REF] Lions | Quelques méthodes de résolution des problèmes aux limites nonlinéaires[END_REF], [START_REF] Temam | Navier-Stokes Equations.Theory and Analysis[END_REF]). One will show it Theorem 3.1 The problem (N S) has at least a solution u ∈ V. Moreover, u checks the estimate [START_REF] Amrouche | Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension[END_REF].

Proof: To show the existence of u, one constructs the approximate solutions of the problem (N S) by the Galerkin method and then thanks to the arguments of compactness, one makes a passage to the limit.

i) For each fixed integer m ≥ 1, one defines an approximate solution u m of (N S) by

u m = m i=1 g im w i , with g im ∈ R ((u m , w i )) + b (u m , u m , w i ) = λn, w i Γ 1 , i = 1, ..., m (17)
where V m = w 1 , ..., w m vector spaces spanned by the vectors w 1 , ..., w m and {w i } is an Hilbertian basis of V which is separable. Let us note that (17) is equivalent to:

∀v ∈ V m , ((u m , v)) + b (u m , u m , v) = λ +1 -1 v 1 (1, y) dy. (18)
With an aim to establish the existence of the solutions of the problem u m , the operator as follows is considered

P m : V m → V m u -→ P m (u) defined by ∀u, v ∈ V m , ((P m (u) , v)) = ((u, v)) + b (u, u, v) -λ +1 -1 v 1 (1, y) dy.
Let us note initially that P m is continuous and

∀u ∈ V, b (u, u, u) = 0.
Indeed, thanks to the Green formula, one has

b (u, u, u) = -1 2 Ω |u| 2 div u dx + 1 2 Γ (u.n) |u| 2 dσ = 0.
But, div u = 0 in Ω and

Γ (u.n) |u| 2 dσ = Γ 0 (u.n) |u| 2 dσ + Γ 1 (u.n) |u| 2 dσ.
since the external normal to Γ 0 is opposed to that of Γ 1 and u ∈ V .

Thanks to Brouwer Theorem, there exists u m satisfying (18) and

u m H 1 (Ω) ≤ λC (Ω) . (19)
ii) We can extract a subsequence u ν such that As for the Stokes problem, one shows the existence of p ∈ L 2 (Ω) , unique except for an additive constant, such that

                                       -∆u + (u.∇) u+∇p = 0 in Ω, div u = 0 in Ω, u = 0 on Γ 2 , u| Γ 1 = u | Γ 0 .
It is checked finally that

∂u ∂x | Γ 1 = ∂u ∂x | Γ 0 , p |Γ 1 = p |Γ 0 -λ.
Remark 2 i) Theorem 3.1 of problem (N S) takes place in three dimension.

ii) One can show that the solution (u, p) belongs to H 2 (Ω) 2 × H 1 (Ω).

2 ΩFigure 1 .

 21 Figure 1. Geometry of channel

u 1 v 1 ( 1 ,

 111 ν ⇀ u weakly in V, and thanks to the compact imbedding of V in L 2 (Ω)2 , we obtain∀v ∈ V , ((u, v)) + b (u, u, v) = λ +1 -y) dy.

+1-1 v 1 (1, y) dy. y x


Erwan Livolant

Pascale Minet

Thomas Watteyne

The Cost of Installing a 6TiSCH Schedule

Keywords: Low-Power Wireless Mesh Networks, IEEE802.15.4e TSCH, 6TiSCH, CoAP, CoMI, OCARI

Scheduling in an IEEE802.15.4e TSCH (6TiSCH) low-power wireless mesh network can be done in a centralized or distributed way. When using centralized scheduling, a scheduler computes a communication schedule, which then needs to be installed into the network. This can be done using standards such CoAP and CoMI, or using a custom protocol such as OCARI. In this paper, we compute the number of messages installing and updating the schedule takes, using both approaches, on a realistic example scenario. The cost of using today's standards is high. In some cases, a standards-based solution requires approximately 4 times more messages to be transmitted in the network, than when using a custom protocol. This paper makes three simple recommended changes to the standards which, when integrated, reduce the cost of a standardsbased solution by 18% to 74%. Since they are still being developed, these recommendations can easily be integrated into the standards.

Introduction

Industrial low-power wireless mesh network applications have strong requirements in terms of latency, energy efficiency and reliability. To cope with these requirements, the IEEE802.15.4e amendment [START_REF]IEEE: IEEE Standard for Local and metropolitan area networks-Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs) Amendment 1: MAC Sublayer. IEEE Std[END_REF] introduces the Time Slotted Channel Hopping (TSCH) mode. In a TSCH network, nodes are synchronized, and time is cut into timeslots, each typically 10 ms long. All communication is orchestrated by a communication schedule, which indicates to each node what to do in each slot: transmit, listen or sleep. This schedule can be built to enable collision-free communication, yielding predictable behavior, ultra-high reliability and years of battery lifetime.

A schedule consists of a number of timeslots which continuously repeat over time. An example schedule is depicted in Fig. 2 with 9 timeslots and 3 logical channels. The index of a timeslot (the x-axis of the matrix in Fig. 2) is called slotOffset. The channelOffset represents the communication channel (the y-axis of the matrix in Fig. 2).

Building the schedule consists in assigning a source node, a destination node and a channel to cells in the schedule. Installing the schedule into the network means indicating to each node the list of cells it is involved in, either as transmitter or as receiver. Each cell is represented by a tuple [slotOffset, channelOffset, nodeAddress, linkType ] (linkType indicates whether it is a transmit -TX -and/or receive -RX -cell).

We want to compare the number of packets it takes a central scheduler to install and update a schedule, using the different approaches listed in Section 2. We are particularly interested in comparing standards-based and custom-built protocols. The goal is not to explore edge cases, but rather to take an example representative enough that we can learn lessons and made recommendations.

We consider the topology depicted in Fig. 1. The network first consists of 12 nodes where Node 1 is the root of the network. Arrows represent the links that are used for communication and that therefore need to appear in the schedule. The goal is to install the schedule from Fig. 2 into the network. When node 13 is added, the scheduler computes the schedule depicted in Fig. 3, in which 12 cells differ. The goal is then to update the schedule in the network. Several centralized scheduling algorithms exist. This paper does not recommend one or the other, nor does it attempt to survey them. Rather, we use a particular scheduling algorithm, MODESA [START_REF] Soua | MODESA: An Optimized Multichannel Slot Assignment for Raw Data Convergecast in Wireless Sensor Networks[END_REF], and measure the number of packets to install the schedule once it is computed. The choice of MODESA is, as far as this paper is concerned, arbitrary.

We focus on two approaches to install the schedule: using standards (detailed in Section 2.2), and using a custom protocol (detailed in Section 2.3). By executing both approaches on a representative example scenario, we show that using a standards-based approach can cost 4 times more frames than using a custombuilt protocol. In Section 3, we make three simple recommended changes to the standards which reduces the overhead of the standards-based approach by 18% to 74% percent.

Approaches to Install the Schedule

We consider different approaches to install a schedule: either using the CoAP and CoMI standards (Section 2.2), or using a custom protocol called OCARI (Section 2.3). Both use IEEE802.15.4 [START_REF]IEEE: IEEE Standard for Local and metropolitan area networks-Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs)[END_REF] as the underlying physical layer. Section 2.1 first details the notation used.

Notation

We define the following notations:

-Depth(u), the depth of node u in the topology.

-Child(u), the set of children of node u.

-T rans(u), the number of user data frames transmitted by node u, including both the ones generated by node u itself and the ones received from its children and forwarded. -P , the number of nodes which are not leaf nodes.

-B(u), number of CoAP blocks sent to node u.

-B sched , number of CoAP blocks needed to broadcast the complete schedule.

-Let N f ield number of fields updated per cell.

-ScheduleN umber, version of the schedule, incremented each time the schedule is computed.

CoAP and CoMI: a Standards-Based Approach

6TiSCH is an active IETF working group which standardizes how to build and maintain a TSCH communication schedule [START_REF] Dujovne | 6TiSCH: Deterministic IP-enabled Industrial Internet (of Things)[END_REF]. While 6TiSCH supports both centralized and distributed scheduling, this paper focuses on the former. The protocol stack considered by 6TiSCH consists of IEEE802.15.4e [START_REF]IEEE: IEEE Standard for Local and metropolitan area networks-Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs) Amendment 1: MAC Sublayer. IEEE Std[END_REF] (physical and link layers), 6LoWPAN [START_REF] Hui | Compression Format for IPv6 Datagrams over IEEE 802.15.4-Based Networks[END_REF], RPL [START_REF] Winter | RPL: IPv6 Routing Protocol for Low-Power and Lossy Networks[END_REF] (network layer ), CoAP [START_REF] Bormann | Block-wise Transfers in CoAP[END_REF][START_REF] Shelby | The Constrained Application Protocol (CoAP)[END_REF][START_REF] Van Der Stok | Patch Method for Constrained Application Protocol (CoAP)[END_REF] and CoMI [START_REF] Van Der Stok | CoAP Management Interface[END_REF] (application layer ). The resulting protocol stack is depicted in Fig. 4. Table 1 depicts the format of an IEEE802.15.4 frame which encapsulates those protocols. The Maximum Transmission Unit (MTU) at the IEEE802.15.4 [START_REF]IEEE: IEEE Standard for Local and metropolitan area networks-Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs)[END_REF] PHY layer is 127 bytes. MAC header and footer require a total of 29 bytes in the context of the IEEE802.15.4e TSCH [START_REF]IEEE: IEEE Standard for Local and metropolitan area networks-Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs) Amendment 1: MAC Sublayer. IEEE Std[END_REF] 4 bytes. The MAC payload can contain up to 98 bytes. At the Network layer, the protocol requires 19 bytes for the IP compressed header (2 bytes), the Hop limit (1 byte) and the Source and Destination addresses coded on 8 bytes each.

At the Transport layer, the payload is reduced by 2 bytes corresponding to the UDP compressed header and the destination and source port coded together in 1 byte. At the Application layer, the size of the applicative payload depends on the CoAP options used. 4 bytes of header and 1 byte of payload marker are required. The URI targeting the resource is defined as follows: /mg/6t/hash where /mg/6t/ is the main path to the 6top management resources and hash is a 30bit hash (encoded on 4 bytes) defining the rest of the path towards the target resource. We look at three options for using CoAP: without fragmentation, less than 16 fragments and with than 4096 fragments. In any case, we get a useful payload whose size is less than 61 bytes. For the Block option, it is specified in CoAP that the block size should be a power of two. As a consequence, the only possibility with this payload size is a block size of 32 bytes. As a conclusion, in the 127-byte IEEE802.15.4 frame, only 32 bytes are available for the actual encoding of the schedule, as the remainder of the frame is occupied by the headers of the different standards.

The standards do not indicate the example approach to install the schedule. We consider three approaches, which we call "Single", "PATCH" and "Broadcast". Each is detailed below.

Updating a Single Field at a Time ("Single"). The central scheduler issues a separate confirmable CoAP POST message to write each field of each cell it installs into a node. This solution assumes that for any network node a route to reach it from the scheduler node is known by all nodes within the route. Fig. 5 shows an example CoAP POST to set nodeAddress = 3 for the cell at slotOffset = 3 and channelOffset = 1 in the schedule.

REQ:

POST url: coap://<ip>:5683/mg/6t/cellList/nodeAddress/?slotOffset=3&channelOffset=1 body: 3 The central scheduler needs to install a cell on both communicating neighbors. For each field, the scheduler issues a CoAP CON and receives CoAP ACK. Since each cell contains N f ield fields, the scheduler sends N f ield CoAP CON messages and receives N f ield CoAP ACK acknowledgments. When sent to node u, each of these messages travels over Depth(u) hops. Moreover, node u is involved in T rans(u) transmissions and v∈Child(u) T rans(v) receptions. This results in the total number of messages in [START_REF] Al Agha | Cross-layering in an Industrial Wireless Sensor Network: Case Study of OCARI[END_REF].

numF rames Single = 2 • N f ield • u =sink Depth(u)   T rans(u) + v∈Child(u) T rans(v)   (1)
Table 2 presents the number of messages required for first installing the schedule (see Fig. 2) computed for a network of 12 nodes, then updating this schedule taking into account the new node 13 (see Fig. 3). When the schedule is updated, node 2 has two cells where two fields are modified and two other cells with only one field changed. Hence in Table 2, the element at line 2 and column "Cells * N f ield" contains 2*2 + 2*1. Sending a PATCH to Each Node ("PATCH"). The scheduler contacts each node once and transfers a list of cells that must be updated, encoded as a CoAP PATCH. This allows it to send only the differences between the current schedule and the new one. An example CoAP PATCH which modifies the nodeAddress and linkType fields of the cell with slotOffset = 1 and channelOffset = 2 is presented in Fig. 6.

REQ: PATCH url: coap://<ip>:5683/mg/6t/cellList body: [ { "op": "replace", "path": "/nodeAddress?slotOffset=1&channelOffset=2", "value": 4 }, { "op": "replace", "path": "/linkType?slotOffset=1&channelOffset=2", "value": 1 } ] Fig. 6. A CoAP PATCH modifying the nodeAddress and linkType fields of the cell.

When the payload of the PATCH is too long for a single frame, CoAP Block is used for application-layer fragmenting. B(u) blocks are transmitted to node u, the scheduler receives an acknowledgment per block, as recommended in [START_REF] Van Der Stok | CoAP Management Interface[END_REF]. The resulting total number of messages is given in [START_REF] Bormann | Concise Binary Object Representation (CBOR)[END_REF].

numF ramesP AT CH = 2 • u =sink B(u) • Depth(u) (2)
Table 3 presents the number of messages required for installing and updating the schedule computed for our illustrative network (see Fig. 1). Detailed CoAP PATCH messages for installing the schedule can be found in [START_REF] Livolant | The Cost of Installing a New Communication Schedule in a 6TiSCH Low-Power Wireless Network using CoAP[END_REF]. Broadcasting the Complete Schedule ("Broadcast"). The scheduler broadcasts the complete schedule over CoAP. The schedule is represented as a CBORencoded [START_REF] Bormann | Concise Binary Object Representation (CBOR)[END_REF] JSON document. Each node u filters the cells it is involved in as transmitter or receiver. An example of broadcast with the complete schedule is proposed in Fig. 7 where each tuple is [slotOffset, channelOffset, source node, destination node].

To ensure that each node correctly receives the schedule, the scheduler uses CoAP Observe to monitor the value of the schedule number on each node. Each time a new schedule is pushed to a node, the scheduler expects to receive a CoAP Observe notification, confirming the successful reception of the schedule update by the node.

At network initialization, the scheduler issues an CoAP Observe request on each of the nodes. Broadcasting the schedule requires B sched blocks of 32 bytes to be broadcast into the network, resulting in P • B sched frames, assuming ideal flooding. Each time a schedule is installed, the waves of CoAP Observe notifications account for u =sink Depth(u) messages. The total number of frames to install a schedule is given in (3); δ P is equal to 1 when at least one new node has joined the network since the last transmission CoAP Observe notification, 0 otherwise.

numF rames Broadcast = P • Bsched + u =sink Depth(u) + δP • P (3)
The length of the CBOR transcription of the JSON document describing our schedule (see Fig. 7) is 149 bytes. This CBOR transcription is divided into 5 fragments of 32 bytes, hence Bsched = 5. In our example topology, the number of parents broadcasting the schedule is P = 4 (node 1, 2, 3, 4) and the sum of depths of nodes is u =sink Depth(u) = 19. For this first schedule, as all nodes just joined the network δ P = 1. Finally, msg(Broadcast) = 4×5+19+1×4 = 43. For the second schedule depicted in Fig. 3, the length of the CBOR transcription of the JSON document describing our schedule is 159 bytes. The detailed CoAP POST for the broadcast of this second schedule can be found in [START_REF] Livolant | The Cost of Installing a New Communication Schedule in a 6TiSCH Low-Power Wireless Network using CoAP[END_REF]. As in the previous case, this CBOR transcription is divided into 5 fragments of 32 bytes, hence Bsched = 5. The number of parents broadcasting the schedule is the same P = 4 but now the sum of depths of nodes is u =sink Depth(u) = 21. Node 13 just joined the network, hence δ P = 1. Finally, msg(Broadcast) = 4 × 5 + 21 + 1 × 4 = 45.

OCARI, a Custom Protocol

An alternative is to use OCARI [START_REF] Al Agha | Cross-layering in an Industrial Wireless Sensor Network: Case Study of OCARI[END_REF], a custom-built non-standards-based protocol. We compare a standards-based approach to it to provide a lower bound on the number of packets needed to install a schedule.

OCARI is compliant with IEEE802.15.4, but not with upper-layer protocols such as 6LoWPAN or CoAP. Each OCARI node is assigned a 2-byte identifier, unique in the network and given during its association.

OCARI schedules activities in the network in a cycle organized in four periods, as depicted in Fig. 8. First, OCARI synchronizes the network in a collisionfree multi-hop way during the [T0,T1] period. All nodes periodically send beacons to maintain this synchronization. The period [T1,T2] is dedicated to control traffic used to collect network characteristics in order to compute a schedule for user data. Period [T2,T3] allows user data gathering. Finally, period [T3,T0'] is a sleep period, all nodes sleep to save energy. Fig. 9 summarizes the format of an OCARI beacon. For the topology depicted in Fig. 1, the first schedule (see Fig. 2) to install is 24 tuples long, the second one (see Fig. 3) is 26 tuples long. The scheduler hence needs to transmit 24 × 7 = 168 bytes for the first schedule, 26 × 7 = 182 bytes for the second. Since in this typical case the maximum payload available for cell tuples in a beacon is 80 bytes, up to 11 cell tuples can be transported in a single beacon. If the schedule contains more than 11 cells, the full schedule is fragmented across different beacons. The entire schedule hence need 3 beacons to be transferred, or a total of 3 × 4 = 12 messages, since in our example P = 4 parent nodes.

Details about OCARI can be found in [START_REF] Al Agha | Cross-layering in an Industrial Wireless Sensor Network: Case Study of OCARI[END_REF].

Table 4 summarizes the total number of messages to install and update the schedule. The detailed computation of the number of messages can be found in [START_REF] Livolant | The Cost of Installing a New Communication Schedule in a 6TiSCH Low-Power Wireless Network using CoAP[END_REF]. It shows that a standard-based approach is less efficient than a custombuilt protocol by a factor of 4 or more.

Simple changes to the standards allow them to be much more efficient. This section lists three simple optimizations which, when applied, yield a reduction in number of messages between 18% and 74% as depicted in Table 4.

Use Short MAC Addresses

In today's 6TiSCH standards, only 64-bit long MAC addresses are used. Implementing an association mechanism would enable a coordinator (typically the root of the network) to assign a 16-bit address unique in the network to each device. This association mechanism would save 12 bytes in each frame. The payload at the MAC layer can then be extended to 110 bytes and the maximum applicative payload could be equal to 73 bytes, allowing a 64-byte CoAP Block size. Table 5 presents the number of messages required for installing and updating the schedule computed for our illustrative network (Fig. 1). We see a dramatic saving for this PATCH method compared to the previous results in Section 2.2.

With the method of broadcasting a schedule to all nodes in the network, the savings in term of the number of messages is also significant. In our example topology, we obtain msg(Broadcast) = 4 × Bsched + 19 + 1 × 4 = 35 messages with Bsched = 3 for the installing the schedule. For updating the schedule, we obtain msg(Broadcast) = 4 × Bsched + 21 + 1 × 4 = 37 messages with Bsched = 3.

Per Table 4, using short MAC addresses reduces the number of frames by 45% (respectively 50%) to Install (respectively Update) the schedule when using the PATCH approach, and 19% (respectively 18%) when using the Broadcast approach. Fig. 10 depicts the schedule description using the CellId coding in the con-Fig. 10. A CoAP POST broadcasting the first schedule using CellId.

Per Table 4, using a CellId rather than a [slotOffset, channelOffset ] tuple saves an additional 34% (respectively 31%) when installing (respectively updating) the schedule using the PATCH approach. Its impact is negligible when using the Broadcast approach.

Shorter PATCH Operators

CoAP PATCH [START_REF] Van Der Stok | Patch Method for Constrained Application Protocol (CoAP)[END_REF], even if if were designed for constrained devices, still uses the multi-character operators "op", "path", "value", "replace". Reducing those to the shorter operators "o", "p", "v", "rpl" would be syntactically equivalent, but more efficient. An example resulting CoAP PATCH is described in Fig. 11.

Table 7 presents the number of messages required for installing and updating the schedule computed for our illustrative network (Fig. 1) with the recommended shorter operators.

The CBOR representation of the JSON document above is 26% shorter than the CBOR representation of the same JSON document with longer operator string. According to 28% (respectively 19%) when installing (respectively updating) the schedule using the PATCH approach. Its impact is negligible when using the Broadcast approach.

Conclusion

This paper discusses the efficiency of installing and updating a communication schedule in a 6TiSCH network with a central scheduler. Not surprisingly, it shows that a set of (generic) standards-based protocols is less efficient than a custombuilt protocol. It shows how simple optimizations to the standards can reduce the number of frames by up to 74%. It is, to the best of our knowledge, the first work to provide a system-wide analysis of the different protocols involved in a 6TiSCH network, and highlight the inefficiencies when "putting them all together". We believe this paper makes a strong contribution to the standardization activities at the IETF, in particular in the 6TiSCH, RPL and 6lo working group. It makes a strong (and quantified) case for adding an association step to the 6TiSCH protocol suite (which requires an additional protocol but reduces the overhead by up to 50%), and reducing the length to the CoAP PATCH operators (a trivial change which saves up to 28% overhead).

Our future work will focus on distributed scheduling in 6TiSCH networks. This paper is an example of the collaboration between the IETF and the academic world which the newly created Thing-to-Thing Research Group is fostering.

Fig. 1 .

 1 Fig. 1. Logical network topology.

Fig. 5 .

 5 Fig. 5. A CoAP POST addressing the nodeAddress value of a cell.

Fig. 7 .

 7 Fig. 7. A CoaP POST broadcasting the complete schedule.

Fig. 8 .

 8 Fig. 8. Cycle provided by OCARI

Fig. 9 .

 9 Fig. 9. Packet format when using the OCARI custom protocol.

Table 1 .

 1 mode1 . The Auxiliary Security header is encoded with 2 bytes and the Message Integrity Code (MIC) is coded with Packet format when using the standards-based approach. Numbers indicate the number of bytes in the fields.

		Fig. 4. The 6TiSCH Protocol Stack.
		Frame	Seq	Dest	Dest	Src	Aux Sec
		Control	Num	PAN	Address	Address Header MIC Payload FCS
	IEEE802.15.4e
	TSCH	2	1	2	8	8	2	4	98	2
			Hop	Src	Dest
		LOWPAN IPHC	limit	Address	Address		Payload
	6loWPAN
	-IP	2	1	8	8		79
			Port src +
		LOWPAN NHC	dest			Payload
	6loWPAN
	-UDP	1	1			77
				Payload		Payload
		Header	URI	marker		without block
	COAP
	without frag	4	11	1			61
				Option delta Option Delta Option Payload		Payload
		Header	URI	+ Length	extended	value	marker	with block
	COAP with
	≤ 16 frag	4	11	1	1	1	1		32
	COAP with
	≤ 4096 frag	4	11	1	1	2	1		32

Table 2 .

 2 Total number of messages required for installing and updating the schedule with the Single method.

			Install		Update
	Node Cells N f ield Messages Cells * N f ield Messages
	2	8	4	64	2*2 + 2*1	8 + 4
	3	9	4	72	3*1	6
	4	7	4	56	3*1 + 2*4	6 + 16
	5	2	4	32	2*2	16
	6	1	4	16	1*1	4
	7	1	4	16	1*1	4
	8	1	4	16	1*1	4
	9	1	4	16	1*1	4
	10	2	4	32	1*1	4
	11	1	4	16	0	0
	12	1	4	16	1*2	8
	13	-	-	-	1*4	16
	Total	-	-	352	-	100

Table 3 .

 3 Total number of messages required for installing and updating the schedules with the PATCH method.

			Install			Update
	Node CBOR bytes Blocks Messages CBOR bytes Blocks Messages
	2	1049	33	66	397	13	26
	3	918	29	58	208	7	14
	4	918	29	58	533	17	34
	5	263	9	36	269	9	36
	6	132	5	20	67	3	12
	7	132	5	20	70	3	12
	8	132	5	20	67	3	12
	9	132	5	20	67	3	12
	10	263	9	36	70	3	12
	11	132	5	20	0	0	0
	12	132	5	20	136	5	20
	13	-	-	-	132	5	20
	Total	-	-	374	-	-	210

Table 4 .

 4 Total number of messages traveling in the network for installing and updating the schedule, with none, some or all optimizations.

	Approach	Optimizations	Install Relative Absolute Update Relative Absolute
			Schedule Gain	Gain	Schedule Gain	Gain
	Single	None	352	N.A.	N.A.	100	N.A.	N.A.
		None	374	N.A.	N.A.	206	N.A.	N.A.
	PATCH + Short addresses (3.1)	206	45%	45%	104	50%	50%
		+ CellId (3.2)	136	34%	64%	72	31%	65%
		+ PATCH syntax (3.3)	98	28%	74%	58	19%	72%
		None	43	N.A.	N.A.	45	N.A.	N.A.
	Broadcast + Short addresses (3.1)	35	19%	19%	37	18%	18%
		+ CellId (3.2)	35	0%	19%	37	0%	18%
	Custom	None	12	N.A.	N.A.	12	N.A.	N.A.

Table 4 ,

 4 using shorter PATCH operators saves an additionalFig. 11. A CoAP PATCH with shorter operators.

	REQ:
	PATCH
	url: coap://<ip>:5683/mg/6t/cellList
	body: [
	{
	"o": "rpl",
	"p": "/nodeAddress?CellId=18",
	"v": 4
	},
	{
	"o": "rpl",
	"p": "/linkType?CellId=18",
	"v": 1
	}
	]

Table 7 .

 7 Total number of messages required for installing and updating the schedules with the PATCH method with shorter operators.

		without node 13			with node 13
	Node CBOR bytes Blocks Messages CBOR bytes Blocks Messages
	2	537	9	18	203	4	8
	3	468	8	16	96	2	4
	4	468	8	16	275	5	10
	5	133	3	12	96	2	8
	6	68	1	4	32	1	4
	7	70	1	4	33	1	4
	8	68	1	4	32	1	4
	9	68	1	4	32	1	4
	10	135	3	12	32	1	4
	11	68	1	4	0	0	0
	12	68	1	4	32	1	4
	13	-	-	-	68	1	4
	Total	-	-	98	-	-	58

[START_REF] Vilajosana | Minimal 6tisch configuration[END_REF] indicates that data messages must provide in their MAC header 64-bit addresses for the destination and the source.

More efficient CellId Representation

According to the IEEE802.15.4e standard [START_REF]IEEE: IEEE Standard for Local and metropolitan area networks-Part 15.4: Low-Rate Wireless Personal Area Networks (LR-WPANs) Amendment 1: MAC Sublayer. IEEE Std[END_REF], slotOffset and channelOffset are each encoded on 2 bytes. Since there are only 16 channels available for a IEEE802.15.4 radio operating at 2.4GHz, only 4 bits are needed to encode the channelOffset. Moreover, since the slotframe length in an IEEE802.15.4e TSCH network is rarely longer than 1000 timeslots (10 s when using 10 ms timeslots), channelOffset can be encoded using 12 bits. a single 16-bit number called "CellId" can encode slotOffset and channelOffset, in which the 4 most significant bits represent the channelOffset and the remaining 12 bits the slotOffset. Table 6 presents the number of messages required for installing and updating the schedule computed for our illustrative network (see Fig. 1). We notice that this simple coding mechanism brings a greater gain compared to the previous results in Section 2.2.