Laurent Debreu

Eugene Kazantsev

Optimal control of grids and schemes for the inertial gravity waves equation

Variational data assimilation technique is applied to a simple bidimensional wave equation that simulates propagation of internal gravity waves in the ocean in order to control grids and numerical schemes. Grid steps of the vertical grid, Brunt-Vaisala frequency and approximation of the horizontal derivative were used as control parameters either separately or in the joint control. Obtained results show that optimized parameters may partially compensate errors committed by numerical scheme due to insufficient grid resolution.

Introduction

Contemporary ocean general circulation models are usually discretized by finite differences on a grid with rectangular cells. This discretization simplifies the model and accelerates its integration. Most oceanic numerical models use low-order space-time schemes to offer the best simplicity and efficiency. Indeed, the robustness of the scheme allows us to refine the grid keeping the computational cost constant. This possibility was considered as important during long time because the lack of resolution was the principal source of model errors.

However, advances in computer technique, ocean models are now capable to resolve finer and finer scales along both horizontal and vertical axes.

The direct consequence of this achievement consists in including of additional physical processes into resolved model physics. Thus, significant part of the internal gravity waves spectrum could be potentially represented by actually employed computational grids. This implies the necessity to review conventional numerical choices for the space-time integration of the model. In particular, current low-order schemes and regular grids may be responsible for dispersion errors and wrong energy transfers.

Conception of the grid and the choice of numerical scheme require, however, a special work in order to bring together somewhat contradictory needs of computational simplicity and correct reproduction of physical processes. This conception admits a certain freedom in choice of both schemes and grids. Possibility to chose the approximation order ( [START_REF] Demange | Numerical representation of internal waves propagation[END_REF]), conventional or adaptive grids [START_REF] Debreu | AGRIF: Adaptive Grid Refinement in Fortran[END_REF], explicit or implicit schemes ( [START_REF] Shchepetkin | An adaptive, courant-number-dependent implicit scheme for vertical advection in oceanic modeling[END_REF]), finite elements ( [START_REF] Danilov | A finite-element ocean model: principles and evaluation[END_REF][START_REF] Iakovlev | On the simulation of temperature and salinity fields in the arctic ocean[END_REF]) of finite volumes ( [START_REF] Engwirda | High-order accurate finite-volume formulations for the pressure gradient force in layered ocean models[END_REF]) approaches are discussed and analyzed in numerous papers.

An alternative approach discussed in this paper suppose to ask the model's opinion about "optimal" discretization and scheme that will be able to compensate error committed by limited resolution of the model grid. The word "optimal" is used here in the sense of the variational control methods as numerical scheme that realizes a minimum of the specially constructed cost function.

In [START_REF] Kazantsev | Optimal boundary conditions for orca-2 model[END_REF][START_REF] Kazantsev | Optimized boundary conditions at staircase-shaped coastlines[END_REF], this approach has been applied to identification of lateral boundary conditions in different configurations of NEMO ( [START_REF] Madec | Nemo ocean engine. Technical Report 27[END_REF]). It has been shown that extending the set of control variables beyond the initial value control has certain advantages that help to bring the solution closer to observations not only during the assimilation stage, but also during the forecast.

In this paper, we intend to analyse the influence of numerical schemes and grids on the model flow in a simple, academic configuration of two dimensional equation that simulates internal gravity waves in a periodical domain and to propose a way of optimization of the vertical grid and horizontal derivative approximation by variational data assimilation technique. Internal waves propagate in the oceanic interior in the water of variable density. Due to their important role in the global distribution of physical mixing and mass transport, proper representation of internal wave dynamics in numerical models is extremely important especially in modern high resolution models.

As well as in [START_REF] Kazantsev | Optimal boundary conditions for orca-2 model[END_REF], tangent linear and adjoint codes, necessary for variational approach, have been obtained by the AD Tapenade described in [START_REF] Hascoët | Tapenade 2.1 user's guide[END_REF]. Particular utility of automatic differentiation in the case of parameter optimization is related to the fact that the derivative of the model with respect to internal parameters is two or three times longer (as well as in terms of the development, the number lines of the code and the necessary CPU time) than the derivative used to control the initial point of the model (see [START_REF] Kazantsev | Optimal boundary discretisation by variational data assimilation[END_REF] for details).

Model setup

In order to study numerical representation of inertial gravity waves we consider a two-dimensional system proposed in [START_REF] Demange | Numerical representation of internal waves propagation[END_REF]:

∂u q ∂t + g ∂p q ∂x = 0, ∂p q ∂t + 1 gλ q ∂u q ∂x = 0 ( 1
)
where g is the gravity acceleration. Variables u q and p q are supposed to be spectral coefficients in the decomposition by vertical modes M q (z) which are the eigenvectors of the Sturm-Liouville problem:

- ∂ 2 ∂z 2 N -2 ∂ 2 M q ∂z 2 = λ q M q , ∂M q ∂z z=0 = ∂M q ∂z z=-H = 0 (2)
with N(z) is the Brunt-Vaisala frequency that is assumed to be always positive. Zonal velocity u(x, z, t) and pressure deviation from the hydrostatic balance p(x, z, t) can be calculated using coefficients u q and p q as

u(x, z, t) = N -1 q=0 u q (x, t)M q (z), p(x, z, t) = ρg N -1 q=0 p q (x, t)M q (z). (3)
The influence of the numerical representation on the solution has been studied in [START_REF] Demange | Numerical representation of internal waves propagation[END_REF]. Two principal consequences of finite dimensional discretisation of the model ( 1),(2) consist in wrong eigenvalues λ q due to numerical solution of (2) and wrong approximation of the wave speed due to discrete horizontal grid.

In this paper we apply variational methods in order to correct these errors or reduce their influence on the solution. As usual, we define a set of parameters that are allowed to vary during the procedure of the minimization of the specially constructed cost function that is supposed to measure the quality of the numerical scheme. The cost function represent the weighted norm of difference between the model solutions on the high and low resolution grids.

Among these parameters we shall focus on the discretization in x coordinate of the system (1) and numerical solution of the equation (2) that is used for the spectral transformation in z coordinate. In particular, we start from the optimization of the vertical grid trying to bring the solution closer to the reference one, obtained on the high resolution grid.

We consider Charney-Philips vertical grid (fig. 1) introduced in [START_REF] Charney | Numerical integration of the quasigeostrophic equations for barotropic and simple baroclinic flows[END_REF] . . .

M 1 M 2 M 3 M nz M nz-1 M nz-2 dz 1 dz 2 dz 3 dz nz-1 dz nz-2 dz nz-3 dz w 1 dz w 2 dz w 3 dz w nz-1 dz w nz-2 dz w nz
N -2 1 dz 1 dz w 1 M 1 - N -2 1 dz 1 dz w 1 M 2 = λM 1 - N -2 k dz k-1 dz w k M k-1 + 1 dz w k N -2 k+1 dz k + N -2 k dz k-1 M k - N -2 k+1 dz k dz w k M k+1 = λM k ∀k : 2 ≤ k ≤ nz -1 (4)
-

N -2 nz-1 dz nz-1 dz w nz M nz-1 + N -2 nz-1 dz nz-1 dz w nz M nz = λM nz
The total depth of the ocean is divided by nz intervals of equal length dz w . We consider that eigenvectors M are defined at the middle points of each interval while their derivatives, that are related to the density and vertical velocity, are determined at the intervals boarders. Boundary conditions are prescribed at the boarders of the first and the last interval.

Numerical resolution of the system (4) approximates well eigenvectors but underestimates eigenvalues. In particular simplified case, when the Brunt-Vaisala frequency is constant, we can easily resolve the continuous problem (2) and get: [START_REF] Demange | Numerical representation of internal waves propagation[END_REF] shows that all eigen vectors are approximated with the precision of the floating point arithmetics, but largest eigenvalues of (4) have much lower values than corresponding theoretical ones. An example for the resolution nz = 51 is shown in fig. 2A (two lower lines corresponding to the constant N). One can see that the square root of the largest numerically calculated eigenvalue is 1.5 times lower than its theoretical estimates. So far, λ participates explicitly in the system (1) and determinates the wave velocity of the mode M q as c q = 1/ λ q , we can estimate the numerical error due to discretization in the vertical coordinate z committed in the solution of ( 4)

λ theor k = π(k -1) N H 2 , M theor k = cos π(k -1)z H Comparison avec numerical solution of
δc q = λ theor k λ num k (5)
In the more complex configuration, we consider a variable Brunt-Vaisala frequency N which is defined by the hyperbolic profile

N = N(z) = -2m/s z -200m
where

-4000m ≤ z ≤ 0. ( 6
)
This formula allows N to vary from 5 × 10 -4 s -1 at the first node near the bottom to 1 × 10 -2 s -1 at the last node near the surface. In this case, eigenvalues can not be obtained theoretically and approximations of exact values are calculated on the five times finer grid. First 51 of total 251 eigenvalues are considered as the reference ones. The comparison of the reference and numerically calculated eigenvalues on the coarse grid is also shown in fig. 2A (two upper lines corresponding to the variable N). As one can see in this plot, the variable N slightly increases the reference eigenvalues and completely modifies the error committed on the coarse grid. The error changes sign and the square root of the largest numerical eigenvalue becomes two times bigger than the reference one.

In addition to the error induced by the vertical discretization, the wave speed is also perturbed by finite resolution in x coordinate. If we substitute a wave u = e iπ(ωt+kx) into the equation (1), we get the dispersion relationship: ω/k = c = 1/ λ q . However, if we apply a numerical scheme in order to discretize this equation in x, we get another dispersion relationship. Thus, if we use leap-frog scheme for the discretization in time and some centraldifferences for derivatives in x

u n+1 i,q -u n-1 i,q 2τ = -g c 1 u p n i-2,q + c 2 u p n i-1,q + c 3 u p n i,q + c 4 u p n i+1,q h p n+1 i,q -p n-1 i,q 2τ = - 1 gλ q c 1 p u n i-1,q + c 2 p u n i,q + c 3 p u n i+1,q + c 4 p u n i+2,q h (7)
that realizes either second (with c = 0, -1, 1, 0) or fourth (with c = 1/24, -27/24, 27/24, -1/24) order approximation of the derivative, we get approximated dispersion relationships

ω 2 = arccos((4τ 2 cos(πkh) + λh 2 -4τ 2 )/(λh 2 )) 2πτ ω 4 = 1 2τ - arccos 54τ 2 cos(2πkh) -τ 2 cos(3πkh) -783τ 2 cos(πkh) -144λh 2 + 730τ 2 144λh 2 2πτ
Corresponding wave velocities c = ω/k are shown in fig. 2B for the resolution nx = nz = 51 assuming that the eigenvalue λ is equal to one. We see that both numerical approximations of the horizontal derivative underestimate the wave velocity. Consequently, we can state that numerically calculated eigenvalues λ q in the case N = const increase the velocity c q when q increases while the discretization in x results in a lower velocity with bigger k. This fact ensures the existence of appropriate k and q for which errors in wave velocity are compensated by each other.

However, the numerical wave velocity remains wrong for other combinations of k and q and the wave becomes either too slow for big k and small q, or too rapid for small k and big q. In the case of variable N, both numerical errors slow the wave speed down and numerical errors committed in x and z discretizations enhance each other.

In order to obtain an optimized wave speed for the particular composition of different waves determined by initial conditions, we shall solve a variational problem trying to minimize a cost function that measures the distance between a numerical solution and a presumably exact one.

We construct the cost function for the system (1) as

J = nt n=1 nz k=1 nx i=1 (u n i,k -ūn i,k ) 2 + (p n i,k -pn i,k ) 2 ( 8
)
where ū is supposed to be the reference model solution, obtained on a much finer grid. The sum is made in the physical space z rather than in over all vertical modes q. To search for a minimum of the cost function, we use its gradient with respect to control parameters. Tangent and adjoint models necessary to calculate the gradient, have been automatically generated by the Tapenade software presented in [START_REF] Hascoët | Tapenade 2.1 user's guide[END_REF]. This software analyses the source code of the model and produces the codes of the tangent model and of its adjoint. The minimization procedure used in all experiments below is based on the limited memory quasi-Newton method [START_REF] Gilbert | Some numerical experiments with variable storage quasi-newton algorithms[END_REF].

We perform either a separate or a joint control of the discretizations in both x and z directions. We consider the cost function [START_REF] Iakovlev | On the simulation of temperature and salinity fields in the arctic ocean[END_REF] that is supposed to depend on the set of parameters to be optimized. Among parameters in this set we try to use the following • grid steps in the vertical discretisation dz k , dz w k used in the Sturm-Liouville problem (4),

• Brunt-Vaisala frequency N k that is supposed to be known with a limited accuracy and allowed to vary within these limits,

• coefficients c 1 , c 2 , c 3 , c 4 in the approximation of the horizontal derivative in (7),

• joint optimization of all parameters above.

We do not intend to optimize gravity acceleration g assuming it is known exactly, nor parameters of time forwarding scheme.

The problem ( 7), ( 4) is considered in a square box of L = 500 km length and H = 4 km depth with periodical boundary conditions in x and Neumann conditions (see ( 2)) in z. The square is discretized with a regular low-resolution grid that counts nx = 102, nz = 51 nodes. Time step is equal to τ = 24 minutes.

Two kinds of starting points are considered: just two distinctive modes in x and in z u(x, z, 0) = 1.6 × 10 -5 cos(16πx/L) cos(2πz/H) p(x, z, 0) = 4.2 × 10 -2 sin(6πx/L) cos(7πz/H) [START_REF] Kazantsev | Optimal boundary discretisation by variational data assimilation[END_REF] or a more complex composition of long waves:

u(x, z, 0) = 8 Kx=1 12 Kz=1 4 × 10 -3 √ K z K x cos(2K x πx/L) cos(K z πz/H) p(x, z, 0) = 8 Kx=1 12 Kz=1 0.5 √ K z K x sin(2K x πx/L) cos(K z πz/H) (10)
The first kind of initial conditions (9) ensures two separate frequencies clearly visible on the energy evolution plot in fig. 3 As the most complex configuration we consider the same sum of multiple waves in the initial state accompanied by variable Brunt-Vaisala frequency N which is defined by a hyperbolic profile [START_REF] Gilbert | Some numerical experiments with variable storage quasi-newton algorithms[END_REF].

We don't introduce short waves in the solution supposing that a model with more realistic physics would dissipate these waves. The reference data ū, p for the cost function [START_REF] Iakovlev | On the simulation of temperature and salinity fields in the arctic ocean[END_REF] have been generated by the same model on the 5 times finer grid nx = 502, nz = 255. These dimensions of the fine grid ensure that for any node of the coarse grid there exists a node on the fine grid at the same position. This allows us to avoid interpolations and to calculate the cost function [START_REF] Iakovlev | On the simulation of temperature and salinity fields in the arctic ocean[END_REF] performing the sum over all nodes of the coarse grid taking ū, p at the corresponding nodes of the fine grid.

Assimilation window in chosen as 20 days, that corresponds approximately 4 periods of the wave, initiated by [START_REF] Kazantsev | Optimal boundary discretisation by variational data assimilation[END_REF]. In order to see the behavior of the solution beyond the assimilation window, the cost function value calculated over 200 days period is also examined. This is particularly important when the composition of waves prescribed in the initial conditions has a longer period of the resulting wave than the assimilation window.

Several assimilation experiments are performed for the model with these two kinds of initial points with different controlled parameters. Equally spaced x and z grids with either second or fourth order approximation of derivatives are considered as initial guess in all minimization experiments. The cost function [START_REF] Iakovlev | On the simulation of temperature and salinity fields in the arctic ocean[END_REF] One can see that in the conventional configuration, fourth order approximation provides a solution that is slightly closer to the reference high resolution one. The principal source of the error lies in the difference of the wave speeds, that's why the fourth order approximation gives better results on the 20 days interval: the second order scheme has already induced the significant phase difference between the reference and approximated solutions while the fourth order scheme succeeded to keep a small phase difference. However, on the long time interval the phase difference becomes bigger that gives comparable cost function values obtained with both schemes.

Optimization of the vertical grid (either grid steps or Brunt-Vaisala frequency) improves the solution in the simple case of evolution of two modes. The trajectory on the low-resolution grid becomes indistinguishable from the reference one due to exact compensation of errors in the two wave speeds of each mode. However, in a more complex case of multiple modes, partial compensation of numerical errors is only possible on a short time interval within the assimilation window. Out of the window, low resolution model with second order approximation of the spatial derivative and optimized vertical grid exhibits even bigger cost function value than on the conventional grid. Better results can be obtained optimizing vertical grid with fourth order derivative approximation. The cost function on the long interval is reduced, however, 8 times with respect to conventional grid.

Optimizing the coefficients of horizontal derivative discretization helps us to get the solution that exactly coincides with the reference one in the case of propagation of two waves. In the case when multiple waves exist in the initial conditions, the cost function is also reduced, but remains bigger than with optimized vertical grid.

Better result is obtained in the joint optimization of both vertical grid steps and derivative approximation. The cost function shows lowest values as within the assimilation window and beyond the window.

Similar situation is observed in the case with variable N. Optimization of the vertical grid provides lower cost function than optimization of the horizontal derivative. The lowest value is also obtained in the joint control of all parameters. The influence of the optimization can be viewed on the potential energy plot showed on fig. 5. We plot the evolution during the last 10 days (between 190th and 200th days) for three model runs: the reference trajectory of the high resolution model, low resolution model with conventional vertical grid and fourth order approximation of the horizontal derivatives and lowresolution model with optimized both vertical grid, Brunt-Vaisala frequency and derivatives.

In the case of the constant N, long waves approximated on the conventional grid are a little faster than the reference ones (see fig. 2B) due to numerical error in eigenvalues which is not completely compensated by horizontal derivative approximation. This results in the displacement to the left of peaks of energy. Contrary to that, variable N ensures smaller numerical wave velocity (eigenvalues λ are overestimated in this case) and the peak of energy are moved to the right with respect to the high resolution reference trajectory. One can see that for both constant and variable N the variational method has succeeded to compensate the error in the wave speed and all peaks of the potential energy coincide with the reference ones. Unfortunately, variational control does not allow to compensate the error in the peaks amplitude, that explains a non-null final cost function value in all experiments.

We should note, that separate optimization of the vertical grid or coefficients for the derivative in x does not allow to retrieve the reference wave speed. This is illustrated in fig. 6 where the potential energy for trajectories on the optimized vertical grid and optimized coefficients c m schemes in cases of the constant (A) and variable N are compared with the reference solution during the same time interval as in fig. 5. One can see, in the experiment with constant N (fig. 6A) even the optimized vertical grid shows the solution with a little bit faster waves than the reference while the waves produced by the model with optimized c m are too slow. In the experiment with variable N (fig. 6B), both optimizations result in lower wave speed with respect to the reference one.

To illustrate the the evolution of two-dimensional fields we plot the p(x, z, t = 200 days) obtained in the most complex case with variable N in fig. 7. Three fields are compared: obtained by the reference high resolution model fig. 7A, by the optimized model on the coarse grid (B) and by the model on low resolution conventional grid (C). One can see that spatial pattern of the optimized solution is quite close to the reference one, but the waves amplitude is not sufficient to fit the solution on the fine grid. Low resolution model without optimization exhibits both insufficient amplitude and wrong spatial pattern.

Optimized coefficients of approximation of the horizontal derivative a A B C

Figure 7: Final (t=200 days) p field in simulations with variable N: reference high resolution (A), low resolution with optimised parameters (B) and low resolution with conventional scheme (C).

shown in the table 2. One can see that only a little modification with respect to a conventional fourth order scheme is made by the optimization. All coefficients sets approximate the derivative multiplied by some coefficient rather than the derivative itself. Significant difference between optimized and conventional coefficients is observed in cases of variable N and two modes evolution.

To understand this phenomenon, we calculate the modifications of the wave speed imposed by the optimized schemes and plot them in fig. 8. Taking into account that the solution contains long waves only, we restrict the plotting area by low wave-numbers.

One can see the principal influence of the optimization on the resulting wave velocity. In the simplest case of two modes evolution, coefficients c have undergone the strongest modification. This fact can be explained by comparison of two figures: fig. 8 and fig. 2. The purpose of the optimization is to compensate the wave speed error committed by numerical approximation [START_REF] Demange | Numerical representation of internal waves propagation[END_REF]. Errors in just two eigenvalues λ 2 and λ 7 require a correction in this case. It becomes to be possible: modified approximation of the horizontal derivative reduces the wave speed at K x = 3 to compensate an excessive speed determined bu numerical λ 7 and keeps the wave speed at K x = 8 close to the exact speed because numerical λ 2 is also close to the analytical value. The consequence of that is the vanishing value of the cost function with optimized approximation of the horizontal derivative.

When multiple long waves are presented in the initial point and N is constant, optimization choose to slightly reduce the speed for all wave-numbers in order to compensate the error of numerically calculated eigenvalues. In the case of variable Brunt-Vaisala frequency N, the error in the wave velocity due to λ is more important for long waves (see fig. 2B), that's why the slowing the waves down by the optimized derivative in x is more important also. 9. One can see quite different modification of the grid steps. In the case of constant frequency N, multiple nodes of the vertical grid are displaced to several meters from their initial position. The nodes displacement reaches 14 meters near the bottom and near the surface of the ocean where the grid become coarser.

In the case of variable N, the grid steps are modified near the surface only. The first grid cell of the w grid (see fig. 1) double its initial length displacing the second node deeper in the ocean while the node of the M grid is moved toward the surface.

Control of the Brunt-Vaisala frequency gives a similar results. Resulting cost function values are quite close after separate optimizations as grid steps and N. Joint control of these two sets of parameters does not improve the results. It is obvious because the increasing of parameters number does not increase the number of controlled degrees of freedom. Optimizing one of them or another one we control the same coefficients in the problem (4) and get the same modification of the eigenvalue problem. Optimized frequency in experiments with simple control of N is not shown because it has the same pattern as optimized grid steps.

It should be noted that joint or separate control of vertical grid steps and the Brunt-Vaisala frequency results in a small modification of the eigenvalues themself. The control of parameters of the Sturm-Liouville problem (4) influences principally on the eigenvectors and especially on the second half of them corresponding to largest eigenvalues.

Conclusion

We have applied variational control methods to optimization of the vertical grid and numerical scheme of the simple bidimensional wave equation that simulates propagation of internal gravity waves in the ocean. Grid steps of the vertical grid, Brunt-Vaisala frequency and approximation of the horizontal derivative were used as controlled parameters either separately or in the joint control. Obtained results show that optimized parameters may partially compensate errors committed by numerical scheme due to insufficient grid resolution.

Optimal vertical grid steps and coefficients in horizontal derivative approximation found in the variational control procedure allow us to get the model solution that is rather close to the solution of the reference model. The error in the wave velocity on the coarse grid is mostly compensated in experiments with joint control of parameters while the error in the wave amplitude occurs to be more difficult to correct. Separate optimization of either horizontal or vertical parameters seems to be less efficient. The wave speed can be corrected in simple cases when few modes are present, but remains wrong in more complex cases with numerous modes in the solution or variable frequency N.

We can note also the importance of using of high order schemes for discretization of spatial derivatives in the optimization experiments. As it is shown in the table 1, if the horizontal derivative is approximated with the second order, optimization of the vertical grid leads to bigger cost function value on long time scales.

Optimization of the horizontal derivative gives us a numerical scheme that does not approximate a derivative. In general, some coefficient appears in front of the derivative to modify the wave velocity when it is necessary to compensate its excess due to underestimated eigenvalues of the Sturm-Liouville problem [START_REF] Danilov | A finite-element ocean model: principles and evaluation[END_REF].

Analyzing optimal grid steps of the vertical grid, we can see that they may be in a disagreement with requirements of other model physics. Thus, bigger grid steps near the bottom correspond in general to conventional grids used in ocean circulation models because deep ocean currents are slow laminar. However, coarser grid near the surface, obtained in the optimization procedure, is in evident contradiction with the necessity to finely resolve the surface boundary layer. Consequently, additional analysis of obtained optimized parameters from the point of view of they agreement with the model is necessary.

Figure 1 :

 1 Figure 1: Vertical grid and grid step notations.

AFigure 2 :

 2 Figure 2: Square roots of exact and caclulated eigenvalues (2) for cases with constant and variable Brunt-Vaisala frequency N (A) and the numerical error in the wave speed (B)

  . One of them, with period wave defined by initial conditions for p, the other one, with period L √ λ 2 8 × 86400 = 0.57 days, is determined by u(t = 0). The second kind represents a sum of waves which length exceeds L/8 in x and H/6 in z. These waves are weighted to ensure lower contribution of shorter waves. This initial point provides a solution with numerous waves moving with different wave velocities as shown in fig.4.

BFigure 3 :

 3 Figure 3: Two modes: evolution of energy from the starting point (9) (A) and the field of p(x, z, t = 20 days) (B)

Figure 5 :

 5 Figure 5: Potential energy of the reference model, conventional and alloptimized schemes in cases of the constant (A) and variable N (B).

Figure 6 :

 6 Figure 6: Potential energy of the reference model, optimized vertical grid and optimized coefficients c m schemes in cases of the constant (A) and variable N (B).

Figure 8 :

 8 Figure 8: Wave velocities defined by conventional and optimized derivative discretizations.

AFigure 9 :

 9 Figure 9: Optimized grid steps of the vertical grid in experiments with the constant (A) and variable N (B).

  value after 100 iterations is shown in the table 1.

				A
	Energy x 10 -3		Long Waves. Energy.
	21.00						Kinetic
	20.00						Potential
	19.00
	18.00
	17.00
	16.00
	15.00
	14.00
	13.00
	12.00
	11.00
	10.00
	9.00
	8.00
	7.00
	6.00
	5.00
	4.00
	3.00
	2.00
	1.00
	0.00
	-1.00	0.00	5.00	10.00	15.00	20.00	time
	Experiments with more iterations have been performed with no significant
	improvement of results.

B

Figure

4

: Set of long waves: evolution of energy from the starting point (10) (A) and the field of p(x, z, t = 20 days) (B)

192.00 194.00 196.00 198.00 200.00

		A			B
	Energy x 10 -3	Const.N Potential Energy.		Energy x 10 -3	Variab.N Potential Energy.
	18.00 19.00 20.00 21.00	Reference Conv.Ord.4 Optim.	22.00 23.00 24.00 25.00	Optimized Conv.Ord.4 Reference
	17.00			21.00
	16.00			20.00
	15.00			19.00
	14.00			18.00
	13.00			17.00
	12.00			16.00
	4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00			7.00 8.00 9.00 10.00 11.00 12.00 13.00 14.00 15.00
	3.00			6.00
	2.00			5.00
	1.00			4.00
	190.00 192.00 194.00 196.00 198.00 200.00	time	190.00	time
			10

190.00 192.00 194.00 196.00 198.00 200.00 B

		A
	Energy x 10 -3	Potential Energy.	Energy x 10 -3	Potential Energy.
	19.00 20.00	Reference Opt. dz	26.00 28.00	Opt dz Reference
	18.00	Opt.coef		Opt.coef
	17.00		24.00
	16.00		22.00
	15.00
	14.00		20.00
	12.00 13.00		18.00
	11.00		16.00
	9.00 10.00		14.00
	8.00		12.00
	6.00 7.00		10.00
	5.00		8.00
	3.00 4.00		6.00
	2.00		4.00
	1.00	time	2.00	time

190.00 192.00 194.00 196.00 198.00 200.00

Table 1 .

 1 Cost function J (8) with conventional and optimized schemes.

	Exp.	Order	Two Modes		Set of Long Waves
	Conventional Opt. dz, dz w Opt. N Conventional Opt. dz, dz w Opt. N Opt. c m	2 2 2 4 4 4 -	J (20) 1.85 × 10 -1 5.43 × 10 -5 6.10 × 10 -4 161.1 J (200) J (20) 19.4 560.2 4.45 × 10 -4 6.30 × 10 -3 164.0 16.7 54.86 1.89 × 10 -2 9.19 × 10 -7 9.31 × 10 -6 5.58 7.07 × 10 -5 1.58 × 10 -3 7.42 0 0 15.74	J (200) 98 847 114 287 115 591 42 099 4 447 5 606 11 752
	Opt. dz, dz w , N, c m	-	0	0	3.40	2 453
	Variable Brunt-Vaisala frequency N
	Conventional	4	-	-	478.4	70 754
	Opt. dz, dz w	4	-	-	23.1	29 759
	Opt. c m	-	-	-	376.9	60 172
	Opt. dz, dz w , N, c m	-	-	-	7.75	15 635

Table 2 .

 2 Optimized coefficients c m in[START_REF] Hascoët | Tapenade 2.1 user's guide[END_REF].

	Conventional fourth order approximation
	1.125 -1.125 -0.0417	0.0417
	Two Modes, Optimized c m
	c u 1.292 -1.292 -0.103	0.103
	c p 1.110 -1.110 -0.037	0.037
	Long waves, Const. N, Optimized c m
	c u 1.110 -1.110 -0.038	0.038
	c p 1.114 -1.114 -0.038	0.038
	Long waves, Const.N, Optimized all param.
	c u 1.102 -1.102 -0.038	0.038
	c p 1.110 -1.110 -0.036	0.036
	Long waves, Var. N, Optimized all param.
	c u 1.059 -1.059 -0.033	0.033
	c p 1.100 -1.100 -0.032	0.032
	Results of optimization of the vertical grid are shown in fig.


F Pajot

Y Atik

C Evesque

S Lefranc

B Leriche

J-P Torre

B Bélier

N Marsot

L Dumoulin

L Bergé

M Piat

E Bréelle

D Prêle

A Benoit

C Hoffmann

T Durand

P Camus

D Santos

Yong Jin

M Giard

Large bolometer arrays with superconducting NbSi sensors for future space experiments

New techniques in microelectronics allow to build large arrays of bolometers filling the focal plane of submillimeter and millimeter telescopes. The expected sensitivity increase is the key for the next generation of space experiments in this wavelength range. Superconducting bolometers offer currently the best prospects in terms of sensitivity and multiplexed readout. We present here the developments led in France based on NbSi alloy thermometers. The manufacturing process of a 23 pixel array and the test setup are described.

INTRODUCTION

The design of current space missions dedicated to submillimeter and millimeter broadband or low spectral resolution observations is based on direct detectors limited by the photon noise of the incoming radiation in a diffraction limited beam. Bolometers are the most sensitive detectors for this purpose. They are associated to feedhorns and arranged in groups covering a fraction only of the focal plane area. It is the case in the HFI/Planck and SPIRE/Herschel instruments.

The study of the inflation phase of the universe requires large improvements of sensitivity in the measurement of the Cosmic Microwave Background polarized emission. The ESA Cosmic Vision and NASA Beyond Einstein programs includes such missions.

An increase factor of 10 to 100 is required in sensitivity with respect to current missions. This cannot be obtained by increasing the integration time, already counted in years for a whole sky survey. The only solution is a full coverage of the focal plane by large contiguous detector arrays of 10 000 pixels or more.

We present here a french collaborative effort in the development of such arrays. The DCMB (Développement Concerté de Matrices de Bolomètres) R&D program is supported by CNES (Centre National d'Etudes Spatiales), the CNRS (Centre National de la Recherche Scientifique) and the participating Universities. The subsystems concerned by the R&D program are: the thermal architectures of the bolometers arrays, the thermometers, the coupling with the optical radiation and the readout electronics [START_REF] Ph | Low temperature NbSi thin film bolometers on Silicon Nitride membranes low temperature for bolometer applications[END_REF]. Two types of thermometers are being studied, based on Niobium Silicon alloys: high impedance (Anderson isolator) or superconducting alloys. We will develop here the superconducting case (Nb fraction larger than 0.13). Superconducting bolometers technology provides sensitive detectors (below 10 -17 WHz -1/2 ) and allows the use of a convenient multiplexed readout based on SQUIDs (2).

ARRAYS MANUFACTURING

The thermal architecture (figure 1) and thermometers are developped in common for both types of thermometers in the microelectronics facility IEF/MINERVE of Paris Sud-11 University at Orsay. Nb and NbSi are deposited in dedicated evaporators and co-evaporators of CSNSM/Orsay.

1.1

Thermometers The thermometric sensor (figure 2) is composed of a film of NbSi (Niobium Silicon) 100 nm thick. It is co-evaporated by irradiating two targets of Nb and Si simultaneously. The mixing ratio x of the Nb x Si x-1 thermometer is adjusted in order to obtain the goal transition temperature [START_REF] Ukibe | Fabrication of large NbSi bolometer arrays for CMB applications[END_REF]. In order to lower (below 1 Ω) the average resistance at the superconducting transition, an interleaved comb geometry is used.

Fig. 1(color online) 23 pixels array architecture

The superconducting thermometer is voltage biased at the normalsuperconducting transition. This transition is very steep, so the total power and consequently the temperature remains quasi-constant when the input radiation varies. This effect is the well know "strong negative electro-thermal feedback" (4) (5). This negative feedback also strongly decreases the time constant of the system. A typical NbSi thermometer (10 mm x 10 mm) transition curve was previously measured [START_REF] Lefranc | Superconducting NbSi thermometers for use in TES devices[END_REF]. The design is scaled down to 0.8 mm x 0.8 mm for the 23 pixels array.

Contact pads

A metal layer is deposited to make the electric interface between the tracks and the contacts. Gold offers the advantage of a compatibility with niobium connections and also a good aptitude for connections with the external readout system. A layer of gold (1000Å-1500Å) is thus deposited on the ends of the Nb tracks. Readout Multiplexed readout will be performed using a 4K SiGe ASIC associated with SQUIDs. Development of the ASIC is described by D. Prele et al. (this conférence). SQUIDs mux and amplifiers procurement is currently in discussion with Supracon (Jena).

1.5

Coupling with radiation For the first realisation, the radiation will be absorbed by a standard resistive layer or a grid adapted to the vacuum impedance. The long term goal is to couple to the incoming radiation by means of antenna [START_REF] Kuo | Antenna-coupled TES bolometers for CMB polarimetry[END_REF].

ARRAYS TEST AND CHARACTERISATION

NbSi alloy is a new material for TES design. It requires a full characterisation and validation before using it to produce large arrays. The uniformity of the superconducting transition temperature and slope is being checked on a 23 pixels array for which only the thermometers are built (figure 3). The test bench, based on a 300 mK mini-fridge system, is being developped at IAS. In parallel, a second bench is being set up to characterize the noise properties of a single NbSi superconducting thermometer. It is based on a commercial SQUID system from Star Cryoelectronics, and a 300 mK mini-fridge and a thermal stage regulated between 300 and 500 mK. Measurements are under way.

CONCLUSION

The process required to build the thermometer array have been set up. The testing phase can now begin, for a single pixel for noise measurement on one side, and for a 23-pixels array for thermometer homogeneity measurement on the other side. Once this phase is passed, we will be able to continue the integration of a complete array with optical coupling.

Fig. 2 :

 2 Fig. 2: (color online)NbSi superconducting thermometer. The leads comb structure is Nb, and the square NbSi.

1 . 3

 13 Microfabrication processThe steps of microfabrication of the superconducting bolometers are as follows:1. Deposition of membranes material by PECVD (SiO 2 + Si 3 N 4 : SiO2/SiN/SiO2 = 290/230/100 nm) 2. Nb x Si 1-x co-evaporation (x=15.55%, 1000 Å) 3. Nb evaporation (500 Å) 4. Au evaporation 5. Silicon deep etching 1.4

Fig. 3 :

 3 Fig. 3: (color online)23 pixels thermometers only array


Farzaneh Pakzad
email: farzaneh@aptira.com

Marius Portmann
email: marius@ieee.org

Thierry Turletti

Thierry Parmentelat

Mohamed Naoufal Mahfoudi

Walid Dabbous

R2Lab Testbed Evaluation for Wireless Mesh Network Experiments

We have provided critical evaluations of new potential testbeds for the evaluation of SDN-based WMNs. We evaluated the R2Lab wireless testbed platform at INRIA Sophia-Antipolis, France. This testbed has 37 customisable wireless devices in an anechoic chamber for reproducible research in wireless WiFi and 4G/5G networks. Our work presents the first initial evaluation of the testbed for wireless multi-hop experiments, using traditional WMN routing protocols. Our results demonstrate the potential for SDN experiments. We believe this is an important contribution in its own right, since experimental validation is a key research methodology in this context, and trust in the validity of experimental results is absolutely critical.

I. INTRODUCTION

R2Lab [START_REF]R2Lab testbed[END_REF] is a wireless testbed platform located at IN-RIA 1 , Sophia-Antipolis, France. This platform is part of the FIT federation, which provides an open large-scale, high performance testing infrastructure for performing experiments on systems and applications on wireless and sensor communications. R2Lab is an open testbed located in an anechoic chamber composed of 37 customisable wireless devices, along with USRP (Universal Software Radio Peripheral) [START_REF] Usrp | [END_REF] nodes and commercial LTE phones for accomplishing reproducible research in WiFi and 4G/5G cellular networks.

R2Lab is equipped with a range of software tools that allow the control of the wireless nodes remotely through a ssh gateway. Each user can reserve the whole testbed for his/her experiment and can take full control of all the wireless devices. The user can run his/her own customised Operating System (OS) on each node in order to perform experiments on customised systems. After loading the OS on a node, it is accessible via ssh with administrative privileges and ready for the user to configure the available resources, such as nodes, USRPs and phones.

The nodes are positioned in a grid layout, as illustrated in Figure 1. Each node is equipped with 3 wired interfaces used for remote power and reset management, control and a data channel dedicated to experimentation. This separation of 1 Institute for Research in Computer Science and Automation. The launch of the R2Lab platform was in November 2016 [START_REF]inaugural meeting R2Lab testbed[END_REF], and consequently there has been limited use and evaluation of the testbed. In particular, there have been no wireless multi-hop or WMN experiments conducted on the platform, and our work provides the first basic evaluation of the platform for this purpose. In this paper we present the results of our experiments, which consist of a comparison of two widely used WMN routing protocols, i.e. OLSR and BATMAN. Our work, which was conducted during a 5-week research visit at INRIA, validates the R2Lab testbed as a suitable platform for wireless multi-hop experiments, with a great potential for SDN-based WMN experiments.

The rest of the paper is organised as follows. In Section II we discuss key related works, i.e. wireless testbeds. Section III explains our basic experiments to evaluate the suitability of R2Lab for multi-hop wireless networks. Section IV discusses the more complex evaluation of R2Lab by considering Wireless Mesh Networks routing experiments. Finally, Section V provides a summary and conclusions for this work.

II. RELATED WORK -WIRELESS TESTBEDS

There exists a wide range of work on the design and evaluation of wireless network testbeds and testbed platforms. Here, we give a very brief summary of some of the key works, to provide the context of our own work.

ORBIT is an Open Access Research Testbed for Next-Generation Wireless Networks (ORBIT) [START_REF] Orbit | [END_REF], [START_REF] Raychaudhuri | Overview of the orbit radio grid testbed for evaluation of next-generation wireless network protocols[END_REF], founded in 2003 for conducting reproducible wireless experiments. The architecture of ORBIT is a two-tier system consisting of a laboratory-based wireless network emulator and a field trial network. This allows the experimenter to perform the basic experiments on the emulator, which addresses the problem of reproducibility, while at the same time providing the opportunity for the user to evaluate the performance of applications and protocols in real-world networks. The ORBIT lab emulator consists of a large number of static 802.11x wireless nodes laid out in a grid. This radio grid emulator provides facilities for the user to reproduce wireless network experiments with a specified topology for quantitative evaluation of different protocols and applications. The user can have full access and control to wireless nodes, such as installing their own OS and software packages, rebooting, etc. The user can move to the field trail network in order to support the validation of the results obtained on the emulator. In order to create a multihop network in ORBIT, it is suggested to use MAC address filtering or noise generation. The former cannot omit the contention and interference between multiple senders and the latter is limited in the topologies that can be achieved. ORBIT has a strong experiment control and management capabilities. One of its shortcomings is the lack of control of background noise due to the fact that wireless nodes are not placed in a anechoic chamber.

Emulab [START_REF] White | An integrated experimental environment for distributed systems and networks[END_REF] is another open access large scale platform for running experiments in computer networking and distributed systems. Emulab has a variety of features, including support for arbitrary network topologies, full control of nodes with arbitrary OS and configuration, and support for both WiFi as well as SDR experiments, using USRPs. Emulab also supports integration with other testbeds such as Planent LAB. As in the case of ORBIT, the Emulab wireless nodes are not isolated from other wireless networks, so the signal from other wireless networks can interfere with any Emulab experiments.

WHYNET (Wireless HYbrid NETwork) [START_REF] Zhou | Whynet: a hybrid testbed for large-scale, heterogeneous and adaptive wireless networks[END_REF] is another large-scale hybrid platform for implementing heterogeneous wireless technologies such as WiFi, cellular networks, sensors, etc. It is a hybrid wireless testbed combining hardware testbeds, simulation and emulation. This combination provides the ability to take advantage of the benefits of the different experiment types, i.e. the realism of physical testbeds with the scalability, flexibility and repeatability of simulation and emulation experiments.

In contrast to these above mentioned testbed platforms, R2Lab provides a wireless hardware testbed that can avoid background noise and interference of other nearby networks, via the use of a anechoic chamber, which provides RF isola- We will now present our validation of the R2Lab for evaluation of WMN routing protocols.

III. R2LAB EXPERIMENTS

For our experiments to evaluate the suitability of R2Lab for wireless multi-hop experiments, we considered two WMN routing protocols, namely Optimised Link State Routing (OLSR) [START_REF] Clausen | Optimized link state routing protocol (olsr)[END_REF] and Better Approach To Mobile Ad hoc Networking (BATMAN) [START_REF] Neumann | Better Approach To Mobile Ad-hoc Networking[END_REF].

• OLSR is a link state routing protocol where every node maintains topology information about the entire network. OLSR uses two types of control packets; HELLO and Topology Control (TC) messages. HELLO messages are used by a node to find its one-hop and two-hop neighbours. By discovering these neighbours, each node chooses a set of Multi Point Relays (MPR) based on its one-hop neighbours advising a best route to two-hops neighbours. MPRs aim to reduce the overhead of sending link state information in the whole network. Nodes regularly send TC packets with the information about their neighbourhood and the state of the links between them. MPR nodes forward the TC packets throughout the network. Considering the information in the TC packets, other nodes create a map of the network topology. In OLSR, nodes use a shortest path algorithm to calculate a path towards a destination. OLSR supports both the hop count as well as the ETX routing metric.

• BATMAN is a relatively new proactive routing protocol where each node only maintains information about the best next hop node for each destination, instead of the entire network topology as in the case of OLSR. Consequently, BATMAN reduces the amount of control traffic flowing in the network that can allow lower CPU usage and lower battery consumption of the mesh nodes. The BATMAN routing protocol operates as follows: -BATMAN maintains a table of the number of OGM messages received from each originator and via which one-hop neighbour the messages were received. The one-hop neighbour via which the largest number of OGM messages from a particular originator was received is considered as the best next hop for this destination.

We have evaluated these two WMN routing protocols in terms of Packet Delivery Ratio (PDR) and end-to-end latency in two different network scenarios, one without and one with the addition of interference.

As mentioned before, the R2Lab testbed consists of 37 nodes. Each node includes the state-of-the-art motherboard including an Intel Core i7-2600 processor, 4GB RAM, 240 GB SSD, and is equipped with 2 wireless interfaces, with Atheros 802.11 93xx a/b/g/n and/or Intel 5300 chips, and with 3 antennas each. The WiFi mode used in this experiment was 802.11a. Figure 1 shows the location of the nodes, which are distributed in a roughly 90m 2 area. The distance between nodes is about 1m in each direction, with some exceptions near the columns that are supporting the room. We installed Ubuntu Linux 16.04 on each of the nodes.

The first challenge that we had to address was the creation of multi-hop topology. Using the default settings, each node can see every other node, which results in only single-hop paths. By minimising the transmission power to 0dBm and setting the transmission rate to a fixed 54 Mbps, we were able to reduce the transmission range. We then ran ping between every node pair in the topology to find all the active (single-hop) links. We then chose a subset of 10 nodes, which maximises the multihop nature of the network. The resulting topology is shown in Figure 3. For example, for node 1 to reach node 37, packets need to traverse at least 3 hops, i.e. via the following path: 1 → 19 → 27 → 37. All our experiments have been conducted using this topology.

We also measured the achievable throughput for different source destination pairs using iperf. We ran iperf with an offered load that guarantees link saturation. We measured the throughput of the network paths between node 1 and every other node. The results of our measurements are shown in Figure 4. We see that for short distances (i.e. one-hop), as shown on the x-axis, the achieved throughput is between 18 and 20 Mbps, which is close to the maximum achievable throughput for 54 Mbps WiFi OFDM. Once the distance between the source and destination increases to two hops, the throughput decreases to roughly half (i.e. around 9 Mbps), due to the fact that radio interfaces operate in half-duplex mode and cannot receive and send data simultaneously. When the path length reaches three hops, the throughput decreases further, to a value of about 3.15 Mbps. This result reflects the expected behavior of wireless multi-hop networks [START_REF] Draves | Comparison of routing metrics for static multi-hop wireless networks[END_REF], [START_REF] Abolhasan | Real-world performance of current proactive multi-hop mesh protocols[END_REF], and provides a basic validation of the R2Lab testbed for the use of wireless multi-hop experiments.

In the following we will consider more complex validation experiments, and for this we will consider the WMN routing protocols BATMAN and OLSR, as mentioned earlier.

IV. WMN ROUTING EXPERIMENTS

In this section, we considered two basic experiment scenarios, a case with no interference and one with interference. We evaluated BATMAN and OLSR in terms of the end-to-end latency and packet delivery ratio for these two scenarios.

In our experiments, we used olsrd version 0.6.6.2 from olsr.org [START_REF]Optimized Link State Routing Protocol (OLSR), OLSR[END_REF] with the default configuration, with the Link Quality (LQ) extensions enabled. This means that the Expected Transmission Count (ETX) was used as the routing metric [START_REF] De Couto | A high-throughput path metric for multi-hop wireless routing[END_REF].

For BATMAN, we used batmand version 0.3.2-17 on each node from open-mesh.org [START_REF] Neumann | Better Approach To Mobile Ad-hoc Networking[END_REF]. As mentioned before, BATMAN

A. No Interference

As an initial experiment, we wanted to discover the paths that both BATMAN and OLSR establish in our network topology. For this, we ran each of the two protocols on all nodes (consecutively) and we used the traceroute command to establish all the routes for all source destination node pairs. As a simple example, we showed how BATMAN and OLSR choose routes between node 1 as a source and all other nodes as destination in the network. As a representative example, Figure 5 shows the established routes between node 1 an all other nodes in the network for OLSR (Figure 5(a)) and BATMAN (Figure 5(b)).

One hop paths are indicated with black dashed lines, blue lines indicate 2-hop paths and red lines 3-hop paths respectively. Since some paths overlap, we added labels for each path, which indicate the sequence of nodes, in order to increase clarity. For example, in Figure 5(a) if node 1 wants to reach node 37 using OLSR, packets will go through nodes 4 and 27. This has been shown with the label (1-4-27-37). In the case of BATMAN, as shown in Figure 5(b), the corresponding path is (1-5-33-37). It is clear from our experiments that OLSR and BATMAN can choose different routes in the exact same network, in some cases with significant differences in path length. This is not surprising, since both protocols vary significantly in regards to their approach to route establishment.

After having considered the basic route establishment of the two WMN routing protocols, we considered their performance in terms of latency and packet delivery ratio (PDR). In order to measure the end-to-end latency of the paths established by both OLSR and BATMAN, we used ping (5000 measurements) between every source destination node pair.

As before, we used paths with node 1 as the source as a representative example, in particular, we considered the following source destination pairs, which represent a 1-hop, 2-hop and 3-hop path respectively: [START_REF]R2Lab testbed[END_REF]19), [START_REF]R2Lab testbed[END_REF]27) and [START_REF]R2Lab testbed[END_REF]37). Figure 6 shows a box and whisker plot of measured RTT values for these 3 paths, both for OLSR and BATMAN. The plot shows the median, the maximum and minimum, as well as the four quartiles.

In Figure 6, the y-axis shows the RTT value of the routes in ms and the x-axis shows the source-destination pairs and the corresponding routing protocol. As expected, the RTT value increases with the path length. We see a roughly similar result for both OLSR and BATMAN for the three source destination pairs, despite the fact they chose different paths. This is as expected, since the path length is identical, and the link quality of the corresponding hops is also similar.

We also measured the Packet Delivery Ratio (PDR) for all the network paths. However, due to lack of mobility and significant levels of interference, we did not observe any packet loss, and all paths achieved a PDR value of 100%. To consider a more interesting scenario with packet loss, we looked at a scenario where interference is artificially generated by a node in the R2Lab testbed. We will explore this in the following section.

B. With Interference

As mentioned before, some nodes on the R2Lab platform are equipped with a USRP device. We used one of these devices to generate noise in the network. In particular, we used the USRP installed on node 11, and with the help of the "uhd siggen" Linux command we generated Gaussian random noise output with 70dB gain. This noise generating node is located between nodes 1 and 19 and above node 12, as shown in Figure 1. So we excepted these three nodes, and corresponding links, to be affected by the interface.

First, we were interested to see how the interference has an impact on the established routes between our three selected source destination node pairs, i.e. [START_REF]R2Lab testbed[END_REF]19), [START_REF]R2Lab testbed[END_REF]27) and [START_REF]R2Lab testbed[END_REF]37). The result is shown in Figure 7. If we consider the path between nodes 1 and 19 for example, we can see that the one-hop path in the scenario without interference has now been replaced with a 5-hop path, going via the following nodes: 1, 4, 5, 33, 27, 19, as shown in Figure 7.

We performed the same experiments to measure path RTT and PDR, as we have done in the case without interference. Figure 8 shows the RTT results, considering the paths from node 1 to nodes 19, 27 and 37 respectively, as in our previous scenario without interference. We see a significant increase in RTT values overall, with some values of well above 1000 ms. (Note that the y-axis (RTT) is in logarithmic scale here.) This is due to two main reasons. The first is the significant increase in path length, as discussed above. The second reason is the lower quality of wireless links, which results in a higher number of packet retransmission. We observe that both OLSR and BATMAN are similarly impacted by the increase in RTT due to interference. Again, this is as expected, since both protocols aim to avoid low quality paths, and therefore establish significantly longer paths.

We also measured the PDR on these paths, as in our previous scenario without interference. Figure 9 shows the results. The graph shows the average of 10 experiment runs, with the 95% confidence interval.

The x-axis shows the considered source-destination pair. The y-axis shows the PDR value in percent. In contrast to the scenario, where we did not have an interference, we now observe paths with less than 100% PDR. Overall, BATMAN and OLSR perform similarly, with the exception of source destination pair [START_REF]R2Lab testbed[END_REF]19), where the path chosen by OLSR achieves a PDR of below 40%, while BATMAN achieves a PDR of greater than 90%. This can be explained by the different approaches of considering link/path quality by the two protocols, i.e. OLSR uses link state routing with the ETX metric, while BATMAN does consider path quality based on the best one-hop neighbour. Overall, the achieved results are as expected, and hence validate the R2Lab testbed as a valid platform for wireless multi-hop experiments.

V. CONCLUSIONS

We have presented a first experimental validation of the R2Lab wireless testbed platform, in regards to wireless multi hop experiments. In particular, we have performed a basic evaluation of the OLSR and BATMAN WMN routing protocols, in terms of latency and PDR. We considered a scenario without interference, and a scenario where we injected interference, and we observed the impact on the route selection for both protocols. Overall, the obtained results give us great confidence about the suitability of the R2Lab testbed for experiments in Wireless Mesh Networking. Given the separation of the data plane and control plane in the R2Lab architecture, we believe it provides a great potential for experiments for SDN-based WMNs which remains to be explored in our future work.

Fig. 1 .

 1 Fig. 1. Ground Plan Layout of Nodes

Fig. 2 .

 2 Fig. 2. R2lab Located in an Anechoic Chamber

Fig. 3 .

 3 Fig. 3. Our Customised Mesh Topology

Fig. 4 .

 4 Fig. 4. Throughput vs. Distance over Different Number of Hops

Fig. 5 .

 5 Fig. 5. Established Route Between Node 1 as Source and Other Nodes as Destinations

Fig. 6 .

 6 Fig. 6. Distribution of RTT Values without Interference

Fig. 7 .

 7 Fig. 7. Established Route Between Node 1 and Other Nodes After Applying Interference

Figure 7 (

 7 a) shows the result of OLSR and the routes established by BATMAN are shown in Figure 7(b). The black dashed line represents one-hop paths, blue represents two-hop paths, red represents 3-hop paths, green represents 4-hop paths, and finally orange represents 5hop paths. We can see the injection of interference results in the establishment of longer paths, since both OLSR (with the ETX routing metric) and BATMAN avoid shorter but lower quality paths.

Fig. 8 .Fig. 9 .

 89 Fig. 8. Distribution of RTT Values with Interference

ACKNOWLEDGMENT

The authors would like to acknowledge the financial support through an "Australian Government Research Training Program Scholarship" and the University of Queensland Graduate School for the Graduate School International Travel Award (GSITA).


Mokhtaria Belkheira

Douniazad El

Jean-Marc Pons

Cyril Bressy
email: cyril.bressy@univ-amu.fr

Feature Syn thesis

Keywords: triazoles, organocatalysis, organoclick reaction, click reaction, chemoselectivity

A highly chemoselective bis-triazole synthesis based on a sequence organocatalyzed click reaction/copper-catalyzed click reaction is described in this paper. A range of bis-azides react with various ketones using proline catalysis through the aryl azide moiety while the alkyl azide one remains available for a metal-catalyzed triazole synthesis.

The discovery of new chemoselective transformations is a stimulating research area in organic chemistry. 1 These reactions present the precious advantage to avoid the masking of functional group while another one is transformed. The development of such transformations represents a crucial challenge in multi-step synthesis to privilege short synthetic sequence but could also be highly valuable for sequential cross-coupling, 2 multicatalytic reactions, 3 or in ligation in various reaction media. [START_REF] Sletten | For an excellent review on the concept of bio-orthogonal chemistry[END_REF] In this last field, the preparation of triazoles played a pioneer and pivotal role (Scheme 1). Since the independent reports by the groups of Meldal 5 and Sharpless 6 about the highly regioselective copper-catalyzed alkyne-azide Huisgen cycloaddition 7 (CuAAC) under very mild conditions, the synthesis of this aromatic nitrogenated heterocycle became an intensive field of research. Indeed this reaction abusively named 'click reaction' in reference to the Sharpless concept, 8 is used as a powerful tool in material sciences 9 as well as in chemical biology. 10 The regioselectivity could be reversed towards the synthesis of the 1,5-disubstituted cycloadduct using ruthenium-catalyzed 11 (RuAAC) or base-catalyzed 12 reaction. Strain-promoted cycloadditions (SPAAC) were elegantly used in a context of bioorthogonal ligation by the Bertozzi's group 13 based on an initial report by Wittig. 14 More recently organocatalyzed triazole synthesis, 15 based on the cycloaddition between enamine, 16 enolate, 17 or nitroolefin 18 with azide, emerged. In 2016, Dehaen's group developed an efficient three-component triazole synthesis between a ketone, a primary amine, and pnitrophenyl azide (Scheme 1). 19

Scheme 1 Main triazole syntheses

Thermal 1,3-Dipolar Cycloaddition (Huisgen) Copper-Catalyzed Alkyne-Azide Cycloaddition -CuAAC (Meldal, Sharpless)

Ruthenium-Catalyzed Alkyne-Azide Cycloaddition -RuAAC (Fokin)

Strain-Promoted Alkyne-Azide Cycloaddition -SPAAC (Wittig, Bertozzi) OrganoClick (Ramachary, Wang, Bressy)

Multi-components Triazole Synthesis

(Dehaen) R + N 3 R' N N N R R' + N N N R' R ∆ R + N 3 R' N N N R R' Cu cat. R + N 3 R' N N N R' R Ru cat. R + N 3 R' R N N N R' O R + N 3 R' amine cat. or DBU cat. R N N N R' O R + N 3 NO 2 + R'-NH 2 R N N N R' then H +
Few studies examined the chemoselectivity of bis-triazole 20 synthesis through alkyne-azide cycloaddition using either diyne or diazide partners (Scheme 2). Several strategies were adopted to use chemoselectively diynes by protection of one alkyne with bulky group 21 or by activation and/or deactivation of alkynes by neighboring effect. 22 Recently Wright and Couty developed a bis-triazole synthesis using an azide bearing a latent alkyne functionality. 23 Even fewer studies were conducted on the use of diazides. An efficient double CuAAC was performed, by Zhu's group, on dissymmetrical diazides featuring a chelating group. 24 In this paper, we describe the unprecedented chemoselective coupling between a ketone 1 and an alkyne 4 with a dissymmetrical diazide 2 using an organoclick and a click reaction (Scheme 3). In contrast to previous approaches, this strategy does not require protection/deprotection steps.

First, we determined the order of the sequence: organoclick, then click reactions, or the reverse. We planned to use diazide 2a bearing an aryl azide and a benzyl azide moieties to evaluate the chemoselectivity of each step (Scheme 4). The CuAAC approach, performed under Zhu's conditions 25 with 1 equivalent of propargyl alcohol (4a), exhibits a mod- This document was downloaded for personal use only. Unauthorized distribution is strictly prohibited.

Biographical Sketches

Mokhtaria

M. Belkheira et al.

Feature Syn thesis

erate chemoselectivity in favor of the benzyl azide moiety (73%) leading to the formation of triazole 7. Nevertheless significant amounts of triazole 6 were obtained through the cycloaddition involving the aryl azide function (24%). Notably, only 3% of bistriazole 8 was formed under these conditions. Using 5 equivalents of the same propargyl alcohol (4a) led to the formation of monotriazole 7 in 62% yield, but this was accompanied by a larger amount of bistriazole 8.

The organoclick reaction, previously developed by our group, 16c was tested with cyclohexanone (1a) and diazide 2a (Scheme 5). The proline-catalyzed reaction, run with di-azide 2a and 1 equivalent of cyclohexanone (1a), led exclusively to triazole 3aa in 56% yield. The presence of regioisomer 9 or bistriazole 10 was not detected even with larger amounts of ketone (5 equiv). An excess of ketone raised the yield of aryltriazole 3aa up to 71%. Scheme 5 Evaluation of the organoclick reaction chemoselectivity using cyclohexanone (1a) and diazide 2a

These preliminary results exhibited the complete chemoselectivity of the organoclick reaction in favor of aryl azides. As a consequence, the coupling sequence should have begun with the organoclick reaction and be followed by the click reaction. The scope of this sequence was explored using the diazides, ketones, and alkynes depicted in Figure 1.

The scope of the organoclick reaction between diazides and ketones was first examined (Scheme 6). Cyclohexanone (1a) was coupled with diazides 2a-e leading to the azidotriazoles 3aa-ae. The formation of azidoanilines 11a-e was observed during the course of the reaction resulting from a selective reduction of the aromatic azide function. The amount of this side-product (yields between parentheses) seems to increase when the two azide functions become closer (para < meta < ortho). Other cyclic and acyclic ke- tones 1b-d were successfully used leading in some cases, 1c and 1d, to dissymmetrical triazoles 3ca and 3da resulting from a complete regioselectivity. Complete chemoselectivity was also observed for the formation, with yields from 37% to 82%, of azidotriazoles 3 reported in Scheme 6.

3 N N N R N N N N N R CuAAC R' N N R' R 3 Si CuAAC N 3 R Deprotection CuAAC N 3 R' N N N R R 3 Si N N N R N N N R' O N H N 3 R CuAAC N 3 R' ∆ O N H N N N R O N H N N N R N N N R' From diazidoalkynes N N N R' N 3 CuAAC R N N N R N N N R Unmasking alkyne then CuAAC N 3 R' latent alkyne FG FG Scheme 3 Bis-triazole synthesis through organoclick-click sequence O R ( ) m Unactivated Ketones N 3 N 3 ( ) n Dissymmetrical Diazides N H CO 2 H cat. R ( ) m N N N N ( ) n N N R' Bis-triazoles R ( ) m N N N N 3 ( ) n

Azidotriazoles

Having these azidotriazoles resulting from the organoclick reaction in hand, we examined the click step of the sequence using copper acetate, as catalyst, in tert-butyl alcohol at room temperature. The bis-triazoles 5 obtained, in excellent yields, through this last step are represented in Scheme 7. The position of the alkyl azide moiety does not seem to influence the efficiency of this step.

Finally, a one-pot procedure, combining sequentially the two triazole syntheses without isolation of the azidotriazole intermediate 3ba was elaborated (Scheme 8). Following the organoclick reaction, dichloromethane was removed under vacuo and changed for t-BuOH before the ad- Scheme 6 Scope of organoclick reaction with diazides; yields in parentheses refer to the side-product 11 formed in the reaction

N H CO 2 H (20 mol%) CH 2 Cl 2 (1.5 mL) 80 °C, MW 1 h 2a-e (1 mmol) 1a-d (2 mmol) + + 3aa-xx N 3 N 3 ( ) n 11a-e H 2 N N 3 ( ) n O R ( ) m R ( ) m N N N N 3 ( ) n N N N N 3 N N N N 3 N N N N 3 N N N N 3 N N N N 3 Me 3ac 50% (40%) 3aa 63% (7%) 3ab 56% (14%) 3ad 62% (19%) 3ae 53% (7%) N N N N 3 N N N N 3 Me Me N N N N 3 N N N N 3 Me Me 3bc 56% (23%) 3ba 82% (13%) 3ca 61% (13%) 3da 37% (10%)

Scheme 7 Scope of bis-triazoles

+ 3aa-xx (0.2 mmol) R ( ) m N N N N 3 ( ) n R' R ( ) m N N N N ( ) n N N R' 5aaa-xxx 4a-e (0.3 mmol) Cu(OAc) 2 (5 mol%) t-BuOH, rt, 15 min-10 h N N N N R' = CH 2 OH R' = Ph R' = CH 2 NMeTs R' = (CH 2 ) 2 OH R' = CO 2 Me N N R' 5aaa 86% 5aab 98% 5aac 99% 5aad 95% 5aae 99% N N N N N N R' R' = Ph R' = (CH 2 ) 2 OH 5abb 98% 5abd 79% N N N 5aca 97% N N N OH N N N N N N Ph Me 5aeb 81% N N N 5adb 99% N N N Ph Me Me N N N N N N OH 5daa 94% N N N Me Me N N N R' 5caa 89% 5cac 78% R' = CH 2 OH R' = CH 2 NMeTs N N N N N N Ph 5bab 99% N N N N N N OH 5bca 89%
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M. Belkheira et al.

Feature Syn thesis

dition of copper acetate. Bis-triazole 5bab was formed in slightly lower yield (65%) than through the two-pot sequence (81% over 2 steps).

In conclusion, we have developed a new chemo-and regioselective synthesis of bis-triazoles based on an organoclick-click sequence. It allows the coupling of a ketone and an alkyne through a diazide linker. The organoclick step, combining the ketone and the diazide, furnishes a triazole by reaction with the aryl azide with complete regio-and chemoselectivities. The click step was revealed to be efficient and robust on a large variety of alkynes. In addition, we had shown the possibility to perform the one-pot synthesis by sequential addition of catalytic system and reagent.

Anhydrous CH 2 Cl 2 , Et 2 O, toluene, THF, and DMSO were obtained from the Solvent Purification System BRAUN MB-SPS800. All experiments using anhydrous solvents were performed under inert atmosphere of argon, using dried apparatus and employing standard techniques for handling air-sensitive materials. All reagents were obtained from commercial suppliers, unless otherwise stated. Flash chromatography was performed using Merck 40-63 μm particle-sized silica gel (230-400 mesh) and eluted with EtOAc in petroleum ether (PE, bp 40-60 °C) or with MeOH in EtOAc, unless otherwise specified. Analytical TLC was performed using 0.20 mm silica gel 60 plates. Visualization was achieved under a UVP mineralight UVGL-58 lamp, and by developing the plates with various stains (e.g., p-anisaldehyde, phosphomolybdic acid). Melting points were determined with a Büchi Melting-point B-450 apparatus and are not corrected. High-resolution mass spectra (HRMS) were obtained from a QStar Elite spectrometer (Applied Biosystems SCIEX) by electrospray ionization. 1 H NMR spectra were recorded at 300 MHz on a Bruker AC-300 and at 400 MHz on a Bruker AC-400 spectrometer using as internal standard the residual CHCl 3 signal for 1 H NMR (δ = 7.26). 13 C NMR spectra were similarly recorded at 75.47 MHz on a Bruker AC-300 and at 100.61 MHz on a Brucker AC-400 spectrometer, using a broadband decoupled mode with the multiplicities obtained using a JMOD or DEPT sequence and as internal standard the CDCl 3 solvent signal for 13 C NMR (δ = 77.0). Chemical shifts (δ) are reported in parts per million (ppm). Standard abbreviations are used for denoting the multiplicities. Coupling constants (J) are reported in hertz (Hz).

Diazides 2a, 16c , 2b, 26 and 2c 27 are known compounds. For the preparation of diazides 2d and 2e, see the Supporting Information.

Azidotriazoles 6 and 7, and Bistriazole 8

1-Azido-4-azidomethylbenzene (2a; 34.80 mg, 0.2 mmol), propargyl alcohol (4a; 11.2 mg, 0.2 mmol), and t-BuOH (0.50 mL) were mixed under argon. A 0.53 M aqueous solution (19 μL) of Cu(OAc) 2 was then added and the resulting mixture stirred for 4 days at r.t. The reaction mixture was then diluted with CH 2 Cl 2 (50 mL) before quenching with H 2 O (10 mL). The organic phase was dried (MgSO 4 ) and the solvent removed under vacuo. Flash chromatography (EtOAc/MeOH 20:1) of the crude product led to azidomethylphenyltriazole 6 (11.1 mg, 24%), azidophenylmethyltriazole 7 (33.7 mg, 73%), and bistriazole 8 (1.7 mg, 3%).

Azidomethylphenyltriazole 6

White solid; mp 59 °C; R f = 0.56 (EtOAc/PE 10:1). C), 122.9, 120.9, 120.1 (2 C), 54.9, 54.8, 51.9.

Azidotriazoles 3xy; General Procedure for the Organoclick Reaction

Ketone 1 (2 mmol) and diazide 2 (1 mmol) were dissolved in CH 2 Cl 2 (1.5 mL) in a sealed tube, followed by the addition of proline (20 mol%). The stirred mixture was then heated to 80 °C in a Micro-Wave reactor (CEM Discover -1-300 W oven). After 1 h, the solvent was removed under vacuo and the crude product submitted to flash chromatography.

1-[4-(Azidomethyl)phenyl]-4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]triazole (3aa)

Prepared from cyclohexanone (1a) and 1-azido-4-azidomethylbenzene (2a); yield: 160 mg (63%); brown solid; 63 °C; R f = 0.42 (EtO-Ac/PE 2:3). Prepared from 1a and 1-azido-3-azidomethylbenzene (2b); yield: 137 mg (56%); brown solid; mp 75 °C; R f = 0.30 (EtOAc/PE 2:3). 1 H NMR (400 MHz, CDCl 3 ): δ = 7.55 (d, J = 6.0 Hz, 1 H), 7.54 (t, J = 6.0 Hz, 1 H), 7.52 (t, J = 3.3 Hz, 1 H), 7.39 (d, J = 6.0 Hz, 1 H), 4.45 (s, 2 H), 2.85 (t, J = 5.7 Hz, 2 H), 2.76 (t, J = 5.9 Hz, 2 H), 1.89 (m, 4 H).

1-[2-(Azidomethyl)phenyl]-4,5,6,7-tetrahydro-1H-benzo-[d][1,2,3]triazole (3ac)

Prepared from 1a and 1-azido-2-azidomethylbenzene (2c); yield: 127 mg (50%); orange solid; mp 86 °C; R f = 0.40 (EtOAc/PE 1:4).

1-[4-(2-Azidoethyl)phenyl]-4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]triazole (3ad)

Prepared from 1a and 1-azido-4-(2-azidoethyl)benzene (2d); yield: 166 mg (62%); viscous yellow oil; R f = 0.36 (EtOAc/PE 2:3).

1-[4-(1-Azidoethyl)phenyl]-4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]triazole (3ae)

Prepared from 1a and 1-azido-4-(1-azidoethyl)benzene (2e); yield: 142 mg (53%); yellow solid; mp 73 °C; R f = 0.41 (EtOAc/PE 2:3). 1 H NMR (400 MHz, CDCl 3 ): δ = 7.58 (d, J = 8.5 Hz, 2 H), 7.48 (d, J = 8.5 Hz, 2 H), 4.70 (q, J = 6.8 Hz, 1 H), 2.84 (t, J = 6.1 Hz, 2 H), 2.75 (t, J = 6.1 Hz, 2 H), 1.88 (m, 4 H), 1.58 (d, J = 6.8 Hz, 3 H).

1-[4-Azidomethyl)phenyl]-1,4,5,6,7,8-hexahydrocyclohepta[d][1,2,3]triazole (3ba)

Prepared from cycloheptanone (1b) and 1-azido-4-azidomethylbenzene (2a); yield: 219 mg (82%); yellow solid; mp 48 °C; R f = 0.40 (EtO-Ac/PE 1:2).

1-[4-(Azidomethyl)phenyl]-5-ethyl-4-methyl-1H-[d][1,2,3]triazole (3da)

Prepared from pentan-3-one (1d) and 1-azido-4-azidomethylbenzene (2a); yield: 90 mg (37%); yellow oil; R f = 0.26 (EtOAc/PE 1:2). 1 H NMR (300 MHz, CDCl 3 ): δ = 7.46 (m, 4 H), 4.45 (s, 2 H), 2.67 (q, J = 7.6 Hz, 2 H), 2.37 (s, 3 H), 1.07 (t, J = 7.6 Hz, 3 H).

Bistriazoles 5xyz; General Procedure

Azidotriazole 3 (0.2 mmol) and alkyne derivative 4 (0.3 mmol) were dissolved under argon at r.t. in t-BuOH (0.5 mL) and then an aq 0.53 M Cu(OAc) 2 solution (19 μL, 5 mol%) was added. The reaction mixture was then stirred at the same temperature (from 15 min to 10 h) until complete disappearance of the starting triazole 3 (reaction followed by TLC). The mixture was then diluted with CH 2 Cl 2 (50 mL) and the organic layer was washed with H 2 O (10 mL). The organic layer was dried (MgSO 4 ) and the volatiles were removed under vacuo. Bis-triazole 5 was then purified by flash chromatography on silica gel.

1-{[4-(4,5,6,7-Tetrahydro-1H-benzo[d][1,2,3]triazol-1-yl)benzyl]-1H-1,2,3-triazol-4-yl}methanol (5aaa)

Prepared from azidotriazole 3aa and propargyl alcohol (4a); yield: 53 mg (86%); yellow solid; mp 179 °C; R f = 0.12 (EtOAc/MeOH 20:1). Prepared from azidotriazole 3aa and ethynylbenzene (4b); yield: 70 mg (98%); white solid; mp 158 °C ; R f = 0.39 (EtOAc/PE 3:2). 1 H NMR (400 MHz, CDCl 3 ): δ = 7.82 (d, J = 7.5 Hz, 2 H), 7 .73 (s, 1 H), 7.59 (d, J = 8.3 Hz, 2 H), 7.45 (d, J = 8.3 Hz, 2 H), 7.42 (t, J = 7.5 Hz, 2 H), 7.32 (t, J = 7.5 Hz, 1 H), 5.65 (s, 2 H), 2.82 (t, J = 6.0 Hz, 2 H), 2.72 (t, J = 6.1 Hz, 2 H), 1.86 (m, 4 H).

N,4-Dimethyl-N-({1-[4-(4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]triazol-1-yl)benzyl]-1H-1,2,3-triazol-4-yl}methyl)benzenesulfonamide (5aac)

Prepared from azidotriazole 3aa and N-tosyl-N-methylpropargylamine (4c); yield: 94 mg (99%); white solid; mp 153 °C; R f = 0.30 (EtO-Ac/PE 3:1). 1 H NMR (300 MHz, CDCl 3 ): δ = 7.68 (d, J = 8.0 Hz, 2 H), 7.60 (d, J = 8.5 Hz, 2 H), 7.59 (s, 1 H), 7.42 (d, J = 8.5 Hz, 2 H), 7.32 (d, J = 8.0 Hz, 2 H), 5.58 (s, 2 H), 4.31 (s, 2 H), 2.82 (t, J = 4.5 Hz, 2 H), 2.75 (t, J = 6.0 Hz, 2 H), 2.71 (s, 3 H), 2.44 (s, 3 H), 1.88 (m, 4 H).

2-{1-[4-(4,5,6,7-Tetrahydro-1H-benzo[d][1,2,3]triazol-1-yl)benzyl]-1H-1,2,3-triazol-4-yl}ethanol (5aad)

Prepared from azidotriazole 3aa and but-3-ynol (4d); yield: 62 mg (95%); yellow solid; mp 114 °C; R f = 0.16 (EtOAc/MeOH 20:1). 1 H NMR (300 MHz, CDCl 3 ): δ = 7.57 (d, J = 8.5 Hz, 2 H), 7.40 (d, J = 8.5 Hz, 2 H), 7.39 (s, 1 H), 5.58 (s, 2 H), 3.95 (t, J = 5.8 Hz, 2 H), 2.95 (t, J = 5.8 Hz, 2 H), 2.83 (t, J = 4.7 Hz, 2 H), 2.73 (t, J = 4.7 Hz, 2 H), 1.87 (m, 4 H), 1.85 (br s, 1 H, OH).

1-{4-[1-(4-Phenyl-1H-1,2,3-triazol-1-yl)ethyl]phenyl}-4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]-triazole (5aeb)

Prepared from azidotriazole 3ae and ethynylbenzene (4b); yield: 60 mg (81%); white solid; mp 133 °C; R f = 0.30 (EtOAc/PE 3:2). Prepared from azidotriazole 3ba and ethynylbenzene (4b); yield: 73 mg (99%); white solid; mp 179 °C; R f = 0.29 (EtOAc/PE 3:2).

N-({1-[4-(6,6-Dimethyl-4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]triazol-1-yl)benzyl]-1H-1,2,3-triazol-4-yl}methyl)-N,4-dimethylbenzenesulfonamide (5cac)

Prepared from azidotriazole 3ca and N-tosyl-N-methylpropargylamine (4c); yield: 79 mg (78%); yellow solid; mp 154 °C; R f = 0.20 (EtOAc/PE 3:2). Prepared from azidotriazole 3da and propargyl alcohol (4a); yield: 56 mg (94%); white solid; mp 132 °C; R f = 0.18 (EtOAc/MeOH 20:1). 1 H NMR (300 MHz, CDCl 3 ): δ = 7.56 (s, 1 H), 7.41 (s, 4 H), 5.60 (s, 2 H), 4.79 (s, 2 H), 2.67 (q, J = 7.6 Hz, 2 H), 2.35 (s, 3 H), 1.85 (br s, 1 H, OH), 1.06 (t, J = 7.6 Hz, 3 H).

One-Pot Preparation of Bistriazole 5bab

Cycloheptanone (1b; 224 mg, 2 mmol) and 1-azido-4-azidomethylbenzene (2a; 174 mg, 1 mmol) were dissolved in CH 2 Cl 2 (1.5 mL) in a sealed tube, followed by the addition of proline (23 mg, 20 mol%). The stirred mixture was then heated to 80 °C in a Micro-Wave reactor (CEM Discover -1-300 W oven). After 1 h, the solvent was removed under vacuo. The crude azidotriazole 3ba formed was mixed with ethynylbenzene (4b; 153 mg, 1.5 mmol) in t-BuOH (2.5 mL) under argon at r.t., followed by the addition of an aq 0.53 M Cu(OAc) 2 solution (95 μL, 5 mol%). The reaction mixture was then stirred at the same temperature for 7 h until complete disappearance of the starting triazole 3ba (reaction followed by TLC). The mixture was then diluted with CH 2 Cl 2 (150 mL) and the organic layer was washed with H 2 O (20 mL). The organic layer was dried (MgSO 4 ) and the volatiles were removed under vacuo. Bis-triazole 5bab (239 mg, 65%) was obtained pure by flash chromatography on silica gel.

  Belkheira studied at the Université d'Oran, Es-Sénia in Algeria where she obtained her Magister's degree in 2003 under the supervision of Prof. Douniazad El Abed. In 2008, she joined as doctoral student in the group of Prof. Cyril Bressy and Prof. Jean-Marc Pons in Marseille, France where she developed a new organocatalyzed triazole synthesis. She obtained her Ph.D. in 2012 and was nominated Maître de Conférences at the Université Tahri-Mohamed in Béchar-Algeria. Douniazad El Abed studied at the Université d'Oran Es-Sénia where she obtained her DEA under the supervision of Prof. Fernand Texier. She then entered as an Assistant at the Université d'Oran Es-Sénia and defended a Thèse de Doctorat de 3 ème Cycle in 1977. In 1982, she joined the group of Prof. Maurice Santelli in the Université d'Aix-Marseille and defended a Thèse d'Etat in 1986. Back in Algeria, she worked on the extraction of natural products and synthesis of heterocycles (triazoles and triazolines). Since 1989 she holds a position of Professor at Uni-versité d'Oran Es-Sénia. Her research focuses on synthesis of nitrogenous bioactive molecules obtained under environmentally-friendly conditions that comply with the principles of green chemistry by synthesis or extraction. Jean-Marc Pons studied at the Université de Provence in Marseille where he obtained his Ph.D. in 1982 under the supervision of Prof. Maurice Santelli. He then entered the CNRS as Chargé de Recherches and de-fended a Thèse d'Etat in 1987 on low valent transition metal complexes in organic synthesis. In 1988, he spent a year as a post-doctoral fellow in the group of Prof. Philip Kocienski at Southampton, where he worked on the total synthesis of natural products. He was dean of the faculty of sciences of Aix-Marseille Université from 2008 to 2017. He is currently involved in organocatalyzed transformations. Cyril Bressy studied at the Université Claude Bernard in Lyon where he obtained his Ph.D. in 2004 under the supervision of Prof. Olivier Piva. He then joined as a post-doctoral researcher in the group of Prof. Mark Lautens in Toronto, Canada where he developed a novel variant of the Catellani reaction. Back in France, he worked on a total synthesis project in Paris at Ecole Supérieure de Physique et Chimie Industrielle (ESPCI-Paris-Tech) with Prof. Janine Cossy. From 2006 he held a position of Maître de Conférences at Aix-Marseille Université (AMU). He obtained in 2012 a Habilitation à Diriger les Recherches (HDR) and was promoted as full professor in 2015 at AMU. His research focuses on total synthesis using desymmetrization strategies and organocatalyzed transformations.

Scheme 2

 2 Scheme 2 Chemoselective strategies to obtain bis-triazoles

Scheme 4

 4 Scheme 4 Evaluation of the click reaction chemoselectivity using propargyl alcohol (4a) and diazide 2a

Figure 1

 1 Figure 1 Partners of the organoclick-click sequence

1 H 7 1 H 13 C 8 1 H

 1711381 NMR (400 MHz, DMSO-d 6 ): δ = 8.69 (s, 1 H), 7.94 (d, J = 8.5 Hz, 2 H), 7.59 (d, J = 8.5 Hz, 2 H), 5.33 (t, J = 5.5 Hz, 1 H, OH), 4.61 (d, J = 5.5 Hz, 2 H), 4.55 (s, 2 H). 13 C NMR (100 MHz, DMSO-d 6 ): δ = 149.1, 136.3, 136.0, 129.8 (2 C), 120.9, 120.1 (2 C), 54.9, 52.8. HRMS: m/z [M + H] + calcd for C 10 H 10 N 6 O: 231.0989; found: 231.0997. Azidophenylmethyltriazole Yellow solid; mp 64 °C; R f = 0.45 (EtOAc/MeOH 20:1). NMR (400 MHz, DMSO-d 6 ): δ = 7.98 (s, 1 H), 7.37 (d, J = 8.5 Hz, 2 H), 7.12 (d, J = 8.5 Hz, 2 H), 5.55 (s, 2 H), 5.17 (t, J = 5.5 Hz, 1 H, OH), 4.49 (d, J = 5.5 Hz, 2 H). NMR (100 MHz, DMSO-d 6 ): δ = 148.3, 139.2, 133.0, 129.7 (2 C), 122.6, 119.3 (2 C), 54.9, 51.9. Bistriazole White solid; mp 83 °C; R f = 0.10 (EtOAc/MeOH 20:1). NMR (400 MHz, DMSO-d 6 ): δ = 8.67 (s, 1 H), 8.09 (s, 1 H), 7.91 (d, J = 8.5 Hz, 2 H), 7.52 (d, J = 8.5 Hz, 2 H), 5.66 (s, 2 H), 5.32 (br s, 1 H, OH), 5.19 (br s, 1 H, OH), 4.60 (s, 2 H), 4.51 (s, 2 H). 13 C NMR (100 MHz, DMSO-d 6 ): δ = 149.1, 148.3, 136.4, 136.3, 129.3 (2

1 H

 1 NMR (300 MHz, CDCl 3 ): δ = 7.60 (d, J = 8.6 Hz, 2 H), 7.47 (d, J = 8.6 Hz, 2 H), 4.44 (s, 2 H), 2.85 (t, J = 6.2 Hz, 2 H), 2.75 (t, J = 6.1 Hz, 2 H), 1.88 (m, 4 H).

13 CScheme 8 4b

 138 Scheme 8 One-pot three-component reaction

13 C

 13 NMR (100 MHz, CDCl 3 ): δ = 144.1, 137.2, 137.3, 132.0, 130.0, 128.0, 122.6, 122.5, 54.1, 22.7, 22.4, 21.9, 21.8. HRMS: m/z [M + H] + calcd for C 13 H 14 N 6 : 255.1353; found: 255.1356.

13 C

 13 NMR (100 MHz, CDCl 3 ): δ = 144.1, 141.5, 136.6, 132.0, 127.5 (2 C), 123.3 (2 C), 60.4, 22.7, 22.4, 21.9, 21.8, 21.6. HRMS: m/z [M + H] + calcd for C 14 H 16 N 6 : 269.1509; found: 269.1509.

13 C

 13 NMR (75 MHz, CDCl 3 ): δ = 143.0, 136.7, 136.0, 132.0, 129.1 (2 C), 123.4 (2 C), 54.1, 35.5, 35.4, 19.1, 31.2, 27.7 (2 C). HRMS: m/z [M + H] + calcd for C 15 H 18 N 6 : 283.1666; found: 283.1666.

13 C

 13 NMR (75 MHz, CDCl 3 ): δ = 140.6, 136 .9, 136.7, 135.3, 129.0 (2 C), 125.6 (2 C), 54.0, 16.2, 13.1, 10.4. HRMS: m/z [M + H] + calcd for C 12 H 14 N 6 : 243.1353; found: 243.1343.

1 H

 1 NMR (300 MHz, DMSO-d 6 ): δ = 8.09 (s, 1 H), 7.65 (d, J = 8.7 Hz, 2 H), 7.51 (d, J = 8.7 Hz, 2 H), 5.68 (s, 2 H), 4.52 (d, J = 5.4 Hz, 2 H), 4.05 (br s, 1 H, OH), 2.73 (t, J = 5.5 Hz, 2 H), 2.69 (t, J = 5.5 Hz, 2 H), 1.78 (m, 4 H).

13 C

 13 NMR (75 MHz, DMSO-d 6 ): δ = 148.4, 143.0, 136 .7, 136.0, 132.3, 129.1 (2 C), 123.2 (2 C), 123.0, 55.0, 52.0, 22.1, 21.9, 21.3, 20.7. HRMS: m/z [M + H] + calcd for C 16 H 18 N 6 O: 311.1615; found: 311.1610.This document was downloaded for personal use only. Unauthorized distribution is strictly prohibited.M. Belkheira et al.

13 C

 13 NMR (100 MHz, CDCl 3 ): δ = 148.4, 144.2, 137.1, 135.2, 132.0, 130.3, 129.0 (2 C), 128.8 (2 C), 128.3, 125.7 (2 C), 123.5 (2 C), 119.6, 53.4, 22.6, 22.3, 21.9, 21.8. HRMS: m/z [M + H] + calcd for C 21 H 20 N 6 : 357.1822; found: 357.1812.

13 C

 13 NMR (75 MHz, CDCl 3 ): δ = 144.21, 144.20, 137.2, 134.8, 134.1, 132.0, 129.8 (2 C), 129.1 (2 C), 127.4 (2 C), 123.5 (2 C), 123.0, 53.5, 45.6, 34.9, 22.7, 22.3, 21.9 (2 C), 21.5. HRMS: m/z [M + H] + calcd for C 24 H 27 N 7 O 2 S: 478.2020; found: 478.2010.

13 C

 13 NMR (75 MHz, CDCl 3 ): δ = 144.3, 144.2, 137.1, 135.3, 132.0, 129.0 (2 C), 123.5 (2 C), 121.7, 61.4, 53.3, 28.7, 22.6, 22.3, 21.9, 21.8. HRMS: m/z [M + H] + calcd for C 17 H 20 N 6 O: 325.1771; found: 325.1773. Prepared from azidotriazole 3aa and methyl propynoate (4e), yield: 67 mg (99%); white solid; mp 147 °C; R f = 0.23 (EtOAc/PE 4:1).

  Prepared from azidotriazole 3ab and ethynylbenzene (4b); yield: 70 mg (98%); white solid; mp 125 °C; R f = 0.34 (EtOAc/PE 2:1).

  Prepared from azidotriazole 3ab and but-3-ynol (4d); yield: 51 mg (79%); white solid; mp 138 °C; R f = 0.14 (EtOAc/MeOH 20:1).

  Prepared from azidotriazole 3ac and propargyl alcohol (4a); yield: 60 mg (97%); white solid; mp 132 °C; R f = 0.20 (EtOAc/MeOH 20:1).

  Prepared from azidotriazole 3ad and ethynylbenzene (4b); yield: 73 mg (99%); yellow solid; mp 204 °C; R f = 0.25 (EtOAc/PE 5:2).

1 H

 1 NMR (400 MHz, CDCl 3 ): δ = 7.82 (dd, J = 7.5, 1.4 Hz, 2 H), 7.72 (s, 1 H), 7.57 (d, J = 8.5 Hz, 2 H), 7.46 (d, J = 8.5 Hz, 2 H), 7.41 (t, J = 7.5 Hz, 2 H), 7.32 (tt, J = 7.5, 1.4 Hz, 1 H), 5.94 (q, J = 7.1 Hz, 1 H), 2.83 (t, J = 5.3 Hz, 2 H), 2.72 (t, J = 4.3 Hz, 2 H), 2.08 (d, J = 7.1 Hz, 3 H), 1.87 (m, 4 H).

13 C

 13 NMR (100 MHz, CDCl 3 ): δ = 148.0, 144.2, 140.5, 137.0, 132.0, 130.4, 128.8 (2 C), 128.2, 127.6 (2 C), 125.7 (2 C), 123.5 (2 C), 118.3, 59.7, 22.7, 22.4, 21.9, 21.8, 21.3. HRMS: m/z [M + H] + calcd for C 22 H 22 N 6 : 371.1979; found: 371.1969. M. Belkheira et al.

1 H

 1 NMR (300 MHz, CDCl 3 ): δ = 7.68 (d, J = 8.3 Hz, 2 H), 7.61 (s, 1 H), 7.58 (d, J = 8.0 Hz, 2 H), 7.42 (d, J = 8.3 Hz, 2 H), 7.33 (d, J = 8.0 Hz, 2 H), 5.58 (s, 2 H), 4.31 (s, 2 H), 2.84 (t, J = 6.0 Hz, 2 H), 2.72 (s, 3 H), 2.53 (s, 2 H), 2.44 (s, 3 H), 1.67 (t, J = 6.0 Hz, 2 H), 1.03 (s, 6 H). 13 C NMR (75 MHz, CDCl 3 ): δ = 144.4, 144.3, 143.7, 137.3, 137.2, 134.8, 134.2, 129.8 (2 C), 129.1 (2 C), 127.4 (2 C), 123.6 (2 C), 123.0, 53.6, 45.6, 35.5 (2 C), 35.0, 31.2, 27.7 (2 C), 21.5, 19.1.1-[2-(5,6,7,8-Tetrahydrocyclohepta[d][1,2,3]triazol-1(4H)-yl)benzyl]-1H-1,2,3-triazol-4-yl)methanol (5bca)Prepared from azidotriazole 3bc and propargyl alcohol (4a); yield: 58 mg (89%); white solid; mp 143 °C; R f = 0.26 (EtOAc/MeOH 20:1). 1 H NMR (300 MHz, CDCl 3 ): δ = 7.53-7.46 (m, 3 H), 7.21-7.19 (m, 1 H), 5.36 (s, 2 H), 4.72 (s, 2 H), 2.95 (t, J = 5.7 Hz, 2 H), 2.70 (br s, 1 H, OH), 2.42 (t, J = 5.7 Hz, 2 H), 1.83-1.60 (m, 6 H).

13 C

 13 NMR (75 MHz, CDCl 3 ): δ = 147.9, 147.3, 137.0, 134.6, 132.3, 130.8, 130.6, 129.6, 127.5, 122.4, 56.2, 49.6, 30.7, 27.0 (2 C), 26.7, 24.2.

13 C

 13 NMR (75 MHz, CDCl 3 ): δ = 148.5, 140.8, 137.1, 136.0, 135.3, 129.0 (2 C), 125.8 (2 C), 121.8, 56.4, 53.4, 16.2, 13.0, 10.4. HRMS: m/z [M + H] + calcd for C 15 H 18 N 6 O: 299.1615; found: 299.1617.

(2-Azidomethyl)phenyl]-1,4,5,6,7,8-hexahydrocyclohep- ta[d][1,2,3]triazole (3bc)

  calcd for C 14 H 16 N 6 : 269.1509; found: 269.1505.

	1-[Prepared from 1b and 1-azido-2-azidomethylbenzene (2c); yield: 150
	mg (56%); yellow solid; mp 77 °C; R f = 0.29 (EtOAc/PE 1:3).
	1 H NMR (300 MHz, CDCl 3 ): δ = 7.56 (m, 2 H), 7.49 (m, 1 H), 7.23 (d, J =
	7.6 Hz, 1 H), 4.21 (s, 2 H), 2.98 (t, J = 5.6 Hz, 2 H), 2.54 (t, J = 5.6 Hz, 2
	H), 1.78 (m, 6 H).
	13 C NMR (75 MHz, CDCl 3 ): δ = 147.0, 137.0, 135.0, 133.3, 130.4, 130.2,
	129.1, 127.7, 50.4, 30.9, 27.2 (2C), 26.9, 24.4.
	HRMS: m/z [M + H] + calcd for C 13 H 14 N 6 : 269.1509; found: 269.1512.
	1-[4-
	1 H NMR (300 MHz, CDCl 3 ): δ = 7.48 (d, J = 8.5 Hz, 2 H), 7.43 (d, J = 8.5
	Hz, 2 H), 4.44 (s, 2 H), 2.96 (t, J = 5.7 Hz, 2 H), 2.74 (t, J = 5.6 Hz, 2 H),
	1.76 (m, 6 H).
	13 C NMR (75 MHz, CDCl 3 ): δ = 147.5, 136.7, 136 .4, 135.6, 129.0 (2 C),
	125.7 (2 C), 54.0, 30.8, 27.2 (2 C), 27.0, 24.8.

HRMS: m/z [M + H] +

(Azidomethyl)phenyl]-6,6-dimethyl-4,5,6,7-tetrahydro-1H- benzo[d][1,2,3]triazole (3ca)

	Prepared from 3,3-dimethylcyclohexanone (1c) and 1-azido-4-azi-
	domethylbenzene (2a); yield: 172 mg (61%); yellow solid: mp 87 °C;
	R f = 0.32 (EtOAc/PE 2:3).
	1 H NMR (300 MHz, CDCl 3 ): δ = 7.56 (d, J = 8.3 Hz, 2 H), 7 .47 (d, J = 8.3
	Hz, 2 H), 4.43 (s, 2 H), 2.83 (t, J = 6.4 Hz, 2 H), 2.51 (s, 2 H), 1.66 (t, J =
	6.4 Hz, 2 H), 1 .03 (s, 6 H).

  : m/z [M + H] + calcd for C 17 H 20 N 6 O: 325.1771; found: 325.1772.

	1 H NMR (400 MHz, DMSO-d 6 ): δ = 7.99 (s, 1 H), 7.59 (m, 3 H), 7.42 (m,
	1 H), 5.67 (s, 2 H), 4.68 (t, J = 5.3 Hz, 1 H, OH), 3.62 (td, J = 6.7, 5.3 Hz, 1
	H), 2.76 (t, J = 6.7 Hz, 2 H), 2.70 (m, 4 H), 1.78 (m, 4 H).
	13 C NMR (100 MHz, DMSO-d 6 ): δ = 144.8, 143.1, 138.0, 136.5, 132.2,
	130.0, 127.9, 122.7, 122.2, 122.1, 60.2, 51.9, 29.0, 22.1, 21.8, 21.3,
	20.8.

HRMS

4-(6,6-Dimethyl-4,5,6,7-tetrahydro-1H-benzo[d][1,2,3]triazol- 1-yl)benzyl]-1H-1,2,3-triazol-4-yl)methanol (5caa)

  H NMR (300 MHz, DMSO-d 6 ): δ = 8.09 (s, 1 H), 7.63 (d, J = 8.5 Hz, 2 H), 7 .52 (d, J = 8.5 Hz, 2 H), 5.68 (s, 2 H), 5.19 (t, J = 5.6 Hz, 1 H, OH), 4.52 (d, J = 5.6 Hz, 2 H), 2.69 (t, J = 6.4 Hz , 2 H), 2.54 (s, 2 H), 1.59 (t, J = 6.4 Hz, 2 H), 0.95 (s, 6 H). [M + H] + calcd for C 18 H 22 N 6 O: 339.1928; found: 339.1919.

	1 H NMR (300 MHz, CDCl 3 ): δ = 7.82 (dd, J = 8.2, 1.5 Hz, 2 H), 7.75 (s, 1
	H), 7.51-7.43 (m, 4 H), 7.40 (m, 2 H), 7.33 (tt, J = 7.5, 1.2 Hz, 1 H), 5.66
	(s, 2 H), 2.95 (t, J = 5.7 Hz, 2 H), 2.71 (t, J = 5.8 Hz, 2 H), 1.87-1.68 (m, 6
	H).
	13 C NMR (75 MHz, CDCl 3 ): δ = 148.5, 147.6, 136.8, 135.9, 135.6, 130.3,
	128.9 (2 C), 128.8 (2 C), 128.3, 125.9 (2 C), 125.7 (2 C), 119.6, 53.5,
	30.8, 27.1, 26.9 (2 C), 24.7.
	HRMS: m/z [M + H] + calcd for C 22 H 22 N 6 : 371.1979; found: 371.1968.
	1-[Prepared from azidotriazole 3ca and propargyl alcohol (4a); yield: 60
	mg (89%); brown solid; mp 197 °C; R f = 0.24 (EtOAc/MeOH 20:1).
	13 C NMR (75 MHz, DMSO-d 6 ): δ = 148.4, 141.2, 136.7, 135.9, 132.2,
	129.2 (2 C), 123.2 (2 C), 122.9, 55.0, 52.0, 34.8, 34.1, 18.5, 30.6, 27.2 (2
	C).

1

HRMS: m/z

© Georg Thieme Verlag Stuttgart • New York -Synthesis 2018, 50, 4254-4262

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Acknowledgment

The authors thank Aix-Marseille Université and the Centre National de la Recherche Scientifique (CNRS) for funding.

Supporting Information

Supporting information for this article is available online at https://doi.org/10.1055/s-0037-1610192. S u p p o r t i n g I n f o r m a t i o n S u p p o r t i n g I n f o r m a t i o n