A Liné

J.-C Gabelle

J Morchain

D Anne-Archard

F Augier

On POD analysis of PIV measurements applied to mixing in a stirred vessel with a shear thinning fluid

Keywords: Mixing, Particle image velocimetry, Proper orthogonal decomposition, Eigenvalue spectrum, Dissipation rate, Stirred tank

P.O.D. technique is applied to 2D P.I.V. data in the field of hydrodynamics in a mixing tank with a Rushton turbine and a shear thinning fluid. Classical eigen-value spectrum is presented and phase portrait of P.O.D. coefficients are plotted and analyzed in terms of trailing vortices. A spectrum of dissipation rate of kinetic energy is introduced and discussed. Length scales associated to each P.O.D. modes are proposed.

Introduction

Advanced experimental techniques such as particle image velocimetry (P.I.V.) are available nowadays, and they generate a huge amount of data, in terms of instantaneous velocity fields (in a plane as far as 2D P.I.V. is concerned). Indeed, advanced data processing tools are required to extract more and more information from this source of experimental data. Proper orthogonal decomposition (P.O.D.) or Karhunen-Loeve method that is known to be efficient to isolate coherent structures from a series of instantaneous velocity fields [START_REF] Berkooz | Proper orthogonal decomposition in the analysis of turbulent flows[END_REF] will be used in this paper. P.O.D. is a modal decomposition of instantaneous velocity fields, the modes being orthogonal to each others. It can be expressed as:

-→ V k (x, y, z, t) = N I=1 -→ V (I) k (x, y, z, t) = N I=1 a (I) k (t) -→ ˚(I) (x, y, z) (1)
where -→ V k is the kth instantaneous event of velocity field measurement and -→ V -→ ˚(I) (x, y, z) is a spatial eigenfunction of mode I and it is independent of time (or instantaneous event k). Modes can be derived from the Fredholm eigenvalue integral equation, adapted by [START_REF] Sirovich | Turbulence and the dynamics of coherent structures[END_REF] as:

˝R(x, y, z, x ′ , y ′ , z ′ ) -→ ˚(I) (x ′ , y ′ , z ′ ) dx ′ dy ′ dz ′ = (I) -→ ˚(I) (x, y, z) (2)
where R is the cross-correlation tensor and (I) is the eigenvalues. Following this approach, the eigenvalues are expressed in m 5 /s 2 in a full 3D analysis. Indeed, one can point out that the dimension corresponds to a kinetic energy (m 2 /s 2 ) times a volume (dx dy dz). In 2D PIV problem, the eigenvalues are expressed in m 4 /s 2 , corresponding to a kinetic energy (m 2 /s 2 ) times a surface (dx dz). Usually eigenvalue sequences are ranked in decreasing order. Clearly, this ordering of eigenvalues ensures that the first P.O.D. modes are the most energetic.

Given its eigenvalue, (1) and its eigenfunction -→ ˚(I) (x, y, z), it is possible to reconstruct the velocity field associated to the mode 1 ( --→ V

(1) k (x, y, z, t)) which will be confirmed to represent the mean flow. Considering modes 2 and 3, [START_REF] Oudheusden | Phase-resolved characterization of vortex shedding in the near wake of a square-section cylinder at incidence[END_REF] have shown that, when their P.O.D. eigenvalues are close to each other, the POD modes are coupled: usually, they correspond to orthogonal components of a periodic process (coherent structures induced by the blades of the impeller). Higher modes correspond to lower eigenvalues that are associated to smaller scale structures. [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF] have shown that the eigenvalue spectrum (plot of eigenvalues versus mode number I) exhibit I -11/9 trend in the inertial subrange of turbulence. This trend was confirmed by De [START_REF] De Angelis | Drag reduction by polymers in turbulent channel flows: energy redistribution between invariant empirical modes[END_REF] and [START_REF] Hoisiadas | Viscoelastic effects on higher order statistics and on coherent structures in turbulent channel flow[END_REF] (among others). The analysis of the different modes will be addressed in this paper, in the field of mixing in a stirred tank with a Rushton turbine and a shear thinning fluid. In addition, a spectrum of dissipation rate of kinetic energy associated to each mode I, similar to the spectrum of eigenvalues, will be presented and discussed. Characteristic length scales (wave length) associated to each mode will be also proposed.

As highlighted by many authors, P.O.D. eigenfunctions present flow-field-like structures and their analysis must be driven with care. However, the significance of P.O.D. modes can become clearer by the detailed analysis of P.O.D. coefficients a (I) k . Some relation between two modes I and J can be tracked down by plotting the two coefficients series [a (I) , a (J) ) plane. Such a plot is called a 2D projection of the phase portrait. Hydrodynamic coherent structures can be revealed when characteristic patterns appear in this plane, such as circle, ellipse or Lissajous figures that can be associated to periodic or intermittent structures. In particular, circular shapes can suggest cyclic variations of modes I and J related to vortices or coherent structures. In this case, (a (I) , a (J) ) series can be modeled as sine-cosine functions with same period. Such phase portraits will be plotted in this paper in order to emphasize the coupling between modes 2 and 3, associated to trailing vortices generated by the blades of the impeller.

(I) k , a (J) k ], k = 1, N in the (a
P.O.D. was applied to hydrodynamics in mixing tank by [START_REF] Moreau | Proper orthogonal decomposition for the study of hydrodynamics in a mixing tank[END_REF], [START_REF] Doulgerakis | Large Scale Vortex and Strain Dynamics in Mixing Vessels and Implications for Macro-mixing Enhancement[END_REF], and [START_REF] Gabelle | Experimental determination of the shear rate in a stirred tank with non-newtonian fluid[END_REF]. Snapshot P.O.D. studies were reviewed by [START_REF] Tabib | Analysis of dominant flow structures and their flow dynamics in chemical process equipment using snapshot proper orthogonal decomposition technique[END_REF] in the field of Chemical Engineering. The goal of this paper is to contribute to extract information from 2D P.I.V. data by P.O.D. technique in the field of mixing. Classical P.O.D. data processing is performed in terms of eigenvalue problem, spectrum of eigenvalues and phase portrait analysis. New data processing is proposed in terms of dissipation rates and wavelengths associated to each mode I.

Materials and methods

Mixing tank

2D-P.I.V. measurements were performed in a 70 L stirred tank equipped with four equally spaced baffles. A Rushton turbine (D/T = 0.33) was used. The vessel was standard (T = 0.45 m); the impeller clearance was C = T/3. More information about the mixing tank dimensions is provided in [START_REF] Gabelle | Experimental determination of the shear rate in a stirred tank with non-newtonian fluid[END_REF].

P.I.V. system

The P.I.V. system used in this study consisted of a laser and an image acquisition system provided by LaVision (LaVision GmbH, Goettingen, Germany). The system included a laser Nd-Yag (Quantel, 10 Hz, 200 × 2 mJ), a synchronization system and a charge-coupled-device (CCD) camera (Imager Intense, 12 bits, 1376 × 1040 pixels). Fast Fourier transform (FFT) cross correlation was used to interrogate the two images, which were divided into interrogation area (32 × 32 or 16 × 16 pixels). For all experiments, 500-700 image pairs were recorded and statistical convergence of the velocity was checked. In order to study the influence of spatial resolution on the vicinity of the impeller, two spatial resolutions x (0.7-1 mm) were tested.

The delay between each frame in an image pair was chosen in relation to the impeller rotational speed. Four zones in the vessel were investigated: one in the impeller steam, two above and one below. POD was only carried out for the image at the impeller level.

Working fluid

The shear thinning fluid Zetag7587 (BASF, Ludvigshafen, Germany) was used. It offers good transparency even at high concentration (0.4%). Rheological measurements were carried out using a Haake Mars III rheometer (ThermoHaake, Germany). Flow curves of Zetag7587 solutions are presented in Fig. 1 for shear rates ranging from 0.001 to 1000 s -1 . Rheological data can be fitted to the model of Ostwald de Waele (power law) according to:

a = K n-1 (3)
with K the fluid consistency factor and n the flow index. The rheological parameters are reported in Table 1. Zetag7587 is known to be visco-elastic [START_REF] Escudier | Fully developed turbulent flow of Newtonian liquids through a square duct[END_REF]. However, the choice of Zetag7587 at a high concentration 0.4% was made because at high concentration the viscosity is high; it is thus easier to measure turbulent flow characteristics close to the impeller, the turbulent microscales being larger at relatively low Reynolds number and the spatial resolution of PIV measurement needed to catch the physics being larger.

Experimental conditions

The impeller speed N is 205 rpm. The global shear rate MO is estimated from Metzner-Otto correlation:

MO = k S N (4)
where the constant k S depends on the impeller type (k S = 11.5 for a Rushton turbine). The shear rate is thus equal to 40 s -1 . Given the rheological law of the shear thinning fluid, the apparent viscosity is 0.15 Pa s. The Reynolds number is thus equal to 530, corresponding to the transition region between laminar and turbulent regime. The volume averaged value of the dissipation rate of kinetic energy ε can be estimated; it is equal to ε = 0.2 m 2 /s 3 or W/kg. The associated value of global shear rate is = 37 s -1 . Indeed, the flow is turbulent close to the impeller and less and less turbulent far from it. The P.I.V. data were acquired in a domain close to the impeller tip (6 cm × 6 cm), ensuring turbulent flow conditions as it will be shown in the results presentation and discussion.

Data processing

Proper orthogonal decomposition (P.O.D.) is a linear procedure, which decomposes a set of instantaneous velocity fields into a modal base (see Section 3.1). We propose in this work to investigate P.O.D. in terms of reconstruction of instantaneous velocity fields in a mixing tank. This methodology allows separating the organized and the turbulent motions without the necessity to collect angular phased data [START_REF] Moreau | Proper orthogonal decomposition for the study of hydrodynamics in a mixing tank[END_REF]. POD is carried out using snapshot method with all the sets of instantaneous velocity fields directly issued from the measurements. The result is then an orthonormal basis of eigenfunctions and associated eigenvalues. The instantaneous velocity field can be projected on each POD eigenfunction. From these projections it is possible to get P.O.D. coefficients and to reconstruct the velocity field (see Section 3.2). Dissipation rate of kinetic energy can be estimated from the 2D velocity field (Section 3.3). Finally, characteristic length scales may also be determined from the PIV experiments.

Snapshot method

P.I.V. measurements are performed in the (x,z) plane, on a regular mesh with L rows and C columns, leading to NP = L C points per plane. Each instantaneous velocity field measurement constitutes a snapshot of the flow. The statistical analysis is performed on N snapshots taken in the same plane (P.I.V. data).

The number of grid points being LC, one obtains the matrix of instantaneous velocity vector data as:

--→ V

(1)

k x =       -→ V k (x 1 , z 1 ) -→ V k (x 1 , z 2 ) -→ V k (x 1 , z C ) -→ V k (x 2 , z 1 ) -→ V k (x 2 , z 2 ) -→ V k (x 2 , z C ) -→ V k (x L , z 1 ) -→ V k (x L , z 2 ) -→ V k (x L , z C )       (5)
where k is the index of the instantaneous event (k = 1,N). This matrix of vectors can be reshaped to build a vector -→ V k with 2LC rows as follows:

-→ V k =                 u k (x 1 , z 1 ) u k (x 2 , z 1 ) u k (x L , z C ) w k (x 1 , z 1 ) w k (x 2 , z 1 ) w k (x L , z C )                 (6)
The snapshot method adopted in this eigenvalue problem was proposed by [START_REF] Sirovich | Turbulence and the dynamics of coherent structures[END_REF]. This method is based on the snapshot matrix M corresponding to the N instantaneous velocity fields. The matrix M can be expressed as

M =                 u 1 (x 1 , z 1 ) u 2 (x 1 , z 1 ) ... u N (x 1 , z 1 ) u 1 (x 2 , z 1 ) u 2 (x 2 , z 1 ) ... u N (x 2 , z 1 ) ... u 1 (x L , z C ) u 2 (x L , z C ) ... u N (x L , z C ) w 1 (x 1 , z 1 ) w 2 (x 1 , z 1 ) ... w N (x 1 , z 1 ) w 1 (x 2 , z 1 ) w 2 (x 2 , z 1 ) ... w N (x 2 , z 1 ) ... w 1 (x L , z C ) w 2 (x L , z C ) ... w N (x L , z C )                 (7)
The matrix M has 2LC rows and N columns, each column of the matrix M representing the kth event of the instantaneous velocity field. The auto-covariance tensor R can be derived as:

R = 1 N M.M T =    u 2 (x 1 , z 1 ) . . . w 2 (x L , z C )    (8)
The Fredholm integral eigenvalue problem is represented in the 2D domain ˝of measurement by:

˝R (x, z, x ′ , z ′ ) -→ ˚(I) (x ′ , z ′ ) dx ′ dz ′ = (I) -→ ˚(I) (x, z) (9)
where (I) is the Ith eigenvalue and -→ ˚(I) x(x, z) is the Ith associated eigenfunction. The correlation tensor can be expressed in the orthonormal basis as a diagonal matrix. The eigenvalues of R, (I) R , are expressed in m 2 /s 2 and are defined as:

(I) R = (I) /(dx dz).
The dimension of M . M T matrix is (2LC) 2 whereas the dimension of M T . M matrix is N 2 . The eigenfunctions of M . M T are -→ ˚(I) (x, z); they can be simply related to the eigenvalues of

M T . M, noted --→ ˚′(I) (x, z), as: --→ ˚′(I) (x, z) = M T -→ ˚(I) (x, z) or -→ ˚(I) (x, z) = M --→ ˚′(I) (x, z) (10)
This must be kept in mind in order to save computing time since M.M T and M T .M have identical eigenvalues, but may have significantly different sizes if N 2 ≪ (2LC) 2 . In addition, the spatial modes constitute an orthonormal basis, thus:

-→ ˚(I) .

-→ ˚(J) = ı IJ (11)

Modal decomposition of the velocity field

Each instantaneous velocity field or snapshot of the flow can be expanded in a series of P.O.D. modes. The P.O.D. coefficients a

(I)
k are obtained by projecting each instantaneous velocity field on the Ith eigenfunction -→ ˚(I) :

a (I) k = -→ V k . -→ ˚(I) (12)
The Ith velocity vector

-→ V (I)
k corresponding to the Ith decomposition is given by:

-→ V (I) k = a (I) k -→ ˚(I) (13)
One can thus reconstruct each instantaneous velocity field given by Eq. ( 1). One can easily derive:

1 N N k=1 a (I) k a (J) k = (I) R ı IJ (14)
This relation shows that each mode makes an independent contribution to the total kinetic energy.

Modal decomposition of the dissipation rate of kinetic energy

The rate of viscous dissipation of kinetic energy is defined as:

ε = 2 a S : S ( 15
)
where a is the apparent viscosity of the fluid and S is the strain rate (or stretching) tensor, defined as:

(I) ∝ -11/3 (16)
Considering the decomposition of the velocity field in the sum of Ith components, one can associate to each Ith component of the velocity V

S (I) ij,k = a (I) k 2 • ∂˚( I) i ∂x j + ∂˚( I) j ∂x i (17)
Accounting for Eqs. ( 11) and ( 14), one can thus derive the Ith component of the rate of viscous dissipation of kinetic energy:

ε (I) = a ( ) (I) R 2 3 i=1 3 j=1 ∂˚( I) i ∂x j + ∂˚( I) j ∂x i 2 (18)
It is interesting to realize that the total dissipation rate of kinetic energy is the sum of the Ith component of the rate of viscous dissipation of kinetic energy: ε(x, z, t) = N I=1 ε (I) (x, z, t). Recall that the eigenvalue I is independent of space. It represents the contribution of each mode to the total kinetic energy in the plane of measurement. One can also estimate the dissipation rate of kinetic energy ε (I) averaged in the plane:

ε (I) = 1 LC L l=1 C k=1 ε (I) (x l , z k ) (19)
It is thus possible to draw a plot similar to the spectrum of eigenvalues; it is the spectrum of dissipation rate of kinetic energy (plot of dissipation rate ε (I) associated to mode I versus mode number I). This spectrum will be presented and discussed.

In addition, the strain rate of small scales of turbulence is large and can be expressed as s ′ ij ≈ u/. Recalling Eq. ( 17) and considering that the P.O.D. coefficient a

(I)
k have the dimension of a velocity (m/s), the inverse of a length scale could be associated to the gradients of the eigen-function -→ ˚(I) . This will be analyzed in depth in the discussion.

Reshaped eigenfunctions

The eigenfunctions are vectors with 2LC rows. These vectors can be reshaped to form a (L,C) matrix of local vectors:

-→ ˚(I) k =             ˚x,k (x 1 , z 1 ) ˚x,k (x 2 , z 1 ) ˚x,k (x L , z C ) ˚z,k (x 1 , z 1 ) ˚z,k (x 2 , z 1 ) ˚z,k (x L , z C )             -→ ˚(I) k =     -→ ˚k(x 1 , z 1 ) -→ ˚k(x 1 , z 2 ) -→ ˚k(x 1 , z C ) -→ ˚k(x 2 , z 1 ) -→ ˚k(x 2 , z 2 ) -→ ˚k(x 2 , z C ) -→ ˚k(x L , z 1 ) -→ ˚k(x L , z 2 ) -→ ˚k(x L , z C )     (20)
The reshaped eigenfunction can be plotted in the plane (x,z). It corresponds to a 2D vector field, each vector -→ ˚k(x i , z j ) having a horizontal ˚x,k (x i , z j ) and a vertical ˚z,k (x i , z j ) component.

Comments on 2D P.I.V. measurements

The flow field is 3D in the mixing tank. However, the analysis performed with 2D-P.I.V. is limited to a 2D plane of measurement. In terms of mean flow, it is clear that the tangential flow is not measured nor taken into account in the present analysis. Indeed, the goal of the study is 2-folds: one the one hand, it is to extract from the measurements the organized motion induced by the impeller rotation. It was shown in previous studies [START_REF] Moreau | Proper orthogonal decomposition for the study of hydrodynamics in a mixing tank[END_REF], among others) that POD can enable to decompose the organized motion from the instantaneous measurement, without collecting angular phased data.

It was also shown by many authors that 2D measurement in a vertical plane enable to extract trailing vortices induced by a Rushton turbine. One the other hand, the goal of this study is to analyze turbulent scales. In terms of turbulence, it was also shown by many authors that the measurement of instantaneous velocity field in a single plane of measurement was sufficient to estimate the local dissipation rate of turbulent kinetic energy as far as the spatial resolution of the PIV system is high enough. This is related to the isotropy of smallest scales of turbulence coupled to the efficiency of the mixing tank process. Keeping this in mind, we must be aware of the fact that the tangential mean flow is not measured and that associated large scales are not measured too. In terms of POD analysis, the 2D limitation of PIV measurements may affect the largest scales of motion (intermediate modes of POD decomposition).

Results

The eigenvalues correspond to the contribution of each mode to the total kinetic energy in the 2D domain of measurement (Section 4.1). Mode 1 is structurally close to the mean (statistical average) velocity field (Section 4.2). When two of the following modes are close to each other in terms of eigenvalues, the energy brought by these modes is the same. Such modes may reveal large scale coherent structures (Sections 4.4 and 4.5). Higher modes may be related to turbulence (Sections 4.5 and 4.6).

Eigen-value spectrum

The eigen-value spectrum is plotted in Fig. to the total kinetic energy in the plane of measurement. One can see that the first mode explains more than 80% of the total kinetic energy; it is characteristic of mean flow contribution. One can observe that modes 2 and 3 and modes 4 and 5 are close to each other. They respectively contribute to 3-5% and 0.6-0.8% of the total kinetic energy. The upper modes contribute to lower levels of kinetic energy. One can notice that

Turbulent motion

Following the decomposition method, a reconstruction of the turbulence is based on all the modes higher than 6:

----→ V (turb) k (x, z, t) = N I=6 -→ V (I) k (x, z, t) = N I=6 a (I) k (t) -→ ˚(I) (x, z) (29)
The probability density function of horizontal (u ′ k ) component of the turbulent velocity vector given by Eq. ( 29) is plotted in Fig. 9a, in a point located in the jet of the impeller. The pdf are significantly different from the previous ones; a Gaussian function based on the mean and rms values of this distribution is plotted and fits perfectly the data.

Kinetic energy dissipation rate spectrum

As afore-mentioned, one can derive the Ith component of the rate of viscous dissipation of kinetic energy averaged in the plane, ε (I) , expressed by Eq. ( 19). It is thus possible to plot the distribution of rate of viscous dissipation associated to each I mode versus the mode number, similarly to the eigen-value spectrum (Fig. 2). This kinetic energy dissipation rate spectrum is plotted in Fig. 9b. One can notice that upper modes follow a linear trend in log-log plot (-5/9); this trend is related to the -11/9 slope of the eigen-value spectrum plotted in Fig. 2, as it will be explained later in the discussion.

Discussion

Eigenvalue spectrum

The eigen-value spectrum exhibits a -11/9 slope in log-log plot of (I) versus the eigen-value number I. This result was explained by [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF] arguing that it is a characteristic of inertial range of turbulence. As exposed by these authors, in turbulent flows, the turbulent kinetic energy (tke) can be related to the energy density spectrum of velocity E V (K), per vector wave number K, as:

tke = 1 ϑ u 2 dv = E V (K) dK (30)
Clearly, the dimension of the energy density spectrum of velocity E V (K) is dim[E V ] = l 5 /t 2 or l 2 /t 2 times l 3 . It corresponds to a kinetic energy in a volume. In the inertial range of turbulence, assuming turbulence isotropy, one can write the energy density spectrum of velocity E V per vector wave number in terms of the energy density spectrum of velocity E S () per scalar wave number : E V = E S ()/4 2 . The trend of the energy density spectrum of velocity E S () per scalar wave number is well known E S () ∝ ε 2/3 -5/3 . One can thus derive the energy density spectrum of velocity E V () per vector wave number as [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF] considered then the POD analysis. As aforementioned in Section 3.1, the eigenvalue (I) is expressed in m 5 /s 2 in a full 3D analysis, similarly to the energy density spectrum of velocity E V per vector wave number. Thus, [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF] considered that each eigen value is a generalization of the energy density spectrum of velocity E V , carrying the same physical dimensions. It follows (I) ∝ -11/3 . In addition, [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF] stipulated that the wave number may be related to the eigenvalue number as ∝ I 1/3 . Thus, they obtained the -11/9 trend (I) ∝ I -11/9 . This trend is observed in our data processing (Fig. 2). This result is surprising at first appearance, since the Reynolds number of the flow in the tank is low (it is equal to 530), corresponding to the "transition" region of turbulence. It is known that local isotropy only exist if the local Reynolds number is large enough. The first argument is that the global Reynolds number is low but the hydrodynamics is heterogeneous in the tank. The level of turbulence is higher close to the impeller (where the data where acquired) than far from it. Secondly, Knight and Sirovich have shown that an inertial range can exist at very modest Reynolds numbers. Such -11/9 trend was also observed in the same mixing tank with another shear thinning fluid (with yield stress, Carbopol, [START_REF] Gabelle | Experimental determination of the shear rate in a stirred tank with non-newtonian fluid[END_REF] but this trend was limited to the higher Reynolds number.

E V ∝ ε 2/3 -11/3 .
One must also highlight that the -11/9 trend corresponds to modes larger than 20. This relatively large value can be explained by the limitation of 2D measurement of a 3D velocity field. As aforementioned, the velocity component normal to the 2D plane of measurement is not measured. In addition, the choice of Zetag fluid was dictated in order to limit the turbulence level close to the impeller; it was thus possible to measure almost all the turbulent scales. Therefore, the

Dissipation rate spectrum

One can extend the development proposed by [START_REF] Knight | Kolmogorov inertial range for inhomogeneous turbulent flows[END_REF] to the analysis of the spectrum of kinetic energy dissipation rate, plotted in Fig. 9b. This spectrum exhibits a linear trend in log-log plot with a slope -5/9. This result can be explained as follows. The spectrum of the dissipation in scalar wave number can be expressed as:

ε = D() d (31)
The dissipation density spectrum can be related to the energy density spectrum by D() ≈ 2 E() ∝ -5/3 . Considering once more that the scalar wave number is related to the eigenvalue number by ∝ I 1/3 , we obtain the trend -5/9 that is observed in the plot of ε (I) , normalized by the total dissipation rate (summation for all modes). This total dissipation rate is equal to 0.42 W/kg, almost twice the volume averaged value of the dissipation rate of kinetic energy ε which is equal to 0.2 W/kg.

Here again, the -5/9 trend corresponds to modes larger than 20. However, the main issue of this work is to explain the -5/9 trend of the dissipation spectrum for large values of modes, corresponding to turbulent motion.

Wave number associated to the mode I

The last question is related to the assumption stating that ∝ I 1/3 . Recall first that the reconstruction process is given by Eq. ( 1). The dimension of the coefficient a ) x (x, z), ˚(I) z (x, z)] are the horizontal and vertical components of the eigen-function -→ ˚(I) (x, z), one can express the cross-correlation function A xx of the horizontal velocity V x between (x,z) and (x + L,z) as: After derivation, based on the reconstruction process and on Eq. ( 14), one obtains:

A xx (L) = V x (x, z)V x (x + L, z) = 1 N N k=1 V x,k (x, z)V x,k (x + L, z) (32)
A xx (L) = V x (x, z)V x (x + L, z) = N I=1 (I) R ˚(I) x (x, z)˚( I) x (x + L, z) (33)
In other words, for each mode I, one can define a crosscorrelation function

A (I) xx in terms of A (I) xx (L) = (I) ˚(I) x (x, z)˚( I)
x (x + L, z) or in non-dimensional form:

R (I) xx (L) = A (I) xx (L) (I) R = ˚(I) x (x, z)˚( I) x (x + L, z) (34)
A Taylor development at the origin gives

R (I) xx (L) = 1 - x 2 2 ∂˚( I) x ∂x 2 = 1 - (I)2 x x 2 (35)
Thus, a wave number can easily be related to the spatial gradient of the eigen-function -→ ˚(I) (x, z). Indeed, the kinetic energy dissipation rate associated to each mode I was expressed by Eq. ( 18). Dimensionally, one can consider a wave number associated to each I mode and defined by: This wave numbers can easily be derived from the eigenfunction gradients. In fact this wave number varies in the plane of measurement. Since the eigenvalue is an average of the kinetic energy in the plane of measurement associated to mode I and ε (I) is also averaged in the plane of measurement, one can estimate an average of the wave number in the plane (I) similarly to Eq. ( 19).

This averaged wave number (I) is normalized by the wave number of the first mode and plotted versus the eigenvalue number I in Fig. 10. The curve I ∝ (I) 3 is also plotted. The normalized averaged wave numbers issued from the eigen-functions gradients behaves as the wave number power 1/3, as stipulated by Knight and Sirovich. The definition of the wave number associated to the Ith mode, (I) , is a generalization of this simplified development given by Eq. ( 35). Further analysis should be done on the pdf of gradients of eigen functions in the plane of measurement for each I mode to better understand the physical meaning of this scale. The higher modes are related to turbulence. The -11/9 trend of the eigen-value spectrum is verified. The spectrum of dissipation rate of kinetic energy is plotted and the -5/9 trend is shown and explained. Length scales associated to each mode are determined and plotted versus mode number. The power 3 trend is shown. The goal of this paper was to show the different trends (-11/9 trend of the eigen-value spectrum, -5/9 for the spectrum of dissipation rate of kinetic energy, and 3 for the length scales associated to each mode) corresponding to turbulent motions. Clearly these trends are limited to modes larger than 20. The analysis of the full 3D flow should be completed to improve the understanding and the quality of both measurement and processing of the P.O.D. modes lower than 20.

k

  is the Ith component of the P.O.D. decomposition. For each mode I, the P.O.D. method generates temporal coefficients a an instantaneous (k) or temporal (t) scalar (called P.O.D. coefficient of P.O.D. mode I); it is independent of space.

Fig. 1 -

 1 Fig. 1 -Apparent viscosity of Zetag7587 solutions versus shear rate.

  Fig. 2 -Eigen value spectrum.

Fig. 3 -

 3 Fig. 3 -First POD mode (a) eigen function, (b) pdf of normalized POD coefficient, and (c) comparison of first mode to mean flow.

Fig. 7 -

 7 Fig. 7 -Modes 4 and 5: (a) portrait of phase, (b) pdf of normalized POD coefficient of mode 4, and (c) pdf of normalized POD coefficient of mode 5.

k

  is constant in the domain and only depends on the time whereas the eigen-function -→ ˚(I) (x, z) is constant with time but it varies in the spatial coordinates (x,z). Consequently, any length-scale associated to the Ith mode can only be related to the eigen-function -→ ˚(I) (x, z) and cannot be related to the P.O.D. coefficient which is constant for each mode in the plane. It is thus foreseeable that the wave number associated to each I mode depends on the eigen-function -→ ˚(I) (x, z) and/or on its gradients. If (V x ,V z ) are the horizontal and vertical components of the velocity vector -→ V , and [˚(

I

Fig. 8 -

 8 Fig. 8 -Portrait of phase: (a) base 2-4, (b) base 2-5, (c) base 3-4, and (d) base 3-5.

Fig. 9 -

 9 Fig. 9 -Turbulence (a) pdf of turbulent velocity fluctuations and (b) spectrum of dissipation rate of kinetic energy.

Fig. 10 -

 10 Fig. 10 -Wave number versus eigen mode number. 6. Conclusion P.O.D. technique was applied to 2D P.I.V. data. Each instantaneous velocity field can be expanded in a series of P.O.D. modes. The first mode is associated to mean flow. The

Table 1 -

 1 Rheological parameters fitted to the power law equation.

	Fluid	n	K (Pa s n )	Shear rate
				range (s -1 )
	Zetag7587 0.1%	0.422	0.30	1-350
	Zetag7587 0.2%	0.376	0.74	1-600
	Zetag7587 0.4%	0.349	1.65	1-1000

Acknowledgement

This work was supported by the French Environment and Energy Management Agency (ADEME) and IFP Energies Nouvelles.

upper modes follow a linear trend in log-log plot (-11/9); this trend will be explained later in the discussion.

First mode analysis

The reshaped eigen-function --→ ˚(1) (x, z) corresponding to mode 1 is plotted in Fig. 3a. It looks like a velocity field but it is not, since the eigen-functions have no dimension. Indeed, the instantaneous velocity field associated to the first mode is given by

for each k realization of the velocity. One can plot in Fig. 3b the probability density function of the coefficients a

(1)

k . The coefficients a

(1) k were normalized by the square root of the first eigen-value

R expressed in m 2 /s 2 . The first remark is that the pdf is centered on the unity value; the second remark is the dispersion of the N realizations a

(1)

k is almost constant. One can thus reconstruct the velocity field associated to the first mode as:

One can compare this velocity field (reconstructed with the first mode) to the statistical average of the instantaneous velocity field, classically defined as:

The vertical profile of the horizontal component of the velocity field reconstructed with the first mode --→ V

(1) k (x, z, t) is compared to the vertical profile of the horizontal component the mean flow -→ V (x, z, t) in Fig. 3c, at a radial position close to the impeller tip. The two profiles are identical. One can thus confirm that the velocity field associated to the first mode is a steady-state field corresponding to the mean flow.

Organized motion associated to modes 2 and 3

As aforementioned, the eigen-values

(2)

R and

(3)

R have similar orders of magnitude, suggesting that they may correspond to organized motions. This result is expected in mixing with Rushton turbine and may correspond to trailing vortices induced by the impeller blades. The eigen-functions corresponding to mode 2, --→ ˚(2) (x, z), and 3, --→ ˚(3) (x, z), are plotted in Fig. 4a andb. These two modes exhibit vortices, almost symmetrical with respect to the plane of the impeller (z = 0), suggesting some organization in the structure of these modes. Once again, these plots are not velocity fields. The velocity field associated to the sum of these modes can be written:

In order to highlight the link between modes 2 and 3, the k realizations of P.O.D. coefficients a (3) k ) plane is thus obtained. The set of data show an organized shape; for sake of simplicity, it will be considered as an ellipse. The coefficients can thus be related by the expression:

where the eigen-values are expressed in m 2 /s 2 . The periodic nature of these coherent structures associated with modes 2 and 3 could be expressed as sinusoidal variations a (2) (ϕ k ) and a (3) (ϕ k ) as shown in previous papers [START_REF] Ducci | Decomposition of flow structures in stirred reactors and implications for mixing enhancement[END_REF][START_REF] Doulgerakis | On the manifestation and nature of macroinstabilities in stirred vessels[END_REF][START_REF] Gabelle | Experimental determination of the shear rate in a stirred tank with non-newtonian fluid[END_REF]:

where ϕ k is the phase angle. The parameters a (2) (ϕ k ) and a (3) (ϕ k ) have the dimension of velocities (m/s). It is thus possible to reconstruct the temporal evolution of the organized motion associated to modes 2 and 3: (2) R and a

(3) k /

(3) R . The distributions are centered on the origin and their shape can be related to sinusoidal distributions, as expected from the previous analysis.

Organized motion associated to modes 4 and 5

As aforementioned, the eigen-values (4) R and

(5)

R have also similar orders of magnitude and may thus contribute to organized motions. The eigen-functions corresponding to mode 4, --→ ˚(4) (x, z), and mode 5, --→ ˚(5) (x, z), are plotted in Fig. 6a andb. Compared to modes 2 and 3, modes 4 and 5 show again vortical structures but the length-scales seem to be roughly half the length-scales of modes 2 and 3. Similarly to the previous analysis of modes 2 and 3, the k realizations of a (5) R were plotted in Fig. 7b andc, the eigen values being expressed in m 2 /s 2 . Fig. 7a exhibits a strong relation between modes 4 and 5 whereas the pdf distributions reveal noisy sinusoidal distributions. In order to go in deeper depth, the modes 4 and 5 have been plotted versus modes 2 and 3 in Fig. 8a-d. 2D projections of the phase portrait are thus plotted for different bases. The organized structures revealed by these plots exhibit two lobes (contrary to the previous portrait of phases plotted in Fig. 5, with only one lobe) and can be analyzed as follows:

• the relation between mode 2 and mode 4 (Fig. 8a) corresponds roughly to a plot of sin(2ϕ k -(/2)) versus cos(ϕ k ), that is -cos(2ϕ k ) versus cos(ϕ k );

• the relation between mode 2 and mode 5 (Fig. 8b) corresponds roughly to a plot of sin(2ϕ k ) versus cos(ϕ k );

• the relation between mode 3 and mode 4 (Fig. 8c) corresponds roughly to a plot of sin(2ϕ k + (/2)) versus cos(ϕ k ), that is cos(2ϕ k ) versus cos(ϕ k );

• the relation between mode 2 and mode 5 (Fig. 8b) corresponds again to a plot of sin(2ϕ k ) versus cos(ϕ k ).

Modes 4 and 5 follow thus a sinusoidal trend, with a period which is half the period of modes 2 and 3. It is thus possible to reconstruct the temporal evolution of the organized motion associated to modes 4 and 5:

---→ V

The structures revealed by modes 2 to 5 can be compared to vortex shedding behind bluff bodies that have been investigated by many workers since the pioneering paper of [START_REF] Roshko | On the wake and drag of bluff bodies[END_REF]. Similar Lissajous figures were obtained in their works.


Gurvan Jodin
email: bgurvan.jodin@laplace.univ-tlse.fr

Johannes Scheller
email: scheller@mit.edu

Éric Duhayon
email: ceric.duhayon@laplace.univ-tlse.fr

Jean-François Rouchon
email: rouchon@laplace.univ-tlse.fr

Michael Triantafyllou

Marianna Braza
email: emarianna.braza@imft.fr

An experimental platform for surface embedded SMAs in morphing applications

Keywords: Morphing, Shape-memory Alloys, Modeling

This article will address the modeling and control of surface embedded shape memory alloys (SMAs) for the camber modification of a hybrid morphing airfoil. An analytical model will be derived. The results of this models will be discussed and compared to the experiments. The advantages of this modeling approach will be highlighted and alternatives will be briefly revisited. This discussion will figure into the utility of these models in the sizing of a full scale prototype of a SMA actuated active trailing edge of an airfoil. Throughout this article the prototype specifications are detailed and the design choices will be discussed. Performance improvements stemming from the inherent nature of the SMAs will be analyzed. It will be shown in this article that through the use of forced convection the overall cycle time can be reduced.

Introduction

The rigid airfoil geometries of today's aircrafts are usually the result of a design compromise optimizing the shape only for some parts of flight. Control surfaces are used to adapt the shape for different parts of the mission profile. However, due to the introduction of discontinuities in the shape these control surfaces usually have poor aerodynamic performance and efficiency. The use of deformable or morphing structures has the potential to resolve this issue. Smart-materials hold great potential in achieving the goal of an entirely deformable airfoil structure [START_REF] Ursache | Morphing Winglets for Aircraft Multi-Phase Improvement[END_REF]. The RTRA supported research program DYNAMORPH and the research platform SMARTWING aim at improving the performance of micro-air-vehicles in realistic environments via electro-active morphing. During the course of this project a prototype NACA4412 wing was developed with embedded Shape memory alloys (SMAs) and trailing-edge Macro-fiber composite actuators enabling both large deformations (~10% of the chord) at limited frequency (<1Hz) and small deformations at high frequencies (≤100Hz) [START_REF] Scheller | A combined smart-materials approach for nextgeneration airfoils[END_REF]. The characteristics of the SMAs, which were actuated using current heating, make it especially suitable to optimize the shape of the wing and to control the flight [START_REF] Calkins | Shape Memory Alloy Based Morphing Aerostructures[END_REF][START_REF] Lexcellent | Shape-memory Alloys Handbook[END_REF]. The highfrequent but low amplitude piezoelectric technology on the other hand is useful to produce trailingedge vortex breakdown [START_REF] Scheller | Trailing-edge dynamics of a morphing NACA0012 aileron at high reynolds number by high-speed PIV[END_REF]. Wind-tunnel tests proved the capacities of the design with respect to both the high-frequency, low-amplitude and low-frequency, high-amplitude actuation. However, the tests also showed the need to further characterize the SMA based high-amplitude actuation before moving to a full-scale prototype. To this end a simple model was created resembling one part of the activated airfoil skin.

Different concepts for SMA based camber control can be found in the literature. Elzey et al. [START_REF] Dana M Elzey | A bio-inspired high-authority actuator for shape morphing structures[END_REF] used for example a flexible vertebra structure inside of an airfoil in order to generate both variable camber and twist actuation. They successfully demonstrated the bending capacity of their approach (> 50% of the chord length) however their model lacked in rigidity for a real-world application. Similarily, Musloff [START_REF] Musolff | Formgedaechtnislegierungen[END_REF] also achieved significant displacements in his thesis for both 2D and 3D deformations. Manzo and Garcia [START_REF] Manzo | Demonstration of an in-situ morphing hyperelliptical cambered span wing mechanism[END_REF] constructed a HyperElliptical Cambered Span (HECS) morphing wing capable of achieving around 25% of spanwise bending within ≈ 3 s. The HECS wing shape was linearly discretized allowing the pulley mechanism to act on fixed sections. Windtunnel experiments were carried out using the morphing HECS wing and it was shown that the SMA based mechanism is capable of overcoming significant aerodynamic loads. Pankonien and Friswell [START_REF] Pankonien | Synergistic smart morphing aileron: aero-structural performance analysis[END_REF][START_REF] Pankonien | SMART MATERIAL WING MORPHING FOR UNMANNED AERIAL VEHICLES[END_REF] combined different SMA actuators with a trailing edge Macro Fiber Composite (MFC) actuatued flexure box to achieve a synergistic morphing overcoming the inherent limitations of the different individual materials. They combined a SMA actuated hinge with a flexure box actuated using piezoelectric MFC actuators. Whereas these concepts certainly demonstrated the capacities of the SMA based camber control the design choices led to a limitation of the overall rigidity and actuation frequency of the wing. In order to address these issues our concept integrates the SMAs directly in the surface of the airfoil skin adding to its rigidity. In addition the integration of a forced convection control concept holds to potential to increase the actuators maximum frequency.

This work is developed as follows: in a first part the experimental setup is described, then an analytical model based on the works of Elahini [START_REF] Elahinia | Effect of system dynamics on shape memory alloy behavior and control[END_REF] is introduced to estimate the achievable deflection of the system. The experimental results are then shown and compared to the analytical model. Finally the results are discussed and a short conclusion is drawn.

Experimental setup

The experimental setup is illustrated in Figure 1 and the actual experiment is shown in Figure 2. The SMA actuators which are embedded on the surface of the aluminum substrate are activated at different current intensities using a Delta SM 400-AR-8 Power Supply Unit (PSU). The Pulse-Width Modulated (PWM) control signal for both the PSU and the solenoid valve controlling the forced convection is generated using an Arduino Mega 2560 which also collects the strain gauge and temperature measurements. In order to channel the forced convection the embedded SMAs are encapsulated inside of a silicone tube which was selected due to its high temperature resistance as well as its large flexibility. The influence of the silicone tube in the mechanical behavior of the system is considered to be negligible as the thickness of the silicone is smaller than the thickness of the aluminum beam and the Young modulus of the silicone is three orders of magnitude smaller than that of aluminum.

Figure 2: Experiment

The overall displacement is measured using a Mitutoyo linear gage. The measurement synchronization is done using a control PC running Matlab. The strain, voltage and current measurements are acquired at an acquisition frequency of 8 Hz. Due to the limited temporal dynamics of the SMA actuators this low acquisition frequency was estimated to be sufficient. The temperature data was re-sampled using a linear approximation algorithm in order to adapt the measurements to the acquisition frequency of the Arduino.

Analytical model

As previously mentioned, the derived actuator model is based on the phenomenological SMA model developed by Elahini [START_REF] Elahinia | Effect of system dynamics on shape memory alloy behavior and control[END_REF]. Due to the fact that phenomenological models are based on easily measured engineering quantities they are favored in engineering applications. In these models the phase-transformation dynamics are described by means of internal variables and the interdependence of these variables is modeled using kinetics equations.

The structural behavior can be estimated using the quasi-static Euler-Bernoulli beam theory modeling the contraction and extension of the SMA actuator as a moment applied at a distance r of the beams neutral layer. In this case the deflection of the beam can be written as

g 2 EI σArL = δ 2 ( 1
)
where δ is the deflection at the linkage between the SMA and the aluminium substrate, σ is the stress in the SMA, A is the SMA's cross-sectional area, r is the distance between the substrate's neutral layer and the SMA, L is the length of the substrate from the fixation to the linkage and E and Ig are the substrate's Young Modulus and area moment of inertia respectively. This allows us to define an equivalent spring constant keq for the use in the simulation model. To simulate the prototype's response due to the SMA's actuation the previously described structural model will be combined with the phenomenological SMA model as defined by Elahini [START_REF] Elahinia | Effect of system dynamics on shape memory alloy behavior and control[END_REF]. The simulation is created using the Matlab/Simulink platform. The block diagram of the combined SMA-structural model is shown in Figure 3. The model forms an algebraic loop as the SMA's behavior is dependent on both stress and temperature of the actuator. Hence, an iterative solution has to be conducted. The parameters were mainly taken from previous experiments as well as from the literature [START_REF] Elahinia | Effect of system dynamics on shape memory alloy behavior and control[END_REF][START_REF] Duval | Conception et mise en oeuvre d'un système d'actionneurs AMF répartis pour le contrôle de forme électroactif de voilures aéronautiques[END_REF] and can be seen in the table above. The equations describing the heat transfer, the phase transformation, the structural dynamics as well as the relationship between stress, strain and temperature in the SMA will only be briefly repeated below for a thorough explanation please refer to Elahini [START_REF] Elahinia | Effect of system dynamics on shape memory alloy behavior and control[END_REF]. Heat transfer. The SMAs are activated using Joule heating. The differential equation governing the SMA's heat transfer is given in the Equation below. It combines Joule heating with natural convection. In this Equation msma is the mass per unit length of the SMA, Ac is the surface area of the wire, I is the current applied, cp is the specific heat, T is the temperature of the SMA, T∞ is the ambient temperature and hc is the coefficient of heat convection. For the purpose of this simulation the resistance R of the wire is assumed constant. Phase transformation. The phase transformation in the actuator is governed by the martensite fraction, which has to be known for each instant in time and can be calculated knowing the temperature, the stress and their derivatives. Due to the SMA's hysteretic nature the description of their phase transformation behavior depends on the previous state of the system. In case the system was previously in the austenite state and is being transformed into the martensite state the transformation is called the forward transformation (see Equation 3). In the inverse case so called reverse transformation is taking place as described by Equation 4. In both cases the phase transformation is expressed in the form of the time derivative of the martensite fraction ζ.

Ṫ = 1 m SMA c p ( I 2 R-h c A c (T -T ∞ )) (2)
ζ = ζ A -1 2 sin ( A m (T -M f + B m σ)( A m Ṫ + B m σ)) (3)
ζ = -ζ M 2 sin ( A a (T -A s +B a σ)( A a Ṫ + B a σ)) (4)
The variables ζM and ζA represent the martensite fraction at the beginning of the A → M and M → A transformation respectively, Aa, Am, Ba and Bm can be calculated via Equation 5knowing the material properties given in the table below. As, Af are the start and finish temperature of austenite creation and respectively Ms, Mf are the start and finish temperature of martensite creation.

Variations on temperature linked to stress are represented through cm and cm.

A a = π A s -A f A m = π M s -M f B a = -A a c a B m = -A m c m ¿ (5)
Constitutive equation. The relationship between the different variables governing the actuators behavior is given as Equation 6, where E(ζ) is the Young modulus, ε is the strain, θ is the thermoelastic tensor and Ω(ζ) the transformation tensor.

σ= E( ζ ) ε+θ Ṫ +Ω( ζ ) ζ (6)
It should be kept in mind however that the model assumes a fully detwinned martensite and hence does not model the detwinning process.

Results

After this very brief introduction to the analytical modeling procedure this paper will now focus on the experimental results. A measurement cycle is show in Figure 4. The measurement cycle is repeated every 45 s. The SMAs are heated during 9.8 s at different current intensities and depending on whether cooling via forced convection is used or not a forced convection cooling cycle of 20 s follows the heating. These heating and cooling times are relevant with system dynamics because cooling is slower than heating. The remaining dead times allow the system stabilization over the cycles. In order to compare the different results the measurements are averaged over the active cycle from the beginning of the heating process to the end of the forced convection cooling. The results of this averaging process are shown in Figure 5 for the measurements without forced convection and in Figure 6 for the measurements with forced convection. Comparing these two graphs two things stick out: first as expected the forced significantly improves the cycle time allowing for faster actuation and second the measurements at higher current intensities have a significantly larger standard deviation and a lower maximum achievable deflection. Whereas this seems counterintuitive it is actually more due to the thermal characteristics of the system than due to the SMAs characteristics. Taking a look at the measurement cycle at 6 A with and without forced convection clearly show that even during the relatively long cycle time the SMA is not completely cooled down hence at the beginning of the subsequent cycle not the entire potential of the actuators can be realized. Furthermore, as mentioned in [START_REF] Jayender | Modelling and gain scheduled control of shape memory alloy actuators[END_REF] the higher starting temperature might even be counter-productive due to the higher convection at higher temperatures.

Hence, as the time dynamics depend largely on the initial temperature a control systems is necessary in order to implement this control an indicator of the trailing edge deflection is necessary. Whereas different publications either use a force-based [START_REF] Featherstone | Improving the speed of shape memory alloy actuators by faster electrical heating[END_REF] or a resistance/inductance [START_REF] Kim | Sensorless displacement estimation of a shape memory alloy coil spring actuator using inductance[END_REF][START_REF] Ma | Position control of shape memory alloy actuators with internal electrical resistance feedback using neural networks[END_REF] based approach a strain-gauge position feedback was selected in this paper due to the relative simplicity of implementation as well as the large forces during deformation and the low cost of the sensors. Comparing the averaged measurement values to the averaged measured strain in mV at the output of the amplifier we can see that this type of simple sensor is indeed capable to accurately represent the deformation achieved by the prototype. Figure 9 depicts the displacement vs. strain response of the system for different current intensities. Alternatively one can also choose to compare the measured voltage to the actual strain using the displacement data and the geometry of the clamped free beam, but as the final goal is to control the tip displacement a simple calibration with respect to the tip displacement was selected. Whereas these measurements show a near linear response between the measured tip displacement and the measured voltage it also becomes evident that a saturation occurs once the displacement surpasses 20 mm. In order to avoid this kind of saturation an a priori knowledge of the achievable displacement is necessary. Hence, the next section will briefly compare the achieved displacement during the measurements with the results obtained using the combination of the phenomenological model developed by Elahini [START_REF] Elahinia | Effect of system dynamics on shape memory alloy behavior and control[END_REF] and the structural behavior.

Result comparison.

Figure 10 shows the comparison of the analytically obtained results and the experimental results with forced convection. A good correspondence both quantitatively and qualitatively can be observed between the analytical and experimental results. However, whereas during the heating section the simulation results follow the trajectory of the experiment a larger difference can be observed during the cooling portion of the cycle.

In order to account for this behavior a more thorough investigation of the temporal dynamics of the experimental setup is necessary especially regarding the heat exchange with or without forced convection in both a constraint and unconstrained environment. In other words the temperature exchange in the silicone tubes has to be more deeply analyzed. Furthermore for a better understanding of the SMAs properties the variation of the SMA properties during repeated cycles should be taken into account. However, even though the simulation results show room for improvement it is notable that even using this simple simulation model as well as the SMAs

Conclusion

This paper presented a simple experimental prototype for the analysis of the behavior or SMA actuators. The goal was to generate an extensible platform for characterization of surface embedded SMAs allowing to implement different control approaches as well as both forced and natural constraint convection cooling. It was shown that using forced convection the cycle time can be significantly reduced by up to 30 %. Furthermore, due to temperature variation of the SMA during the cycle in the natural constraint convection case the use of forced convection also allows for smaller standard deviation and larger average temperature. Furthermore, a simple low-cost position feedback was introduced in the experimental platform. The simplicity of the linear relationship between displacement and measured strain was highlighted. However, it was shown that care has to be taken as saturation can occur especially at larger values of deflection. Finally, in order to gain an apriori knowledge of the displacement, a combination of a phenomenological SMA model with a simplified structural model was introduced and it was shown that however simple this model a good quantitative and qualitative correspondence during the heating cycle can be obtained. However, it was also shown that due to the added complexity by thermally constraining the actuators a more indepth analysis of the thermal behavior of the platform is necessary.

In summary this paper showed a simple platform for evaluating different SMA control strategies characterized its deflection behavior both using forced and natural convection and also developed a mean to analyze the platform using a simulation model. This allows in a next step to create and validate different control approaches enabling potentially larger cycle times.

Figure 1 :

 1 Figure 1: Experimental setup

Figure 3 :

 3 Figure 3: Block diagram of the combined SMA-structural model

Figure 4 :

 4 Figure 4: Averaged displacement vs. time without forced convection. The grey envelope represents onetime the standard deviation averaged over three iterations.

Figure 5 :

 5 Figure 5: Averaged displacement vs. time with forced convection. The grey envelope represents one-time the standard deviation averaged over three iterations.

Figure 6 :Figure 7 :Figure 3 :

 673 Figure 6: Measurement cycle at 6A without forced convection (black: tip displacement, blue cooling cycle (when active) and red heating cycle) Figure 7: Measurement cycle at 6 A with forced convection (black: tip displacement, blue cooling cycle (when active) and red heating cycle)

Figure 8 :

 8 Figure 8: Displacement vs. strain response of the actuated flat plate


Leila Amgoud
email: amgoud@irit.fr

Srdjan Vesic
email: vesic@irit.fr

Two roles of preferences in argumentation frameworks

In this paper, we show that preferences intervene twice in argumentation frameworks: i) to compute standard solutions, and ii) to refine those solutions. The two roles are independent and obey to distinct postulates. After introducing and studying the postulates, we provide an example of a formal framework which models the two roles and verifies all the proposed postulates.

Introduction

An argumentation framework (AF) consists of a set of arguments and an attack relation among them. Arguments are evaluated using an acceptability semantics. This amounts to compute acceptable sets of arguments, called extensions. The attack relation is at the heart of all existing semantics. An attacker wins unless the attacked argument is defended by "good" arguments. Since [START_REF] Simari | A mathematical treatment of defeasible reasoning and its implementation[END_REF], it has been argued that arguments may not have the same strength and some of them may be stronger or preferred to others. Consequently, several attempts have been made in the literature for taking into account preferences in argumentation frameworks (e.g. [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF]).

Besides, preferences play a key role in nonmonotonic reasoning [START_REF] Brewka | Preferences and nonmonotonic reasoning[END_REF]. They are used in order to narrow down the number of possible belief sets of a base theory. To say it differently, from a given base theory, a first set of standard solutions (belief sets) is computed, then a subset of those solutions (called preferred solutions) is chosen on the basis of available preferences. Thus, preferences refine the standard solutions.

In this paper, we show that preferences intervene twice in an argumentation framework. They are mandatory for: i) computing the standard solutions of an AF, and then ii) for narrowing the number of those solutions. The first role of preferences may not take into account all the available preferences. It focuses only on those which conflict with the attacks; such attacks are said critical. The idea is that an attack may fail if the attacked argument is stronger than its attacker. Ignoring this issue may lead to counter-intuitive standard solutions. The first role has largely been discussed in existing literature while the second one has only been pointed out recently in [START_REF] Dimopoulos | Extending argumentation to make good decisions[END_REF]. However, the difference between the two roles is still obscure. In this paper, we clarify the distinction between the two roles, and show that they are completely independent since none of them can be modeled by the other one. We propose postulates that should be satisfied by any preference-based argumentation framework. Some of them concern the first role while others concern the refinement role. Those postulates confirm again that the two roles are different. We propose a particular framework in which both roles are modeled. The properties of this framework are investigated.

The paper is structured as follows: We start by recalling Dung's AF, then we discuss informally the two roles of preferences. The two next sections propose postulates that guide the definition of 'approaches' for each role. Then, we propose a particular framework which considers both roles. Before concluding, we compare our contribution with existing works. Due to lack of space, the proofs are not included in the paper.

Basics of Argumentation

The abstract argumentation framework proposed in [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF] Different acceptability semantics for evaluating arguments have been proposed in the same paper [START_REF] Dung | On the acceptability of arguments and its fundamental role in nonmonotonic reasoning, logic programming and n-person games[END_REF]. Each semantics amounts to define sets of acceptable arguments, called extensions. The following definition recalls them.

Definition 2 (Stable semantics) Let F = (A, R) be an AF and B ⊆ A.

-B is an admissible set iff it is conflict-free and defends all its elements. -B is a complete extension iff it is conflict-free and contains all arguments it defends.

-B is a preferred extension iff it is a maximal (for set inclusion) admissible set.

-B is a grounded extension iff it is a minimal (for set inclusion) complete set.

-B is a stable extension iff it is a preferred set that attacks any element in A \ B.

Let Ext(F ) be the set of extensions of F under a given semantics.

Preferences in Argumentation: Informal Discussion

In what follows, we assume that F = (A, R) is an arbitrary argumentation framework where A is finite. Let ≥ be a binary relation that expresses preferences between arguments of A. For instance, an argument may be preferred to another if it is grounded on more certain information, or if it promotes a more important value. Throughout the paper, the relation ≥ ⊆ A × A is assumed to be a preorder (i.e. reflexive and transitive). Let us now analyze the role that preferences between arguments can play in an argumentation framework. We will discuss different critical examples.

Example 1 (Cont):

If we assume that a > b and c > d, then the stable extension {a, c} is better than {b, d} since each element of the latter is weaker than an element of the former. Thus, F 2 would have only {a, c} as its preferred stable extension.

It is worth mentioning that preferences in Example 1 refine the results obtained in the standard case. Indeed, the set of preferred solutions is a subset of the set of the standard ones. Preferences play here exactly the role described in nonmonotonic logic formalisms. Let us now consider a different example.

Example 2 Let F 2 = (A, R) be s.t. A = {a, b} and aRb. F 2 has one stable extension: the set {a}. Now, if we assume that b > a, it is clear that the standard solution cannot be refined and {a} is the preferred solution of the framework. What happened here is that the preferred argument is rejected when computing the standard solution. Thus, there is no way to apply the preference of b over a. However, is it intuitive to still consider {a} as an extension of F 2 ? The answer is certainly no as illustrated next. Assume that F 2 is is built over a knowledge base K = {x} and a set of defeasible rules D = {⇒ y; y ⇒ ¬x} as in ASPIC system [START_REF] Amgoud | Towards a consensual formal model: inference part[END_REF]. Let a :⇒ y; y ⇒ ¬x and b : x. If the attack relation is the one which allows to undermine a premise of another argument, then a undermines b in its premise x while b does not undermine a since it has no premise. If now we assume that x is more certain than both ⇒ y; y ⇒ ¬x, then it is natural to keep b and to reject a. To put it differently, the preferred solution of F 2 would be the extension {b}.

Contrarily to Example 1, the use of preferences in Example 2 completely modifies the original set of extensions. Consequently, the preferred solutions of a framework are not necessarily a subset of the standard ones. This is not surprising since preferences in this case are used in order to compute the standard solutions. Thus, {b} is a standard solution. Preferred solutions refine the standard ones. In this example, {b} is the only standard solution, thus it is also the unique preferred solution.

It is also worth mentioning that when preferences are used for computing the standard solutions of an argumentation framework, not all available preferences are exploited. Only those which conflict with the attacks, as in Example 2, are used. Consequently, the result which is returned may need to be refined as shown in the following example. This framework has one stable extension which is {a, c}. Assume that b > c, d > a and b > e. Note that only b > e conflicts with the attack relation since eRb. Thus, only this preference is taken into account for computing the two standard solutions {a, c} and {b, d}. Consequently, the two remaining preferences can be used in order to refine the standard result and to prefer the extension {b, d}.

In sum, two roles of preferences are distinguished:

1. To weaken the critical attacks (i.e. the attacks which conflict with the preferences) in an AF, and thus to compute intuitive standard solutions. 2. To refine the standard solutions computed in the first role.

Example 2 shows that a refinement does not solve the problem of critical attacks and Example 3 shows that the first role is not sufficient and its results may need to be refined as the first role does not exploit all the available preferences.

Handling critical attacks

The aim of this section is to propose the basic postulates that any preference-based argumentation framework (PAF) should satisfy. We focus here on the use of preferences for computing the standard solutions, thus for modeling the first role of preferences.

Definition 4 (PAF) A preference-based argumentation framework (PAF) is a tuple T = (A, R, ≥) where A is a set of arguments, R is an attack relation and ≥ is partial or total preorder on A.

Note that we do not show how arguments are evaluated in such a PAF. In fact, we do not focus on a particular approach, but we propose postulates that any approach should satisfy. Before presenting those postulates, let us first define formally critical attacks.

Definition 5 (Critical attack) Let

T = (A, R, ≥) be a PAF. An attack (b, a) ∈ R is critical iff a > b.
The role of preferences which consists of handling critical attacks has already been identified in the literature, namely in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Amgoud | Generalizing stable semantics by preferences[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF]. While all these approaches agree that a strong argument may be accepted if it is attacked by a weaker argument, they disagree on whether the weak attacker should be rejected or not. Let us say it differently, in Example 2, the works [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF] return one stable extension which contains both the attacker and the attacked argument, that is the set {a, b}. This extension violates one of the basic requirements of acceptability semantics, the conflict-freeness of extensions. In [START_REF] Amgoud | Generalizing stable semantics by preferences[END_REF], the authors have argued that this is undesirable since the intuition behind an extension is that it encodes a 'coherent position'. This coherence is captured by the notion of conflict-freeness in acceptability semantics. That is why it is at the heart of all semantics. The authors have then proposed an alternative solution in which the argument a is rejected and the only stable extension of the framework F 2 is {b}. In this paper, we argue that the extensions of an argumentation framework should be conflict-free, otherwise the whole theory of argumentation collapses. We propose four basic postulates that should be satisfied by any approach for preference-based argumentation that models the first role of preferences. The first postulates states that the extensions of a PAF should be conflict-free.

Postulate 1 (Conflict-freeness) Let T = (A, R, ≥) be a PAF and Ext(T ) it set of extensions. Each extension E ∈ Ext(T ) should be conflict-free wrt R.

The second postulate says that when there are no critical attacks, then the output of the PAF should coincide with that of a system without preferences. The reason is that we suppose that a PAF is built over a well-founded basic system (i.e. the system constructed only from a pair (A, R)). The third postulate shows how to privilege a strong argument over a weak attacker.

Postulate 3 (Critical attacks) Let

T = (A, R, ≥) be a PAF and a, b ∈ A. Let E 1 , E 2 be two conflict-free (wrt R) subsets of A s.t. E 1 = E ∪ {a} and E 2 = E ∪ {b}. If aRb and b > a, then E 1 / ∈ Ext(T ).
The last postulate states that attacks should win when they are not critical. Works in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF], proceed by removing critical attacks from an argumentation graph and applying Dung's semantics on the remaining sub-graph. It is easy to show that when there are no critical attacks, the two graphs coincide. Property 1. Let F = (A, R) be an AF, ≥ ⊆ A × A, and

F ′ = (A, R r ) be such that R r = R \ {aRb s.t. b > a}. If ∄a, b ∈ A s.t. aRb and b > a, then R = R r .
It can be shown that that such an approach violates the conflict-freeness in some cases when the attack relation is not symmetric, and the third postulate (for example for admissible semantics), while it satisfies Postulates 2 and4. This means that when the attack relation is symmetric, all the postulates are verified. However, the following example shows that the result may still need to be refined. The extensions of this PAF are {a, b} and {c, d}. However, {a, b} is clearly preferred to {c, d}. Thus, the frameworks developed in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF][START_REF] Modgil | Reasoning about preferences in argumentation frameworks[END_REF] do not take into account the second role of preferences even when the attack relation is symmetric.

Proposition 1 Let T = (A, R, ≥) be a PAF s.t. Ext(T ) = Ext((A, R r )) where R r = R \ {aRb s.
In the recent paper ( [START_REF] Amgoud | Generalizing stable semantics by preferences[END_REF]) an approach has been proposed which verifies all postulates.

Proposition 3

The class of PAFs defined in [START_REF] Amgoud | Generalizing stable semantics by preferences[END_REF] verifies Postulates 1 -4.

Refining AFs by preferences

Until now, we have studied the first role of preferences. We have particularly shown that "some" preferences should be taken into account for computing the standard solutions of an argumentation framework. Examples 1 and 3 show that standard solutions may need to be narrowed down using the remaining preferences. What is worth noticing is that a refinement amounts to compare subsets of arguments. In Example 1, the so-called democratic relation, d , is used for comparing the two sets {a, c} and {b, d}:

Let E, E ′ ⊆ A. E d E ′ iff ∀x ′ ∈ E ′ \ E, ∃x ∈ E \ E ′ s.t. x > x ′ .
Relation d is not unique and different relations can be used as shown next. Let us now define the basic properties that such a relation should satisfy. The first property ensures that the refinement relation is a preorder, that is reflexive and transitive. Note that these are the basic properties of any preference relation.

Postulate 5 (Preorder) Let A be a set of arguments. A refinement relation on P(A) is a preorder (reflexive and transitive).

The second property ensures that the relation privileges sets that contain strong arguments (wrt).

Postulate 6 (Privileging strong arguments) Let

T = (A, R, ≥) be a PAF, a, b ∈ A and E 1 , E 2 ∈ P(A). If E 1 = E ∪ {a} and E 2 = E ∪ {b} and a > b, then E 1 E 2 .

Notation:

The maximal elements of a set S wrt. a given relation are defined as follows:

Max(S, ) = {E ⊆ S | ∄E ′ ⊆ S s.t. E ′ ≻ E}.
Property 2. The democratic relation verifies the two postulates 5 and 6.

In this section, we propose a particular framework which models both roles of preferences and verifies all the postulates introduced in this paper. The framework follows two steps: at the first step, it computes the standard solutions by handling correctly the available critical attacks. These solutions are then refined using an appropriate refinement relation. In order to make the paper easy to read, we will call PAF the framework which computes the standard solutions and rich PAF the one which refines the results of the PAF.

Definition 6 (Rich PAFs) A rich PAF is a tuple T = (A, R, ≥, ) where A is a set of arguments, R ⊆ A × A is an attack relation, ≥ ⊆ A × A is a (partial or total) preorder and ⊆ P(A) × P(A) is a relation which verifies Postulates 5 and 6. The extensions of T (under a given semantics) are elements of Max(S, ), where S is the set of extensions (under the same semantics) of the PAF (A, R, ≥).

In what follows, we propose a new approach that handles correctly critical attacks (i.e. which satisfies the four postulates introduced in section 4). We exploit for that a simple result that is proved recently in [START_REF] Amgoud | Generalizing stable semantics by preferences[END_REF]. In that paper, the authors have proposed a new approach for taking into account preferences and which prevents the shortcomings of existing ones, namely the problem of conflicting extensions. The basic idea is to integrate preferences in the definition of semantics. A refinement of stable semantics is defined as a dominance relation which compares sets of arguments. The best sets wrt that relation are the extensions of the PAF. In that paper, the authors have shown that all their extensions are conflict-free and Postulates 2, 3 and 4 as satisfied as well. They have also shown an important result for semantics that refine stable one with preferences. The result says that the extensions of their approach (i.e. the best sets wrt the dominance relation) are exactly the stable extensions of the basic argumentation framework in which each critical attack is inverted. In what follows, we show that this idea can be generalized to any acceptability semantics.

The idea of inverting the arrows of critical attacks in an argumentation graph allows to take into account the preference (between the two arguments involved in a critical attack) and in the same time the conflict between the two arguments of the attack is represented. The intuition behind this is that an attack between two arguments represents two things: i) an incoherence between the two arguments (in logic-based systems, it captures inconsistency between the supports of the two arguments), and ii) a kind of preference determined by the direction of the attack. Thus, in our approach, the direction of the arrow represents a real preference between arguments. Moreover, the conflict is kept between the two arguments. Dung's acceptability semantics are then applied on the modified graph. In our approach, standard solutions are computed by the following preference-based framework. From Definition 7, it is clear that if a PAF has no critical attacks, then the repaired framework coincides with the basic one. Property 3. Let T = (A, R, ≥) be a PAF. If T has no critical attacks, then R r = R.

Definition 7 (PAF) A preference-based argumentation framework (PAF) is a tuple T = (A, R, ≥) where A is a set of arguments, R ⊆ A × A is an attack relation and ≥ is a preorder on

This property shows also that when a PAF has no critical attacks, then preferences do not play any role in the evaluation process.

Our approach does not suffer from the drawback of existing ones. Indeed, it delivers conflict-free extensions of arguments. Thus, it satisfies Postulate 1.

Proposition 4 Let T = (A, R, ≥) be a PAF and E 1 , . . . , E n its extensions under a given semantics. For all i = 1, . . . , n, E i is conflict-free wrt. R.

The next result confirms that our approach is well-founded in the sense that acceptable arguments are defended by "good" arguments. Moreover, it verifies the orderings between the attack relation and the preference relation, meaning that it verifies Postulates 3 and 4.

Proposition 5 Let T = (A, R, ≥) be a PAF.

-For each admissible set E of T , it holds that (∀x ∈ E) (∀x ′ / ∈ E) if (x ′ Rx and not (x > x ′ )) or (xRx ′ and x ′ > x) then (∃y ∈ E) s.t. (yRx ′ and not (x ′ > y)) or (x ′ Ry and y > x ′ ). -For each stable extension E of T , it holds that (∀x ′ / ∈ E) (∃x ∈ E) s.t. (xRx ′ and not (x ′ > x)) or (x ′ Rx and x > x ′ ).
The fact of inverting the arrows of critical attacks in an argumentation graph does not affect the status of arguments that are not related to the arguments of those attacks. This means that our approach has no side effects. Before presenting the formal result, let us first give a useful definition. Definition 8 Let F = (A, R) be an AF and a, b ∈ A. The arguments a and b are related in F iff there is exists a finite sequence a 1 , . . . , a n of arguments such that a 1 = a, a n = b and for all i = 1, . . . , n -1, either (a i , a i+1 ) ∈ R or (a i+1 , a i ) ∈ R.

Proposition 6 Let T = (A, R, ≥) be a PAF. For all a ∈ A s.t. ∄b, c ∈ A s.t. (b, c) ∈ R is a critical
attack and a is related with b, it holds that:

-Status(a, (A, R)) = Status(a, (A, R r )) (under preferred and grounded semantics). -If (A, R) and (A, R r ) both have at least one stable extension, then Status(a, (A, R))

= Status(a, (A, R r )) (under this semantics).

Our approach privileges the strongest arguments. Indeed, we show that these arguments are skeptically accepted when they are not conflicting. If such a strong argument is not skeptically accepted, then it is for sure attacked (wrt. R) by another strongest argument. Before presenting the formal result, let us define the strongest arguments (or the top elements) wrt. a relation ≥.

Definition 9 (Maximal elements) Let O be a set of objects and ≥⊆ O×O is a (partial or total) preorder. The maximal elements

of O wrt. ≥ are Max(O, ≥) = {o ∈ O | ∄o ′ ∈ O s.t. o ′ > o}. Property 4. Let T = (A, R, ≥) be a PAF s.t ≥ is complete 1 . -If Max(A, ≥) is conflict-free (wrt. R), then ∀a ∈ Max(A, ≥) :
• a is skeptically accepted in T wrt. preferred and grounded semantics.

• if T has at least one stable extension, then a is skeptically accepted wrt. stable semantics. -If a is not skeptically accepted (under preferred or grounded semantics), or there exists at least one stable extension and a is not skeptically accepted, then ∃b ∈

Max(A, ≥) s.t. (b, a) ∈ R.
The following result shows that when the preference relation ≥ is a linear order (i.e. reflexive, antisymmetric, transitive and complete), then the corresponding PAF has a unique stable/preferred extension. Moreover, this extension is computed in O(n 2 ) time where |A| = n. It is clear that in this case, there is no need to refine the result.

Proposition 7 Let T = (A, R, ≥) be a PAF s.t. R is irreflexive and ≥ is a linear order.
-T has exactly one stable extension.

-Stable, preferred and grounded extensions of T coincide.

-

If |A| = n, then this extension is computed in O(n 2 ) time.
Let us now see what happens in case the attack relation is symmetric. The following result shows that our approach returns the same results as the approach developed in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF]. This means that inverting the arrows or removing them will lead to the same result.

Property 5. Let T = (A, R, ≥) be a PAF where R is symmetric. Extensions of T coincide with extensions of (A, R ′ ) (under the same semantics) where

R ′ = {(a, b)|(a, b) ∈ R and¬(b > a)}.
We can also show that when the attack relation is symmetric, the extensions of a PAF are a subset of those of its basic framework. This means that preferences filter the extensions. However, the result is not optimal since it may need to be refined again as shown in Example 4.

Proposition 8 Let T = (A, R, ≥) be a PAF where R is symmetric. If E ⊆ A is a preferred (stable) extension of system T then E is a preferred (stable) extension of (A, R).

Recall that this result is not true in case the attack relation is not symmetric as shown in Example 2.

The following result characterizes the extensions of (A, R) that are discarded in a PAF when R is symmetric. The idea is that an extension is discarded iff some argument outside it is strictly preferred to any arguments of that extension with which it conflicts. Property 6. Let T = (A, R, ≥) be a PAF s.t. R is symmetric, and E ⊆ A. E is a stable extension of (A, R) but not of

T iff ∃x ′ / ∈ E s.t. ∀x ∈ E, if xRx ′ , then x ′ > x.
When the attack relation is symmetric and irreflexive, the corresponding PAF is coherent (i.e. its preferred and stable extensions coincide) and it has at least one stable extension.

Proposition 9 Let T = (A, R, ≥) be a PAF. If R is symmetric and irreflexive, then:

-T is coherent.

-T has at least one stable extension.

Until now, we have proposed a particular framework for handling the first role of preferences. From now on, we will use the democratic relation for refining the results of this framework. Recall that this relation verifies the two postulates 5 and 6.

We will now show that when the preference relation ≥ is a linear order, then the democratic relation does not change the output of the underlying PAF.

Property 7. Let T = (A, R, ≥, ) be a rich PAF and S be the set of extensions (under a given semantics) of the repaired framework (A, R r ). If R is irreflexive and ≥ is a linear order, then Max(S, ) = S holds for stable, preferred, grounded and complete semantics.

It is also easy to show that when a rich PAF has no critical attacks, then its extensions are a subset of the extensions of its basic version (i.e. without preferences). Property 8. Let T = (A, R, ≥, ) be a rich PAF s.t. R has no critical attacks. Preferred (stable) extensions of T are exactly the elements of Max(S, ) where S is the set of all preferred (stable) extensions of the AF (A, R).

Example 1 (Cont):

Let us use the democratic relation d . In F 1 , there is no critical attacks (R r = R). The extensions of the rich PAF are Max({{a, c}, {b, d}}, d ) = {{a, c}}. Thus, {a, c} is the unique stable extension.

Example 2 (Cont):

The repaired framework of F 2 is ({a, b}, R r ) where bRa. Thus, the PAF has one stable extension {b} which is the only extension of the rich PAF: Max({{b}}, d ) = {{b}}.

Example 3 (Cont):

Recall that the repaired framework of F 3 has two stable extensions: {a, c} and {b, d}. Moreover, Max({{a, c}, {b, d}}, d ) = {{a, c}}. Thus, {a, c} is the unique stable extension of the rich PAF that uses the democratic relation.

Related work

Introducing preferences in argumentation frameworks goes back to the paper by Simari and Loui in [START_REF] Simari | A mathematical treatment of defeasible reasoning and its implementation[END_REF]. In that work, the authors have defined an AF in which arguments are built from a propositional knowledge base. The arguments grounded on specific information are considered as stronger than the ones built from more general information. This preference is used to solve dilemmas between any pair of conflicting arguments. Thus, it is used for handling critical attacks. The idea of this paper has been generalized in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF] then in [START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF] to any AF and to any preference relation. Unfortunately, the approach followed in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF] delivers correct results only when the attack relation is symmetric. When the attack relation is not symmetric, the approach suffers from two main drawbacks: the first is that it may return conflicting extensions as shown in Example 1 since it may put two conflicting arguments in the same extension. One of these arguments is clearly undesirable. The second drawback is a consequence of the first one. Indeed, since an undesirable argument may be accepted, then all the arguments that are defended by this argument are accepted as well at the detriment of good ones. Let us illustrate this issue on the following example. Assume that a > b. The approach in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF] gets the framework depicted in the right side of the same figure. Its grounded extension is the set {a, b, d}. This result is incorrect for two reasons: The first one is that the two arguments a and b cannot be both accepted.

The second reason is that the argument b (which should be rejected) defends d against c, leading thus to an undesirable result. Indeed, d is defended by a "bad" argument! It is easy to check that our approach returns {a, c} as the grounded extension and rejects the two other arguments: i.e. b and c.

Our approach overcomes the limits of the one proposed in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF]. Moreover, it is more general since it models even the second role of preference (i.e. the refinement). Recently, in [START_REF] Amgoud | Repairing preference-based argumentation systems[END_REF], the authors have pointed out the first limit of the approach followed in [START_REF] Amgoud | A reasoning model based on the production of acceptable arguments[END_REF][START_REF] Bench-Capon | Persuasion in practical argument using value-based argumentation frameworks[END_REF], namely the violation of conflict-freeness. They have proposed a new approach for handling critical attacks where preferences are introduced at a semantics level. As shown in this paper, the approach developed in [START_REF] Amgoud | Repairing preference-based argumentation systems[END_REF] satisfies the four rationality postulates. However, it completely neglects the second role of preferences, i.e. refinement. Another work which handles correctly the problem of critical attacks is that proposed in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF]. In that paper, Prakken has proposed a logic-based instantiation of Dung's framework in which three kinds of attacks are considered: rebuttal, assumption attack and undercut. For each relation, the author has found a way to avoid the problem of critical attack and ensured conflict-free extensions. We think that our work is more general since we solved the problem at an abstract level. This avoid the user who wants to use another attack relation to look for new ways to avoid conflicting extensions. Moreover, our approach is axiomatic, meaning that it is well founded. It is also worth mentioning that in [START_REF] Prakken | An abstract framework for argumentation with structured arguments[END_REF], the second role of preferences is neglected. To the best of our knowledge, the only work on refinement is that appeared in [START_REF] Dimopoulos | Extending argumentation to make good decisions[END_REF]. The authors have proposed a particular refinement relation in case of stable semantics. In this sense, our work is more general since it accepts any refinement relation. Moreover, there is no restriction to particular semantics. Finally, we would like to mention the work done in [START_REF] Kaci | Refined preference-based argumentation framworks[END_REF]. In this paper, the author made a survey of the critics presented in [START_REF] Amgoud | Repairing preference-based argumentation systems[END_REF][START_REF] Dimopoulos | Extending argumentation to make good decisions[END_REF] against existing approaches for PAFs. The author concluded that one should use a symmetric attack relation in order to avoid the problem of conflicting extensions and then to refine the result with the preference relation already mentioned in [START_REF] Dimopoulos | Extending argumentation to make good decisions[END_REF]. The first suggestion is certainly not realistic, especially in light of new results in the literature stating that symmetric relations should be avoided in logic-based argumentation systems.

Conclusion

This paper has studied deeply the difference between the two roles that preferences may play in an AF. We have shown that preferences intervene both for computing what is called standard solutions in nonmonotonic reasoning formalisms and for refining that result, and choosing a subset of those solutions. We have shown that the two roles are completely independent and should be taken into account in two steps. Main postulates that any approach modeling each role have been proposed. Finally, we have developed a particular framework that considers both roles. The framework satisfies the proposed postulates and its properties show that it is well-founded.

Definition 1 (

 1 consists of a set of arguments and an attack relation between them. Argumentation framework) An argumentation framework (AF) is a pair F = (A, R), where A is a set of arguments and R ⊆ A × A is an attack relation. The notation aRb means that a attacks b.

Example 1 Definition 3

 13 Let us consider the AF F 1 depicted below. It has two stable extensions: {a, c} and {b, d}. Let F = (A, R) be an AF and E 1 , . . . , E n its extensions (under a given semantics). Let a ∈ A. a is skeptically accepted iff there exists at least one extension and ∀i = 1, . . . , n, a ∈ E i . a is credulously accepted iff ∃i = 1, . . . , n s.t. a ∈ E i . a is rejected iff ∀i = 1, . . . , n, a / ∈ E i . Let Status(a, F ) be a function that returns the status of an argument a in F .Example 1 (Cont):The four arguments a, b, c, d are credulously accepted in F 1 .

  For arguments a and b, writing a ≥ b (or (a, b) ∈ ≥) means that a is at least as strong as b. The relation > is the strict version of ≥. Indeed a > b iff a ≥ b and not (b ≥ a).

Example 3

 3 Let us consider the argumentation framework F 3 , which is depicted below.

Postulate 2 (

 2 Recovering existing semantics) Let T = (A, R, ≥) be a PAF and F = (A, R) its basic version. If there are no critical attacks in T , then Ext(T ) = Ext(F ) where Ext(F ) is the set of the extensions of F under a given semantics.

Postulate 4 (

 4 Normal attacks) Let T = (A, R, ≥) be a PAF and a, b ∈ A. Let E 1 , E 2 be two conflict-free (wrt R) subsets of A s.t. E 1 = E ∪ {a} and E 2 = E ∪ {b}. If aRb and not(bRa) and not(b > a), then E 2 / ∈ Ext(T ).

Proposition 2

 2 t. b > a}. Then, T verifies Postulates 2 and 4. When the attack relation is symmetric, Postulates 1 and 3 are verified. Let T = (A, R, ≥) be a PAF s.t. Ext(T ) = Ext(F ) where F = (A, R r ). If R is symmetric, then T verifies Postulates 1 and 3.

Example 4

 4 Let A = {a, b, c, d}, R = {(a, c), (c, a), (a, d), (d, a), (b, c), (c, b), (b, d), (d, b)} and a > c, b > d.

Example 1 (

 1 Cont): Let us consider again F 1 and assume that a ≈ b and c > d. According to relation d , the two extensions {a, c} and {b, d} are incomparable. However, since a ≈ b and c > d, it is clear that one could prefer {a, c} to {b, d}.

  A. The extensions of T under a given semantics are the extensions of the argumentation framework (A, R r ), called repaired framework, under the same semantics with: R r = {(a, b)|(a, b) ∈ R and not (b > a)} ∪ {(b, a)|(a, b) ∈ R and b > a}.

Example 5

 5 Let us consider the AF depicted in the left side of the figure below.

A relation ≥ on a set A is complete iff for all a, b ∈ A, a ≥ b or b ≥ a.