A Riveiro

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R Comesaña

J Pou

STUDY OF MELT FLOW DYNAMICS AND INFLUENCE ON QUALITY FOR CO 2 LASER FUSION CUTTING

The understanding of melt flow dynamics during fusion laser cutting is still a topic of great importance because this determines the quality characteristics of the processed workpiece. Despite the complexity of the experimental study of the physical processes involved in this technique, fusion laser cutting can be visualized during the processing of glass because this material absorbs the laser radiation provided by a CO 2 laser but shows transparency to the visible radiation. Then, we present in this work the results of the study of the melt flow dynamics during laser cutting of glass. For different experimental conditions, the dynamics of the cutting front and its complete geometry (front wall inclination), and the evolution of the melt along the cut edge have been analysed by using a high speed video camera to study the process. It was found a phenomena no previously reported in the literature concerning to the ability of the plasma plume formed during the process. This can displace the normal shock wave (MSD) commonly formed in the inlet kerf being and can affect to the assist gas flow into the kerf.

On the other hand, the analysis of the recorded images allows determining the amount of molten material along the cut edge but also the direction and the velocity of the melt. Relevant processing parameters affecting to the flow of molten material were assessed. These results are used as a basis to explain the different processes involved in the generation of dross, a typical imperfection appeared in laser cutting.

INTRODUCTION

Laser cutting is one of the most widespread applications of lasers and a wellestablished and effective method of cutting a wide range of materials, mainly metals.

Different variants of the laser cutting technology involve (I) inert-gas fusion cutting, (II) reactive-gas fusion cutting (III) vaporization cutting, (IV) scribing and, (V) controlled fracture (1). However, the preferred techniques to cut metals are inert-gas or reactivegas fusion cutting processes. These are based on the simultaneous and coaxial action of a high-intensity laser beam and an inert or reactive gas jet onto the workpiece.

In this process, the laser beam melts the workpiece along its full depth and produces a cutting front which travels through the workpiece. The material on the cutting front is partially melted and evaporated by the laser beam, and then removed by the assist gas jet from the cut slot. This technique involves multiple and interdependent physical processes that affect the laser cutting results. Among others we can cite, the beam absorption in the cutting front (which depends on the laser wavelength, the temperature of the melt, the presence of oxides, the polarization state of the laser radiation, etc.), the fluid dynamics of the assist gas and the melt flow, the oxidation reaction of the melt, etc. (2-6). The practice of laser cutting posses a number of basic problems; one of the most important is the understanding of the reasons for defect formation (such as dross) and its suppression. This fact becomes essential for cutting some materials for demanding applications. Theoretical predictions of the laser cut quality is very complicated, requiring complex and sophisticated modelling approaches due to the great variety of parameters involved. For this reason, an experimental evaluation of the specific conditions of laser cutting of each material is needed to optimize the results in terms of cut quality.

An alternative and complementary approach to modelling is the visualization of the process. There are quite a few precedents to the visualization of laser cutting due to the intrinsic difficulties of filming a hot melt moving at high speed. First filming studies of laser cutting were conducted by the group of Prof. Yoshiaki Arata at Osaka University, Japan in the late 1970's (7, 8). They observed the cutting front with the aid of high-speed imaging during the CO 2 laser cutting of mild steel using oxygen. They observed the cutting front normally (at an angle of 30º downwards to the workpiece surface) and complemented these results also filming the lateral motion of the cutting front by cutting in the workpiece edge. They found that cyclical combustions occurred faster than by the direct action of the laser beam for low cutting speeds resulting in the formation of striations. However, the melt flow along the cut edge was not considered in their work; in fact, the understanding of this feature is essential to determine the resulting cut quality of the cut edge. On the other hand, the experimental arrangement of cutting in the workpiece edge is somehow unrealistic.

Poprawe and König (9) visualized the cutting process using a CCD camera coaxial to the laser beam for monitoring and control purposes. The motion of the melt was monitored but could not be resolved during the flow along the cutting edge. Yudin and Kovalev (10) observed laterally the process by cutting along the workpiece edge of a low melting point alloy sheet and putting a glass sheet to simulate the process in a more realistic way than the early work of Prof. Arata. They visualized the cutting front and the striation structure. However, they were cutting two different materials at the same time to be able to simulate the cutting process. In all these studies there is a lack of a detailed investigation on the dynamics of the melt flow, especially studies designed to reveal how the processing parameters affect to the motion of the molten material and to the cut quality results, such as dross formation.

The objective of this work is the visualization of the laser cutting process of glass in continuous wave (CW) mode (i.e.: the energy provided by the laser beam is kept constant with time). The first part of this paper is devoted to the visualization of the cutting process and to the description of the relationship between the main processing parameters and the cutting front dynamics and melt flow pattern along the cut edge. A comprehensive survey of the phenomena occurred during laser fusion cutting is presented because the understanding of this feature is essential to determine the resulting cut quality. In the second part, we will establish the mechanism that explains how the dross is formed according to the different melt flow patterns previously described. These results are discussed taking into account the physical mechanisms governing the process and an insight, based on a similarity approach, on the possible transfer of the obtained results to the process of laser cutting of metals has been provided.

EXPERIMENTAL SETUP

The experimental set-up used for the experiments is shown in Fig. 1a. All experiments were performed with a 3.5kW CO 2 slab laser (Rofin-Sinar), the laser mode being a TEM 00 . The laser beam was focused onto the surface of the workpiece using a 127 mm focal length lens. Tests were conducted in continuous wave (CW) and varying one factor at a time. The ranges of cutting parameter are summarized in Table 1. A commercial cutting head incorporating a conical converging coaxial nozzle with a 2 mm exit diameter was used to supply the assist gas jet into the interaction zone. Argon was used as inert assist gas.

In order to visualize the cutting process, we have followed a similar approach to that described by Arata et al. (11,[START_REF] Arata | Fundamental phenomena in high power CO 2 laser (Report II) : Vacuum laser welding[END_REF] and Jin and Li (13) to observe the keyhole shape during laser deep penetration welding. As depicted in Fig. 1, we have used a soda-lime silicate glass of the 'float glass' type, 6 mm thick, as workpiece. This material strongly absorbs the laser radiation emitted by the CO 2 laser but is transparent to the visible radiation; therefore, a realistic and very clear picture of the process can be obtained. The survey was performed using a CMOS fast camera (Photron FASTCAM-1024 PCI) recording at 10,000 frames per second, a shutter speed of 1/402000 s, and with contrast illumination using a halogen lamp behind the sample faced towards the camera. The maximum resolution achieved was 128x272 for this frame rate. The camera was placed about 180 mm from the workpiece and positioned in the transverse direction to the cutting direction, as represented in Fig. 1a, to observe the behaviour of the cutting front and the flow of molten material along the cutting edge. In order to record fine details of the cutting process a macro objective (Sigma EX DG Macro) with a F2.8 maximum aperture was used.

The camera was connected to a PC in order to record and process the images of the cutting procedure. The influence of the processing parameters on the cutting front geometry was determined by means of the evaluation of the cutting front inclination (see Fig. 1b). The slope of the cutting front was assessed in several points and a mean inclination was calculated. On the other hand, two main parameters were evaluated to characterize the flow of molten material along the cutting edge in selected images of the process: the flow direction and the extension of the molten material flowing along the cut edge. These parameters were measured using the free image processing and analysis software ImageJ on selected frames in recorded high-speed videos.

RESULTS AND DISCUSSION

CUTTING FRONT GEOMETRY

Fig. 2 shows the typical aspect of the cut kerf during the laser cutting process.

As can be observed, the profile of the cutting front does not match that of the laser beam. This becomes curved in the lower part depending on the processing parameters.

As depicted in Fig. 2b, and quantified in Table 2, the cutting front exhibits a larger curvature when the cutting speed is increased, mainly, in the lower part. Due to the fact that the laser beam was focused close to the surface of the workpiece, the irradiance in the lower part of the cutting front is lower than in the upper part, resulting in a curved profile. The observed dependence of the cutting front inclination is in good agreement with theoretical predictions reported in the literature (14, 15). However, additional influences on the inclination, not considered in previous works, were observed. As summarized in Table 2, cutting front inclination also depends on the laser power, cutting speed, and on the assist gas pressure. In order to take into account the combined action of the laser power and cutting speed we have used the linear energy density. The inclination of the cutting front is seen to increase when the linear energy density decreases (see Table 2). However, an increase of the assist gas pressure increments the inclination probably due to the energy losses consequence of the higher removal of molten material.

MOLTEN MATERIAL DYNAMICS

The interaction of the laser beam with the cutting front produces a thin film of molten material as observed in Fig. 2a. The high brightness of this melt indicates the superior temperature in this region. The melt is subjected to the removal action of the assist gas by means of two mechanisms: the pressure gradient and the shear stress along the cutting front. Both mechanisms tend to remove the molten material in a direction parallel to the cutting front (Z-direction as depicted in Fig. 2) (2). On the other, hand, as observed in Fig. 2a, the thickness of this film is not uniform, probably as a consequence of Kelvin-Helmholtz instabilities produced by the high speed assist gas stream adjacent to the molten film (16).

In laser cutting, the laser beam has enough irradiance, typically in a range from 10 6 to 10 8 W/cm 2 to partially evaporate the material on the cutting front (1, 17).

Therefore, the vapour is ejected perpendicularly to the surface of this cutting front.

Evaporation produces a uniform pressure on the molten material generated in cutting front; this action, so-called recoil pressure (p s ), depends on the temperature of the molten material according to the Eq. (1) (18):

0 exp v v s B s s B m L p A k T T ⎛ ⎞ = - ⎜ ⎟ ⎝ ⎠ ( 1
)
where A is a factor which depends on the ambient pressure, B is an evaporation constant, T s the temperature of the molten material on the cutting front, m v the mass of the vapour molecule, L v the latent heat of vaporization and k B the Boltzmann constant.

The recoil pressure ejects the molten material along the cut edge in a direction perpendicular to the cutting front (Y-direction) in Fig. 2. As shown in Fig. 3, flow streams ejected from the cutting front, and with higher temperature than the rest of the melt, flow along the cut edge. Occasionally, the molten material is accelerated from the cutting front after an energetic explosion, probably due volumetric evaporation into the melt (see Fig. 4) (19).

The velocity of the molten material ejected from the cutting front was estimated using the bright flow streams periodically ejected from the cutting front (as seen in Fig. 3). Several measurements were accomplished and an average value was extracted and assumed to be the flow velocity of the molten material. The ejection velocity of molten material from the cutting front by the recoil pressure was found to depend on the processing conditions because, as shown in Eq. ( 1), this force depends on the temperature of the molten material. Furthermore, this velocity varies as a function of the studied region in the cutting front. This indicates the existence of differences in the temperature as a function of the considered region of the cutting front. Under all tested processing conditions, the molten material generated in the upper part of the cutting front is mainly dragged downwards by the assist gas. In this part of the cutting front the effect of the recoil pressure is surpassed by the action of the gas. On the opposite, the recoil pressure in the lower part of the cutting front is higher. This is in accordance with the predicted increment in temperature in the lower part of the cut front which, in turns, produces a more intense evaporation regime (20, 21). Flow velocities of molten material around v y =500 mm/s estimated for a laser power of P=1400 W and a cutting speed of v c =500 mm/min. However, if the cutting speed is increased to v c =1000 mm/min, the ejection velocity rises up to v y =2300 mm/s. The dependence of the ejection velocity with the cutting speed is a consequence of the predicted increment in the cutting front temperature with the cutting speed (14, 22) which promotes a more intense evaporation and therefore a higher recoil pressure as deduced from Eq. ( 1) (18). On the other hand, the ejection velocity was also revealed to be dependent on the laser power, being increased with this parameter also due to the increment in the temperature and evaporation intensity.

Finally, the analysis of the processes does not reveal cyclic combustion waves in the cutting front as in the process of laser cutting of steel using oxygen (see for example

Refs. (23-25)) because the assist gas cannot produce a further oxidation on the molten glass

FLOW OF MOLTEN MATERIAL ALONG THE CUT EDGE

After being ejected backwards by the recoil pressure (in the Y-direction), the molten material flows along the cut edge only subjected to the sweeping action of the assist gas. Then, the direction of the main force acting on the melt changes and the gas propels the melt in a direction almost parallel to the cutting front (Z-direction) because the gas flow in the kerf is inclined. As described by Kovalev et al. (26), the streamlines for the assist gas are substantially deflected from the nozzle centreline into the kerf due to the interaction of the jet with the cutting front. Furthermore, the use of a converging coaxial nozzle can induce the phenomenon of gas flow separation from the cut front and the formation of a reverse flow into the kerf (26-29). This produces a significant decrease in the flow velocity of the assist gas and a reduction in the removal of molten material by the gas.

If the melt is not completely extracted by the assist gas from the cut kerf, it will solidify on the cut edge resulting in cuts with poor finishing and a noticeable heat affected zone (HAZ). An interesting parameter to determine the ability of the gas to remove the molten material from the cut edge is the extension of molten material flowing along it. As expected, the longer the extension of molten material on the cut edge, the poorer the surface finishing of the cut edge due to the higher tendency to the presence of resolidified material, dross and a large HAZ. Experimental tests allow us to determine the influence of processing parameters on the extension of the molten material flowing along the cut edge (see Table 3). In one hand, the extension is increased with the laser power because the rate of melting is higher and the ejection backwards the cutting front due to the high recoil pressure too. Then, these actions surpass the reduced removal of molten material by the assist gas, consequence of its separation from the cutting front. Also it is noticed that the extension of molten material is increased with the cutting speed, probably due to two reasons. First, the rate of melting is promoted by the heat conduction loses which rise with the cutting speed (14, 30); second, the higher temperatures reached on the cutting front, when the cutting speed is increased, produces a more intense evaporation (14). Consequently, the higher recoil pressure expels more molten material backwards the cutting front. Combined action of these effects promotes a larger amount of molten material on the cut edge (for a constant removal action of the assist gas). This generates cuts with poor finishing because the amount of resolidified material will be larger.

On the other hand, visualization results confirm the primary role of the assist gas on the process. The increment of the assist gas pressure tends to remove a larger amount of molten material and confine it too near to the cutting front. Hence, the amount of resolidified material decreases and the cut quality is improved as experimentally confirmed in cutting different materials (see for example Ref. (31)). However, it is necessary to considerer the increment in the strength of the normal shock wave produced with the increment in the assist gas pressure. As observed in Fig. 5a by means of shadowgraph imaging, a normal shock wave, called Mach Shock Disk (MSD), is formed on the workpiece, principally, as consequence of the normal impingement of the jet and due to the superior dimensions of the jet as compared to the cut slot. This shock wave is a discontinuity that reduces the velocity of the gas and, in consequence, the removal of molten material. As observed in Fig. 5b, the increment in the stand-off distance decreases the normal shock wave but increases the extension of molten material on the cut edge. Despite several works explain the formation of this normal shock wave and the consequences of this phenomenon on the removal of molten material during laser cutting, this is the first report of experimental observation of this phenomenon during a real laser cutting process.

On the other hand, as observed in Figs. 3 and5, the vaporized material in the cutting front becomes ionized and a plume of plasma emerges from the upper face of the workpiece. Despite, the experimental tests were performed in CW mode, the plasma plume exhibits a periodic nature as clearly depicted in Fig. 3. Experiments show that at low assist gas pressure (p=3 bar), the plume is generated and disappears, whereas at higher pressures (p=8 bar) the plume is constantly present, but the height of the plume fluctuates. In both cases, the frequency of ignition-extinction and the oscillation in height was found to be around f=680 Hz under the tested processing conditions. The influence of the assist pressure on the plume dynamics can be attributable to the swirls generated in the upper face of the workpiece due to the normal impingement of the jet.

These swirls produce a higher exposition time of the assist gas to the laser radiation and to the vaporization products formed in the cut kerf; therefore, the trend to ionization is increased resulting in a constant presence of the plasma plume. This non-periodic nature of the plume indicates the unsteady character of the laser cutting in CW; this must be considered a quasi-steady process. On the other hand, it was found that the plasma plume is able to displace the MSD up to 140 μm as depicted in Fig. 6.

This conclusion was corroborated observing the non-oscillatory character of the shock wave when the laser radiation was interrupted. As observed in Fig. 7, the position of the MSD remains stable when the cutting process stops and the laser beam does not interacts with the cutting front. Hence, despite the unsteady behaviour of an impinging jet flow structure (see for example Refs. (32, 33)), the imperceptible deviation in the position of the normal shock wave with regard to the workpiece when the laser is interrupted confirms the primary role of the plasma plume on the motion of the shock.

This fact, no previously reported in the literature, makes the MSD to oscillate with the same frequency than the plasma. Despite it is asserted that the effect of plasma is negligible in CW mode cutting (34), this oscillation of the plume can affect to the dynamics of the assist gas into the cut kerf because the shock wave pattern inside the cutting kerf is modified. Moreover, the separation point of the assist gas can fluctuate with this frequency. Therefore, this can affect the removal of molten material by the gas, inducing periodic oscillations on the flow of molten material along the cut edge and cutting front which could explain some unsolved questions regarding the striation formation during laser cutting using inert gases.

Analysis of recorded images also indicates that the flow direction of the molten material along the cut edge takes place at an angle of θ=30-50º with respect to the cutting front depending on the assist pressure. At low gas pressure (p=3 bar in this work), the molten material flows at an angle of θ=50º (see Fig. 8a); however, if the supplying pressure is higher (p=8 bar), the molten material flows with less inclination, at an angle of θ=30º (see Fig. 8b). This modification in the trajectory reduces the amount of melt flowing along the cut edge and the thermal input to the workpiece. This is the reason of the improvement in cut quality experienced during the increase in the assist gas pressure, as corroborated by different authors (31, 35).

It is necessary to take into account the influence of the laser power on the flow velocity of molten material along the cut edge. A detailed survey of the process reveals a faster motion of the molten material for high laser power as summarized in Table 4.

The increment in the laser power makes more fluid the melt due to the reduction of the kinematic viscosity with the temperature; hence, this is removed at higher rate by the assist gas. On the other hand, the increment in the assist gas pressure also promotes the removal of molten material; as reported in Table 4, the flow velocity of molten material along the cut edge increases with the pressure because the viscous friction and the pressure gradient arise (2). In summary, an increment in the assist gas pressure is necessary to remove a larger amount of molten material although it is also necessary a minimum laser power to make it enough fluid.

Finally, behind the region of molten material on the cut edge is a quasi-solidified layer of molten material. This material is part of the molten material which previously was flowing along the cut edge and was not removed by the assist gas. This material flows at low velocity due to its high viscosity and its location at a region of the cut edge too far from the direct action of the assist gas. The analysis of the recorded images reveals a velocity flow around 4.5 mm/s for a laser power of 800 W, a cutting speed of 500 mm/min, and an assist pressure of 8 bar. It is worth to compare this velocity with that of the molten material under the same processing conditions (see Table 4) which is approximately 25 times higher. This suggests that the cooling of the molten material makes very difficult to completely remove it from the cut edge and obtain clean cuts.

Therefore, it is very important to remove the molten material as soon as possible from the cut edge or avoid as much as possible a high extension of molten material on the cut edge to obtain cuts with good quality (low HAZ, low dross and low resolidified material).

DROSS FORMATION

Dross formation is one of the typical imperfections generated on the cut edges.

The process is a consequence of the agglomeration of molten material which flows along the cut edge in the lower wedge of it. The sequence of images presented in Fig. 9 reveals the formation process of this imperfection. As observed, a drop of molten material is generated in the lower part of the cutting front by the molten material flowing along the cut edge. The drop size is higher as more molten material arrives. On the other hand, the previously cited separation of the assist gas flow from the cutting front, produces a thin film of molten material flowing along the cut edge. This film induces the formation of more drops behind the drop previously formed near to the cutting front (see Fig. 10). Then, these drops can merge. The drop closer to the cutting front was seen to move in a direction contrary to the cutting front propagation probably due to the surface tension gradients induced by the temperature gradient. Due to the fact that the temperature diminishes along the trailing edge of the cut edge, backwards the cutting front, the lower surface tension in the hot surface of the molten material near to the cutting front pulls the liquid backwards the cutting front to the cold region.

It is worth to mention that the motion of the drop is more probably produced due to surface tension gradients than to the action of Y-component of the assist gas, because the analysis of the images depicted in Fig. 10 reveals a slow motion of the drops. The velocity of these were estimated by means of the analysis of the images. An average value of the velocity around 0.03-0.6 m/s was found. However, the Y-component of the assist gas, which may produce this motion of the drop, can be estimated around 0 V cosϕ =40-170 m/s (being V 0 the average velocity of the assist gas and φ the deflection of the gas jet with regard to the nozzle centreline). This velocity was estimated taking into consideration an average value V 0 =340 m/s for the gas jet and an inclination of the gas similar to that of the cutting front inclination (φ varying from 6.5º up to 30,4º as reflected in Table 2). A simple comparison of the orders of magnitude suggests that the motion of the drop should be faster if it is a direct consequence of the Y-component of the assist gas. Then, it is probably consequence of surface tension gradients.

While the drops are solidifying, they are subjected to the shear action of the gas which can induce three main processes on them: deformation, breaking-up or detachment. The occurrence of each phenomenon primarily depends on the Weber number (We) for the drop (36), which is a measure of the relative importance of the fluid's inertia ρu m 2 as compared to the counteracting capillary pressure σ/d:

2 m (T )u (T )d We (T ) ρ σ = (2)
where, ρ(T) is the density of the molten material at the corresponding temperature in the cut edge, u m (T) the velocity of the molten material induced by the shear action of the gas (which depends on the value of the viscosity of the molten material at the corresponding temperature in the cut edge), d the drop diameter and σ(T) the surface tension of the molten material at the corresponding temperature in the cut edge.

If the capillary forces become comparable with the inertia of the melt (We ≈ 1), the separation of melt flow from the lower part of the cutting front is not total and tendency to dross formation is very high. The appearance of dross is related to small values of the Weber number. In order to have dross free cuts, the Weber number of the melt, in the lower part of the cutting front, must exceed some critical value. The values for the critical Weber number available from the literature are scattered over a very wide range of We from 2.2 to 99.6. (37). However, it is generally agreed that the minimum critical Weber number increases with the viscosity of the drop; this dependence is usually expressed through the Ohnesorge number:

Oh d μ ρσ = (3)
where, μ is the viscosity of the molten material at the corresponding temperature in the cut edge (38). Then, as the viscosity of the melt increases, while the molten material is solidifying, the momentum that the assist gas transfers to the melt must be very high to break up the drop and produce dross free cuts.

For Weber numbers in a range from the unit to the critical value, the drop only tends to be deformed and elongated and can be partially break up. Under this regime, the presence of dross is also constant.

The visualization of the dross formation process, under different processing conditions, reveals that the predominance of these processes primarily depends on the distance of the drop on the cut edge to the cutting front. This is because the molten material is hotter when is nearer to the cutting front than in positions far away from it.

Thus, the physical properties, namely, the viscosity, density, surface tension and consequently the velocity of the melt, vary as a function of the position of the drop on the cut edge. Finally, the variation of these properties modify the Weber number of the melt in different locations of the lower part of the cut edge. It is expected that near to the cutting front, the higher temperature of the melt increases the Weber number because the velocity of the melt is higher (due to its lower viscosity) and the surface tension is lower. Therefore, the molten material tends to be broken-up (see Fig. 11) and small droplets can be formed as observed in Fig. 12. This process, primarily occurring near to the cutting front, produces small particles with different size. This process can be identified as the main responsible for the well-known formation of micro-particles, observed during laser cutting of steel, ranging from 70 to 400 μm during laser cutting (39).

On the other hand, far from the cutting front, the lower velocity of the melt (due to the higher viscosity) and the higher surface tension makes the value Weber number very low. Then, the molten material principally tends to be elongated as observed in Fig. 9 and the presence of dross is persistent.

From these results we can assert that the separation of the assist gas from the cutting front, which produces a film of melt along a large area on the cut edge, tends to produce dross because the molten material far away from the cutting font has Weber number close to the unit. Therefore, we must confine the molten material near to the cutting front in order to have a melt with large Weber numbers and to obtain dross free cuts. An option to produce this confinement is promoting the impact of the assist gas jet onto the cutting front being as close to 90º as possible. Therefore, the assist gas jet should be injected by means of an oblique nozzle with regard to the laser beam.

COMPARISON TO METAL CUTTING

Finally a question on similarities can be arisen. Despite the results presented in this paper have been focussed on the laser cutting of glass, some common features with the process of cutting metals are evident. As observed in the cut edges depicted in Fig. 13, a clear parallelism between the cut characteristics of metals and glass can be draw.

Both kinds of materials exhibit common features such as the generation of striations and dross. The extension of results can be roughly done by means of the comparison of the thermophysical properties of metals and glass, namely the following:

1. Despite metals have a smaller absorption length as compared to glasses, metals can compensate this by a much higher thermal conductivity (for example, the thermal conductivity of soda lime glass is k g =1.7 W/mK, while for typical metals such as aluminium is σ al =247 W/mK and for steels σ s =52 W/mK (40)).

Then, the order of magnitude of the molten material generated in the cutting front by the action of the laser beam would be similar.

2. Furthermore, the high viscosity of the molten glass, which embarrasses the ejection of the molten material from the kerf, is also observed when metals cutting. Although metals have a small viscosity, their high thermal conductivity produces a rapid cooling of the molten material and consequently an increment in their viscosity; this being comparable, up to certain extent, to the case of cutting glass.

3. However, the smaller viscosity of metals as compared to glass (neglecting the influence of the thermal conductivity), induces lower Reynolds numbers for glass from μ g =10 5 to 10 9 P while for metals ranges from μ m =10 -3 to 10 -1 P). Therefore, the turbulent characteristics of the flow of molten material should be more pronounced in the case of cutting metals than glass because the Reynolds number will be, in principle, higher during the processing of metals than glass.

This assertion can be roughly validated comparing the more turbulent flow of molten material observed during laser cutting of metals (see for example Refs.

(7-10)) than in the case of cutting glass, as previously presented.

4. On the other hand, the surface tension of soda-lime glass is one order of magnitude lower than metals (for example, the surface tension of soda lime glass is σ g =315 mN/m at 1573 K, while for typical metals such as aluminium is σ al =1070 mN/m at 933 K and for iron σ fe =1900 mN/m at 1573 K). Then, the Weber number for glass because the density of glasses and metals have the same order of magnitude (for example, the density of soda lime glass is ρ g =2.5 g/cm 3 , while for typical metals such as aluminium is ρ al =2.7 g/cm 3 at 933 K and for mild steel and stainless steel ρ msteel =7.85 g/cm 3 , ρ ssteel =8.03 g/cm 3 respectively). Then, the lower Weber number of glasses as compared to that of metals indicates a higher tendency for the deformation of the molten droplets. This can be corroborated by the elongated character of the dross as compared to the less pronounced dross present in the case of metals. Therefore, this is the main difference between dross formation for metals and glass; however, the general overview of the process observed in this work for glass could be transferred to metal cutting.

In summary, most of the results obtained in this paper, regarding the physical mechanisms governing the flow of molten material and the dross formation, should be valid for metal cutting under certain restrictions.

CONCLUSIONS

Fundamental phenomena during high power CO 2 laser cutting of glass were dynamically observed and the flow of molten material during the process was analysed.

Furthermore, the influence of the flow of molten material on the quality results, mainly, the formation of dross during the process, was dynamically assessed.

The visualization experiments were performed during laser cutting of glass because the material strongly absorbs the 10.6 μm laser radiation but it is transparent to the visible radiation. Then, it is possible to clearly observe the process.

The main conclusions from the visualization experiments can be summarized as follows:

(i)

The cutting front geometry varies as a function of the processing parameters in order to balance the different energy contributions.

(ii)

The material melted in the cutting front was seen to flow along the cut edge.

The recoil pressure ejects the molten material backwards the cutting front. The intensity of the force depends on the temperature of the cutting front.

Estimations on the ejection velocity of the molten glass were provided by means of image analysis.

(iii) After being ejected, the molten material flows along the cut edge at an angle of 30º-50º with regard to the cutting front. The flow pattern also depends on the processing conditions. The assist gas pressure was seen to reduce the amount of molten material flowing along the cut edge. However, the intensity of the MSD was seen to increase.

(iv)

The plume of ionized material was observed to oscillate with a frequency around 680 Hz. An interesting issue is that, these oscillations can produce variations on the position of the MSD which will affect to the assist gas flow into the kerf and to the removal of molten material. This result has a strong relevance during high pressure laser cutting.

(v)

The formation of dross was seen to be a cyclical process. A drop of molten material is formed in the lower wedge of the cut edge. If the shear action of the gas is not able to detach the drop, this can solidify producing cuts with dross.

(vi)

The detachment of the drop, and in consequence the obtaining of dross free cuts, was seen to occur at a critical Weber number. Large Weber numbers ensures the detachment of the molten material.

(vii)

The process of dross generation was seen to be amplified by the flow of a large amount of molten material on the cut edge.

(viii)

As the Weber number of the drops vary as a function of the distance from the cutting front due to the variation on the temperature of the molten material, different phenomena takes place. Near to the cutting front, drops tend to breakup producing the particulate emissions typical in laser cutting. However, drops far from the cutting front tend to be elongated due to the higher viscosity of the molten material.

Finally, under a basis of similarity analysis it is suggested that the results obtained in this work could be transferred, up to certain extent, to the laser cutting of metals.

ACKNOLEDGEMENTS

The authors wish to thank to technical staff from CACTI (University of Vigo) for his help with sample characterization. This work was partially supported by the Spanish

New drop

Elongated drop previously formed

Figure 12 Image showing the formation of a small particle due to the rupture of molten material (Processing conditions: P=1400 W, v c =500 mm/min, Argon, p=8 bar).

  is the density, u the flow velocity, d a characteristic dimension and μ the dynamic viscosity of the molten material and the subscript g refers to glass) than for metals

Figure 1 .

 1 Figure 1. a) Experimental setup used for the visualization of the cutting process (view from above) and b) parameters evaluated in a typical recorded image.

Figure 2 .

 2 Figure 2. a) Scheme showing the typical image of the cutting front and b) evolution of the cutting front geometry as a function of the cutting speed (Processing conditions: P=1400 W, p=8 bar).

Figure 3

 3 Figure 3 Image sequence showing the ejection of molten material from the cutting front and along the cut edge (Processing conditions: P=1400 W, vc=1400 mm/min, Argon, p=8 bar).

Figure 4

 4 Figure 4 High speed video imaging showing the existence of regions with explosive evaporation probably due to boiling. (Processing conditions: P=1400 W, v c =1000 mm/min, Argon, p=3 bar).

Figure 5

 5 Figure 5 Examples of the MSD formation and weakeing for two stand-off distances; a) z=1,5 mm y b) z=3 mm (Processing conditions: P=1400 W, v c =1000 mm/min, Argon, p=8 bar).

Figure 6

 6 Figure 6 Examples of displacement of the MSD by the plasma plume (Processing conditions: P=1400 W, v c =1000 mm/min, argon, p=8 bar).

Figure 7

 7 Figure 7 Example of the non-oscillatory character of the MSD when the laser radiation is interrupted (Processing conditions: P=1400 W, v c =1000 mm/min, argon, p=8 bar).

Figure 8

 8 Figure 8 Selected images of the cut front which reveals the difference in the flow direction as a function of the assist gas pressure: a) θ=50º (p=3 bar), b) θ=30º (p=8 bar) (Processing conditions: P=1400 W, v c =1000 mm/min, Argon).

Figure 9

 9 Figure 9 Image sequence showing the formation of dross: a) formation of a droplet, b) growing, c) merging of the droplet with previous molten material d) complete merging and e) formation of another droplet (Processing conditions: P=1400 W, v c =1000 mm/min, Argon, p=3 bar).

Figure 10

 10 Figure 10Image of the formation of drops of molten material behind the cutting front which are elongated by the assist gas (Processing conditions: P=1400 W, v c =1000 mm/min, Argon, p=3 bar).

Figure 11

 11 Figure 11 Image sequence showing the rupture of molten material and the formation of small droplets in the lower part of the cutting front (Processing conditions: P=1400 W, v c =1400 mm/min, Argon, p=8 bar).

Figure 12

 12 Figure[START_REF] Arata | Fundamental phenomena in high power CO 2 laser (Report II) : Vacuum laser welding[END_REF] Image showing the formation of a small particle due to the rupture of molten material (Processing conditions: P=1400 W, v c =500 mm/min, Argon, p=8 bar).

Figure 13

 13 Figure[START_REF] Jin | An experimental study on the keyhole shapes in laser deep penetration welding[END_REF] Typical aspect of the cut edge of workpiece of a) a glass and b) an aluminium-copper alloy using argon as assist gas.

Figure 1 .Figure 2 Figure 3 Figure 4 Figure 5 Figure 7 Figure 8 Figure 9

 12345789 FIGURES

Figure 10

 10 Figure 10 Image of the formation of drops of molten material behind the cutting front which are elongated by the assist gas (Processing conditions: P=1400 W, v c =1000 mm/min, Argon, p=3 bar).

Table 2 .

 2 Mean inclination of the cutting front as a function of the main processing parameters.

	Government, (CICYT MAT2006-10481 and FPU program AP2006-03500 grant), and
	by	Xunta	de	Galicia	(PGIDIT05TMT30302PR,	PR405A2002/11-0,
	INCITE081R303002ES and INCITE08PXIB303225PR).

Table 3 .

 3 Extension of the molten material flowing along the cut edge as a function of the processing parameters.

	P=3 bar	P=8 bar

Table 4 .

 4 Flow velocity of molten material along the cut edge as a function of the laser power and the assist gas pressure

	Laser Power (W)	V (mm/s)
		p=3 bar	p=8 bar
	800	90	110
	1400	700	1400

TABLES

Figure 13 Typical aspect of the cut edge of of a) a glass plate and b) an aluminium-copper alloy plate using argon as assist gas.

mm mm


Assaghir Zainab

Girardin Philippe
email: girardin@colmar.inra.fr

Napoli Amedeo
email: napoli@loria.fr

Fuzzy logic approach to represent and propagate imprecision in agri-environmental indicator assessment

Keywords: Possibility theory, fuzzy set, fusion, maximal coherent subset, interval analysis, indicator

The indicator of groundwater contamination developed and used in agriculture is calculated from data available in the field or data estimated by an expert. The modeling of this indicator generally requires a large number of parameters whose measure is imprecise. Several information sources provide information about the same imprecise quantities which have to be combined for defining what is called an "indicator of groundwater contamination" (Igro). This indicator estimates the impact of cultivation practices on the groundwater contamination. In this paper, we explore a possibilistic information fusion method by using the notion of maximal coherent subsets to represent the imprecisions of multisource variables of the indicator. We also calculate the bounds of this indicator, and we propagate imprecision by using an interval analysis. Finally, we present the indicator's results for pesticides applied on different crops.

Introduction

Data used in application domains are often imperfect: imprecise, uncertain, incoherent... Some data come from information sources (database, expert) and others are approximative measurements. When multiple sources -such as a database, an expert, a website, a book-deliver information about some unknown quantity, aggregating this information can be a tedious task, especially when information are incoherent. Actually, there are many approaches to specify the impact of input parameter imprecision of a model, such as statistical sampling methods, the Monte-Carlo method [START_REF] Metropolis | The monte carlo method[END_REF], and bayesian method. These methods require either a significant number of data and a lot of computation time, or an exact definition of the statistical properties of the input parameters. There exists also alternative approaches based on fuzzy set and possibility theory [START_REF] Zadeh | Fuzzy sets[END_REF][START_REF] Dubois | Théorie des possibilités[END_REF] to express, in a non-probabilistic sense, imprecision of parameters. Each approach represents parameters imprecision in a specific way. When, we need to aggregate data from multiple sources, fuzzy approaches show more flexibility in the treatment of incoherent information. Recently, Destercke et al [START_REF] Destercke | Fusion using maximal coherent subset[END_REF] explored a possibilistic information fusion using the notion of maximal coherent subsets to synthesize information from several incoherent data. In this paper, we will use this method to synthesize an information from several information sources to calculate variables used in indicator assessment. An indicator measures a certain aspect of the agrosystem. It is proposed to help farmers to improve the environmental sus-tainability of their agricultural practices and to reduce the pollution of environment such as indicator of groundwater contamination I gro . It estimates the possibility for a pesticide to reach groundwater through leaching. The modeling and the assessment of an indicator generally require a large number of parameters whose measure is imprecise such as the variable "soil depth", that indicates how thick the soil cover is. Multiple sources provide information about the imprecise quantities which compose the indicator I gro , for example, pesticide characteristics such as pesticide half-life DT 50. The calculation of I gro is based on a system of decision rules using fuzzy sets [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF]. Authors in [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF] do not take into account the imprecision in parameters of the indicator. In this paper, we present an original work that, at our knowledge, has not been carried out until now. We propose here an analysis of imprecision of the indicator, especially variables used and their nature. We are interested in imprecision of parameters provided by several sources. Then, we use imprecision for evaluating the amplitude and the effect of imprecision on the value of the indicator I gro . Firstly, we change the input values of variables in the process of calculation. Then, we use intervals of values in place of single values of input variables in the calculation process of the indicator I gro . We also explore a fusion method, using the notion of maximal coherent subset, for summarizing information about the quantity given by several sources [START_REF] Destercke | Fusion using maximal coherent subset[END_REF]. Finally, we calculate a variation interval for the indicator I gro given by the input intervals of the variables. The paper is divided as follows: the first section presents the indicator of groundwater contamination I gro , its definition, structure and data used. Theoretical preliminaries are introduced in the third section. Then, the fourth section describes the fusion method used and the way how maximal coherent subsets can be used to obtain intervals of the imprecise variables; we also explore a way to compute the lower and upper bounds of the indicator. Then, the fifth section presents the indicator of groundwater contamination, its definition and its structure. Finally, the sixth section shows the results of the indicator for four pesticides applied in different crops.

Application description

In this section, we define the indicator I gro , its structure and data used in the process of calculation.

The risk of groundwater contamination I gro

Definition and data used

The indicator module I gro reflects the potential of a pesticide to reach groundwater through leaching and to affect its potential use as a source of drinking water. I gro as proposed by Van der Werf and Zimmer [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF] depends on four input variables : (1) pesticide leaching potential (GUS); (2) position ; (3) leaching risk, and (4) toxicity of the pesticide for humans (based on Acceptable Daily Intake). Variable "pesticide leaching potential" (GUS) provides an estimation of the risk of leaching of the compound. This potentiel is calculated by the formula:

GU S = log(DT 50) * (4 -log(koc)) (1)
where variable DT50 is the pesticide half-life and "koc" is the organic-carbone partition coefficient. Variable DT50 is the time required for the pesticide concentration under defined conditions to decrease to 50% of the amount in application. Variable "koc" is the organic-carbon constant that describes the tendency of a pesticide to bind to soil particles. Variable "position" is the position of application of the pesticide (on the crop, on the soil, in the soil). This position is the interception rate of active ingredient per leaf area of studied culture. It is obtained from variable "soil cover". Variable "soil cover" is a value thats depends on the leaf surface of the culture. The variation of "soil cover" over time is fixed.

Then, the position depends on the date of application by the farmer. The variable "leaching risk" depends on characteristics of the soil. In I gro , the estimation of soil "leaching risk" is given by experts on a scale between 0 (minor leaching risk) and 1 (major leaching risk). The "position" and "leaching risk" take values between 0 and 1. The variable "Acceptable Daily Intake" (ADI) reflects chronic toxicity from humans. The pesticide properties (DT50, koc, and ADI) are given by several heterogeneous information sources such as Agritox (database, France), RIVM (database, Netherlands), Pesticide Manual (book, england), Dabene (expert, France), etc . . . . For each pesticide, these infomation sources give fuzzy intervals for each characteristic. Let us consider the following example: Three sources (Pesticide manuel: a book, Agritox: a database, Dabene: an expert) provide information, about the variable "DT50" (pesticide half-life) of "I gro ", in term of two intervals Int 1 and Int 2 . The information is summarized in Table 2 and represented on Figure 1. The interval Int 2 represents the most plausible values for the variable DT50.

The structure of the indicator of groundwater contamination I gro

The indicator of groundwater contamination I gro is calculated on a scale between 0 (maximal risk) and 10 (minimal risk). These values are calculated according to fuzzy if-then rules and to the degree of membership of the input variables to fuzzy subsets. The mechanism is explained below.

To calculate the indicator I gro , we need to aggregate expertise knowledge and variables heterogeneous (qualitatives and quantitatives), authors in [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF] are used fuzzy control [START_REF] Driankov | An introduction to fuzzy control[END_REF] for achieving this two goals. For all input variables, two fuzzy subsets F (Favourable, i.e. the sets of values giving rise to acceptable environmental effect) and U (Unfavourable, i.e. the set of values giving rise to unacceptable environmental effect) are defined. Table 3 shows favourable and unfavourable limits for input variables of I gro , which are extracted from literature or based on human expert knowledge. For each class (favourable, unfavourable, and transition interval which corresponds to values between favourable and unfavourable limits ), we have defined a membership function on knowledge of experts. The membership of values of input variables takes any value in the interval [0,1]. The value 0 represents complete nonmembership and the value 1 represents complete membership. Values between 0 and 1 are used to represent partial membership. The membership function is defined in such a way that the value of an input variable either belongs fully to one of the two fuzzy subsets (favourable, unfavourable), or partially to both. In the latter case, the value is within a transition interval. We used membership functions that are sinus shaped in the transition interval, as they provide smoother variations of the output values than membership functions that are linear in the transition interval [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF]. For example, experts classify the leaching potential GUS into three classes. They affect pesticides classified as "leacher" (i.e. GUS > 2.8 or called unfavourable subset) a membership value of 1 for the fuzzy subset U and membership value of 0 for the fuzzy subset F. Pesticides classified as "no-leacher" (GUS < 1.8 or called favourable subset) are given a membership value of 0 for the subset U and a membership value of 1 for the fuzzy subset F. The class of borderline compounds (1.8 < GUS < 2.8) falls within a transition interval where the membership value for F decreases from 1 (GUS = 1.8) to 0 (GUS= 2.8), and the membership value of U increases from 0 to 1 (see Fig. 2). The favourable (respectively unfavourable) membership function "F-Function" (respectively "U-Function") of GUS is the degree for GUS to be in the favourable subset (respectively unfavourable subset). We have also "U-Function" = 1 -"F-Function". The function of subset favourable of GUS is 0.5 + 0.5 cos(π(GUS -1.8)). If variable GUS has a value of 2.5, then "F-Function" = 0.16 and "U-Function" = 0.84. The calculations are carried out according to a set of if-then rules. The experts attribute values between 0 and 10 for each rule.They consider firstly -what determines the leaching-the properties of organic matter (molecule), so they use variable GUS. If the variable GUS is favourable (GUS < 1.8), then there is no problem. The molecule does not present the characteristics of leaching. If variable GUS is unfavourable (GUS > 2.8), the experts are interested in the other variables of indicator. When a molecule is able to leach, it is able to reach the soil. Then, the experts use variable "position" of application. If the variable "position" is favourable, experts set a value to 9 for indicating the risk of leaching, but it is not too important. Then the environment is taken into account through the variable "leaching risk" (is the molecule leaching or not?, is the soil sensitive to the leaching?) and, finally they set on the variable "ADI". Fig. 3 shows the if-then rules of I gro . For example, if all input variables of I gro are F then conclusion is 10. If all input variables of I gro are U then conclusion is 0.

For each rule, we obtain the truth value by applying the minimum operator on the set of membership of the rule. For the first rule, if all input variables are F, the memberships of

Problem statement

In this paper, we consider the indicator of groundwater contamination I gro , which depends on parameters: GUS, DT50, "koc", ADI, "leaching risk" and "position". It is calculated by Van der Werf and others in [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF] as I gro = We are interested here in multisource variable DT50, koc, ADI. Assume that the set of n information sources provide n fuzzy intervals about these variables (see Table 2). We have to search the most plausible values to calculate the indicator. We use here the support of fuzzy interval (α-cut with α = 0 i.e. the interval Int 1 in Table 2). We use the notion of maximal coherent subset into a possibilistic fusion to synthesize a final result for multisource variables. These results combined with values of others variables will be used to compute the bounds (lower and upper) of the indicator I gro . In this paper, ∀x ∈ R, x -and x + represent the lower and upper bounds of interval where x can vary.

Possibility theory, fuzzy sets

In this section, we briefly summarize basic concepts of possibility theory, which are needed for understanding this paper. Fuzzy sets were firstly introduced by Zadeh [START_REF] Zadeh | Fuzzy sets[END_REF] as a possible way to handle uncertainty in processing incertain or imprecise data, and to control expert knowledge. This theory allows the notion of graduation to express whether an element belongs to a set (For more details, see [START_REF] Zadeh | Fuzzy sets[END_REF][START_REF] Dubois | Théorie des possibilités[END_REF][START_REF] Bouchon-Meunier | La logique floue[END_REF]).

Possibility theory

Possibility theory was introduced in 1978, in connection with fuzzy set theory, to allow reasoning to be carried out on imprecise or vague knowledge, making it possible to deal with uncertainties on knowledge [START_REF] Dubois | Théorie des possibilités[END_REF][START_REF] Zadeh | Fuzzy sets as a basis for a theory of possibility[END_REF]. The basic tool of possibility theory is possibility distribution. A possibility distribution is equivalent to the definition of a normalized fuzzy membership function. We are interested in this theory, in this paper, because we need to synthesize information of a variables which come from various sources. When we want to synthetize the value of a variable (i.e. approximate the value of the variable depending a set of a different sources (This process is called "information fusion" (see here after). In such a situation, possibility theory can be used to handle this kind of problems. This theory presents many choices of fusion operators in different contexts.

Information fusion

Definition 1 Information fusion consists of merging, or exploiting conjointly, several sources of information for answering questions of interest and make proper decisions.

A large set of information fusion operators has been designed in the possibility theory framework [START_REF] Dubois | Théorie des possibilités[END_REF]. They can be split into three subsets, according to the behavior of operators:

• Conjunctive type operators: it is the equivalent of a set intersection. It makes the assumption that all sources are reliable (i.e. all sources give information with high confidence), and usually results in very precise information.

If there is an incoherence between variable values (i.e. detected by domain expert), then the result of the conjunction becomes poorly reliable, or even empty.

• Disjunctive type operators: it is the equivalent to a set union. It makes the assumption that at least one source is reliable. The result of a disjunctive operator can be considered as very reliable.

• Trade-off type operators: they are compromise operators between conjunctive and disjunctive operators. They are typically used when sources are partially conflicting (i.e. in contradiction). As its name indicates, such an operator tries to make a trade-off between disjunction and conjunction for achieving a good balance between informativeness and reliability.

Methodology

Fusion based on maximal coherent subsets to assess imprecision of multi-source variables

Information sources which supply the information about imprecise variables (DT50, koc, ADI) are heterogeneous (book, expert, database). They come from different places (France, USA, Netherlands), then the type of soil, weather and laboratory conditions are not similar. Furthermore, we ignore the reliability of sources (they do not supply a degree of confidence with the value). They provide data in term of a fuzzy interval (or fuzzy number). Also, no information about conflict and relations between sources is available. Then, we can not use a conjunctive rule (respectively disjunctive and compromise rule). We need a fusion method which take into account all information given by sources (i.e. without discarding any source). The notion of maximal coherent subset is a natural way for achieving this goal. The fusion method using the notion of maximal coherent subset consists firstly inn applying a conjunctive operator into each non-conflicting subset of sources, then a disjunctive operator between those partial results. With such a method, as much precision as possible is gained when we do not ignore any sources [START_REF] Rescher | On inference from inconsistent premises[END_REF][START_REF] Benferhat | Reasoning in inconsistent startified knowledge bases[END_REF]. We will explain in details how this approach applies to support of fuzzy intervals of variables of I gro (see the section 4.3).

Computing maximal coherent subsets of intervals

In this section, we describe how we can obtain values of multisource variable using the fusion method based on maximal coherent subset of intervals, which are give by information sources. Let us consider n intervals,

I i = [a i , b i ], (i = 1, . . . , n).
(i) = a if ci = a k and type(i) = b if ci = b k ; for i = 1 to 2n -1 do if type(i) = a then Add source k to Ij t.q. ci = a k ; if type(i + 1) = bthen Add k to List (Ij); j = j + 1; else Remove source k from Ij such as ci = b k ;
Algorithm 1, that finds maximal coherent subsets, was introduced by Dubois and others in [START_REF] Dubois | Knowledge-driven versus data-driven logics[END_REF]. The algorithm 1 is linear in the number of intervals, and thus computationally efficient. The algorithm is based on an increasing sorting of the interval end-points into a sequence (c i ) i=1,...,2n . Each time and only then, an element c i of type upper bound which is an upper bound of an interval, followed by an element c i+1 of type lower bound wich is a lower bound of an interval meet, a maximally coherent set is obtained.

Example of application

We consider the variable DT50 (pesticide half-life) provided by three information sources (see Table 2): a French expert (Dabene), a French database (Agritox), and an England book (Pesticide Manual). Using algorithm 1 on the level α = 0 (i.e. the support Int 1 of fuzzy intervals in Table 2), we find two maximal coherent subsets K 1 = {Dabene, Pesticide Manual} and K 2 = {Agritox, Pesticide Manual}. After applying the maximal coherent subset method, the result of the variable DT50 is:

(Dabene ∩ Pesticide Manual) ∪ (Agritox ∩ Pesticide Manual) = [21,42] ∪ [46,53].

Computation method to the fuzzy weighted average

The objective of this section is to compute the bounds of the indicator I gro . The function

f (s 1 , s 2 , . . . , s n , ω 1 , . . . , ω n ) = n i=1 ωisi n i=1 ωi
, where ω i is a fuzzy interval and s i ∈ R, is called

ISBN: 978-989-95079-6-8
IFSA-EUSFLAT 2009 fuzzy weighted average by Dong and Wong [START_REF] Dong | Fuzzy weighted averages and implementation of the extension principle[END_REF]. Authors in [START_REF] Dong | Fuzzy weighted averages and implementation of the extension principle[END_REF] proposed an algorithm to compute the fuzzy weighted average. Their algorithm is based on the α-cut representation of fuzzy sets and combinatorial interval analysis. Subsequently, Liou and Wang [START_REF] Liou | Fuzzy weighted average: An improved algorithm[END_REF] suggested an improved fuzzy weighted average algorithm to simplify the computational process. Afterwards, Lee and Park [START_REF] Lee | An efficient algorithm for fuzzy weighted average[END_REF] have proposed an even more efficient algorithm for fuzzy weighted average by reducing the number of arithmetic operations to O(n log n). The main idea of Lee and Park in [START_REF] Lee | An efficient algorithm for fuzzy weighted average[END_REF] is to sort the s i variables such as for all i < j, we have s i ≤ s j . In this order, we can find a rank k with 1 < k < n such as for all i ≤ k, the function f is decreasing with respect to ω i , and for all i > k, the function f is increasing with respect to ω i . Then, we can compute the lower and the upper bounds of the fuzzy weighted average in accordance with [START_REF] Fortin | Gradual numbers and their application to fuzzy interval analysis[END_REF],which are bounds of I gro , as:

f - k = k j=1 (ω j ) + .s j + n j=k+1 (ω j ) -.s j k j=1 (ω j ) + + n j=k+1 (ω j ) - ( 2
)
f + k = k j=1 (ω j ) -.s j + n j=k+1 (ω j ) + .s j k j=1 (ω j ) -+ n j=i+1 (ω k ) + (3)
Where ω - i and ω + i are respectively the lower and upper bounds of ω i . The equation (2) (respectively (3)) represent the lower (upper) bound of the level k and k = 1, . . . , n. Then, we determinate the fuzzy interval of indicator such as introduced by Fortin and others in [START_REF] Fortin | Gradual numbers and their application to fuzzy interval analysis[END_REF]:

[ min k=1,...,n (f - k ), max k=1,...,n (f + k )] (4)

Results

In this application when computing I gro , about fifteen information sources are used such as ARS (USA), Agritox (France), Pesticide manual (England)... These sources provide information based on a fuzzy interval or on a fuzzy number. In this paper, we use the support of fuzzy intervals given by the sources. The decision rules do not change. For each variable, we replace the input value by an input interval. We propagate the imprecision into computation indicator to calculate a fuzzy interval of indicator. For a pesticide application, an indicator of groundwater contamination score is calculated. The application considers the indicator of groundwater contamination I gro in several sites with different crops. The positions of application of pesticides is shown in Table 4. Its value is calculated as: "position = soil coverd by the crop 100 ". The "soil covered" by the crop is estimated by the user of the systems [START_REF] Bockstaller | Mode de calcul des indicateurs agri-environnementaux de la méthode indigo[END_REF]. The variable "leaching risk" set the value of 0.9 [START_REF] Bockstaller | Mode de calcul des indicateurs agri-environnementaux de la méthode indigo[END_REF]. Pesticide properties (DT50, koc, and ADI) given by several information source are calculated by fusion based on maximal coherent subsets. Table 5 shows the results of pesticide characteristics. The values shown in Table 5 are convexified to simplify the calculation. By using the interval analysis [START_REF] Edgar | Interval Analysis[END_REF] and the equation 1, we compute the fuzzy interval of variable GUS, such as:

GUS -= log(DT50 -) × (4 -log(koc + )) (5)
GUS + = log(DT50 + ) × (4 -log(koc -))

where ([DT50 -,DT50 + ]) and ([koc -,koc + ]) are fuzzy intervals of variables DT50 and Koc.

f ([a, b]) = [f (a), f(b)] if f is increasing [f (b), f(a)] if f is decreasing (7)
Then, we calculate the truth intervals of the sixteen decision rules of I gro . These truth intervals can be computed by minimum extension applied on intervals such as 4). The Table 6 shows the results of I gro for different pesticides. The fuzzy interval in Table 6 represent the exact interval when the indicator is varying. It represents also the support, of the fuzzy interval of the indicator I gro , that can obtain from all fuzzy intervals of input variables. The results presented in Table 6 show a lot of imprecision in some cases. We take example of the organic matter "glyphosate": the indicator obtained by application of glyphosate on the soil varies between 0 and 10. This imprecision is due to the variable GUS -which varies in [0.73,2.97]that reach limits fixed by experts in Table 3 (i.e (lower limit) < (favourable limit), and (upper limit) > (unfavourable limit)). We note that the indicator varies in [0,10], when we have one of the input variables has a lower (respectively upper) limit exceeds favourable (respectively unfavourable) limit. In the calculation of only one value of indicator I gro [START_REF] Van Der Werf | An indicator of pesticide environmental impact based on a fuzzy expert system[END_REF], the role of the variable GUS is more important than the role of others variables. In this paper, all variables have the same influence in the final result.

Discussion and conclusions

In this paper, we introduced the calculus of the indicator of groundwater contamination I gro . We try to control the imprecisions of input variables by using a possibilistic fusion method based on the maximal coherent subsets. Then, we use the notion of interval analysis to propagate the imprecision in the process of calculus of indicator. A fusion method for merging fuzzy subsets, based on the notion of maximal coherent subsets, is proposed to represent the imprecision of input parameters of the indicator of groundwater contamination. This notion appears as a very natural way to conciliate two objectives, gaining information and considering all the conflicts between sources. The method of fusion is simple: it can be applied without any additional information, and its computational complexity remains affordable. The way is summarizes information is conceptually attractive: maximal coherent subsets are the best we can do in the presence of conflict.

We have also described a way to calculate the bounds of indicator. The method of fuzzy interval arithmetic used in this paper is also simple. The support of the fuzzy number of intervals is calculated. Also, we can calculate the membership function of indicators by computing several values of α-cut of input variables. This method allows to obtain exact fuzzy profiles of indicator with reducing time complexity as in the sampling case. It still remains to validate the fusion method using the notion of maximal coherent subset in contrast with other fusion rules. We have to calculate the fuzzy interval for all α-cut to obtain the distribution of I gro . We plane to use the methodology described in this paper (representing by fuzzy intervals the imprecision and propagating this imprecision by interval analysis) to calculate other indicators such as indicator of pesticide in environment which has a process of calculation similar to process of I gro .

Figure 1 :

 1 Figure 1: An example of values of DT50 given by sources, α-cut with α = 0.4 and the support of data given by Dabene

Figure 2 :

 2 Figure 2: F-Function and U-Function function of variable GUS

Figure 3 :

 3 Figure 3: If-then rules of the indicator I gro (I gro has 2 4 = 16 rules)

16 i=1 ωisi 16 i=1

 1616 ωi , where ω i depends of variables of I gro , and s i ∈ [0, 10].

  min([a, b], [c, d]) = [min(a, c), min(b, d)]. Then, we use the equations (2) and (3) to compute respectively the lower and upper bounds of indicator for the sixteen α-cuts established by the fuzzy input membership intervals. Finally, we obtain the final interval of indicator by the equation (

Table 1

 1 shows variables used to calculate I gro .

Table 1 :

 1 All variables for the I gro and their description.

	Variables	Description
	Pesticide characteristics (quantitative variables)
	GUS	pesticide leaching potential
	DT50	pesticide half-life
	koc	organic-carbon partition
	ADI	Acceptable Daily Intake
	Environmental characteristics (qualitative variables)
	Leaching risk Quantity leaching
	Application characteristics (quantitative variables)
	Position	Position of application

Table 2 :

 2 Example information from sources Source Int 1 Int 2 Dabene [10,42] [20,35] Agritox [46,53] 50 Pesticide Manual [21,56] [35,40]

Table 3 :

 3 Favourable and unfavourable limits of input variables

	Variable	Favourable limit Unfavourable limit
	DT50	1	30
	GUS	1.8	2.8
	ADI	1	10 -4
	Position	1	0
	Leaching risk 0	1

  This method allows to find every maximal subsets of sources. A maximal subset is obtained when i∈[n] I i = ∅, where [n] represents the set of 2 n subsets of the set {1, . . . , n}. Then, we apply the union of these partial results (i.e. j i∈[n] I i ).

	Algorithm 1: Maximal coherent subset of intervals
	Input: n intervals

Output: List of m maximal coherent subsets Ij List=∅;j = 1;Ij = ∅; Order in an increasing order {ai, i = 1, . . . , n} ∪ {bi, i = 1, . . . , n}; Rename them {ci, i = 1, . . . , 2n}with type

Table 4 :

 4 Values of the variables of application condition

	Pesticide name Position soil covered
	Chloridazone	0.9	90
	Glyphosate	0.75	75
	Nicosulfuron	0.1	10
	Isoproturon	0.1	10

Table 5 :

 5 Results of pesticide characteristics

	Pesticide name GUS	DT50	koc	ADI
	Chloridazone	[2.53,3.19] [21,46] 120	0.025
	Glyphosate	[0.73,2.97] [18,47] [167,2640] 0.3
	Nicosulfuron	[2.78,5.39] [15,43] [5,43]	0.4
	Isoproturon	[2.68,3.03] [22,28] [80,99.99] 0.4
	Afterwards, we compute the membership intervals of input
	variables of indicator. For each input variable that varies in
	[a, b] and has a membership function f , we compute the mem-
	bership interval by using the function:

Table 6 :

 6 Results of indicator Pesticide name Interval of I gro

	Chloridazone	[4.3,9.82]
	Glyphosate	[0,10]
	Nicosulfuron	[4.42, 4.45]
	Isoproturon	[4.42,5.27]