Mohamad Nazar
email: mohamad.nazar@univ-eiffel.fr

Ali Ibrahim
email: ali.ibrahim@univ-eiffel.fr

Zoubir Khatir
email: zoubir.khatir@univ-eiffel.fr

Nicolas Degrenne
email: n.degrenne@fr.merce.mee.com

Zeina Al Masry
email: zeina.al.masry@ens2m.fr

Remaining Useful Lifetime estimation for Electronic Power modules using an analytical degradation model

Power electronic modules undergo electro-thermal stresses due to power losses that lead to several kinds of degradations, and finally to failure. In order to prevent power electronic module failure, one should assess its state of health in realtime operation. For this purpose, Prognostics and Health Management (PHM) approach could be a promising tool for reliability evaluation for an IGBT device. The Insulated Gate Bipolar Transistor is a three-terminal power semiconductor device used as an electronic switch which combines high efficiency and fast switching. In this paper, an analytical model is proposed that describes the metallization to wirebond contact resistance. This model computes the crack length and then using online measurement the rate of crack propagation is computed and used to predict the future crack lengths with the assumption of linearity in crack propagation. The main failure mechanism of the IGBT device in this paper is the crack propagation in the wire-bond interconnection. The usual aging indicator of such damages is the voltage between the collector and the emitter of current (𝑉 𝐶𝐸 ) that increases with degradation. The analytical model is related to this indicator and it is based on the contact resistance theory and constriction current lines. The proposed model is hence used to build a prognostics model for estimating the remaining useful lifetime (RUL) of IGBT power modules. The prognostics model is illustrated using aging data coming from accelerated power cycling tests with different stress conditions. The first cycling test is under the conditions of stress duration 𝑡 𝑜𝑛 = 3sec and junction temperature swing amplitudes 𝛥𝑇 𝑗=110 °𝐶 and the second cycling test is under 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐 and Δ𝑇 𝑗 = 90°𝐶. Results show a prognostics capability.

INTRODUCTION

The reliability of power electronic devices is important for most power electronic systems. Electronic components, such as IGBTs, assembled in power modules are subjected to high environmental and functional stresses (ambient temperature, vibrations, etc.). All these factors have a strong impact on device reliability and lifetime. From an economic point of view, predictive maintenance may be less detrimental than preventive maintenance performed at fixed time intervals. Consequently, the use of lifetime prognostics tools can be necessary. The problem consists of the health state prediction of power modules in operation to be able to schedule maintenance before the failure of the equipment.

In power electronic systems, two main kinds of models can be built to achieve prognostics: the damage accumulation models and the condition based prediction models. The damage accumulation models can estimate the lifetime of components operating under given usage conditions. The effect of past usage is explicitly modeled by summing up the partial damages. At present, such models, do not specifically represent the physics of degradation to estimate the state of a specific system. An example of this model is the Coffin Manson [START_REF] Manson | thermal stress low-cycle fatigue[END_REF]) and Paris's law [START_REF] Yang | Physics-of-failure lifetime prediction models for wire bond interconnects in power electronic modules[END_REF]. Research is ongoing to improve these types of models, for example, N. Dornic found a relation between the 𝑁 𝑓 (number of cycles until failure) with the stress duration 𝑡 𝑜𝑛 and junction temperature swing amplitudes 𝛥𝑇 𝑗 [START_REF] Dornic | Stress-Based Model for Lifetime Estimation of Bond Wire Contacts Using Power Cycling Tests and Finite-Element Modeling[END_REF]. The second type, which is conditionbased prediction models, is based on the measurement and extrapolation of a precursor to failure. It assesses the behavior of individual units not only on their usage history but also on expected future load profiles. In other words, it utilizes a degradation model that predicts the future state based on input that describes the current system state and the expected load levels on the system. These component-specific models are generally adaptable and can be developed based on the physics of the system like [START_REF] Degrenne | Diagnostics and Prognostics of Wire-Bonded Power Semi-Conductor Modules subject to DC Power Cycling with Physically-Inspired Models and Particle Filter[END_REF][START_REF] Hu | Cost-Effective Prognostics of IGBT Bond Wires with Consideration of Temperature Swing[END_REF] or can be learned from run to failure data using data-driven methods like [START_REF] Sreenuch | Probabilistic Monte-Carlo Method for Modelling and Prediction of Electronics Component Life[END_REF][START_REF] Alghassi | Prognostics and Health Management of Power Electronics[END_REF][START_REF] Liu | Remaining useful life estimation of insulated gate biploar transistors (IGBTS) based on a novel volterra K-nearest neighbor optimally pruned extreme learning machine (VKOPP) model using degradation data[END_REF]. T. Sreenuch used the Probabilistic Monte-Carl o method for modeling and online prediction of the remaining useful life (RUL) in an electronic component [START_REF] Sreenuch | Probabilistic Monte-Carlo Method for Modelling and Prediction of Electronics Component Life[END_REF]. His study was based on the 𝑉 𝐶𝐸 data collected from the electronic component, also Zhen Liu used the Machine learning method to predict RUL for IGBT semiconductors using degradation data [START_REF] Liu | Remaining useful life estimation of insulated gate biploar transistors (IGBTS) based on a novel volterra K-nearest neighbor optimally pruned extreme learning machine (VKOPP) model using degradation data[END_REF]. The model built in this paper as highlighted in the red circle in Fig. 1 is classified as a physics-based model in the condition based prediction category. Based on the previous studies done in this field, this paper presents a new analytical model that describes the degradation in the IGBT module. Then, using this model, a prognostics model is built to predict the RUL.

The remainder of the paper is organized as follows. Section 2 starts by defining different types of degradations that occur in IGBT devices due to aging. Several aging tests were done in SATIE lab that shows the evolution of 𝑉 𝐶𝐸 with the number of cycles. Section 3 presents an analytical model describing the evolution of the dominant degradation that takes place in this module, based on the theory of contact resistance. An online prognostics model is then built-in Section 4 to predict the RUL. The steps for building the model are illustrated. This model is validated by experimental results with different stress conditions. Finally, a conclusion is provided in Section 5.

AGING AND DEGRADATION

During the aging of power electronic devices, different types of degradations can take place as shown in Fig. 2. The degradation types are: wire-bond degradation that usually leads to heel cracks, wire-bond lift-off due to crack propagation in the wire-bond interconnection with the metallization, top metallization degradation due to grains recrystallization, die-attach degradation that leads to cracks at the die bottom connection, cracks in the metallization ceramic interface, cracks in the ceramic itself, and substrate attach that leads to solder cracks. In the IGBT module, wire-bond lift-off is a dominant degradation. For this reason, the focus in this paper will be in modeling the crack propagation leading to the lift-off. Fig.3(b). presents crack propagation in the wire-bond interconnection with the metallization pad, and Fig. 3(a) presents a lifted wire-bond. Fig. 3: a) wire-bond lift-off [START_REF] Ciappa | Selected failure mechanisms of modern power modules[END_REF], b) crack propagation in the wire-bond interconnection with the metallization [START_REF] Degrenne | Diagnostics and Prognostics of Wire-Bonded Power Semi-Conductor Modules subject to DC Power Cycling with Physically-Inspired Models and Particle Filter[END_REF].

Since pure analytical models of damage processes are very complicated to build, failure prediction needs an assist from experimental data. Thus, it is important to collect information of an aging indicator in the form of a degradation profile that covers failure data that will assist in the prognostics approach.

A way to see the 𝑉 𝐶𝐸 evolution with degradation due to thermal fatigue is obtained by accelerated aging tests like power cycling. This type of test is done in the SATIE Lab. Fig. 4 shows experimental data for 4 IGBTs of the same module type until failure. As data from the field are not easy to obtain, these types of accelerated results will be used hereafter for model validation. The dominant degradation observed in these accelerated aging tests was the lift-off due to crack propagation. Hence, we can assume that the voltage variation in the wire-bond contact is the same as the voltage variation between the collector and the emitter.

In Fig. 4, the experimental accelerated conditions used are from power pulses with 3sec heating time and 6sec cooling time with power value that leads to 110°C temperature variation.

Fig. 4: Collector-Emitter voltage variation, and its zones due to types of degradation.

Based on the type of degradation in Fig. 4, we divided the lifetime line into 2 zones. Zone 1 is much longer than zone 2; it covers more than 80% of the device's lifetime. It has a progressive evolution in 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸 due to crack propagation in the wire-bond metallization interconnection, which leads to a decrease in the contact area between wire-bonds and metallization. At the end of this zone, the wire-bonds' lift-off starts. Zone 2 is when the wire-bonds start to lift-off where the failure takes place quickly. The End of Life (EoL) criterion for the failure of these devices is when the Δ𝑉 𝐶𝐸 /𝑉 𝐶𝐸 reaches 5%. Since zone 1 covers more than 80% of the lifetime, and since in zone 2 failure occurs quickly due to liftoff, then it is more important as the first step to model and to predict the degradation in zone 1 than zone 2.

Based on the importance of zone 1, an analytical physicsbased model is done to describe the impact of crack propagation on the contact resistance.

CONTACT RESISTANCE ANALYTICAL MODEL

In this section, an analytical model is built that estimates the increase of the crack length at the wire-bond contact from the increase in the 𝑉 𝐶𝐸 . This model is then used to predict the RUL, assuming a mean speed linear crack propagation. In comparison to literature, a study done by [START_REF] Degrenne | Diagnostics and Prognostics of Wire-Bonded Power Semi-Conductor Modules subject to DC Power Cycling with Physically-Inspired Models and Particle Filter[END_REF]) presents a physical model that relates the crack propagation to on-state voltage. Using a specific algorithm, this model is used to predict the RUL with an accuracy of less than ±10%. This model assumes that crack propagation is linear. However, Hu [START_REF] Hu | Cost-Effective Prognostics of IGBT Bond Wires with Consideration of Temperature Swing[END_REF]) built a physical model that relates the crack propagation to the voltage using the Finite Element Method and taking into consideration nonlinear crack propagation. This model took into account also the effects of temperature variation and metallization degradation on the on-state voltage increase.

The model in this paper combines two different models. The first one describes the constriction resistance at the bond contact due to the crack from the wire-bond side. The second one gives the spreading resistance in the metallization due to the current lines flowing from the die, collected by the metallization pad, and going through the bonding wire. The spreading of the current lines is also affected by the crack length. The model is simplified to consider a disc as a surface contact between a cylindrical bond-wire and the metallization as depicted in Fig. 5. Based on Holm's relation [START_REF] Holm | Electric Contacts[END_REF], the constriction resistance in one member is defined as:

𝑅 𝐶 = 𝜌 4𝑎 (1)
Equation ( 1) represents the effect of "𝜌" resistivity of the material), and "𝑎" (radius of the contact area), on "𝑅 𝑐 " which is the constriction resistance in the wire-bond side. In this study, we assume the resistivity is constant, which makes the constriction resistance as a function of the radius only.

On the other side, as shown in Fig. 6 the resistance of a thin layer (metallization) between 𝑟 = 𝑎 and 𝑟 = 𝑏 connected to a wire-bond through a contact area of radius "𝑎" as shown in Fig. 6 is:

𝑅 = 𝜌 2𝜋𝑡 ( 𝑏 2 𝑏 2 -𝑎 2 𝑙𝑛 ( 𝑏 𝑎 ) - 1 2 ) (2.1) Equation (2.1) is demonstrated in Appendix A.
Where "𝑡" is the thickness of the layer, and "𝑏" is the considered overall radius. In this equation, the parameters "𝑏", "𝑡" and "𝜌" are constant. Hence, the resistance of the thin film will be only a function of "𝑎". The constriction resistance of the metallization member under the contact is neglected in comparison with the metallization resistance from 𝑟 = 𝑎 to 𝑟 = 𝑏. Equation (2.1) doesn't include the resistance increase in the metallization due to resistivity increase and/or structure change (recrystallization and grain grooving). Knowing that the resistance of metallization increases with aging due to power cycling as mentioned in [START_REF] Zhao | A study on the effect of microstructure evolution of the aluminum metallization layer on its electrical performance during power cycling[END_REF]. Based on that, a factor "𝑤" is multiplied to the resistance of Equation 2.1. No additional measurements were done to fix "𝑤", however, it is assumed to be equal to 2 based on the study of [START_REF] Zhao | A study on the effect of microstructure evolution of the aluminum metallization layer on its electrical performance during power cycling[END_REF] that carries almost the same conditions of power cycling tests used in this study. Other different papers also prove that this value could be increased up to 7 based on the conditions used in the aging tests [START_REF] Arab | Experimental investigations of trench field stop IGBT under repetitive short-circuits operations[END_REF].

𝑅 = 𝑤𝜌 2𝜋𝑡 ( 𝑏 2 𝑏 2 -𝑎 2 𝑙𝑛 ( 𝑏 𝑎 ) - 1 2 ) (2.2)
It is necessary to be mentioned that the wire-bond contact shape in IGBT is ellipsoidal. However, Holms book [START_REF] Holm | Electric Contacts[END_REF] mentioned that for ellipse contact shape the difference can be neglected if the multiplication of the ellipses' major and minor semi-axis is equal to 𝑎 2 (which is our situation). Moreover, the dimensions of our metallization are not circular. However, in our study "b" is assessed experimentally. That will give an approximation of the value b, which in any way has a slight effect on the equation's results.

The first model presented by Eq. ( 1) shows the effect of "𝑎" change on the constriction resistance, and the second model represented by Eq. ( 2) shows the effect of "𝑎" on the metallization's resistance.

The other values of parameters in Eq. ( 1) & (2.2) based on the structure are 𝑏 = 2 𝑚𝑚, radius initial value 𝑎 0 = 332 𝜇𝑚, 𝑡 = 4𝜇𝑚, and 𝜌 = 3.8 * 10 -8 Ωm.

Fig. 7: Effect of contact area radius "a" change in Eq. ( 1) & (2.1) on the resistance.

In Fig. 7, the effect of radius changes of the bond contact on both resistances, due to crack propagation, is shown. It is observed that with a decrease in the radius, the two resistances increase. It can be observed that the metallization resistance remains very high compared to that of constriction except when the contact is reduced to a very small radius. The sensitivity of resistance to radius change increases with radius decrease, and it increases in high slope at the last few micrometers.

By adding these two models, we can build a new model that describes the effect of crack propagation (contact area decrease) on the resistance between wire-bond and metallization.

𝑅(𝑎) = 𝜌 2 ( 1 2𝑎 + 𝑤 𝜋𝑡 ( 𝑏 2 𝑏 2 -𝑎 2 𝑙𝑛 ( 𝑏 𝑎 ) - 1 2 )) (3)
The new model represented by Eq. (3) will be used in the prognostics methodology to predict the remaining voltage variation at each measurement. Besides, as mentioned before, the variation of voltage calculated from Eq. (3) equals the variation of collector-emitter voltage (𝛥𝑉 𝐶𝐸 ). The approach is more detailed in the next section.

PROGNOSTICS APPROACH

The previous model is then used in the prognostics approach to estimate RUL. Data of four failed IGBTs Fig. 4, done by accelerated power cycling test, were used for testing the prediction. The methodology of prognostics starts with measuring the 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸 value after a short time of aging to let the degradation take place and the crack propagate. Then, use this value to calculate the contact area radius "𝑎 1 " using Eq. ( 4)-( 6). After that, we use the initial contact area radius "𝑎 0 " to compute the rate of crack propagation (𝐾 1 ) using Eq. ( 7). This rate of crack propagation is used to predict the contact area radius with the assumption of linear crack propagation using Eq. ( 9).

At first, we have the following equation:

𝑅(𝑎 0 ) & 𝑅(𝑎 𝑚 ) are calculated using Eq. (3).

After "𝑎 𝑚 " is calculated using Equations ( 4)-( 6). we use it to compute "𝐾 𝑚 " which is the rate of crack propagation using Eq. ( 7):

𝐾 𝑚 = 𝑎 𝑚 -𝑎 𝑚-1 𝑛 𝑚 -𝑛 𝑚-1 (7)
Using 𝐾 𝑚 in Eq. ( 8), 𝐾 𝑚 ̅̅̅̅ is calculated using Eq. ( 8).

𝐾 𝑚 ̅̅̅̅ = (𝐾 1 + ⋯ + 𝐾 𝑚 ) 𝑚 (8)
Where 𝐾 𝑚 ̅̅̅̅ is the average crack propagation rate. Using 𝐾 𝑚 ̅̅̅̅ in Eq. ( 9), gives the predicted values of 𝑎 𝑛 until the crack propagation completes. By compensating the predicted values of 𝑎 𝑛 in Eq. ( 4)-( 6), we will have the first predicted 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 's curve.

𝑎 𝑛 = K m ̅̅̅̅ 𝑛 + 𝑎 0 (9)
"n" represents the number of cycles for the prediction process at each measurement m. Which means that at each new measurement "m", a prediction curve that predicts 𝛥𝑉 𝐶𝐸 / 𝑉 𝐶𝐸,𝑖𝑛𝑖 is formed. This prediction curve stops when the contact area radius "𝑎 𝑛 " reaches zero. For instance, for the first prediction at m=1, 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 is measured at 8 kilocycles and is equal to 0.16% as shown in Table 1. This value is inserted in Equations (4-6) to calculate the area contact radius 𝑎 1 . Then, by having 𝑎 0 and 𝑎 1 in Eq. ( 7) 𝐾 1 is calculated. Then the average rate of crack propagation 𝐾 1 ̅̅̅ is computed using Eq. ( 8), which is the same as 𝐾 1 for the first prediction.

Then 𝐾 1 ̅̅̅ is used in Eq. ( 9) to predict the remaining radiuses until "a" reaches zero. These values of radiuses predicted are then inserted in Equations (4-6) to calculate the predicted 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 . The first prediction can be observed in Fig. 8. After repeating the previous process for several measurements and plotting the predicted 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 , Fig. 9 could be observed. Fig. 9 can show the predicted curves for 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 at several numbers of cycles. The experimental results in Fig. 9 refer to a device aging under 𝛥𝑇 = 110°𝐶 and 𝑡 𝑜𝑛 = 3 𝑠𝑒𝑐. The experimental data used are filtered with an average moving filter on 3 consecutive points. As it can be observed in Fig. 9, the prediction was done to estimate the remaining 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 curve at each step of measurement. The conventional failure criteria assumption on the collector-emitter voltage indicator is when the 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 reaches a 5% increase. However, it is important to note that this threshold is quickly achieved in the last part of aging, which is wire-bond lift-off (zone 2 Fig. 4). In Fig. 9 the experimental data represented by the dashed black curve is used to validate the predictions. The first and third predictions (8k.cycles, 16k.cycles) are less acceptable than the rest of the curves. This is due to the primary construction of the degradation propagation rate. However, this rate is then tuned after each new measurement due to computing the mean of crack propagation rates, and as you observe for the rest six predictions, the estimated 𝛥𝑉 𝐶𝐸 /𝑉 𝐶𝐸,𝑖𝑛𝑖 is agreeable and can be relied on. It is shown that the variation of 𝑎 𝑚 is almost linear with the number of cycles. This verifies the assumption of linear crack propagation with time for the conditions of 𝛥𝑇 = 110°𝐶 𝑎𝑛𝑑 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐.

One way to assess the model's robustness is by applying it to different stress conditions. For that, the physical model is also applied to data aged in different conditions. This is probably due to an error in experimental data measurement as observed at first 30 k.cycles. After that, the prediction curves start to trend upward as the experimental curve. A percentage of error can be observed in Fig. 11. This error, from an industrial point of view, would prevent failures and the corresponding economic losses since the expected predicted failure is before the real failure.

The contact area radius "𝑎 𝑚 " decrease related to the last aging test conditions can be observed in Fig. 12. The linearity of 𝑎 𝑚 decrease under 𝛥𝑇 = 90°𝐶 condition is less achieved, especially after the 35 kcycles where before that there exist errors in experimental data measured as shown in the dashed black curve in Fig. 11. Fig. 13 and Fig. 14 present the predicted RUL and Real RUL change with the number of cycles for the two different test conditions applied in this study. They start where the first prediction curve took place, except for Fig. 14 the curve starts from 32 k.cycles due to the errors observed in the experimental data at the first 30 k.cycles. The values of predicted RULs could be also verified in Fig. 9 and Fig. 11 where the RUL in this study presents the wire-bond's lift-off.

In both figures, the predicted RUL underestimates the failure which could mean that no additional costs the company will pay to failure consequences.

CONCLUSION

In this study, a physics-based analytical model is extracted that relates crack propagation to the aging indicator 𝑉 𝐶𝐸 .

Since 𝑉 𝐶𝐸 is monitored in real-time operation, then the crack length at each measurement can be calculated. Using the new calculated crack length, the rate of crack propagation is then computed and used to predict the crack lengths until the crack is completed, assuming a linear crack propagation. These predicted crack lengths give the predicted collector-emitter voltage variation using the model built. The predicted 𝑉 𝐶𝐸 identify the end of zone 1, which forms 80% of the total lifetime of the IGBT semiconductor device. It is interesting to be mentioned that the rate of crack propagation is averaged at each new measurement and used for the next prediction. The importance of this work lies under the prediction of the important part of the lifetime in the IGBT. For the evaluation of this prognostics approach, some metrics such as the prognostics horizon and 𝛼 -𝜆 accuracy are evaluated [START_REF] Saxena | Metrics for evaluating performance of prognostic techniques[END_REF]. The proposed model has a good 𝛼 -𝜆 accuracy within 10 % error for the first test condition (ΔT=110°C) while a bad 𝛼 -𝜆 accuracy for the second test condition (ΔT=90°C). The same for the prognostics horizon. This means that this model looses its reliability when changing the aging test conditions. For these reasons, the model should be improved in further studies. One should think about developing a hybrid model by combining the proposed one with a data-driven model.

APPENDIX A

𝐽 0 is the vertical current density in the die. 𝐼 1 is the total current from Die under the contact where 𝐼 1 = 𝐽 0 𝜋𝑎 2 . 𝐼 2 is the total current from the Die outside the contact where 𝐼 2 = 𝐽 0 𝜋(𝑏 2 -𝑎 2 ), and 𝐼 = 𝐼 1 + 𝐼 2 .

Based on the current law:

𝑖(𝑟) -𝑖(𝑟 + 𝑑𝑟) = 𝐽 0 2𝜋𝑟𝑑𝑟 (10)
"𝑖(𝑟)" is the current going out from the cell laterally and it is at distance 𝑟 from the center of the metallization as shown in Fig. 12. "𝑖(𝑟 + 𝑑𝑟)" is the current flowing inside the cell laterally and it is at a distance 𝑟 + 𝑑𝑟. And 𝐽 0 2𝜋𝑟𝑑𝑟 represents the current flowing upward from the die to the metallization going inside the cell.

Equation ( 10) is integrated from "𝑟" to "𝑏" knowing that 𝑖(𝑏) = 0. The following equation will be formulated:

𝑖(𝑟) = -𝜋𝐽 0 (𝑏 2 -𝑟 2 ) ( 11
)
Where 𝑖(𝑟) is the lateral current in the metallization. where E is the electric field in the radial direction and 𝜎 is the electrical conductivity of the material. By compensating 𝐸 = -𝑑𝑉/𝑑𝑟 and Eq.( 11) in Eq.( 12) the following equation is formed for voltage variation with r:

𝑑𝑉 = 𝜌𝐽 0 2𝑡 𝑏 2 -𝑟 2 𝑟 𝑑𝑟 (13)
Then, Eq. ( 13) is integrated from "𝑎" to "𝑏" knowing that voltage at b is 𝑉(𝑏) = 0 to have the following voltage equation:

𝑉(𝑎) = -𝜌𝐽 0 2𝑡 (𝑏 2 𝑙𝑛 𝑏 𝑎 - 1 2 (𝑏 2 -𝑎 2 )) (14)
The equivalent resistance of metallization seen between 𝑟 = 𝑎 and 𝑟 = 𝑏, can be written as:

𝑅 met. = (V(b) -V(a)) I 2 (15)
Then

𝑅 𝑚𝑒𝑡. = 𝜌 2𝜋𝑡 ( 𝑏 2 𝑏 2 -𝑎 2 𝑙𝑛 𝑏 𝑎 - 1 2 ) (16)

APPENDIX B

It is mentioned in the Prognostics Approach section that 11 wire-bonds are degrading and 4 are not degrading assuming that the 11 wire-bonds have the same rate of degradation. Fig. 13 presents a top view image for the IGBT device, where it is shown the wire-bonds distribution on the metallization of IGBT (red rectangles) and of the diode (yellow rectangle). You can observe that the wire-bond distribution on the IGBT is divided into two zones. Zone 1 contains 6 wire-bonds connected to the metallization, while zone 2 contains 9 wirebonds connected to the metallization, knowing that zone 1 is wider than zone 2. This motivates and allures us to study the current distribution in the different wire-bonds after being collected in the metallization. For a deeper discussion, Fig. 14 is introduced to facilitate understanding of the current distribution in the wire-bonds. Fig. 14 shows the current vertically directing upward to be collected in the metallization (black colored). This metallization is very thin (≃ 4 𝜇𝑚) compared to the wirebond radius (≃ 350𝜇𝑚) which will make the current highly prefer to pass through wire-bonds than the metallization, letting themselves pass through less distance of metallization as possible. Based on this theory, the current collected in metallization in part "d" will continue to pass through the 6 wire-bonds of zone 𝑍 1 , avoiding the metallization path of part "c". The percentage of current collected in part "d" is 45% of the whole current and this is due to the 45% area of the whole metallization assuming that the current is uniformly collected in the metallization. In part "c", the current will also continue to pass through the 6 wire-bonds of zone 𝑍 1 , and this is due to the less resistive path. Since, if the current of part c passes through the 9 wire-bonds, they are forced to pass also through the metallization of part "a", which is a high resistive path. The percentage of current collected in part "c" is 11%. Finally, in part "b" the current will continue to pass through the 9 wire-bonds of zone 𝑍 2 , avoiding passing through extra metallization path in part "c". Metallization in part "b" collects 44% of the whole current. Based on the previous study, 56% of the current will pass through the 6 wire-bonds and 44% will pass through the 9 wire-bonds.

Let 𝐼 𝑆 be the current in each of the 9 wire-bonds of zone 𝑍 2 , and 𝐼 𝐵 the current in each of the 6 wire-bonds of zone 𝑍 1 assuming that the current is uniformly distributed in each zone based on the symmetrical wire bonding on the metallization. Then we have:

𝐼 𝐵 = 0.56 6 𝐼 = 0.0933𝐼 (17)
The same for 𝐼 𝑆 :

𝐼 𝑆 = 0.44 9 𝐼 = 0.0488𝐼

Where 𝐼 is the total current. By compensating the total current of Eq. ( 17) in the total current of Eq. ( 18), the following equations are formed:

𝐼 𝑆 = 0.524𝐼 𝐵 => 9𝐼 𝑆 = 4.71𝐼 𝐵 ≃ 5𝐼 𝐵 (19) Equation ( 19) means that the current passing in the 9 wirebonds of zone 𝑍 2 can be distributed in 5 wire-bonds to have in each one of these 5 the same current density as the 6 wirebonds in zone 𝑍 1 . This gives the ability to assume that the current 𝐼 𝑆 is passing through 5 wire-bonds. Thus, it can be also assumed that the crack is just in these 5 wire-bonds. It is also assumed that the crack propagation in the 11 degrading wire-bond is the same.

Fig. 1 :

 1 Fig.1: Prognostics models classification.

Fig. 2 :

 2 Fig.2: Degradation types in IGBT semiconductors.

Fig. 5

 5 Fig.5presents wire-bond connected to metallization showing the constriction resistance at the contact surface of radius due to the connection between the members and the crack propagation. The constriction resistance is the resistance produced due to the current constricted to pass through the contact area. It can be observed that the equipotential lines start to be not parallel due to current lines constructed in the contact area.

Fig. 6 :

 6 Fig.6: Flow of current from wire-bond to metallization to Die.

  𝐶𝐸,𝑖𝑛𝑖 " is the initial collector-emitter voltage. "𝐼" is the direct current used. "𝛥𝑅(𝑎) 15𝑊 " is the resistance variation of 15 parallel wire-bonds connected to metallization. Eleven of them are degrading and four are not. This degradation distribution is due to the structural wire bonding in the device used, which leads the current to take less resistive paths. This study is explained in Appendix B.𝛥𝑅(𝑎) 15𝑊 = 𝑅(𝑎 𝑚 ) 15𝑊 -𝑅(𝑎 0 ) 15𝑊 (4.2)𝑅(𝑎 0 ) 15𝑊 represents the resistance of 15 wire-bonds before the degradation start, and 𝑅(𝑎 𝑚 ) 15𝑊 represents the resistance of the 15 wire-bonds at the 𝑚 𝑡ℎ measurement after degradation took place. Where:

Fig. 8 :

 8 Fig.8: Voltage variation percentage of the first prediction at 8 k.cycles compared to Experimental data.

Fig. 9 :

 9 Fig.9: Prediction of the relative increase in 𝑉 𝐶𝐸 for IGBT device under the conditions of 𝛥𝑇 = 110°𝐶 & 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐.

Fig. 10 :

 10 Fig.10: Contact area radius measured (𝑎 𝑚 ) decrease with the number of cycles for the aging tests done under 𝛥𝑇 = 110°𝐶 𝑎𝑛𝑑 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐. At each mth measurement done a new contact area radius (𝑎 𝑚 ) is computed. Fig.10 presents the 𝑎 𝑚 variation with time.

Fig. 11 :

 11 Fig.11: Prediction of the relative increase in 𝑉 𝐶𝐸 for IGBT device under the conditions of 𝛥𝑇 = 90°𝐶 & 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐.The conditions used in Fig.11are for an aging test under 𝛥𝑇 = 90°𝐶 and 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐. These tests, as observed, last much more than tests under 𝛥𝑇 = 110°𝐶. The lifetime of the first test is between 35 and 40 k.cycles. However, the lifetime of the next aging test is around 100 k.cycles. This is due to the thermo-mechanical stress effect produced by high temperatures on the degradation. As observed in Fig.11the first and second curves are far from the experimental curve. This is probably due to an error in experimental data measurement as observed at first 30 k.cycles. After that, the prediction curves start to trend upward as the experimental curve. A percentage of error can be observed in Fig.11. This error, from an industrial point of view, would prevent failures and the corresponding economic losses since the expected predicted failure is before the real failure.

Fig. 12 :

 12 Fig.12: Contact area radius measured (𝑎 𝑚 ) decrease with the number of cycles for the aging tests done under 𝛥𝑇 = 90°𝐶 𝑎𝑛𝑑 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐.

Fig. 13 :

 13 Fig.13: Predict RUL change when applied on aging tests done under𝛥𝑇 = 110°𝐶 and 𝑡 𝑜𝑛 = 3𝑠𝑒𝑐.

Fig. 12 :

 12 Fig.12: current distribution from metallization to Die and a zoomed part facilitates the understanding of current integration. Let 𝐽(𝑟) be the lateral current density in the metallization at 𝑟. 𝐽(𝑟) = 𝑖(𝑟) 2𝜋𝑟𝑡 = 𝜎𝐸(𝑟) (12)

Fig. 13 :

 13 Fig.13: Top image for the IGBT and the diode showing the metallization and wire-bonds distribution on it.

Fig. 14 :

 14 Fig.14: Schematic side view for the IGBT that helps in understanding the current distribution in the different wirebonds.

Table 1 .

 1 Voltage % measurements with respect to number of cycles. (5 measurements are presented to clarify the process).

		Number of cycle (k.cycles)	Δ𝑉 𝐶𝐸 𝑉 𝐶𝐸	%
	First prediction m=1	8	0.16
	m=2	12	0.23
	m=3	16	0.29
	m=4	20	0.43
	…	…	…
	m=9	34.5	1.58

  Zeina Al Masry received her Ph.D. in Applied Mathematics from the University of Pau and Pays de l'Adour in France in 2016. Since 2017, she joined ENSMM (Ecole Nationale Supérieure de Mécanique et des Microtechniques) in Besançon France as an Associate Professor. She is doing her research activities at the FEMTO-ST institute in the Prognostics and Health Management (PHM) research group. Her research works concern stochastic processes and applied statistics for PHM applications.

Mohamad Nazar et al. This is an open-access article distributed under the terms of the Creative Commons Attribution 3.0 United States License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

BIOGRAPHIES


Mouhamadou Mansour Mbow
email: mouhamadou-mansour.mbow@g-scop.fr

Christelle Grandvallet
email: christelle.grandvallet@univ-grenoble-alpes.fr

Frédéric Vignat

Philippe René Marin
email: philippe.marin@grenoble-inp.fr

Nicolas Perry
email: nicolas.perry@u-bordeaux.fr

Franck Pourroy
email: franck.pourroy@grenoble-inp.fr

Frederic Vignat
email: frederic.vignat@grenoble-inp.fr

Philippe Rene Marin

Frederic Vignat

Philippe Rene Marin

Mathematization of experts knowledge: example of part orientation in additive manufacturing

Keywords: Expert knowledge, Quantitative representation model, Desirability function, Action rule, Additive manufacturing, CAM, Part orientation

come   L'archive ouverte pluridisciplinaire

Introduction

Nowadays, companies are trying more and more to capitalize and organize their employees' knowledge in order to share it for better performance. This activity is referred to as knowledge management (KM) and has become a vital part of successful businesses as demonstrated by [START_REF] Tan | Linkage between knowledge management and manufacturing performance: A structural equation modeling approach[END_REF] through a work examining the efect of KM on manufacturing performance. Indeed, capturing, structuring and transmitting the workers knowledge, is a good way to assist activities like decision-making, problem solving, dynamic learning or strategic planning [START_REF] Gupta | An exploration of knowledge management techniques[END_REF]. The knowledge management methodologies may vary from one company to another depending on their objective use of the created or captured knowledge [START_REF] Davenport | Successful knowledge management projects[END_REF]. Various authors worked on how leading KM would positively inluence the competitiveness of manufacturing companies (Gunasekaran and 1 3

Ngai 2007; de Pablos 2002; [START_REF] Singh | Survey of knowledge management practices in Indian manufacturing industries[END_REF][START_REF] Paiva | Focusing information in manufacturing: A knowledge management perspective[END_REF]. Globally, it has been seen as an important element that maintains the sustainability and competitive advantage of an organization. In industries, the management of knowledge is critical to all processes of the value chain. Especially, the domain of product manufacturing requires intensive knowledge, from product design to production and quality inspection.

To manufacture a product implies to deliver at the right time and at an optimized cost in the required speciications, such as the right shape, dimensions, roughness, or material properties. To do so, multiple manufacturing strategies (for instance toolpaths) and parameters have to be set, which is the aim of CAM (Computer Aided Manufacturing) operations. Nevertheless, making the right choices requires a lot of knowledge and high expertise from the CAM engineer.

Regarding milling technologies, engineers rely on some commercial CAM software based on the following approach: feature recognition from the part CAD model; association of a process composed of one or several operations related to the identiied features; setting of parameters related to these operations [START_REF] Mourtzis | Computeraided manufacturing[END_REF]. The automation of CAM operations mainly relies on expert systems based on if/then rules that make the CAM software hardly scalable. Moreover, relevant feature recognition often remains challenging in many situations [START_REF] Albert | Feature recognition-the missing link to automated CAM[END_REF][START_REF] Corney | The CAD/ CAM interface: a 25-year retrospective[END_REF].

For more recent additive manufacturing technologies, CAM software are even less developed. They do not ofer any assistance to the user for certain operations. On the other hand, some other operations that are sometimes fully automated (i.e. support generation) do not leave any control to the user.

Our research aims at deining various means of assistance for CAM operations. The target is to provide the CAM engineer with a decision assistant that would be based on the expert knowledge of the domain, and that would help him/her to make the right choices among numerous parameters to be set. In this context, the present paper proposes a structured approach for transforming manufacturing rules, gathered from a panel of experts and expressed in their natural language, into a quantitative mathematical model. This model will make it possible to assess in a simple way the extent to which the diferent manufacturing rules are satisied when decisions are made about the CAM parameters.

In the remainder of this document, background and scope related to this research are presented in "Background and scope" section. Then, a case study of additive manufacturing part orientation that will be used for deining and illustrating our modelling approach is presented. The four main steps of this modelling approach are afterwards detailed in "Modeling approach" section. "Application to part orientation" section presents the modelling of some additive manufacturing rules and the application to a part geometry. Finally, "Conclusion" section summarizes the results and provides concluding perspectives.

Background and scope

Background

The knowledge of CAM experts is the starting point this work. It is thus important to state what is meant by knowledge. Although some researchers consider that knowledge can be objectiied and directly managed, for instance with information systems, the position that we adopt in this paper is more in line with [START_REF] Wilson | The nonsense of "knowledge management[END_REF] who claims that knowledge "involves the mental processes of comprehension, understanding and learning that go on in the mind and only in the mind". In addition, [START_REF] Nonaka | Perspective-Tacit knowledge and knowledge conversion: Controversy and advancement in organizational knowledge creation theory[END_REF] make the distinction between explicit knowledge, which is "accessible through consciousness", and tacit knowledge that is more rooted in action, skills, movement, or senses. Knowledge is seen as a continuum between these two dimensions.

At the opposite side, information is deined as "everything outside the mind that can be manipulated in any way" [START_REF] Wilson | The nonsense of "knowledge management[END_REF]. When available and formalized in documentation, information can be extracted for further processing (such as classiication, storage, sharing or reuse).

Knowledge elicitation is "the process of collecting from a human source of knowledge, information that is thought to be relevant to that knowledge" [START_REF] Cooke | Varieties of knowledge elicitation techniques[END_REF]. Various elicitation techniques, either based on observations, interviews or process tracing, have been widely used and studied in the literature [START_REF] Milton | Knowledge acquisition in practice: A step-bystep guide[END_REF]. These techniques make it possible for the experts to explain their knowledge, and hence to get some relevant information on this knowledge, that can afterwards be shared and managed. During knowledge elicitation, experts might become more conscious about certain parts of their knowledge. According to [START_REF] Nonaka | The knowledge-creating company: How Japanese companies create the dynamics of innovation[END_REF], it is believed that some tacit knowledge moves towards the explicit dimension, making it possible to get information about this knowledge).

Knowledge structuring implies to classify it into, for instance, general and domain speciic knowledge, concrete and abstract knowledge, formal and informal knowledge, conceptual and procedural knowledge [START_REF] De Jong | Types and qualities of knowledge[END_REF]. While conceptual knowledge is valuable for the understanding of a speciic domain, procedural knowledge is of prime importance in order to support action, and in particular to provide a CAM assistance.

Recent elicitation activities led in the manufacturing domain have enabled the formalization of experts' knowledge thanks to speciic elicitation tools and techniques. When experimenting the use of an inluence matrix in a collective debate, [START_REF] Grandvallet | From elicitation to structuration of additive manufacturing knowledge[END_REF] managed to capture crucial knowledge about manufacturing practices, and classiied it into diferent categories. Thereby, deinitions, examples, inluences, state rules were identiied as four sub-categories of conceptual knowledge, and action rules that were part of procedural knowledge. The latter, action rules (ARs), are of primary interest in this work. Indeed, action rules can drive the process of setting multiple parameters. They are expressed in a natural language and integrate action verbs. In computational systems, they can be translated into diferent forms. For instance, they can take the form of a condition with if/then statement in expert systems [START_REF] Zhou | An equivalent generating algorithm to model fuzzy Petri net for knowledge-based system[END_REF], they can be a basis for a reward system in machine learning algorithms, or they can be expressed in the form of a speciic mathematical function.

Desirability functions prove to be suitable for the mathematical transformation of ARs. Desirability functions (DF) allow the conversion of diferent scales of quality measures into 0 to 1 objective functions [START_REF] Trautmann | On the distribution of the desirability index using Harrington's desirability function[END_REF]. These functions were irst introduced by Harrington, then, [START_REF] Derringer | Simultaneous optimization of several response variables[END_REF] proposed a modiication to show their ability to transform multi variate responses into only one single aggregated output. Diferent types of these functions are available in the literature (from the simplest to the most sophisticated) [START_REF] Costa | Desirability function approach: A review and performance evaluation in adverse conditions[END_REF], and have been used combined with the response surface methodology or fuzzy logic to solve manufacturing parameter optimization problems [START_REF] Datta | Desirability function approach for solving multi-objective optimization problem in submerged arc welding[END_REF][START_REF] Aggarwal | Optimization of multiple quality characteristics for CNC turning under cryogenic cutting environment using desirability function[END_REF][START_REF] Kim | Simultaneous optimization of mechanical properties of steel by maximizing exponential desirability functions[END_REF][START_REF] Singh | Optimization of bead geometry of submerged arc weld using fuzzy based desirability function approach[END_REF][START_REF] Al-Refaie | Solving dynamic systems with multi-responses by integrating desirability function and data envelopment analysis[END_REF]. Figure 1 shows examples one sided and two sided desirability functions. Another advantage with this approach is the possibility to incorporate the decision maker preferences such as giving priority levels or weights to each response, for multiple response problems [START_REF] Costa | Desirability function approach: A review and performance evaluation in adverse conditions[END_REF]. Equation 1, shows an example using geometrical weighted mean proposed by [START_REF] Derringer | Simultaneous optimization of several response variables[END_REF], where w i represent the weights given to each function.

Three types of functions can be found, "the larger the better", "the nominal the best" or the "smaller the better". In this work, only the larger the best is used i.e. the most desirable case corresponds to a desirability index near the upper bound target U and the less desirable corresponds to that near the lower bound L . The formulation of desirability functions has evolved through years with a variety of proposals. However, for the sake of simplicity, this study will consider the formulation proposed by [START_REF] Derringer | Simultaneous optimization of several response variables[END_REF].

In the present context, DFs make it possible to quantify the satisfaction level of an action rule related to some variations of given variables. The use of desirability functions usually consists of identifying irst the most important factors ( ŷ ) inluencing the output response of the system of interest. Then, those are transcribed into desirability indices ( d ) (each value of the identiied factor will have an image d ranging from "0" to "1") by experimental means or by using any expertise. In the present work, these factors or quantiiable variables ( ŷ ) are directly referred in the action rules either explicitly or implicitly. In the latter case, the variable is inferred from the fundamental relationship between the constitutive concepts of the AR. For this purpose, a relational analysis of the rules content must be led in order to extract the variable ŷ and its threshold values allowing the mapping with desirability.

This short review of literature is summarized by Fig. 2 which shows the positioning of the paper.

On the right side of the diagram in Fig. 2, experts have more or less conscious knowledge that allow them to take relevant decisions regarding the choice of CAM parameters. Elicitation techniques make it possible to go to the information side since they provide information on this knowledge, and more especially in the form of action rules that can be (1)

D = d w1 1 * d w2 2 * d w3 3 * ⋯ * d wi i (2) ∑ w i = 1
Fig. 1 Example of desirability functions used to drive the CAM activities. These action rules can then be reworked through a mathematization process, using for instance desirability functions, in order to get quantitative representations. The term mathematization refers to the action of introducing principles and methods specific to the mathematical sciences into a field of knowledge that did not belong to it. It also refers to the act of giving a mathematical formulation to something (CNRTL -CNRS). It is an act of interpreting or expressing mathematically, or the state of being considered or explained mathematically (Collins dic.).

Thereafter, once integrated within a CAM environment, those quantitative representations will make it possible to evaluate easily and quickly the satisfaction or compliance level of each one of the action rules for a set of values of the CAM parameters. These evaluations will be useful for the engineers to make relevant decisions about these parameters.

Although it takes time to achieve this process, and skilled operators might consider more eicient to make decision based on their own knowledge, the proposed approach has the following advantages:

• The task of elicitation and mathematization of the company knowledge is only done once, • The approach makes it possible to develop a companyspeciic shared expertise, • Decisions are justiied based on measurable criteria.

Scope of the paper

The present paper aims at bridging the gap between the action rules, expressed in natural language, and a mathematical representation of these rules. The research question states as follows: which methodology based on desirability functions could be deined in order to get in a systematic way, quantitative representations of action rules that drive the CAM activities?

The proposed methodology is based on a study case about part orientation in AM. In this context, knowledge elicitation has been conducted with industry experts [START_REF] Grandvallet | Eight action rules for the orientation of additive manufacturing parts in powder bed fusion: An industry practice[END_REF]. The resulting rules are used as the starting point of the work presented in this paper.

Study case: part orientation in EBM manufacturing

Manufacturing process

Electron Beam Melting (EBM) is one of the Powder Bed Fusion technologies (PBF) commonly used in Additive Manufacturing. The manufactured part is created layer by layer by melting metal powder with an electron beam. Figure 3 shows a typical PBF manufacturing process. Following the design step and preceding the manufacturing step, the CAM step encompasses various operations that aim at preparing the build ile, including the settings of manufacturing parameters. For PBF technologies, this includes the selection of optimal build orientation, the creation of support structures for overhanging surfaces, the nesting on the build platform. But irst and foremost, part orientation is one of the most important steps of CAM operations.

Action rules for part orientation in EBM AM

Part orientation is basically the setting of two angles that will deine how the part should be oriented regarding the building plate of the machine (start plate). Nevertheless, it remains a complex and critical activity as it highly afects the quality, cost, production and inishing time of the part. Usually, operators arbitrarily orient parts without being able to explain why they chose a given orientation. Various authors in the literature have investigated the computation of part orientation [START_REF] Frank | Expert system-based selection of the preferred direction of build for rapid prototyping processes[END_REF][START_REF] Delfs | Optimized build orientation of additive manufactured parts for improved surface quality and build time[END_REF][START_REF] Pandey | Part deposition orientation studies in layered manufacturing[END_REF][START_REF] Das | Optimum part build orientation in additive manufacturing for minimizing part errors and support structures[END_REF][START_REF] Rattanawong | A volumetric approach to part-build orientations in rapid prototyping[END_REF][START_REF] Zhang | Build orientation optimization for multi-part production in additive manufacturing[END_REF][START_REF] Qin | Determination of optimal build orientation for additive manufacturing using Muirhead mean and prioritised average operators[END_REF]. Typically, the computation of optimal orientation usually consists in selecting a set of criteria to fulill, then evaluating those on geometries of interest. The most encountered aspects consist of minimizing: the support volume or area, the build time, the build height, the material consumption, the projected area and the distortions.

Today more and more commercial software are used to compute these criteria on parts as indicated in Table 1. These tools are globally using physical parameters of diferent scales (e.g. support volume, part height, etc.) that cannot be compared to each other. Most of them propose optimum solutions computed by assigning weights to the aforementioned dimensional physical parameters. These weights remain the only values on which the operator can intervene, as the algorithms used by the commercial tools are black boxes. Beyond the criteria computation, some of the tools allow the deinition of surface category (3DXpert, Netfabb and Magics). Only one software (Amphyon) allows a heatmap presentation of the criteria values over the possible orientations to support decision-making. Moreover, the criteria deined by the software editor cannot be evolving with the expertise of the company; they do not ofer to experts the possibility to add new orientation rules or criteria.

As mentioned in "Scope of the paper" section, the originality of the approach proposed in this paper consists in starting from expert knowledge for building a set of rules and dimensionless criteria applied to speciic part entities, so as to support decision in part orientation.

In an industrial context, [START_REF] Grandvallet | Eight action rules for the orientation of additive manufacturing parts in powder bed fusion: An industry practice[END_REF] used the LIT (Limited Information Task) technique in order to capture main concepts involved in the execution of part orientation. Eight action rules were constructed from statements of industry experts working on powder-based additive manufacturing. Six of them have been more deeply studied for mathematization. They are listed and explained hereafter: overhanging surfaces need support structures during the manufacturing process to get the surface anchored and prevented from distortion or warping [START_REF] Hussein | Advanced lattice support structures for metal additive manufacturing[END_REF]. When support structures are used, their removal leaves marks that usually need inishing or pol- ishing. This rule intends to minimize the marks (thus, overhanging surfaces) on surfaces for which machining will not be performed. An overhanging surface is characterized by its normal vector angle with the negative vertical as shown in Fig. 4 ( = ⃗ n, -⃗ z ). Usually, support structures are only required for surface angles less than a speciied value. This value depends on the technology, the material and the machine. In PBF, a tool is used to remove the support structures after manufacturing, for this reason, the accessibility needs to be considered. Here, by making the assumption that the support is diicult to remove when the surface in question has another one in front of it, the removability depends on the space between the two surfaces. A surface is therefore quali-ied as a SSDR when its accessibility is limited by surrounding surfaces.

Usually, action rules can easily be interpreted or even completed by human reasoning. However, to do so with computational systems, mathematical formal models have to be established. To this end, the next section presents the proposed modeling approach that allows interpreting the action rules content into desirability functions.

Modeling approach

The proposed approach consists in translating action rules expressed in natural language into fuzzy functions. The main aim is to provide a mathematical image as close as possible to the objective of a given action rule. To achieve this, desirability function based modeling is used. This modeling approach has got the advantage of normalizing variables of diverse natures into dimensionless values between zero and one.

Figure 5 summarizes the general steps of the methodology. The irst step (1) is to codify the action rule in order to put it in the form of an objective (if necessary) using a common grammar. The second step proceeds to a mathematization of the action rule by performing the following steps: (2) Deine the evaluation concept, (3) Construct the desirability function, (4) Deine a combination operator. In (2) a relational analysis of AR content is carried out to ind out the quantiiable variables ŷ and their related evolution laws that will be used for the construction of desirability functions in (3). The inal step (4) consists of deining how the overall desirability is calculated on the whole part. It has to be noted that the experts may intervene in the whole modeling process.

Codiication of action rules

Action rules captured from expertise are expressed in diferent forms with diferent grammatical structures. This diversity makes it diicult to deine and handle a general modeling process. For that reason, this work proposes to codify the ARs in the form of objectives using a common grammar made of words and classiied in diferent categories of concepts related with the domain of interest.

For additive manufacturing, [START_REF] Kim | A design for additive manufacturing ontology to support manufacturability analysis[END_REF] recently proposed a classiication that includes these categories: Design Features, Manufacturing Features, Geometry Parameter, Material Parameter, Process Parameter and Feature Constraint. The Design Features are linked with the part topology (e.g. cylinder, spheres, surfaces, holes…), while the Manufacturing Features are those related to additive manufacturing itself (e.g. support structures, powder, layer …). The Geometry Parameters correspond to characteristics that a Design Feature includes; for instance, a cylinder might have the following feature characteristics: height, diameter, area and orientation.

The proposition is not completely in line with the rules described "Action rules for part orientation in EBM AM" section. In fact, industrial practitioners diferentiate the part geometric entities (shape entities) and gives them diferent attributes. Among those are the post-machined surfaces, the priority surfaces and the machining datum surfaces. A given surface can have multiple attributes (e.g. a non-machined surface can also be a priority surface). Action rules then apply to a subset of the part shape entities identiied by their attributes.

Finally, the following categories have been identified: Shape entity, Attribute, Evaluation Concept and Action (Fig. 6). Given an action rule, its composition is such that the action applies to one or more evaluation concepts. Each evaluation concept is associated to shape entities identiied by the given attribute.

Part

This concept clusters the diferent classes of information related to the geometry in question. The class Shape is deined by the distinct entities (volume, surfaces, edges and vertices) constituting the part's topology. As a complement, the class Specifications (corresponding to the global product speciications) indicates the extrinsic and intrinsic characteristics of the Shape entities. For instance, this class provides information about the roughness or the tolerances on the dimensions of a surface or between surfaces. The class Attributes is used to classify the Shape entities according to their expected quality or post processing need (e.g. machined surfaces). Figure 7a shows a summary of a part composition. A non-exhaustive list of attributes including hole, non-machined, machining datum and priority are deined based on the ARs formulated by experts. It has to be noted that the attributes may or may not exist in the action rule. They point to the involved Shape entities as illustrated in Fig. 7a by the dashed lines.

Evaluation concept

The evaluation concepts are used for assessing the impact of an action on a part of interest. Evaluation concepts are the pivotal base of the decision-making through an AR since they allow evaluating and understanding the consequences of an action onto the inal requirements namely, the quality, cost and processing time. In the AR, each evaluation concept is linked to a shape entity by quantiiable variables (discussed in "Identify the variable ŷ linking the shape and the evaluation concept" section). The identiied evaluation concepts for the list of rules given in the "Action rules for part orientation in EBM AM" section are reported in Fig. 7b.

Action

The Actions are introduced to normalize the ARs in order to explicitly express them as objectives by using the following word units: minimize, maximize, avoid or require. They make it possible to transform an action rule into an objective to reach. Minimize and maximize terms are used when intermediary cases are accepted and, avoid and require are used when the non-acceptable cases (zero desirability) cannot be compensated (i.e. orientation with non-acceptable local situation has to be excluded).

Fig. 6 Example of action rule's concepts codiication

Codiication

Each action rule implicitly includes an objective to complete. The codiication intends to explicitly show the objective of the action rule by replacing its action verb with one of the Actions deined in Fig. 7c. For instance, the rule "Orient machining datum surfaces (MDS) out of horizontal" seeks to get the maximum MDS possible out of horizontal position. Therefore, the codiied version of this example could be either "Maximize MDS out of horizontal" or "Require MDS out of horizontal" depending on the wanted severity. It should be noted that the codiication step is not always necessary because action rules are sometimes directly expressed in the form of objectives. For instance, in the aforementioned list, rules (a), (b), (e) and (f) are already codiied.

Deinition of the evaluation concept

In the literal expression of action rules, there are implicit laws involving the necessary understanding of the interrelations linking the concepts. [START_REF] Beissner | Using and selecting graphic techniques to aquire structural knowledge[END_REF] deine the relation between the concepts as structural knowledge. The process of building the structural knowledge might require the preliminary scientiic comprehension of interactions between the involved parameters and the intervention of experts to establish threshold values.

The idea in this step is to perform a mathematization of the action rules to represent the relationship between the concepts by following three main steps. As discussed in its deinition, an evaluation concept is linked to the speciied shape entity by quantiiable variables. Therefore, the variable ŷ linking the shape entity and the evaluation concept is irst identiied by experts; second, the way to measure ŷ on the part is determined. Finally, the calculation of the evaluation concept evolution according to the variable ŷ is established. The expected outputs are ŷ , the way to measure it and the function linking ŷ with the evaluation concept (Fig. 8).

Identify the variable ŷ linking the shape and the evaluation concept

This process involves the manufacturing experts and CAD experts' knowledge. The aim is to identify the evaluation concept input variables ( ŷ ). These variables are selected with CAD experts as far as they have to be measured from the shape entities. In fact, each shape entity has some characteristic parameters that can be used as variables. For instance for a surface, the angle, the length of its perimeter, the area or the projected area, etc. can be identiied. The parameter among those, that its more with the evaluation concept will be used as the variable ŷ . Diferent opinions might arise among experts for the identiication of ŷ . The part shadow onto the machine plate for which diferent types of measures have been identiied is one illustration (refer to Fig. 9). The measurement of part shadow can be the area of the projection, the maximum dimension of the projection, and the projection itself can be the silhouette (Fig. 9a), the contour (Fig. 9b) or the bounding box (Fig. 9c). An iterative discussion with experts helped to inally converged on the solution (b) of Fig. 9. Indeed, in PBF it can be foreseen to put another part inside the hole of the shadow in order to save place.

Determine how to measure the variable ŷ on the part

According to the part's involved entity, the way to measure the variable might difer. For some cases, a speciic discretization method for local evaluation is required. For instance, to ind the local angle ŷ of a complex surface by its normal, it is necessary to tessellate the latter to transform it into triangular lat elemental surfaces. Helpful geometrical operations on shape entities for the evaluation of desirability are reported in Table 2.

Find out how to calculate the evaluation concept according to the variable ( ŷ)

This establishes the relationship between the ŷ and the evalu- ation concept. This analysis is one of the most important steps that allow going from a conceptual view to a process view of the ARs. The expected output for each identiied variable is a relational function (RF) showing the evolution of the evaluation concept according to ŷ.

For instance, given the codiied action rule depicted in Fig. 6, a codiication according to the discussed categories irstly allows identifying the concepts on which the analysis is carried out. The attribute non-machined indicates that the AR does not apply to all of the surfaces of the geometry of interest. For the identiication of the linking variable ( ŷ ), the response is the inclination angle ( ) of the surfaces as the law of evolution of overhang (evaluation concept) depends on the surface orientation (explanation in "Action rules for part orientation in EBM AM" section). A tessellation is performed to measure the elementary of the involved surfaces. Then, the evolution of the overhang according to the inclination angle can be inferred in Fig. 10. The threshold values 1 and 2 must be indicated by the experts.

Thereafter, the desirability functions are constructed based on the established structural knowledge. As shown in Fig. 10, the RFs are mapped between 0 and 1, but for some evaluation concepts, a fuzziication is necessary (refer to section). In case of more than one evaluation concept, the mathematical considerations in "Mathematical combination of various evaluation concepts" section are to carry out logical combinations.

Fuzziication Evaluation concepts can be of diferent types; they can be either non-quantiiable notions (e.g. deformation risks) or quantiiable notions (e.g. part shadow). In the irst case, the relational diagram is built based on the importance degree of the evaluation concept according to the variable.

In the second case, the relation is dictated by the maximum and minimum values of the quantiiable evaluation concept. For these reasons, a fuzziication is led to scale the evaluation concepts between 0 and 1. This consists in mapping their maximum and minimum possible values with values deined in the interval [0, 1].

For a variable ŷ , there is a function associating to each of its values a real m ∈ [0, 1] deined by: This deinition describes the relation between the evaluation concept and the variable linking it to the involved shape entity of the part. For instance, if the shadow area is considered, the maximum and minimum values are not limited between zero and one. Therefore, the fuzziication is done such that the maximum possible shadow area for this part is assigned the value 1 and the minimum the value 0.

Mathematical combination of various evaluation concepts

The previous illustration (Fig. 10) shows a simple case for which, there is a single variable, however, in some cases, more than one variable can exist depending on the number of evaluation concepts. Logical operations such as "and" and "or" bind the latter; for instance, the rule (f) in "Action rules for part orientation in EBM AM" section, for which, there is an "and" operator. Hence, some mathematical notions are introduced to make possible the combination of multiple desirability functions. The properties used here are the generalization of properties used in fuzzy logic theory introduced by [START_REF] Zadeh | Fuzzy sets[END_REF]. A few deinitions such as the negation of a RF, the intersection or union of multiple RFs are given in "Appendices" section.

Construction of desirability functions

As described in the Introduction, the desirability functions make it possible to translate any variable into normalized positive values in the [0, 1] interval. For the CAM opera- tions, the purpose of using expert knowledge is to compare diferent scenarios of parameter combination for the sake of quality, cost or time optimization (main drivers for experts to deine evaluation concepts). Thus, here, the desirability approach makes it possible to see any of the experts' rules in the same map of comparable dimensionless values.

The desirability function of each action rule is constructed based on the relational function from the mathematization and its relative action. For minimize and avoid, the corresponding desirability function is the opposite of the relational function (Eq. 4). Then, for maximize and require actions, the desirability function will be equivalent to the relational function (Eq. 5).

(3)

m = f (ŷ) (4) DF = RF = 1 -f (ŷ)
In the example of Fig. 10, the variable linking the part involved shape entity (surface) and the evaluation concept (overhanging criteria) is the angle ( ) between the surface and the built direction. This angle measures the surface orientation relative to the Z build direction (see Fig. 4). The need of support depends on this angle. In fact supports are needed for overhanging surface i.e. with low values. The unit relational function describes when a surface is overhanging and needs support i.e. the angle θ is less than 1 and when the surface is progressively not overhanging i.e. tending to be greater than 2 . To get the desirability function representing the action rule, the related action "minimize" is used. Owing to the fact that this action rule prescribes to minimize the overhanging non-machined surfaces, the situation in which those are overhanging is not desirable and the situation is desirable. Therefore, the desirability function in Fig. 11 is obtained by inverting the relational function in Fig. 10.

This function associates to each value of the variable of the considered surface a satisfaction value ranging from 0 to 1 (the best values being those near 1).

Deinition of combination operator

Analyzing various desirability values simultaneously is not a trivial manual task. When the surfaces of a part are considered for instance, the involved AR satisfaction level has to be discretely evaluated on each surface. Specially, if discretization has been performed on the shape, it becomes necessary to combine the local desirability values. Figure 12 shows an example of tessellated part made of over 16,000 triangles; this number of entity is impossible to handle manually when comparing scenarios. For such reasons, it is better combining the local desirability values to get a composite one by using an adequate averaging operator.

To evaluate the global desirability of a given geometry, the desirability is irst calculated locally on each of the n elemental facets obtained by tessellation (Fig. 12), then, calculated for the whole geometry itself by averaging on (5) DF = RF = f (ŷ) The result in Eq. 6 is by deinition the geometric mean weighted by the reals r i . Similarly, the Eq. 7 indicates the weighted arithmetic mean which has been demonstrated to be greater than the geometric one [START_REF] Qi | New proofs of weighted power mean inequalities and monotonicity for generalized weighted mean values[END_REF]. When the weights r i are equal, Eqs. 6 and 7 become equivalent to clas- sic means. Here, the r i are linked with a characteristic param- eter of the involved shape element. They can for example be relative surface area:

r i = A i A total
. This would give more importance to big surfaces at the expense of the smaller ones. It has been noticed that experts commonly deine weights with facet area. Exploring diferent possibilities is part of future works.

The choice of the averaging operation depends on the Action prescribed in the AR. As aforementioned, the categories minimize and maximize show that intermediary values are accepted, thus they will use a compensatory operation i.e. the weighted arithmetic average as it refers to a sum.

On the other hand, avoid and require categories do not accept the zero desirability cases and will use the geometric weighted average (due to the product, the compensation efects are relatively low). This means that even if only one of the

(6) D = n ∏ i=1 d r i i (7) D = n ∑ i=1 r i d i
elemental entities i has a null desirability, the whole geometry of n entities will have an aggregated weak desirability regard- less the other values.

Note that the discretization parameters afect the combined values very slightly. For the triangulation case, diferent parameters have been tested, but the combined desirability remained constant regardless the number of triangle. For complex parts with massive number of surfaces, the computation time might be long; its minimization should be considered in future developments.

Application to part orientation

Modeling the action rules

This part presents an example of knowledge modeling process about one of the most important CAM steps in additive manufacturing, which is the orientation of parts in the machine's build space. As suggested by the modeling approach, the irst step is the codiication of all action rules. Then, from a relational analysis of the content, a variable linking the part and the evaluation concept is inferred. Finally, the desirability functions are built and for each one, the corresponding evaluation formulae or averaging operator is provided. Results are summarized in Table 3.

Action rule (f) is a special case as it has two evaluation concepts thus, two link variables. The desirability of a SSDR is given by the intersection of the support structures requirement ( A ) and support removal diiculty existence ( B ). A mathemati- cal formulation of the rule and its equivalent can be written this way:

The corresponding relational functions are depicted in Fig. 13.

By applying the De Morgan law given in Eq. 14, it follows:

Then, the equivalent desirability function f c of the rule will be given by the maximum between ( 1 -f A ) and ( 1 -f B ) (Fig. 14).

Experts must indicate the threshold values in all of the desirability functions. Note that if the threshold values are equal (e.g. d 1 = d 2 ), the functions will become binary functions that can only take the values 0 or 1.

Example of application to a geometry

Problem description

Consider the bracket depicted in Fig. 15; the aim is to determine an optimal orientation around the X and Y axes with 3. Their corresponding threshold values are given in Table 4. The threshold values are proposed by the experts according to their machine capabilities. Some of them depend on the part geometry. For instance, for the part shadow rule, the thresholds are maximum (A max ) and minimum (A min ) area of part shadow (for the given part).

The green surfaces are post machined, thus, no matter if they have support structures. In fact, the machining price is already expected on those surfaces, that's why, the experts state that it's better putting support structures on them to save the non-machined surfaces (which include all of the other surfaces). Priority and datum surfaces are highlighted in red and yellow respectively. The surfaces with potential support diicult to remove are computed by a dedicated algorithm, since their manual selection would be diicult to achieve. The evaluation of the ARs on the geometry is made using a C ++ program based on Opencascade libraries (for geometry processing). This tool makes it possible to easily add Attributes to the shape entities. The rotations around X-axis (Rx) range from 0° to 180° and those around Y-axis (Ry) range from 0° to 360°. The results displayed in the form of response surfaces.

Results

Figure 16 shows for action rule (a) a total of 648 diferent desirability values of the bracket's orientations. The efect of the type of action (e.g. minimize) can be noticed on some of the results for which, aggregation on the whole part has been carried out (refer to "Deinition of combination operator" section). That is the case for (b) and (c), that include "minimize" and "maximize" respectively. For this speciic geometry, (b) is highly compensated. Replacing the use of "minimize" by "avoid" would change the result as no compensation is allowed. This is illustrated by the ARs (d) and (f) for which, a zero local desirability automatically causes a zero global desirability.

Each AR's desirability is bounded by maximum and minimal values reported in Table 5. A set of rotations may produce conlicts between the action rules. For instance, at Rx = 0° and Ry = 50° the deformation risks, the overhanging non-machined surfaces and the datum surfaces return good desirability values (0.717, 0.893 and 0.785 respectively), whereas all the other ARs have a poor desirability less than 0.3 (Fig. 17i). Another scenario at Rx = 40° and Ry = 120° would result in (b), (d), (e) and (f) greater than 0.7 and (a) and (c) are lower than average (Fig.

For most of the scenarios, the rule (c) does not return a good desirability while the others are suicient. In fact, the surfaces involved in this rule cannot be simultaneously vertical because of their perpendicularity. One maximum desirability of (c) is reached for Rx = 90° and Ry = 0°, but the rules (d) and (f) would be minimal (Fig. Several diferent other scenarios can also be analyzed. The one in Fig. 17iiii is an example that would be recommended for manufacturing the part in question. The rule (c) is not high, but is more than half of its maximum value.

Obviously, owing to this variety of solutions, it is not trivial to choose a point complying all of the AR simultaneously. However, one can make trade-ofs by selecting a pair of angles for which the individual desirability reach an acceptable value. Alternatively, an aggregation of all of the responses as in the example of Eq. 1 can be useful to tackle this. One can also favor the use of optimization algorithms to ind out constrained solutions. Extensive study aiming at proposing tools to facilitate the decision-making would be necessary.

As mentioned in the Introduction, action rules relect in some ways criteria related to quality, cost and time. The quantitative evaluation enabled by the proposed knowledge modeling approach gives thus to the CAM user a big picture of how the extent of those criteria are respected.

Conclusion

This paper proposes a modeling approach to enable the transformation of elicitation data, more speciically the mathematical transformation of action rules into desirability functions. To achieve this quantitative representation process, evaluation concepts must be deined, calculated and combined thanks to the identiication of variables and relational functions. In the framework of additive manufacturing, the elicitation of experts' knowledge has led to the identiication of a list of action rules used by CAM engineers for part orientation, which is a key step in build preparation.

According to the modelling approach, in a irst step each action rule is standardized through a simple grammar by the use of four action verbs (namely minimize, maximize, avoid, require, which explicitly show the objective of the rule) associated with an evaluation concept applied to some shape entities. Following that, a mathematization of the action rules content is done by performing a relational analysis that provides the variable of the evaluation concept measurable from the shape. Then the mathematical expression of the evaluation concept is determined. The step of analyzing the relations between shapes entities and evaluation concept is complex; it requires the intervention of experts or a prior understanding of the domain of interest.

The desirability function corresponding to each action rule is built based on the variable and the mathematical function.

Through the modelling approach, mathematical considerations regarding the combination of multiple relational functions are provided.

As shown by the illustrative part, the present quantitative representation approach gives a big view of the satisfaction level of the expert rules. The results clearly show the robustness of the approach for which the aim is not to decide for the CAM operator but to assist his/her decision-making. Whilst the choice of an orientation is still manual, the orientation evaluation model is not arbitrary as it is based on desirability functions that represent experts' knowledge.

The way the CAM operator uses the proposed evaluation concepts to make its decision is of key importance. In future works, the way these results should be presented to the CAM operator to support his decision has to be studied. Several possibilities have already been proposed, such as:

• showing him the set of desirability side by side for all possible orientations (response surfaces), • calculating and representing a weighted combination of the desirability values, • limiting the orientation possibilities by selecting a threshold for each desirability.

The proposed mathematical models can also be integrated into optimization algorithms. How this multi-objective optimization can be done still have to be studied.

Another perspective would be to analyze how the orientation evaluation could inluence the design of a shape through topology optimization. To perform shape optimization in a design process with this orientation evaluation used among other optimization criteria is also part of the project COFFA in which the current research has been conducted.

Fig. 2

 2 Fig. 2 Knowledge transformation

( a )

 a Minimize part shadow on start plate: the projected area of the part on the build platform (start plate) must be minimal to allow the maximum nesting(Chergui et al. 2018;[START_REF] Wang | Production planning for cloud-based additive manufacturing-A computer vision-based approach[END_REF]. The extent of the projected area varies with the orientation of the geometry. (b) Minimize total overhanging non-machined surfaces:

  (c) Orient part priority surfaces (PPS) close to vertical: priority surfaces constitute the most important surfaces of a given part. In PBF, surfaces oriented vertically have better quality, thus priority surfaces are oriented as close as possible to the vertical. (d) Orient machining datum surfaces (MDS) out of horizontal: machining datum surfaces allow the positioning of the part for post processing. If they are positioned horizontally (close to 0° or close to 180°), they might exhibit geometrical distortions. (e) Minimize shape deformation risks: parts made with PBF technologies are usually subject to deformations linked with the solidiication and cooling process of the fused metal layers. According to experts, if a certain number of long layers are stacked, there may be risk of deformation. Changing the orientation of a part modiies the coniguration of the layers through a diferent slicing. Therefore, this issue is to date mainly solved by inding the convenient building orientation. Note that only shrinkage defect (Vo et al. 2018; Ghaoui et al. 2020) is concerned with this rule. (f) Avoid support structures and support removal difficulty on surfaces with potential support difficult to remove (SSDR): this rule suggests to take into account the support removal diiculty.

Fig. 4

 4 Fig. 4 Illustration of overhanging surface

Fig. 7

 7 Fig. 7 Categories of concepts: a part classiication; b evaluation concepts; c actions

Fig. 9

 9 Fig. 9 Illustration of part shadow measurement possibilities

Fig. 10

 10 Fig. 10 Relational function between the evaluation concept (overhang) and the linking variable ( ŷ = )

Fig. 11

 11 Fig. 11 Desirability function

  Fig. 12 Tessellated geometry (n Facets)

  d. Orient machining datum surfaces out of horizontal Codiication: Require machining datum surfaces out of horizontal Shape entities: machining datum surfaces Evaluation concept: out of horizontal Link variable: surface angle ( ) Geometrical Operation: tessellation Averaging Operator: geometric mean weighted by relative area e. Minimize shape deformation risks Codiication: not required Shape entities: volume Evaluation concept: deformation risks Link variable: number of stacked long layers ( n) Geometrical Operation: slicing Averaging Operator: weighted arithmetic mean f. Avoid support structures and support removal difficulty on surfaces with potential support difficult to remove Multiple desirability function case developed below Codiication: not required Shape entities: surfaces with potential support diicult to remove Evaluation concepts: support structures, support removal diiculty Link variables: surface angle ( ), distance face/face ( d) Geometrical Operation: tessellation Averaging Operator: geometric mean weighted by relative area

Fig. 13

 13 Fig. 13 Relational functions of surfaces with potential support diicult to remove: a support structure requirements; b support removal diiculty

Fig. 15 Fig

 15 Fig. 15 Example of bracket

Fig. 17

 17 Fig. 17 Comparison of desirability values for the scenario: i Rx = 0° Ry = 50°; ii Rx = 40° Ry = 120°; iii Rx = 90° Ry = 0°; iiii Rx = 0° Ry = 150°

Table 1

 1 Criteria used by commercial part orientation software including speciic functionalities Software

	Criteria

Table 2

 2

	Geometrical operation for variable assessment	Shape entity	Request	Geometrical operation	Discretization
		Volume	Shadow area or length	Projection	-
			Size (length, width, height)	Bound-box	-
			Layer area	Contour illing (triangles, pixels)	Slicing
			Layer max/min length	Multi-directional hatching	Slicing
		Surface	Shadow area or length	Projection	-
			Surface angle	Normal vector mapping	Tessellation
			Surface area	Triangular illing	Tessellation

Table 3 (

 3 continued)

Acknowledgements This work beneited from the support of the Project COFFA ANR-17-CE10-0008 of the French National Research Agency (ANR) and DP Research Institute KAM Lab Project.

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1 3 For a given scenario, the intersection C (associated with m c ) of two relational functions A and B deined by the functions f A ŷA = m A and f B ŷB = m B respectively, cor- responds to the logical "and" operation. Its corresponding function is given by:

Deinition 3 Union For a given scenario, the union C (associated with m c ) of A and B deined by the relational functions f A ŷA = m A and f B ŷB = m B respectively, is the logical "or" operation for which the corresponding function is:

Deinition 4 De Morgan Laws

Example of pseudocode for surface angle desirability computation:


Fernando Pereira

Valérie Locurto

Virginie Ouya

Didier Graillot

Ana Cameirao

Julie Cazal

Jean-Michel Herri

Training project "Management of polluted sites and soils and mineral waste processing/recycling"

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Conclusions

TRAINING PROJECT "MANAGEMENT OF POLLUTED SITES AND SOILS AND MINERAL WASTE PROCESSING/RECYCLING"

The area of depollution is undergoing significant growth since 2000. The rehabilitation of polluted sites and soils was worth some 200 M€ in 2000200 M€ in , approached 500 M€ in 2010200 M€ in and reached 560 M€ in 2012200 M€ in (ADEME, 2013)). After a two-year economic slowdown, a market analysis study (Xerfi institute, 2015) projected that the turnover of the companies concerned should grow of considerable form in the future. The regulatory backgrounds (in perpetual evolution) of the cutting back of raw materials (especially, metals) and the land resource (often situated in urban areas) optimization bring real growth opportunities to the sector.

Context Description of the training

Methodologies for contaminated-site characterization

• Diagnoses implementation to fully understand the routes of exposure to harmful substances.

Risks study

• Pollutant transfer processes in soils and underlying aquifers (bio-indication, etc.). • Management tool for polluted sites and soils according to the use. • Experimental modeling/simulation.

Regulatory, territorial, social and economic approaches

• Organization and spatialization of the information. • Tools for environmental impact assessment. • Land and regulatory aspects.

• Financing instruments.

Objectives

The training program will be specifically designed to answer to an identified expertise demand regarding mineral waste treatment schemes. It will provide tools to solve any shortcoming or challenge linked to the processing and recycling of contaminated mineral matrices (soils, sludge, sediments, excavated soil, byproducts and waste) and will be aimed at developing teachings useful in proposing sustainable, integrated and economically rational solutions that address the full range issues posed by these materials since their on-site management until their evacuation, treatment and reuse. All this taking into account the environmental and sanitary impacts that their implementation, treatment and reuse cause.

Training expert engineers in the area of the management of polluted sites and able to:

• Manage project portfolios linked to the processing/recycling of contaminated mineral matrices (soils, sludges, sediments, by-products, etc.), • Develop a circular economy with a sustainable development approach, • Apply the engineering of environment to the sites remediation.

1 -Mines Saint-Étienne -Centre "Sciences des Processus Industriels et Naturels" (SPIN) -"Département Procédés pour l'Environnement et les Géo-ressources" (PEG) -Saint-Étienne (France), 2 -Mines Saint-Étienne -"Direction de la Recherche et de l'Innovation" (DRI) -Saint-Étienne (France), 3 -CNRS ("Centre National de la Recherche Scientifique") UMR ("Unité Mixte de Recherche") EVS ("Environnement, Ville et Société") 5600 -Lyon (France), 4 -CNRS UMR LGF ("Laboratoire Georges Friedel") -Saint-Étienne (France) [Contact : fernando.pereira@mines-stetienne.fr].

Fernando PEREIRA (1, 3) Valérie LOCURTO (2) Virginie OUYA (2) Didier GRAILLOT (1, 3) Ana CAMEIRÃO (1, 4) Julie CAZAL (2) Jean-Michel HERRI (1, 4)