PSI - Issue 5

S. Sahnoun et al. / Procedia Structural Integrity 5 (2017) 997–1004 H.Halloua et al / Structural Integrity Procedia 00 (2017) 000 – 000

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The equation governing the heat transfer by conduction in the material is expressed in the cartesian coordinate system by the following relation:

= ( ) + ( ) + ( )

(1)

Initial and boundary conditions that translate natural convection and radiation between the sample surfaces and the external environment are translated by the following equations: − ( , , , ) | = = + ℎ ( − ( , , , )) (2) − ( , , , ) | =0 = ℎ ( − ( , , 0, )) (3) − ( , , , ) | =0 = − ( , , , ) | = = 0 (4) − ( , , , ) | =0 = − ( , , , ) | = = 0 (5) ( , , , )| =0 = (6) The resolution of the heat diffusion equation with these boundary conditions by finite element software gave the controlled surface temperature distribution (Fig. 2). We reported in Figure 3 the temperature variations with time of the heated region center (controlled by the laser) for coating thicknesses ranging from 0.4 to 3 mm. We can distinguish a correlation between the coatings thicknesses and the controlled zone temperature value; In fact, the temperature reaches important values, passing through maximums for small thicknesses of the substrate. We will use neural networks to model the relationship between the temperature and the coating thickness to apply it in the non-destructive thermal control. Therefore, a network inputs pre-treatment by principal components and the structure optimization with the initial weights will be presented in the following sections. Artificial neural networks are highly connected systems of elementary processors that mimic biological neurons. By learning, the network is impregnated with the conclusions to be given in a new situation. The neural networks general architecture consists of representing neurons in successive layers. If it is theoretically possible to build networks with a very large number of layers, a three-layer network architecture with a single hidden layer using sigmoidal type activation functions is usually sufficient to approximate any nonlinear function (Cybenko (1989)). Supervised learning has been chosen because it is most suitable for functions approximation. In most cases, neural network training is formulated as an optimization problem by seeking the synaptic weight matrix W which minimizes 4. Artificial neural networks

Fig. 2 : Simulated temperature field results of a sample with a 3mm thickness at the time t=0.2s on the sample's upper side

Fig. 3 : Temperatures according to coating thicknesses Ep