PSI - Issue 23

Petr Opěla et al. / Procedia Structural Integrity 23 (2019) 221 – 226 Petr Opěla et al. / Structural Integrity Procedia 00 (2019) 000 – 000

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In the first case, for the approximation of each parameter, an individually customized multi-layer feed-forward artificial neural network (ANN) has been assembled – see general scheme in Fig. 2. Independent variables (temperature, strain rate) were connected via set of artificial neurons (computational units) with the dependent variable (always one of the studied parameters). Feed-forward of functional signal (i.e. calculation of the network output) has been generally described by Krenker et al. (2011). A high-accuracy response of each proposed ANN was provided by the adaptation procedure. Based on this process, in the case of each examined parameter, various network architectures (differing in the number of hidden neurons and corresponding transfer functions) have been tested in the sense of network approximation and prediction accuracy. Error response (evaluated as a mean squared error – MSE) of tested architectures was always minimized via network training. For this purpose, the heuristic minimization algorithm, proposed by Levenberg (1944) and modified by Marquardt (1963), was applied in cooperation with the back-propagation of error signal proposed by Rumelhart et al. (1986). Resulting ANN architectures of the examined parameters have following mutual attributes: one hidden layer, log-sigmoid transfer function for hidden neurons and pure line function for the output neuron. Ideal number of hidden neurons was, however, for each parameter individual: ε p (2), σ p (4), σ ss (4), c (2) and s (2).

Fig. 2. General scheme of the proposed artificial neural networks: the black-blue sphere – artificial neurons (computational units); the red lines – synaptic weights (material constants). In the second case, each parameter was approximated via a universal predictive relationship Opěla et al. (2016) : (3) The variable y ( ̇ , T , p ) represent described parameter (i.e. ε p , σ p , σ ss , c or s ) while p 1 (various), p 2 (-), p 3 (K) and p 4 (K − 1 ) are material constants. Optimal values of these constants were for each specific parameter achieved via minimization of performance function (MSE) – the genetic algorithm (GA) was proposed to deal with this issue, see Fig. 3. This evolutionary heuristic algorithm is based on the generating of high amount of possible solutions (called as individuals) – each individual consist of values of the searched material constants (genes). Individuals are generated as a set (population) in each iteration step of the proposed algorithm. Individuals of new population are always produced on the basis of individuals of previous population via three operators (selection, crossover and mutation). The GA runs ad infinitum until the predefined conditions are achieved (minimum MSE or maximum iterations). For the detailed information about the GA procedure see Mitchell (1998). Resulting material constants of the Eq. 3 are displayed in Tab. 1.     2 3 1 1 4 p T , , y T p p  exp p p T        