Issue 68

B. Spisák et alii, Frattura ed Integrità Strutturale, 68 (2024) 296-309; DOI: 10.3221/IGF-ESIS.68.20

a previously isolated data set. If these are within a given error range, the network can be simulated with the measured data, from which the GTN parameter set for the material can be determined. From the above, in the first step, it is necessary to specify a matrix of damage parameters in order to generate the input data set. During the optimization process, these ranges represent the constraints. For this purpose, a range for each GTN parameter was defined based on the literature. Among the GTN parameters, q 1 , q 2 and S n (variance) were given fixed values, thus changing a total of 5 parameters in the optimization task. Eight force-displacement data pairs were selected to compare the simulated and experimental force-displacement curves. Since the force-displacement curve is relatively horizontal in the region where the GTN parameters affect the curve, displacement values were taken for given force values.

Figure 5: Built-up of artificial neural network. The usage of the ANN to determine the optimal GTN parameters was introduced in more detail in reference [11] and [12], here only the most important information are included. Two-layer feedforward network with sigmoid hidden neurons and linear output neurons was used. As the data complexity is simple only one hidden layer was applied. The ANN was cross validated; thus the 90 sample was split into training and testing sets. The neurons in the input layer correspond to the displacements associated to the chosen 8 forces. 8 values were chosen to describe sufficiently the force-displacement curve. The size of the output layer was fixed as the neurons in the output layer represent the damage parameters to be identified (f n , f c , f f ,  n , f n ). The number of hidden neurons is commonly chosen close to the number of input data. To determine the correct number of hidden neurons, a trial-and-error approach was used. For different numbers of hidden neurons, the ANN was trained ten times, and the value of the errors was calculated. Then, for the final ANN, the smallest number of hidden neurons that still gave satisfactory results was chosen. The ANN (8-6-5) has a structure consisting of 8 neurons in the input layer, 6 neurons in the hidden layer and 5 neurons in the output layer. The

build-up of the artificial neural network is shown in Fig. 5. Finally, the optimized GTN parameters are listed in Tab. 2.

Name

Parameter

Value range

Yield surface multiplier Yield surface multiplier Initial void volume fraction Critical void volume fraction Failure void volume fraction Mean strain for nucleation

q 1 q 2

1.5

1

0.0008 0.1591 0.4035 0.1803

f 0 f c f f

 n S n

Standard deviation

0.05

0.0099

Volume fraction for void nucleation

f n

Table 2: Optimized values of GTN parameters.

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