PSI - Issue 79

Davide D’Andrea et al. / Procedia Structural Integrity 79 (2026) 283–290

288

R 2 =1- ∑( y i -y î ) 2 ∑( y i -y̅ ) 2

(2)

n ⋅∑ | n i=1

y i -y î |

MAE= 1

(3)

The coefficient of determination R 2 quantifies the proportion of variance in the observed data that is explained by the model, thereby indicating how well the model fits the validation subset. The MAE provides a measure of the average absolute deviation between model predictions and actual data, reflecting the typical size of the prediction error. Differential Evolution algorithm was applied to the predictive model to estimate the global minimum of the objective function and identify the parameter combinations that achieved it. The proposed geometric configuration results to be coherent with what was suggested in previous study by Gaiser et al., (2015), who first observed how these variables influence fracture mechanic parameters. Table 3 reports the optimal configuration that minimizes the target output, along with the performance metrics of the predictive model. A coefficient of determination of R 2 = 0.7728 and a MAE of 0.0371 were obtained. The optimal configuration was subsequently analysed via FEM to assess the predictive accuracy of the model, yielding an output of 0.2390. Compared to the predicted value of 0.2581, this corresponds to a relative error of 7.4%, indicating a satisfactory agreement between the model’s estimation and the numerical simulation.

Table 3 . Optimal configuration obtained by Differential Evolution and predictive model’s performance. [ ] [ ] [ ] MAE Estimated Output [//] Calculated Output [//] Error [%] 0.7728 0.0371 0.2581 0.2390 7.4 133 77 261

Figure 5 presents the response surfaces of the target output. As the output depends on three independent variables, its complete behaviour cannot be visualized simultaneously in three dimensions; therefore, the trends are shown pairwise. In Figure 5-a, the response surface is plotted as a function of the dimple position and the bi-material notch opening angle. The surface exhibits a minimum as the angle approaches 135°, while the influence of the dimple position result to be negligible at higher angles. Conversely, its contribution becomes significantly more pronounced at lower angles, around 90°, where the output decreases as the Cu increase values approach the lower bound of their domain.

(a)

(b)

Figure 5. Response surfaces obtained by Random Forest predictive model.

In Figure 5- b, the response surface is plotted as a function of the dimple’s geometric characteristics, which exhibit comparable Spearman rank coefficients. A minimum is observed for dimple ’s positions tending to 50 µm combined with dimple radii approaching the upper limit of the domain. In both response surfaces the third variable, which is