Issue 78

C. Popa et alii, Fracture and Structural Integrity, 78 (2026) 118-128; DOI: 10.3221/IGF-ESIS.78.07

interpretation would be that points close to the red line indicate accurate predictions while those far from the line indicate larger prediction errors. We can see that most points cluster reasonably close to the line. The reported value (R 2 =0.6058) means that approximately 60.6% of the variation in the target variable is explained by the model on the training data. Similar observations can be made for the testing set, for unseen test data (fig. 6b). Interestingly, the test R² (0.6421) is slightly higher than the training R². This is a positive sign because it suggests no obvious overfitting, stable model behaviour, and good generalization. Although the obtained coefficient of determination (R² ≈ 0.65) indicates that the model explains approximately 65% of the variance, we consider this level of accuracy appropriate for the intended purpose of the proposed methodology. The objective of the regression model is not to replace finite element analysis, but to provide a fast preliminary prediction of the optimal shape parameters during the early design stage. The prediction error remains sufficiently low to identify the general trends and relative influence of the design variables, substantially reducing the number of computationally expensive numerical simulations required. The third plot (Training vs Testing Performance) shown in Fig, 6c compares the three metrics R², RMSE, and MAE for both training and testing datasets. As discussed before, the R² values are very similar; the same thing can be observed for the other two metrics, both RMSE and MAE are almost identical. This is in accordance with the previous observations. The next two plots (figs. 6d and 6e) display residual values for the training and testing sets. The first one shows residuals centered around zero with a slightly larger spread at higher predicted values. This suggests that the linear model captures the main trend; however, it should be noted that prediction uncertainty increases slightly for larger values. Likewise, the second residual plot shows a similar spread. This indicates no obvious systematic bias, suggesting that the linear relationship is reasonably appropriate. Finally, the last plot is a histogram of residuals, Fig. 6.f. It can be observed that residual distributions peak near zero, are reasonably symmetric, and contain only a few outliers. This implies that predictions are generally unbiased, errors are distributed randomly, and the model does not systematically overpredict or underpredict.

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Figure 6: Multilinear regression results for optimal shape prediction

Based on the outcomes of the machine-learning analysis the heat map of S parameter for the three design parameters of the generated data are presented in Fig. 7. In this case the maximum value of S parameter was obtained for 6 mm distance, a 60° angle, and 0.8 mm radius. This prediction is close to the one provided by numerical simulation, with a small difference for notch radius 0.8 mm instead 1 mm. The difference between the predicted radius values (0.8 mm from ML and 1 mm from numerical simulation) is relatively small, and is not considered mechanically significant. The predicted optimum remains very close to the optimum obtained by the numerical simulations, indicating that the regression model successfully captures the overall design trend. Given the moderate prediction accuracy of the linear regression model (R² ≈ 0.65), a small

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