## 3 Results
### Validation on Machine-Made Parts  
The third step of our application scenario demonstration covers the validation of the derived ART sorting algorithms for defect detection (see Figure 1). This is pointed out on the two dozen machine-made validation parts and their measurement data. Before applying the final classification models to the machine-made parts, we derived some illustrative visualizations for the separation task between the good parts and the damaged ones. The related idea is to predict the eigenfrequencies of a part from the individual component structure data being variable within a population from part to part, meaning from the full geometric information in the case examined here. The deviations between the predicted and the actual eigenfrequencies of an undamaged part should be close to zero, provided that the frequency predictions are reliable. In contrast, large deviations in the frequencies can indicate that the actual eigenfrequencies of a part are affected by a structural impact, such as a macroscopic defect. The MATLAB(r) function fitlm (fit linear regression model) with the model option quadratic was selected for the eigenfrequency predictions. The eight geometric parameters of the part type served as predictor variables. The model calculations were performed on the data of the defect-free 5000 virtual training parts, featuring uniformly distributed geometric variations in the extended ranges compared to the basic population. Thereby, the eigenfrequency data were transformed to the measurement world with the correction factors, but no additional noise was applied. The measured geometric dimensions of the machine-made parts were put into the final regression functions. Finally, the deviations of the resulting eigenfrequency predictions versus the ART measured frequencies were calculated. Figure 13 visualizes the derived eigenfrequency deviations depending on the part number and quality state. The \(y\)-axis of each diagram indicates the mean eigenfrequency deviation, which is averaged from the absolute percentage deviations of the measured eigenfrequencies relative to the associated regression model predictions. The first diagram includes all of the 24 machine-made validation parts before implementing the through-hole defects. Three parts (no. 2, 8, and 20) show a significantly larger mean eigenfrequency deviation, which indicates that they are affected by a structural anomaly, but which is unknown in detail. These three parts have already been identified as suspicious in terms of the deviations between the measured and related simulated eigenfrequencies, as described in our previous publication. The available data, especially the similar behavior in cross-comparison, imply that a human mistake may have been made in production, such as the use of a material from another batch, despite an agreement to the opposite with the manufacturer. The further diagrams of Figure 13 once again contain the mean  
eigenfrequency deviation values of the defect-free states, but the suspicious parts were left out, due to reasons of visualization. In addition, the corresponding deviation values derived after cutting or enlarging a through-hole in some parts are plotted. There is an overlap in the case of a 1 mm defect size. Nevertheless, for the defect sizes of 1.5 mm or larger, there is a clear separation between the good and defective conditions, indicating that an ART defect detection based on the undamaged virtual training parts or synthetic training data alone is achievable. Finally, the classification models from the previous subsection were applied to the machine-made validation parts. The three suspicious parts were classified as defective in the majority, although they did not (yet) feature a through-hole, but rather a different anomaly. Figure 14 presents the classification decisions for the parts in terms of the TNR and TPR, indicating the amount of correct classifications. Thereby, the suspicious parts were excluded, resulting in a reduced amount of 21 good parts and 13 defective parts.