## 4 Demonstration and Validation of the Simulation-Based Generation of ART Training Data
Figure 11 shows further analogous results for geometric parameter \(h_{\text{B}}\). Again, results considering a training data adaption for measured eigenfrequencies, and exclusively virtual results as reference, are visualized. We performed the analyses for all 8 geometric parameters and evaluated the results by Pearson correlation coefficient, since no relevant systematic deviations or offset could be seen in the diagrams. This coefficient measures the linear relationship and takes values near 1, the more similar the actual dimensions and the estimated values are (assuming no other complicated relationship). We found values of 0.998 and 0.995 for the correlation coefficient regarding \(d_{\text{R1}}\) and \(h_{\text{B}}\). In general, we found only one value for the correlation coefficient much worse than 0.97 (around 0.945 regarding parameter \(l_{\text{R}}\)). Therefore, the procedure using synthetic training data showed very satisfying results. Figure 10: Results of the eigenfrequency-based estimation regarding geometric parameter \(d_{\text{R1}}\) based on synthetic, simulation-based training data including adaption to measurement. Figure 9: Results of the eigenfrequency-based estimation regarding geometric parameter \(d_{\text{R1}}\) based on synthetic, simulation-based training data without adaption to measurement. The findings above confirm that ART training data generated mainly by simulation techniques in combination with only a few real parts can yield very satisfying results. It should be emphasized that only five real parts were used as reference or for adjustment for the application example, whereas the conventional approach with an experimentally generated data set would have required the metrological characterization of dozens or hundreds of parts. The geometry estimates determined using measured eigenfrequencies deviate only slightly more from the actual geometric values than associated estimates based on simulated frequency values. Overall, very satisfactory results were obtained using the machine-made parts' measured eigenfrequencies. It is worth mentioning that within a series production there may be effects like variations in material properties due to different batches. This is essential for an ART application, because such changes may be reflected in the natural vibration behavior and would normally require a renewed training procedure or a new training data generation. The presented simulation-based approach is advantageous here. For the application example demonstrated above, instead of a totally new generation of training data, our novel approach allows easy (e.g., multiplicative) re-adjustments. In total, with these positive validation results, it is concluded that the novel approach is a suitable procedure requiring the experimental characterization of only a few real reference parts and which significantly reduces the effort associated with efficient and reliable acoustic component testing. Of course, our approach is not limited to the estimation of the geometry of samples, which was used as an application example within this study. Moreover, synthetic training data could be used in any ART application as mentioned in the following conclusion. Regardless of the specific application, synthetic data generated with the help of simulations are linked with many advantages. Simulation techniques can help to significantly reduce the experimental effort regarding an ART training procedure for any ART task. It is possible to react quickly to batch influences, or the scatter ranges of training data can be selected wide enough to cover batch jumps. In addition, it is also possible to react quickly to design changes of components or to create a data basis for acoustical quality inspection even before series production is launched. In principle, synthetic training data also allow for the application of more complicated and maybe more powerful analysis algorithms due to the potentially huge amount of simulation data compared to experimental data collection.