PSI - Issue 78

Ivan Roselli et al. / Procedia Structural Integrity 78 (2026) 128–136

133

Accordingly, the latent variable z can be written as follows: = + × (3) The number and type of layers, the number of latent dimensions, the type of training, and other internal parameters are essential for the CVAE configuration (Kingma and Welling (2014)). More details on the used CVAE configuration can be found in Palumbo et al. (2025). Training of CVAE was implemented in order to reconstruct signals of the undamaged structure (training on recordings of the WNDC test executed before the seismic shakes). In substance, we have trained the CVAE at DI = 0. All successive WNDC tests at DI higher than 0 were also reconstructed by CVAE. MSE measures how good is the approximation of the signal reconstructed by CVAE in terms of error with regards to the true input signal. ORSR is the ratio in decibels between the magnitudes of the input of the CVAE and its reconstructed signal. Pollastro et al. (2023) and De Angelis et al. (2024) considered MSE and ORSR as effective metrics for discriminating undamaged from damaged scenarios. Values of MSE and ORSR were calculated for each sample for all WNDC tests (all damage cases). Each damage case is associated to a DI value calculated by modal analysis of vibration data. For each DI case MSE and ORSR values provided a DI cluster. Finally, the centroids of each DI cluster in the MSE-ORSR were computed in order to extract the distance of each centroid from the centroid corresponding to the training dataset (DI = 0). 5. Results The implemented CVAE-based procedure can provide a reconstruction of the vibration data time histories. Three examples of reconstructed signals are illustrated in Fig. 3. In particular, Fig. 3a shows the reconstruction of vibration data in undamaged conditions (DI = 0). As the CVAE algorithm was trained with DI = 0, it is well expected that the reconstruction in Fig. 3a provides the best approximation of the original data (true data). Consequently, the case DI = 0 provided the lowest values of MSE and ORSR. In Fig. 3b and Fig. 3c two examples of reconstruction in the case of limited (DI = 0.257) and remarkable (DI = 0.875) damages, respectively, are displayed. As expected, the approximation of the true data gradually deteriorates with the increase of DI values, because they represent progressively different conditions from the training dataset (undamaged case). In practice, as it is reasonable, the higher the DI value of the damage case, the higher the MSE and the ORSR. In Fig. 4 the clusters representing the results of simulations in the MSE-ORSR plane for each considered DI case (DI clusters) is shown. The achieved DI clusters did not result completely separated. However, most points of each DI cluster seem to be placed in distinctive positions, quite distant from the undamaged case points.

Fig. 3. Examples of data reconstruction by CVAE: (a) signal at DI = 0; (b) signal at DI = 0.25671 and (b) signal at DI = 0.87505. Abscissa of time histories is in recorded frames (200 fps).