## 3 Method
### Algorithmic Implementation
In this initial example, we follow the assumption that the excitation source, the features and the sensors have quasi zero size. While this could be modeled for the excitation \begin{table}
\begin{tabular}{c c c} \hline \hline
**Young’s Modulus \(E\)** & **Poisson Ratio \(\
u\)** & **Density \(\rho\)** \\ \hline \(70\times 10^{9}\)\(\frac{\mathrm{N}}{\mathrm{m}^{2}}\) & 0.33 & 270 \(\frac{\mathrm{kg}}{\mathrm{m}^{3}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material properties. and sensors, this is not possible for the features, which need to be modeled with finite size. The features in the presented example are therefore modeled with non-zero size, which introduces errors. However, these errors are small, if the ratio between the size of the feature and the distance to the feature is small. For a non-point-wise excitation, the spatial dimension of the excitation will still be small compared to the size of the wave propagation distances and, therefore, non-point-wise excitations are also expected to be possible. When seeking to model arbitrary excitation sources, i.e., not symmetric or anti-symmetric, we can always express these via superposition of a symmetric and anti-symmetric counterpart, as the behavior of the plate is expected to be linear-elastic. Two different excitation scenarios are applied. Firstly, in a training phase, a sudden impulse is applied as excitation, where the loading at the first time step is non-zero, while for all further time steps, the loading is zero. This is possible, as an impulse induces a broadband excitation across all discrete frequencies, thus allowing for the derivation of the FRFs and transmissibility functions for all frequencies of interest. For the testing phase, a Hann-window pulse is applied, as commonly used in SHM applications [13; 28]. The Hann-window pulse is described as follows:

\[f(t)=\left\{ \begin{array}{ll}(1-cos(2\pi t\frac{f_{c}}{c}))(sin(2\pi tf_{c}) )&if\quad 0<t<\frac{c}{f_{c}}\\ 0&else\end{array} \right. \tag{20}\]

with \(f_{c}\) denoting the carrier frequency of the pulse and \(c\) the number of cycles. A Hann-window excitation in frequency domain shows a narrow main lobe with flat side lobes. The frequency range covered by the main lobe can be evaluated through the following [93]:

\[f_{c}\pm 2\frac{f_{c}}{c} \tag{21}\]

A narrow and distinct frequency range allows for dealing with limited dispersion of the wave packet. For the following examples, the used Hann-windows will have a carrier frequency \(f_{c}=300\,\mathrm{kHz}\) and \(c=5\) cycles, resulting in a frequency range of interest between \(180\,\mathrm{kHz}\) and \(420\,\mathrm{kHz}\). In this frequency range, the \(S_{0}\)-mode is nearly non-dispersive, while the \(A_{0}\)-mode is significantly dispersive (Figure 1). ### Offline Phase

In the offline phase, the plate should be configured in order to learn the scattering behavior efficiently. The setup of the plate is shown in Figure 7, while the mesh details in the area of the features are illustrated in Figure 8. As two different features (a through hole with \(5\,\mathrm{mm}\) radius and an elliptical notch with \(5\,\mathrm{mm}\) radius and \(0.5\,\mathrm{mm}\) depth) are investigated in what follows, a total of six simulations are needed. These simulations cover the three cases of a pristine plate, a plate with a through hole, and a plate with a notch, where for each case, a simulation with symmetric and anti-symmetric excitation, respectively, is needed. In total, 808 sensors are assumed to be the positions where the response is to be reconstructed by the delivered ROM. More specifically, we assume the sensor placed in an arrangement of 8 circles around the feature, with 101 sensors per circle. For these simulations, the time step size is set at \(2.5\times 10^{-8}\,\mathrm{s}\), slightly less than the critical time step for the regular mesh. The simulations are run for 7999 time steps or a total of \(2\times 10^{-4}\)-\(2.5\times 10^{-8}\,\mathrm{s}\).