## 3 Analyses and Results
### Finite Element Part: Parametric Analysis about Airflow Resistivity and Periodicity Dimension
\[\text{TL}=10\text{log}_{10}(\frac{1}{4}|T_{11}+\frac{T_{12}}{\rho_{0}c_{0}}+ \rho_{0}c_{0}T_{21}+T_{22}|^{2}), \tag{13}\]  
\[\text{with} \begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix} = \begin{bmatrix}\cos(kd)&\text{j}\sin(kd)Z_{c}\\ \frac{\text{j}\sin(kd)}{Z_{c}}&\cos(kd)\end{bmatrix} . \tag{14}\]  
A parametric analysis is performed for all possible combinations of two chosen sets of foam airflow resistivity and unit cell dimension values. In particular, 12 airflow resistivities and 10 unit cell dimensions are selected (refer to Table 1 for the ranges), both on a logarithmic scale, for a total of 120 different test cases; for each of them, the periodic resonance frequency (Figure 3) is analytically calculated through the approach illustrated in Section 3, while the TL percentage increase (Figure 4) related the meta-core solution, \begin{table}
\begin{tabular}{c c} \hline \hline
**Density**, \(\rho_{0}\) [kg/m\({}^{3}\)] & 1.213 \\ \hline
**Speed of sound**, \(c_{0}\) [m/s] & \(3.42\times 10^{2}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Properties of air. Figure 2: Example of FE geometry with five unit cells along the foam core thickness; the foam is green–colored, while the layers of air are blue–colored. compared to its homogeneous counterpart at fixed total thickness, is numerically estimated in the COMSOL environment using Equations (12) and (13). For example, considering a couple of \(\sigma\) and \(d\) values respectively equal to 5004 kg/(sm\({}^{3}\)) and 0.0127 m, from Figure 3 a resonance frequency of around 13,000 Hz (the analytical value is 12,735 Hz) may be found, while from Figure 4 an increase in the TL amplitude peak of around 150% (the FE value is 151%) may be estimated; instead, considering a couple of \(\sigma\) and \(d\) values respectively equal to 17,985 kg/(sm\({}^{3}\))] and 0.0189 m, from Figure 3, a resonance frequency of around 8000 Hz (the analytical value is 7689 Hz) may be found, while from Figure 4, an increase in the TL amplitude peak of around 50% (the FE value is 40%) may be estimated. Machine Learning Part: Prediction of the Transmission Loss Increase at the Resonance Frequency (DBM Model)  
The objective of this Section is to analyze the performance capabilities of Gaussian processes when predicting the TL percent increase of meta-materials (with respect to the equivalent homogeneous configuration). As already stated in Section 2.2, the predictions  
Figure 4: Transmission Loss peak amplitude behavior as a function of unit cell dimension and foam airflow resistivity. Figure 3: Transmission Loss peak frequency behavior as a function of unit cell dimension and foam airflow resistivity. on new instances are given by Equation (7), and the estimation of the hyper-parameters is carried out with an optimization algorithm. Therefore, once the co-variance function is decided, Gaussian processes do not need any user-defined choice other than the number of training examples. Figure 1 summarizes the several steps occurring in this phase. Since the generation of both a numerical and/or experimental training set would be time consuming and financially expensive, there is an attempt to carry out the prediction of new instances by using the lowest number of training examples as possible; such a prediction would be considered satisfying if the percent error between the numerical and predicted peak reduction is below 5%. Moreover, in order to demonstrate the robustness of the method with the chosen number of examples, in the problem at hand, the predictions are made with three different training sets, called TS1, TS2, and TS3. To this aim, the 120 test cases numerically obtained are used as pool from which to extract training and test examples. In particular, the complete observation of each sample is made of four features: Airflow resistivity, unit cell dimension, and resonance frequency as input, while the TL peak decrease is, obviously, the output.