## 3 Analyses and Results
### Finite Element Part: Parametric Analysis about Airflow Resistivity and Periodicity Dimension  
This Section provides the details of the FE model used to simulate the meta-material, creating both the training and test sets: The outcomes of the analytical design phase, for the DBM model, or the material parameters, for the JCA model, are provided to the FE software, which returns TLs of a homogeneous and a meta-core medium, allowing to evaluate the TL peak percent increase. These results, along with the airflow resistivity and unit cell dimension are then passed to the machine learning algorithm. The details of the FE simulations (geometry, mesh size, number of elements, and number of nodes) used for modeling do not change whether the DBA or JCA model is used; the only difference is in the type and number of input parameters. Moreover, the foam characteristics (airflow resistivity, porosity, unit cell dimension, tortuosity, viscous, and thermal characteristic lengths), which are variable in order to generate the training and test sets, are restrained into intervals of engineering reasonability of common industrial foams. These intervals are summarized in Table 1. The first columns reports the foam parameter, the second column lists the value intervals, and the third columns indicates the model for which they are used as input features in the machine learning phase. The FE model is constituted by a layer of foam, with embedded periodic inclusions with the shape of cylindrical holes with perfectly rigid walls (Figure 2); the choice of the inclusion geometry represent an example, and may eventually impact the amplitude and/or the shape of the TL resonance peak, but not its frequency (since this only depends on the foam properties and on the dimension of the periodicity). For what concerns the FE implementation, the module \"Pressure Acoustics and Frequency Domain\" of COMSOL MultiPhysics is used both as a modeling environment and numerical solver. For all structures presented in this work, the mesh consists of tetrahedral elements, generated through physics-controlled algorithms that are pre-implemented in the software. Nevertheless, the authors verified that the maximum element size of each HR meshed is always lower than 1/4 of the minimum wavelength \(\lambda\), which leads to an average number of elements through models with different sizes approximately equal to 14,250. Since the analyses are carried out considering an excitation consisting of a normal incidence plane wave acting on a layer of air (whose properties are reported in Table 2) along the \(z\)-axis, the only fundamental boundary condition to apply is the so-called Perfectly Matched Layer (PML) on the very bottom face of the models. Indeed, this represents an artificial absorbing layer for wave equations, commonly used to truncate computational regions in numerical methods in order to simulate problems with open boundaries; this allows the PML to strongly absorb outgoing waves from the interior of a computational region, without reflecting them back into the interior. On the contrary, boundary conditions applied on faces normal to the \(x\)- and \(y\)- axes are not relevant for analyses with the above-mentioned kind of excitation, since under this condition the waves do not have propagating components along those directions; thus, for the sake of computational simplicity, a Sound Hard Boundary Wall (SHBW) boundary condition is applied herein. Eventually, for different angles of excitation, proper periodic conditions should be used instead. For a plane wave configuration at normal incidence \(\theta=\phi=0\), and thus oriented towards the negative direction of \(z\)-axis, the TL is computed as reported in Equation (12):  
\[\text{TL}=10\text{log}_{10}\frac{\Pi_{incident}}{\Pi_{transmitted}}. \tag{12}\]  
In the context of this work, the TLs related to the homogeneous (which, obviously, has no inclusions) and the meta-core models are respectively estimated through a TMM implementation [43] of Equations (13) and (14), and an FE solution of Equation (12).