## 4 Discussion  
Filtering Function of the Optimal Unit Cell Design Combinations in Each Considered Unit Cell Configuration  
This study utilized data on bandgaps and key features of the widest bandgap (i.e., size and centroid; see Supplementary Materials) for all normalized design combinations analyzed in the previous parametric analysis. The objective of this work is to filter and identify optimal design combinations for each unit cell configuration. Both the bandgaps' data and bandgaps' key features were normalized. By varying the sizes of unit cells in these normalized design combinations, we determined their corresponding frequency ranges of actual bandgaps. Ultimately, the optimal design within each unit cell configuration was selected based on its ability to cover a substantial portion of the audible frequency range (i.e., 20 Hz to 20 kHz) with its bandgaps. The performance evaluation metric for the unit cells' bandgaps in terms of covering the range of audible frequencies is referred to as bandgap coverage. Bandgap coverage represents the fraction that indicates how much of the audible frequency range (i.e., 20 Hz to 20 kHz) is encompassed by a unit cell's bandgaps. The proposed filtering approach in this study determines the optimal unit cell design combination for each configuration through a series of steps, as shown in Figure 12. Firstly, the previously collected normalized bandgaps' data and key features from the parametric study were used as input. Secondly, a comprehensive sweep of unit cell sizes (\(C\)) ranging from 10 mm to 50 mm was applied to obtain the actual frequency ranges of the bandgaps for each unit cell design combination. It is important to highlight that the unit cell sizes (\(C\)) were discretized into 401 variants ranging from 10 mm to 50 mm. Considering these 401 unit cells sizes in the 505 normalized design combinations led to 202,505 designs. The actual bandgaps of 202,505 designs were efficiently computed from the stored normalized bandgaps of their corresponding normalized design combinations that share the same non--dimensional geometrical parameters (computations happen within a couple of minutes) instead of going back and utilizing [PERSON]--[PERSON]'s theorem computations, which will require unrealistic computational time. For instance, a Basic--Periodic unit cell with non--dimensional geometrical parameters of \(\frac{L}{C}=0.2\), \(\frac{d}{C}=0.1\) with unit cell size (\(C\)) of 30 mm will have the same normalized bandgaps as the Basic--Periodic unit cell with the same non--dimensional geometrical parameters but with a different unit cell size (e.g., \(C=50\) mm). Thirdly, a design criterion was established to exclude any unit cells whose cylindrical holes are located too close to the boundary layer thicknesses (i.e., vorticity boundary layer thickness and entropy boundary layer thickness). Excluding these designs was crucial because the presence of closely spaced cylindrical holes near the boundary layer thicknesses leads to a medium flow behavior that encompasses not only the acoustic mode, but also includes the vorticity mode and entropy mode [42]. As a result, the governing equation for sound wave propagation solved in this paper (i.e., Equation (1)) is invalid in this specific scenario. It is essential to note that although having medium flow with additional vorticity mode and entropy mode may be favorable in attenuation, accounting them will lead into solving the coupled system for the specific designs with narrow cylindrical holes, which is a computationally expensive process. \[l_{vor}=\left(\frac{2\mu}{\omega\rho}\right)^{\frac{1}{2}} \tag{8}\]  
Equation (8) [42] provides a representation of the vorticity boundary layer thickness. In this equation, \(\mu\) represents the kinematic viscosity of air and \(\rho\) refers to the density of air. The variable \(\omega\) denotes the frequency at which the air is propagating.