## 2 Materials and Methods
### Bandgaps Determination through Utilization of FE--Bloch's Theorem Method
The Periodic Boundary Condition can be defined as  
\[P(x+C)=P(x)e^{i\mathbf{e}.\mathbf{k}} \tag{4}\]  
Equation (4) depicts the application of Periodic Boundary Conditions to one of the faces aligned with the direction of tessellation in 1D acoustic metamaterials along the x--axis. Here, \(C\) represents the size of the unit cell, \(\mathbf{e}\) stands for the lattice vector, and \(\mathbf{k}\) denotes the wave vector. The lattice vector (\(\mathbf{e}\)) signifies structural periodicity by indicating that repeating translations of multiple unit cells using this vector will yield a corresponding periodic structure. The lattice vector for the considered 1D acoustic metamaterial is (\(\mathbf{e}=Ci+0j+0\mathbf{k}\)) for the Basic--Periodic, Tapered--Diverging, and Tapered--Converging metamaterials and (\(\mathbf{e}=2 Ci+0j+0\mathbf{k}\)) for the Semi--Periodic metamaterial. Consequently, it is necessary to determine wave vectors (\(\mathbf{k}\)). These are determined by computing reciprocal vectors based on lattices and defining their relationship within the 1 st Brillouin zone [41]. The reciprocal vector (\(\mathbf{b}\)) is expressed as (\(\mathbf{b}=\frac{2\pi}{C}i+0j+0\mathbf{k}\)) for the Basic--Periodic, Tapered--Diverging, and Tapered--Converging metamaterials, and (\(\mathbf{b}=\xi i+0j+0\mathbf{k}\)) for the Semi--Periodic Metamaterial, and it plays a crucial role in determining the first Irreducible Brillouin zone, as shown in Figure 6. Figure 6 depicts the first Irreducible Brillouin zone for the examined one--dimensional periodic metamaterial in the \(\Gamma-X\) direction (i.e., direction of tessellation). The first Irreducible Brillouin zone encompasses the permissible wave vectors through which sound waves can propagate within this periodic structure, specifically within our considered one--dimensional periodic metamaterials. It is worth noting that the first Brillouin zone corresponding to the structures that are one--dimensional periodic is a line in the reciprocal space, as illustrated in Figure 6. This work considered a total of 21 points corresponding to wave vectors (\(\mathbf{k}\)), obtained by sweeping across the first Irreducible Brillouin zone along the \(\Gamma-X\) line. The eigenvalue problem, described by Equation (3), will be computed for each specific Periodic Boundary Condition within the first Irreducible Brillouin zone. This process is repeated 21 times to account for each design combination and their respective Periodic Boundary Conditions. To solve these eigenvalue problems, the finite element software ABAQUS (ABAQUS 2023) is utilized. The EQUATION function in ABAQUS will incorporate the Periodic Boundary Conditions, while the LANSCOZ SOLVER will compute the eigenfrequencies. In this study, we used acoustic elements (A3D10) as elements comprising of tetrahedron element with 10 nodes. Additionally, a mesh sensitivity analysis was conducted to determine accurate results independent of variations in meshing density. Our findings indicated that when using a maximum mesh size equivalent to 3 mm and a  
Figure 6: First Irreducible Brillouin zone. Note: The first Brillouin zone for one—dimensional periodic structures is a line in the reciprocal space. minimum mesh size proportional to 0.05 times the hole diameter (i.e., the hole diameter in the uniform cross--sectional hole of the Basic--Periodic unit cell (\(d\)) and the smallest diameter (\(d_{2}\)) in the non--uniform cross--sectional hole of the Semi--Periodic, Tapered Diverging, and Tapered Converging unit cells) maintained accuracy during simulation.