## 2 Finite Element Modeling of Interconnection Metamaterial
### Stopband Analysis of Interconnected Metamaterials  
In this subsection, we discuss the stopband analysis of interconnected metamaterials. Figure 1 shows a conceptual diagram of the interconnected metamaterial. Figure 1a shows the size of the interconnected metamaterial. The host structure is an aluminum plate with 1T thickness, and the resonator part is PLA plastic. The mass of the resonator is determined by \(d_{1}\) and \(h_{1}\), while the stiffness of the resonator is determined by \(d_{2}\) and \(h_{2}\). The parameter \(w_{beam}\) affects the interaction of each resonator. As \(d_{1}\) and \(h_{1}\) increase, the mass of the resonator increases, and the frequency width of the stopband can be changed. The stiffness of the metamaterial depends on the radius and height of the neck beam. By changing of \(d_{2}\) and \(h_{2}\), the stiffness of resonator can be changed. The geometrical parameters used in this work are shown in Table 1.  
As shown in Figure 1b, the interconnected part of the resonator (\(\partial\Omega_{\text{res,shell}}\)) and host structure (\(\partial\Omega_{\text{host,shell}}\)) are composed of shell elements, and the remaining resonator (\(\Omega_{\text{res,solid}}\)) domain comprises solid elements. At the solid-shell coupling boundary (\(\Omega_{\text{solid-shell}}\)), the solid and shell elements share the same displacement. \[\mathbf{u}_{\text{shell}}-\mathbf{u}_{\text{solid}}=\mathbf{0}\text{ at }\partial\Omega_{\text{solid-shell}}. \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value (mm)** \\ \hline \(d_{1}\) & 25 \\ \(d_{2}\) & 3 \\ \(h_{1}\) & 15 \\ \(h_{2}\) & 15 \\ \(w_{beam}\) & 2.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Geometrical parameters of the interconnected metamaterial. Figure 1: Conceptual diagram of an interconnected metamaterial; (a) dimension and material of host structure (yellow) and resonator (violet); (b) layout of a unit-cell structure and element type for each component. Table 2 shows the mechanical properties of aluminum and PLA plastic that make up the host structure and resonator. Figure 2 describes the Floquet-Bloch boundary condition for stopband analysis. Figure 2a shows a schematic diagram explaining the Floquet-Bloch boundary condition. The multi-celled structure is a periodic structure in which the unit-cell structure is infinitely repeated along with the \(a_{1}\) and \(a_{2}\) directions. The unit-cell structure has a length of L in the x- and y-directions, and each unit structure comprises a host structure and a resonator. Each resonator is linked with adjacent resonators. Because of the spatial periodicity of the multi-celled structure, the unit-cell structure follows the Bloch theorem. The Floquet-Bloch boundary condition can be expressed as follows:  
\[\mathbf{u}(\mathbf{r}_{A}) =\mathbf{u}(\mathbf{r}_{B}+\mathbf{a}) \tag{2}\] \[=\mathbf{u}(\mathbf{r}_{B}+\mathbf{a}_{1}+\mathbf{a}_{2})\] \[=\mathbf{u}(\mathbf{r}_{B})e^{j(k_{x}\mathbf{a}_{1}+k_{y}\mathbf{a}_{2})}\]  
where \(\mathbf{u}(\mathbf{r}_{A})\) and \(\mathbf{u}(\mathbf{r}_{B})\) are the displacement responses at points A and B, respectively. The function \(e^{j(k_{x}\mathbf{a}_{1}+k_{y}\mathbf{a}_{2})}\) represents the spatial periodicity, expressed by the wave vector \(\mathbf{k}\). The Floquet-Bloch boundary condition was applied to consider the spatial periodicity of this metamaterial. According to Equation (2), the three left/top boundaries (\(\Gamma_{\text{T,host}}\), \(\Gamma_{\text{RT,host}}\), \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Resonator** & **Host Structure** \\ \hline Type & PLA plastic & Aluminum \\ Elastic modulus (GPA) & 4.16 & 71.89 \\ Poisson’s ratio (1) & 0.403 & 0.284 \\ Density (kg/m\({}^{3}\)) & 1400 & 2669 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Mechanical properties of the resonator and host structure. Figure 2: Schematic diagram describing (**a**) the Floquet–Bloch boundary condition of the unit-cell structure and each boundary of (**b**) the host structure and (**c**) resonator.