## 2 Finite Element Modeling of Interconnection Metamaterial
### Stopband Analysis of Interconnected Metamaterials
In this study, we considered the unit-cell structure in the form of a square of length L. Thus, the IBZ has the shape of a triangle made up of three points, \(\Gamma(0,0)\), \(X\big{(}\frac{\pi}{L},0\big{)}\), and \(\text{M}\big{(}\frac{\pi}{L},\frac{\pi}{L}\big{)}\). The dispersion diagram of the interconnected metamaterial represents the set of iterative solutions for the eigenvalue problem in Equation (5) with the wave-vector domain. Figure 4 shows the dispersion diagram and Bloch modes calculated from stopband analysis for the unit structure of the interconnected metamaterial. The Bloch mode \(\omega\) of the interconnected metamaterial is divided into the host structure dominant Bloch mode, \(\omega_{\text{host}}\), which contributes to the transmission of the bending vibration of the host structure, and the resonator dominant Bloch mode, \(\omega_{\text{res}}\), which does not directly contribute to the transmission of bending vibration. Therefore, the Bloch mode for the stopband which reduces the flexural vibration is determined by the host structure dominant Bloch mode, which is determined by the modal strain energy, as follows:  
\[ \begin{array}{c}\omega=\omega_{\text{host}}\;(E_{\text{res}}\leq E_{\text{ host}})\\ \omega=\omega_{\text{res}}\;(E_{\text{res}}>E_{\text{host}})\end{array}  \tag{6}\]  
where \(E_{\text{host}}\) and \(E_{\text{res}}\) are the modal strain energies calculated for the host structure and resonator, respectively. The modal strain energies calculated in the host structure and resonator can be expressed as follows [17]. \[ \begin{array}{c}E_{\text{host}}=\iint_{\partial\Omega_{\text{host,shell}}} \frac{1}{2}\textbf{\Psi}_{j}^{\text{T}}\textbf{K}\textbf{\Psi}_{j}\text{d} \partial\Omega\\ E_{\text{res}}=\iint_{\partial\Omega_{\text{res,shell}}}\frac{1}{2}\textbf{ \Psi}_{j}^{\text{T}}\textbf{K}\textbf{\Psi}_{j}\text{d}\partial\Omega+\iint_{ \Omega_{\text{res,solid}}}\frac{1}{2}\textbf{\Psi}_{j}^{\text{T}}\textbf{K} \textbf{\Psi}_{j}\text{d}\Omega\end{array}  \tag{7}\]  
where \(\textbf{\Psi}_{j}\) is the displacement vector in the jth Bloch mode. As mentioned in the description of Figure 1, \(\partial\Omega_{\text{host,shell}}\), \(\partial\Omega_{\text{res,shell}}\) and \(\Omega_{\text{res,solid}}\) are the shell boundaries of the host structure, interconnecting shell, and the solid domain of the resonator, respectively. As shown in Figure 4b,c, the modal strain energy of the host structure (\(\partial\Omega_{\text{host,shell}}\)) is larger than that of the resonator because the deformation of the host structure is dominant. Thus, the Bloch modes in Figure 4b,c are the host-structure-dominant modes. The Bloch modes in  
Figure 3: The irreducible Brillouin zone (IBZ) in the wave vector domain of a unit-cell structure. Figure 4d,e are the resonator-dominant Bloch modes because the modal strain energy of the resonator (\(\partial\Omega_{\text{res,shell}}+\Omega_{\text{res,solid}}\)) is relatively larger than that of the host structure. Details on extracting the host-structure-dominant Bloch mode are described in the Appendix A. The stopband for attenuating the flexural vibration can be determined by using the host-structure-dominant Bloch mode. As shown in Figure 4a, the interconnected metamaterial proposed in this study has a stopband from 763.48 to 1613.9 Hz. In this frequency range, the flexural vibration of the host structure with multi-celled metamaterials cannot be propagated because of the effect of the local resonance of each resonator.