## 2 Finite Element Modeling of Interconnection Metamaterial
### Stopband Analysis of Interconnected Metamaterials
\(\Gamma_{\text{R,host}}\)) of the host structure in Figure 2b are represented by five remaining boundaries (\(\Gamma_{\text{LT,host}}\), \(\Gamma_{\text{L,host}}\), \(\Gamma_{\text{LB,host}}\), \(\Gamma_{\text{B,host}}\), \(\Gamma_{\text{RB,host}}\)), and the two left/bottom boundaries (\(\Gamma_{\text{L,res}}\), \(\Gamma_{\text{B,res}}\)) of the resonator are represented by two right/top boundaries (\(\Gamma_{\text{R,res}}\), \(\Gamma_{\text{T,res}}\)). According to Equation (2), each boundary condition is expressed as follows [9; 10; 11]:  
\[ \begin{array}{l}\mathbf{u}_{\text{R,host}}=\mathbf{u}_{\text{L,host}}e^{ \mathbf{i}_{\text{R}}L}\\ \mathbf{u}_{\text{T,host}}=\mathbf{u}_{\text{B,host}}e^{\mathbf{i}_{\text{R}} L}\\ \mathbf{u}_{\text{RB,host}}=\mathbf{u}_{\text{LB,host}}e^{\mathbf{i}_{\text{R}} L}\\ \mathbf{u}_{\text{LT,host}}=\mathbf{u}_{\text{LB,host}}e^{\mathbf{i}_{\text{R}} L}\\ \mathbf{u}_{\text{RT,host}}=\mathbf{u}_{\text{LB,host}}e^{\mathbf{i}(k_{x}L+k_{y}L)} \\ \mathbf{u}_{\text{R,res}}=\mathbf{u}_{\text{L,res}}e^{\mathbf{i}_{\text{R}} L}\\ \mathbf{u}_{\text{T,res}}=\mathbf{u}_{\text{B,res}}e^{\mathbf{i}_{\text{R}} L}\end{array}  \tag{3}\]  
where \(\mathbf{u}_{\text{host}}\) and \(\mathbf{u}_{\text{res}}\) are the displacements of the host structure and resonator, respectively. The constant \(L\) is the length of the unit-cell structure. Equation (3) can be expressed in matrix form as follows:  
\[\text{where }\mathbf{I}(\mathbf{k})=\left[ \begin{array}{ccccc}-e^{ \mathbf{i}_{\text{R}}L}&1&0&0&\\ 0&0&-e^{\mathbf{i}_{\text{R}}L}&1&\mathbf{0}_{(2\times 4)}&\mathbf{0}_{(2 \times 4)}\\ \mathbf{0}_{(2\times 4)}&&-e^{\mathbf{i}_{\text{R}}L}&1&0&0\\ \mathbf{0}_{(3\times 4)}&&\mathbf{0}_{(3\times 4)}&&-e^{\mathbf{i}_{\text{R}}L}&1& 0&0\\ \mathbf{0}_{(3\times 4)}&&\mathbf{0}_{(3\times 4)}&&-e^{\mathbf{i}_{\text{R}}L}&0&1&0\\ &&-e^{\mathbf{i}_{\text{R}}L}&0&0&1\end{array} \right] \tag{4}\]  
where \(\mathbf{\tilde{u}}\) is the displacement vector comprising the displacement responses of the host structure and resonator. According to the Lagrange multiplier method, the eigenvalue problem for the augmented system can be expressed as follows:  
\[ \begin{array}{c}\left[\mathbf{K}_{\text{aug}}-\omega^{2}\mathbf{M}_{\text {aug}}\right]\mathbf{u}_{\text{aug}}=\mathbf{0}\\ \text{where }\mathbf{K}_{\text{aug}}=\left[\begin{array}{cc}\mathbf{K}& \mathbf{I}(\mathbf{k})^{T}\\ \mathbf{I}(\mathbf{k})&\mathbf{0}\end{array}\right]\text{, }\mathbf{M}_{\text{aug}}=\left[\begin{array}{cc}\mathbf{M}& \mathbf{0}\\ \mathbf{0}&\mathbf{0}\end{array}\right]\text{, }\mathbf{K}_{\text{aug}}=\left[\begin{array}{c} \mathbf{u}\\ \lambda\end{array}\right]\end{array}  \tag{5}\]  
where \(\mathbf{K}_{\text{aug}}\) and \(\mathbf{M}_{\text{aug}}\) are the augmented stiffness and mass matrices, respectively. The variable \(\omega\) and vector \(\mathbf{u}\) are the natural frequency and displacement of the wave vector \(\mathbf{k}\big{(}k_{x},k_{y}\big{)}\), respectively. The vector \(\mathbf{u}_{\text{aug}}\) is an augmented displacement vector comprising the displacement vector \(\mathbf{u}\) and the Lagrange multiplier \(\lambda\). Substituting \(\mathbf{I}(\mathbf{k})\) in Equations (4) and (5), the stopband of the interconnected metamaterial can be calculated by using the iterative eigenvalue problem for each wave vector. The iterative solution obtained from Equation (5) indicates the modal frequencies for the variation in the wave vector. The modal frequency is the Bloch mode for propagating the vibration. In contrast, flexural waves are not propagated in the stopband, which is a frequency band in which the vibrating mode does not exist for all wave vectors. Considering the spatial periodicity of the interconnecting metamaterial, Equation (4) only calculates the wave vector domain for the unit structure, not the entire periodic structure. According to the symmetry of the unit-cell structure, the wave vector domain of the unit-cell structure can be reduced to the irreducible Brillouin zone (IBZ) defined in Figure 3 [2].