## 2 Materials and Methods
### Analytical Approach
#### 2.2.2 Band Structure Solution
By assembling the matrix using the number of each unit, the finite element equation of the piezoelectric phononic crystal can be obtained:  
\[\mathbf{M}_{uu}\ddot{\mathbf{u}}+\mathbf{K}_{uu}\mathbf{u}+\mathbf{K}_{u\varphi }\mathbf{\varphi}=0, \tag{18}\]  
\[U^{\varepsilon}=\int\limits_{V_{1}}\frac{1}{2}S^{\mathrm{T}}TdV+\int\limits_{ V_{2}}\frac{1}{2}S^{\mathrm{T}}TdV-\int\limits_{V_{2}}\frac{1}{2}EDdV. \tag{10}\]0  
where \(\mathbf{u}\) is the nodal displacement vector of the finite element discrete system and \(\varphi\) represents the nodal potential vector. \(\mathbf{M}_{uu}\) and \(\mathbf{K}_{uu}\) are the mass matrix and the elastic stiffness matrix of the finite discrete system. \(\mathbf{K}_{u\varphi}\) and \(\mathbf{K}_{qu}\) are the elastic coupling stiffness matrixes of the piezoelectric material as well as \(\mathbf{K}_{\varphi\varphi}\) is the electrical stiffness matrix of the piezoelectric material. For the convenience of description, the proposed piezoelectric phononic crystal unit cell is simplified as a square lattice, as shown in \[U^{\varepsilon}=\int\limits_{V_{1}}\frac{1}{2}S^{\mathrm{T}}TdV+\int\limits_{ V_{2}}\frac{1}{2}S^{\mathrm{T}}TdV-\int\limits_{V_{2}}\frac{1}{2}EDdV. \tag{10}\]1  
The Bloch Floquet conditions are applied on the boundary of the square lattice, which are \(l_{1}\), \(l_{2}\), \(l_{3}\), and \(l_{4}\). In addition, \(c_{1}\)-\(c_{4}\) represent the four vertices of the square lattice. Thus, Equations (18) and (19) could be rewritten as:  
\[U^{\varepsilon}=\int\limits_{V_{1}}\frac{1}{2}S^{\mathrm{T}}TdV+\int\limits_{ V_{2}}\frac{1}{2}S^{\mathrm{T}}TdV-\int\limits_{V_{2}}\frac{1}{2}EDdV. \tag{10}\]2  
\[U^{\varepsilon}=\int\limits_{V_{1}}\frac{1}{2}S^{\mathrm{T}}TdV+\int\limits_{ V_{2}}\frac{1}{2}S^{\mathrm{T}}TdV-\int\limits_{V_{2}}\frac{1}{2}EDdV. \tag{10}\]3  
where  
\[ \begin{array}{c}\mathbf{T}_{u}^{\mathrm{H}}\mathbf{M}_{uu}\mathbf{T}_{u}= \overline{\mathbf{M}}_{uu}\\ \mathbf{T}_{u}^{\mathrm{H}}\mathbf{K}_{uu}\mathbf{T}_{u}=\overline{\mathbf{K} }_{uu}\\ \mathbf{T}_{u}^{\mathrm{H}}\mathbf{K}_{u\varphi}\mathbf{T}_{\varphi}=\overline{ \mathbf{K}}_{u\varphi}.\\ \mathbf{T}_{u}^{\mathrm{H}}\mathbf{K}_{\varphi u}\mathbf{T}_{u}=\overline{ \mathbf{K}}_{\varphi u}\\ \mathbf{T}_{\varphi}^{\mathrm{H}}\mathbf{K}_{\varphi\varphi}\mathbf{T}_{\varphi }=\overline{\mathbf{K}}_{\varphi\varphi}\end{array}  \tag{22}\]  
In Equation (22), \(\mathbf{T}_{u}^{\mathrm{H}},\mathbf{T}_{u}\), \(\mathbf{T}_{\varphi}^{\mathrm{H}}\), and \(\mathbf{T}_{\varphi}\) represent Bloch Floque periodicity. By substituting Equation (22) to Equations (20) and (21), so Equations (20) and (21) can be reformulated:  
\[\overline{\mathbf{M}}_{uu}\ddot{\overline{\mathbf{u}}}+\overline{\mathbf{K}}_{ uu}\overline{\mathbf{u}}+\overline{\mathbf{K}}_{u\varphi}\overline{\mathbf{ \varphi}}=0, \tag{23}\]  
\[\overline{\mathbf{K}}_{qu}\overline{\mathbf{u}}+\overline{\mathbf{K}}_{ \varphi\varphi}\overline{\mathbf{\varphi}}=0, \tag{24}\]  
thus, Equation (24) can be expressed in the following form:  
\[\overline{\mathbf{\varphi}}=-\overline{\mathbf{K}}_{\varphi\varphi}^{-1} \overline{\mathbf{K}}_{pu}\overline{\mathbf{u}}. \tag{25}\]  
Figure 2: The classifications of nodes when using Bloch Floquet conditions. Combining Equation (23) and Equation (25) yields:  
\[\overline{\mathbf{M}}_{uu}\overline{\mathbf{u}}+\mathbf{\Theta}\overline{\mathbf{ u}}=0, \tag{26}\]  
where  
\[\mathbf{\Theta}=(\overline{\mathbf{K}}_{uu}-\overline{\mathbf{K}}_{u\varphi} \overline{\mathbf{K}}_{\varphi\varphi}^{-1}\overline{\mathbf{K}}_{\varphi u}). \tag{27}\]  
Equation (26) represents a linear eigenvalue problem. By solving the equation, the real band structure of the proposed piezoelectric phononic crystal can be drawn.