## 3 Results and Discussion  
### Band Structure Analysis  
The band structure is shown in Figure 3. One can see that the proposed piezoelectric phononic crystal produces multiple broadband complete bandgaps. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Material Parameters**} & \multicolumn{2}{c}{**Piezoelectric Material**} & \multicolumn{2}{c}{**Elastic Materials**} \\ \cline{2-5}  & **PZT-5H** & **Iron** & **Epoxy** \\ \hline \(\rho\) (kg/m\({}^{3}\)) & 7500 & 7870 & 1200 \\ \(E_{m}\) (Pa) & - & 200 \(\times\) 10\({}^{9}\) & 2.8 \(\times\) 10\({}^{9}\) \\ \(v\) & - & 0.29 & 0.39 \\ C\({}_{11}\) (GPa) & 127.205 & - & - \\ C\({}_{12}\) (GPa) & 80.212 & - & - \\ C\({}_{13}\) (GPa) & 84.670 & - & - \\ C\({}_{22}\) (GPa) & 127.205 & - & - \\ C\({}_{23}\) (GPa) & 84.670 & - & - \\ C\({}_{33}\) (GPa) & 117.436 & - & - \\ C\({}_{44}\) (GPa) & 22.989 & - & - \\ C\({}_{55}\) (GPa) & 22.989 & - & - \\ C\({}_{66}\) (GPa) & 23.474 & - & - \\ \(e_{15}\) (C/m\({}^{2}\)) & 17.035 & - & - \\ \(e_{24}\) (C/m\({}^{2}\)) & 17.035 & - & - \\ \(e_{31}\) (C/m\({}^{2}\)) & \(-\)6.623 & - & - \\ \(e_{32}\) (C/m\({}^{2}\)) & \(-\)6.623 & - & - \\ \(e_{33}\) (C/m\({}^{2}\)) & 23.240 & - & - \\ \(e_{11}\) (F/m) & 1.509 \(\times\) 10\({}^{-8}\) & - & - \\ \(e_{22}\) (F/m) & 1.509 \(\times\) 10\({}^{-8}\) & - & - \\ \(e_{33}\) (F/m) & 1.269 \(\times\) 10\({}^{-8}\) & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material constants of the elastic and piezoelectric components. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{1}{c}{**Parameters**} & \multicolumn{1}{c}{**Descriptions**} & \multicolumn{1}{c}{**Value**} \\ \hline \(b_{1}\) (mm) & The width of the convex structures & 20 \\ \(b_{2}\) (mm) & The length of the convex structures & 40 \\ \(x_{1}\) (mm) & The side length of the square hollow & 30 \\ \(x_{2}\) (mm) & The side length of the internal hard material & 56 \\ \(x_{3}\) (mm) & The side length of the piezoelectric material & 80 \\ \(x_{4}\) (mm) & The side length of the outer hard material & 100 \\ \(h\) (mm) & The thickness of the phononic crystal & 16.67 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Geometric characteristics of the established piezoelectric phononic crystal. The first bandgap is between the 6 th band and the 7 th band, whose range starts from 2268.89 Hz to 3356.36 Hz, and the width is 1087.47 Hz. Subsequently, the second bandgap with a width of 1247.74 Hz is created between the 7 th and 8 th bands, and its frequency range varies from 3547.67 Hz to 4795.41 Hz. The widths of the third bandgap and the fourth bandgap are 690.33 Hz and 532.34 Hz, and these bandgaps are located between the 8 th and 9 th bands as well as between the 9 th and 10 th bands. Among the four bandgaps, the second bandgap is the widest, with a width of 1247.74 Hz, while the fourth bandgap is the narrowest, with a width of 532.34 Hz, which is only half the width of the second bandgap. To understand the formation mechanism of the bandgaps, the vibration modes of the upper and lower boundaries of the bandgaps are explored. In Figure 3, the upper and lower boundaries of the bandgaps are marked with the letters A-H, and their frequencies are 2268.89 Hz, 3356.36 Hz, 3547.67 Hz, 4795.41 Hz, 6204.55 Hz, 6894.88 Hz, 7345.23 Hz, and 7877.57 Hz, respectively. The corresponding modes are displayed in Figures 4 and 5. Figure 4: Vibration modes of the first and second bandgap edges: (**a**) Point A; (**b**) Point B; (**c**) Point C; (**d**) Point D.  
Figure 3: Band structure of the proposed piezoelectric phononic crystal. The vibrations of the upper and lower boundaries of the first bandgap are mainly concentrated in the four corners of the cube. To be specific, for the lower boundary, the vibration is confined to the \(x\)-\(y\) plane, while the upper boundary vibrates up and down along the \(z\)-axis. The adjacent corners move in anti-phase along the \(z\)-axis, whereas the opposite corners vibrate in phase along the \(z\)-axis.