# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 3 Numerical Results and Discussion  
### The Validation of Transmittance and Phase Shift  
The effects of control gain (\(g\)) and \(p\)-values on the phase shift and transmission coefficient were investigated separately for an operating frequency of 6 kHz. The two-dimensional basic cell shown in Figure 2 was constructed in COMSOL (plane strain module) to calculate the transmission coefficient and phase shift of the adaptive unit cell, and the structural parameters are shown in Table 1. The substrate of the adaptive unit cell is made of aluminum materials whose surfaces are covered with piezoelectric materials. The density, elastic modulus, and Poisson's ratio of aluminum materials are \(\rho_{\text{s}}\) = 2700 kg/m\({}^{3}\), \(E_{\text{p}}\) = 70 GPa, and \(\upsilon\) = 0.33. A piezoelectric material with mass density \(\rho_{\text{p}}\) = 7700 kg/m\({}^{3}\), elasticity constant c\({}_{11}\) = 70.6 Gpa, piezoelectric constant \(d_{31}\) = \(-\)12.6374 C/m\({}^{2}\), and dielectric constant \(\varepsilon_{33}\) = 1.59 \(\times\) 10\({}^{-8}\) F/m was adopted in the present work [50]. A unit stress is applied in the \(-\) z direction to the leftmost interface of region I. The grid size is one 140 th of the wavelength. The choice of meshing size has been validated.  
The effects of \(g\) on the wave velocity of sub-cell 1 and sub-cell 2 were investigated according to Equation (12) and are shown in Figure 3a. It can be observed from Figure 3a that the velocity of flexural waves rises as \(g\) increases. Therefore, the value of \(g\) was chosen to be 100 for sub-cell 1 and 0 for sub-cell 2, respectively. The phase shift and transmittance of adaptive unit cells when \(p\) varied from 0 to 1 were also examined with this prerequisite. The results obtained by the TMM were verified by comparison with those derived from the finite element method (FEM). To perform the FEM calculations, a location was selected for the phase and transmittance coefficient calculation in region IV, which was 240 mm away from the line connecting regions III and IV. It is worth noting that the absolute value of the phase can be altered if another point at a different location in the transmittance region is selected. However, the difference between the phase shift for the adaptive unit cell is always the same.  
Figure 3b shows the transmission coefficient and phase shift versus the value of \(g\) when \(p\) = 0.5, and Figure 3c shows the phase shift and transmittance with respect to varying \(p\) as \(g\) was chosen as 100 for sub-cell 1 and 0 for sub-cell 2. In Figure 3b,c, the red solid line indicates the transmission coefficient resolved by the TMM, and the cyan dashed line indicates the numerical solution to the FEM. The black solid line and the dark yellow dashed line indicate the phase shifts obtained from the TMM and FEM calculations, respectively. As can be seen in Figure 3b, there is a very small error in the transmission coefficients calculated using the TMM and FEM; in fact, the maximum difference in values is within 0.05, while the phase gradient variation is achieved by adjusting the phase. We think that this produces a small effect on the tectonic metasurface. As can be seen in Figure 3b,c, the adaptive unit cell fails to cover a phase shift ranging from 0 to 2\(\pi\) when changing the control gain, but it can achieve this range with the change in \(p\). According to the GSL, we used the adaptive unit cells to compose the column elastic metasurface by varying the parameters of the adaptive cells in order to effectively change the phase of each adaptive unit cell. We exploited this feature to construct a passive modulation metasurface by varying the adaptive unit cells of \(p\) and further varying \(g\) for active modulation based on it.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \(h_{0}\) & \(h_{1}\) & \(h_{2}\) & \(h_{s1}\) & \(h_{s2}\) & \(h_{p}\) & \(L\) & \(w_{1}\) \\ \hline
3 mm & 3 mm & 1 mm & 2.6 mm & 0.6 mm & 0.2 mm & 64 mm & 5 mm \\ \hline \end{tabular}
\end{table}
Table 1: Geometry of the adaptive unit cell.