# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 3 Numerical Results and Discussion
### Results of Abnormal Transmittance  
Based on the generalized Snell's law, different abnormal refraction strategies can be conducted by adjusting the adaptive unit cells of the metasurface. Additionally, it can be seen from Section 3.1 that the control gain (\(g\)) and \(p\)-values have an effect on the propagation of the flexural wave. Finite element simulation of flexural wave propagation was conducted using COMSOL Multiphysics software 6.0. Three-dimensional elements are adopted to discretize the incident field, the metasurface, and the transmitted field, whose size accounts for one 180 th of the wavelength. The choice of meshing size has been validated. In order to achieve the abnormal refraction of flexural waves, 40 adaptive unit cells were utilized to form the metasurface, with a slit width \(w_{2}\) of 1.98 mm between adaptive unit cells. Perfectly matched layers (PMLs) were arranged around the boundaries of the whole plate to eliminate the influence of reflected waves from boundaries (PML acts as a near-ideal wave absorber). A z-direction displacement \(-1\times\exp(-((y-0)/\cos 0)^{2}/60^{2})\) (mm) was applied at the boundary between the left PML and the incident plate to excite the flexural-wave Gaussian beam, and the wave in the plane surrounded by the black dashed line on the left side in Figure 4 represents the Gaussian beam. The wavelengths of flexural waves corresponding to \(f=6\) kHz and \(f=7\) kHz are 69.94 mm and 64.75 mm, respectively, within the 3 mm thick aluminum plate, satisfying the sub-wavelength requirement. Firstly, the abnormal transmission of passively modulated metasurfaces was realized by changing \(p\). Two different abnormal refraction angles, \(\theta_{t}=\)30\({}^{\circ}\) and 38.7\({}^{\circ}\), with operating frequency \(f=6\) kHz were designed as the targets. The \(g\) values for sub-cells 1 and 2 were assumed as 100 and 0, respectively, and the designed values of \(p\) were set to match the  
Figure 4: The out-of-plane displacement diagrams (z-direction) for flexural wave propagation with passive modulation by changing \(p\) at \(f=6\) kHz: the design refraction angles are (**a**) 30\({}^{\circ}\) and (**b**) 38.7\({}^{\circ}\). Figure 3: (**a**) The phase velocity curve varying with \(g\) for \(f=6\) kHz. (**b**) The transmission coefficient and phase shift as \(g\) increases while \(p=0.5\). (**c**) The transmission coefficient and phase shift with respect to varying \(p\) at 6 kHz when \(g\) is 100 in sub-cell 1 and 0 in sub-cell 2.  
requirement of phase shift ranging from 0 to 2\(\pi\). The \(p\)-values of the \(R_{\text{t}}\)th adaptive cell units (\(R_{i}=1-40\)) for different refraction angles in the case of \(f=6\) kHz are listed in Table 11 (\(\theta_{\text{t}}=30^{\circ}\), \(38.7^{\circ}\)). Full-wave simulations were performed, as shown in Figure 4a,b, from which it can be seen that the vertically incident Gaussian beam can be deflected on demand via the metasurface, realizing the abnormal transmission of the flexural waves. Obviously, the propagation of the transmitted wave matches very well with the theoretical design route. Then, regarding the active modulation of flexural waves, the geometrical parameters of the unit cells were fixed, and abnormal transmission was achieved by varying the \(g\) from the negative proportional feedback control strategy applied in the sub-cells. The target angles for transmitted waves are assumed to be \(20^{\circ}\) and \(38.7^{\circ}\) when the incident frequency is 6 kHz, and at 7 kHz, these angles are assumed to be \(20^{\circ}\), \(30^{\circ}\), and \(38.7^{\circ}\), respectively. Additionally, the used values of \(g\) for the active modulation are provided in Table 12, and the active tunability performance is shown in Figure 5. The simulated angles and target angles basically coincide with each other, which proves the effectiveness and reliability of the active metasurface based on the piezoelectrical effects and negative proportional feedback control technique.