# Active Tunable Elastic Metasurface for Abnormal Flexural Wave Transmission
## 3 Numerical Results and Discussion
### Realization of Focus Functionality  
In this subsection, the focusing functionality of flexural waves via the passively modulated and actively modulated metasurface is detailed separately. For a positively incident flexural wave, the phase distribution along the \(x\)-axis is calculated using Equation (17) within different operating frequencies and focusing point locations. The phase distribution is symmetrically distributed about the \(y\)-axis, and thus only 20 adaptive unit cells at one side of the \(y\)-axis need to be calculated. In the FEM simulation, a unit displacement along the \(\text{-}z\) direction is applied to the left side of the metasurface for the excitation of the flexural wave. We designed passively modulated metasurfaces, which are expected to realize two focal points, (100 mm, 0) and (120 mm, 0), with different focal lengths at a functional frequency of 6 kHz by varying the \(p\)  
Figure 5: The out-of-plane displacement diagrams (z-direction) for flexural wave propagation with active modulation by changing \(g\) at \(f=6\) and 7 kHz: the design refractive angles are (**a**) \(20^{\circ}\), (**b**) \(38.7^{\circ}\), (**c**) \(20^{\circ}\), (**d**) \(30^{\circ}\), and (**e**) \(38.7^{\circ}\), respectively. and their design values, as shown in Table A1. Figure 6a,b show the numerically calculated out-of-plane displacement plots (\(z\)-direction) with the designed focal points (100 mm, 0) and (120 mm, 0) marked with black dashed crosses. It can be clearly seen that the flexural wave is incident vertically on the left side, and the transmitted wave on the right side produces a focusing phenomenon, with a clear focusing phenomenon at the position of the design focal point. To further demonstrate the flexural wave focusing effect obtained with our designed metasurface, the normalized displacement amplitude on the straight line \(x=x_{0}\) where the focal point is located was calculated. Figure 7a,b show the normalized displacement amplitude on the line of \(x=x_{0}\) where the design focus is located, in which the blue solid line is the displacement amplitude at \(x=x_{0}\) and the horizontal blue dashed line denotes the displacement amplitude of the incident wave. From Figure 7, it can be seen that the vibration amplitude at the focal point of the design target is several times larger than the displacement amplitude of the incident wave, and the highest point of the amplitude is at \(y=0\) mm. Apart from passive adjustment, an actively metasurface was also designed to achieve the focusing of flexural waves. In this case, different focuses at different operating frequencies were required without changing the geometry of the metasurface. Table A3 presents the design values of \(g\) used in the sub-cells for achieving active wave focusing. From Figure 8a,b, it can be seen that the displacements of transmitted wave fields are concentrated in the positions (80 mm, 0) and (120 mm, 0), which is in accordance with the  
Figure 6: The out-of-plane displacement diagrams (\(z\)-direction) for flexural wave propagation with passive modulation by changing \(p\) at \(f=6\) kHz. The design focal positions were (**a**) (100 mm, 0) and (**b**) (120 mm, 0). Figure 7: The normalized amplitude (\(z\)-component) of the displacement field along the straight line (**a**) \(x_{0}=100\) mm and (**b**) \(x_{0}=120\) mm when performing focusing (the vertical coordinate is the displacement amplitude). design focal points at 6 kHz. Similarly, at another frequency of 7 kHz, three sets of focal points, (80 mm, 0), (100 mm, 0), and (120 mm, 0), were selected, and the verified performance is shown in Figure 8c-e. Figure 9a-e correspond sequentially to the normalized displacement amplitude on the line \(x=x_{0}\), where the focus is located in Figure 8a-e. As shown in Figure 9, the displacement amplitude at the focusing point far exceeds that of the surrounding region. Figure 8: The out-of-plane displacement diagrams (z-direction) for flexural wave propagation with active modulation by changing \(g\) at \(f=6\) and 7 kHz.