# Radial Gradient Seismic Metamaterials with Ultra-Low Frequency and Ultra-Wide Band Gap
## 4 Radial Gradient Seismic Metamaterial Structure after Optimization of Supercell
### The Attenuation Effect of OS-RGSM Structure on Surface Wave  
In order to further study and optimize the attenuation effect of OS-RGSM on seismic surface waves, finite element simulation calculations were performed using COMSOL Multiphysics 5.5. In theory, the infinite periodic unit cell structure has ideal filtering characteristics. However, in practical engineering applications, the number of unit cells is  
Figure 11: Displacement map of seismic Lamb wave attenuation of OS-RGSM structure. (**a**) The stress field diagram of the OS-RGSM on the lamb wave barrier at 5.7 Hz. (**b**) The stress field diagram of the OS-RGSM on the lamb wave barrier at 1.3 Hz. always limited. In order to verify the effectiveness of OS-RGSM to suppress seismic surface waves, the dynamic response of the three-dimensional finite-period OS-RGSM system is numerically simulated. The three-dimensional finite-period system with three-period radial continuation is shown in Figure 12. On line A, a source that vibrates along the x direction on the free surface is used to introduce surface waves, and a perfect matching layer is set on the bottom and both sides of the three-dimensional model structure to reduce reflection. The surface perpendicular to the x-direction is set as a periodic boundary condition so that the results are closer to reality. The main view of the three-dimensional period designed in this paper is shown in Figure 12a, and the top view is shown in Figure 12b. In Figure 12a,b, the soil base layer is represented by gray shadow, and the perfect matching layer is represented by purple. The distance between the source of the OS-RGSM and the right side of the optimized supercell barrier is set to 38.5a, the depth of the soil base along the z direction is set to 23.5a, the length of the soil base along the x direction is set to 62.5a, and the width along the y direction is set to 25a. In order to effectively prevent the influence of factors such as wave reflection on the results, the thickness of the perfect matching layer is set to 4a. The calculated attenuation effect of the finite three-dimensional periodic composite system on the seismic surface wave can be observed from Figure 13a. In the ultra-low-frequency range of 0-9 Hz, the system optimized by the OS-RGSM barrier has a significant attenuation effect of more than 50% in the range of 3.5 Hz to 9 Hz compared with the barrier-free system. In order to further study the effect of optimizing the metamaterial barrier on the attenuation of seismic surface waves, the displacement diagrams at two frequencies in the band gap are calculated, as shown in Figure 13b,c. From the diagram, it can be observed that when the seismic surface wave reaches the OS-RGSM barrier, most of the surface wave is absorbed by the structure acting as a vibrator in the barrier, and a small  
Figure 12: The attenuation effect of OS-RGSM on seismic surface waves. (**a**) The side view of OS-RGSM barrier attenuation of seismic surface waves; (**b**) top view of OS-RGSM barrier attenuation seismic surface wave. amount is converted into a body wave, which dissipates in the soil base and plays a good damping effect. Furthermore, this paper studies the acceleration-time response of the OS-RGSM barrier under the action of an EI-Centro seismic wave. Figure 14a is the acceleration time curve of the EI-Centro seismic wave, which is converted into the acceleration curve after Fourier change, as shown in Figure 14b; in the three-dimensional model simulating OS-RGSM attenuation seismic wave, the perfect matching layer is applied on the side and bottom of the soil to eliminate the interference of the external environment on the structure and ensure the accuracy of the results. In the designed model, the grid is divided into regions, and the transient calculation time is not long set to \(1/60\)f\({}_{\text{max}}\). The calculation results show that the OS-RGSM barrier has a good attenuation effect on seismic waves.