## 2 Modeling and Simulation Analysis of Plate Embedded with Two-Dimensional ABH Array
### Analysis of Energy Propagation in Plate Embedded with ABH Array
The top view of the plate embedded with the two-dimensional ABH array is schematically shown in Figure 6. The distance from the plate center to the reference cross-section is 0.21 m. In order to study the relationship between the vibration energy density and ABH radius, particularly for ABH(i), (ii) and (iii) in Figure 6, nodes at 19 different circular sections with radii from 10 mm to 60 mm are investigated. Besides, the nodes at the circular section with a radius of 11.22 mm (radius of the central platform) are also selected and their average vibration power density expressed as Equation (6) are calculated. Model parameters are set as Sections 2.1 and 2.2 and the plate is under BC 1. \[P_{ad}=\frac{\sum p_{di}}{N_{s}} \tag{6}\]  
where \(P_{ad}\) is the average vibration power density; \(p_{di}\) represents the structural intensity at the \(i\)th node, as shown in Equation (4), and \(N_{s}\) is the total number of nodes at the selected section. Then, the ratios of average vibration power density at different circular sections of an ABH to that at the reference cross-section of the plate are calculated, and by using this ratio, the focalization characteristics of the vibration energy at circular sections with different radii in the ABH can be analyzed. To investigate the vibration energy distribution with ABH radius under excitation frequency 1000 Hz, 1050 Hz and 1100 Hz, and the influence of natural frequencies, average vibration power density ratios of different circular sections of ABH(i), ABH(ii) and ABH(iii) at the excitation frequency of 1000 Hz, 1002 Hz, 1050 Hz, 1072 Hz, 1088 Hz, and 1100 Hz are shown in Figures 7-12 (1002.0 Hz, 1072.1 Hz, and 1088.0 Hz are natural frequencies of the ABH plate). As shown in Figures 7-12, the average vibration power density ratios at different circular sections in the ABHs are much greater than unity. Particularly, the maximum ratio value can reach hundreds when the excitation frequency is near the natural frequency of the plate, which indicates a significant energy focalization effect. Additionally, several other observations can be made. First, a peak value in the radius range of 10 mm to 20 mm exists for each curve, and more peaks can be observed in Figures 11 and 12. That indicates that the vibration energy focalized in the ABH center is not the largest, which may be caused by the central plateau. As vibration response peak values at different positions are related to excitation frequency and nearby mode shapes, at different excitation frequencies, vibration response is various so that the numbers of peaks are different as well. Second, for all excitation frequencies, the energy focalization of ABH(i) and ABH(iii) is more effective than that of ABH(ii). In addition, the ratios are relatively low when the radius of circular section is larger than 30 mm. That is, energy focalization effect is more obvious within a 30 mm radius for every ABH pit designed in this work, which provides a reference for using damping layers more effectively. Figure 6: Top view of the plate with embedded two-dimensional ABH array. Figure 8: Average vibration power density ratios at excitation frequencies of 1002 Hz. Figure 7: Average vibration power density ratios at excitation frequencies of 1000 Hz. Figure 9: Average vibration power density ratios at excitation frequencies of 1050 Hz. most of the energy is focalized within a limited area. Therefore, these factors should be concerned in further research to better design the plate embedded with the ABH array.