## 2 Modeling and Simulation Analysis of Plate Embedded with Two-Dimensional ABH Array
### Analysis of Energy Propagation in Plate Embedded with ABH Array  
According to the simulation data of the nodes within each ABH area (ABH(i), ABH(ii) and ABH(iii)) and Equation (4), the structural intensity \(I_{x}\), \(I_{y}\) and \(I_{z}\) can be calculated. To reflect the power flow propagation characteristics in the x-y plane, a series of power flow transmission graphs can be obtained by using \(I_{x}\) and \(I_{y}\) in MATLAB. In this section, two boundary conditions (BC 1: right and left edges of the plate are fixed, BC 2: all edges are fixed) are considered, as shown in Figure 5. For the plate embedded with the ABH array, as shown in Figure 3b, under two different boundary conditions and at different excitation frequencies (1000 Hz, 1050 Hz, and 1100 Hz), power flow transmission graphs of ABH(i), (ii) and (iii) in the \(x\)-\(y\) plane are shown in Figures 11-13 in Appendix A. The length of arrows in these graphs represents the relative magnitude of power flow density (power flow per unit area) vector, while the orientation of arrows indicates the direction of power flow. Figure 4: Vibration energy comparison among three ABH plate configurations. (**a**) 5–5000 Hz, (**b**) 5–1050 Hz. Figures A1-A3 show that the vibration power flows to the ABH area obviously when passing through. For the ABH plate with both boundary conditions, power flow density vectors near the central plateau of ABH are obviously larger than those in other positions, especially in the uniform part of the plate. The power flow transmission graphs with different boundary conditions are similar at the frequency of 1000 Hz and 1100 Hz. However, at 1050 Hz, the power flow density vectors in locations other than the central plateau with BC 2 is much larger than those of the ABH plate with BC 1, which may be due to the natural frequency. The modal analysis results show that, for the plate with BC 1, its natural frequencies near 1050 Hz are 1029.1 Hz and 1053.3 Hz; however, the natural frequencies of the plate with BC 2 near 1050 Hz are 1046.4 Hz and 1053.9 Hz. That is, for the plate with BC 2, the excitation frequency 1050 Hz is between two close natural frequencies, which leads to overall vibration of the whole plate. However, even in that case, the ABH area still gathers a certain amount of energy. That is, the vibration energy is focused on the central area of the ABH structure and the ABH effect is significant in various excitation frequencies and boundary conditions. According to the structural intensity \(p_{di}\) calculated at nodes within ABH areas by Equation (4), the structural intensity contours of ABH(i), (ii) and (iii) with different boundary conditions and at different excitation frequencies can be drawn, as shown in Figures A4-A6 in Appendix A. Besides, structural intensity contours of ABHs in corresponding positions of the plate with two boundary conditions at 1000 Hz, 1050 Hz and 1100 Hz are shown in Figures A7 and A8 in Appendix A. Figures A4-A6 show that high structural intensity areas appear near the central plateau for every ABH pit, which indicates that vibration energy can indeed be focused near the central areas even if the ABH pit is imperfect. In this way, the damping materials pasted on such high energy density areas can generate an efficient energy dissipation. Additionally, the area of high energy density is similar in size under both boundary conditions. As shown in Figures A7 and A8, although energy focalization is realized in every ABH pit, the amount of energy focused is related to the position of ABH. For the plate model under BC 1, most of the vibration energy is focalized to the surrounding four ABHs except in the case of 1100 Hz. However, for the plate model under BC 2, most of the vibration energy is focalized to the two ABHs on the lower side except in the case of 1000 Hz. That is, under certain excitation force, the energy focalization effect of the plate embedded with the ABH array is related to the excitation frequency, the boundary condition and the ABH position, which should be studied further.