## 2 Modeling and Simulation Analysis of Plate Embedded with Two-Dimensional ABH Array
### Energy Realization Analysis of Different ABH Configurations  
The power flow method is very useful for the investigation of energy propagation in a structure [17,18], and it can be used to describe the energy distribution and propagation. The energy focalization in the ABH areas can be analyzed by using the structural power flow method [15]. Structure power flow represents the energy dissipation ability of the structure per unit time. The instantaneous power flow is defined as the product of the force and the velocity in the same direction and phase, and for FE simulation, the stress state at any point in the elastic body can be represented by six stress components, namely, normal stress \(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\), and shear stress \(\tau_{xy}\), \(\tau_{yz}\), \(\tau_{zx}\). Structural intensity defined as power flow per unit area can be expressed as [19]:  
\[I_{n}=-\frac{1}{2}\text{Re}(\sigma_{n}v_{n}^{*}+\tau_{n1}v_{1}^{*}+\tau_{n2}v_ {2}^{*}) \tag{3}\]  
where \(\sigma_{n}\) represents normal stress in the normal direction; \(\tau_{n1}\) and \(\tau_{n2}\) are shear stress in the direction of 1 and 2, \(v_{n}^{*}\), \(v_{1}^{*}\) and \(v_{2}^{*}\) indicate complex conjugate of velocities in the normal \(n\), 1 and 2 direction, respectively. By FE simulation of the plate embedded with ABHs (Figure 3) excited by a harmonic force, the nodal velocity vector \(\{v\}=\left\{v_{x},v_{y},v_{z}\right\}\) and the nodal displacement vector \(\{u\}=\left\{u_{x},u_{y},u_{z}\right\}\) (where \(\{u\}\) is \(\{v\}/i\omega\)) can be calculated. Then, for a single solid element, the structural intensity of direction \(x\), \(y\) and \(z\) can be expressed as:  
\[ \begin{array}{l}I_{x}=-\frac{\omega}{2}\text{Im}(\sigma_{x}u_{x}{{}^{*}}+ \tau_{xy}u_{y}{{}^{*}}+\tau_{xz}u_{z}{{}^{*}})\\ I_{y}=-\frac{\omega}{2}\text{Im}(\sigma_{y}u_{y}{{}^{*}}+\tau_{yx}u_{x}{{}^{*} }+\tau_{yz}u_{z}{{}^{*}})\\ I_{z}=-\frac{\omega}{2}\text{Im}(\sigma_{z}u_{z}{{}^{*}}+\tau_{zx}u_{x}{{}^{*} }+\tau_{zy}u_{y}{{}^{*}})\\ p_{di}=\sqrt{I_{x}^{2}+I_{y}^{2}+I_{z}^{2}}\end{array}  \tag{4}\]  
where \(I_{x}\), \(I_{y}\) and \(I_{z}\) indicate the structural intensity in the \(x\), \(y\) and \(z\) directions, respectively; \(p_{di}\) is the modulus of structural intensity at a certain node; \(\omega\) is the circular frequency, \(u_{x}{{}^{*}}\), \(u_{y}{{}^{*}}\) and \(u_{z}{{}^{*}}\) indicate complex conjugate of displacement in the normal \(x\), \(y\) and \(z\) directions, respectively. Real and imaginary parts of values of stress components \(\sigma_{x}\), \(\sigma_{y}\), \(\sigma_{z}\), \(\tau_{xy}\), \(\tau_{yz}\), \(\tau_{zx}\) and values of displacement components \(u_{x}{{}^{*}}\), \(u_{y}{{}^{*}}\), \(u_{z}{{}^{*}}\) in the frequency domain can be obtained from FE results. Then, the structural intensity of each node in the ABH central plateau can be calculated. Higher vibration energy means higher dissipation efficiency when damping layers are coated. Therefore, the energy focalization ability can describe the performance of different ABH plates in Figure 3. The vibration energy gathered in the ABH central plateau as defined in Equation (5) is calculated in the frequency band of 10-5000 Hz, the results are shown in Figure 4.  
\[W_{pf}=S_{c}\frac{\sum{p_{di}}}{N_{c}} \tag{5}\]  
where \(W_{pf}\) represents the vibration energy (average power flow) in a certain area per unit time; \(p_{di}\) represents structural intensity at the \(i\)th node; \(N_{c}\) is the number of total nodes at ABH central plateau and \(S_{c}\) is the area of the central plateau. The cut-on frequency of the big ABH is 559.5 Hz and that of the small ABH is 2790.5 Hz according to Equation (2). Figure 4a shows that, in the frequency band of 200-2200 Hz, the overall level of vibration energy of the central plateau of the big ABH is the highest, followed by the plate embedded with the ABH array and the plate with a small ABH.