## 3 Finite element model (FEM) Analysis of Sound Transmission Loss of the ABH Circular Plate
### FEM Modeling of TL Analysis  
The geometric parameters of the model used for finite element analysis are shown in Table 1. The diameter and thickness of the circular plate are 400 mm and 4 mm respectively, the power index of ABH is \(m=2\), the diameter of the ABH region is 60 mm, and the truncation thickness is 0.5 mm, which are the same as shown in Figure 1.  
Figure 3 shows the FEM model of the ABH circular plate and the vibro-acoustic coupled model for TL analysis. The ABH circular plate was meshed by solid elements and tetrahedral meshes, and the mesh size of the uniform region is 2 mm. In order to improve the calculation accuracy, the grid of the ABH region has a basic size of 1 mm and encrypts the mesh in the small curved area with a minimum size of 0.2 mm; thereby, each wavelength contains at least 10 elements to ensure calculation accuracy. The acoustic model has a size of \(0.6\times 0.6\times 0.02\) m. The calculated frequency has a corresponding relationship with the grid cell size. For the linear finite element model, the side length of the largest cell must be less than 1/6 of the wavelength corresponding to the highest calculated frequency, that is, \(L\leq c/6f_{\text{max}}\). In this paper, the calculation frequency of the transmission loss is 0-10 kHz, so the cell length of the mesh should not be greater than 5.6 mm. Finally, the mesh size of the acoustic finite element model of this paper was selected as 4 mm.  
The FEM simulations were performed using the commercial software LMS Virtual.Lab. The acoustic model below the plate in Figure 3b represented the reverberation chamber, and the acoustic model above the plate represented the anechoic chamber. The ABH region of the circular plate was on the side close to the anechoic chamber, and the boundary condition of the periphery of the circular plate was clamped. A reverberation sound source was generated in the reverberation chamber, and the sound was transmitted to the anechoic chamber through the ABH plate, and finally passed through the outer wall surface of the anechoic chamber and radiated outward without reflection. The faces of the two acoustic meshes close to the structural mesh were coupled with the corresponding faces of the structural mesh and the remaining faces of the two acoustic meshes were respectively applied with a diffuse sound field on the side of the reverberation chamber and a zero reflection surface on the side of the anechoic chamber by the Automatic Matched Layer (AML) method. The transmission  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\epsilon\) & **Power Index** & \(d\) **(mm)** & \begin{tabular}{c} **Uniform** \\ **Thickness (mm)** \\ \end{tabular} & \begin{tabular}{c} **Truncation** \\ **Thickness (mm)** \\ \end{tabular} & \
\begin{tabular}{c} **ABH Diameter** \\ **(mm)** \\ \end{tabular} &
\begin{tabular}{c} **Plate Diameter** \\ **(mm)** \\ \end{tabular} \\ \hline \(3.89\times 10^{-3}\) & 2 & 1.5 & 4 & 0.5 & 60 & 400 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric parameters of the ABH circular plate.  
Figure 3: Transmission loss (TL) analysis model of an ABH plate: (**a**) structural model of the ABH plate; (**b**) vibro-acoustic coupling model.  
loss of the ABH plate can be obtained by computing the sound power incident on the ABH circular plate and radiated from the plate to the anechoic chamber.  
The material parameters of the ABH circular plate are shown in Table 2. The first-order resonant frequency and the coincidence frequency of the ABH circular plate were equal to approximately 350 Hz and 2900 Hz according to Equations (11) and (12). Therefore, the calculated frequency of 0-10 KHz selected in this paper covers all characteristic frequency regions from the stiffness-controlled region to the damping-controlled region.