## 3 Modeling of ABH Plate-Cavity Coupling System  
Numerical methods are used to establish a series of models of a rectangular elastic plate-cavity coupling system embedded with 2D ABHs. For a general vibro-acoustic coupling system, the vibro-acoustic coupling equation in matrix form is as follows:  
\[\left(\left[ \begin{array}{cc}M_{\mathrm{s}}&0\\ -A^{T}&M_{f}\end{array} \right]\cdot\left\{ \begin{array}{c}\overset{...}{u_{s}} \\ \overset{...}{p}\end{array} \right\}+\left[ \begin{array}{cc}B_{s}&0\\ 0&B_{f}\end{array} \right]\cdot\left\{ \begin{array}{c}\overset{...}{u_{s}}\\ \overset{...}{p}\end{array} \right\}+\left[ \begin{array}{cc}K_{s}&A\\ 0&K_{f}\end{array} \right]\right)\cdot\left\{ \begin{array}{c}u_{s}\\ \overset{...}{p}\end{array} \right\}=\left\{ \begin{array}{c}P_{s}\\ P_{f}\end{array} \right\} \tag{4}\]  
where \(M\) is the mass matrix, \(K\) is the stiffness matrix, \(B\) is the damping matrix, subscript \(s\) represents the partitions of the structure, subscript \(f\) represents the fluid degrees of freedom, \(P\) is the loading vector, \(u\) is the freedom degrees of displacements of the structure and pressure, \(p\) is the freedom degrees of displacements of the fluid, and \([A]\) is the coupling matrix between the fluid and structure degrees of freedom at the wetted interface. In the vibro-acoustic coupling calculation, the structural modal participation factor and the acoustic modal participation factor can represent the contribution of the structural modes and the acoustic modes to the vibro-acoustic coupling response, respectively. The structural modal participation factor \([\chi_{s}]\) and the acoustic modal participation factor \(\left[\chi_{f}\right]\) can be calculated by  
\[[\chi_{s}]=-\omega^{2}\Big{[}\Phi_{f}\Big{]}[Z_{2}]\Big{[}a^{T}\Big{]}[\Lambda _{s}] \tag{5}\]  
\[\Big{[}\chi_{f}\Big{]}=\Big{[}\Phi_{f}\Big{]}\Big{[}\Lambda_{f}\Big{]} \tag{6}\]  
where \(\Big{[}\Phi_{f}\Big{]}\) are the uncoupled, undamped rigid-walled acoustic modes, \([\Lambda_{s}]\) is a diagonal matrix such that \([\Lambda_{s_{\mathrm{i}}}]=q_{s_{\mathrm{i}}}\), \(\Big{[}\Lambda_{f}\Big{]}\) is a diagonal matrix such that \(\Big{[}\Lambda_{f_{\mathrm{ii}}}\Big{]}=q_{f_{\mathrm{i}}}\), \(\{q_{s}\}\) is the modal amplitude for the structure, \(\Big{\{}q_{f}\Big{\}}\) is the modal amplitude for the fluid, \(a=\Phi_{s}^{T}A\Phi_{f}\), and \([\Phi_{s}]\) are the uncoupled, undamped structural mode shapes, and \(Z_{2}\) can be expressed as  
\[[Z_{2}]=\Big{[}-\omega^{2}\Big{[}m_{f}\Big{]}+i\omega\Big{[}b_{f}\Big{]}+ \Big{[}k_{f}\Big{]}\Big{]}^{-1} \tag{7}\]  
where \(m_{f}=\Phi_{f}^{T}M_{f}\Phi_{f}\), \(b_{f}=\Phi_{f}^{T}B_{f}\Phi_{f}\), \(k_{f}=\Phi_{f}^{T}K_{f}\Phi_{f}\). Figure 2: The geometric parameters of the proposed ABH plate. The closed plate-cavity coupling system was composed of the ABH plate designed in Section 2 and five other rectangular plates with uniform thickness, and the thickness of the uniform rectangular plates were 2 mm. The vibro-acoustic FEM model of the plate-cavity coupling system is shown in Figure 3. The ABH rectangular plate and the acoustic cavity were meshed using the second-order tetrahedral unit C3D10. The uniform region of the ABH plate and five other rectangular plates had a grid size of 4 mm, while the ABH region had a grid size of 2 mm. The mesh in the ABH region's tiny curved portion was encrypted to provide a seamless transition from small cells to big cells, and the minimum cell size during encryption was 0.1 mm. For the acoustic mesh of the cavity, the side length of the largest element needed to be less than 1/6 of the wavelength corresponding to the highest calculation frequency. The grid size of the acoustic FEM model established in this paper was 10 mm, so the highest acoustic calculation frequency could reach 5 kHz. The material of the plates was 1060 AL, the density was 2710 kg/m\({}^{3}\), the Poisson's ratio was 0.33, the Young's modulus was 69 GPa, and the damping loss factor was 0.005.