## 2 Modeling and Simulation Analysis of Plate Embedded with Two-Dimensional ABH Array  
### Finite Element Model of Plate Embedded with ABH Array  
According to existing machining capability, an ideal ABH structure is impossible to be realized. Therefore, typical two-dimensional ABH structures usually contain a hole or a circular plateau of constant thickness at the center of the indentation [12], that is, there must be a residual thickness \(h_{1}\) at the bottom of the ABH indentation. The profile of a two-dimensional ABH indentation is schematically shown in Figure 2. In this paper, two-dimensional ABHs are embedded in a uniform plate with thickness of \(h_{2}~{}=~{}0.004\) m and the geometry dimension of the plate is 0.5 m \(\times\) 0.5 m \(\times\) 0.004 m (length \(\times\) width \(\times\) thickness). The outer radius of ABH is \(x_{2}\), the thickness of the central plateau (residual thickness) is \(h_{1}\), its radius is \(x_{1}\) and the power index \(m\) is set as 2.2, then the variable \(\varepsilon\) can be calculated. The thickness profile of ABH can be described as:  
\[h(x)=\left\{ \begin{array}{l}h_{1},x\leq x_{1}\\ \varepsilon x^{2},x_{1}\leq x\leq x_{2}\end{array} \right. \tag{1}\]  
Cut-on frequency [6] is an indicative frequency at which the wavelength of the incident bending waves starts to be equal to the characteristic dimension (the diameter in this case) of the ABH pit. It can be defined as:  
\[f_{cut-on}=\frac{\pi h}{2(R_{ABH})^{2}}\sqrt{\frac{E_{\rm p}}{12\rho_{p}(1- \
u_{p}{}^{2})}} \tag{2}\]  
Figure 2: A typical two-dimensional ABH indentation. where \(h\) represents the thickness of the uniform part of the plate; \(R_{ABH}\) denotes the radius of the ABH indentation; \(E_{\text{p}}\) is the [PERSON]'s modulus of the plate material; \(\rho_{p}\) represents mass density and \(v_{p}\) is the Poisson's ratio. According to Equation (2), an ABH indentation with a larger radius will have lower cut-on frequency and may have better energy focalization performance at lower frequency. To investigate the potential advantages of the ABH array at lower frequency, 3 plate models are established, as shown in Figure 3. That is, a plate with a single ABH (Figure 3a), a plate with an array of five ABHs (Figure 3b) and a plate with a big ABH (Figure 3c). Parameters \(h_{1}\) and \(h_{2}\) of the big ABH (Figure 3c) are the same as those of the small ABH (Figure 3a). In addition, Radius \(x_{2}\) of the big ABH is 0.1342 m to ensure that the plate with a big ABH has the same mass as the plate embedded with 5 ABHs. Commercial finite element code Abaqus is adopted for the computation of vibration response in the plate models subject to a harmonic point force with a value of 1 N and normal to the surface, as shown in Figure 3. The ABH plates are discretized using C3D8R solid elements in ABAQUS, which is a first-order element with 8 nodes. The frequency range is from 5 to 5000 Hz. Finer meshes are used in the central area of the ABHs to ensure more than ten elements per local wavelength at twice the maximum frequency of interest and the element size is from 0.1 mm to 1 mm. Moreover, the resulting element numbers are 1,084,720, 1,420,592 and 1,104,189, and the resulting DOFs are 8,411,952, 11,913,756 and 9,088,884 for the three ABH plates, respectively. The geometric parameters of the small ABH and the big ABH in this simulation are described in Table 1. The left and right edges of the plate model are fixed and the material properties of the plate used in this paper are shown in Table 2.  
As the ABH array structure is symmetrical, this paper focused on three typical ABHs in Figure 3b, named ABH(i), ABH(ii) and ABH(iii), respectively. The center of ABH(ii) is located on the center of the whole plate structure, and the coordinates of ABH(i) are \((x_{a},y_{a})=(0.12\text{ m},0.12\text{ m})\) relative to ABH(ii), while the relative coordinates of ABH(iii) are \((0.12\text{ m},-0.12\text{ m})\). The harmonic point force is applied at \((0.12\text{ m},-0.21\text{ m})\) relative to ABH(ii).