# A Single-Phase Lightweight Double-Leaf Multi-Stage Acoustic Black Hole Model of Metamaterial
## 3 Calculation Results and Discussion  
### Complex Energy Band Structure and Vibration Modes  
The partial differential equation module of the coefficient form of the finite element software is calculated. In theory, due to the complexity of the calculation, the material model satisfies the finite conditions such as the isotropy, linear elasticity, and homogeneity of media in order to simplify the calculation. However, these conditions are usually not satisfied for real materials. Therefore, the further influence of these conditions on material properties should be further analysed in practical engineering applications. The complex energy band structures of the SDM-ABH are shown in Figure 2a-c when N = 1, 2, and 3, respectively. Observing the complex energy band structure of the SD1-ABH metamaterial in Figure 2a, it is found that there are three dispersion curves in the real part of the complex energy band in the range of 0-2000 Hz, and only one flexural wave bandgap (243 Hz-672 Hz) is generated, with a generalised width of \((f_{\text{cut}}-f_{\text{st}})/f_{\text{st}}=1.765\)[35], and the bandgap percentage is 21%. For studying the band gap formation mechanism of SD1-ABH metamaterials, the special point vibration modes of the real part of the curve in the complex energy band are calculated, as shown in Figure 3. Observing Figure 3, it is found that the vibrational modes at point A1 (243 Hz) of the initial frequency of the first flexural wave are concentrated in the uniform beams on two sides of the cell and the non-tip area of the first stage ABH, and the whole region presents local antisymmetric flexural vibration, and at this time, the first bandgap is opened. The vibrational modes at point B1 (672 Hz) at the cut-off frequency of the bandgap of the first flexural wave presents local torsional vibrational modes, and the Bloch wave in the periodic structure couples to the local resonance modes making the energy localised in the ABH mid-region, which hinders the propagation of the elastic wave; in addition, the two sides of the structure show torsional vibration modes with smaller displacements, which is due to the complex multiple elastic scattering phenomenon of the wave in the structure that starts to show between the periodic structures [36]. Observation of the imaginary part of the complex energy band in Figure 2a reveals the existence of an asymmetric semicircular arc with smooth frequency variation starting at 0 Hz, which is a typical feature of the coupling of local resonance and Bragg scattering [37]. Thus, the first flexural wave band gap of the SD1-ABH metamaterial is generated by the coupling of local resonance and Bragg scattering. In addition, there is a part of semicircular arc in the imaginary part of the complex energy band exists outside the bandgap, which belongs to the non-zero order diffraction wave. At the same time, the evanescent wave of swift to propagatable wave is realised at the cut-off frequency B1 point [29]. Figure 2: Complex energy band structure. (**a**) N = 1, i.e., single-phase double-leaf one-stage acoustic black hole (SD1-ABH) metamaterial. (**b**) N = 2, i.e., single-phase double-leaf two-stage acoustic black hole (SD2-ABH) metamaterial. (**c**) N = 3, i.e., single-phase double-leaf three-stage acoustic black hole (SD3-ABH) metamaterial. Red represents flexural wave dispersion curves and blue represents longitudinal wave dispersion curves. The structure is SD2-ABH metamaterial when N = 2, and the complex energy band structure is shown in Figure 2b. Observing the real part of the complex energy band in Figure 2b, it is found that there are six dispersion curves below 2000 Hz, forming three flexural bandgaps as well as one longitudinal bandgap.