# A Single-Phase Lightweight Double-Leaf Multi-Stage Acoustic Black Hole Model of Metamaterial
## 6 Conclusions  
In this paper, the finite element method is applied to calculate the complex energy band structure and the vibrational modes of the special points in the real part of the complex energy band of single-phase double-leaf multi-stage acoustic black hole (SDM-ABH) metamaterials, to explore the mechanism of the generation of the low-frequency wide bandgap, and then the reasons for the enhancement of the low-frequency bandgap attenuation are revealed by the vibrational modes. Finally, vibrational transmission, and vibrational displacement fields of the special points in the imaginary part of the complex energy band, and the effects of the structural parameters on the characteristics of the bandgap, as well as the effects of the number of stages on the trapped region, are discussed. (1) The finite element method is applied to calculate the complex energy band structure of SDM-ABH metamaterials, and it is found that in the range below 2000 HZ, compared with the conventional double-leaf acoustic black hole metamaterials (with a bandgap  
Figure 14: Frequency response analysis of metamaterial interfaces consisting of 12 unit cells: (**a**) schematic diagram of the computational analysis of interfacial response of different cells; (**b,d,f**) represent the flexural wave trapping regions of different interfaces at N = 1, 2, and 3; (**c,e,g**) represent the longitudinal wave trapping regions of different interfaces at N = 1, 2, and 3. The red colour represents the trapped wave region. share of 21%), the single-phase double-leaf two-grade acoustic black hole (SD2-ABH) metamaterials can widen the bandgap percentage to 60.9% with a mass reduction of 10%, and single-phase double-leaf three-grade acoustic black hole (SD3-ABH) metamaterials widen the bandgap percentage to 63.7% with a mass reduction of 13%. And the maximum attenuation in the first flexural wave bandgap of SD2-ABH and SD3-ABH increases to 1.4 and 1.6 times, respectively, compared with the conventional double-leaf acoustic black hole metamaterials. (2) By calculating the vibrational modes at special points in the real part of the complex energy band, it is found that an increase in the amount of stages causes an increase in the amount of sub-ABH structures in which the ABH effect occurs, although the mass decreases, which results in a significant increase in the width and attenuation of the first flexural bandgap. Moreover, an increase in the number of stages causes the emergence of the second flexural and longitudinal wave bandgaps. Further, the vibrational modes at special points and the vibrational displacement fields at special frequencies in the imaginary part of the complex energy band indicate that the phenomena of displacement aggregation and displacement mutation have occurred at several sub-ABHs of the SDM-ABH metamaterial. (3) In addition, the effects of the ABH truncation thickness t and power exponent m on the band gap of the first flexural wave are investigated. With the increase in the truncation thickness t, the ABH effect of the sub-ABHs is gradually weakened, resulting in the first flexural wave cut-off frequency moving faster to high frequencies, and the ABH effect disappears when t > 2 mm, resulting in the first flexural wave cut-off frequency moving to low frequencies. In addition, as the power exponent m increases, the ABH effect of the sub-ABHs increases and the local resonance stiffness decreases, resulting in the first flexural wave bandgap moving to lower frequencies. Finally, the interfacial frequency response was calculated, and it is found that the trapping effect of SDM-ABH metamaterial on elastic waves gradually increased with the increase in the number of stages. This study extends the study of ABH structures to the field of multi-stage metamaterials. Through this study, we can obtain a low-frequency as well as wide bandgap with strong attenuation and lightweight, making it potentially useful for lightweight vibration and noise reduction applications.