TY - JOUR

T1 - A two-dimensional wave and Rayleigh–Ritz method for complex dispersion in periodic arrays of circular damped acoustic black holes

AU - Deng, Jie

AU - Xu, Yuxin

AU - Guasch, Oriol

AU - Gao, Nansha

AU - Tang, Liling

AU - Chen, Xu

N1 - Funding Information:
This work is supported by the Fundamental Research Funds for the Central Universities, China (Grant No. G2022KY05106 ), National Natural Science Foundation of China (Grant Nos. 11704314 and 52171323 ), and the China Postdoctoral Science Foundation (Grant No. 2018M631194 ). The third author gratefully acknowledges the support of the Generalitat de Catalunya (Departament de Recerca i Universitats) through grant 2021 SGR 01396 awarded to the HER group.
Publisher Copyright:
© 2023 Elsevier Ltd

PY - 2023/10/1

Y1 - 2023/10/1

N2 - Two-dimensional arrays of circular acoustic black holes (ABHs) on plates are known to exhibit many surprising properties such as bandgap formation, scattering, topological edge states and negative refraction and bi-refraction, to name a few. These effects can be deduced from the analysis of dispersion relations. However, to date, most of the work on periodic ABH arrays has only dealt with real dispersion curves that do not adequately reproduce their behavior. The role of the damping layer and intrinsic losses, which is essential to ABHs, is ignored. This can be accounted for by complex dispersion. In this paper, we extend the wave and Rayleigh–Ritz method (WRRM) previously developed for periodic ABHs in beams to address complex dispersion in two-dimensional periodic ABH arrays. This poses several challenges. On the one hand, in the WRRM the solution of the equations of motion is expanded in terms of the basis of the nullspace determined by the periodic boundary conditions. Although this only involves point connections for beams, in the two-dimensional case continuous line constraints are present and care must be taken to find the appropriate nullspace. To obtain the complex dispersion surface and complex equi-frequency contours of two-dimensional periodic ABH arrays we then need to set the eigenvalue problem in the k(ω) approach. For a fixed angular frequency ω∈R and a direction of propagation, θ, whose tangent is the quotient of the unknown wavenumber components in the y and x directions, we derive polynomial and transcendental eigenvalue problems for the complex wavenumber k∈ℂ×ℂ as a function of θ. The complex dispersion surface and equi-frequency contours of the periodic ABH arrays can be obtained by solving these eigenvalue problems with an iteration method. The two-dimensional WRRM is validated by comparison with finite element simulations. Finally, we apply it to calculate the complex dispersion of an infinite periodic strip and of an infinite periodic array of circular ABHs. In the latter case, collimation effects are observed. The transmissibility of finite ABH strips and arrays is also presented for comparison.

AB - Two-dimensional arrays of circular acoustic black holes (ABHs) on plates are known to exhibit many surprising properties such as bandgap formation, scattering, topological edge states and negative refraction and bi-refraction, to name a few. These effects can be deduced from the analysis of dispersion relations. However, to date, most of the work on periodic ABH arrays has only dealt with real dispersion curves that do not adequately reproduce their behavior. The role of the damping layer and intrinsic losses, which is essential to ABHs, is ignored. This can be accounted for by complex dispersion. In this paper, we extend the wave and Rayleigh–Ritz method (WRRM) previously developed for periodic ABHs in beams to address complex dispersion in two-dimensional periodic ABH arrays. This poses several challenges. On the one hand, in the WRRM the solution of the equations of motion is expanded in terms of the basis of the nullspace determined by the periodic boundary conditions. Although this only involves point connections for beams, in the two-dimensional case continuous line constraints are present and care must be taken to find the appropriate nullspace. To obtain the complex dispersion surface and complex equi-frequency contours of two-dimensional periodic ABH arrays we then need to set the eigenvalue problem in the k(ω) approach. For a fixed angular frequency ω∈R and a direction of propagation, θ, whose tangent is the quotient of the unknown wavenumber components in the y and x directions, we derive polynomial and transcendental eigenvalue problems for the complex wavenumber k∈ℂ×ℂ as a function of θ. The complex dispersion surface and equi-frequency contours of the periodic ABH arrays can be obtained by solving these eigenvalue problems with an iteration method. The two-dimensional WRRM is validated by comparison with finite element simulations. Finally, we apply it to calculate the complex dispersion of an infinite periodic strip and of an infinite periodic array of circular ABHs. In the latter case, collimation effects are observed. The transmissibility of finite ABH strips and arrays is also presented for comparison.

KW - Acoustic black holes

KW - Complex dispersion curves

KW - Damping

KW - Periodic arrays

KW - Wave and Rayleigh–Ritz method

UR - http://www.scopus.com/inward/record.url?scp=85161659889&partnerID=8YFLogxK

U2 - 10.1016/j.ymssp.2023.110507

DO - 10.1016/j.ymssp.2023.110507

M3 - Article

AN - SCOPUS:85161659889

SN - 0888-3270

VL - 200

JO - Mechanical Systems and Signal Processing

JF - Mechanical Systems and Signal Processing

M1 - 110507

ER -