Homogenization of the vibro-acoustic transmission on periodically perforated elastic plates with arrays of resonators

Based on our previous work, we propose a homogenized model of acoustic waves propagating through periodically perforated elastic plates with metamaterial properties due to embedded arrays of soft elastic inclusions serving for resonators. Such structures enable to suppress the acoustic transmission for selected frequency bands. Homogenization of the vibro-acoustic fluid-structure interaction problem in a 3D complex geometry of the transmission layer leads to effective transmission conditions prescribed on the acoustic metasurface associated with the mid-plane of the Reissner-Mindlin plate. Asymptotic analysis with respect to the layer thickness, proportional to the plate thickness and to the perforation period, yields an implicit Dirichlet-to-Neumann operator defined on the homogenized metasurface. An efficient method is proposed for computing frequency-dependent effective parameters involved in the homogenized model of the layer. These can change their signs, thus modifying the acoustic impedance and the effective mass of the metasurface. The global problem of the acoustic wave propagation in a waveguide fitted with the plate is solved using the finite element method. The homogenized interface allows for a significant reduction of the computational model. Numerical illustrations are presented.

[1]  Caglar Oskay,et al.  Spectral variational multiscale model for transient dynamics of phononic crystals and acoustic metamaterials , 2020 .

[2]  Georges Griso,et al.  Multiscale Modeling of Elastic Waves: Theoretical Justification and Numerical Simulation of Band Gaps , 2008, Multiscale Model. Simul..

[3]  A. Marciniak-Czochra,et al.  A Rigorous Derivation of the Equations for the Clamped Biot-Kirchhoff-Love Poroelastic Plate , 2012, 1211.6456.

[4]  B. Schweizer.  Effective Helmholtz problem in a domain with a Neumann sieve perforation , 2020 .

[5]  Houssem Haddar,et al.  Approximate models for wave propagation across thin periodic interfaces , 2012 .

[6]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[7]  Vladimír Lukes,et al.  Multiscale finite element calculations in Python using SfePy , 2018, Adv. Comput. Math..

[8]  G. Shmuel,et al.  Dynamic homogenization of composite and locally resonant flexural systems , 2018, Journal of the Mechanics and Physics of Solids.

[9]  J. Willis,et al.  On cloaking for elasticity and physical equations with a transformation invariant form , 2006 .

[10]  Vladimír Lukes,et al.  Homogenization of the acoustic transmission through a perforated layer , 2010, J. Comput. Appl. Math..

[11]  N. Peake,et al.  Wave scattering by an infinite cascade of non-overlapping blades , 2020 .

[12]  P. Sheng,et al.  Acoustic metasurface with hybrid resonances. , 2014, Nature materials.

[13]  Peter Hansbo,et al.  A Nitsche-type Method for Helmholtz Equation with an Embedded Acoustically Permeable Interface , 2015, 1511.09363.

[14]  V. P. Smyshlyaev,et al.  Propagation and localization of elastic waves in highly anisotropic periodic composites via two-scale homogenization , 2009 .

[15]  Takayuki Yamada,et al.  Optimum design of a multi-functional acoustic metasurface using topology optimization based on Zwicker’s loudness model , 2018 .

[16]  Acoustic metasurface by layered concentric structures , 2020 .

[17]  Eduard Rohan,et al.  Numerical simulation of acoustic band gaps in homogenized elastic composites , 2009 .

[18]  Eduard Rohan,et al.  Elastodynamics of strongly heterogeneous periodic plates using Reissner‐Mindlin and Kirchhoff‐Love models , 2016 .

[19]  Justin T. Webster,et al.  Weak solutions for a poro-elastic plate system , 2021, Applicable Analysis.

[20]  Xavier Claeys,et al.  High order asymptotics for wave propagation across thin periodic interfaces , 2013, Asymptot. Anal..

[21]  Pai Peng,et al.  An ultra-thin acoustic metasurface with multiply resonant units , 2019, Physics Letters A.

[22]  Yong Li,et al.  Acoustic perfect absorbers via Helmholtz resonators with embedded apertures. , 2019, The Journal of the Acoustical Society of America.

[23]  Huanyang Chen,et al.  Reversal of transmission and reflection based on acoustic metagratings with integer parity design , 2019, Nature Communications.

[24]  Vladimír Lukes,et al.  Homogenization of the vibro-acoustic transmission on perforated plates , 2019, Appl. Math. Comput..

[25]  E. Rohan,et al.  Homogenized Perforated Interface in Acoustic Wave Propagation – Modeling and Optimization , 2011 .

[26]  Jensen Li,et al.  Willis Metamaterial on a Structured Beam , 2018, Physical Review X.

[27]  Takayuki Yamada,et al.  Topology optimization for hyperbolic acoustic metamaterials using a high-frequency homogenization method , 2018, Computer Methods in Applied Mechanics and Engineering.

[28]  B. Liang,et al.  Wavefront manipulation by acoustic metasurfaces: from physics and applications , 2018, Nanophotonics.

[29]  Eduard Rohan,et al.  Modelling response of phononic Reissner-Mindlin plates using a spectral decomposition , 2015, Appl. Math. Comput..

[30]  Graeme W Milton,et al.  On modifications of Newton's second law and linear continuum elastodynamics , 2007, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[31]  Luca Placidi,et al.  Wave propagation in relaxed micromorphic continua: modeling metamaterials with frequency band-gaps , 2013, 1309.1722.

[32]  E. Rohan,et al.  Shape optimization of phononic band gap structures using the homogenization approach , 2016, 1601.02902.

[33]  Jean-Jacques Marigo,et al.  Two-scale homogenization to determine effective parameters of thin metallic-structured films , 2016, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[34]  Anne-Sophie Bonnet-Ben Dhia,et al.  MATHEMATICAL ANALYSIS OF THE ACOUSTIC DIFFRACTION BY A MUFFLER CONTAINING PERFORATED DUCTS , 2005 .

[35]  Jean-Jacques Marigo,et al.  Homogenization models for thin rigid structured surfaces and films. , 2016, The Journal of the Acoustical Society of America.

[36]  Doina Cioranescu,et al.  The periodic unfolding method for perforated domains and Neumann sieve models , 2008 .