Complex periodic structures inherit spectral properties from the constituent parts of their unit cells, chiefly their spectral band gaps. Exploiting this intuitive principle, which is made precise in this work, means spectral features of periodic systems with very large unit cells can be predicted without numerical simulation. We study a class of difference equations with periodic coefficients and show that they inherit spectral gaps from their constituent elements. This result shows that if a frequency falls in a band gap for every constituent element then it must be in a band gap for the combined complex periodic structure. This theory and its instantaneous utility is demonstrated in a series of vibro-acoustic and mechanical examples.
Les structures périodiques complexes héritent des propriétés spectrales des éléments constitutifs de leurs cellules de base, principalement de leurs trous spectraux. L’exploitation de ce principe intuitif, qui est précisé dans ce travail, permet de prédire les caractéristiques spectrales des systèmes périodiques avec de très grandes cellules de base sans avoir recours aux simulations numériques. Nous étudions une classe d’équations aux différences avec des coefficients périodiques et montrons qu’elles héritent des écarts spectraux de leurs éléments constitutifs. Ce résultat montre que si une fréquence tombe dans une bande interdite pour chaque élément constitutif, elle doit se trouver dans une bande interdite pour la structure périodique complexe combinée. Cette théorie et son utilité immédiate sont illustrées dans une série d’exemples vibro-acoustiques et mécaniques.
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Mots-clés : TMatrice de transfert, spectre de Bloch, trace, métamatériau, cristal phononique
Lucas Dunckley 1; Bryn Davies 1
CC-BY 4.0
@article{CRMECA_2024__352_G1_143_0,
author = {Lucas Dunckley and Bryn Davies},
title = {Hierarchical band gaps in complex periodic systems},
journal = {Comptes Rendus. M\'ecanique},
pages = {143--156},
publisher = {Acad\'emie des sciences, Paris},
volume = {352},
year = {2024},
doi = {10.5802/crmeca.249},
language = {en},
}
Lucas Dunckley; Bryn Davies. Hierarchical band gaps in complex periodic systems. Comptes Rendus. Mécanique, Volume 352 (2024), pp. 143-156. doi : 10.5802/crmeca.249. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.249/
[1] Homogenization and two-scale convergence, SIAM J. Math. Anal., Volume 23 (1992) no. 6, pp. 1482-1518 | DOI | MR | Zbl
[2] Colloquium: Topological band theory, Rev. Mod. Phys., Volume 88 (2016) no. 2, 021004 | DOI
[3] Wave Propagation in Periodic Structures: Electric Filters and Crystal Lattices, Dover Publications, New York, 1953 | Zbl
[4] et al. Surface-plasmon-enhanced transmission through metallic film perforated with fractal-featured aperture array, Appl. Phys. Lett., Volume 90 (2007) no. 25, 251914 | DOI
[5] Acoustic Metamaterials: Negative Refraction, Imaging, Lensing and Cloaking, Springer Series in materials science, 166, Springer, Dordrecht, 2012
[6] Symmetry-induced quasicrystalline waveguides, Wave Motion, Volume 115 (2022), 103068 | DOI | MR | Zbl
[7] Super band gaps and periodic approximants of generalised Fibonacci tilings, Proc. R. Soc. A., Volume 480 (2024) no. 2285, 20230663 | DOI | MR
[8] Green’s Functions in Quantum Physics, Springer Series in Solid-State Sciences, 7, Springer, 2006 | DOI
[9] Ultrasonic metamaterials with negative modulus, Nature Mater., Volume 5 (2006) no. 6, pp. 452-456 | DOI
[10] The Fibonacci quasicrystal: Case study of hidden dimensions and multifractality, Rev. Mod. Phys., Volume 93 (2021) no. 4, 045001 | DOI | MR
[11] Photonic Crystals, Molding the Flow of Light, Princeton University Press, 1995 | Zbl
[12] Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett., Volume 50 (1983) no. 23, pp. 1870-1872 | DOI | MR
[13] Cantor spectrum for an almost periodic Schrödinger equation and a dynamical map, Phys. Lett. A, Volume 102 (1984) no. 4, pp. 145-148 | DOI
[14] An Overview of Periodic Elliptic Operators, Bull. Am. Math. Soc., Volume 53 (2016) no. 3, pp. 323-414 | DOI | MR | Zbl
[15] Topological equivalence between the Fibonacci quasicrystal and the Harper model, Phys. Rev. Lett., Volume 109 (2012) no. 11, 116404 | DOI
[16] Design and characterization of bubble phononic crystals, Appl. Phys. Lett., Volume 95 (2009) no. 17, 171904 | DOI
[17] Localised point defect states in asymptotic models of discrete lattices, Q. J. Mech. Appl. Math., Volume 66 (2013) no. 3, pp. 289-316 | DOI | MR | Zbl
[18] Geometrically navigating topological plate modes around gentle and sharp bends, Phys. Rev. B, Volume 98 (2018) no. 18, 184105 | DOI | Zbl
[19] Negative refraction in quasicrystalline multilayered metamaterials, J. Mech. Phys. Solids, Volume 124 (2019), pp. 282-298 | DOI | MR
[20] Waves in one-dimensional quasicrystalline structures: dynamical trace mapping, scaling and self-similarity of the spectrum, J. Mech. Phys. Solids, Volume 119 (2018), pp. 83-103 | DOI | MR
[21] On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. Lond., Ser. A, Volume 462 (2006) no. 2074, pp. 3027-3059 | DOI | MR | Zbl
[22] Wave Propagation: From Electrons to Photonic Crystals and Left-Handed Materials, Princeton University Press, 2008 | DOI | Zbl
[23] Disorder persistent transparency within the bandgap of a periodic array of acoustic Helmholtz resonators, J. Appl. Phys., Volume 117 (2015) no. 10, 104902 | DOI
[24] Metamaterials and negative refractive index, Science, Volume 305 (2004) no. 5685, pp. 788-792 | DOI
[25] Propagation of nonlinear acoustic waves in a tunnel with an array of Helmholtz resonators, J. Fluid Mech., Volume 244 (1992), pp. 55-78 | DOI | Zbl
[26] Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys., Volume 56 (1989), pp. 525-531 | DOI | MR | Zbl
[27] Resonant transmission of microwaves through subwavelength fractal slits in a metallic plate, Phys. Rev. B, Volume 72 (2005) no. 15, 153406 | DOI
[28] Subwavelength acoustic energy harvesting via topological interface states in 1D Helmholtz resonator arrays, AIP Adv., Volume 11 (2021) no. 1, 015241 | DOI
[29] Symmetry and degeneracy of phonon modes for periodic structures with glide symmetry, J. Mech. Phys. Solids, Volume 122 (2019), pp. 244-261 | DOI | MR | Zbl
[30] Topological interface modes in local resonant acoustic systems, Phys. Rev. B, Volume 98 (2018) no. 1, 014110 | DOI
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