Hierarchical band gaps in complex periodic systems

Lucas Dunckley; Bryn Davies

doi:10.5802/crmeca.249

Article de recherche

Hierarchical band gaps in complex periodic systems
[Trous spectaux hiérarchiques dans les systèmes périodiques complexes]

Lucas Dunckley ¹ ; Bryn Davies ¹

¹ Department of Mathematics, Imperial College London, London, UK

Comptes Rendus. Mécanique, Volume 352 (2024), pp. 143-157.

Résumé
Abstract

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.

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.

Reçu le : 2024-01-26
Accepté le : 2024-03-23
Publié le : 2024-06-07

DOI : 10.5802/crmeca.249

Keywords: Transfer matrix, Bloch spectrum, trace, metamaterial, phononic crystal
Mot clés : TMatrice de transfert, spectre de Bloch, trace, métamatériau, cristal phononique

Affiliations des auteurs :

Lucas Dunckley ¹ ; Bryn Davies ¹

¹ Department of Mathematics, Imperial College London, London, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Hierarchical band gaps in complex periodic systems},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {143--157},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {352},
     year = {2024},
     doi = {10.5802/crmeca.249},
     language = {en},
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Lucas Dunckley; Bryn Davies. Hierarchical band gaps in complex periodic systems. Comptes Rendus. Mécanique, Volume 352 (2024), pp. 143-157. doi : 10.5802/crmeca.249. https://comptes-rendus.academie-sciences.fr/mecanique/articles/10.5802/crmeca.249/

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