A High-Frequency Homogenization Approach Near the Dirac Points in Bubbly Honeycomb Crystals
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Published:2020-09-22
Issue:3
Volume:238
Page:1559-1583
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ISSN:0003-9527
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Container-title:Archive for Rational Mechanics and Analysis
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language:en
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Short-container-title:Arch Rational Mech Anal
Author:
Ammari HabibORCID, Hiltunen Erik Orvehed, Yu Sanghyeon
Abstract
AbstractIn (Ammari et al. in SIAM J Math Anal. arXiv:1811.03905), the existence of a Dirac dispersion cone in a bubbly honeycomb phononic crystal comprised of bubbles of arbitrary shape is shown. The aim of this paper is to prove that, near the Dirac points, the Bloch eigenfunctions is the sum of two eigenmodes. Each eigenmode can be decomposed into two components: one which is slowly varying and satisfies a homogenized equation, while the other is periodic across each elementary crystal cell and is highly oscillating. The slowly oscillating components of the eigenmodes satisfy a system of Dirac equations. Our results in this paper prove for the first time a near-zero effective refractive index near the Dirac points for the plane-wave envelopes of the Bloch eigenfunctions in a sub-wavelength metamaterial. They are illustrated by a variety of numerical examples. We also compare and contrast the behaviour of the Bloch eigenfunctions in the honeycomb crystal with that of their counterparts in a bubbly square crystal, near the corner of the Brillouin zone, where the maximum of the first Bloch eigenvalue is attained.
Funder
Swiss Federal Institute of Technology Zurich
Publisher
Springer Science and Business Media LLC
Subject
Mechanical Engineering,Mathematics (miscellaneous),Analysis
Reference38 articles.
1. Ammari, H., Fitzpatrick, B., Gontier, D., Lee, H., Zhang, H.: Sub-wavelength focusing of acoustic waves in bubbly media. Proc. A. 473(2208), 20170469, 2017 2. Ammari, H., Fitzpatrick, B., Gontier, D., Lee, H., Zhang, H.: Minnaert resonances for acoustic waves in bubbly media. Annal. l’Inst. Henri Poincaré C Anal. Nonlinéaire 35, 1975–1998, 2018 3. Ammari, H., Fitzpatrick, B., Gontier, D., Lee, H., Zhang, H.: A mathematical and numerical framework for bubble meta-screens. SIAM J. Appl. Math. 77, 1827–1850, 2017 4. Ammari, H., Fitzpatrick, B., Kang, H., Ruiz, M., Yu, S., Zhang, H.: Mathematical and Computational Methods in Photonics and Phononics, Mathematical Surveys and Monographs, vol. 235. American Mathematical Society, Providence 2018 5. Ammari, H., Fitzpatrick, B., Lee, H., Orvehed Hiltunen, E., Yu, S.: Honeycomb-lattice Minnaert bubbles. SIAM J. Math. Anal. (to appear). arXiv:1811.03905.
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