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Phononic Band Structures and Transmission Coefficients: Methods and Approaches

  • J. O. Vasseur
  • Pierre A. Deymier
  • A. Sukhovich
  • B. Merheb
  • A.-C. Hladky-Hennion
  • M. I. Hussein
Chapter
Part of the Springer Series in Solid-State Sciences book series (SSSOL, volume 173)

Abstract

The purpose of this chapter is first to recall some fundamental notions from the theory of crystalline solids (such as direct lattice, unit cell, reciprocal lattice, vectors of the reciprocal lattice, Brillouin zone, etc.) applied to phononic crystals and second to present the most common theoretical tools that have been developed by several authors to study elastic wave propagation in phononic crystals and acoustic metamaterials. These theoretical tools are the plane wave expansion method, the finite-difference time domain method, the multiple scattering theory, and the finite element method. Furthermore, a model reduction method based on Bloch modal analysis is presented. This method applies on top of any of the numerical methods mentioned above. Its purpose is to significantly reduce the size of the final matrix model and hence enable the computation of the band structure at a very fast rate without any noticeable loss in accuracy. The intention in this chapter is to give to the reader the basic elements necessary for the development of his/her own calculation codes. The chapter does not contain all the details of the numerical methods, and the reader is advised to refer to the large bibliography already devoted to this topic.

Keywords

Reciprocal Lattice Band Structure Calculation Phononic Crystal Plane Wave Expansion Bravais Lattice 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976)Google Scholar
  2. 2.
    M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141–143 (1993)CrossRefGoogle Scholar
  3. 3.
    J.O. Vasseur, B. Djafari-Rouhani, L. Dobrzynskiand, P.A. Deymier, Acoustic band gaps in fibre composite materials of boronnitride structure. J. Phys. Condens Matter 9, 7327–7341 (1997)CrossRefGoogle Scholar
  4. 4.
    ZhilinHou, Xiujun Fu, and Youyan Liu, Singularity of the Bloch theorem in the fluid/solid phononic crystal. Phys. Rev. B 73, 024304–024308 (2006)Google Scholar
  5. 5.
    J.O. Vasseur, P.A. Deymier, A. Khelif, P. Lambin, B. Djafari-Rouhani, A. Akjouj, L. Dobrzynski, N. Fettouhi, J. Zemmouri, Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: a theoretical and experimental study. Phys. Rev. E 65, 056608 (2002)CrossRefGoogle Scholar
  6. 6.
    B. Manzanares-Martinez, F. Ramos-Mendieta, Surface elastic waves in solid composites of two-dimensional periodicity. Phys. Rev. B 68, 134303 (2003)CrossRefGoogle Scholar
  7. 7.
    C. Goffaux, J.P. Vigneron, Theoretical study of a tunable phononic band gap system. Phys. Rev. B 64, 075118 (2001)CrossRefGoogle Scholar
  8. 8.
    Y. Tanaka, Y. Tomoyasu, S.I. Tamura, Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch. Phys. Rev. B 62, 7387 (2000)CrossRefGoogle Scholar
  9. 9.
    J.O. Vasseur, P.A. Deymier, B. Djafari-Rouhani, Y. Pennec, A.-C. Hladky-Hennion, Absolute forbidden bands and waveguiding in two-dimensional phononic crystal plates. Phys. Rev. B 77, 085415 (2008)CrossRefGoogle Scholar
  10. 10.
    C. Charles, B. Bonello, F. Ganot, Propagation of guided elastic waves in 2D phononic crystals. Ultrasonics 44, 1209(E) (2006)CrossRefGoogle Scholar
  11. 11.
    C. Croënne, E.D. Manga, B. Morvan, A. Tinel, B. Dubus, J. Vasseur, A.-C. Hladky-Hennion, Negative refraction of longitudinal waves in a two-dimensional solid-solid phononic crystal. Phys. Rev. B 83, 054301 (2011)CrossRefGoogle Scholar
  12. 12.
    P. Lambin, A. Khelif, J.O. Vasseur, L. Dobrzynski, B. Djafari-Rouhani, Stopping of acoustic waves by sonic polymer-fluid composites. Phys. Rev. E 63, 066605 (2001)CrossRefGoogle Scholar
  13. 13.
    G. Mur, Absorbing boundary conditions for the finite difference approximation of the time-domain electromagnetic field equations. IEEE Trans. Electromagn. Compatibility 23, 377 (1981)CrossRefGoogle Scholar
  14. 14.
    A. Taflove, Computational electrodynamics: the finite difference time domain method (Artech House, Boston, 1995)Google Scholar
  15. 15.
    B. Merheb, P.A. Deymier, M. Jain, M. Aloshyna-Lesuffleur, S. Mohanty, A. Berker, R.W. Greger, Elastic and viscoelastic effects in rubber/air acoustic band gap structures: a theoretical and experimental study. J. Appl. Phys. 104, 064913 (2008)CrossRefGoogle Scholar
  16. 16.
    B. Merheb, P.A. Deymier, K. Muralidharan, J. Bucay, M. Jain, M. Aloshyna-Lesuffleur, R.W. Greger, S. Mohanty, A. Berker, Viscoelastic effect on acoustic band gaps in polymer-fluid composites. Model. Simul. Mater. Sci. Eng. 17, 075013 (2009)CrossRefGoogle Scholar
  17. 17.
    M. Kafesaki, E.N. Economou, Multiple-scattering theory for three-dimensional periodic acoustic composites. Phys. Rev. B. 60, 11993 (1999)CrossRefGoogle Scholar
  18. 18.
    Z. Liu, C.T. Chan, P. Sheng, A.L. Goertzen, J.H. Page, Elastic wave scattering by periodic structures of spherical objects: theory and experiment. Phys. Rev. B. 62, 2446 (2000)CrossRefGoogle Scholar
  19. 19.
    I.E. Psarobas, N. Stefanou, A. Modinos, Scattering of elastic waves by periodic arrays of spherical bodies. Phys. Rev. B. 62, 278 (2000)CrossRefGoogle Scholar
  20. 20.
    J. Mei, Z. Liu, J. Shi, D. Tian, Theory for elastic wave scattering by a two-dimensional periodical array of cylinders: an ideal approach for band-structure calculations. Phys. Rev. B 67, 245107 (2003)CrossRefGoogle Scholar
  21. 21.
    P. Langlet, A.-C. Hladky-Hennion, J.N. Decarpigny, Analysis of the propagation of plane acoustic waves in passive periodic materials using the finite element method. J. Acoust. Soc. Am. 95, 1792 (1995)Google Scholar
  22. 22.
    J.O. Vasseur, A.-C. Hladky-Hennion, B. Djafari-Rouhani, F. Duval, B. Dubus, Y. Pennec, Waveguiding in two-dimensional piezoelectric phononic crystal plates. J. Appl. Phys. 101, 114904 (2007)CrossRefGoogle Scholar
  23. 23.
    L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953)Google Scholar
  24. 24.
    K. Busch, G. von Freymann, S. Linden, S.F. Mingaleev, L. Tkeshelashvili, M. Wegener, Periodic nanostructures for photonics. Phys. Rep. 444, 101–202 (2007)CrossRefGoogle Scholar
  25. 25.
    O. Sigmund, J.S. Jensen, Systematic design of phononic band-gap materials and structures by topology optimization. Philos. Trans. R. Soc. Lond. A361, 1001–1019 (2003)Google Scholar
  26. 26.
    O.R. Bilal, M.I. Hussein, Ultrawidephononic band gap for combined in-plane and out-of-plane waves. Phys. Rev. E 84, 065701(R) (2011)CrossRefGoogle Scholar
  27. 27.
    R.L. Chern, C.C. Chang, R.R. Hwang, Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration. Phys. Rev. E 68, 026704 (2003)CrossRefGoogle Scholar
  28. 28.
    D.C. Dobson, An efficient method for band structure calculations in 2D photonic crystals. J. Comput. Phys. 149, 363–376 (1999)CrossRefGoogle Scholar
  29. 29.
    S.G. Johnson, J.D. Joannopoulos, Photonic crystals: putting a new twist on light. Opt. Express 8, 173 (2001)CrossRefGoogle Scholar
  30. 30.
    T.W. McDevitt, G.M. Hulbert, N. Kikuchi, An assumed strain method for the dispersive global-local modeling of periodic structures. Comput. Methods Appl. Mech. Eng. 190, 6425–6440 (2001)CrossRefGoogle Scholar
  31. 31.
    M.I. Hussein, G.M. Hulbert, Mode-enriched dispersion models of periodic materials within a multiscale mixed finite element framework. Finite Elem. Anal. Des. 42, 602–612 (2006)CrossRefGoogle Scholar
  32. 32.
    M.I. Hussein, Reduced Bloch mode expansion for periodic media band structure calculations. Proc. R. Soc. Lond. A465, 2825–2848 (2009)Google Scholar
  33. 33.
    Q. Guo, O.R. Bilal, M.I. Hussein, Convergence of the reduced Bloch mode expansion method for electronic band structure calculations,” in Proceedings of Phononics 2011, Paper PHONONICS-2011-0176, Santa Fe, New Mexico, USA, May 29–June 2, 2011, pp. 238–239Google Scholar
  34. 34.
    M.I. Hussein, Dynamics of banded materials and structures: analysis, design and computation in multiple scales, Ph.D. Thesis, University of Michigan, Ann Arbor, USA, 2004.Google Scholar
  35. 35.
    O. Døssing, IMAC-XIII keynote address: going beyond modal analysis, or IMAC in a new key. Modal Anal. Int. J. Anal. Exp. Modal Anal. 10, 69 (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • J. O. Vasseur
    • 1
  • Pierre A. Deymier
    • 2
  • A. Sukhovich
    • 3
  • B. Merheb
    • 2
  • A.-C. Hladky-Hennion
    • 1
  • M. I. Hussein
    • 4
  1. 1.Institut d’Electronique, de Micro-électronique et de Nanotechnologie, UMR CNRS 8520Cité ScientifiqueVilleneuve d’Ascq CedexFrance
  2. 2.Department of Materials Science and EngineeringUniversity of ArizonaTucsonUSA
  3. 3.Laboratoire Domaines Océaniques, UMR CNRS 6538, UFR Sciences et TechniquesUniversité de Bretagne OccidentaleBrestFrance
  4. 4.Department of Aerospace Engineering SciencesUniversity of Colorado BoulderBoulderUSA

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