Skip to main content
SpringerLink
Log in

Phononic Band Structures and Transmission Coefficients: Methods and Approaches

  • Chapter
  • First Online:
Acoustic Metamaterials and Phononic Crystals

Part of the book series: Springer Series in Solid-State Sciences ((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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    The same mode selection concept, but in the context of a multiscale two-field variational method, was presented in [31, 34].

  2. 2.

    The concept of modal analysis is rooted in the idea of extracting a reduced set of representative information on the dynamical nature of a complex system. This practice is believed to have originated by the Egyptians in around 4700 b.c. in their quest to find effective ways to track the flooding of the Nile and predict celestial events [35].

References

  1. N.W. Ashcroft, N.D. Mermin, Solid State Physics (Saunders College, Philadelphia, 1976)

    Google Scholar 

  2. M. Sigalas, E.N. Economou, Band structure of elastic waves in two dimensional systems. Solid State Commun. 86, 141–143 (1993)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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. 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)

    Article  CAS  Google Scholar 

  6. B. Manzanares-Martinez, F. Ramos-Mendieta, Surface elastic waves in solid composites of two-dimensional periodicity. Phys. Rev. B 68, 134303 (2003)

    Article  Google Scholar 

  7. C. Goffaux, J.P. Vigneron, Theoretical study of a tunable phononic band gap system. Phys. Rev. B 64, 075118 (2001)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  10. C. Charles, B. Bonello, F. Ganot, Propagation of guided elastic waves in 2D phononic crystals. Ultrasonics 44, 1209(E) (2006)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  14. A. Taflove, Computational electrodynamics: the finite difference time domain method (Artech House, Boston, 1995)

    Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  17. M. Kafesaki, E.N. Economou, Multiple-scattering theory for three-dimensional periodic acoustic composites. Phys. Rev. B. 60, 11993 (1999)

    Article  CAS  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  19. I.E. Psarobas, N. Stefanou, A. Modinos, Scattering of elastic waves by periodic arrays of spherical bodies. Phys. Rev. B. 62, 278 (2000)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  23. L. Brillouin, Wave Propagation in Periodic Structures (Dover, New York, 1953)

    Google Scholar 

  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)

    Article  Google Scholar 

  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. 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)

    Article  Google Scholar 

  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)

    Article  CAS  Google Scholar 

  28. D.C. Dobson, An efficient method for band structure calculations in 2D photonic crystals. J. Comput. Phys. 149, 363–376 (1999)

    Article  Google Scholar 

  29. S.G. Johnson, J.D. Joannopoulos, Photonic crystals: putting a new twist on light. Opt. Express 8, 173 (2001)

    Article  CAS  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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. 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. 238–239

    Google Scholar 

  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. 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 

Download references

Author information

Authors and Affiliations

  1. Institut d’Electronique, de Micro-électronique et de Nanotechnologie, UMR CNRS 8520, Cité Scientifique, 59652, Villeneuve d’Ascq Cedex, France

    J. O. Vasseur & A.-C. Hladky-Hennion

  2. Department of Materials Science and Engineering, University of Arizona, Tucson, AZ, 85721, USA

    Pierre A. Deymier & B. Merheb

  3. Laboratoire Domaines Océaniques, UMR CNRS 6538, UFR Sciences et Techniques, Université de Bretagne Occidentale, Brest, France

    A. Sukhovich

  4. Department of Aerospace Engineering Sciences, University of Colorado Boulder, Boulder, CO, 80309, USA

    M. I. Hussein

Authors
  1. J. O. Vasseur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pierre A. Deymier
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. A. Sukhovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. B. Merheb
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A.-C. Hladky-Hennion
    View author publications

    You can also search for this author in PubMed Google Scholar

  6. M. I. Hussein
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. O. Vasseur .

Editor information

Editors and Affiliations

  1. , Dept. of Materials Science and Engineeri, University of Arizona, East James E. Rogers Way 1235, Tucson, 85721-0012, Arizona, USA

    Pierre A. Deymier

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Vasseur, J.O., Deymier, P.A., Sukhovich, A., Merheb, B., Hladky-Hennion, AC., Hussein, M.I. (2013). Phononic Band Structures and Transmission Coefficients: Methods and Approaches. In: Deymier, P. (eds) Acoustic Metamaterials and Phononic Crystals. Springer Series in Solid-State Sciences, vol 173. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31232-8_10

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-31232-8_10

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-31231-1

  • Online ISBN: 978-3-642-31232-8

  • eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)

Publish with us

Policies and ethics

Search

Navigation