Band Structure Calculations by Modal Analysis

  • Mahmoud I. Hussein
Conference paper
Part of the IUTAM Bookseries book series (IUTAMBOOK, volume 26)


In this paper we present a new paradigm by which modal analysis – which is well established in engineering structural dynamics – is applied to band structure calculations for phononic crystals, or periodic media in general. Our method, which we refer to as reduced Bloch mode expansion (RBME), is essentially an expansion employing a natural basis composed of a selected reduced set of Bloch eigenfunctions. This reduced basis is selected within the Irreducible Brillouin Zone at high symmetry points determined by the crystal structure and group theory (and possibly at additional related points). At each of these high symmetry points, a number of Bloch eigenfunctions are selected up to the frequency range of interest for the band structure calculations. Since it is common to initially discretize the problem at hand using some choice of basis, reduced Bloch mode expansion constitutes a secondary expansion using a set of Bloch eigenvectors, and hence keeps and builds on any favorable attributes a primary expansion approach might exhibit. We report phonon band structure calculations by the proposed method showing up to two orders of magnitude reduction in computational effort with negligible loss in accuracy.


Photonic Crystal Modal Analysis Band Structure Calculation Phononic Crystal Irreducible Brillouin Zone 
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  1. .
    Brillouin, L. Wave Propagation in Periodic Structures. Dover, New York (1953).Google Scholar
  2. .
    Ho, K.M. et al.: Existence of a photonic gap in periodic dielectric structures. Phys. Rev. Lett. 65 3152-3155 (1990).CrossRefGoogle Scholar
  3. .
    Pendry, J.B., MacKinnon, A.: Calculation of photon dispersion-relations. Phys. Rev. Lett. 69 2772-2775 (1992).CrossRefGoogle Scholar
  4. .
    Yang, H.Y.D.: Finite difference analysis of 2-D photonic crystals. IEEE T. Microw. Theory 44 2688-2695 (1996).CrossRefGoogle Scholar
  5. .
    Dobson, D.C.: An efficient method for band structure calculations in 2D photonic crystals. J. Comput. Phys. 149 363-76 (1999).CrossRefGoogle Scholar
  6. .
    Axmann, W., Kuchment, P.: An efficient finite element method for computing spectra of photonic and acoustic band-gap materials - I. Scalar case. J. Comput. Phys. 150 468-481 (1999).CrossRefGoogle Scholar
  7. .
    Busch, K. et al.: Periodic nanostructures for photonics. Phys Reports 444 101-202 (2007).CrossRefGoogle Scholar
  8. .
    Burger, M. et al.: Inverse problem techniques for the design of photonic crystals. IEICE T. Electron. E87C 258-265 (2004).Google Scholar
  9. .
    Chern, R.L., Chang, C.C. and Hwang, R.R.: Large full band gaps for photonic crystals in two dimensions computed by an inverse method with multigrid acceleration. Phys. Rev. E 68 026704 (2003).Google Scholar
  10. 0.
    Johnson, S.G. and Joannopoulos, J.D.: Photonic crystals: Putting a new twist on light. Opt. Express 8 173 (2001).CrossRefGoogle Scholar
  11. 1.
    McDevitt, T.W., Hulbert, G.M. and Kikuchi, N.: An assumed strain method for the dispersive global-local modeling of periodic structures. Comput. Method. Appl. M. 190 6425 (2001).CrossRefGoogle Scholar
  12. 2.
    Nagai, G., Fish, J. and Watanabe, K.: Stabilized nonlocal model for dispersive wave propagation in heterogeneous media. Comput. Mech. 33 144 (2004).CrossRefGoogle Scholar
  13. 3.
    Hussein, M.I. 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
  14. 4.
    Hussein, M.I., Hulbert, G.M.: Mode-enriched dispersion models of periodic materials within a multiscale mixed finite element framework. Finite Elem. Anal. Des. 42 602-612 (2006).CrossRefGoogle Scholar
  15. 5.
    Døssing, O.: IMAC-XIII keynote address: Going beyond modal analysis, or IMAC in a new key. Modal Analysis. Int. J. Anal. Exp. Modal Analysis 10 69 (1995).Google Scholar
  16. 6.
    Hussein, M.I., Reduced Bloch mode expansion for periodic media band structure calculations. arXiv:0807.2612v4 (2008).Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2010

Authors and Affiliations

  1. 1.Department of Aerospace Engineering SciencesUniversity of Colorado at BoulderBoulderUSA

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