Band Structure Calculations by Modal Analysis
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.
KeywordsPhotonic Crystal Modal Analysis Band Structure Calculation Phononic Crystal Irreducible Brillouin Zone
Unable to display preview. Download preview PDF.
- .Brillouin, L. Wave Propagation in Periodic Structures. Dover, New York (1953).Google Scholar
- .Burger, M. et al.: Inverse problem techniques for the design of photonic crystals. IEICE T. Electron. E87C 258-265 (2004).Google Scholar
- .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
- 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
- 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
- 6.Hussein, M.I., Reduced Bloch mode expansion for periodic media band structure calculations. arXiv:0807.2612v4 (2008).Google Scholar