Archives of Acoustics, 42, 4, pp. 735–742, 2017

Using PWE/FE Method to Calculate the Band Structures of the Semi-Infinite PCs: Periodic in x-y Plane and Finite in z-direction

Denghui QIAN
Nanjing University of Aeronautics and Astronautics

Zhiyu SHI
Nanjing University of Aeronautics and Astronautics

This paper introduces the concept of semi-infinite phononic crystal (PC) on account of the infinite periodicity in x-y plane and finiteness in z-direction. The plane wave expansion and finite element methods are coupled and formulized to calculate the band structures of the proposed periodic elastic composite structures based on the typical geometric properties. First, the coupled plane wave expansion and finite element (PWE/FE) method is applied to calculate the band structures of the Pb/rubber, steel/epoxy and steel/aluminum semi-infinite PCs with cylindrical scatters. Then, it is used to calculate the band structure of the Pb/rubber semi-infinite PC with cubic scatter. Last, the band structure of the rubbercoated Pb/epoxy three-component semi-infinite PC is calculated by the proposed method. Besides, all the results are compared with those calculated by the finite element (FE) method implemented by adopting COMSOL Multiphysics. Numerical results and further analysis demonstrate that the proposed PWE/FE method has strong applicability and high accuracy.
Keywords: semi-infinite phononic crystal; coupled plane wave expansion and finite element method; band structure
Full Text: PDF


Åberg M., Gudmundson P. (1997), The usage of standard finite element codes for computation of dispersion relations in materials with periodic microstructure, Journal of the Acoustical Society of America, 102, 4, 2007–2013.

Benchabane S., Khelif A., Rauch J.Y., et al. (2006), Evidence for complete surface wave band gap in a piezoelectric phononic crystal, Physical Review E: Statistical Nonlinear & Soft Matter Physics, 73, 2, 95–104.

Cao Y., Hou Z., Liu Y. (2004a), Convergence problem of plane-wave expansion method for phononic crystals, Physics Letters A, 327, 2–3, 247–253.

Cao Y., Hou Z., Liu Y. (2004b), Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals, Solid State Communications, 132, 8, 539–543.

Eslami M.R., Hetnarski R.B., Ignaczak J., Noda N., Sumi N., Tanigawa Y. (2013), Variational Principles of Elastodynamics. In: Theory of Elasticity and Thermal Stresses. Solid Mechanics and Its Applications, Vol. 197, pp. 127–149. Springer, Dordrecht.

Hirsekorn M., Delsanto P.P., Batra N.K., et al. (2004), Modelling and simulation of acoustic wave propagation in locally resonant sonic materials, Ultrasonics, 42, 1, 231–235.

Hou Z., Fu X., Liu Y. (2004), Calculational method to study the transmission properties of phononic crystals, Physical Review B: Condensed Matter, 70, 1, 2199–2208.

Hsu J.C., Wu T.T. (2006), Efficient formulation for band-structure calculations of two-dimensional phononic-crystal plates, Physical Review B: Condensed Matter, 74, 74, 2952–2961.

Hsu J.C., Wu T.T. (2007), Lamb waves in binary locally resonant phononic plates with two-dimensional lattices, Applied Physics Letters, 90, 20, 201904–201904-3.

Kushwaha M.S, Halevi P. (1997), Stop bands for cubic arrays of spherical balloons, Journal of the Acoustical Society of America, 101, 1, 619–622.

Kushwaha M.S., Halevi P., Martínez G., Dobrzynski L., Djafari-Rouhani B. (1994), Theory of acoustic band structure of periodic elastic composites, Physical Review B: Condensed Matter, 49, 4, 2313–2322.

Li S., Chen T., Wang X., Li Y., Chen W. (2016), Expansion of lower-frequency locally resonant band gaps using a double-sided stubbed composite phononic crystals plate with composite stubs, Physics Letters A, 380, 25–26, 2167–2172.

Li Y., Chen T., Wang X., Xi Y., Liang Q. (2015), Enlargement of locally resonant sonic band gap by using composite plate-type acoustic metamaterial, Physics Letters A, 379, 5, 412–416.

Liu Z., Chan C.T., Sheng P., Goertzen A.L., Page J.H. (2000a), Elastic wave scattering by periodic structures of spherical objects: Theory and experiment, Physical Review B, 62, 4, 2446–2457.

Liu Z., Zhang X., Mao Y., et al. (2000b), Locally resonant sonic materials, Science, 289, 5485, 1734–1736.

Ma J., Hou Z., Assouar B.M. (2014), Opening a large full phononic band gap in thin elastic plate with resonant units, Journal of Applied Physics, 115, 9, 093508–093508-5.

Mei J., Liu Z., Shi J., Decheng T. (2003), Theory for elastic wave scattering by a two-dimensional periodical array of cylinders: An ideal approach for band-structure calculations, Physical Review B, 67, 24, 841–845.

Mohammadi S., Eftekhar A.A., Hunt W.D., Adibi A. (2009), High-Q micromechanical resonators in a two-dimensional phononic crystal slab, Applied Physics Letters, 94, 5, 051906–051906-3.

Orris R.M., Petyt M. (1974), A finite element study of harmonic wave propagation in periodic structures, Journal of Sound & Vibration, 33, 2, 223–236.

Oudich M., Li Y., Assouar B.M, Hou Z. (2010), A sonic band gap based on the locally resonant phononic plates with stubs, New Journal of Physics, 12, 2, 201–206.

Qian D., Shi Z. (2016), Bandgap properties in locally resonant phononic crystal double panel structures with periodically attached spring-mass resonators, Physics Letters A, 380, 41, 3319–3325.

Qian D., Shi Z. (2017a), Bandgap properties in simplified model of composite locally resonant phononic crystal plate, Physics Letters A, 381, 40, 3505–3513.

Qian D., Shi Z. (2017b), Using PWE/FE method to calculate the band structures of the semi-infinite beam-like PCs: periodic in z-direction and finite in x-y plane, Physics Letters A, 381, 17, 1516–1524.

Sigalas M., Economou E.N. (1993), Band structure of elastic waves in two dimensional systems, [J]. Solid State Communications, 86, 3, 141–143.

Sigalas M., Kushwaha M.S., Economou E.N., Kafesaki M., Psarobas I.E., Steurer W. (2005), Classical vibrational modes in phononic lattices: theory and experiment, Zeitschrift für Kristallographie – Crystalline Materials, 220, 9, 765–809.

Sigalas M.M., Economou E.N. (1992), Elastic and acoustic wave band structure, Journal of Sound and Vibration, 158, 2, 377–382.

Sigalas M.M., Garcıa N. (2000), Theoretical study of three dimensional elastic band gaps with the finite-difference time-domain method, Journal of Applied Physics, 87, 6, 3122–3125.

Wang G., Wen J., Liu Y., Wen X. (2004), Lumped-mass method for the study of band structure in two-dimensional phononic crystals, Physical Review B, 69, 18, 1324–1332.

Wang G., Wen J., Wen X. (2005), Quasi-one-dimensional phononic crystals studied using the improved lumped-mass method: Application to locally resonant beams with flexural wave band gap, Physical Review B, 71, 10, 4302.

Wang G., Wen X., Wen J., Liu Y. (2006), Quasi-One-Dimensional Periodic Structure with Locally Resonant Band Gap, Journal of Applied Mechanics, 43, 1, 167–170.

Wu T., Wu L.C., Huang Z.G. (2005), Frequency band-gap measurement of two-dimensional air/silicon phononic crystals using layered slanted finger interdigital transducers, Journal of Applied Physics, 97, 9, 094916–094916-7.

Xiao W, Zeng G.W., Cheng Y.S. (2008), Flexural vibration band gaps in a thin plate containing a periodic array of hemmed discs, Applied Acoustics, 69, 3, 255–261.

Xiao Y., Wen J., Wen X. (2012), Flexural wave band gaps in locally resonant thin plates with periodically attached spring–mass resonators, Journal of Physics D: Applied Physics, 45, 19, 195401–195412(12).

Yan P., Vasseur J.O., Djafari-Rouhani B., Dobrzyński L., Deymier P.A. (2010a), Two-dimensional phononic crystals: Examples and applications, Surface Science Reports, 65, 8, 229–291.

Yan Z.Z., Zhang C., Wang Y.S. (2010b), Wave propagation and localization in randomly disordered layered composites with local resonances, Wave Motion, 47, 7, 409–420.

Yu D., Liu Y., Wang G., Zhao H, Qiu J. (2006), Flexural vibration band gaps in Timoshenko beams with locally resonant structures, Journal of Applied Physics, 100, 12, 124901–124901-5.

Zhang X., Liu Z., Liu Y., Wu F. (2003), Elastic wave band gaps for three-dimensional phononic crystals with two structural units, Physics Letters A, 313, 5, 455–460.

Zhao H.J., Guo H.W., Gao M.X, Liu R.Q., Deng Z.Q. (2016), Vibration band gaps in double-vibrator pillared phononic crystal plate, Journal of Applied Physics, 119, 1, 377.

DOI: 10.1515/aoa-2017-0076

Copyright © Polish Academy of Sciences & Institute of Fundamental Technological Research (IPPT PAN)