Bandgap calculation of two-dimensional mixed solid–fluid phononic crystals by Dirichlet-to-Neumann maps

A numerical method based on the Dirichlet-to-Neumann (DtN) map is presented to compute the bandgaps of two-dimensional phononic crystals, which are composed of square or triangular lattices of circular solid cylinders in a fluid matrix. The DtN map is constructed using the cylindrical wave expansion in a unit cell. A linear eigenvalue problem, which depends on the Bloch wave vector and involves relatively small matrices, is formulated. Numerical calculations are performed for typical systems with various acoustic impedance ratios of the solid inclusions and the fluid matrix. The results indicate that the DtN-map based method can provide accurate results for various systems efficiently. In particular it takes into account the fluid–solid interface conditions and the transverse wave mode in the solid component, which has been proven to be significant when the acoustic impedance of the solid inclusions is close to or smaller than that of the fluid matrix. For systems with an acoustic impedance of the inclusion much less than that of the matrix, physical flat bands appear in the band structures, which will be missed if the transverse wave mode in the solid inclusions is neglected.

[1]  Yih-Hsing Pao,et al.  Diffraction of elastic waves and dynamic stress concentrations , 1973 .

[2]  E. Yablonovitch,et al.  Inhibited spontaneous emission in solid-state physics and electronics. , 1987, Physical review letters.

[3]  Ping Sheng,et al.  Scattering And Localization Of Classical Waves In Random Media , 1990 .

[4]  J. Pendry,et al.  Calculation of photon dispersion relations. , 1992, Physical review letters.

[5]  Eleftherios N. Economou,et al.  Elastic and acoustic wave band structure , 1992 .

[6]  B. Djafari-Rouhani,et al.  Acoustic band structure of periodic elastic composites. , 1993, Physical review letters.

[7]  P. Halevi,et al.  Band‐gap engineering in periodic elastic composites , 1994 .

[8]  Steven G. Johnson,et al.  Photonic Crystals: Molding the Flow of Light , 1995 .

[9]  P. Halevi,et al.  Giant acoustic stop bands in two‐dimensional periodic arrays of liquid cylinders , 1996 .

[10]  Eleftherios N. Economou,et al.  Multiple-scattering theory for three-dimensional periodic acoustic composites , 1999 .

[11]  M. Sigalas,et al.  Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: The aluminum in mercury case , 2000 .

[12]  Zhengyou Liu,et al.  Point defect states in two-dimensional phononic crystals , 2001 .

[13]  Sylvain Ballandras,et al.  Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal , 2003 .

[14]  Jing Shi,et al.  Theory for elastic wave scattering by a two-dimensional periodical array of cylinders: An ideal approach for band-structure calculations , 2003 .

[15]  Z. Hou,et al.  Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals , 2004 .

[16]  Zhengyou Liu,et al.  The layer multiple-scattering method for calculating transmission coefficients of 2D phononic crystals , 2005 .

[17]  Yue-Sheng Wang,et al.  Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals , 2006 .

[18]  Ya Yan Lu,et al.  Photonic bandgap calculations with Dirichlet-to-Neumann maps. , 2006, Journal of the Optical Society of America. A, Optics, image science, and vision.

[19]  Ya Yan Lu,et al.  Computing photonic band structures by Dirichlet-to-Neumann maps: The triangular lattice , 2007 .

[20]  Chuanzeng Zhang,et al.  Wavelet Method for Calculating the Defect States of Two-Dimensional Phononic Crystals , 2008 .

[21]  Ya Yan Lu,et al.  Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps , 2008, J. Comput. Phys..

[22]  Yuesheng Wang,et al.  Application of Dirichlet-to-Neumann Map to Calculation of Band Gaps for Scalar Waves in Two-Dimensional Phononic Crystals , 2011 .