Analysis of Band Structure for 2D Acoustic PhononicStructure by BEM and the Block SS Method

By its very basis, the block Sakurai-Sugiura (SS) method may be ap- plied to solving nonlinear eigenvalue problem formulated by boundary integral equation. In this paper, the methodology of BEM combined with the block SS method is applied to the computation of band structures of phononic structures for acoustic problems. Since the conventional boundary integral equation yields the real fictitious eigenfrequencies when it is applied to an exterior problem of scatters, Burton-Miller's method is employed to remove the real fictitious eigenfrequen- cies resulted by the boundary of the scatterers. The application of the method is demonstrated through analyses of band structures of homogenous and composite structures.

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

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

[3]  Mitsuhisa Sato,et al.  A parallel method for large sparse generalized eigenvalue problems using a GridRPC system , 2008, Future Gener. Comput. Syst..

[4]  Mihail M. Sigalas,et al.  Defect states of acoustic waves in a two-dimensional lattice of solid cylinders , 1998 .

[5]  T. Reinecke,et al.  Boundary-element calculations of electromagnetic band-structure of photonic crystals , 1998 .

[6]  Toru Takahashi,et al.  Three Dimensional Acoustic Shape Sensitivity Analysis by Means of Adjoint Variable Method and Fast Multipole Boundary Element Approach , 2011 .

[7]  Masato Kobayashi,et al.  Application of the Sakurai‐Sugiura projection method to core‐excited‐state calculation by time‐dependent density functional theory , 2008, J. Comput. Chem..

[8]  R. Martínez-Sala,et al.  Sound attenuation by sculpture , 1995, Nature.

[9]  S. Atluri,et al.  Directly Derived Non-Hyper-Singular Boundary Integral Equations for Acoustic Problems, and Their Solution through Petrov-Galerkin Schemes , 2004 .

[10]  Takayuki Yamada,et al.  Eigenvalue analysis for 2D acoustic problem by BEM with block SS method , 2011 .

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

[12]  A. Modinos,et al.  Scattering of elastic waves by periodic arrays of spherical bodies , 2000 .

[13]  Han Xiao-Yun,et al.  Finite difference time domain method for the study of band gap in two-dimensiona l phononic crystals , 2003 .

[14]  G. F. Miller,et al.  The application of integral equation methods to the numerical solution of some exterior boundary-value problems , 1971, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[15]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[16]  P A Deymier,et al.  Phononic crystal with low filling fraction and absolute acoustic band gap in the audible frequency range: a theoretical and experimental study. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Yukihiro Tanaka,et al.  Band structure of acoustic waves in phononic lattices: Two-dimensional composites with large acoustic mismatch , 2000 .

[18]  Tetsuya Sakurai,et al.  A filter diagonalization for generalized eigenvalue problems based on the Sakurai-Sugiura projection method , 2008, J. Comput. Appl. Math..

[19]  T. Sakurai,et al.  A projection method for generalized eigenvalue problems using numerical integration , 2003 .

[20]  Abdelkrim Khelif,et al.  Transmission and dispersion relations of perfect and defect-containing waveguide structures in phononic band gap materials , 2003 .

[21]  Hiroto Tadano,et al.  A numerical method for nonlinear eigenvalue problems using contour integrals , 2009, JSIAM Lett..

[22]  Leslie Greengard,et al.  A new integral representation for quasi-periodic fields and its application to two-dimensional band structure calculations , 2010, J. Comput. Phys..

[23]  Non-Hyper-Singular Boundary Integral Equations for Acoustic Problems, Implemented by the Collocation-Based Boundary Element Method , 2004 .

[24]  Yuesheng Wang,et al.  Boundary element method for bandgap computation of photonic crystals , 2012 .

[25]  M. Kushwaha.  Stop-bands for periodic metallic rods: Sculptures that can filter the noise , 1997 .

[26]  Eleftherios N. Economou,et al.  Band structure of elastic waves in two dimensional systems , 1993 .

[27]  M. Makela,et al.  Phononic band structure in a mass chain , 1995 .

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