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Application of the generalized multipole technique in band structure calculation of two-dimensional solid/fluid phononic crystals. (English) Zbl 1334.74027

Summary: A multiple monopole method based on the generalized multipole technique is presented for the calculation of band structures of two-dimensional mixed solid/fluid phononic crystals. In this method, the fields are expanded by using the fundamental solutions with multiple origins. Besides the sources used to expand the wave fields, an extra monopole source is introduced as the external excitation. By varying the frequency of the excitation, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and sweeping the boundary of the irreducible first Brillouin zone, the band structure of the phononic crystals can be obtained. The method can consider the fluid-solid interface conditions and the transverse wave mode in the solid component strictly. Some typical examples are illustrated to discuss the accuracy of the present method.

MSC:

74E15 Crystalline structure
78A48 Composite media; random media in optics and electromagnetic theory
76A15 Liquid crystals
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[1] Ludwig, The generalized multipole technique, Computer Physics Communications 68 (1-3) pp 306– (1991) · doi:10.1016/0010-4655(91)90205-Y
[2] Ballisti, The multiple multipole method (MMP) in electro and magnetostatic problems, IEEE Transactions on Magnetics 19 (6) pp 2367– (1983) · Zbl 0626.65132 · doi:10.1109/TMAG.1983.1062871
[3] Hafner, The Generalized Multipole Technique Computational Electromagnetics (1990)
[4] Wriedt, Generalized Multipole Techniques for Electromagnetic and Light Scattering pp 5– (1999) · doi:10.1016/B978-044450282-7/50014-2
[5] Reutskiy, The method of fundamental solutions for Helmholtz eigenvalue problems in simply and multiply connected domains, Engineering Analysis with Boundary Elements 30 (3) pp 150– (2006) · Zbl 1195.65204 · doi:10.1016/j.enganabound.2005.08.011
[6] Chen, Eigensolutions of multiply connected membranes using the method of fundamental solutions, Engineering Analysis with Boundary Elements 29 (2) pp 166– (2005) · Zbl 1182.74249 · doi:10.1016/j.enganabound.2004.10.005
[7] Tsai, The method of fundamental solutions for eigenproblems in domains with and without interior holes, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Science 462 (2069) pp 1443– (2006) · Zbl 1149.35339 · doi:10.1098/rspa.2005.1626
[8] Mei, Multiple-scattering theory for out-of-plane propagation of elastic waves in two-dimensional phononic crystals, Journal of Physics: Condensed Matter 17 (25) pp 3735– (2005)
[9] Li, Bandgap calculation of two-dimensional mixed solid-fluid phononic crystals by Dirichlet-to-Neumann maps, Physica Scripta 84 pp 055402– (2011) · Zbl 1264.82143 · doi:10.1088/0031-8949/84/05/055402
[10] Zhen, Bandgap calculation for mixed in-plane waves in 2D phononic crystals based on Dirichlet-to-Neumann map, Acta Mechanica Sinica 28 pp 1143– (2012) · Zbl 1293.74060 · doi:10.1007/s10409-012-0092-9
[11] Eremin, The discrete source method for investigating three-dimensional electromagnetic scattering problems, Electromagnetics 13 (1) pp 1– (1993) · doi:10.1080/02726349308908326
[12] Anastassiu, Efficient preconditioning of the method of auxiliary sources (MAS) for cylindrical scatterers of quasi-circular cross-section, Open Electrical & Electronic Engineering Journal 2 pp 50– (2008) · doi:10.2174/1874129000802010050
[13] Leviatan, Analysis of electromagnetic scattering using arrays of fictitious sources, IEEE Transactions on Antennas and Propagation 43 (10) pp 1091– (1995) · doi:10.1109/8.467645
[14] Tayeb, Combined fictitious-sources-scattering-matrix method, Journal of the Optical Society of America A 21 (8) pp 1417– (2004) · doi:10.1364/JOSAA.21.001417
[15] Kawano, Numerical analysis of 3-D scattering problems using the Yasuura method, IEICE Transactions on Electronics 79 (10) pp 1358– (1996)
[16] Hafner, Application of the multiple multipole (MMP) method to electrodynamics, COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering 4 (3) pp 137– (1985) · Zbl 0626.65133 · doi:10.1108/eb010007
[17] Hafner, Multiple multipole program computation of periodic structures, Journal of the Optical Society of America A 12 (5) pp 1057– (1995) · doi:10.1364/JOSAA.12.001057
[18] Moreno, Band structure computations of metallic photonic crystals with the multiple multipole method, Physical Review B 65 (15) pp 155120– (2002) · doi:10.1103/PhysRevB.65.155120
[19] Smajic, Automatic calculation of band diagrams of photonic crystals using the multiple multipole method, Applied Computational Electromagnetics Society Journal 18 (3) pp 172– (2003)
[20] Imhof, Multiple multipole expansions for acoustic scattering, The Journal of the Acoustical Society of America 97 (2) pp 754– (1995) · doi:10.1121/1.412122
[21] Imhof, Acousto-elastic multiple scattering: a comparison of ultrasonic experiments with multiple multipole expansions, The Journal of the Acoustical Society of America 101 (4) pp 1836– (1997) · doi:10.1121/1.418221
[22] Ammari, Layer potential techniques in spectral analysis. II. Sensitivity analysis of spectral properties of high contrast band-gap materials, Multiscale Modeling & Simulation 5 (2) pp 646– (2006) · Zbl 1115.81072 · doi:10.1137/050646287
[23] Ammari, Asymptotic analysis of high-contrast phononic crystals and a criterion for the band-gap opening, Archive for Rational Mechanics and Analysis 193 (3) pp 679– (2009) · Zbl 1170.74023 · doi:10.1007/s00205-008-0179-4
[24] Ammari, Mathematical Surveys and Monographs, in: Layer Potential Techniques in Spectral Analysis (2009) · doi:10.1090/surv/153
[25] Shi, Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique, Applied Mathematics and Mechanics 34 (9) pp 1123– (2013) · doi:10.1007/s10483-013-1732-6
[26] Kushwaha, Acoustic band structure of periodic elastic composites, Physical Review Letters 71 (13) pp 2022– (1993) · doi:10.1103/PhysRevLett.71.2022
[27] Kushwaha, Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders, Applied Physics Letters 69 (1) pp 31– (1996) · doi:10.1063/1.118108
[28] Vasseur, Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide, Zeitschrift für Kristallographie 220 (9-10) pp 829– (2005) · doi:10.1524/zkri.2005.220.9-10.829
[29] Khelif, Guiding and bending of acoustic waves in highly confined phononic crystal waveguides, Applied Physics Letters 84 (22) pp 4400– (2004) · doi:10.1063/1.1757642
[30] Li, Bandgap calculations of two-dimensional solid-fluid phononic crystals with the boundary element method, Wave Motion 50 (3) pp 525– (2013) · doi:10.1016/j.wavemoti.2012.12.001
[31] Li, Dispersion relations of a periodic array of fluid-filled holes embedded in an elastic solid, Journal of Computational Acoustics 20 (4) pp 125004– (2012) · Zbl 1360.76266 · doi:10.1142/S0218396X12500142
[32] Yan, Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals, Physical Review B 74 (22) pp 224303– (2006) · doi:10.1103/PhysRevB.74.224303
[33] Sigalas, Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: the aluminum in mercury case, Applied Physics Letters 76 pp 2307– (2000) · doi:10.1063/1.126328
[34] Cao, Finite difference time domain method for band-structure calculations of two-dimensional phononic crystals, Solid State Communications 132 (8) pp 539– (2004) · doi:10.1016/j.ssc.2004.09.003
[35] Vekua, New Methods for Solving Elliptic Equations (1967)
[36] Bogdanov, Generalized Multipole Techniques for Electromagnetic and Light Scattering pp 143– (1999) · doi:10.1016/B978-044450282-7/50019-1
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