Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique

doi:10.1007/s10483-015-1938-7

Abstract

Abstract: A numerical method, the so-called multiple monopole (MMoP) method, based on the generalized multipole technique (GMT) is proposed to calculate the band structures of in-plane waves in two-dimensional phononic crystals, which are composed of arbitrarily shaped cylinders embedded in a solid host medium. To find the eigenvalues (eigenfrequencies) of the problem, besides the sources used to expand the wave fields, an extra monopole source is introduced which acts as the external excitation. By varying the excitation frequency, the eigenvalues can be localized as the extreme points of an appropriately chosen function. By sweeping the frequency range of interest and the boundary of the irreducible first Brillouin zone (FBZ), the band structures can be obtained. Some typical numerical examples with different acoustic impedance ratios and with inclusions of various shapes are presented to validate the proposed method.

Key words: phononic crystal, generalized multipole technique (GMT), eigenvalue problem, multiple monopole (MMoP) method, fluid-solid interaction condition, band structure

2010 MSC Number:

O175.9
O735

Zhijie SHI, Yuesheng WANG, Chuanzeng ZHANG. Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique. Applied Mathematics and Mechanics (English Edition), 2015, 36(5): 557-580.

TrendMD

References

[1] Kushwaha, M. S., Halevi, P., Martinez, G., Dobrzynski, L., and Djafarirouhani, B. Acoustic band structure of periodic elastic composites. Physical Review Letters, 71, 2022-2025 (1993)
[2] Kushwaha, M. S. and Halevi, P. Giant acoustic stop bands in two-dimensional periodic arrays of liquid cylinders. Applied Physics Letters, 69, 31-33 (1996)
[3] Yuan, J. H., Lu, Y. Y., and Antoine, X. Modeling photonic crystals by boundary integral equations and Dirichlet-to-Neumann maps. Journal of Computational Physics, 227, 4617-4629 (2008)
[4] Tanaka, Y., Tomoyasu, Y., and Tamura, S. Band structure of acoustic waves in phononic lattices: two-dimensional composites with large acoustic mismatch. Physical Review B, 62, 7387 (2000)
[5] Sainidou, R., Stefanou, N., and Modinos, A. Widening of phononic transmission gaps via Anderson localization. Physical Review Letters, 94, 205503 (2005)
[6] Zhao, Y. C., Zhao, F., and Yuan, L. B. Numerical analysis of acoustic band gaps in twodimensional periodic materials. Journal of Marine Science and Application, 4, 65-69 (2005)
[7] Li, F. L. and Wang,Y. S. Application of Dirichlet-to-Neumann map to calculation of band gaps for scalar waves in two-dimensional phononic crystals. Acta Acustica United with Acustica, 97, 284-290 (2011)
[8] Yan, Z. Z. and Wang, Y. S. Wavelet-based method for calculating elastic band gaps of twodimensional phononic crystals. Physical Review B, 74, 224303 (2006)
[9] Yan, Z. Z., Zhang, C., and Wang, Y. S. Wave propagation and localization in randomly disordered layered composites with local resonances. Wave Motion, 47, 409-420 (2010)
[10] Vasseur, J. O., Deymier, P. A., Beaugeois, M., Pennec, Y., Djafari-Rouhani, B., and Prevost, D. Experimental observation of resonant filtering in a two-dimensional phononic crystal waveguide. Zeitschrift für Kristallographie, 220, 829-835 (2005)
[11] Rupp, C. J., Evgrafov, A., Maute, K., and Dunn, M. L. Design of phononic materials/structures for surface wave devices using topology optimization. Structural and Multidisciplinary Optimization, 34, 111-121 (2007)
[12] Sigalas, M. M. and Economou, E. N. Elastic and acoustic wave band structure. Journal of Sound and Vibration, 158, 377-382 (1992)
[13] Wu, T. T., Huang, Z. G., and Lin, S. Surface and bulk acoustic waves in two-dimensional phononic crystal consisting of materials with general anisotropy. Physical Review B, 69, 094301 (2004)
[14] Kafesaki, M. and Economou, E. N. Multiple scattering theory for 3D periodic acoustic composites. Physical Review B, 60, 11993-12001 (1999)
[15] Mei, J., Liu, Z. Y., Shi, J., and Tian, D. Theory for elastic wave scattering by a two dimensional periodical array of cylinders: an ideal approach for band-structure calculations. Physical Review B, 67, 245107 (2003)
[16] Zhen, N., Li, F. L., Wang, Y. S., and Zhang, C. Bandgap calculation for plane mixed waves in 2D phononic crystals based on Dirichlte-to-Neumann map. Acta Mechanica Sinica, 28, 1143-1153 (2012)
[17] Zhen, N., Wang, Y. S., and Zhang, C. Surface/interface effect on band structures of nanosized phononic crystals. Mechanics Research Communications, 46, 81-89 (2012)
[18] Zhen, N.,Wang, Y. S., and Zhang, C. Bandgap calculation of in-plane waves in nanoscale phononic crystals taking account of surface/interface effects. Physics E, 54, 125-132 (2013)
[19] Axmann, W. and Kuchment, P. An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: I, scalar case. Journal of Computational Physics, 150, 468-481 (1999)
[20] Li, J. B., Wang, Y. S., and Zhang, C. Dispersion relations of a periodic array of fluid-filled holes embedded in an elastic solid. Journal of Computational Acoustics, 20, 1250014 (2012)
[21] Wang, G., Wen, J., Liu, Y., and Wen, X. Lumped-mass method for the study of band structure in two-dimensional phononic crystals. Physical Review B, 69, 184302 (2004)
[22] Li, F. L., Wang, Y. S., Zhang, C., and Yu, G. L. Boundary element method for bandgap calculations of two-dimensional solid phononic crystals. Engineering Analysis with Boundary Elements, 37, 225-235 (2013)
[23] Li, F. L., Wang, Y. S., Zhang, C., and Yu, G. L. Bandgap calculations of two-dimensional solidfluid phononic crystals with the boundary element method. Wave Motion, 50, 525-541 (2013)
[24] Sigalas, M. M. and García, N. Importance of coupling between longitudinal and transverse components for the creation of acoustic band gaps: the aluminum in mercury case. Applied Physics Letters, 76, 2307-2309 (2000)
[25] Wu, Z. J., Wang, Y. Z., and Li, F. M. Analysis on band gap properties of periodic structures of bar system using the spectral element method. Waves in Random and Complex Media, 23, 349-372 (2013)
[26] Wu, Z. J., Li, F. M., andWang, Y. Z. Study on vibration characteristics in periodic plate structures using the spectral element method. Acta Mechanica, 224, 1089-1101 (2013)
[27] Shi, Z. J., Wang, Y. S., and Zhang, C. Z. Band structure calculation of scalar waves in twodimensional phononic crystals based on generalized multipole technique. Applied Mathematics and Mechanics (English Edition), 34, 1123-1144 (2013) DOI 10.1007/s10483-013-1732-6
[28] Leviatan, Y. and Boag, A. Analysis of electromagnetic scattering from dielectric cylinders using a multifilament current model. IEEE Transactions on Antennas and Propagation, 35, 1119-1127 (1987)
[29] Hafner, C. The Generalized Multipole Technique for Computational Electromagnetics, Artech House, Boston, 157-266 (1990)
[30] Reutskiy, S. Y. The method of fundamental solutions for Helmholtz eigenvalue problems in simply and multiply connected domains. Engineering Analysis with Boundary Elements, 30, 150-159 (2006)
[31] Pao, Y. H. and Mao, C. C. Diffraction of Elastic Waves and Dynamic Stress Concentration, Adam Hilger, U.K. (1973)
[32] Hafner, C. Post-Modern Electromagnetics: Using Intelligent Maxwell Solvers, John Wiley and Sons, New York (1999)
[33] Tayeb, G. and Enoch, S. Combined fictious sources-scattering-matrix method. Journal of the Optical Society of America A, 21, 1417-1423 (2004)
[34] Moreno, E., Erni, D., and Hafner, C. Band structure computations of metallic photonic crystals with the multiple multipole method. Physical Review B, 65, 155120 (2002)
[35] Smajic, J., Hafner, C., and Erni, D. Automatic calculation of band diagrams of photonic crystals using the multiple multipole method. Applied Computational Electromagnetics Society Journal, 18, 172-180 (2003)
[36] Bogdanov, E. G., Karkashadze, D. D., and Zaridze, R. S. The method of auxiliary sources in electromagnetic scattering problems. Generalized Multipole Techniques for Electromagnetic and Light Scattering (ed. Wriedt, T.), Elsevier, Amsterdam, 143-172 (1999)
[37] Goffaux, C. and Vigneron, J. P. Theoretical study of a tunable phononic band gap system. Physical Review B, 64, 075118 (2001)