Skip to main content
SpringerLink
Log in

Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique

  • Published:
Applied Mathematics and Mechanics Aims and scope Submit manuscript

Abstract

A multiple monopole (or multipole) method based on the generalized multipole technique (GMT) is proposed to calculate the band structures of scalar waves in two-dimensional phononic crystals which are composed of arbitrarily shaped cylinders embedded in a host medium. In order to find the eigenvalues of the problem, besides the sources used to expand the wave field, an extra monopole source is introduced which acts 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 is obtained. Some numerical examples are presented to validate the proposed method.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

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)

    Article  Google Scholar 

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

    Article  MATH  Google Scholar 

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

    Article  Google Scholar 

  4. 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  6. Wang, G., Wen, J. H., and Han, X. Y. Finite difference time domain method for the study of band gap in two-dimensional phononic crystals (in Chinese). Physics Letters, 52, 1943–1947 (2003)

    Google Scholar 

  7. Yan, Z. Z. and Wang, Y. S. Wavelet-based method for calculating elastic band gaps of two-dimensional phononic crystals. Physical Review B, 74, 224303 (2006)

    Article  Google Scholar 

  8. Yan, Z. Z. and Wang, Y. S. Wavelet-based method for computing elastic band gaps of one-dimensional phononic crystals. Science in China Series G: Physics, Mechanics and Astronomy, 50, 622–630 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Yan, Z. Z., Wang, Y. S., and Zhang, C. Z. Wavelet method for calculating the defect states of two-dimensional phononic crystals. Acta Mechanica Solida Sinica, 21, 104–109 (2008)

    Article  Google Scholar 

  10. Yan, Z. Z., Wang, Y. S., and Zhang, C. Z. A method based on wavelets for band structure analysis of phononic crystals. Computer Modeling in Engineering and Sciences, 38, 59–87 (2008)

    MathSciNet  Google Scholar 

  11. Kafesaki, M. and Economou, E. N. Multiple-scattering theory for three-dimensional periodic acoustic composites. Physical Review B, 60, 11993–12001 (1999)

    Article  Google Scholar 

  12. Mei, J., Qiu, C., and Liu, Z. Multiple-scattering theory for out-of-plane propagation of elastic waves in two-dimensional phononic crystals. Journal of Physics: Condensed Matter, 17, 3735–3757 (2005)

    Article  Google Scholar 

  13. Axmann, W. and Kuchment, P. An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: scalar case. Journal of Computational Physics, 150, 468–481 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, J. B., Wang, Y. S., and Zhang, C. Z. Dispersion relations of a periodic array of fluid-filled holes embedded in an elastic solid. Journal of Computational Acoustics, 20, 1250014 (2012)

    Article  MathSciNet  Google Scholar 

  15. Wang, Y. F., Wang, Y. S., and Su, X. X. Large band gaps of two-dimensional phononic crystals with cross-like holes. Journal of Applied Physics, 110, 113520 (2011)

    Article  Google Scholar 

  16. Li, F. L., Wang, Y. S., Zhang, C. Z., 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)

    Article  MathSciNet  Google Scholar 

  17. Wu, Y. M. and Lu, Y. Y. Dirichlet-to-Neumann map method for analyzing interpenetrating cylinder arrays in a triangular lattice. Journal of the Optical Society of America B, 25, 1466–1473 (2008)

    Article  Google Scholar 

  18. 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)

    Article  Google Scholar 

  19. Li, F. L., Wang, Y. S., and Zhang, C. Z. Bandgap calculation of two-dimensional mixed solid-fluid phononic crystals by Dirichlet-to-Neumann maps. Physica Scripta, 84, 055402 (2011)

    Article  Google Scholar 

  20. Zhen, N., Li, F. L., Wang, Y. S., and Zhang, C. Z. Bandgap calculation for plane mixed waves in 2D phononic crystals based on Dirichlte-to-Neumann map. Acta Mechanica Sinica, 28, 1143–1153 (2012)

    Article  MathSciNet  Google Scholar 

  21. Hou, X. H., Deng, Z. C., and Zhou, J. X. Symplectic analysis for wave propagation in one dimensional nonlinear periodic structures. Applied Mathematics and Mechanics (English Edition), 31(11), 1371–1382 (2010) DOI 10.1007/s10483-010-1369-7

    Article  MathSciNet  MATH  Google Scholar 

  22. Hou, X. H., Deng, Z. C., Zhou, J. X., and Liu, T. Q. Symplectic analysis for elastic wave propagation in two-dimensional cellular structures. Acta Mechanica Sinica, 26, 711–720 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  23. Leviatan, Y., Li, P. G., Adams, A. J., and Perini, J. Single-post inductive obstacle in rectangular waveguide. IEEE Transactions on Microwave Theory and Techniques, MTT-31, 1806–1812 (1983)

    Google Scholar 

  24. 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)

    Article  Google Scholar 

  25. Leviatan, Y. Analytic continuation considerations when using generalized formulations for scattering problems. IEEE Transactions on Antennas and Propagation, 38, 1259–1263 (1990)

    Article  Google Scholar 

  26. Ludwig, A. The generalized multipole technique. Computer Physics Communications, 68, 306–314 (1991)

    Article  Google Scholar 

  27. Ballisti, R. and Hafner, C. The multiple multipole method (MMP) in electro and magnetostatic problems. IEEE Transactions on Magnetics, 19, 2367–2370 (1983)

    Article  Google Scholar 

  28. Hafner, C. and Klaus, G. Application of the multiple multipole (MMP) method to electrodynamics. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 4, 137–144 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hafner, C. and Ballisti, R. The multiple multipole (MMP) method. COMPEL: The International Journal for Computation and Mathematics in Electrical and Electronic Engineering, 2, 1–7 (1983)

    Article  MATH  Google Scholar 

  30. Hafner, C. The Generalized Multipole Technique for Computational Electromagnetics, Artech House, Boston, 157–266 (1990)

    Google Scholar 

  31. Eremin, Y. A., Orlov, N. V., and Sveshnikov, A. G. Models of electromagnetic scattering problems based on discrete sourcesmethod. Generalized Multipole Techniques for Electromagnetic and Light Scattering (ed. Wriedt, T.), Elsevier, Amsterdam, 39–80 (1999)

    Chapter  Google Scholar 

  32. Cheng, C. H. The Generalized Multipole Technique in Spectral Domain for the Underground Electromagnetic Scattering Problems (in Chinese), Ph.D. dissertation, Southeast University, Nanjing (1993)

    Google Scholar 

  33. Fairweather, G. and Karageorghis, A. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, 9, 69–95 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  34. Karageorghis, A. The method of fundamental solutions for the calculation of the eigenvalues of the Helmholtz equation. Applied Mathematics Letters, 14, 837–842 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Chen, J. T., Chen, I. L., and Lee, Y. T. Eigensolutions of multiply connected membranes using the method of fundamental solutions. Engineering Analysis with Boundary Elements, 29, 166–174 (2005)

    Article  MATH  Google Scholar 

  36. Reutskiy, S. Y. The method of fundamental solutions for eigenproblems with Laplace and biharmonic operators. Computers, Materials and Continua, 12, 177–188 (2005)

    Google Scholar 

  37. 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)

    Article  MATH  Google Scholar 

  38. Tsai, C. C., Young, D. L., Chen, C. W., and Fan, C. M. The method of fundamental solutions for eigenproblems in domains with and without interior holes. Proceedings of The Royal Society A: Mathematical, Physical and Engineering Sciences, 462, 1443–1466 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  39. Reutskiy, S. Y. The method of external sources (MES) for eigenvalue problems with Helmholtz equation. Computer Modeling in Engineering and Sciences, 12, 27–39 (2006)

    MATH  Google Scholar 

  40. Reutskiy, S. Y. The methods of external and internal excitation for problems of free vibrations of non-homogeneous membranes. Engineering Analysis with Boundary Elements, 31, 906–918 (2007)

    Article  MATH  Google Scholar 

  41. Tayeb, G. The method of fictitious sources applied to diffraction gratings. Applied Computational Electromagnetics Society Journal, 9, 90–100 (1994)

    Google Scholar 

  42. Tayeb, G. and Enoch, S. Combined fictious sources-scattering-matrix method. Journal of the Optical Society of America A, 21, 1417–1423 (2004)

    Article  Google Scholar 

  43. 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)

    Chapter  Google Scholar 

  44. Kaklamani, D. I. and Anastassiu, H. T. Aspects of the method of auxiliary sources (MAS) in computational electromagnetics. IEEE Antennas and Propagation Magazine, 44, 48–64 (2002)

    Article  Google Scholar 

  45. Hafner, C. Post-modern Electromagnetics: Using Intelligent Maxwell Solvers, John Wiley and Sons, New York (1999)

    Google Scholar 

  46. 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)

    Article  Google Scholar 

  47. Moreno, E., Erni, D., and Hafner, C. Modeling of discontinuities in photonic crystal waveguides with the multiple multipole method. Physical Review E, 66, 036618 (2002)

    Article  Google Scholar 

  48. 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)

    Google Scholar 

  49. Hafner, C. MMP computation of periodic structures. Journal of the Optical Society of America, 12, 1057–1067 (1995)

    Article  Google Scholar 

  50. Imhof, M. G. Multiple multipole expansions for acoustic scattering. Journal of the Acoustical Society of America, 97, 754–763 (1995)

    Article  MathSciNet  Google Scholar 

  51. Imhof, M. G. Multiple multipole expansions for elastic scattering. Journal of the Acoustical Society of America, 100, 2969–2979 (1996)

    Article  Google Scholar 

  52. Chen, J. T., Huang, C. X., and Chen, K. H. Determination of spurious and multiplicities of true eigenvalues using the real-part dual BEM. Computational Mechanics, 24, 41–51 (1999)

    Article  MATH  Google Scholar 

  53. Burton, A. J. and Miller, G. F. The application of integral equation methods to the numerical solution of some exterior boundary-value problem. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 323, 201–210 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  54. Vekua, I. N. New Methods for Solving Elliptic Equations, North-Holland Publishing Company, Amsterdam (1967)

    MATH  Google Scholar 

  55. Hafner, C. and Smajic, J. Efficient and accurate boundary methods for computational optics. Lecture Note Series of the IMS, NUS, World Scientific Publishing Company, Singapore, 1–76 (2005)

    Google Scholar 

  56. Gurtin, M. E. and Murdoch, A. I. A continuum theory of elastic material surfaces. Archive for Rational Mechanics and Analysis, 57, 291–323 (1975)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institute of Engineering Mechanics, Beijing Jiaotong University, Beijing, 100044, P. R. China

    Zhi-jie Shi  (史志杰) & Yue-sheng Wang  (汪越胜)

  2. Department of Civil Engineering, University of Siegen, D-57068, Siegen, Germany

    Chuan-zeng Zhang  (张传增)

Authors
  1. Zhi-jie Shi  (史志杰)
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yue-sheng Wang  (汪越胜)
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Chuan-zeng Zhang  (张传增)
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yue-sheng Wang  (汪越胜).

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 51178037 and 10632020), the German Research Foundation (DFG) (Nos. ZH 15/11-1 and ZH 15/16-1), the International Bureau of the German Federal Ministry of Education and Research (BMBF) (No. CHN 11/045), and the National Basic Research Program of China (No. 2010CB732104)

Rights and permissions

Reprints and permissions

About this article

Cite this article

Shi, Zj., Wang, Ys. & Zhang, Cz. Band structure calculation of scalar waves in two-dimensional phononic crystals based on generalized multipole technique. Appl. Math. Mech.-Engl. Ed. 34, 1123–1144 (2013). https://doi.org/10.1007/s10483-013-1732-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10483-013-1732-6

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

Search

Navigation