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RBF collocation method and stability analysis for phononic crystals

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Abstract

The mesh-free radial basis function (RBF) collocation method is explored to calculate band structures of periodic composite structures. The inverse multi-quadric (MQ), Gaussian, and MQ RBFs are used to test the stability of the RBF collocation method in periodic structures. Much useful information is obtained. Due to the merits of the RBF collocation method, the general form in this paper can easily be applied in the high dimensional problems analysis. The stability is fully discussed with different RBFs. The choice of the shape parameter and the effects of the knot number are presented.

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References

  1. Kushwaha, M. S., Halevi, P., Dobrzynski, L., and Djafari-Rouhani, B. Acoustic band structure of periodic elastic composites. Physical Review Letters, 71, 2022–2025 (1993)

    Article  Google Scholar 

  2. Sigalas, M. M. Elastic wave band gap and defect states in two-dimensional composites. Journal of the Acoustical Society of America, 101, 1256–1261 (1997)

    Article  Google Scholar 

  3. Ni, Z. Q., Zhang, Z. M., Han, L., and Zhang, Y. Study on the convergence of plane wave expansion method in calculation the band structure of one dimensional typical phononic crystal. Opoelectronics and Advanced Materials—Rapid Communications, 6, 87–90 (2012)

    Google Scholar 

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

  5. Liu, Z. Y., Chan, C. T., Sheng, P., Goertzen, A. L., and Page, J. H. Elastic wave scattering by periodic structures of spherical objects: theory and experiment. Physical Review B, 62, 2446–2457 (2000)

    Article  Google Scholar 

  6. 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)

    Article  Google Scholar 

  7. Yan, Z. Z., Wang, Y. S., and Zhang, C. Z. Wavelet method for calculating the defect states of two-dimensional phononic crystals. CMES-Computer Modeling in Engineering and Sciences, 38, 59–87 (2008)

    MathSciNet  Google Scholar 

  8. 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, 7387–7392 (2000)

    Article  Google Scholar 

  9. Khelif, A., Djafari-Rouhani, B., Vasseur, J. O., Deymier, P. A., Lambin, P., and Dobrzynski, L. Transmittivity through straight and stublike waveguides in a two-dimensional phononic crystal. Physical Review B, 65, 174308 (2002)

    Article  Google Scholar 

  10. Khelif, A., Choujaa, A., Djafari-Rouhani, B., Wilm, M., Ballandras, S., and Laude, V. Trapping and guiding of acoustic waves by defect modes in a full-band-gap ultrasonic crystal. Physical Review B, 68, 214301 (2003)

    Article  Google Scholar 

  11. Khelif, A., Choujaa, A., Benchabane, S., Djafari-Rouhani, B., and Laude, V. Guiding and bending of acoustic waves in highly confined phononic crystal waveguides. Applied Pysics Letters, 84, 4400–4402 (2004)

    Article  Google Scholar 

  12. Lu, T. J., Gao, G. Q., Ma, S. L., Feng, J., and Kim, T. Acoustic band gaps in two-dimensional square arrays of semi-hollow circular cylinders. Science in China Series E: Technological Sciences, 52, 303–312 (2009)

    Article  MATH  Google Scholar 

  13. Hu, X. H., Chan, C. T., and Zi., J. Two-dimensional sonic crystals wit. Helmholtz resonators. Physical Review E, 71, 055601 (2005)

    Article  Google Scholar 

  14. Hsieh, P. F., Wu, T. T., and Sun, J. H. Three-dimensional phononic band gap calculations using the FDTD method and a PC cluster system. IEEE Transactions on Ultrasonics, Ferroelectrics, and Frequency Control, 53, 148–158 (2006)

    Article  Google Scholar 

  15. Su, X. X. and Wang, Y. S. A postprocessing method based on high-resolution spectral estimation for FDTD calculation of phononic band structures. Physica B: Condensed Matter, 405, 2444–2449 (2010)

    Article  Google Scholar 

  16. 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)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, J. B., Wang, Y. S., and Zhang, C. Tuning of acoustic band gaps in phononic crystals with Helmholtz resonators. Journal of Vibration and Acoustics, 135(3), 031015 (2013)

    Article  Google Scholar 

  18. Stoica, P. and Moses, R. Spectral Analysis of Signals, Prentice Hall, New Jersey (2005)

    Google Scholar 

  19. Golberg, M. A. and Chen, C. S. The method of fundamental solutions for potential, Helmholtz and diffusion problems. Computational Engineering, WIT Press, Boston, 103–176 (1998)

    MATH  Google Scholar 

  20. Chen, W. and Tanaka, M. A meshless, integration-free, and boundary-only RBF technique. Computers and Mathematics with Applications, 43, 379–391 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hu, H. Y., Chen, J. S., and Hu, W. Weighted radial basis collocation method for boundary value problems. International Journal for Numerical Methods in Engineering, 69, 2736–2757 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sarra, S. A. Adaptive radial basis function methods for time dependent partial differential equations. Applied Numerical Mathematics, 54, 79–94 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Chen, C. S., Fan, C.M., and Wen, P. H. The method of approximate particular solutions for solving certain partial differential equations. Numerical Methods for Partial Differential Equations, 28, 506–522 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  24. Hart, E. E., Cox, S. J., and Djidjeli, K. Compact RBF meshless methods for photonic crystal modelling. Journal of Computational Physics, 230, 4910–4921 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  25. Kosec, G. and Sarler, B. Solution of phase change problems by collocation with local pressure correction. Computer Modeling in Engineering and Sciences, 25, 191–216 (2009)

    MathSciNet  MATH  Google Scholar 

  26. Zheng, H., Zhang, C. Z., Wang, Y. S., Sladek, J., and Sladek, V. A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals. Journal of Computational Physics, 305, 997–1014 (2016)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, 100081, China

    Chunqiu Wei & Zhizhong Yan

  2. Beijing Key Laboratory on MCAACI, Beijing Institute of Technology, Beijing, 100081, China

    Chunqiu Wei & Zhizhong Yan

  3. Department of Civil Engineering, University of Siegen, Siegen, D-57076, Germany

    Hui Zheng & Chuanzeng Zhang

Authors
  1. Chunqiu Wei
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  2. Zhizhong Yan
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  3. Hui Zheng
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  4. Chuanzeng Zhang
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Corresponding author

Correspondence to Zhizhong Yan.

Additional information

Project support by the National Natural Science Foundation of China (Nos. 11002026 and 11372039), the Beijing Natural Science Foundation (No. 3133039), and the Scientific Research Foundation for the Returned Overseas Chinese Scholars (No. 20121832001)

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Wei, C., Yan, Z., Zheng, H. et al. RBF collocation method and stability analysis for phononic crystals. Appl. Math. Mech.-Engl. Ed. 37, 627–638 (2016). https://doi.org/10.1007/s10483-016-2076-8

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  • DOI: https://doi.org/10.1007/s10483-016-2076-8

Key words

Chinese Library Classification

2010 Mathematics Subject Classification

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