Skip to main content
SpringerLink
Log in

A gradient-based optimization method for the design of layered phononic band-gap materials

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

Phononic materials with specific band-gap characteristics at desired frequency ranges are in great demand for vibration and noise isolation, elastic wave filters, and acoustic devices. The attenuation coefficient curve depicts both the frequency range of band gap and the attenuation of elastic wave, where the frequency ranges corresponding to the none-zero attenuation coefficients are band gaps. Therefore, the band-gap characteristics can be achieved through maximizing the attenuation coefficient at the corresponding frequency or within the corresponding frequency range. Because the attenuation coefficient curve is not smooth in the frequency domain, the gradient-based optimization methods cannot be directly used in the design optimization of phononic band-gap materials to achieve the maximum attenuation within the desired frequency range. To overcome this difficulty, the objective of maximizing the attenuation coefficient is transformed into maximizing its Cosine, and in this way, the objective function is smoothed and becomes differentiable. Based on this objective function, a novel gradient-based optimization approach is proposed to open the band gap at a prescribed frequency range and to further maximize the attenuation efficiency of the elastic wave at a specific frequency or within a prescribed frequency range. Numerical results demonstrate the effectiveness of the proposed gradient-based optimization method for enhancing the wave attenuation properties.

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  3. Liu, Z., Zhang, X., Mao, Y., Zhu, Y., Yang, Z., Chan, C. and Sheng, P., Locally resonant sonic material. Science, 2000, 289: 1734–1736.

    Article  Google Scholar 

  4. Martinsson, P.G. and Movchan, A.B., Vibrations of lattice structures and phononic band gaps. Quarterly Journal of Mechanics and Applied Mathematics, 2003, 6(1): 45–64

    Article  MathSciNet  Google Scholar 

  5. Sigalas, M.M., Kushwaha, M.S., Economou, E.N., Kafesaki, M. and Psarobas, I.E., Classical vibrational modes in phononic lattices: theory and experiment. Zeitschrift fur Kristallographie, 2005, 220: 765–809.

    Google Scholar 

  6. Zhang, K., Deng, Z.C., Xu, X.J., et al. Sympletic analysis for wave propagation of hierarchical honeycomb structures. Acta Mechanica Solida Sinica, 2015, 28(3): 294–304.

    Article  Google Scholar 

  7. Jensen, J.S., Phononic band gaps and vibrations in one- and two- dimensional mass-spring structures. Journal of Sound and Vibration, 2003, 266(5): 1053–1078.

    Article  Google Scholar 

  8. Jensen, J.S., Topology optimization problem for reflection and dissipation of elastic waves. Journal of Sound and Vibration, 2007, 301: 319–340.

    Article  Google Scholar 

  9. Halkjær, S., Sigmund, O. and Jensen, J.S., Inverse design of phononic crystals by topology optimization. Zeitschrift fur Kristallographie, 2005, 220(9–10): 895–905.

    Google Scholar 

  10. Gazonas, G.A., Weile, D.S. and Wildman, R., Genetic algorithm optimization of phononic band gap structures. International Journal of Solids and Structures, 2006, 43(18–19): 5851–5866.

    Article  Google Scholar 

  11. Rupp, C.J., Evgrafov, A., Maute, K. and Dunn, M., Design of phononic materials/structures for surface wave devices using topology optimization. Structural and Multidisciplinary Optimization, 2007, 32(2): 111–121.

    Article  MathSciNet  Google Scholar 

  12. Romero-García, Sánchez-Pérez, J.V., Sánchez-Pérez, L.M., et al., Hole distribution in phononic crystals: Design and optimization. Journal of the Acoustical Society of America, 2009, 125(6): 3774–3783.

    Article  Google Scholar 

  13. Dong, H.W., Su, X.X., Wang, Y.S., et al., Topology optimization of two-dimensional asymmetrical phononiccrystals. Physics Letters A, 2014, 378(4): 434–441.

    Article  Google Scholar 

  14. Hedayatrasa, S., Abhary, K. and Uddin, M., Numerical study and topology optimization of 1D periodic bio-material phononic crystal plates for band gaps of low order Lamb waves. Ultrasonics, 2015, 57: 104–124.

    Article  Google Scholar 

  15. Zhong, H.L., Wu, F.G. and Yao, L.N., Application of genetic algorithm in optimization of band gap of two-dimensional phononic crystals. Acta Physica Sinica, 2006, 55(1): 275–280 (in Chinese).

    Google Scholar 

  16. Liu, Z.F., Wu, B. and He, C.F., Band-gap optimization of two-dimensional phononic crystals based on genetic algorithm and FPWE. Wave Random Complex Media, 2014, 24: 286–305.

    Article  MathSciNet  Google Scholar 

  17. Liu, Z.F., Wu, B. and He, C.F., The characteristics of optimal two-dimensional solid/solid phononic crystals. Chinese Journal of Solid Mechanics, 2015, 36(4): 283–289 (in Chinese).

    Google Scholar 

  18. Dong, H.W., Su, X.X., Wang, Y.S. and Zhang, C., Topological optimization of two dimensional phononic crystals based on the finite element method and genetic algorithm. Structural and Multidisciplinary Optimization, 2014, 50(4): 593–604.

    Article  MathSciNet  Google Scholar 

  19. Dong, H.W., Su, X.X. and Wang, Y.S., Multi-objective optimization of two-dimensional porous phononic crystals. Journal of Physics D: Applied Physics, 2014, 47(15): 155301.

    Article  Google Scholar 

  20. Hussein, M.I., Hulbert, G.M. and Scott, R.A., Tailoring of wave propagation characteristic in periodic structures with multilayer unit cells. In: Proceedings of the 17th American Society of Composites Technical Conference, West Lafayette, Indiana, CRC Press, Boca Raton, Florida, 2002.

  21. Hussein, M.I., Hamzak, K., Hulbert, G.M., Scott, R.A. and Saitou, K., Multiobjective evolutionary optimization of periodic layered materials for desired wave dispersion characteristics. Structural and Multidisciplinary Optimization, 2006a, 31(1): 60–75

    Article  Google Scholar 

  22. Hussein, M.I., Hulbert, G.M. and Scott, R.A., Dispersive elastodynamics of 1D banded materials and structures: analysis. Journal of Sound and Vibration, 2006b, 289(4–5): 779–806.

    Article  Google Scholar 

  23. Hussein, M.I., Hulbert, G.M. and Scott, R.A., Dispersive elastodynamics of 1D banded materials and structures: design. Journal of Sound and Vibration, 2007a, 307(3–5): 865–893.

    Article  Google Scholar 

  24. Hussein, M.I., Hamza, K. and Hulbert, G.M., Optimal synthesis of 2D phononic crystals for broadband frequency isolation. Wave Random Complex, 2007b, 17(4): 491–510

    Article  MathSciNet  Google Scholar 

  25. Cox, S.J. and Dobson, D.C., Maximizing band gaps in two dimensional photonic crystals. SIAM Journal of Applied Mathematics, 1999, 59(6): 2108–2120.

    Article  MathSciNet  Google Scholar 

  26. Dahl, J., Jensen, J.S. and Sigmund, O., Topology optimization for transient wave propagation problems in one dimension—Design of filters and pulse modulators. Structural and Multidisciplinary Optimization, 2008, 36: 585–595.

    Article  MathSciNet  Google Scholar 

  27. Huang, Y., Liu, S.T. and Zhao, J., Optimal design of two-dimensional band-gap materials for uni-directional wave propagation. Structural and Multidisciplinary Optimization, 2013, 48: 487–499.

    Article  MathSciNet  Google Scholar 

  28. Park, J.H., Ma, P.S. and Kim, Y.Y., Design of phononic crystals for self-collimation of elastic waves using topology optimization method. Structural and Multidisciplinary Optimization, 2015, 51(6): 1199–1209.

    Article  MathSciNet  Google Scholar 

  29. Sigmund, O. and Jensen, J.S., Systematic design of phononic band gap materials and structures by topology optimization. Philos Trans R Soc Lond, 2003, 361: 1001–1019.

    Article  MathSciNet  Google Scholar 

  30. Halkjasr, S., Sigmund, O. and Jensen, J.S., Maximizing band gaps in plate structures. Structural and Multi-disciplinary Optimization, 2006, 32(4): 263–275.

    Article  Google Scholar 

  31. EI-Sabbagh, A., Akl, W. and Baz, A., Topology optimization of periodic Mindlin plates. Finite Elements in Analysis and Design, 2008, 44: 439–449.

    Article  MathSciNet  Google Scholar 

  32. Diaz, A.R., Haddow, A.G. and Ma, L., Design of band-gap grid structures. Structural and Multidisciplinary Optimization, 2005, 29(6): 418–431.

    Article  Google Scholar 

  33. Kevin, L.M., Michael, J.L. and Massimo, R., Topology design and optimization of nonlinear periodic materials. Journal of the Mechanics and Physics of Solids, 2013, 61: 2433–2453.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. State Key Laboratory of Structural Analysis for Industrial Equipment, Department of Engineering Mechanics, Dalian University of Technology, 116024, Dalian, China

    Yu Huang, Shutian Liu & Jian Zhao

Authors
  1. Yu Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Shutian Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian Zhao
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shutian Liu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 11502043, 11332004 and 11402046), the Fundamental Research Funds for the Central Universities Of China (DUT15ZD101) and the 111 Project (B14013).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, Y., Liu, S. & Zhao, J. A gradient-based optimization method for the design of layered phononic band-gap materials. Acta Mech. Solida Sin. 29, 429–443 (2016). https://doi.org/10.1016/S0894-9166(16)30245-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1016/S0894-9166(16)30245-2

Key Words

Search

Navigation