A novel method to study the phononic crystals with fluid–structure interaction and hybrid uncertainty

  • X. Y. Lin
  • Eric Li
  • Z. C. HeEmail author
  • Y. Wu
Original Paper


Traditional finite element methods for the computation of the response of phononic crystals (PCs) with fluid-structure interaction (FSI) generally suffer from the dispersion error in the simulation, and the unavoidable uncertainties due to the manufactural errors and the material properties deviation. Therefore, it is important to develop an efficient numerical method to quantify the physical response of PCs with FSI. This paper presents a novel hybrid uncertain mass-redistributed finite element method (HUMR-FEM) to determine the uncertainty response of PCs with FSI. In this method, the MR-FEM is used to handle the FSI in PCs, which can minimize the dispersion error. The uncertainty of PCs is treated as random uncertainty with bounded distribution parameter instead of the precise values, and the response uncertainties are transformed into the deterministic computations of the extreme bounds of the statistical characteristics. Influences of the hybrid uncertainty on the physical responses in the design of PCs with FSI are discussed, and the accuracy and efficiency of the proposed method are validated through several numerical examples.



The project was supported by the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (Grant No. 51621004) and the Natural Science Foundation of China (Grant No. U1864207), the Opening Project of the Guangxi Key Laboratory of Automobile Components and Vehicle Technology of Guangxi University of Science and Technology (No. 2017GKLACVTKF01) and Guangxi Science and Technology Project (No. 2017AA10104), the opening project of the Hunan Provincial Key Laboratory of Vehicle Power and Transmission System (No. VPTS201903).


  1. 1.
    Liu, Z., Zhang, X., Mao, Y., Zhu, Y.Y., Yang, Z., Chan, C.T., Sheng, P.: Locally resonant sonic materials. Science 289(5485), 1734 (2000). CrossRefGoogle Scholar
  2. 2.
    Lai, Y., Wu, Y., Sheng, P., Zhang, Z.-Q.: Hybrid elastic solids. Nat. Mater. 10, 620 (2011). CrossRefGoogle Scholar
  3. 3.
    Li, Q.Q., He, Z.C., Li, E.: Dissipative multi-resonator acoustic metamaterials for impact force mitigation and collision energy absorption. Acta Mech. 230(8), 2905–2935 (2019). CrossRefGoogle Scholar
  4. 4.
    Li, Y., Wei, P., Wang, C.: Dispersion feature of elastic waves in a 1-D phononic crystal with consideration of couple stress effects. Acta Mech. 230(6), 2187–2200 (2019). MathSciNetCrossRefGoogle Scholar
  5. 5.
    Zhang, B., Yu, J.G., Wang, Y.C., Li, L.J., Zhang, X.M.: Complete guided wave modes in piezoelectric cylindrical structures with fan-shaped cross section using the modified double orthogonal polynomial series method. Acta Mech. 230(1), 367–380 (2019). MathSciNetCrossRefGoogle Scholar
  6. 6.
    Wu, Y., Lai, Y., Zhang, Z.-Q.: Elastic metamaterials with simultaneously negative effective shear modulus and mass density. Phys. Rev. Lett. 107(10), 105506 (2011). CrossRefGoogle Scholar
  7. 7.
    Zhu, R., Liu, X.N., Hu, G.K., Sun, C.T., Huang, G.L.: Negative refraction of elastic waves at the deep-subwavelength scale in a single-phase metamaterial. Nat. Commun. 5(5), 5510 (2014)CrossRefGoogle Scholar
  8. 8.
    Kaina, N., Lemoult, F., Fink, M., Lerosey, G.: Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials. Nature 525(7567), 77 (2015)CrossRefGoogle Scholar
  9. 9.
    Oh, J.H., Seung, H.M., Kim, Y.Y.: Doubly negative isotropic elastic metamaterial for sub-wavelength focusing: design and realization. J. Sound Vib. 410, 169–186 (2017). CrossRefGoogle Scholar
  10. 10.
    Zigoneanu, L., Popa, B.-I., Cummer, S.A.: Three-dimensional broadband omnidirectional acoustic ground cloak. Nat. Mater. 13, 352 (2014). CrossRefGoogle Scholar
  11. 11.
    Zhang, G.Y., Gao, X.L., Ding, S.R.: Band gaps for wave propagation in 2-D periodic composite structures incorporating microstructure effects. Acta Mech. 229(10), 4199–4214 (2018). MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kulkarni, P.P., Manimala, J.M.: Realizing passive direction-bias for mechanical wave propagation using a nonlinear metamaterial. Acta Mech. 230(7), 2521–2537 (2019). MathSciNetCrossRefGoogle Scholar
  13. 13.
    Laubie, H., Monfared, S., Radjaï, F., Pellenq, R., Ulm, F.-J.: Disorder-induced stiffness degradation of highly disordered porous materials. J. Mech. Phys. Solids 106, 207–228 (2017). MathSciNetCrossRefGoogle Scholar
  14. 14.
    Chen, N., Yu, D., Xia, B., Liu, J., Ma, Z.: Interval and subinterval homogenization-based method for determining the effective elastic properties of periodic microstructure with interval parameters. Int. J. Solids Struct. 106–107, 174–182 (2017). CrossRefGoogle Scholar
  15. 15.
    Li, E., He, Z.C., Hu, J.Y., Long, X.Y.: Volumetric locking issue with uncertainty in the design of locally resonant acoustic metamaterials. Comput. Methods Appl. Mech. Eng. 324, 128–148 (2017). MathSciNetCrossRefGoogle Scholar
  16. 16.
    He, Z.C., Hu, J.Y., Li, E.: An uncertainty model of acoustic metamaterials with random parameters. Comput. Mech. 62(5), 1023–1036 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Sukhovich, A., Jing, L., Page, J.H.: Negative refraction and focusing of ultrasound in two-dimensional phononic crystals. Phys. Rev. B 77(1), 014301 (2008). CrossRefGoogle Scholar
  18. 18.
    Zhang, S., Yin, L., Fang, N.: Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett. 102(19), 194301 (2009). CrossRefGoogle Scholar
  19. 19.
    Chen, J., Xia, B., Liu, J.: A sparse polynomial surrogate model for phononic crystals with uncertain parameters. Comput. Methods Appl. Mech. Eng. 339, 681–703 (2018). MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wu, J., Zhang, Y., Chen, L., Luo, Z.: A Chebyshev interval method for nonlinear dynamic systems under uncertainty. Appl. Math. Model. 37(6), 4578–4591 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bernard, B.P., Owens, B.A.M., Mann, B.P.: Uncertainty propagation in the band gap structure of a 1D array of magnetically coupled oscillators. J. Vib. Acoust. 135(4), 041005-041005-041007 (2013). CrossRefGoogle Scholar
  22. 22.
    Xia, B., Yu, D., Liu, J.: Hybrid uncertain analysis of acoustic field with interval random parameters. Comput. Methods Appl. Mech. Eng. 256, 56–69 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Elishakoff, I., Elettro, F.: Interval, ellipsoidal, and super-ellipsoidal calculi for experimental and theoretical treatment of uncertainty: which one ought to be preferred? Int. J. Solids Struct. 51(7), 1576–1586 (2014). CrossRefGoogle Scholar
  24. 24.
    He, Z.C., Wu, Y., Li, E.: Topology optimization of structure for dynamic properties considering hybrid uncertain parameters. Struct. Multidiscip. Optim. 57(2), 625–638 (2018). MathSciNetCrossRefGoogle Scholar
  25. 25.
    Kafesaki, M., Economou, E.N.: Multiple-scattering theory for three-dimensional periodic acoustic composites. Phys. Rev. B 60(17), 11993–12001 (1999). CrossRefGoogle Scholar
  26. 26.
    Shi, Z., Wang, Y., Zhang, C.: Band structure calculations of in-plane waves in two-dimensional phononic crystals based on generalized multipole technique. Appl. Math. Mech. 36(5), 557–580 (2015). MathSciNetCrossRefGoogle Scholar
  27. 27.
    Axmann, W., Kuchment, P.: An efficient finite element method for computing spectra of photonic and acoustic band-gap materials: i. scalar case. J. Comput. Phys. 150(2), 468–481 (1999). MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Liu, Y., Gao, L-t: Explicit dynamic finite element method for band-structure calculations of 2D phononic crystals. Solid State Commun. 144(3), 89–93 (2007). CrossRefGoogle Scholar
  29. 29.
    Li, F.-L., Wang, Y.-S., Zhang, C., Yu, G.-L.: Bandgap calculations of two-dimensional solid-fluid phononic crystals with the boundary element method. Wave Motion 50(3), 525–541 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Zheng, H., Zhang, C., Wang, Y., Chen, W., Sladek, J., Sladek, V.: A local RBF collocation method for band structure computations of 2D solid/fluid and fluid/solid phononic crystals. Int. J. Numer. Methods Eng. 110(5), 467–500 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zheng, H., Zhang, C., Wang, Y., Sladek, J., Sladek, V.: A meshfree local RBF collocation method for anti-plane transverse elastic wave propagation analysis in 2D phononic crystals. J. Comput. Phys. 305, 997–1014 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Li, E., He, Z.C., Wang, G., Liu, G.R.: An efficient algorithm to analyze wave propagation in fluid/solid and solid/fluid phononic crystals. Comput. Methods Appl. Mech. Eng. 333, 421–442 (2018). MathSciNetCrossRefGoogle Scholar
  33. 33.
    Li, F.-L., Wang, Y.-S., Zhang, C., Yu, G.-L.: Boundary element method for band gap calculations of two-dimensional solid phononic crystals. Eng. Anal. Bound. Elem. 37(2), 225–235 (2013). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    He, Z.C., Li, E., Liu, G.R., Li, G.Y., Cheng, A.G.: A mass-redistributed finite element method (MR-FEM) for acoustic problems using triangular mesh. J. Comput. Phys. 323, 149–170 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Yao, L., Huang, G., Chen, H., Barnhart, M.V.: A modified smoothed finite element method (M-SFEM) for analyzing the band gap in phononic crystals. Acta Mech. 230(6), 2279–2293 (2019). MathSciNetCrossRefGoogle Scholar
  36. 36.
    Li, E., He, Z.C., Jiang, Y., Li, B.: 3D mass-redistributed finite element method in structural-acoustic interaction problems. Acta Mech. 227(3), 857–879 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Li, E., He, Z.C., Xu, X., Zhang, G.Y., Jiang, Y.: A faster and accurate explicit algorithm for quasi-harmonic dynamic problems. Int. J. Numer. Methods Eng. 108(8), 839–864 (2016). MathSciNetCrossRefGoogle Scholar
  38. 38.
    Li, E., He, Z.C., Zhang, Z., Liu, G.R., Li, Q.: Stability analysis of generalized mass formulation in dynamic heat transfer. Numer. Heat Transf. Part B Fundam. 69(4), 287–311 (2016). CrossRefGoogle Scholar
  39. 39.
    Li, E., He, Z.C.: Development of a perfect match system in the improvement of eigenfrequencies of free vibration. Appl. Math. Model. 44, 614–639 (2017). MathSciNetCrossRefGoogle Scholar
  40. 40.
    Chadil, M.-A., Vincent, S., Estivalèzes, J.-L.: Accurate estimate of drag forces using particle-resolved direct numerical simulations. Acta Mech. 230(2), 569–595 (2019). MathSciNetCrossRefGoogle Scholar
  41. 41.
    Liu, G.-R., Trung, N.: Smoothed Finite Element Methods. CRC Press, Boca Raton (2016)CrossRefGoogle Scholar
  42. 42.
    Wang, G., Wen, J., Liu, Y., Wen, X.: Lumped-mass method for the study of band structure in two-dimensional phononic crystals. Phys. Rev. B 69(18), 184302 (2004). CrossRefGoogle Scholar
  43. 43.
    Li, E., He, Z.C., Wang, G., Jong, Y.: Fundamental study of mechanism of band gap in fluid and solid/fluid phononic crystals. Adv. Eng. Softw. 121, 167–177 (2018). CrossRefGoogle Scholar
  44. 44.
    Long, X.Y., Jiang, C., Han, X.: New method for eigenvector-sensitivity analysis with repeated eigenvalues and eigenvalue derivatives. AIAA J. 53(5), 1226–1235 (2015). CrossRefGoogle Scholar
  45. 45.
    Kwon, Y.W., Bang, H.: The Finite Element Method Using MATLAB, 2nd edn. CRC Press, Inc., Boca Raton (2000)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Austria, part of Springer Nature 2019

Authors and Affiliations

  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle BodyHunan UniversityChangshaPeople’s Republic of China
  2. 2.School of Science, Engineering and DesignTeesside UniversityMiddlesbroughUK
  3. 3.Guangxi Key Laboratory of Automobile Components and Vehicle TechnologyGuangxi University of Science and TechnologyLiuzhouPeople’s Republic of China

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