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

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Abstract

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.

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References

  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). https://doi.org/10.1126/science.289.5485.1734

    Article  Google Scholar 

  2. Lai, Y., Wu, Y., Sheng, P., Zhang, Z.-Q.: Hybrid elastic solids. Nat. Mater. 10, 620 (2011). https://doi.org/10.1038/nmat3043

    Article  Google Scholar 

  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). https://doi.org/10.1007/s00707-019-02437-4

    Article  Google Scholar 

  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). https://doi.org/10.1007/s00707-019-02395-x

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s00707-018-2266-4

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.107.105506

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.jsv.2017.08.027

    Article  Google Scholar 

  10. Zigoneanu, L., Popa, B.-I., Cummer, S.A.: Three-dimensional broadband omnidirectional acoustic ground cloak. Nat. Mater. 13, 352 (2014). https://doi.org/10.1038/nmat3901

    Article  Google Scholar 

  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). https://doi.org/10.1007/s00707-018-2207-2

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/s00707-019-02415-w

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.jmps.2017.05.008

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.ijsolstr.2016.11.022

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.cma.2017.06.005

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/s00466-018-1548-y

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1103/PhysRevB.77.014301

    Article  Google Scholar 

  18. Zhang, S., Yin, L., Fang, N.: Focusing ultrasound with an acoustic metamaterial network. Phys. Rev. Lett. 102(19), 194301 (2009). https://doi.org/10.1103/PhysRevLett.102.194301

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.cma.2018.05.001

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.apm.2012.09.073

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1115/1.4023821

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.cma.2012.12.016

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ijsolstr.2014.01.010

    Article  Google Scholar 

  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). https://doi.org/10.1007/s00158-017-1769-2

    Article  MathSciNet  Google Scholar 

  25. Kafesaki, M., Economou, E.N.: Multiple-scattering theory for three-dimensional periodic acoustic composites. Phys. Rev. B 60(17), 11993–12001 (1999). https://doi.org/10.1103/PhysRevB.60.11993

    Article  Google Scholar 

  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). https://doi.org/10.1007/s10483-015-1938-7

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1006/jcph.1999.6188

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.ssc.2007.08.014

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.wavemoti.2012.12.001

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1002/nme.5366

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.jcp.2015.10.020

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.cma.2018.01.006

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1016/j.enganabound.2012.10.003

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1016/j.jcp.2016.07.025

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s00707-019-02396-w

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1007/s00707-015-1496-y

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1002/nme.5233

    Article  MathSciNet  Google Scholar 

  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). https://doi.org/10.1080/10407790.2015.1104215

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.apm.2017.02.013

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1007/s00707-018-2305-1

    Article  MathSciNet  Google Scholar 

  41. Liu, G.-R., Trung, N.: Smoothed Finite Element Methods. CRC Press, Boca Raton (2016)

    Book  Google Scholar 

  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). https://doi.org/10.1103/PhysRevB.69.184302

    Article  Google Scholar 

  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). https://doi.org/10.1016/j.advengsoft.2018.04.014

    Article  Google Scholar 

  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). https://doi.org/10.2514/1.J053362

    Article  Google Scholar 

  45. Kwon, Y.W., Bang, H.: The Finite Element Method Using MATLAB, 2nd edn. CRC Press, Inc., Boca Raton (2000)

    MATH  Google Scholar 

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Acknowledgements

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).

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Authors and Affiliations

  1. State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, Hunan University, Changsha, 410082, People’s Republic of China

    X. Y. Lin, Z. C. He & Y. Wu

  2. School of Science, Engineering and Design, Teesside University, Middlesbrough, UK

    Eric Li

  3. Guangxi Key Laboratory of Automobile Components and Vehicle Technology, Guangxi University of Science and Technology, Liuzhou, 545006, People’s Republic of China

    Z. C. He

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  1. X. Y. Lin
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Lin, X.Y., Li, E., He, Z.C. et al. A novel method to study the phononic crystals with fluid–structure interaction and hybrid uncertainty. Acta Mech 231, 321–352 (2020). https://doi.org/10.1007/s00707-019-02530-8

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