Smart Structures and Systems

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Bayesian in-situ parameter estimation of metallic plates using piezoelectric transducers

  • Asadi, Sina (New Technologies Research Center, Amirkabir University of Technology (Tehran Polytechnic)) ;
  • Shamshirsaz, Mahnaz (New Technologies Research Center, Amirkabir University of Technology (Tehran Polytechnic)) ;
  • Vaghasloo, Younes A. (Department of Mechanical Engineering, Amirkabir University of Technology (Tehran Polytechnic))
  • Received : 2020.02.22
  • Accepted : 2020.09.11
  • Published : 2020.12.25
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Abstract

Identification of structure parameters is crucial in Structural Health Monitoring (SHM) context for activities such as model validation, damage assessment and signal processing of structure response. In this paper, guided waves generated by piezoelectric transducers are used for in-situ and non-destructive structural parameter estimation based on Bayesian approach. As Bayesian approach needs iterative process, which is computationally expensive, this paper proposes a method in which an analytical model is selected and developed in order to decrease computational time and complexity of modeling. An experimental set-up is implemented to estimate three target elastic and geometrical parameters: Young's modulus, Poisson ratio and thickness of aluminum and steel plates. Experimental and simulated data are combined in a Bayesian framework for parameter identification. A significant accuracy is achieved regarding estimation of target parameters with maximum error of 8, 11 and 17 percent respectively. Moreover, the limitation of analytical model concerning boundary reflections is addressed and managed experimentally. Pulse excitation is selected as it can excite the structure in a wide frequency range contrary to conventional tone burst excitation. The results show that the proposed non-destructive method can be used in service for estimation of material and geometrical properties of structure in industrial applications.

Keywords

References

  1. Ablitzer, F., Pezerat, C., Genevaux, J.M. and Begue, J. (2014), "Identification of stiffness and damping properties of plates by using the local equation of motion", J. Sound Vib., 333(9), 2454-2468. https://doi.org/10.1016/j.jsv.2013.12.013.
  2. Agrahari, J.K. and Kapuria, S. (2016), "A refined Lamb wave time-reversal method with enhanced sensitivity for damage detection in isotropic plates", J. Intell. Mater. Syst. Struct., 27(10), 1283-1305. https://doi.org/10.1177/1045389X15590269.
  3. Akgoz, B. and Civalek, O. (2017), "Effects of thermal and shear deformation on vibration response of functionally graded thick composite microbeams", Compos. Part B Eng., 129, 77-87. https://doi.org/10.1016/j.compositesb.2017.07.024.
  4. Alexanderian, A. (2013), "On spectral methods for variance based sensitivity analysis", Probab. Surv., 10, 51-68. https://doi.org/10.1214/13-PS219.
  5. Ambrozinski, L., Packo, P., Pieczonka, L., Stepinski, T., Uhl, T. and Staszewski, W. (2015), "Identification of material properties-efficient modelling approach based on guided wave propagation and spatial multiple signal classification", Struct. Control Health Monit., 22(7), 969-983. https://doi.org/10.1002/stc.1728.
  6. Arani, A.J. and Kolahchi, R. (2016), "Buckling analysis of embedded concrete columns armed with carbon nanotubes", Comput. Concrete, Int. J., 17(5), 567-578. https://doi.org/10.12989/cac.2016.17.5.567.
  7. Araujo, A.L., Soares, C.M.M. and De Freitas, M.M. (1996), "Characterization of material parameters of composite plate specimens using optimization and experimental vibrational data", Compos. Part B Eng., 27(2), 185-191. https://doi.org/10.1016/1359-8368(95)00050-X.
  8. Araujo, A.L., Soares, C.M.M., De Freitas, M.M., Pedersen, P. and Herskovits, J. (2000), "Combined numerical-experimental model for the identification of mechanical properties of laminated structures", Compos. Struct., 50(4), 363-372. https://doi.org/10.1016/S0263-8223(00)00113-6.
  9. Araujo, A.L., Soares, C.M.M., Herskovits, J. and Pedersen, P. (2002), "Development of a finite element model for the identification of mechanical and piezoelectric properties through gradient optimisation and experimental vibration data", Compos. Struct., 58(3), 307-318. https://doi.org/10.1016/S0263-8223(02)00192-7
  10. Araujo, A., Mota Soares, C., Herskovits, J. and Pedersen, P. (2009), "Visco-piezo-elastic parameter estimation in laminated plate structures", Inverse Probl. Sci. Eng., 17(2), 145-157. https://doi.org/10.1080/17415970802082732.
  11. Argyris, C., Chowdhury, S., Zabel, V. and Papadimitriou, C. (2018), "Bayesian optimal sensor placement for crack identification in structures using strain measurements", Struct. Control Health Monit., 25(5), e2137. https://doi.org/10.1002/stc.2137.
  12. Asadi, S., Sepehry, N., Shamshirsaz, M. and Vaghasloo, Y. (2017), "Implementation of a novel efficient low cost method in structural health monitoring", Smart Mater. Struct., 26(5), 055032. https://doi.org/10.1088/1361-665X/aa6b65.
  13. Attia, A., Tounsi, A., Bedia, E. and Mahmoud, S. (2015), "Free vibration analysis of functionally graded plates with temperature-dependent properties using various four variable refined plate theories", Steel Compos. Struct., Int. J., 18(1), 187-212. https://doi.org/10.12989/scs.2015.18.1.187.
  14. Auzins, J. and Skukis, E. (2015), "Robust optimization approach for mixed numerical/experimental identification of elastic properties of orthotropic composite plates", Proceedings of the VI International Conference on Computational Methods for Coupled Problems in Science and Engineering, Venice, Italy, May.
  15. Bales, B., Petzold, L., Goodlet, B.R., Lenthe, W.C. and Pollock, T.M. (2018), "Bayesian inference of elastic properties with resonant ultrasound spectroscopy", J. Acous. Soc. Am., 143(1), 71-83. https://doi.org/10.1121/1.5017840.
  16. Banerjee, B. (2016), "Elastic parameter identification of plate structures using modal response: An ECE based approach", J. Eng. Mech., 142(1), 04015059. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000970.
  17. Battaglia, G., Di Matteo, A., Micale, G. and Pirrotta, A. (2018), "Vibration-based identification of mechanical properties of orthotropic arbitrarily shaped plates: Numerical and experimental assessment", Compos. Part B Eng., 150, 212-225. https://doi.org/10.1016/j.compositesb.2018.05.029.
  18. Bhalla, S., Gupta, A., Bansal, S. and Garg, T. (2009), "Ultra low-cost adaptations of electro-mechanical impedance technique for structural health monitoring", J. Intell. Mater. Syst. Struct., 20(8), 991-999. https://doi.org/10.1177/1045389X08100384.
  19. Bochud, N., Laurent, J., Bruno, F., Royer, D. and Prada, C. (2018), "Towards real-time assessment of anisotropic plate properties using elastic guided waves", J Acoust. Soc. Am., 143(2), 1138-1147. https://doi.org/10.1121/1.5024353.
  20. Civalek, O., Uzun, B., Yayli, M.O. and Akgoz, B. (2020), "Size-dependent transverse and longitudinal vibrations of embedded carbon and silica carbide nanotubes by nonlocal finite element method", Eur. Phys. J. Plus, 135(4), 381. https://doi.org/10.1140/epjp/s13360-020-00385-w.
  21. Cuadrado, M., Pernas-Sanchez, J., Artero-Guerrero, J. and Varas, D. (2020), "Model updating of uncertain parameters of carbon/epoxy composite plates using digital image correlation for full-field vibration measurement", Measurement, 2020, 107783. https://doi.org/10.1016/j.measurement.2020.107783.
  22. Deng, Y., Cheng, C., Yang, Y., Peng, Z., Yang, W. and Zhang, W. (2016), "Parametric identification of nonlinear vibration systems via polynomial chirplet transform", J. Vib. Acoust., 138(5), 051014. https://doi.org/10.1115/1.4033717.
  23. Ebrahimi, F., Barati, M.R. and Civalek, O. (2019), "Application of Chebyshev-Ritz method for static stability and vibration analysis of nonlocal microstructure-dependent nanostructures", Eng. Comput., 2019, 1-12. https://doi.org/10.1007/s00366-019-00742-z.
  24. Ebrahimian, H., Astroza, R. and Conte, J.P. (2015), "Extended Kalman filter for material parameter estimation in nonlinear structural finite element models using direct differentiation method", Earthq. Eng. Struct. Dyn., 44(10), 1495-1522. https://doi.org/10.1002/eqe.2532.
  25. Erazo, K. and Nagarajaiah, S. (2018), "Bayesian structural identification of a hysteretic negative stiffness earthquake protection system using unscented Kalman filtering", Struct. Control Health Monit., 25(9), e2203. https://doi.org/10.1002/stc.2203.
  26. Eremin, A., Glushkov, E., Glushkova, N. and Lammering, R. (2015), "Evaluation of effective elastic properties of layered composite fiber-reinforced plastic plates by piezoelectrically induced guided waves and laser Doppler vibrometry", Compos. Struct., 125, 449-458. https://doi.org/10.1016/j.compstruct.2015.02.029.
  27. Gallina, A., Pieczonka, L., Ambrozinski, L., Packo, P., Nazarko, P., Uhl, T. and Waszczyszyn, Z. (2015), "Analysis of Lamb wave dispersion curve sensitivity to material elastic constants in composites", Proceedings of the SPIE Smart Structures and Materials Nondestructive Evaluation and Health Monitoring, California, USA, March.
  28. Gallina, A., Ambrozinski, L., Packo, P., Pieczonka, L., Uhl, T. and Staszewski, W.J. (2017), "Bayesian parameter identification of orthotropic composite materials using Lamb waves dispersion curves measurement", J. Vib. Control, 23(16), 2656-2671. https://doi.org/10.1177/1077546315619264.
  29. Ge, L., Wang, X. and Wang, F. (2014), "Accurate modeling of PZT-induced Lamb wave propagation in structures by using a novel spectral finite element method", Smart Mater. Struct., 23(9), 095018. https://doi.org/10.1088/0964-1726/23/9/095018.
  30. Giurgiutiu, V. (2005), "Tuned Lamb wave excitation and detection with piezoelectric wafer active sensors for structural health monitoring", J. Intell. Mater. Syst. Struct., 16(4), 291-305. https://doi.org/10.1177/1045389X05050106.
  31. Giurgiutiu, V. (2007), Structural Health Monitoring: With Piezoelectric Wafer Active Sensors, Academic Press, California, USA.
  32. Gogu, C., Haftka, R., Le Riche, R., Molimard, J. and Vautrin, A. (2010), "Introduction to the bayesian approach applied to elastic constants identification", AIAA J., 48(5), 893-903. https://doi.org/10.2514/1.40922.
  33. Hall, J.S. and Michaels, J.E. (2011), "Model-based parameter estimation for characterizing wave propagation in a homogeneous medium", Inverse Probl., 27(3), 035002. https://doi.org/10.1088/0266-5611/27/3/035002.
  34. Jaynes, E.T. (2003), Probability Theory: The Logic of Science, Cambridge University Press, Cambridge, UK.
  35. Jia, H., Zhang, Z., Liu, H., Dai, F., Liu, Y. and Leng, J. (2019), "An approach based on expectation-maximization algorithm for parameter estimation of Lamb wave signals", Mech. Syst. Signal Process., 120, 341-355. https://doi.org/10.1016/j.ymssp.2018.10.020.
  36. Lechleiter, A. and Schlasche, J.W. (2017), "Identifying Lamé parameters from time-dependent elastic wave measurements", Inverse Probl. Sci. Eng., 25(1), 2-26. https://doi.org/10.1080/17415977.2015.1132713.
  37. Lee, F.W., Chai, H.K. and Lim, K.S. (2017), "Characterizing concrete surface notch using Rayleigh wave phase velocity and wavelet parametric analyses", Constr. Build. Mater., 136, 627-642. https://doi.org/10.1016/j.conbuildmat.2016.08.145.
  38. Liew, K., Lei, Z. and Zhang, L. (2015), "Mechanical analysis of functionally graded carbon nanotube reinforced composites: A review", Compos. Struct., 120, 90-97. https://doi.org/10.1016/j.compstruct.2014.09.041.
  39. Liu, G., Ma, W. and Han, X. (2002), "An inverse procedure for determination of material constants of composite laminates using elastic waves", Comput. Methods Appl. Mech. Eng., 191(33), 3543-3554. https://doi.org/10.1016/S0045-7825(02)00292-X.
  40. Lu, J., Zhan, Z., Liu, X. and Wang, P. (2018), "Numerical modeling and model updating for smart laminated structures with viscoelastic damping", Smart Mater. Struct., 27(7), 075038. https://doi.org/10.1088/1361-665X/aac623.
  41. Lynch, S.M. (2007), Introduction to Applied Bayesian Statistics and Estimation for Social Scientists, Springer Science & Business Media, New Jersey, USA.
  42. Ma, T., Zhang, Y. and Huang, X. (2014), "A novel approach for stochastic finite element model updating and parameter estimation", Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci., 228(18), 3329-3342. https://doi.org/10.1177/0954406214529945.
  43. Morales‐Valdez, J., Alvarez‐Icaza, L. and Sanchez‐Sesma, F.J. (2018), "Shear building stiffness estimation by wave traveling time analysis", Struct. Control Health Monit., 25(1), e2045. https://doi.org/10.1002/stc.2045.
  44. Murakami, A., Shinmura, H., Ohno, S. and Fujisawa, K. (2018), "Model identification and parameter estimation of elastoplastic constitutive model by data assimilation using the particle filter", Int. J. Num. Anal. Methods Geomech., 42(1), 110-131. https://doi.org/10.1002/nag.2717.
  45. Nevaranta, N., Parkkinen, J., Lindh, T., Niemela, M., Pyrhonen, O. and Pyrhonen, J. (2015), "Online identification of a mechanical system in the frequency domain with short-time DFT", 36(3), 157-165. https://doi.org/10.4173/mic.2015.3.3.
  46. Nichols, J., Link, W., Murphy, K. and Olson, C. (2010), "A Bayesian approach to identifying structural nonlinearity using free-decay response: Application to damage detection in composites", J. Sound Vib., 329(15), 2995-3007. https://doi.org/10.1016/j.jsv.2010.02.004.
  47. Oliveira, É., Maia, N., Marto, A., Da Silva, R., Afonso, F. and Suleman, A. (2016), "Modal characterization of composite flat plate models using piezoelectric transducers", Mech. Syst. Signal Process., 79, 16-29. https://doi.org/10.1016/j.ymssp.2016.02.046.
  48. Olivier, A. and Smyth, A.W. (2018), "A marginalized unscented Kalman filter for efficient parameter estimation with applications to finite element models", Comput. Methods Appl. Mech. Eng., 339, 615-643. https://doi.org/10.1016/j.cma.2018.05.014.
  49. Pabisek, E. and Waszczyszyn, Z. (2015), "Identification of thin elastic isotropic plate parameters applying guided wave measurement and artificial neural networks", Mech. Syst. Signal Process., 64, 403-412. https://doi.org/10.1016/j.ymssp.2015.04.007.
  50. Pagnotta, L. (2008), "Recent progress in identification methods for the elastic characterization of materials", Int. J. Mech., 2(4), 129-140.
  51. Pant, S. (2014), Lamb Wave Propagation and Material Characterization of Metallic and Composite Aerospace Structures for Improved Structural Health Monitoring (SHM), Carleton University Ottawa, Ottawa, Canada.
  52. Pedersen, P., Araujo, A., Soares, C. and Herskovits, J. (2005), "An Inverse Method for Parameter Estimation in Active Laminated Structures", Proceedings of the 6th World Congresses of Structural and Multidisciplinary Optimization, Rio de Janeiro, Brazil, May.
  53. Perie, J.N. and Passieux, J.C. (2020), "Special issue on advances in digital image correlation (DIC)", Adv. Digit. Image Correl., 10(4), 1530. https://doi.org/10.3390/app10041530.
  54. Pirboudaghi, S., Tarinejad, R. and Alami, M.T. (2018), "Damage detection based on system identification of concrete dams using an extended finite element-wavelet transform coupled procedure", J. Vib. Control, 24(18), 4226-4246. https://doi.org/10.1177/1077546317722428.
  55. Pollock, P., Yu, L., Sutton, M., Guo, S., Majumdar, P. and Gresil, M. (2014), "Full-field measurements for determining orthotropic elastic parameters of woven glass-epoxy composites using off-axis tensile specimens", Exp. Tech., 38(4), 61-71. https://doi.org/10.1111/j.1747-1567.2012.00824.x
  56. Raghavan, A. and Cesnik, C.E. (2005), "Finite-dimensional piezoelectric transducer modeling for guided wave based structural health monitoring", Smart Mater. Struct., 14(6), 1448. https://doi.org/10.1088/0964-1726/14/6/037.
  57. Rahmatabadi, D., Shahmirzaloo, A., Farahani, M., Tayyebi, M. and Hashemi, R. (2019), "Characterizing the elastic and plastic properties of the multilayered Al/Brass composite produced by ARB using DIC", Mater. Sci. Eng. A, 753, 70-78. https://doi.org/10.1016/j.msea.2019.03.002.
  58. Reed, H., Leckey, C.A., Dick, A., Harvey, G. and Dobson, J. (2018), "A model based bayesian solution for characterization of complex damage scenarios in aerospace composite structures", Ultrasonics, 82, 272-288. https://doi.org/10.1016/j.ultras.2017.09.002.
  59. Rethore, J., Elguedj, T., Coret, M., Chaudet, P. and Combescure, A. (2013), "Robust identification of elasto-plastic constitutive law parameters from digital images using 3D kinematics", Int. J. Solids Struct., 50(1), 73-85. https://doi.org/10.1016/j.ijsolstr.2012.09.002.
  60. Sanayei, M., Khaloo, A., Gul, M. and Catbas, F.N. (2015), "Automated finite element model updating of a scale bridge model using measured static and modal test data", Eng. Struct., 102, 66-79. https://doi.org/10.1016/j.engstruct.2015.07.029.
  61. Sen, S. and Bhattacharya, B. (2018), "Non-Gaussian parameter estimation using generalized polynomial chaos expansion with extended Kalman filtering", Struct. Safety, 70, 104-114. https://doi.org/10.1016/j.strusafe.2017.10.009.
  62. Shao, Q., Younes, A., Fahs, M. and Mara, T.A. (2017), "Bayesian sparse polynomial chaos expansion for global sensitivity analysis", Comput. Methods Appl. Mech. Eng., 318, 474-496. https://doi.org/10.1016/j.cma.2017.01.033.
  63. Shen, Y. and Giurgiutiu, V. (2016), "Combined analytical FEM approach for efficient simulation of Lamb wave damage detection", Ultrasonics, 69, 116-128. https://doi.org/10.1016/j.ultras.2016.03.019.
  64. Shinozuka, M. and Ghanem, R. (1995), "Structural system identification. II: Experimental verification", J. Eng. Mech., 121(2), 265-273. https://doi.org/10.1061/(ASCE)0733-9399(1995)121:2(265).
  65. Shirzad‐Ghaleroudkhani, N., Mahsuli, M., Ghahari, S.F. and Taciroglu, E. (2018), "Bayesian identification of soil‐foundation stiffness of building structures", Struct. Control Health Monit., 25(3), e2090. https://doi.org/10.1002/stc.2090.
  66. Simsek, M., Kocaturk, T. and Akbas, S.D. (2013), "Static bending of a functionally graded microscale Timoshenko beam based on the modified couple stress theory", Compos. Struct., 95, 740-747. https://doi.org/10.1016/j.compstruct.2012.08.036.
  67. Slonski, M. (2014), "Sequential stochastic identification of elastic constants using Lamb waves and particle filters", Comput. Assist. Methods Eng. Sci., 21(1), 15-26.
  68. Sudret, B. (2008), "Global sensitivity analysis using polynomial chaos expansions", Reliab. Eng. Syst. Safety, 93(7), 964-979. https://doi.org/10.1016/j.ress.2007.04.002.
  69. Tam, J.H., Ong, Z.C., Ismail, Z., Ang, B.C. and Khoo, S.Y. (2017a), "Identification of material properties of composite materials using nondestructive vibrational evaluation approaches: A review", Mech. Adv. Mater. Struct., 24(12), 971-986. https://doi.org/10.1080/15376494.2016.1196798.
  70. Tam, J.H., Ong, Z.C., Lau, C.L., Ismail, Z., Ang, B.C. and Khoo, S.Y. (2017b), "Identification of material properties of composite plates using Fourier-generated frequency response functions", Mech. Adv. Mater. Struct., 2017, 1-10. https://doi.org/10.1080/15376494.2016.1196798.
  71. Vignoli, L.L., Savi, M.A., Pacheco, P.M. and Kalamkarov, A.L. (2019), "Comparative analysis of micromechanical models for the elastic composite laminae", Compos. Part B Eng., 174, 106961. https://doi.org/10.1016/j.compositesb.2019.106961.
  72. Vijayanand, V., Mokhtarishirazabad, M., Peng, J., Wang, Y., Gorley, M., Knowles, D. and Mostafavi, M. (2020), "A novel methodology for estimating tensile properties in a small punch test employing in-situ DIC based deflection mapping", J. Nucl. Mater., 538, 152260. https://doi.org/10.1016/j.jnucmat.2020.152260.
  73. Vishnuvardhan, J., Krishnamurthy, C. and Balasubramaniam, K. (2007a), "Genetic algorithm based reconstruction of elastic constants of orthotropic fibre-reinforced composite plates from ultrasonic velocity data from a single non-symmetry plane", Compos. Part B, 38, 216-227. https://doi.org/10.1016/j.compositesb.2006.06.006
  74. Vishnuvardhan, J., Krishnamurthy, C. and Balasubramaniam, K. (2007b), "Genetic algorithm based reconstruction of the elastic moduli of orthotropic plates using an ultrasonic guided wave single-transmitter-multiple-receiver SHM array", Smart Mater. Struct., 16(5), 1639. https://doi.org/10.1088/0964-1726/16/5/017.
  75. Warner, J.E. and Hochhalter, J.D. (2016), "Probabilistic damage characterization using the computationally-efficient bayesian approach", NASA Langley Research Center Hampton, USA.
  76. Webersen, M., Johannesmann, S., Duchting, J., Claes, L. and Henning, B. (2018), "Guided ultrasonic waves for determining effective orthotropic material parameters of continuous-fiber reinforced thermoplastic plates", Ultrasonics, 84, 53-62. https://doi.org/10.1016/j.ultras.2017.10.005.
  77. Xie, L., Zhou, Z., Zhao, L., Wan, C., Tang, H. and Xue, S. (2018), "Parameter identification for structural health monitoring with extended Kalman filter considering integration and noise effect", Appl. Sci., 8(12), 2480. https://doi.org/10.3390/app8122480.
  78. Xu, C., Yang, Z., Qiao, B. and Chen, X. (2020), "A parameter estimation based sparse representation approach for mode separation and dispersion compensation of Lamb waves in isotropic plate", Smart Mater. Struct., 29(3), 035020. https://doi.org/10.1088/1361-665X/ab6ce7.
  79. Yang, J.N., Pan, S. and Lin, S. (2004). "Identification and tracking of structural parameters with unknown excitations", Am. Control Conf., 5, 4189-4194. https://doi.org/10.23919/ACC.2004.1383965.
  80. Yuen, K.V. and Ortiz, G.A. (2018), "Multiresolution Bayesian nonparametric general regression for structural model updating", Struct. Control Health Monit., 25(2), e2077. https://doi.org/10.1002/stc.2077.
  81. Zhang, Q. and Zhao, J. (2013), "Determination of mechanical properties and full-field strain measurements of rock material under dynamic loads", Int. J. Rock Mech. Mining Sci., 60, 423-439. https://doi.org/10.1016/j.ijrmms.2013.01.005.
  82. Zhang, X., Qiang, B. and Greenleaf, J. (2011), "Comparison of the surface wave method and the indentation method for measuring the elasticity of gelatin phantoms of different concentrations", Ultrasonics, 51(2), 157-164. https://doi.org/10.1016/j.ultras.2010.07.005.
  83. Zhang, F., Xiong, H., Shi, W. and Ou, X. (2016a), "Structural health monitoring of Shanghai Tower during different stages using a Bayesian approach", Struct. Control Health Monit., 23(11), 1366-1384. https://doi.org/10.1002/stc.1840.
  84. Zhang, X., Gao, R.X., Yan, R., Chen, X., Sun, C. and Yang, Z. (2016b), "Multivariable wavelet finite element-based vibration model for quantitative crack identification by using particle swarm optimization", J. Sound Vib., 375, 200-216. https://doi.org/10.1016/j.jsv.2016.04.018.
  85. Zhang, F., Yang, Y., Ye, X., Yang, J. and Han, B. (2019), "Structural modal identification and MCMC-based model updating by a Bayesian approach", Smart Struct. Syst., Int. J., 24(5), 631-639. https://doi.org/10.12989/sss.2019.24.5.631.
  86. Zhou, J., Mita, A. and Mei, L. (2015), "Posterior density estimation for structural parameters using improved differential evolution adaptive Metropolis algorithm", Smart Struct. Syst., Int. J., 15(3), 735-749. https://doi.org/10.12989/sss.2015.15.3.735.
  87. Zou, F. and Aliabadi, M. (2017), "On modelling three-dimensional piezoelectric smart structures with boundary spectral element method", Smart Mater. Struct., 26(5), 055015. https://doi.org/10.1088/1361-665X/aa6664.