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A Critical Assessment of Kriging Model Variants for High-Fidelity Uncertainty Quantification in Dynamics of composite Shells

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Abstract

This paper presents a critical comparative assessment of Kriging model variants for surrogate based uncertainty propagation considering stochastic natural frequencies of composite doubly curved shells. The five Kriging model variants studied here are: Ordinary Kriging, Universal Kriging based on pseudo-likelihood estimator, Blind Kriging, Co-Kriging and Universal Kriging based on marginal likelihood estimator. First three stochastic natural frequencies of the composite shell are analysed by using a finite element model that includes the effects of transverse shear deformation based on Mindlin’s theory in conjunction with a layer-wise random variable approach. The comparative assessment is carried out to address the accuracy and computational efficiency of five Kriging model variants. Comparative performance of different covariance functions is also studied. Subsequently the effect of noise in uncertainty propagation is addressed by using the Stochastic Kriging. Representative results are presented for both individual and combined stochasticity in layer-wise input parameters to address performance of various Kriging variants for low dimensional and relatively higher dimensional input parameter spaces. The error estimation and convergence studies are conducted with respect to original Monte Carlo Simulation to justify merit of the present investigation. The study reveals that Universal Kriging coupled with marginal likelihood estimate yields the most accurate results, followed by Co-Kriging and Blind Kriging. As far as computational efficiency of the Kriging models is concerned, it is observed that for high-dimensional problems, CPU time required for building the Co-Kriging model is significantly less as compared to other Kriging variants.

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References

  1. Mallick PK (2007) Fiber-reinforced composites: materials, manufacturing, and design, 3rd edn. CRC Press, Boca Raton

    Book  Google Scholar 

  2. Baran I, Cinar K, Ersoy N, Akkerman R, Hattel JH (2016) A review on the mechanical modeling of composite manufacturing processes. Arch Comput Methods Eng. doi:10.1007/s11831-016-9167-2

    Google Scholar 

  3. Arregui-Mena JD, Margetts L, Mummery PM (2016) Practical application of the stochastic finite element method. Arch Comput Methods Eng 23(1):171–190

    Article  MathSciNet  MATH  Google Scholar 

  4. Venkatram A (1988) On the use of Kriging in the spatial analysis of acid precipitation data. Atmos Environ (1967) 22(9):1963–1975

    Article  Google Scholar 

  5. Fedorov VV (1989) Kriging and other estimators of spatial field characteristics (with special reference to environmental studies). Atm Environ (1967) 23(1):175–184

    Article  Google Scholar 

  6. Diamond P (1989) Fuzzy Kriging. Fuzzy Sets Syst 33(3):315–332

    Article  MathSciNet  MATH  Google Scholar 

  7. Carr JR (1990) UVKRIG: a FORTRAN-77 program for universal Kriging. Comput Geosci 16(2):211–236

    Article  Google Scholar 

  8. Deutsch CV (1996) Correcting for negative weights in ordinary Kriging. Comput Geosci 22(7):765–773

    Article  Google Scholar 

  9. Cressie NAC (1990) The origins of Kriging. Math Geol 22:239–252

    Article  MathSciNet  MATH  Google Scholar 

  10. Matheron G (1963) Principles of geostatistics. Econ Geol 58(8):1246–1266

    Article  Google Scholar 

  11. Cressie NAC (1993) Statistics for spatial data: revised edition. Wiley, New York

    Google Scholar 

  12. Montgomery DC (1991) Design and analysis of experiments. Wiley, New Jersey

    MATH  Google Scholar 

  13. Michael JB, Norman RD (1974) On minimum-point second-order designs. Technometrics 16(4):613–616

    Article  MathSciNet  MATH  Google Scholar 

  14. Martin JD, Simpson TW (2005) Use of Kriging models to approximate deterministic computer models. AIAA J 43(4):853–863

    Article  Google Scholar 

  15. Lee KH, Kang DH (2006) A robust optimization using the statistics based on Kriging metamodel. J Mech Sci Technol 20(8):1169–1182

    Article  Google Scholar 

  16. Sakata S, Ashida F, Zako M (2004) An efficient algorithm for Kriging approximation and optimization with large-scale sampling data. Comput Methods Appl Mech Eng 193:385–404

    Article  MATH  Google Scholar 

  17. Ryu J-S, Kim M-S, Cha K-J, Lee TH, Choi D-H (2002) Kriging interpolation methods in geostatistics and DACE model. KSME Int J 16(5):619–632

    Article  Google Scholar 

  18. Bayer V, Bucher C (1999) Importance sampling for first passage problems of nonlinear structures. Probab Eng Mech 14:27–32

    Article  Google Scholar 

  19. Yuan X, Lu Z, Zhou C, Yue Z (2013) A novel adaptive importance sampling algorithm based on Markov chain and low-discrepancy sequence. Aerosp Sci Technol 19:253–261

    Article  Google Scholar 

  20. Au SK, Beck JL (1999) A new adaptive importance sampling scheme for reliability calculations. Struct Saf 21:135–138

    Article  Google Scholar 

  21. Kamiński B (2015) A method for the updating of stochastic Kriging metamodels. Eur J Oper Res 247(3):859–866

    Article  MathSciNet  MATH  Google Scholar 

  22. Angelikopoulos P, Papadimitriou C, Koumoutsakos P (2015) X-TMCMC: adaptive Kriging for Bayesian inverse modeling. Comput Methods Appl Mech Eng 289:409–428

    Article  MathSciNet  Google Scholar 

  23. Peter J, Marcelet M (2008) Comparison of surrogate models for turbomachinery design. WSEAS Trans Fluid Mech 3(1):10–17

    Google Scholar 

  24. Dixit V, Seshadrinath N, Tiwari MK (2016) Performance measures based optimization of supply chain network resilience: a NSGA-II + Co-Kriging approach. Comput Ind Eng 93:205–214

    Article  Google Scholar 

  25. Huang C, Zhang H, Robeson SM (2016) Intrinsic random functions and universal Kriging on the circle. Stat Probab Lett 108:33–39

    Article  MathSciNet  MATH  Google Scholar 

  26. Tonkin MJ, Kennel J, Huber W, Lambie JM (2016) Multi-event universal Kriging (MEUK). Adv Water Resour 87:92–105

    Article  Google Scholar 

  27. Kersaudy P, Sudret B, Varsier N, Picon O, Wiart J (2015) A new surrogate modeling technique combining Kriging and polynomial chaos expansions: application to uncertainty analysis in computational dosimetry. J Comput Phys 286:103–117

    Article  MathSciNet  MATH  Google Scholar 

  28. Khodaparast HH, Mottershead JE, Badcock KJ (2011) Interval model updating with irreducible uncertainty using the Kriging predictor. Mech Syst Signal Process 25(4):1204–1226

    Article  Google Scholar 

  29. Nechak L, Gillot F, Besset S, Sinou JJ (2015) Sensitivity analysis and Kriging based models for robust stability analysis of brake systems. Mech Res Commun 69:136–145

    Article  Google Scholar 

  30. Pigoli D, Menafoglio A, Secchi P (2016) Kriging prediction for manifold-valued random fields. J Multivar Anal 145:117–131

    Article  MathSciNet  MATH  Google Scholar 

  31. Wang D, DiazDelaO FA, Wang W, Lin X, Patterson EA, Mottershead JE (2016) Uncertainty quantification in DIC with Kriging regression. Opt Lasers Eng 78:182–195

    Article  Google Scholar 

  32. Jeong S, Mitsuhiro M, Kazuomi Y (2005) Efficient optimization design method using Kriging model. J Aircr 42(2):413–420

    Article  Google Scholar 

  33. Hanefi B, Turalioglu SF (2005) A Kriging-based approach for locating a sampling site: in the assessment of air quality. Stoch Environ Res Risk Assess 19(4):301–305

    Article  MATH  Google Scholar 

  34. Den Hertog D, Kleijnen JPC, Siem AYD (2006) The correct Kriging variance estimated by bootstrapping. J Oper Res Soc 57(4):400–409

    Article  MATH  Google Scholar 

  35. Xavier E (2005) Simple and ordinary multigaussian Kriging for estimating recoverable reserves. Math Geol 37(3):295–319

    Article  MATH  Google Scholar 

  36. Martin JD, Simpson TW (2004) On using Kriging models as probabilistic models in design. SAE Trans J Mater Manuf 5:129–139

    Google Scholar 

  37. Elsayed K (2015) Optimization of the cyclone separator geometry for minimum pressure drop using Co-Kriging. Powder Technol 269:409–424

    Article  Google Scholar 

  38. Thai CH, Do VNV, Nguyen-Xuan H (2016) An improved moving Kriging-based meshfree method for static, dynamic and buckling analyses of functionally graded isotropic and sandwich plates. Eng Anal Bound Elem 64:122–136

    Article  MathSciNet  Google Scholar 

  39. Yang X, Liu Y, Zhang Y, Yue Z (2015) Probability and convex set hybrid reliability analysis based on active learning Kriging model. Appl Math Model 39(14):3954–3971

    Article  MathSciNet  Google Scholar 

  40. Gaspar B, Teixeira AP, Guedes SC (2014) Assessment of the efficiency of Kriging surrogate models for structural reliability analysis. Probab Eng Mech 37:24–34

    Article  Google Scholar 

  41. Huang X, Chen J, Zhu H (2016) Assessing small failure probabilities by AK–SS: an active learning method combining Kriging and subset simulation. Struct Saf 59:86–95

    Article  Google Scholar 

  42. Kwon H, Choi S (2015) A trended Kriging model with R 2 indicator and application to design optimization. Aerosp Sci Technol 43:111–125

    Article  Google Scholar 

  43. Sakata S, Ashida F, Zako M (2008) Kriging-based approximate stochastic homogenization analysis for composite materials. Comput Methods Appl Mech Eng 197(21–24):1953–1964

    Article  MATH  Google Scholar 

  44. Luersen MA, Steeves CA, Nair PB (2015) Curved fiber paths optimization of a composite cylindrical shell via Kriging-based approach. J Compos Mater 49(29):3583–3597

    Article  Google Scholar 

  45. Qatu MS, Leissa AW (1991) Natural frequencies for cantilevered doubly curved laminated composite shallow shells. Compos Struct 17:227–255

    Article  Google Scholar 

  46. Qatu MS, Leissa AW (1991) Vibration studies for laminated composite twisted cantilever plates. Int J Mech Sci 33(11):927–940

    Article  Google Scholar 

  47. Chakravorty D, Bandyopadhyay JN, Sinha PK (1995) Free vibration analysis of point supported laminated composite doubly curved shells: a finite element approach. Comput Struct 54(2):191–198

    Article  MATH  Google Scholar 

  48. Dey S, Karmakar A (2012) Free vibration analyses of multiple delaminated angle-ply composite conical shells: a finite element approach. Compos Struct 94(7):2188–2196

    Article  Google Scholar 

  49. Leissa AW, Narita Y (1984) Vibrations of corner point supported shallow shells of rectangular planform. Earthq Eng Struct Dyn 12:651–661

    Article  Google Scholar 

  50. Carrera E (2003) Theories and finite elements for multilayered plates and shells: a unified compact formulation with numerical assessment and benchmarking. Arch Comput Methods Eng 10(3):215–296

    Article  MathSciNet  MATH  Google Scholar 

  51. Hu HT, Peng HW (2013) Maximization of fundamental frequency of axially compressed laminated curved panels with cutouts. Compos B Eng 47:8–25

    Article  Google Scholar 

  52. Ghavanloo E, Fazelzadeh SA (2013) Free vibration analysis of orthotropic doubly curved shallow shells based on the gradient elasticity. Compos B Eng 45(1):1448–1457

    Article  Google Scholar 

  53. Tornabene F, Brischetto S, Fantuzzia N, Violaa E (2015) Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels. Compos B Eng 81:231–250

    Article  Google Scholar 

  54. Fazzolari FA (2014) A refined dynamic stiffness element for free vibration analysis of cross-ply laminated composite cylindrical and spherical shallow shells. Compos B Eng 62:143–158

    Article  Google Scholar 

  55. Mantari JL, Oktem AS, Guedes SC (2012) Bending and free vibration analysis of isotropic and multilayered plates and shells by using a new accurate higher-order shear deformation theory. Compos B Eng 43(8):3348–3360

    Article  MATH  Google Scholar 

  56. Fang C, Springer GS (1993) Design of composite laminates by a Monte Carlo method. Compos Mater 27(7):721–753

    Article  Google Scholar 

  57. Mahadevan S, Liu X, Xiao Q (1997) A probabilistic progressive failure model for composite laminates. J Reinf Plast Compos 16(11):1020–1038

    Google Scholar 

  58. Dey S, Mukhopadhyay T, Spickenheuer A, Adhikari S, Heinrich G (2016) Bottom up surrogate based approach for stochastic frequency response analysis of laminated composite plates. Compos Struct 140:712–772

    Article  Google Scholar 

  59. Dey S, Mukhopadhyay T, Adhikari S (2015) Stochastic free vibration analyses of composite doubly curved shells: a Kriging model approach. Compos B Eng 70:99–112

    Article  Google Scholar 

  60. Dey S, Mukhopadhyay T, Khodaparast HH, Kerfriden P, Adhikari S (2015) Rotational and ply-level uncertainty in response of composite shallow conical shells. Compos Struct 131:594–605

    Article  Google Scholar 

  61. Pandit MK, Singh BN, Sheikh AH (2008) Buckling of laminated sandwich plates with soft core based on an improved higher order zigzag theory. Thin-Walled Struct 46(11):1183–1191

    Article  Google Scholar 

  62. Dey S, Mukhopadhyay T, Sahu SK, Li G, Rabitz H, Adhikari S (2015) Thermal uncertainty quantification in frequency responses of laminated composite plates. Compos B Eng 80:186–197

    Article  Google Scholar 

  63. Dey S, Mukhopadhyay T, Spickenheuer A, Gohs U, Adhikari S (2016) Uncertainty quantification in natural frequency of composite plates: an artificial neural network based approach. Adv Compos Lett (accepted)

  64. Mukhopadhyay T, Naskar S, Dey S, Adhikari S (2016) On quantifying the effect of noise in surrogate based stochastic free vibration analysis of laminated composite shallow shells. Compos Struct 140:798–805

    Article  Google Scholar 

  65. Shaw A, Sriramula S, Gosling PD, Chryssanthopoulos MK (2010) A critical reliability evaluation of fibre reinforced composite materials based on probabilistic micro and macro-mechanical analysis. Compos B Eng 41(6):446–453

    Article  Google Scholar 

  66. Dey S, Mukhopadhyay T, Khodaparast HH, Adhikari S (2016) Fuzzy uncertainty propagation in composites using Gram–Schmidt polynomial chaos expansion. Appl Math Model 40(7–8):4412–4428

    Article  MathSciNet  Google Scholar 

  67. Dey S, Naskar S, Mukhopadhyay T, Gohs U, Spickenheuer A, Bittrich L, Sriramula S, Adhikari S, Heinrich G (2016) Uncertain natural frequency analysis of composite plates including effect of noise: a polynomial neural network approach. Compos Struct 143:130–142

    Article  Google Scholar 

  68. Afeefa S, Abdelrahman WG, Mohammad T, Edward S (2008) Stochastic finite element analysis of the free vibration of laminated composite plates. Comput Mech 41:495–501

    MATH  Google Scholar 

  69. Dey S, Mukhopadhyay T, Adhikari S (2015) Stochastic free vibration analysis of angle-ply composite plates: a RS-HDMR approach. Compos Struct 122:526–553

    Article  Google Scholar 

  70. Loja MAR, Barbosa JI, Mota Soares CM (2015) Dynamic behaviour of soft core sandwich beam structures using Kriging-based layerwise models. Compos Struct 134:883–894

    Article  Google Scholar 

  71. Dey S, Mukhopadhyay T, Khodaparast HH, Adhikari S (2015) Stochastic natural frequency of composite conical shells. Acta Mech 226(8):2537–2553

    Article  MathSciNet  MATH  Google Scholar 

  72. Singh BN, Yadav D, Iyengar NGR (2001) Natural frequencies of composite plates with random material properties using higher-order shear deformation theory. Int J Mech Sci 43(10):2193–2214

    Article  MATH  Google Scholar 

  73. Tripathi V, Singh BN, Shukla KK (2007) Free vibration of laminated composite conical shells with random material properties. Compos Struct 81(1):96–104

    Article  Google Scholar 

  74. McKay MD, Beckman RJ, Conover WJ (2000) A comparison of three methods for selecting values of input variables in the analysis of output from a computer code. Technometrics 42(1):55–61

    Article  MATH  Google Scholar 

  75. Bathe KJ (1990) Finite element procedures in engineering analysis. Prentice Hall Inc., New Delhi

    Google Scholar 

  76. Meirovitch L (1992) Dynamics and control of structures. Wiley, New York

    Google Scholar 

  77. Krige DG (1951) A statistical approach to some basic mine valuation problems on the witwatersrand. J Chem Metall Min Soc S Afr 52:119–139

    Google Scholar 

  78. Hengl T, Heuvelink GBM, Rossiter DG (2007) About regression-Kriging: from equations to case studies. Comput Geosci 33:1301–1315

    Article  Google Scholar 

  79. Matías JM, González-Manteiga W (2005) Regularized Kriging as a generalization of simple, universal, and bayesian Kriging. Stoch Environ Res Risk Assess 20:243–258

    Article  MathSciNet  Google Scholar 

  80. Omre H, Halvorsen KB (1989) The Bayesian bridge between simple and universal kriging. Math Geol 21:767–786

    Article  MathSciNet  MATH  Google Scholar 

  81. Tonkin MJ, Larson SP (2002) Kriging Water Levels with a Regional-Linear and Point-Logarithmic Drift. Ground Water 40:185–193

    Article  Google Scholar 

  82. Warnes JJ (1986) A sensitivity analysis for universal Kriging. Math Geol 18:653–676

    MathSciNet  Google Scholar 

  83. Stein A, Corsten CA (1991) Universal Kriging and cokriging as a regression procedure on JSTOR. Biometrics 47:575–587

    Article  Google Scholar 

  84. Olea RA (2011) Optimal contour mapping using Kriging. J Geophys Res 79:695–702

    Article  Google Scholar 

  85. Ghiasi Y, Nafisi V (2015) The improvement of strain estimation using universal Kriging. Acta Geod Geophys 50:479–490

    Article  Google Scholar 

  86. Li L, Romary T, Caers J (2015) Universal kriging with training images. Spat Stat 14:240–268

    Article  MathSciNet  Google Scholar 

  87. Joseph VR, Hung Y, Sudjianto A (2008) Blind Kriging: a new method for developing metamodels. J Mech Des 130:031102

    Article  Google Scholar 

  88. Hung Y (2011) Penalized blind Kriging in computer experiments. Stat Sin 21:1171–1190

    Article  MathSciNet  MATH  Google Scholar 

  89. Couckuyt I, Forrester A, Gorissen D et al (2012) Blind Kriging: implementation and performance analysis. Adv Eng Softw 49:1–13

    Article  Google Scholar 

  90. Koziel S, Bekasiewicz A, Couckuyt I, Dhaene T (2014) Efficient multi-objective simulation-driven antenna design using Co-Kriging. IEEE Trans Antennas Propag 62:5900–5905

    Article  MathSciNet  Google Scholar 

  91. Elsayed K (2015) Optimization of the cyclone separator geometry for minimum pressure drop using Co-Kriging. Powder Technol 269:409–424

    Article  Google Scholar 

  92. Clemens M, Seifert J (2015) Dimension reduction for the design optimization of large scale high voltage devices using co-Kriging surrogate modeling. IEEE Trans Magn 51:1–4

    Google Scholar 

  93. Perdikaris P, Venturi D, Royset JO, Karniadakis GE (2015) Multi-fidelity modelling via recursive Co-Kriging and Gaussian–Markov random fields. Proc Math Phys Eng Sci 471:20150018

    Article  Google Scholar 

  94. Kamiński B (2015) A method for the updating of stochastic Kriging metamodels. Eur J Oper Res 247:859–866

    Article  MathSciNet  MATH  Google Scholar 

  95. Qu H, Fu MC (2014) Gradient extrapolated stochastic Kriging. ACM Trans Model Comput Simul 24:1–25

    Article  MathSciNet  Google Scholar 

  96. Chen X, Kim K-K (2014) Stochastic Kriging with biased sample estimates. ACM Trans Model Comput Simul 24:1–23

    Article  MathSciNet  MATH  Google Scholar 

  97. Wang K, Chen X, Yang F et al (2014) A new stochastic Kriging method for modeling multi-source exposure-response data in toxicology studies. ACS Sustain Chem Eng 2:1581–1591

    Article  Google Scholar 

  98. Wang B, Bai J, Gea HC (2013) Stochastic Kriging for random simulation metamodeling with finite sampling. In: 39th Design automation conference ASME, vol 3B, p V03BT03A056

  99. Chen X, Ankenman BE, Nelson BL (2013) Enhancing stochastic Kriging metamodels with gradient estimators. Oper Res 61:512–528

    Article  MathSciNet  MATH  Google Scholar 

  100. Chen X, Nelson BL, Kim K-K (2012) Stochastic Kriging for conditional value-at-risk and its sensitivities. In: Proceedings of title proceedings 2012 winter simulation conference. IEEE, pp 1–12

  101. Kennedy M, O’Hagan A (2000) Predicting the output from a complex computer code when fast approximations are available. Biometrika 87:1–13

    Article  MathSciNet  MATH  Google Scholar 

  102. Rivest M, Marcotte D (2012) Kriging groundwater solute concentrations using flow coordinates and nonstationary covariance functions. J Hydrol 472–473:238–253

    Article  Google Scholar 

  103. Biscay Lirio R, Camejo DG, Loubes J-M, Muñiz Alvarez L (2013) Estimation of covariance functions by a fully data-driven model selection procedure and its application to Kriging spatial interpolation of real rainfall data. Stat Methods Appl 23:149–174

    Article  MathSciNet  MATH  Google Scholar 

  104. Putter H, Young GA (2001) On the effect of covariance function estimation on the accuracy of Kriging predictors. Bernoulli 7:421–438

    Article  MathSciNet  MATH  Google Scholar 

  105. Mukhopadhyay T, Chowdhury R, Chakrabarti A (2016) Structural damage identification: a random sampling-high dimensional model representation approach. Adv Struct Eng. doi:10.1177/1369433216630370

    Google Scholar 

  106. Mukhopadhyay T, Dey TK, Chowdhury R, Chakrabarti A (2015) Structural damage identification using response surface based multi-objective optimization: a comparative study. Arabian J Sci Eng 40(4):1027–1044

    Article  MathSciNet  Google Scholar 

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Acknowledgments

TM acknowledges the financial support from Swansea University through the award of Zienkiewicz Scholarship during the period of this work. SC acknowledges the support of MHRD, Government of India for the financial support provided during this work. SA acknowledges the financial support from The Royal Society of London through the Wolfson Research Merit award. RC acknowledges the support of The Royal Society through Newton Alumni Funding.

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  1. College of Engineering, Swansea University, Swansea, UK

    T. Mukhopadhyay & S. Adhikari

  2. Department of Civil Engineering, Indian Institute of Technology Roorkee, Roorkee, India

    S. Chakraborty & R. Chowdhury

  3. Leibniz-Institut für Polymerforschung Dresden e.V., Dresden, Germany

    S. Dey

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Mukhopadhyay, T., Chakraborty, S., Dey, S. et al. A Critical Assessment of Kriging Model Variants for High-Fidelity Uncertainty Quantification in Dynamics of composite Shells. Arch Computat Methods Eng 24, 495–518 (2017). https://doi.org/10.1007/s11831-016-9178-z

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