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Applied Mathematics and Mechanics

, Volume 40, Issue 1, pp 63–84 | Cite as

Uncertainty quantification for stochastic dynamical systems using time-dependent stochastic bases

  • Jinchun Lan
  • Qianlong Zhang
  • Sha Wei
  • Zhike Peng
  • Xinjian Dong
  • Wenming Zhang
Article
  • 29 Downloads

Abstract

A novel method based on time-dependent stochastic orthogonal bases for stochastic response surface approximation is proposed to overcome the problem of significant errors in the utilization of the generalized polynomial chaos (GPC) method that approximates the stochastic response by orthogonal polynomials. The accuracy and effectiveness of the method are illustrated by different numerical examples including both linear and nonlinear problems. The results indicate that the proposed method modifies the stochastic bases adaptively, and has a better approximation for the probability density function in contrast to the GPC method.

Key words

uncertainty quantification stochastic response surface approximation time-dependent orthogonal bases polynomial chaos 

Chinese Library Classification

O213 O313 

2010 Mathematics Subject Classification

60H35 

References

  1. [1]
    ABGRALL, R. and CONGEDO, P. M. A semi-intrusive deterministic approach to uncertainty quantification in non-linear fluid flow problems. Journal of Computational Physics, 235, 828–845 (2013)MathSciNetCrossRefGoogle Scholar
  2. [2]
    SIMON, F., GUILLEN, P., SAGAUT, P., and LUCOR, D. A GPC-based approach to uncertain transonic aerodynamics. Computer Methods in Applied Mechanics and Engineering, 199, 1091–1099 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    SEPAHVAND, K., MARBURG, S., and HARDTKE, H. J. Stochastic free vibration of orthotropic plates using generalized polynomial chaos expansion. Journal of Sound and Vibration, 331, 167–179 (2012)CrossRefzbMATHGoogle Scholar
  4. [4]
    CAPIEZ-LERNOUT, E., SOIZE, C., and MIGNOLET, M. P. Post-buckling nonlinear static and dynamical analyses of uncertain cylindrical shells and experimental validation. Computer Methods in Applied Mechanics and Engineering, 271, 210–230 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    JACQUELIN, E., ADHIKARI, S., SINOU, J. J., and FRISWELL, M. I. Polynomial chaos expansion in structural dynamics: accelerating the convergence of the first two statistical moment sequences. Journal of Sound and Vibration, 356, 144–154 (2015)CrossRefGoogle Scholar
  6. [6]
    ZHANG, J. and ELLINGWOOD, B. Effects of uncertain material properties on structural stability. Journal of Structural Engineering, 121, 705–716 (1995)CrossRefGoogle Scholar
  7. [7]
    SINGH, B. N., IYENGAR, N., and YADAV, D. Effects of random material properties on buckling of composite plates. Journal of Engineering Mechanics, 127, 873–879 (2001)CrossRefGoogle Scholar
  8. [8]
    XIU, D. and KARNIADAKIS, G. E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM Journal on Scientific Computing, 24, 619–644 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    XIU, D. Numerical Methods for Stochastic Computations: A Spectral Method Approach, Princeton University Press, Princeton (2010)CrossRefzbMATHGoogle Scholar
  10. [10]
    WAN, X. and KARNIADAKIS, G. E. Multi-element generalized polynomial chaos for arbitrary probability measures. SIAM Journal on Scientific Computing, 28, 901–928 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    GHANEM, R. G. and SPANOS, P. D. Stochastic Finite Elements: A Spectral Approach, Dover Publications, Inc., New York (2003)zbMATHGoogle Scholar
  12. [12]
    XIU, D. and HESTHAVEN, J. S. High-order collocation methods for differential equations with random inputs. SIAM Journal on Scientific Computing, 27, 1118–1139 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    MA, X. and ZABARAS, N. An adaptive hierarchical sparse grid collocation algorithm for the solution of stochastic differential equations. Journal of Computational Physics, 228, 3084–3113 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    GANIS, B., KLIE, H., WHEELER, M. F., WILDEY, T., YOTOV, I., and ZHANG, D. Stochastic collocation and mixed finite elements for flow in porous media. Computer Methods in Applied Mechanics and Engineering, 197, 3547–3559 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    NEGOITA, C., ZADEH, L., and ZIMMERMANN, H. Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets and Systems, 1, 3–28 (1978)MathSciNetCrossRefGoogle Scholar
  16. [16]
    KLIMKE, A. and WOHLMUTH, B. Computing expensive multivariate functions of fuzzy numbers using sparse grids. Fuzzy Sets and Systems, 154, 432–453 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    MOORE, R. E. Interval Analysis, Prentice-Hall Englewood Cliffs, New York (1966)Google Scholar
  18. [18]
    IMPOLLONIA, N. and MUSCOLINO, G. Interval analysis of structures with uncertain-butbounded axial stiffness. Computer Methods in Applied Mechanics and Engineering, 200, 1945–1962 (2011)CrossRefzbMATHGoogle Scholar
  19. [19]
    YIN, S., YU, D., YIN, H., and XIA, B. Interval and random analysis for structure-acoustic systems with large uncertain-but-bounded parameters. Computer Methods in Applied Mechanics and Engineering, 305, 910–935 (2016)MathSciNetCrossRefGoogle Scholar
  20. [20]
    SOIZE, C. Random matrix theory for modeling uncertainties in computational mechanics. Computer Methods in Applied Mechanics and Engineering, 194, 1333–1366 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    SOIZE, C. Random matrix theory and non-parametric model of random uncertainties in vibration analysis. Journal of Sound and Vibration, 263, 893–916 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    XIU, D. and KARNIADAKIS, G. E. Modeling uncertainty in flow simulations via generalized polynomial chaos. Journal of Computational Physics, 187, 137–167 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    BLATMAN, G. and SUDRET, B. Adaptive sparse polynomial chaos expansion based on least angle regression. Journal of Computational Physics, 230, 2345–2367 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    ROSIĆ, B. V., LITVINENKO, A., PAJONK, O., and MATTHIES, H. G. Sampling-free linear Bayesian update of polynomial chaos representations. Journal of Computational Physics, 231, 5761–5787 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    GERRITSMA, M., VAN DER STEEN, J. B., VOS, P., and KARNIADAKIS, G. Time-dependent generalized polynomial chaos. Journal of Computational Physics, 229, 8333–8363 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    PANUNZIO, A. M., SALLES, L., and SCHWINGSHACKL, C. W. Uncertainty propagation for nonlinear vibrations: a non-intrusive approach. Journal of Sound and Vibration, 389, 309–325 (2017)CrossRefGoogle Scholar
  27. [27]
    NAJM, H. N. Uncertainty quantification and polynomial chaos techniques in computational fluid dynamics. Annual Review of Fluid Mechanics, 41, 35–52 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    TOOTKABONI, M., ASADPOURE, A., and GUEST, J. K. Topology optimization of continuum structures under uncertainty—a polynomial chaos approach. Computer Methods in Applied Mechanics and Engineering, 201, 263–275 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    GHANEM, R. and GHOSH, D. Efficient characterization of the random eigenvalue problem in a polynomial chaos decomposition. International Journal for Numerical Methods in Engineering, 72, 486–504 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    PASCUAL, B. and ADHIKARI, S. Hybrid perturbation-polynomial chaos approaches to the random algebraic eigenvalue problem. Computer Methods in Applied Mechanics and Engineering, 217, 153–167 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    MANAN, A. and COOPER, J. Prediction of uncertain frequency response function bounds using polynomial chaos expansion. Journal of Sound and Vibration, 329, 3348–3358 (2010)CrossRefGoogle Scholar
  32. [32]
    PENG, Y. B., GHANEM, R., and LI, J. Polynomial chaos expansions for optimal control of nonlinear random oscillators. Journal of Sound and Vibration, 329, 3660–3678 (2010)CrossRefGoogle Scholar
  33. [33]
    WAN, X. and KARNIADAKIS, G. E. Long-term behavior of polynomial chaos in stochastic flow simulations. Computer Methods in Applied Mechanics and Engineering, 195, 5582–5596 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  34. [34]
    ORSZAG, S. A. and BISSONNETTE, L. R. Dynamical properties of truncated Wiener-Hermite expansions. Physics of Fluids, 10, 2603–2613 (1967)CrossRefzbMATHGoogle Scholar
  35. [35]
    HEUVELINE, V. and SCHICK, M. A local time-dependent generalized polynomial chaos method for stochastic dynamical systems. Preprint (2011) https://doi.org/10.11588/emclpp.2011.04.11694 Google Scholar
  36. [36]
    BECK, M. H., JÄCKLE, A., WORTH, G., and MEYER, H. D. The multiconfiguration timedependent Hartree (MCTDH) method: a highly efficient algorithm for propagating wavepackets. Physics Reports, 324, 1–105 (2000)CrossRefGoogle Scholar
  37. [37]
    DE LATHAUWER, L., DE MOOR, B., and VANDEWALLE, J. A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21, 1253–1278 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    DE LATHAUWER, L., DE MOOR, B., and VANDEWALLE, J. On the best rank-1 and rank-(r 1, r 2, · · ·, r n) approximation of higher-order tensors. SIAM Journal on Matrix Analysis and Applications, 21, 1324–1342 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Jinchun Lan
    • 1
  • Qianlong Zhang
    • 1
  • Sha Wei
    • 1
  • Zhike Peng
    • 1
  • Xinjian Dong
    • 1
  • Wenming Zhang
    • 1
  1. 1.State Key Laboratory of Mechanical System and Vibration, School of Mechanical EngineeringShanghai Jiao Tong UniversityShanghaiChina

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