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Frontiers of Mechanical Engineering

, Volume 14, Issue 1, pp 33–46 | Cite as

Uncertainty propagation analysis by an extended sparse grid technique

  • X. Y. Jia
  • C. JiangEmail author
  • C. M. Fu
  • B. Y. Ni
  • C. S. Wang
  • M. H. Ping
Open Access
Research Article
  • 218 Downloads
Part of the following topical collections:
  1. Innovative Design and Intelligent Design

Abstract

In this paper, an uncertainty propagation analysis method is developed based on an extended sparse grid technique and maximum entropy principle, aiming at improving the solving accuracy of the high-order moments and hence the fitting accuracy of the probability density function (PDF) of the system response. The proposed method incorporates the extended Gauss integration into the uncertainty propagation analysis. Moreover, assisted by the Rosenblatt transformation, the various types of extended integration points are transformed into the extended Gauss-Hermite integration points, which makes the method suitable for any type of continuous distribution. Subsequently, within the sparse grid numerical integration framework, the statistical moments of the system response are obtained based on the transformed points. Furthermore, based on the maximum entropy principle, the obtained first four-order statistical moments are used to fit the PDF of the system response. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed method, which includes two mathematical problems with explicit expressions and an engineering application with a black-box model.

Keywords

uncertainty propagation analysis extended sparse grid maximum entropy principle extended Gauss integration Rosenblatt transformation high-order moments analysis 

Notes

Acknowledgements

This work was supported by the National Science Fund for Distinguished Young Scholars (Grant No. 51725502), the major program of the National Natural Science Foundation of China (Grant No. 51490662), and the National Key Research and Development Project of China (Grant No. 2016YFD0701105).

References

  1. 1.
    Lee S H, Chen W. A comparative study of uncertainty propagation methods for black-box-type problems. Structural and Multidisciplinary Optimization, 2009, 37(3): 239–253CrossRefGoogle Scholar
  2. 2.
    Wang X, Wang L, Qiu Z. Response analysis based on smallest interval-set of parameters for structures with uncertainty. Applied Mathematics and Mechanics, 2012, 33(9): 1153–1166CrossRefGoogle Scholar
  3. 3.
    Wang X, Wang L, Qiu Z. A feasible implementation procedure for interval analysis method from measurement data. Applied Mathematical Modelling, 2014, 38(9–10): 2377–2397MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Qiu Z P, Wang L. The need for introduction of non-probabilistic interval conceptions into structural analysis and design. Science China. Physics, Mechanics & Astronomy, 2016, 59(11): 114632CrossRefGoogle Scholar
  5. 5.
    Gu X, Renaud J E, Batill S M, et al. Worst case propagated uncertainty of multidisciplinary systems in robust design optimization. Structural and Multidisciplinary Optimization, 2000, 20(3): 190–213CrossRefGoogle Scholar
  6. 6.
    Li M, Azarm S. Multiobjective collaborative robust optimization with interval uncertainty and interdisciplinary uncertainty propagation. Journal of Mechanical Design, 2008, 130(8): 081402CrossRefGoogle Scholar
  7. 7.
    Li G, Zhang K. A combined reliability analysis approach with dimension reduction method and maximum entropy method. Structural and Multidisciplinary Optimization, 2011, 43(1): 121–134MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Jiang Z, Li W, Apley D W, et al. A spatial-random-process based multidisciplinary system uncertainty propagation approach with model uncertainty. Journal of Mechanical Design, 2015, 137(10): 101402CrossRefGoogle Scholar
  9. 9.
    Mazo J, El Badry A T, Carreras J, et al. Uncertainty propagation and sensitivity analysis of thermo-physical properties of phase change materials (PCM) in the energy demand calculations of a test cell with passive latent thermal storage. Applied Thermal Engineering, 2015, 90: 596–608CrossRefGoogle Scholar
  10. 10.
    Li M, Mahadevan S, Missoum S, et al. Special issue: Simulationbased design under uncertainty. Journal of Mechanical Design, 2016, 138(11): 110301CrossRefGoogle Scholar
  11. 11.
    Madsen H O, Krenk S, Lind N C. Methods of structural safety. Mineola: Dover Publications, 2006Google Scholar
  12. 12.
    Wilson B M, Smith B L. Taylor-series and Monte-Carlo-method uncertainty estimation of the width of a probability distribution based on varying bias and random error. Measurement Science & Technology, 2013, 24(3): 035301CrossRefGoogle Scholar
  13. 13.
    Rochman D, Zwermann W, van der Marck S C, et al. Efficient use of Monte Carlo: Uncertainty propagation. Nuclear Science and Engineering, 2014, 177(3): 337–349CrossRefGoogle Scholar
  14. 14.
    Hong J, Shaked S, Rosenbaum R K, et al. Analytical uncertainty propagation in life cycle inventory and impact assessment: Application to an automobile front panel. International Journal of Life Cycle Assessment, 2010, 15(5): 499–510CrossRefGoogle Scholar
  15. 15.
    Xu L. A proportional differential control method for a time-delay system using the Taylor expansion approximation. Applied Mathematics and Computation, 2014, 236: 391–399MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Low B. FORM, SORM, and spatial modeling in geotechnical engineering. Structural Safety, 2014, 49: 56–64CrossRefGoogle Scholar
  17. 17.
    Lim J, Lee B, Lee I. Post optimization for accurate and efficient reliability-based design optimization using second-order reliability method based on importance sampling and its stochastic sensitivity analysis. International Journal for Numerical Methods in Engineering, 2015, 107(2): 93–108MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lee I, Choi K K, Gorsich D. Sensitivity analyses of FORM-based and DRM-based performance measure approach (PMA) for reliability-based design optimization (RBDO). International Journal for Numerical Methods in Engineering, 2010, 82(1): 26–46zbMATHGoogle Scholar
  19. 19.
    Sudret B. Global sensitivity analysis using polynomial chaos expansions. Reliability Engineering & System Safety, 2008, 93(7): 964–979CrossRefGoogle Scholar
  20. 20.
    Kersaudy P, Sudret B, Varsier N, et al. A new surrogate modeling technique combining Kriging and polynomial chaos expansions— Application to uncertainty analysis in computational dosimetry. Journal of Computational Physics, 2015, 286: 103–117MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rajabi M M, Ataie-Ashtiani B, Simmons C T. Polynomial chaos expansions for uncertainty propagation and moment independent sensitivity analysis of seawater intrusion simulations. Journal of Hydrology (Amsterdam), 2015, 520: 101–122CrossRefGoogle Scholar
  22. 22.
    Rahman S, Xu H. A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics. Probabilistic Engineering Mechanics, 2004, 19(4): 393–408CrossRefGoogle Scholar
  23. 23.
    Xu H, Rahman S. A generalized dimension-reduction method for multidimensional integration in stochastic mechanics. International Journal for Numerical Methods in Engineering, 2004, 61(12): 1992–2019CrossRefzbMATHGoogle Scholar
  24. 24.
    Nobile F, Tempone R, Webster C G. A sparse grid stochastic collocation method for partial differential equations with random input data. SIAM Journal on Numerical Analysis, 2008, 46(5): 2309–2345MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Xiong F, Greene S, Chen W, et al. A new sparse grid based method for uncertainty propagation. Structural and Multidisciplinary Optimization, 2010, 41(3): 335–349MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    He J, Gao S, Gong J. A sparse grid stochastic collocation method for structural reliability analysis. Structural Safety, 2014, 51: 29–34CrossRefGoogle Scholar
  27. 27.
    Smolyak S A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Doklady Akademii Nauk SSSR, 1963, 4: 240–243zbMATHGoogle Scholar
  28. 28.
    Novak E, Ritter K. High dimensional integration of smooth functions over cubes. Numerische Mathematik, 1996, 75(1): 79–97MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Novak E, Ritter K. Simple cubature formulas with high polynomial exactness. Constructive Approximation, 1999, 15(4): 499–522MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Bathe K, Wilson E. Stability and accuracy analysis of direct integration methods. Earthquake Engineering & Structural Dynamics, 1972, 1(3): 283–291CrossRefGoogle Scholar
  31. 31.
    Tao J, Zeng X, Cai W, et al. Stochastic sparse-grid collocation algorithm (SSCA) for periodic steady-state analysis of nonlinear system with process variations. In: Proceedings of the 2007 Asia and South Pacific Design Automation Conference. IEEE, 2007, 474–479Google Scholar
  32. 32.
    Jia B, Xin M, Cheng Y. Sparse Gauss-Hermite quadrature filter for spacecraft attitude estimation. In: Proceedings of the 2010 American Control Conference. Baltimore: IEEE, 2010, 2873–2878Google Scholar
  33. 33.
    Petvipusit K R, Elsheikh A H, Laforce T C, et al. Robust optimisation of CO2 sequestration strategies under geological uncertainty using adaptive sparse grid surrogates. Computational Geosciences, 2014, 18(5): 763–778CrossRefGoogle Scholar
  34. 34.
    Chen H, Cheng X, Dai C, et al. Accuracy, efficiency and stability analysis of sparse-grid quadrature Kalman filter in near space hypersonic vehicles. In: Proceedings of Position, Location and Navigation Symposium-PLANS 2014, 2014 IEEE/ION. Monterey: IEEE, 2014, 27–36Google Scholar
  35. 35.
    Kendall M G, Stuart A. The Advanced Theory of Statistics Volume 1: Distribution Theory. London: Charles Griffin & Company, 1958Google Scholar
  36. 36.
    Press WH, Teukolsky S A, Vetterling WT, et al. Numerical Recipes in C. Cambridge: Cambridge University Press, 1996zbMATHGoogle Scholar
  37. 37.
    Ghosh D D, Olewnik A. Computationally efficient imprecise uncertainty propagation. Journal of Mechanical Design, 2013, 135(5): 051002CrossRefGoogle Scholar
  38. 38.
    Ahlfeld R, Belkouchi B, Montomoli F. SAMBA: Sparse approximation of moment-based arbitrary polynomial chaos. Journal of Computational Physics, 2016, 320: 1–16MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Patterson T. Modified optimal quadrature extensions. Numerische Mathematik, 1993, 64(1): 511–520MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Kronrod A S. Nodes and Weights of Quadrature Formulas: Sixteenplace Tables. New York: Consultants Bureau, 1965zbMATHGoogle Scholar
  41. 41.
    Patterson T. The optimum addition of points to quadrature formulae. Mathematics of Computation, 1968, 22(104): 847–856MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Genz A, Keister B D. Fully symmetric interpolatory rules for multiple integrals over infinite regions with Gaussian weight. Journal of Computational and Applied Mathematics, 1996, 71(2): 299–309MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Scarth C, Cooper J E, Weaver P M, et al. Uncertainty quantification of aeroelastic stability of composite plate wings using lamination parameters. Composite Structures, 2014, 116: 84–93CrossRefGoogle Scholar
  44. 44.
    Feinberg J, Langtangen H P. Chaospy: An open source tool for designing methods of uncertainty quantification. Journal of Computational Science, 2015, 11: 46–57MathSciNetCrossRefGoogle Scholar
  45. 45.
    Huang B, Du X. Uncertainty analysis by dimension reduction integration and saddlepoint approximations. Journal of Mechanical Design, 2006, 128(1): 26–33CrossRefGoogle Scholar
  46. 46.
    Gerstner T, Griebel M. Numerical integration using sparse grids. Numerical Algorithms, 1998, 18(3–4): 209–232MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Heiss F, Winschel V. Likelihood approximation by numerical integration on sparse grids. Journal of Econometrics, 2008, 144(1): 62–80MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Jaynes E T. Information theory and statistical mechanics. Physical Review, 1957, 106(4): 620–630MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Phillips S J, Anderson R P, Schapire R E. Maximum entropy modeling of species geographic distributions. Ecological Modelling, 2006, 190(3–4): 231–259CrossRefGoogle Scholar
  50. 50.
    Mohammad-Djafari A. A Matlab program to calculate the maximum entropy distributions. In: Smith C R, Erickson G J, Neudorfer P O, eds. Maximum Entropy and Bayesian Methods. Dordrecht: Springer, 1992, 221–233Google Scholar
  51. 51.
    Yeo S K, Chun J H, Kwon Y S. A 3-D X-band T/R module package with an anodized aluminum multilayer substrate for phased array radar applications. IEEE Transactions on Advanced Packaging, 2010, 33(4): 883–891CrossRefGoogle Scholar
  52. 52.
    Pamies Porras M J, Bertuch T, Loecker C, et al. An AESA antenna comprising an RF feeding network with strongly coupled antenna ports. IEEE Transactions on Antennas and Propagation, 2015, 63 (1): 182–194MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://doi.org/creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the appropriate credit is given to the original author(s) and the source, and a link is provided to the Creative Commons license, indicating if changes were made.

Authors and Affiliations

  • X. Y. Jia
    • 1
  • C. Jiang
    • 1
    Email author
  • C. M. Fu
    • 1
  • B. Y. Ni
    • 1
  • C. S. Wang
    • 2
  • M. H. Ping
    • 1
  1. 1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Vehicle EngineeringHunan UniversityChangshaChina
  2. 2.Key Laboratory of Electronic Equipment Structure Design of Ministry of Education, School of Electro-Mechanical EngineeringXidian UniversityXi’anChina

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