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Non-intrusive polynomial chaos for efficient uncertainty analysis in parametric roll simulations

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Abstract

Monte Carlo analyses are generally considered the standard for uncertainty analysis. While accurate, these analyses can be expensive computationally. Recently, polynomial chaos has been proposed as an alternative approach to the estimation of uncertainty distributions (Hosder et al. A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada, 2006; Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). This approach works by representing the function as a series of orthogonal polynomials; the weights for which can be calculated via several methods. Previous studies have demonstrated the usefulness of this technique for comparatively simple systems such as parametric roll modeled by the Mathieu equation with normally distributed parameter values (Wu et al. Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA, 2011). In the present work, a polynomial chaos method is applied to a nonlinear computational ship dynamics model with normally distributed input parameters. Test cases were selected where parametric roll was expected to potentially occur. The resulting probability distributions are compared to the results of a Monte Carlo analysis. In general, these results demonstrate good agreement between Monte Carlo simulation and polynomial chaos in the absence of capsize with significant computation gains found with polynomial chaos. Overall, we conclude that polynomial chaos is an effective tool for reducing simulation time costs when studying parametric roll, and potentially other ship dynamics phenomena, particularly in the absence of capsize-like bifurcations.

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References

  1. Metropolis N (1987) The beginning of the Monte Carlo method. Los Alamos Sci (1987 Special Issue dedicated to Stanisław Ulam) 15(584):125–130

    MathSciNet  Google Scholar 

  2. Hosder S, Walters RW, Perez R (2006) A non-intrusive polynomial chaos method for uncertainty propagation in CFD simulations. In: 44th AIAA aerospace sciences meeting and exhibit, Reno, Nevada

  3. Wu W, Bulian G, McCue LS (2011) Uncertainty analysis for parametric roll using non-intrusive polynomial chaos. In: Proceedings of the 12th international ship stability workshop, Washington, DC, USA

  4. Wiener N (1938) The homogeneous chaos. Am J Math 60(4):897–936

    Article  MathSciNet  MATH  Google Scholar 

  5. Cameron R, Martin W (1947) The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann Math 48(2):385–392

    Article  MathSciNet  MATH  Google Scholar 

  6. Ghanem R, Spanos P (1991) Stochastic finite elements: a spectral approach. Springer, New York

    Book  MATH  Google Scholar 

  7. Ghanem R (1999) Ingredients for a general purpose stochastic finite element formulation. Comput Methods Appl Mech Eng 168:19–34

    Article  MathSciNet  MATH  Google Scholar 

  8. Xiu D, Karniadakis G (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187(1):137–167

    Article  MathSciNet  MATH  Google Scholar 

  9. Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(42):619–644

    Article  MathSciNet  MATH  Google Scholar 

  10. Lucor D, Xiu D, Karniadakis GE (2001) Spectral representations of uncertainty in simulations: algorithms and applications. In: Proceedings of the international conference on spectral and high order methods, Uppsala, Sweden

  11. Li D, Chen Y, Wenbo L, Zhou C (2011) Stochastic response surface method for reliability analysis of rock slopes involving correlated non-normal variables. Comput Geotech 38:58–68

    Article  Google Scholar 

  12. Isukapalli S, Roy A, Georgopoulos P (1998) Stochastic response surface methods (SRSMs) for uncertainty propagation: application to environmental and biological systems. Risk Anal 18(3):351–363

    Article  Google Scholar 

  13. Box GEP, Wilson KB (1951) On the experimental attainment of optimum conditions. J R Stat Soc Ser B (Methodol) 13(1):1–45

    MathSciNet  MATH  Google Scholar 

  14. Belenky V, Yu HC, Weems K (2011) Numerical procedures and practical experience of assessment of parametric roll of container carriers. Contemporary ideas on ship stability and capsizing in waves. Springer, Dordrecht, pp 295–305

    Book  Google Scholar 

  15. Shin YS, Belenky V, Paulling J, Weeks K, Lin WM, Mctaggart K, Spyrou K, Treakle T, Levadou M, Hutchison B, Falzarano J, Chen H, Letizia L (2004) Criteria for parametric roll of large containerships in longitudinal seas. Trans Soc Naval Archit Mar Eng 112:14–47

    Google Scholar 

  16. Weems K (2009) Introduction and training for LAMP (Large-Amplitude Motion Program). Science Applications International Corporation (SAIC)—advanced systems and technology division

  17. Paulling JR (2006) Parametric rolling of ships—then and now. In: Proceedings of the 9th international conference on stability of ships and ocean vehicles, Rio de Janeiro, Brazil

  18. France W, Levadou M, Treakle T, Paulling J, Michel R, Moore C (2001) An investigation of head-sea parametric rolling and its influence on container lashing systems. In: SNAME annual meeting

  19. France WN, Levadou M, Treakle TW, Paulling JR, Michel RK, Moore C (2003) An investigation of head-sea parametric rolling and its influence on container lashing systems. Ma Technol 40(1):1–19

    Google Scholar 

  20. Hosder S, Walters RW, Balch M (2010) Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics. AIAA J 48(12):2721–2730

    Article  Google Scholar 

Download references

Acknowledgments

The authors are extremely grateful for the advice provided by Mr. Kenneth Weems. The authors would also like to thank Drs. Pat Purtell, Ki-Han Kim, and Thomas Fu at the Office of Naval Research under Grant Number N00014-10-1-0398 and Eduardo Misawa at the National Science Foundation under Grant CMMI 0747973 for providing funding for this work.

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Correspondence to M. D. Cooper.

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Cooper, M.D., Wu, W. & McCue, L.S. Non-intrusive polynomial chaos for efficient uncertainty analysis in parametric roll simulations. J Mar Sci Technol 21, 282–296 (2016). https://doi.org/10.1007/s00773-015-0351-0

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  • DOI: https://doi.org/10.1007/s00773-015-0351-0

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