Advertisement

Journal of Marine Science and Technology

, Volume 21, Issue 2, pp 282–296 | Cite as

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

  • M. D. CooperEmail author
  • W. Wu
  • L. S. McCue
Original article

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.

Keywords

Parametric roll Non-intrusive polynomial chaos Monte Carlo Uncertainty analysis 

Notes

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.

References

  1. 1.
    Metropolis N (1987) The beginning of the Monte Carlo method. Los Alamos Sci (1987 Special Issue dedicated to Stanisław Ulam) 15(584):125–130MathSciNetGoogle Scholar
  2. 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, NevadaGoogle Scholar
  3. 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, USAGoogle Scholar
  4. 4.
    Wiener N (1938) The homogeneous chaos. Am J Math 60(4):897–936MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cameron R, Martin W (1947) The orthogonal development of nonlinear functionals in series of Fourier-Hermite functionals. Ann Math 48(2):385–392MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Ghanem R, Spanos P (1991) Stochastic finite elements: a spectral approach. Springer, New YorkCrossRefzbMATHGoogle Scholar
  7. 7.
    Ghanem R (1999) Ingredients for a general purpose stochastic finite element formulation. Comput Methods Appl Mech Eng 168:19–34MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Xiu D, Karniadakis G (2003) Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys 187(1):137–167MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Xiu D, Karniadakis GE (2002) The Wiener–Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput 24(42):619–644MathSciNetCrossRefzbMATHGoogle Scholar
  10. 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, SwedenGoogle Scholar
  11. 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 38:58–68CrossRefGoogle Scholar
  12. 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):351–363CrossRefGoogle Scholar
  13. 13.
    Box GEP, Wilson KB (1951) On the experimental attainment of optimum conditions. J R Stat Soc Ser B (Methodol) 13(1):1–45MathSciNetzbMATHGoogle Scholar
  14. 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 295–305CrossRefGoogle Scholar
  15. 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 112:14–47Google Scholar
  16. 16.
    Weems K (2009) Introduction and training for LAMP (Large-Amplitude Motion Program). Science Applications International Corporation (SAIC)—advanced systems and technology divisionGoogle Scholar
  17. 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, BrazilGoogle Scholar
  18. 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 meetingGoogle Scholar
  19. 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):1–19Google Scholar
  20. 20.
    Hosder S, Walters RW, Balch M (2010) Point-collocation nonintrusive polynomial chaos method for stochastic computational fluid dynamics. AIAA J 48(12):2721–2730CrossRefGoogle Scholar

Copyright information

© JASNAOE 2015

Authors and Affiliations

  1. 1.Department of Aerospace and Ocean EngineeringVirginia TechBlacksburgUSA
  2. 2.KvaernerHoustonUSA
  3. 3.ASNEAlexandriaUSA

Personalised recommendations