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Quasi random numbers in stochastic finite element analysis

Published online by Cambridge University Press:  17 August 2007

Géraud Blatman ,
Bruno Sudret  and
Marc Berveiller
Géraud Blatman
Affiliation:
EDF R&D, Département Matériaux et Mécanique des Composants, Site des Renardières, 77818 Moret-sur-Loing, France Institut Français de Mécanique Avancée et Université Blaise Pascal, Laboratoire de Mécanique et d'Ingénieries, Campus des Cézeaux, 63175 Aubière Cedex, France
Bruno Sudret
Affiliation:
EDF R&D, Département Matériaux et Mécanique des Composants, Site des Renardières, 77818 Moret-sur-Loing, France
Marc Berveiller
Affiliation:
EDF R&D, Département Matériaux et Mécanique des Composants, Site des Renardières, 77818 Moret-sur-Loing, France
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Abstract

A non-intrusive stochastic finite-element method is proposed for uncertainty propagation through mechanical systems with uncertain input described by random variables. A polynomial chaos expansion (PCE) of the random response is used. Each PCE coefficient is cast as a multi-dimensional integral when using a projection scheme. Common simulation schemes, e.g. Monte Carlo Sampling (MCS) or Latin Hypercube Sampling (LHS), may be used to estimate these integrals, at a low convergence rate though. As an alternative, quasi-Monte Carlo (QMC) methods, which make use of quasi-random sequences, are proposed to provide rapidly converging estimates. The Sobol' sequence is more specifically used in this paper. The accuracy of the QMC approach is illustrated by the case study of a truss structure with random member properties (Young's modulus and cross section) and random loading. It is shown that QMC outperforms MCS and LHS techniques for moment, sensitivity and reliability analyses.

Keywords

Uncertainty propagationstochastic finite-elementsnon intrusive spectral approachpolynomial chaos expansionquasi-random numbersstructural reliabilityPropagation d'incertitudeséléments-finis stochastiquesapproche non intrusivechaos polynomialnombres quasi-aléatoiresfiabilité des structures
Type
Research Article
Information
Mechanics & Industry , Volume 8 , Issue 3: Congrès Mécanique de Grenoble , May 2007 , pp. 289 - 297
Copyright
© AFM, EDP Sciences, 2007

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References

R. Ghanem, P. Spanos, Stochastic finite elements – A spectral approach, Dover Publications, 2 ed., 2003
Soize, C., Ghanem, R., Physical systems with random uncertainties: chaos representations with arbitrary probability measure, SIAM J. Sci. Comp. 26 (2004) 395–410 CrossRef
Ghiocel, D.M., Ghanem, R.G., Stochastic finite element analysis of seismic soil-structure interaction, J. Eng. Mech. (ASCE) 128 (2002) 66–77 CrossRef
Le Maître, O.P., Reagan, M., Najm, H.N., Ghanem, R.G., Knio, O.M., A stochastic projection method for fluid flow, II. Random process, J. Comp. Phys. 181 (2002) 9–44 CrossRef
A. Keese, H. Matthies, Numerical methods and smolyak quadrature for nonlinear stochastic partial differential equations, Technical Report, Technische Universität Braunschweig, 2003
Choi, S.K., Grandhi, R.V, Canfield, R.A., Pettit, C.L., Polynomial chaos expansion with Latin Hypercube Sampling for estimating response variability, AIAA J. 45 (2004) 1191–1198 CrossRef
Choi, S.K., Grandhi, R.V, Canfield, R.A., Structural reliability under non-Gaussian stochastic behavior, Computers and Structures 82 (2004) 1113–1121 CrossRef
M. Berveiller, Éléments-finis stochastiques : approches intrusive et non intrusive pour des analyses de fiabilité, Ph.D. thesis, Université Blaise Pascal, Clermont-Ferrand, 2005
Beran, P.S., Pettit, C.L., Uncertainty quantification of limit-cycle oscillations, J. Comp. Phys. 217 (2006) 217–247 CrossRef
H. Niederreiter, Random number generation and quasi-Monte Carlo methods, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1992
Morokoff, W.J., Caflisch, R.E., Quasi-Monte Carlo integration, J. Comp. Phys. 122 (1995) 218–230 CrossRef
R.Y. Rubinstein, Simulation and the Monte Carlo methods, John Wiley & Sons, 1981
McKay, M.D., Beckman, R.J., Conover, W.J., A comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics 2 (1979) 239–245
P. Malliavin, Stochastic Analysis, Springer, 1997
Sobol', I.M., Sensitivity estimates for nonlinear mathematical models, Math. Modeling & Comp. Exp. 1 (1993) 407–414
A. Saltelli, K. Chan, E.M. Scott, editors, Sensitivity analysis, J. Wiley & Sons, 2000
B. Sudret, Global sensitivity analysis using polynomial chaos expansions, Rel. Eng. Sys. Safety (2007)
Lee, S.H., Kwak, B.M., Response surface augmented moment method for efficient reliability analysis, Struct. Safe. 28 (2006) 261–272 CrossRef
M. Wand, M.C. Jones, Kernel smoothing, Chapman and Hall, 1995
B. Sudret, G. Blatman, M. Berveiller, Quasi random numbers in stochastic finite element analysis – application to global sensitivity analysis, in Proc. 10th Int. Conf. on Applications of Stat. and Prob. in Civil Engineering (ICASP10), Tokyo (2007)