RPCM: a strategy to perform reliability analysis using polynomial chaos and resampling

Abstract

Using stochastic finite elements, the response quantity can be written as a series expansion which allows an approximation of the limit state function. For computational purpose, the series must be truncated in order to retain only a finite number of terms. In the context of reliability analysis, we propose a new approach coupling polynomial chaos expansions and confidence intervals on the generalized reliability index as truncating criterion.RÉSUMÉ. La méthode des éléments finis stochastiques permet d'exprimer la réponse d'un système sous forme d'une série polynomiale appelée chaos polynomial. Numériquement, cette série doit être tronquée pour ne retenir qu'un nombre fini de termes. Dans le cadre de l'analyse de fiabilité, nous proposons une méthode adaptative utilisant les intervalles de confiance sur l'indice de fiabilité généralisé comme critère de troncature du chaos.

Relative Entropy Method Applied for the Fatigue Life Distribution of Carbon Fiber/Epoxy Composites

“…The characteristic function, i.e., the Laplace transform, is the same for Equations (6) and (7). More details of the relationship between Equations (6) and (7) can be found in [17,30,31].…”

“…However, essentially, the fatigue of a material is a statistical problem in the sense that if we want to predicate its behavior, then we should allow some intrinsic uncertainty. Except for the uncertainty of working conditions and measurement, the intrinsic uncertainty origin from dynamic chaos should also be considered [ 5 , 6 , 7 , 8 ]. Thus, a few probability and statistic theory tools are selected to describe the behavior of fatigue for different kinds of materials, such as the frequently used exponential distribution and the Weibull distribution [ 9 , 10 , 11 ].…”

Construction of bootstrap confidence intervals on sensitivity indices computed by polynomial chaos expansion

“…where b S i½α=2 and b S i½1 À α=2 are the α=2 and 1 À α=2 empirical quantiles. This interval does not need any hypothesis on b S i distribution, but needs a lot of re-sampling B (higher than 500, see [16]) in order to approximate these quantiles with a sufficient precision. The setting of this parameter is going to be discussed in Section 4.…”

“…Here an application of bootstrap re-sampling is presented in order to minimize the number of points in the design of experiments, so as to get sensitivity indices from PCE with a fixed level of confidence. The algorithm described here is inspired from a previous work [16], using bootstrap re-sampling in a similar way on reliability indices. Our methodology, summarized in Fig.…”

“…For example, in [11], the authors use a complete quadratic response surface as a meta-model and build confidence intervals by Jack-knife re-sampling around the design point. In [16], a bootstrap re-sampling on the failure probability is used to construct an optimized PCE. Confidence interval constructed by bootstrap re-sampling is also widely used in global sensitivity analysis.…”