Jean-Marc Bourinet

Reliability-based design optimization using kriging surrogates and subset simulation

The aim of the present paper is to develop a strategy for solving reliability-based design optimization (RBDO) problems that remains applicable when the performance models are expensive to evaluate. Starting with the premise that simulation-based approaches are not affordable for such problems, and that the most-probable-failure-point-based approaches do not permit to quantify the error on the estimation of the failure probability, an approach based on both metamodels and advanced simulation techniques is explored. The kriging metamodeling technique is chosen in order to surrogate the performance functions because it allows one to genuinely quantify the surrogate error. The surrogate error onto the limit-state surfaces is propagated to the failure probabilities estimates in order to provide an empirical error measure. This error is then sequentially reduced by means of a population-based adaptive refinement technique until the kriging surrogates are accurate enough for reliability analysis. This original refinement strategy makes it possible to add several observations in the design of experiments at the same time. Reliability and reliability sensitivity analyses are performed by means of the subset simulation technique for the sake of numerical efficiency. The adaptive surrogate-based strategy for reliability estimation is finally involved into a classical gradient-based optimization algorithm in order to solve the RBDO problem. The kriging surrogates are built in a so-called augmented reliability space thus making them reusable from one nested RBDO iteration to the other. The strategy is compared to other approaches available in the literature on three academic examples in the field of structural mechanics.

Rare-event probability estimation with adaptive support vector regression surrogates

Abstract Assessing rare event probabilities still suffers from its computational cost despite some available methods widely accepted by researchers and engineers. For low to moderately high dimensional problems and under the assumption of a smooth limit-state function, adaptive strategies based on surrogate models represent interesting alternative solutions. This paper presents such an adaptive method based on support vector machine surrogates used in regression. The key idea is to iteratively construct surrogates which quickly explore the safe domain and focus on the limit-state surface in its final stage. Highly accurate surrogates are constructed at each iteration by minimizing an estimation of the leave-one-out error with the cross-entropy method. Additional training points are generated with the Metropolis–Hastings algorithm modified by Au and Beck and a local kernel regression is made over a subset of the known data. The efficiency of the method is tested on examples featuring various challenges: a highly curved limit-state surface at a single most probable failure point, a smooth high-dimensional limit-state surface and a parallel system.

The cross-entropy method for reliability assessment of cracked structures subjected to random Markovian loads

The paper investigates the reliability of cracked components subjected to random amplitude loads modeled by discrete-time Markov processes. The proposed approach is able to capture interaction effects between cycles along the random loading sequence, which are of real interest in the damage tolerance design of aircraft structural components. Random fatigue loads are either modeled by discrete-time First-order Markov Chains or hidden Markov chains with continuous state space and their parameters are identified from in-flight data recorded on a fleet of fighter aircrafts. The uncertainties of the initial crack parameters and material properties are not accounted for in this work and some additional simplifying assumptions are made in order to define a tractable problem. The solution strategy for reliability assessment hinges on the cross-entropy method. The application of this method to Markov chains with discrete state space is first presented based on previous works of the literature and the paper then develops its extension to the selected Hidden Markov Model with continuous state space. Several damage tolerance applications are performed to illustrate the relevance and efficiency of the proposed methodology for which the strengths and limitations are finally highlighted.

FORM Sensitivities to Distribution Parameters with the Nataf Transformation

The Nataf transformation has been proven very useful in reliability assessment when marginal distributions are statistically known and linear correlation is sufficient for modeling the dependence between random inputs. Under the assumption that the use of FORM is appropriate for the problem of interest, it is often of importance to quantify how the FORM solution is sensitive to the distribution parameters of the random inputs. Such information can be exploited in different contexts including optimal design under uncertainty. This chapter describes how sensitivities to marginal distribution parameters and linear correlation can be assessed numerically in the context of FORM based on the Nataf transformation. The emphasis is on the accuracy of such sensitivities with no other approximations than the one due to numerical integration. In the presented examples, the accuracy of these sensitivities is assessed w.r.t. reference solutions. The sensitivity to correlation brings useful information which are complementary to those w.r.t. marginal distribution parameters. High sensitivities may be detected such as illustrated in the context of stochastic crack growth based on the Virkler data set.

Reliability-based sensitivity estimators of rare event probability in the presence of distribution parameter uncertainty

Abstract This paper aims at presenting sensitivity estimators of a rare event probability in the context of uncertain distribution parameters (which are often not known precisely or poorly estimated due to limited data). Since the distribution parameters are also affected by uncertainties, a possible solution consists in considering a second probabilistic uncertainty level. Then, by propagating this bi-level uncertainty, the failure probability becomes a random variable and one can use the mean estimator of the distribution of the failure probabilities (i.e. the “predictive failure probability”, PFP) as a new measure of safety. In this paper, the use of an augmented framework (composed of both basic variables and their probability distribution parameters) coupled with an Adaptive Importance Sampling strategy is proposed to get an efficient estimation strategy of the PFP. Consequently, double-loop procedure is avoided and the computational cost is decreased. Thus, sensitivity estimators of the PFP are derived with respect to some deterministic hyper-parameters parametrizing a priori modeling choice. Two cases are treated: either the uncertain distribution parameters follow an unbounded probability law, or a bounded one. The method efficiency is assessed on two different academic test-cases and a real space system computer code (launch vehicle stage fallback zone estimation).

Comparative Study of Kriging and Support Vector Regression for Structural Engineering Applications

AbstractMetamodeling techniques have been widely used as substitutes for high-fidelity and time-consuming models in various engineering applications. Examples include polynomial chaos expansions, n...