Abstract
In this article, a combined use of Latin hypercube sampling and axis orthogonal importance sampling, as an efficient and applicable tool for reliability analysis with limited number of samples, is explored for sensitivity estimation of the failure probability with respect to the distribution parameters of basic random variables, which is equivalently solved by reliability sensitivity analysis of a series of hyperplanes through each sampling point parallel to the tangent hyperplane of limit state surface around the design point. The analytical expressions of these hyperplanes are given, and the formulas for reliability sensitivity estimators and variances with the samples are derived according to the first-order reliability theory and difference method when non-normal random variables are involved and not involved, respectively. A procedure is established for the reliability sensitivity analysis with two versions: (1) axis orthogonal Latin hypercube importance sampling and (2) axis orthogonal quasi-random importance sampling with the Halton sequence. Four numerical examples are presented. The results are discussed and demonstrate that the proposed procedure is more efficient than the one based on the Latin hypercube sampling and the direct Monte Carlo technique with an acceptable accuracy in sensitivity estimation of the failure probability.
Highlights
Reliability analysis and sensitivity analysis should be an important part of any analysis of engineering structures, with (1) reliability analysis providing the probabilities of failure or of unacceptable structural performance due to those uncontrolled random factors and (2) as an importance measure, sensitivity analysis identifying the contributions of random analysis inputs to those probabilities
We further focus on the numerical algorithm employing axis orthogonal importance Latin hypercube sampling (AOILHS) method to analyze the reliability sensitivities
To verify the efficiency and accuracy of the proposed algorithm, it is compared to the other samplingbased methods in this article including the standard Monte Carlo (SMC) method, axis orthogonal importance correlation Latin hypercube sampling (AOICLHS) method, and axis orthogonal importance sampling with the Halton sequence (AOIHalton) and illustrated by a number of numerical examples
Summary
Reliability analysis and sensitivity analysis should be an important part of any analysis of engineering structures, with (1) reliability analysis providing the probabilities of failure or of unacceptable structural performance due to those uncontrolled random factors and (2) as an importance measure, sensitivity analysis identifying the contributions of random analysis inputs to those probabilities. The reliability sensitivities calculated directly by differentiating P^f with respect to u (udenotes the mean, standard deviation of a random variable, or other distribution parameters) may produce big errors for the MPP which is usually not the point maximizing the derivative of joint PDF, namely, ∂f0=∂u, and the samples employed to estimate the failure probability are not suitable to estimate the reliability sensitivities. First considering the variance of reliability sensitivities in the transformed normal space and since the samples generated in normal space are i.i.d, the variance of reliability sensitivities can be estimated by For those problems not involving non-normal random variables, variances of the reliability sensitivities can be calculated according to equations (22) and (23) and written as var. For those problems involving non-normal random variables, variances of the reliability sensitivities can be calculated according to equations (11) and (26) and written as comparison, the failure probabilities (pf ), the reliability indices (b), and sensitivities determined from the provar∂P^f ∂u varÀP^f ju + DuÀP^f ðDuÞ2 "
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