Based on two procedures for efficiently generating conditional samples, i.e. Markov chain Monte Carlo (MCMC) simulation and importance sampling (IS), two reliability sensitivity (RS) algorithms are presented. On the basis of reliability analysis of Subset simulation (Subsim), the RS of the failure probability with respect to the distribution parameter of the basic variable is transformed as a set of RS of conditional failure probabilities with respect to the distribution parameter of the basic variable. By use of the conditional samples generated by MCMC simulation and IS, procedures are established to estimate the RS of the conditional failure probabilities. The formulae of the RS estimator, its variance and its coefficient of variation are derived in detail. The results of the illustrations show high efficiency and high precision of the presented algorithms, and it is suitable for highly nonlinear limit state equation and structural system with single and multiple failure modes.
Sensitivity analysis has been widely applied in a broad range of sciences, for example, chemistry, environmental sciences, aerospace engineering, risk analysis, etc. In engineering design, sensitivity analysis explores the model response, evaluates the accuracy of model, tests the validity of the assumption made and so on. Sensitivity is used to find the rate of change in a model output due to changes in the model inputs in deterministic design, which is usually performed by partial derivative analytically or numerically.
Random uncertainty exists in the model parameter of system widely due to uncontrollable factors. When random uncertainty is considered, sensitivity analysis has different meanings. We assume that the random uncertainty in a design performance is described probabilistically by its mean (μ ), variance (σ ), probability density function (PDF) or cumulative distribution function (CDF), etc. Correspondingly, the sensitivity analysis under uncertainty needs to be performed on the probability characteristics of a model response with respect to the probability characteristics of the model inputs. In general, the reliability sensitivity analysis is to investigate the rate of change in the probability characteristics of the response, especially failure probability P f, due to the probability characteristics of a basic random variable xi , such as ∂Pf/∂μxi∂Pf/∂μxi and ∂Pf/∂σxi∂Pf/∂σxi [1], [2], [3], [4], [5], [6] and [7]. These partial derivatives can objectively describe the effect of distribution parameters on the failure probability. And this paper studies the partial derivative of the failure probability with respect to the distribution parameter of the basic random variable.
Wu [1] and [2] presented a reliability sensitivity method based on the CDF of the structural response variable. The normalized reliability sensitivity coefficient is expressed as an expectation of the partial derivative of the PDF, evaluated over the failure region, wherein the sampling based method can be used to compute the reliability sensitivity.
The reliability sensitivity based on the most probable point (MPP) is very simple, but it depends on the linearization of the limit state function [6] and [7]. The linearization of the nonlinear performance function will lead to a weak precision in the reliability and the reliability sensitivity evaluations.
This paper focuses on the numerical method based on subset simulation (Subsim) to analyze the reliability sensitivity. Subsim [8], [9], [10], [11], [12], [13] and [14] is an efficient simulation to perform the reliability analysis in a progressive manner. Introducing a set of intermediate failure events, Subsim separates the original probability space into a sequence of subsets, and then the small failure probability can be expressed as a product of larger conditional failure probabilities. Markov chain Monte Carlo (MCMC) simulation [8] is used to generate conditional samples that correspond to specified levels of failure probabilities. However, the conditional samples generated by MCMC are dependent in general. These samples are used for statistical averaging as if they are independent and identically distributed (i.i.d.) with some reduction in efficiency. The concept of importance sampling (IS) procedure is employed for generating i.i.d. conditional samples to efficiently calculate the conditional failure probability corresponding to the specified levels of failure probabilities [14]. On the basis of reliability analysis of Subsim/MCMC and Subsim/IS, the RS of the failure probability with respect to the distribution parameter of the basic variable is transformed as a set of RS of conditional failure probabilities with respect to the distribution parameter of the basic variable, and the latter can be evaluated by the conditional samples generated by MCMC simulation and IS. The concept and the implementation of the presented Subsim-based reliability sensitivity are explained in the following sections.
The outline of the paper is as follows. The reliability analysis of Subsim/MCMC and Subsim/IS are described in Section 2. Section 3 explains the concept and the implementation of Subsim-based reliability sensitivity method. In Section 3.1, the basic concept of Subsim/MCMC-based reliability sensitivity method is described in brief. The reliability sensitivity estimator and its variance analysis based on Subsim/IS are discussed in Section 3.2. After the numerical examples verify the feasibility and rationality of the presented method in Section 4, Section 5 concludes with a summary of the main advantages of the presented method.
