Computer Physics Communications 183 (2012) 1728–1743 Contents lists available at SciVerse ScienceDirect Computer Physics Communications www.elsevier.com/locate/cpc Eﬃcient sampling methods for global reliability sensitivity analysis Pengfei Wei a,∗,1 , Zhenzhou Lu a,2 , Wenrui Hao a,3 , Jun Feng b , Bintuan Wang b a Northwestern Polytechnical University, Xi’an, Shaanxi, 710072, China b The First Aircraft Institute, AVIC (Aviation Industry Corporation of China), Xi’an, Shaanxi, China article info abstract Article history: Received 4 January 2012 Received in revised form 29 February 2012 Accepted 17 March 2012 Available online 20 March 2012 Keywords: Global reliability sensitivity analysis Main effect indices Total effect indices Importance sampling Truncated importance sampling An important problem in structure reliability analysis is how to reduce the failure probability. In this work, we introduce a main and total effect indices framework of global reliability sensitivity. By decreasing the uncertainty of input variables with high main effect indices, the most reduction of failure probability can be obtained. By decreasing the uncertainty of the input variables with small total effect indices (close to zero), the failure probability will not be reduced signiﬁcantly. The eﬃcient sampling methods for evaluating the main and total effect indices are presented. For the problem with large failure probability, a single-loop Monte Carlo simulation (MCS) is derived for computing these sensitivity indices. For the problem with small failure probability, the single-loop sampling methods combined with the importance sampling procedure (IS) and the truncated importance sampling procedure (TIS) respectively are derived for improving the calculation eﬃciency. Two numerical examples and one engineering example are introduced for demonstrating the eﬃciency and precision of the calculation methods and illustrating the engineering signiﬁcance of the global reliability sensitivity indices.  2012 Elsevier B.V. All rights reserved. 1. Introduction Two main problems are concerned in structural safety analysis: reliability analysis and sensitivity analysis. Reliability analysis aims at estimating the failure probability (or reliability) of the structure. Sensitivity analysis focuses on measuring the effects of the uncertainty or distribution parameters of the input variables on failure probability [1–4]. In this paper, the reliability sensitivity analysis is concerned. Sensitivity analysis can be classiﬁed into two groups: local sensitivity analysis and global sensitivity analysis. Local reliability sensitivity indices are deﬁned as the partial derivative of the failure probability with respect to the distribution parameters of the input variables [2]. Local sensitivity indices can only reﬂect the local effect of the distribution parameters on the failure probability, but cannot tell us how the uncertainty of the input variables affects the failure probability globally. The global sensitivity indices can help measure the effect of the uncertainty of each input variables on the variance or the distribution of the model outputs of interest. Many global sensitivity analysis techniques are now available. For measuring the effects of the uncertainty of input variables on the variance of model outputs, Sobol, Saltelli and others proposed the variance based global sensitivity indices [5–9]. For measuring the effects of the uncertainty of each input variables on the distribution of model outputs, Borgonovo developed the probability density function (PDF) based (also called moment-independent) global sensitivity indices [10–12]. For measuring the effects of the uncertainty of each input variables on the failure probability, Cui and Li proposed the global reliability sensitivity indices [3,4]. For reducing the computational cost in global sensitivity analysis, Kucherenko, Sobol’ and others proposed the derivative based global sensitivity indices, and established the link between the derivative based indices and Sobol’s variance based global sensitivity indices [13–15]. This work aims at studying the effect of the uncertainty of each input variable on the failure probability, and deciding how to reduce the failure probability by decreasing the uncertainty of the input variables. Generally, the global reliability sensitivity indices deﬁned by Cui [3] and Li [4] can be used for this purpose. But only the main effect indices are deﬁned and computed in their works. In this paper, we standardize Li’s deﬁnition and introduce the main and total effect indices for each input variable. The main effect index of one input variable can tell us how much the contribution is made by the uncertainty of this input variable to the failure probability. By decreasing the uncertainty of the input variables with high main effect indices we can get the most reduction of failure probability. The total effect * Corresponding author. E-mail addresses: wpf0414@163.com (P. Wei), zhenzhoulu@nwpu.edu.cn (Z. Lu), haowr22@163.com (W. Hao). 1 PhD candidate, School of Aeronautics, Mail Box 120. 2 Professor, School of Aeronautics, Mail Box 120. 3 Master, School of Aeronautics, Mail Box 120. 0010-4655/$ – see front matter  2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.cpc.2012.03.014