A variable and mode sensitivity analysis method for structural system using a novel active learning Kriging model

https://doi.org/10.1016/j.ress.2020.107285Get rights and content

Highlights

  • A variable sensitivity analysis index for system reliability analysis is developed.

  • A mode sensitivity analysis index for system reliability analysis is proposed.

  • An ALK-SIS method is proposed to system reliability analysis.

  • Four examples are studied to validate the performance of proposed solving strategy.

Abstract

In the frame of structural system reliability, quantifying the impact degree of random variables and failure modes on system output is indispensable. A generalized variable sensitivity analysis (VSA) index is introduced for measuring the effect of each random variable. Then a mode sensitivity analysis (MSA) index is proposed for quantifying the effect of each failure mode on system failure probability. Additionally, to estimate the system failure probability, VSA and MSA indices efficiently, a novel active learning Kriging model combined with the system importance sampling (SIS) density function is proposed for structural systems. The SIS density function is obtained by kernel density estimation-based importance sampling and weighted coefficients of failure modes, which is employed for generating the candidate population to reduce the computational cost on predicting the enormous candidates. The proposed called ALK-SIS model can precisely and effectively deal with small failure probability events as well as multiple most probable point problems, and estimate the system failure probability and sensitivity indices without calling the performance functions additionally. Four case studies are investigated to demonstrate the significance of proposed VSA and MSA indices on system reliability as well as the efficiency and accuracy of the ALK-SIS.

Introduction

The reliability analysis aims to estimate the failure probability considering variable uncertainty while the global sensitivity analysis pays attention to the effects of input random variables on reliability. The structural reliability analysis and global sensitivity analysis with single failure mode have been studied comprehensively. However, complex engineering mechanical system usually contains multiple failure modes. Researchers call this typical reliability analysis problem as structural system reliability analysis (SRA). It is significant to extend reliability analysis and global sensitivity analysis to structural systems.

For component reliability analysis, lots of methods have been proposed. The approximation analysis method contains first order reliability method (FORM) and second order reliability method (SORM) [1]. The sampling simulation method consists of the Monte Carlo simulation (MCS) method [2], importance sampling (IS) [3], [4], truncated importance sampling (TIS), subset simulation (SS) [5], directional sampling (DS) [6], line sampling (LS) [7], etc. The surrogate model method contains polynomial response surface method [8], polynomial chaos expansion method [9], neural network (NN) [10], support vector machine (SVM) [11], high-dimensional model representation (HDMR) [12] method, Kriging model [13], [14], [15], etc. Additionally, the research in Ref. [13] has demonstrated that the Kriging surrogate model is efficient for structural reliability problems by comparing with the common first- and second-order polynomial regression models with a nonlinear marine structure example.

Among these methods, the active learning Kriging (ALK) method receives the most attention. The ALK method [14], [15] combines the Kriging model with MCS (i.e., AK-MCS) and establishes the Kriging model with active learning functions. Researchers have carried out systematic improvements on ALK method in several directions. To assess small failure probability problems, Echard et al. [16] combined the ALK method with importance sampling to reduce the calculation cost and proposed the AK-IS method. Cadini et al. [17] proposed a metaAK-IS2 algorithm to generate samples in multiple failure regions of low probability. Yang et al. [18] combined the ALK model with kernel-density-estimation-based importance sampling method to evaluate the low failure probability. Yun et al. [19] proposed a modified importance sampling method based on the TIS and combined it with ALK model to proposed a novel strategy to estimate small failure probability. Gaspar et al. [20] proposed an adaptive Kriging model combined with active refinement and a trust region method, which predict the initial failure probability by FORM firstly and verify or improve the initial failure probability by the Monte Carlo simulation with importance sampling based on a Kriging model efficiently. Lv et al. [21] further combined the ALK model with LS and proposed an AK-LS method, which showed great performance to solve engineering reliability problems. Huang et al. [22] proposed an adaptive learning method for small failure probability by combining the ALK model with SS. Ling et al. [23] also combined the adaptive Kriging model with fuzzy simulation (FS) and proposed an efficient AK-FS method. Above methods give great ways for the time-demanding reliability analysis and low failure probability problems.

Meanwhile, various adding point strategies, i.e. active learning function, have been developed. The expected feasibility function (EFF) [14] was first proposed by Bichon et al. for the efficient global reliability analysis algorithm. Then Echard et al. [15] proposed the U function in the famous AK-MCS method. Lv et al. [21] proposed an H function for the AK-LS method. Yang et al. [24] developed an expected risk function (ERF) in the hybrid reliability analysis. Sun et al. [25] proposed a least improvement function (LIF) for the ALK method and applied it to the structural reliability analysis. In general, the EFF and U functions are adopted most. Ref. [26] compared the performance of them. The results reported that there is no distinct advantage or disadvantage between them.

In other aspects, the ALK model is also applied to the time-variant reliability analysis and reliability-based design optimization (RBDO). Dubourg et al. [27] used the adaptive Kriging surrogates and subset simulation to realize the RBDO. Hawchar et al. [28] made a time-variant reliability-based design optimization with the global Kriging surrogate modeling. Ling and Lu [29] proposed an adaptive Kriging with importance sampling for the time-variant hybrid reliability analysis. Xiao et al. [30] developed a novel system active learning Kriging method for system reliability-based design optimization, which defined three system active learning functions and considered the confidence interval of estimated system failure probability in the stopping condition. The blooming development of the ALK model demonstrates the excellent performance in reliability analysis field.

For the application of ALK model in system reliability analysis, researchers implemented lots of improvements on the basis of adaptive Kriging strategy. Bichon et al. [31] constructed a composite Gaussian process model for the system-level formulations based on the efficient global reliability analysis method. Fauriat and Gayton [32] achieved an adaptation of the AK-MCS for the system reliability analysis and proposed an AK-SYS method. Perrin [33] proposed a nested Gaussian process regression-based surrogate models for system conception and reliability analysis. Wei et al. [34] developed a multiple response Gaussian process model for the structural system reliability analysis. Yang et al. [35], [36] proposed a system reliability analysis approach by combining the ALK model with truncated candidate region (ALK-TCR) and adaptive size of candidate points (ALK-TCR-ASCP). As shown in literature, the various sampling techniques, which are widely introduced into the ALK models for component reliability analysis, are not developed to combine with the ALK models for the system reliability analysis. This paper carries out a good attempt to combine the kernel density estimation-based adaptive importance sampling method (KDAIS) with the ALK model for system reliability analysis. The KDAIS is employed to generate the candidate population to avoid the computation waste on estimating the enormous candidate samples and solve the small failure probability events. Construct the system importance sampling (SIS) density function with the weighted importance sampling density functions obtained by KDAIS for different failure modes. Then a novel approach called ALK-SIS is proposed for the structural system reliability analysis and sensitivity analysis.

In previous works, the sensitivity analysis is mainly concentrated on the variable sensitivity analysis (VSA) [37], [38], [39], [40], [41], [42], [43] of component reliability analysis (CRA). Based on the failure probability, the global sensitivity analysis is necessary to measure the effects of uncertainty in random variables on the failure probability. The global sensitivity analysis indices can be divided into three groups: non-parametric index [37], moment-independent index [38] and variance-based indices [39]. Borgonovo [38] proposed a moment independent sensitivity index based on probability density function (PDF) while Liu and Homma [40] proposed another one based on cumulative distribution function (CDF). Based on the Sobol’ indices [39], [41], [42], Saltelli [43] proposed the variance-based indices to evaluate the individual, interaction and total effects of input variables on failure probability. With the development of structural system reliability analysis, several sensitivity indices estimating the effect of random variables and failure modes on the system failure probability are proposed. Lamboni et al. [44] decomposed the multiple outputs by the principal component analysis and proposed a generalized sensitivity index to estimate the effect of input variables on the output responses. Li et al. [45] proposed a multivariate probability integral transformation based moment independent sensitivity index, which further considered the correlation between the variables. Wei et al. [46] proposed the variance-based sensitivity indices for system reliability. Zhou et al. [47] proposed several mode sensitivity analysis (MSA) indices to approximate the effects of failure modes on system failure probability, which is based on the PDFs of the performance functions. However, the variance-based sensitivity analysis measures the uncertainty of output variable with variance only, and cannot consider the whole distribution information of input random variables. The general moment independent sensitivity analysis requires estimating the unconditional and conditional PDF, or the unconditional and conditional failure probability, which causes enormous computational cost. Furthermore, the system reliability analysis is similar with the component reliability analysis, both of which concentrate more on whether the structure or system fails or not. Therefore, the output classification-based generalized sensitivity analysis strategy is also suitable for the structural system reliability analysis. Therefore, by introducing the idea of the generalized sensitivity analysis [48], this paper develops a generalized VSA index for structural systems, which approximates the effect of the random variables on the system failure probability, and proposes a generalized MSA index to estimate the effect of the failure modes on the system failure probability. The VSA and MSA indices in this paper can be estimated by the system failure probability and conditional PDF in failure state of input variable or failure mode, and only a set of samples is utilized to estimate them simultaneously. These two sensitivity indices provide a novel effective term to measure the effects of random variables as well as failure modes on the system failure probability of structural system. Meanwhile, based on the proposed ALK-SIS method, the VSA and MSA indices are approximated without calling the performance functions additionally.

This paper is arranged as follows. In Section 2, based on the derived system failure probability with multiple failure modes, the generalized VSA and MSA indices are developed to measure the effects of variables and failure modes on the system failure probabilities, respectively. In Section 3, we propose an ALK-SIS method for efficiently approximating the system failure probability as well as the VSA and MSA indices. And the solving procedure of ALK-SIS method is summarized in this part. In Section 4, the significance of the proposed VSA and MSA indices as well as the effectiveness of the ALK-SIS method is validated by four examples. The last section concludes this paper and emphasizes the significance of our work.

Section snippets

System reliability analysis

Let gi(x)denote the performance function of the ith failure mode where i=1,2,,mand xrepresents the p-dimensional input random variables. Define the failure region of the ith mode as Fi={x|gi(x)<0}. We assume that the mfailure modes are in series, then the failure region of structural system is FS={x|i=1mgi(x)<0}. Similarly, the failure region of parallel system is FP={x|i=1mgi(x)<0}. For a general system with mfailure modes and rcut sets, the failure region is FG={x|i=1rj=1miFi,j}, where j

Active learning Kriging model

The expression of Kriging model [13], [14], [15] isg(x)=β+z(x)where, g(x)is the function to be estimated, βrepresents the global estimation of the model, i.e. the expectation of the Kriging model, z(x)denotes the Gaussian stochastic process to provide a local deviation.

Given a design of experiments (DoE) containing m samples, the predicted value and variance of the Kriging model at unknown points are given as followsμg(x)=β^+r(θ,x)TR(θ)1(gβ^1)σg2(x)=σ2[1+(1TR(θ)1r(θ,x)1)21TR(θ)11r(θ,x)TR(θ

Numerical example I

A parallel system with two branches is expressed as follows{g1=2x2+exp(0.1x12)+(0.2x1)4g2=4.5x1x2where x1and x2obey independent standard normal distribution. The number of the initial DoE for the Kriging models is N0=12, which is same for the AK-SYS, ALK-TCR and the proposed ALK-SIS. The number of candidates for AK-SYS and ALK-TCR is 107, while that of ALK-SIS generated by importance sampling density function is 3000. And 2000 samples in failure region generated by Markov chain is generated

Conclusion

This paper aims to develop an effective method to estimate the effect of variables and failure modes on failure probability of structural systems. For this goal, a VSA index is introduced to measure the effect of each input variable on the failure probability of structural systems effectively. Meanwhile, a novel MSA index is proposed to quantify the effect of each failure mode on the failure probability. Furthermore, a novel active learning Kriging model combined with Kriging model and system

Declaration of Competing Interest

The authors declare that they have no conflict of interest.

Acknowledgement

This work is sponsored by Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX202030). The authors are grateful for the financial support.

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