Fuzzy importance sampling method for estimating failure possibility
Introduction
Reliability analysis aims at quantitatively measuring the safety degree of the structure in presence of various uncertainties [1], [2], [3], [4]. Traditional reliability analysis is based on the probabilistic theory, where the uncertainty model inputs are handled as random variables, and it is also called probabilistic reliability analysis. In recent decades, numerous computational methods were proposed to estimate the probabilistic reliability, and they can be mainly classified into three groups, i.e., approximate analytical approaches [5], [6], numerical simulation approaches [7], [8], [9], [10] and surrogate model approaches [11], [12]. The approximately analytical approach includes the first/second-order reliability method (FORM/SORM) [5] and some higher-order reliability methods [6]. In these methods, some statistical moments (such as first-order moment, second-order moment and higher-order moments) of model output are computed at first, and then the probabilistic reliability can be approximately evaluated by using these statistical moments according to the basic theory of probability and statistic. Although the approximately analytical approach is simple, it lacks sufficient accuracy for the problems with highly non-linear input-output relationships. Monte Carlo simulation (MCS) is the most fundamental method among several available numerical simulation approaches. It is exact and easy to implement, but the computational burden of MCS for the problems with high probabilistic reliability may be too large to be accepted by the engineers. Hence, in order to improve the sampling efficiency of MCS, some improved numerical simulation approaches were subsequently proposed, such as directional sampling approach [7], importance sampling approach [8], subset simulation method [9], line sampling method [10], etc. Compared with the MCS, these improved numerical simulation methods can greatly reduce the computational cost in probabilistic reliability analysis. In the surrogate model approach, a surrogate model is primarily constructed to replace the actual input-output relationship by using regression and/or classification methods, and then the probabilistic reliability can be estimated easily according to the surrogate model by using some mature simulation approaches. Despite the surrogate model approach has higher computational efficiency than other approaches, it may lose the ability to capture the model behavior for the models which are fundamentally controlled by high-order interactions.
The application of probabilistic reliability analysis needs adequate information to accurately construct the probability density functions (PDFs) or cumulative distribution functions (CDFs) of random model inputs. Unfortunately, in the early design stage of the structure, the sample information is limited, thus the probabilistic reliability analysis is no longer applicable in this situation [13], [14]. In addition, as pointed out in Ref. [14], some vital uncertainties cannot be handled reasonably by probabilistic theory, such as those coming from human behaviors and expertise. As the membership function (MF) of fuzzy variable mainly depends on the personal knowledge [15], [16], less or no samples are needed to construct it. Therefore, fuzzy reliability analysis method can be used to analyze the safety degree of the structure under the above two situations. Based on the fuzzy set theory [17], some fuzzy reliability analysis approaches have been established. Zadeh [18] proposed the concept of failure possibility to quantitatively measure the safety degree of the structure with fuzzy uncertainty. According to the MF of model output, Shrestha and Duckstein [19] introduced an area ratio index, which was defined as the area of MF of model output in the failure domain to the whole area of the MF of model output. Cremona and Gao [20] presented a novel possibilistic reliability analysis theory on the basis of a new possibilistic safety index, which was introduced as the shortest Chebyshev distance from the coordinate origin to the failure domain. Li and coworkers [21] put forward an innovative fuzzy reliability analysis approach by treating the MFs of all the fuzzy model inputs as a group of independent random variables which follow the standard uniform distribution. In this paper, the classical failure possibility is selected to be studied because it not only has been widely applied in numerous engineering fields [22], [23], but also is the fundamental of some other fuzzy reliability measures, such as failure necessity [22] and failure credibility [4].
At present, the computational methods for estimating the failure possibility can be principally divided into two categories, i.e., the optimization based methods and the simulation based methods. The basic idea of the optimization based methods, such as alpha-level membership cut approach [24] and dichotomy approach [13], is to estimate the MF of model output by employing the optimization technique at first, and then compute the failure possibility based on this estimated MF of the model output. The optimization based method is no doubt a good means to propagate the fuzzy uncertainty from the model inputs to the model output, as well as estimate the failure possibility in theory. Nevertheless in practice, local optimal nature and slow computation speed of the optimization technique make this method hard to be widely applied to complex engineering problems. Inspired by the MCS in estimating probabilistic reliability, fuzzy simulation (FS) technique [25] based on the fuzzy transformation is investigated as a more universal method in fuzzy reliability analysis. While employing the FS to estimate failure possibility, a large number of samples of model inputs should be generated at first, and then the corresponding model outputs are evaluated according to the input-output relationship. Next, the maximum of the joint MF with respect to the input samples in the failure domain is considered as the estimate of the failure possibility. Since the optimization process is avoided in FS, FS is more accurate and robust in estimating failure possibility than the optimization based methods [23]. However, the computational cost of FS in handling the problem with complex input-output relationship and/or multidimensional model inputs is too large to be accepted by the engineers.
In order to improve the sampling efficiency of FS in estimating the failure possibility, the fuzzy importance sampling (FIS) method is proposed in this paper. The idea of FIS is to carry out FS with failure samples possessing larger joint MFs because only these samples contribute to the estimation of failure possibility. According to this basic idea, the optimal importance sampling density (OISD) for estimating the failure possibility is deduced as the product of the indicator function of the failure domain and the joint MF of the model inputs in this paper. As the indicator function of the failure domain is involved in the OISD, the explicit expression of OISD is extremely difficult to be obtained. Therefore, inspired by Ref. [8], a technique combining the kernel sampling density [26] and Markov chain simulation [27] is employed to construct an approximate expression of the OISD. At first, the Markov chain simulation is used to generate a sequence of samples asymptotically following the OISD. Then, by employing these samples, a kernel sampling density is constructed as an approximate OISD. Finally, the failure possibility can be accurately and efficiently estimated by using a small number of samples generated from the acquired approximate OISD.
The remainder of this paper is organized as follows. The concept of the failure possibility and its FS solution are introduced in Section 2. The FIS method combining the Markov chain simulation and kernel sampling method, is developed for estimating the failure possibility in Section 3. Three benchmark examples including a mathematical and two engineering examples are employed to compare the accuracy, efficiency and robustness of the proposed FIS and the original FS methods in Section 4. A brief conclusion is summarized in Section 5.
Section snippets
Concept of the failure possibility and its fuzzy simulation solution
Let represent a fuzzy vector containing n independent fuzzy variable . In fuzzy set theory, a single fuzzy variable is described by its MF while the fuzzy vector X is characterized by its joint MF . According to Ref. [22], if all the fuzzy variables are independent, the relationship between and can be represented by,
Assume the input-output relationship of a structure be denoted by
Fuzzy importance sampling method
Although some numerical cases have shown the precision and robustness of FS in estimating the failure possibility, they also indicate the low sampling efficiency and large computational burden of this method [23], especially for the problem with multidimensional inputs and complex input-output relationship. Thus, in order to overcome the shortage of the classical FS, an efficient FIS method is put forward in this section.
Test examples
In this section, a mathematical example and two engineering examples are employed to illustrate the performance of the proposed FIS method in estimating the failure possibility. For the first example, the analytical method is used as a reference solution to compare the accuracy of other methods. For the second and third examples, the results obtained by the FS method with small variation coefficient (the threshold of the variation coefficient is set to 5% in this paper) are considered as the
Conclusion
Although the fuzzy simulation (FS) is an accurate and universal method in estimating the failure possibility of the structure involving fuzzy uncertainty, the heavy computational cost of this method hinders its application in complex engineering problems. Therefore, a novel fuzzy importance simulation (FIS) is proposed to drastically reduce the total model evaluations in estimating the failure possibility through improving the sampling efficiency of the classical FS. According to the estimation
Appendix. Detailed steps of the alpha-level membership cut approach
The main procedure of the alpha-level membership cut approach for estimating failure possibility is illustrated in Fig. 6, and it can be divided into the following three steps.
Step 1. Discretize the membership level of the fuzzy input vector X into p discrete points, i.e., []. On the basis of the fuzzy set theory, X is restricted in the membership interval at any discrete membership level , where and .
Declaration of Competing Interest
We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service and/or company that could be construed as influencing the position presented in, or the review of, the manuscript.
Acknowledgement
This work was supported by the National Natural Science Foundation of China (Grant No. NSFC 52075442), the National Science and Technology Major Project (2017-IV-0009-0046), and the Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (Grant No. CX201934).
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