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An efficient hierarchical fuzzy simulation method for estimating failure possibility

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Abstract

The failure possibility (FP) can reasonably measure the safety degree of the structure under fuzzy uncertainty, and the estimation of FP can be transformed into searching the point with the maximum joint membership function (MF) of fuzzy input vector in the failure domain (also known as fuzzy design point). In the current fuzzy simulation (FS) method, the fuzzy design point is searched in the maximum value region of the fuzzy input vector corresponding to the lowest membership level which is equal to 0 and the computational efficiency is low. In this paper, an efficient hierarchical fuzzy simulation (HFS) method is proposed for estimating FP. In the proposed method, by the nature that the fuzzy design point is a failure point with the maximum joint MF, the fuzzy design point is first searched in the smaller value region of input vector corresponding to the larger membership level, and the membership level is automatically reduced layer by layer to expand the search region until the failure points appear. Compared with the traditional FS method, the proposed HFS method not only guarantees the search accuracy of the fuzzy design point, but also reduces the total search region; thus the computational efficiency is improved. In addition, an adaptive Kriging model is also embedded in the search process of HFS. Since the adaptively updated Kriging model is used to replace the real performance function for recognizing the state of the simulated sample points during the search process, the strategy of combining the Kriging model with HFS method can further improve the search efficiency of the fuzzy design point. The results of examples show that the proposed HFS method is reasonable and efficient.

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Abbreviations

FP:

Failure possibility

FS:

Fuzzy simulation

HFS:

Hierarchical fuzzy simulation

MF:

Membership function

AK:

Adaptive Kriging

RE:

Relative error

\({\varvec{X}}\) :

Fuzzy input vector, and \({\varvec{X}} = {x_{1} ,x_{2} ,...,x_{n} }^{{\text{T}}}\)

\({\varvec{x}}\) :

Realization of \({\varvec{X}}\), and \({\varvec{x}} = {x_{1} ,x_{2} ,...,x_{n} }^{{\text{T}}}\)

\(F\) :

Failure domain, and \(F = \{ g({\varvec{x}}) \le 0\}\)

\(g({\varvec{x}})\) :

Performance function

\(\pi_{f}\) :

Failure possibility

\(N_{\alpha }\) :

The number of layers

\(\rho_{{\varvec{X}}} ( \cdot )\) :

Joint MF of \({\varvec{X}}\)

\(\rho_{{X_{i} }} ( \cdot )\) :

MF of \(X_{i}\)

\(\rho_{{X_{i} }}^{ - 1} ( \cdot )\) :

Inverse function of MF \(\rho_{{X_{i} }} ( \cdot )\)

\({\text{Poss}}\{ \cdot \}\) :

Possibility operator

\(N_{{{\text{call}}}}\) :

The number of calls to performance function

\(\alpha\) :

Membership level

\({\text{Sup}}\{ \cdot \}\) :

Supremum operator

\({\varvec{x}}\user2{(}\alpha \user2{)}\) :

Membership interval vector

\({\varvec{x}}^{L} \user2{(}\alpha \user2{)}\) :

Lower bound of membership interval

\({\varvec{x}}^{U} \user2{(}\alpha \user2{)}\) :

Upper bound of membership interval

\(\gamma\) :

Scale factor and \(\gamma < 1\)

\(g_{K} {(}{\varvec{x}}{)}\) :

Kriging surrogate model of performance function

\(S_{{\varvec{x}}}^{(k)} \,\) :

Candidate sample pool of fuzzy input corresponding to \(\alpha_{k}\) membership level

\(I_{{\text{F}}} ( \cdot )\) :

Indicator function of failure domain

\({\varvec{x}}^{ * }\) :

Fuzzy design point

\(U\user2{(} \cdot \user2{)}\) :

U-learning function

\(\mu_{{g_{K} }} ({\varvec{x}})\) :

Predicted mean of \(g_{K} {(}{\varvec{x}}{)}\)

\(\sigma_{{g_{K} }} ({\varvec{x}})\) :

Prediction standard deviation of \(g_{K} {(}{\varvec{x}}{)}\)

\(T\) :

The training sample set

\({\varvec{x}}_{{{\text{new}}}}\) :

New training sample point

\(N_{{\text{T}}}\) :

The number of training sample points

\(S_{{\varvec{x}}}\) :

Candidate sample pool of fuzzy input corresponding to 0 membership level

\(i\) :

\(i = 1,2,...,N_{k}\); \(N_{k}\) Is the number of samples in the \(k\)-th layer

\(k\) :

\(k = 1,2,...,N_{\alpha }\); \(N_{\alpha }\) Is the number of layers

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Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant no. NSFC 52075442), National Science and Technology Major Project (2017-IV-0009-0046) and Innovation Foundation for Doctor Dissertation of Northwestern Polytechnical University (CX2022018).

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  1. School of Aeronautics, Northwestern Polytechnical University, Xi’an, 710072, Shaanxi, China

    Xia Jiang, Zhenzhou Lu & Kaixuan Feng

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  1. Xia Jiang
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Correspondence to Zhenzhou Lu.

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Jiang, X., Lu, Z. & Feng, K. An efficient hierarchical fuzzy simulation method for estimating failure possibility. Engineering with Computers 39, 3085–3097 (2023). https://doi.org/10.1007/s00366-022-01692-9

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