A Single-Loop Fuzzy Simulation-Based Adaptive Kriging Method for Estimating Time-Dependent Failure Possibility

Abstract

To improve the efficiency of the double-loop fuzzy simulation (DLFS) for estimating the time-dependent failure possibility (TDFP), a single-loop fuzzy simulation (SLFS) is proposed in this paper. In the SLFS, an equivalent transformation formula of TDFP is put forward for the first time, then the estimation of TDFP is transformed into a single-loop fuzzy simulation procedure where the fuzzy inputs and time parameter are sampled in the same level. As only single-loop sampling is needed in the SLFS, the computational complexity and cost of the proposed method are both reduced compared to the DLFS. Subsequently, a single-loop Kriging model based SLFS (ASLK-SLFS) is developed to enhance the performance of the SLFS. Based on the candidate sampling pool of SLFS to sample the fuzzy inputs and the time parameter in the same level, a single Kriging can be more efficiently constructed and updated. To further improve the efficiency of ASLK-SLFS, an improved version is then developed by using a candidate sampling pool reduction strategy. Finally, three examples are employed to illustrate the advantages of the proposed methods. Through the proposed ASLK-SLFS, the safety degree of the time-dependent structure with fuzzy uncertainty can be efficiently evaluated.

Appendix: Brief Introduction of Kriging and U-Learning Function

In Kriging theory, the actual performance function $g({\varvec{X}})$ is considered as a realization of the stochastic field $g_{K} ({\varvec{X}})$ which is written as,

$$ g_{K} \left( {\varvec{X}} \right) = {\varvec{f}}\left( {\varvec{X}} \right){\varvec{\xi}} + {\varvec{Z}}\left( {\varvec{X}} \right) $$

(22)

in which ${\varvec{f}}({\varvec{X}}) = [f_{1} ({\varvec{X}}),f_{2} ({\varvec{X}}), \cdots ,f_{{N_{f} }} ({\varvec{X}})]$ represents the $N_{f}$-dimensional basis function vector of ${\varvec{X}}$, ${\varvec{\xi}} = \left[ {\xi_{1} ,\xi_{2} , \cdots ,\xi_{{N_{f} }} } \right]^{T}$ expresses the $N_{f}$-dimensional regression coefficient vector, and ${\varvec{Z}}\left( {\varvec{X}} \right)$ denotes a stationary Gaussian process with the following statistic characteristics,

$$ \begin{gathered} {\text{Expectation}}\quad E\left( {Z\left( {\varvec{x}} \right)} \right) = 0 \hfill \\ {\text{Variance}}{\kern 1pt} \quad \quad {\text{Var}}\left( {Z\left( {\varvec{x}} \right)} \right) = \sigma_{Z}^{2} \hfill \\ {\text{Covariance}}\quad {\text{cov}} \left[ {Z\left( {{\varvec{x}}_{{j_{1} }} } \right),Z\left( {{\varvec{x}}_{{j_{2} }} } \right)} \right] = \sigma_{Z}^{2} R_{Z} \left[ {Z\left( {{\varvec{x}}_{{j_{1} }} } \right),Z\left( {{\varvec{x}}_{{j_{2} }} } \right)} \right] \hfill \\ \end{gathered} $$

(23)

in which ${\varvec{x}}_{{j_{1} }}$ and ${\varvec{x}}_{{j_{2} }}$ are the $j_{1}$ th and $j_{2}$ th elements among the training sample set, and $R_{Z}$ is the kernel function that determines the smoothness of the Kriging model. In this paper, the commonly used Gaussian kernel function is used in the Kriging model. The detailed process for constructing the Kriging model can refer to Ref. [25].

For any untrained sample x, the Kriging prediction can be expressed as,

$$ g_{K} \left( {\varvec{x}} \right)\sim N\left( {\mu_{{g_{K} }} \left( {\varvec{x}} \right),\sigma_{{g_{K} }}^{2} \left( {\varvec{x}} \right)} \right) $$

(24)

in which N (·) stands for the normal distribution, $\mu_{{g_{K} }} \left( {\varvec{x}} \right)$ represents the mean of the prediction, and $\sigma_{{g_{K} }} \left( {\varvec{x}} \right)$ denotes its standard deviation.

Based on the prediction characteristics of the Kriging model $g_{K} ({\varvec{X}})$, the probability of misidentifying the sign of $g({\varvec{x}})$ by using $g_{K} ({\varvec{x}})$ can be expressed as,

$$ P_{{{\text{mis}}}} = \Phi \left( { - \frac{{\left| {\mu_{{g_{K} }} \left( {\varvec{x}} \right)} \right|}}{{\sigma_{{g_{K} }} \left( {\varvec{x}} \right)}}} \right) = \Phi \left( { - U\left( {\varvec{x}} \right)} \right) $$

(25)

in which $U({\varvec{x}})$ is known as the U learning function, which is defined as,

$$ U\left( {\varvec{x}} \right) = \frac{{\left| {\mu_{{g_{K} }} \left( {\varvec{x}} \right)} \right|}}{{\sigma_{{g_{K} }} \left( {\varvec{x}} \right)}} $$

(26)

Eqs. (25) and (26) indicate that the smaller the value of $U({\varvec{x}})$ is, the higher the probability of misidentifying the sign of $g({\varvec{x}})$. Generally, the sign of the performance function with respect to the sample with U learning function value not less than 2 can be considered to be accurately predicted, since the state of this samples can be correctly identified by the Kriging model with $1 - \Phi ( - 2) = 97.7\%$ probability at least.