An efficient reliability analysis method combining adaptive Kriging and modified importance sampling for small failure probability

Abstract

A vital challenge problem of structural reliability analysis is how to estimate the small failure probability with a minimum number of model evaluations. The Adaptive Kriging combined with Importance Sampling method (AK-IS) which is developed from the adaptive Kriging combined with Monte Carlo simulation (AK-MCS) is a viable method to deal with this challenge. The aim of this paper is to reduce the number of model evaluations of the existing AK-IS algorithm. Firstly, we use a contributive weight function to divide the candidate samples of model input variables generated in AK-IS. The candidate samples are used to select the best next sample to update the Kriging model in AK-IS. Secondly, select the best next sample only in the important area obtained according to the contributive weight value to failure probability to update the Kriging model until the stopping condition is satisfied. Thirdly, use the Kriging model constructed in the important area to predict the other area and update the important area by adding the point with the maximum contributive weight value in the area except the important area ceaselessly until the probability of the accurate identification on the limit state function’s signs (positive limit state value or negative limit state value) of all the importance sampling points satisfies a criterion. Finally, the updated Kriging model is used to estimate the failure probability especially for the small failure probability. The proposed method uses a thought from local to global in order to reduce the computational cost of AK-IS and simultaneously guarantees the accuracy of estimation. A non-linear oscillator system, a roof truss structure and a planar ten-bar structure are analyzed by the proposed method. The results demonstrate the efficiency and accuracy of the proposed method in structural reliability analysis especially for small failure probability.

Appendix 1. Kriging model

The Kriging model is a semi-parametric interpolation technique based on the statistical theory (Sacks et al. 1989) including the parametric linear regression part and the nonparametric stochastic process. For an unknown function g(x), the Kriging model is given as follows:

$$ {g}_K\left(\boldsymbol{x}\right)=\sum \limits_{i=1}^p{B}_i\left(\boldsymbol{x}\right){\beta}_i+\boldsymbol{Z}\left(\boldsymbol{x}\right)={\boldsymbol{B}}^{\mathrm{T}}\left(\boldsymbol{x}\right)\boldsymbol{\beta} +\boldsymbol{Z}\left(\boldsymbol{x}\right) $$

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where B(x) = [B₁(x), B₂(x), …, B_p(x)]^T are the base functions of vector x, β = [β₁, β₂, …, β_p]^T is the regression coefficient vector, and p denotes the number of base functions. Z(x) is a stationary Gaussian process with zero mean and covariance which can be defined as follows:

$$ \mathit{\operatorname{cov}}\left[Z\left({\boldsymbol{x}}^{(i)}\right),Z\left({\boldsymbol{x}}^{(j)}\right)\right]={\sigma}^2\boldsymbol{R}\left(\left({\boldsymbol{x}}^{(i)}\right),\left({\boldsymbol{x}}^{(j)}\right)\right)\kern4.5em i,j=1,\dots, {N}_0 $$

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where N₀ denotes the number of training points.

Define $\boldsymbol{R}=\left[\begin{array}{ccc}R\left(\left({\boldsymbol{x}}^{(1)}\right),\left({\boldsymbol{x}}^{(2)}\right)\right)& \cdots & R\left(\left({\boldsymbol{x}}^{(1)}\right),\left({\boldsymbol{x}}^{\left({N}_0\right)}\right)\right)\\ {}\vdots & \ddots & \vdots \\ {}R\left(\left({\boldsymbol{x}}^{\left({N}_0\right)}\right),\left({\boldsymbol{x}}^{(1)}\right)\right)& \cdots & R\left(\left({\boldsymbol{x}}^{\left({N}_0\right)}\right),\left({\boldsymbol{x}}^{\left({N}_0\right)}\right)\right)\end{array}\right]$, B is a vector of B(X) and g is corresponding vector of the limit state functions calculated at each experimental points (x⁽ⁱ⁾)(i = 1, 2, …, N₀), the unknown β and σ² can be estimated as follows:

$$ \widehat{\boldsymbol{\beta}}={\left({\boldsymbol{B}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{B}\right)}^{-1}{\boldsymbol{B}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{g} $$

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$$ \widehat{\sigma^2}=\frac{1}{N_0}{\left(\boldsymbol{g}-\boldsymbol{B}\widehat{\boldsymbol{\beta}}\right)}^T{\boldsymbol{R}}^{-1}\left(\boldsymbol{g}-\boldsymbol{B}\widehat{\boldsymbol{\beta}}\right) $$

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Therefore, for any unknown point x, the Best Linear Unbiased Predictor of model g_K(x) is shown to be a Gaussian random ${g}_K\left(\boldsymbol{x}\right)\sim N\left({\mu}_{g_K}\left(\boldsymbol{x}\right),{\sigma}_{g_K}\left(\boldsymbol{x}\right)\right)$ where the mean and the variance are given as follows:

$$ {\mu}_{g_K}\left(\boldsymbol{x}\right)={\boldsymbol{B}}^{\mathrm{T}}\left(\boldsymbol{x}\right)\widehat{\boldsymbol{\beta}}+{\boldsymbol{r}}^T\left(\boldsymbol{x}\right){\boldsymbol{R}}^{-1}\left(\boldsymbol{g}-\boldsymbol{B}\widehat{\boldsymbol{\beta}}\right) $$

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$$ {\sigma}_{g_K}^2\left(\boldsymbol{x}\right)={\sigma}^2-\left[{\boldsymbol{B}}^{\mathrm{T}}\left(\boldsymbol{x}\right)\kern0.5em {\boldsymbol{r}}^{\mathrm{T}}\left(\boldsymbol{x}\right)\right]{\left[\begin{array}{cc}\mathbf{0}& {\boldsymbol{B}}^T\\ {}\boldsymbol{B}& \boldsymbol{R}\end{array}\right]}^{-1}\left[\begin{array}{c}\boldsymbol{B}\left(\boldsymbol{x}\right)\\ {}\boldsymbol{r}\left(\boldsymbol{x}\right)\end{array}\right] $$

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where ${\boldsymbol{r}}^{\mathrm{T}}\left(\boldsymbol{x}\right)={\left[R\left(\left(\boldsymbol{x}\right),\left({\boldsymbol{x}}^{(1)}\right)\right),\dots, R\left(\left(\boldsymbol{x}\right),\left({\boldsymbol{x}}^{\left({N}_0\right)}\right)\right)\right]}^{\mathrm{T}}$.