Adaptive subdomain sampling and its adaptive Kriging–based method for reliability and reliability sensitivity analyses

Abstract

Reliability measures the ability that the structure finishes its intended function without failures by taking uncertainties into account. Reliability sensitivity commonly is defined as the partial derivative of the failure probability with respect to the distribution parameter, which is often of great importance for the reliability-based design optimization. In this paper, two improvements and one extension of the subdomain sampling (SS) method are researched. The first improvement is the criterion for adaptively determining the number of subdomains. The second improvement is that based on the first improvement, adaptive Kriging (AK) model is embedded into the modified SS (MSS) method to substitute the actual limit state function to identify the limit states of the samples generated in the MSS method. Through adaptively partitioning the distribution region, the size of candidate sampling pool in each circle of updating process of Kriging model is decreased compared with that in the method with the candidate samples being directly sampled in the whole uncertain distribution region, which improves the efficiency of each circle’s updating process. Then, the MSS-based adaptive Kriging (AK-MSS) method is extended to the reliability sensitivity analysis where no extra model evaluations are needed after the failure probability is assessed by the AK-MSS method. That is to say, the reliability and the reliability sensitivity can be simultaneously estimated by the AK-MSS method. Results of case studies in this paper demonstrate the effectiveness of the AK-MSS method.

Appendix. Kriging model

The Kriging model includes two parts. The first part is the parametric linear regression part, and the second part is the non-parametric stochastic process part. The Kriging model of an unknown function is described as follows (Sacks et al. 1989),

$$ {T}_K\left(\boldsymbol{u}\right)=\sum \limits_{i=1}^p{B}_i\left(\boldsymbol{u}\right){\beta}_i+S\left(\boldsymbol{u}\right)={\boldsymbol{B}}^T\left(\boldsymbol{u}\right)\boldsymbol{\beta} +S\left(\boldsymbol{u}\right) $$

(34)

where $B(u)={\left[{B}_1(u),{B}_2(u),\dots, {B}_p(u)\right]}^T$ are the base functions of vector , is the regression coefficient vector, and is the number of base function. is a stationary Gaussian process with zero mean and covariance which can is defined as follows:

$$ \operatorname{cov}\left[S\left({\boldsymbol{u}}_i\right),S\left({\boldsymbol{u}}_j\right)\right]={\sigma}^2R\left({\boldsymbol{u}}_i,{\boldsymbol{u}}_j\right)\kern0.5em i,j=1,2,\dots, {N}_0 $$

(35)

where ${N}_0$ denotes the number of training points.

Define $\boldsymbol{R}=\left[\begin{array}{ccc}R\left({\boldsymbol{u}}_1,{\boldsymbol{u}}_1\right)& \cdots & R\left({\boldsymbol{u}}_1,{\boldsymbol{u}}_{N_0}\right)\\ {}\vdots & \ddots & \vdots \\ {}R\left({\boldsymbol{u}}_{N_0},{\boldsymbol{u}}_1\right)& \cdots & R\left({\boldsymbol{u}}_{N_0},{\boldsymbol{u}}_{N_0}\right)\end{array}\right]$, B is a vector of B (u) and T is the corresponding vector of the function T (u) calculated at each experiment points u_ii = 1, 2, …, N₀; the unknown β and σ² can be estimated by the following equations, i.e.,

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

(36)

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

(37)

Therefore, the Kriging model and its variance are given as follows:

$$ {T}_K\left(\boldsymbol{u}\right)={\boldsymbol{B}}^{\mathrm{T}}\left(\boldsymbol{u}\right)\hat{\boldsymbol{\beta}}+{\boldsymbol{r}}^{\mathrm{T}}\left(\boldsymbol{u}\right){\boldsymbol{R}}^{-1}\left(\boldsymbol{B}\hat{\boldsymbol{\beta}}\right) $$

(38)

$$ {\sigma}_{g_K\left(\boldsymbol{u}\right)}^2={\sigma}^2\left(1-{r}^{\mathrm{T}}\left(\boldsymbol{u}\right){\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{u}\right)+{\left[{\boldsymbol{B}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{u}\right)-\boldsymbol{B}\left(\boldsymbol{u}\right)\right]}^{\mathrm{T}}{\left({\boldsymbol{B}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{B}\right)}^{-1}{\left[{\boldsymbol{B}}^{\mathrm{T}}{\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{u}\right)-\boldsymbol{B}\left(\boldsymbol{u}\right)\right]}^{\mathrm{T}}\right) $$

(39)

where ${\boldsymbol{r}}^{\mathrm{T}}\left(\boldsymbol{u}\right)={\left[\boldsymbol{R}\left(\boldsymbol{u},{\boldsymbol{u}}_1\right),\dots, \boldsymbol{R}\left(\boldsymbol{u},{\boldsymbol{u}}_{N_0}\right)\right]}^{\mathrm{T}}.$

Replication of results

The original codes of the examples in the Sect. 5 are available in the Supplementary materials.