Adaptive kriging model-based structural reliability analysis under interval uncertainty with incomplete data

Abstract

Uncertainty of quantitative models of input variables and computational model could certainly cause the uncertainties of structural response and structural reliability. Hence, structural reliability analysis requires precise input uncertainty model and highly accurate solving model. However, not all uncertain input variable can be described by an explicit quantitative model in practical engineering. Generally, only incomplete samples of some variables are available in practice. Though Monte Carlo Simulation (MCS) has been used to solve above problem, we are confronted with another trouble due to the expensive two-layer MCSs. Furthermore, approximate methods for reliability assessment would cause confidence problem of reliability. To handle with the challenges, an adaptive kriging (AK) model-based approach is proposed by dividing the two-layer MCSs into two AK models. Simultaneously, a new quantitative model for interval variables is developed to deal with input uncertainty. And a novel learning function improved by H learning function (IH function) is developed with a weight function to enhance the efficiency of constructing kriging models. The IH function not only considers design sites with large uncertainty, but also actively searches for that around the LSF by assigning different weight value for design points. In the proposed approach, the first AK model is constructed for reliability prediction. And the relationship between parameters of input models and reliability is built by the other kriging model using the first one. Credibility assessment will be implemented according to the second model. Since, only the first AK model needs the time-consuming finite element (FE) calculations, the proposed approach could significantly improve the efficiency of confidential reliability analysis without losing accuracy. Several numerical examples are implemented to demonstrate the feasibility and effectiveness of the proposed model.

Appendix

According to Eqs. (4) and (5), the posterior joint PDF of the two parameters can be solved as

$$\begin{aligned} p({\varvec{\mu}},\;\left. {\varvec{r}} \right|x^{*} ) & = \frac{{\left( {2{\varvec{r}}} \right)^{{ - n_{{{\text{initial}}}} }} \times c}}{{\int_{\Sigma } {\left( {2{\varvec{r}}} \right)^{{ - n_{{{\text{initial}}}} }} \times c{\text{d}}{\varvec{\mu}}{\text{d}}{\varvec{r}}} }} \\ & = \frac{{{\varvec{r}}^{{ - n_{{{\text{initial}}}} }} }}{{\int_{\Sigma } {{\varvec{r}}^{{ - n_{{{\text{initial}}}} }} {\text{d}}{\varvec{\mu}}{\text{d}}{\varvec{r}}} }} \\ & = \lambda {\varvec{r}}^{{ - n_{{{\text{initial}}}} }} , \\ \end{aligned}$$

(31)

where constant $\lambda$ is

$$\begin{aligned} \lambda & = \left[ {\int_{\Sigma } {{\varvec{r}}^{{ - n_{{{\text{initial}}}} }} {\text{d}}{\varvec{\mu}}{\text{d}}{\varvec{r}}} } \right]^{ - 1} \\ & = \left[ {\int_{{\left( {x_{\max }^{*} - x_{\min }^{*} } \right)/2}}^{ + \infty } {{\text{d}}{\varvec{r}}\int_{{x_{\max }^{*} - r}}^{{x_{\min }^{*} + r}} {{\varvec{r}}^{{ - n_{{{\text{initial}}}} }} {\text{d}}{\varvec{\mu}}} } } \right]^{ - 1} \\ & = \left[ {\int_{{\left( {x_{\max }^{*} - x_{\min }^{*} } \right)/2}}^{ + \infty } {{\varvec{r}}^{{ - n_{{{\text{initial}}}} }} \left( {x_{\min }^{*} - x_{\max }^{*} + 2{\varvec{r}}} \right){\text{d}}{\varvec{r}}} } \right]^{ - 1} \\ & = \left[ {\frac{1}{{\left( {n_{{{\text{initial}}}} - 1} \right)\left( {n_{{{\text{initial}}}} - 2} \right)}}\frac{{2^{{n_{{{\text{initial}}}} - 1}} }}{{\left( {x_{\max }^{*} - x_{\min }^{*} } \right)^{{n_{{{\text{initial}}}} - 2}} }}} \right]^{ - 1} . \\ \end{aligned}$$

(32)

Thus, the marginal PDFs is shown as

$$\begin{aligned} p(\left. {\varvec{r}} \right|x^{*} ) & = \int {p({\varvec{\mu}},\;\left. {\varvec{r}} \right|x^{*} ){\text{d}}{\varvec{\mu}}} \\ & = \int_{{x_{\max }^{*} - r}}^{{x_{\min }^{*} + r}} {\lambda {\varvec{r}}^{{ - n_{{{\text{initial}}}} }} {\text{d}}{\varvec{\mu}}} \\ & = \frac{{\lambda \left( {2\varvec{r + }x_{\min }^{*} - x_{\max }^{*} } \right)}}{{{\varvec{r}}^{{n_{{{\text{initial}}}} }} }},\quad {\varvec{r}} \ge \left( {x_{\max }^{*} - x_{\min }^{*} } \right)/2, \\ \end{aligned}$$

(33)

$$\begin{aligned} p(\left. {\varvec{\mu}} \right|x^{*} ) & = \int {p({\varvec{\mu}},\;\left. {\varvec{r}} \right|x^{*} ){\text{d}}{\varvec{r}}} \\ & = \left\{ \begin{gathered} \int_{{\varvec{\mu - }x_{\min }^{*} }}^{ + \infty } {\lambda {\varvec{r}}^{{ - n_{{{\text{initial}}}} }} {\text{d}}{\varvec{r}}} \hfill \\ \int_{{\varvec{ - \mu } + x_{\max }^{*} }}^{ + \infty } {\lambda {\varvec{r}}^{{ - n_{{{\text{initial}}}} }} {\text{d}}{\varvec{r}}} \hfill \\ \end{gathered} \right. \\ & = \left\{ {\begin{array}{*{20}c} {\frac{\lambda }{{\left( {n_{{{\text{initial}}}} - 1} \right)\left( {x_{\max }^{*} - {\varvec{\mu}}} \right)^{{n_{{{\text{initial}}}} - 1}} }},} & {{\varvec{\mu}} \le \left( {x_{\min }^{*} + x_{\max }^{*} } \right)/2} \\ {\frac{\lambda }{{\left( {n_{{{\text{initial}}}} - 1} \right)\left( {{\varvec{\mu}} - x_{\min }^{*} } \right)^{{n_{{{\text{initial}}}} - 1}} }},} & {{\varvec{\mu}} > \left( {x_{\min }^{*} + x_{\max }^{*} } \right)/2.} \\ \end{array} } \right. \\ \end{aligned}$$

(34)