Global kriging surrogate modeling for general time-variant reliability-based design optimization problems

Abstract

While design optimization under uncertainty has been widely studied in the last decades, time-variant reliability-based design optimization (t-RBDO) is still an ongoing research field. The sequential and mono-level approaches show a high numerical efficiency. However, this might be to the detriment of accuracy especially in case of nonlinear performance functions and non-unique time-variant most probable failure point (MPP). A better accuracy can be obtained with the coupled approach, but this is in general computationally prohibitive. This work proposes a new t-RBDO method that overcomes the aforementioned limitations. The main idea consists in performing the time-variant reliability analysis on global kriging models that approximate the time-dependent limit state functions. These surrogates are built in an artificial augmented reliability space and an efficient adaptive enrichment strategy is developed that allows calibrating the models simultaneously. The kriging models are consequently only refined in regions that may potentially be visited by the optimizer. It is also proposed to use the same surrogates to find the deterministic design point with no extra computational cost. Using this point to launch the t-RBDO guarantees a fast convergence of the optimization algorithm. The proposed method is demonstrated on problems involving nonlinear limit state functions and non-stationary stochastic processes.

Appendix: Kriging technique

The kriging technique consists in approximating a function G(x) that depends on a vector of input variables x = {x₁, x₂, …, x_m} as a sum of a regression model and a Gaussian stochastic process. This can be expressed as follows:

$$ \widehat{G}\left(\mathbf{x}\right)=\mathbf{f}{\left(\mathbf{x}\right)}^t\boldsymbol{\upbeta} +{\upsigma}_{\mathrm{Z}}^2Z\left(\mathbf{x}\right) $$

(27)

where f(x) = {f₁(x), f₂(x), …, f_p(x)}^t is a vector of regression functions, β is a vector of unknown coefficients and f(x)^tβ is the mean value of the Gaussian process, also known as the trend. ${\upsigma}_{\mathrm{Z}}^2$ is the Gaussian process variance and Z(x) is a stationary Gaussian process with zero mean and unit variance. Z(x) is completely described by its user-defined autocorrelation function R(x⁽ⁱ⁾, x^(j), θ) where θ = {θ₁, θ₂, …, θ_d} represents the vector of unknown correlation parameters to be determined.

Given an experimental design {x⁽ⁱ⁾, G(x⁽ⁱ⁾)}, i = 1, …, n, β and ${\upsigma}_{\mathrm{Z}}^2$ can be approximated as follows (Jones et al., 1998):

$$ \widehat{\boldsymbol{\upbeta}}\left(\boldsymbol{\uptheta} \right)={\left({\mathbf{F}}^t{\mathbf{R}}^{-1}\mathbf{F}\right)}^{-1}{\mathbf{F}}^t{\mathbf{R}}^{-1}\mathbf{G} $$

(28)

and

$$ {\widehat{\upsigma}}_{\mathrm{Z}}^2\left(\boldsymbol{\uptheta} \right)=\frac{1}{n}{\left(\mathbf{G}-\mathbf{F}\boldsymbol{\upbeta } \right)}^t{\mathbf{R}}^{-1}\left(\mathbf{G}-\mathbf{F}\boldsymbol{\upbeta } \right) $$

(29)

where F is a matrix of size n × p and of general term F_ij = f_j(x⁽ⁱ⁾). And G = {G(x⁽¹⁾), G(x⁽²⁾), …, G(x⁽ⁿ⁾)}^t is the response vector.

Note that β and ${\upsigma}_{\mathrm{Z}}^2$ both depend on the correlation parameters θ_i. The optimal values of θ can be determined by maximum likelihood estimation (Marrel et al., 2008):

$$ {\boldsymbol{\uptheta}}^{opt}=\underset{\boldsymbol{\uptheta}}{\mathit{\arg}\;\mathit{\min}}\left(\widehat{\sigma}2{\left(\det \mathbf{R}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$n$}\right.}\right) $$

(30)

Once the kriging parameters are determined, they can be used to predict the model response at a new trial point x^∗ as a Gaussian random variable of mean ${\mu}_{\widehat{G}}\left({\mathbf{x}}^{\ast}\right)$ and variance ${\sigma}_{\widehat{G}}^2\left({\mathbf{x}}^{\ast}\right)$ (Jones et al., 1998) defined as follows:

$$ {\mu}_{\widehat{G}}\left({\mathbf{x}}^{\ast}\right)=\mathbf{f}{\left({\mathbf{x}}^{\ast}\right)}^t\boldsymbol{\upbeta} +\mathbf{r}{\left({\mathbf{x}}^{\ast}\right)}^t{\mathbf{R}}^{-1}\left(\mathbf{G}-\mathbf{F}\boldsymbol{\upbeta } \right) $$

(31)

and

$$ {\sigma}_{\widehat{G}}^2\left({\mathbf{x}}^{\ast}\right)={\upsigma}_{\mathrm{Z}}^2\left(1-\left[\mathbf{f}{\left({\mathbf{x}}^{\ast}\right)}^t\kern0.75em \mathbf{r}{\left({\mathbf{x}}^{\ast}\right)}^t\right]\left[\begin{array}{cc}\mathbf{0}& {\mathbf{F}}^t\\ {}\mathbf{F}& \mathbf{R}\end{array}\right]\left[\begin{array}{c}\mathbf{f}\left({\mathbf{x}}^{\ast}\right)\\ {}\mathbf{r}\left({\mathbf{x}}^{\ast}\right)\end{array}\right]\right) $$

(32)

where r(x^∗) is the correlation vector between x^∗ and the inputs of the experimental design: r_i(x^∗) = R(x⁽ⁱ⁾, x^∗, θ^opt).