An efficient algorithm for time-dependent failure credibility by combining adaptive single-loop Kriging model with fuzzy simulation

Abstract

The time-dependent failure credibility (TDFC) can reasonably measure the safety level of the time-dependent structure under the fuzzy uncertainty, but the direct optimization algorithm to estimate the TDFC requires large computational cost and even results in locally optimal solutions. Therefore, an efficient method is proposed for estimating the TDFC by combining the fuzzy simulation and the single-loop Kriging model. In the proposed method, fuzzy inverse transformation theorem is firstly used to transform the estimation of the TDFC into a sample classification problem, in which the candidate sample pool generated by fuzzy simulation (FS) is classified into the failure group and the safety one. For improving the efficiency of the classification, a Kriging model is adaptively trained by an elaborate U-learning function in the candidate sample pool. After the candidate sample is divided into the failure group and the safety one by the convergent Kriging model, the TDFC can be estimated as a byproduct easily. The innovation of the proposed method includes two aspects: establishing the idea of the fuzzy simulation combined with the single-loop Kriging model to estimate TDFC efficiently and robustly, and designing an elaborate U-learning function to improve the efficiency of training the single-loop Kriging model. The presented examples validate the efficiency of the proposed method under the acceptable precision.

Appendices

Appendix A Common membership functions

The following Table 9 lists several common membership functions which include the normal type, the logarithmic normal type and the Gaussian type, the triangular type, and the trapezoid type (Jia and Lu 2018).

Table 9 Several common membership functions

Appendix B Kriging model

The Kriging model, as an unbiased estimation model with minimum variance, has the characteristics of combining global approximation and local random error. Its effectiveness does not depend on the existence of random errors, and it has a good fitting effect for the problem of high nonlinear degree and local response mutation. Therefore, the Kriging model can be used to approximate the global and local functions (Hurtado 2004). The Kriging model can be approximately expressed as follows,

$$ {g}_K\left(\boldsymbol{X}\right)={p}^T\left(\boldsymbol{X}\right)\boldsymbol{\beta} +z\left(\boldsymbol{X}\right) $$

(24)

where g_K(X) is the unknown Kriging model, and p(X) = {p₁(X), p₂(X), …, p_M(X)}^T is the basis functions of the input variables X_. β = {β₁, β₂, …, β_M}^T is the undetermined coefficients of the corresponding regression function, whose value can be estimated by the known response value; M represents the number of basis functions. z(X)is a Gaussian process with an expectation of 0 and a variance of ${\sigma}_Z^2$ created on the basis of global simulation. The covariance matrix of x_i and x_j can be expressed as,

$$ \operatorname{cov}\left[z\left({\boldsymbol{x}}_i\right),z\left({\boldsymbol{x}}_j\right)\right]={\sigma}_Z^2R\left({\boldsymbol{x}}_i,{\boldsymbol{x}}_j\right) $$

(25)

where R(x_i, x_j)(i, j = 1, 2, .…, m) represents the correlation function of any two sample points. There are various function forms of R(x_i, x_j); Gaussian type (Lophaven et al. 2002) is commonly used, and its expression is shown as follows:

$$ R\left({\boldsymbol{x}}_i,{\boldsymbol{x}}_j\right)=\mathit{\exp}\left[-\sum \limits_{k=1}^m{\theta}_k{\left|{\boldsymbol{x}}_{\mathrm{ik}}-{\boldsymbol{x}}_{\mathrm{jk}}\right|}^2\right] $$

(26)

where θ_k(k = 1, 2, …, m) are the unknown correlation parameters. R is the correlation matrix, which is represented by the following formula:

$$ \boldsymbol{R}=\left(\begin{array}{cccc}R\left({x}_1,{x}_1\right)& R\left({x}_1,{x}_2\right)& \cdots & R\left({x}_1,{x}_m\right)\\ {}R\left({x}_2,{x}_1\right)& R\left({x}_2,{x}_2\right)& \cdots & R\left({x}_2,{x}_m\right)\\ {}\vdots & \vdots & \ddots & \vdots \\ {}R\left({x}_m,{x}_1\right)& R\left({x}_m,{x}_2\right)& \cdots & R\left({x}_m,{x}_m\right)\end{array}\right) $$

(27)

Based on the theory of Kriging, the response value at the unknown point x can be obtained by the following formula:

$$ {g}_K(x)=p(x)\hat{\boldsymbol{\beta}}+{\boldsymbol{r}}^T(x){\boldsymbol{R}}^{-1}\left(g-\boldsymbol{P}\hat{\boldsymbol{\beta}}\right) $$

(28)

where $\hat{\boldsymbol{\beta}}$ is the estimated value of β, g is the column vector formed by the response value of the training samples, P = {p(x₁), p(x₂), .…, p(x_m)}^T is a m × M matrix, and r(x) is the correlation function vector between the training points and the prediction points, which can be expressed as

$$ {\boldsymbol{r}}^T\left(\boldsymbol{x}\right)={\left(\mathrm{R}\left(\boldsymbol{x},{\boldsymbol{x}}_1\right),\mathrm{R}\left(\boldsymbol{x},{\boldsymbol{x}}_2\right),\dots, \mathrm{R}\left(\boldsymbol{x},{\boldsymbol{x}}_m\right)\right)}^T $$

(29)

$\overset{\frown }{\beta }$and the variance estimate ${\sigma}_Z^2$ can be obtained by the following formula:

$$ {\displaystyle \begin{array}{c}\hat{\boldsymbol{\beta}}={\left({\boldsymbol{P}}^T{\boldsymbol{R}}^{-1}\boldsymbol{P}\right)}^{-1}{\boldsymbol{P}}^T{\boldsymbol{R}}^{-1}\boldsymbol{g}\\ {}{\sigma}_Z^2={\left(\boldsymbol{g}-\boldsymbol{P}\hat{\boldsymbol{\beta}}\right)}^T{\boldsymbol{R}}^{-1}\left(\boldsymbol{g}-\boldsymbol{P}\hat{\boldsymbol{\beta}}\right)/m\end{array}} $$

(30)

Relevant parameters θ = [θ₁, θ₂, …, θ_m]^T can be obtained by solving the maximum value of maximum likelihood estimation (MLE) (Jones et al. 1998), namely,

$$ \max F\left(\boldsymbol{\theta} \right)=-\frac{\mathrm{mln}\left({\sigma}_Z^2\right)+\ln \left|\boldsymbol{R}\right|}{2},{\theta}_k\ge 0\left(k=1,2,\dots, m\right) $$

(31)

Kriging model composed of the values of θ obtained by solving the above equation is the proxy model with the best fitting accuracy. Therefore, for any unknown point x, g_K(x) follows a Gaussian distribution, i.e., ${g}_K\left(\boldsymbol{x}\right)\sim N\left({\mu}_K\left(\boldsymbol{x}\right),{\sigma}_K^2\left(\boldsymbol{x}\right)\right)$, where the mean and variance are calculated as follows:

$$ {\mu}_K\left(\boldsymbol{x}\right)={\boldsymbol{p}}^T\left(\boldsymbol{x}\right)\hat{\boldsymbol{\beta}}+{r}^T\left(\boldsymbol{x}\right){\boldsymbol{R}}^{-1}\left(\boldsymbol{g}-\boldsymbol{P}\hat{\boldsymbol{\beta}}\right) $$

(32)

$$ {\sigma}_K^2\left(\boldsymbol{x}\right)={\sigma}_Z^2\left[1-{\boldsymbol{r}}^T(x){\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{x}\right)\right]+{\sigma}_Z^2\left[{\left(\boldsymbol{P}{\boldsymbol{R}}^{-\mathbf{1}}\boldsymbol{r}\left(\boldsymbol{x}\right)-\boldsymbol{p}\left(\boldsymbol{x}\right)\right)}^T{\left(\boldsymbol{P}{\boldsymbol{R}}^{-1}\boldsymbol{P}\right)}^{-1}\left(\boldsymbol{P}{\boldsymbol{R}}^{-1}\boldsymbol{r}\left(\boldsymbol{x}\right)-\boldsymbol{p}\left(\boldsymbol{x}\right)\right)\right] $$

(33)

where the calculation of μ_K(x) and ${\sigma}_K^2(x)$ can be realized by the DACE MATLAB toolbox.

Appendix C Radial basis function neural network

Artificial Neural Networks (ANN) is a mathematical model of algorithm that imitates the behavioral characteristics of animal neural networks and performs distributed parallel information processing (Elhewy et al. 2006). Radial basis function neural network (Matera 1998) is an efficient feed-forward neural network, which has the best approximation performance and global optimal characteristics, and has a simple structure and fast training speed. It is widely used in various fields such as pattern recognition and nonlinear function approximation.

RBF network realizes the non-linear transformation from input space to output space by linear combination of nonlinear basis functions. Radial basis neural networks generally use radial basis function (commonly Gaussian function) as activation function,

$$ R\left({\boldsymbol{x}}_P,{\boldsymbol{c}}_i\right)=\exp \left(-\frac{1}{2{\sigma}^2}{\left\Vert {\boldsymbol{x}}_P-{\boldsymbol{c}}_i\right\Vert}^2\right) $$

(34)

where x_P is the training sample and c_i is the selected center. By self-organizing the central learning method, the network output can be obtained as

$$ {y}_j=\sum \limits_{i=1}^N{w}_{ij}\exp \left(-\frac{1}{2{\sigma}^2}{\left\Vert {\boldsymbol{x}}_P-{\boldsymbol{c}}_i\right\Vert}^2\right)j=1,2,\dots, n $$

(35)

The basis function of the RBF neural network is selected as a Gaussian function, so the variance σ can be obtained by the following formula:

$$ \sigma =\frac{c_{\mathrm{max}}}{\sqrt{2h}} $$

(36)

where c_max is the maximum distance between the selected centers and h is the number of nodes in the hidden layer.

The connection weight of the neuron between the hidden layer and the output layer can be directly calculated by the least square method, and the calculation formula is as follows:

$$ w=\exp \left(\frac{N}{c_{\mathrm{max}}^2}{\left\Vert {\boldsymbol{x}}_P-{\boldsymbol{c}}_i\right\Vert}^2\right) $$

(37)

where N is the number of samples.