Efficient time-variant reliability analysis through approximating the most probable point trajectory

Abstract

Time-variant reliability analysis (TRA) is widely utilized to assess the performance of engineering structures under various time-variant uncertainties. Recently, the time discretization-based TRA (TDTRA) methods have been developed, which can achieve satisfactory accuracy but need to excessively perform most probable point (MPP) searches at many equidistant time instants. To improve the efficiency of TDTRA, this paper proposes a TRA method based on approximating the MPP trajectory, referred to as AMPPT. First, this paper introduces a new concept of the MPP trajectory (MPPT), which is defined as the moving path of the MPP in the U-space when time changes. Then, a one-dimensional Kriging model is constructed to approximate the MPPT by the adaptive sampling method, which only performs MPP searches at several critical time instants. To further improve the computational efficiency, a warm-starting strategy is proposed to accelerate the MPP search. Then, the approximated MPPT is employed to transform the time-variant response into an equivalent Gaussian process. Finally, the spectral decomposition method and Monte Carlo simulation are used to compute the time-variant reliability. Comparative studies on four numerical examples and one practical engineering example of the solid rocket engine shell verify that the proposed AMPPT outperforms TDTRA in terms of both accuracy and efficiency. Test results also indicate that the efficiency gain of the proposed AMPPT comes from not only the reduction in the number of MPP searches but also the acceleration of the MPP search itself.

Appendix. Kriging model

Without loss of generality, the method for constructing the jth (j = 1, 2, …, n + m) component u^{^}_{MPP, j}(t) of the one-dimensional multioutput Kriging model u^{^}_MPP(t) is briefly described as follows.

The Kriging model (Lophaven et al. 2002; Li and Wang 2018, 2019) approximates an unknown function u_{MPP, j}(t) as

$$ {\hat{u}}_{\mathrm{MPP},j}(t)=f(t)+s(t) $$

(42)

where f(t) is a polynomial term of the input parameter t and s(t) is a Gaussian process with zero mean and covariance Cov[s(t_p), s(t_q)]. In this paper, the polynomial term f(t) is treated as a constant μ. The covariance Cov[s(t_p), s(t_q)] of s(t) is calculated by

$$ \mathrm{Cov}\left[s\left({t}_p\right),s\left({t}_q\right)\right]={\sigma}^2R\left({t}_p,{t}_q\right) $$

(43)

where σ² is the variance of s(t) and the R(t_p, t_q) is the correlation coefficient.

The Gaussian function is commonly used in R(t_p, t_q)

$$ R\left({t}_p,{t}_q\right)=\exp \left[-\theta {\left({t}_p-{t}_q\right)}^2\right] $$

(44)

where θ is a parameter that can be determined by the maximum likelihood estimation (Giunta and Watson 1998).

Assume n time-MPP samples {(t_i, u^{^}_MPP(t_i))| i = 1, 2, …, n} are available for training u^{^}_{MPP, j}(t). Denote y = {u_{MPP, j}(t_i)| i = 1, 2, …, n}. The natural logarithm of the likelihood function is defined as

$$ L\left(\theta |\mathbf{y}\right)=-\frac{1}{2}\left[n\ln \left(2\pi \right)+n\ln {\sigma}^2+\ln \left|\mathbf{R}\right|+\frac{{\left(\mathbf{y}-\mathbf{A}\mu \right)}^T{\mathbf{R}}^{-1}\left(\mathbf{y}-\mathbf{A}\mu \right)}{2{\sigma}^2}\right] $$

(45)

where R is a n × n correlation matrix defined by R = [R(t_p, t_q)]_n × n and A is a n × 1 unit vector. By setting the derivatives of (45) with respect to μ and σ² to zero, μ and σ² can be estimated as

$$ {\displaystyle \begin{array}{l}\hat{\boldsymbol{\mu}}=\frac{{\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{y}}{{\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{A}}\\ {}{\sigma}^2=\frac{{\left(\mathbf{y}-\mathbf{A}\mu \right)}^T{\mathbf{R}}^{-1}\left(\mathbf{y}-\mathbf{A}\mu \right)}{n}\end{array}} $$

(46)

Substituting (46) into (45), the unknown parameter θ can be determined by maximizing the likelihood function

$$ \theta =\underset{\theta }{\arg \max}\left(-\frac{n\ln {\sigma}^2+\ln \left|\mathbf{R}\right|}{2}\right) $$

(47)

Once all hyper parameters are obtained, the Kriging model u^{^}_{MPP, j}(t) can be used to predict the jth (j = 1, 2, …, n + m) component of the MPP at an arbitrary time instant t^′:

$$ {\hat{\boldsymbol{\mu}}}_{\mathrm{MPP},j}\left({t}^{\prime}\right)=\mu +{\mathbf{r}}^T{\mathbf{R}}^{-1}\left(\mathbf{y}-\mathbf{A}\hat{\boldsymbol{\mu}}\right) $$

(48)

where r is a correlation vector defined by r = [R(t^′, t₁), R(t^′, t₂), …, R(t^′, t_n)]^T. The variance of the prediction in (48) is given by

$$ {\sigma}_j^2\left({t}^{\prime}\right)={\sigma}^2\left[1-{\mathbf{r}}^T{\mathbf{R}}^{-1}\mathbf{r}+\frac{{\left(1-{\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{r}\right)}^2}{{\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{A}}\right] $$

(49)