Time-variant global reliability sensitivity analysis of structures with both input random variables and stochastic processes

Abstract

The ubiquitous uncertainties presented in the input factors (e.g., material properties and loads) commonly lead to occasional failure of mechanical systems, and these input factors are generally characterized as random variables or stochastic processes. For identifying the contributions of the uncertainties presented in the input factors to the time-variant reliability, this work develops a time-variant global reliability sensitivity (GRS) analysis technique based on Sobol’ indices and Karhunen- Loève (KL) expansion. The proposed GRS indices are shown to be effective in identifying the individual, interaction and total effects of both the random variables and stochastic processes on the time-variant reliability, and can be especially useful for reliability-based design. Three numerical methods, including the Monte Carlo simulation (MCS), the first order envelope function (FOEF) and the active learning Kriging Monte Carlo simulation (AK-MCS), are introduced for efficiently estimating the proposed GRS indices. A numerical example, a beam structure and a ten-bar structure under time-variant loads are introduced for demonstrating the significance of the time-variant GRS analysis technique and the effectiveness of the numerical methods.

Appendices

Appendix A: The OSE method

The use of OSE method avoids solving the eigenvalue problem in (5) by introducing a set of orthogonal functions (Sudret and Der Kiureghian 2000). Let {h _j (t)} ^∞ _j = 1 denote a set of orthogonal functions (e.g., Legendre polynomials), which holds:

$$ {\displaystyle {\int}_0^T{h}_j(t){h}_k(t)\mathrm{d}t}={\delta}_{jk} $$

(A1)

where δ _jk is a Kronecker symbol. Then the random Gaussian process Y _i (t) can be represented by the following expansion:

$$ {Y}_i(t)={\mu}_{Y_i}(t)+{\displaystyle \sum_{j=1}^{\infty }{\chi}_{ij}{h}_j(t)} $$

(A2)

where {χ _ij } ^∞ _j = 1 indicates a set of zero-mean Gaussian random variables, possibly correlated. Based on the orthogonal property of the basic {h _j (t)} ^∞ _j = 1 , the covariance between χ _ij and χ _ik can be derived to be:

$$ {\sum}_{\chi_{ij}{\chi}_{ik}}={\displaystyle {\int}_0^T{\displaystyle {\int}_0^T{C}_{Y_i}\left(t,{t}_1\right){h}_j(t){h}_k\left({t}_1\right)\mathrm{d}t\mathrm{d}{t}_1}} $$

(A3)

Then, given M _i terms left for the approximation, the stochastic process Y _i (t) is approximated by

$$ {Y}_i(t)\cong {\mu}_{Y_i}(t)+{\displaystyle \sum_{j=1}^{M_i}{\chi}_{ij}{h}_j(t)} $$

(A4)

For generating the KL expansion given in (6), the set of random variables ${\left\{{\chi}_{ij}\right\}}_{j=1}^{M_i}$ needs to be transformed to uncorrelated standard Gaussian variables. Suppose the covariance matrix of ${\left\{{\chi}_{ij}\right\}}_{j=1}^{M_i}$ estimated by (A3) is ${\sum}_{\chi_i{\chi}_i}$, then ${\left\{{\chi}_{ij}\right\}}_{j=1}^{M_i}$ can be transformed into a set of independent standard Gaussian variables by many statistical methods such as the spectral decomposition and Mahalanobis transformation (Zuber and Strimmer 2011). Take the former as an example. The spectral decomposition of ${\sum}_{\chi_i{\chi}_i}$ is given by

$$ {\sum}_{\chi_i{\chi}_i}\bullet {\mathtt{D}}_i={\mathtt{D}}_i\bullet \varLambda $$

(A5)

where Λ is the (M _i  × M _i ) diagonal matrix with the diagonal elements being the eigenvalues λ _ik of ${\sum}_{\chi_i{\chi}_i}$ and ${\mathtt{D}}_j$ is a matrix of dimension (M _i  × M _i ) with each column being the corresponding eigenvectors. Then the relation between the random vector χ _i and the independent standard Gaussian vector ξ _i is expressed as:

$$ {\boldsymbol{\upchi}}_i^T={\mathtt{D}}_i\bullet {\mathtt{\varLambda}}^{1/2}\bullet {\boldsymbol{\upxi}}_i^T $$

(A6)

Let {D ^k _ij } ^M _j = 1 denote the coordinates of the k th eigenvector, then from (A6), the j th component of χ _i is given by:

$$ {\chi}_{ij}={\displaystyle \sum_{k=1}^{M_i}{\mathrm{D}}_{ij}^k\sqrt{\lambda_k}{\xi}_{ik}} $$

(A7)

Substituting (A7) into (A4) yields:

$$ {Y}_i(t)\cong {\mu}_{Y_i}(t)+{\displaystyle \sum_{j=1}^{M_i}\left({\displaystyle \sum_{k=1}^{M_i}{\mathrm{D}}_{ij}^k\sqrt{\lambda_k}{\xi}_{ik}}\right){h}_j(t)}={\mu}_{Y_i}(t)+{\displaystyle \sum_{k=1}^{M_i}\sqrt{\lambda_k}{\xi}_{ik}\left({\displaystyle \sum_{j=1}^{M_i}{\mathrm{D}}_{ij}^k{h}_j(t)}\right)} $$

(A8)

Let

$$ {\varphi}_{ik}(t)={\displaystyle \sum_{j=1}^{M_i}{\mathrm{D}}_{ij}^k{h}_j(t)} $$

(A9)

Eq. (A8) can then be rewritten as:

$$ {Y}_i(t)\cong {\mu}_{Y_i}(t)+{\displaystyle \sum_{k=1}^{M_i}\sqrt{\lambda_k}{\xi}_{ik}{\varphi}_{ik}(t)} $$

(A10)

Comparing (6) and (A10), one can find that (A10) is an estimate of the KL expansion of the stochastic process Y _i (t).

Appendix B: Derivation of the distribution parameters required in FOEF method

The mean vector ${\boldsymbol{\upmu}}_{M{X}_i}$ and covariance matrix ${\sum}_{M{X}_i}$ of ${\mathbf{L}}_{M{X}_i}$ required for estimating the main partial variance ${V}_{X_i}$ are derived as ${\boldsymbol{\upmu}}_{M{X}_i}=\left({\boldsymbol{\upmu}}_L,{\boldsymbol{\upmu}}_L\right)$ and ${\sum}_{M{X}_i}={\left({\sigma}_{M{X}_ipq}\right)}_{p,q=1,2,\dots, 2 nt}$, respectively, where

$$ {\sigma}_{M{X}_ipq}=\left\{\begin{array}{l}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_q\right)}}\hfill & \hfill p,q\le nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\beta}_{X_i}^2+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_{q- nt}\right)}}\hfill & \hfill p\le nt,q> nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\beta}_{X_i}^2+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_{p- nt}\right){\varphi}_{lk}\left({t}_q\right)}}\hfill & \hfill p> nt,q\le nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_{p- nt}\right){\varphi}_{lk}\left({t}_{q- nt}\right)}}\hfill & \hfill p,q> nt\hfill \end{array}\end{array}\right. $$

(A11)

The mean vector ${\boldsymbol{\upmu}}_{M{Y}_i}$ and covariance matrix ${\sum}_{M{Y}_i}$ of ${\mathbf{L}}_{M{Y}_i}$ required in (22) for estimating the main partial variance ${V}_{Y_i}$ can be derived as ${\boldsymbol{\upmu}}_{M{Y}_i}={\boldsymbol{\upmu}}_{M{X}_i}$ and ${\sum}_{M{Y}_i}={\left({\sigma}_{M{Y}_ipq}\right)}_{p,q=1,2,\dots, 2 nt}$, respectively, where

$$ {\sigma}_{M{Y}_ipq}=\left\{\begin{array}{l}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_q\right)}}\hfill & \hfill p,q\le nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{k=1}^{M_i}{\beta}_{Y_{ik}}^2{\varphi}_{ik}\left({t}_p\right){\varphi}_{ik}\left({t}_{q- nt}\right)}\hfill & \hfill p\le nt,q> nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{k=1}^{M_i}{\beta}_{Y_{ik}}^2{\varphi}_{ik}\left({t}_{p- nt}\right){\varphi}_{ik}\left({t}_q\right)}\hfill & \hfill p> nt,q\le nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_i}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_{p- nt}\right){\varphi}_{lk}\left({t}_{q- nt}\right)}}\hfill & \hfill p,q> nt\hfill \end{array}\end{array}\right. $$

(A12)

The mean vector ${\boldsymbol{\upmu}}_{M{X}_i{X}_j}$ and covariance matrix ${\sum}_{M{X}_i{X}_j}$ of ${\mathbf{L}}_{M{X}_i{X}_j}$ for estimating the second order partial variance ${V}_{X_i{X}_j}$ by (23) are derived as ${\boldsymbol{\upmu}}_{M{X}_i{X}_j}={\boldsymbol{\upmu}}_{M{X}_i}$ and ${\sum}_{M{X}_i{X}_j}={\left({\sigma}_{M{X}_i{X}_jpq}\right)}_{p,q=1,2,\dots, 2 nt}$, where

$$ {\sigma}_{M{X}_i{X}_jpq}=\left\{\begin{array}{l}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_q\right)}}\hfill & \hfill p,q\le nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\beta}_{X_i}^2+{\beta}_{X_j}^2+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_{q- nt}\right)}}\hfill & \hfill p\le nt,q> nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\beta}_{X_i}^2+{\beta}_{X_j}^2+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_{p- nt}\right){\varphi}_{lk}\left({t}_q\right)}}\hfill & \hfill p> nt,q\le nt\hfill \end{array}\\ {}\begin{array}{cc}\hfill {\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_{p- nt}\right){\varphi}_{lk}\left({t}_{q- nt}\right)}}\hfill & \hfill p,q> nt\hfill \end{array}\end{array}\right. $$

(A13)

The mean vector ${\boldsymbol{\upmu}}_{T{X}_i}$ and covariance matrix ${\sum}_{T{X}_i}$ of ${\mathbf{L}}_{T{X}_i}$ for estimating the total partial variance ${V}_{T{X}_i}$ are derived to be ${\boldsymbol{\upmu}}_{T{X}_i}={\boldsymbol{\upmu}}_L$ and ${\sum}_{T{X}_i}={\left({\sigma}_{T{X}_ipq}\right)}_{p,q=1,2,\dots, nt}$, where

$$ {\sigma}_{T{X}_ipq}={\displaystyle \sum_{l=1,l\ne i}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_q\right)}} $$

(A14)

The mean vector ${\boldsymbol{\upmu}}_{T{Y}_i}$ and covariance matrix ${\sum}_{T{Y}_i}$ of ${\mathbf{L}}_{T{Y}_i}$ for estimating the total partial variance ${V}_{T{\boldsymbol{\upxi}}_i}$ are derived as ${\boldsymbol{\upmu}}_{T{Y}_i}={\boldsymbol{\upmu}}_L$ and ${\sum}_{T{Y}_i}={\left({\sigma}_{T{Y}_ipq}\right)}_{p,q=1,2,\dots, nt}$ with

$$ {\sigma}_{T{Y}_ipq}={\displaystyle \sum_{l=1}^n{\beta}_{X_l}^2}+{\displaystyle \sum_{l=1,l\ne i}^m{\displaystyle \sum_{k=1}^{M_l}{\beta}_{Y_{lk}}^2{\varphi}_{lk}\left({t}_p\right){\varphi}_{lk}\left({t}_q\right)}} $$

(A15)

Appendix C: The Kriging surrogate model

For simplicity, we rewrite the random input vector (X, Y ^', t) as X, and the limit state function as Z = G(X). The basic rationale of Kriging is to assume the limit state function G(X) is a realization of an underlying Gaussian random field, and can be expressed as:

$$ G\left(\mathbf{X}\right)=\mathbf{f}{\left(\mathbf{X}\right)}^T\boldsymbol{\upbeta} +Z\left(\mathbf{X}\right) $$

(A16)

Where β = (β ₁, β ₂, …, β _p )^T is a vector of unknown regression coefficients, f(X) = (f ₁(X), f ₂(X), …., f _p (X))^T is a vector of square integrable functional basis, f(X)^T β is the trend of prediction, Z(X) is a zero-mean stationary Gaussian random field with constant variance σ ² _G . If the trend function is specified as a constant, then (A16) is an ordinary Kriging model. The trend function can also be specified as other types such as linear regression model and higher order polynomial regression models. Given two realizations x and x ^' of the random input vector, the autocovariance function of the Gaussian field is given as:

$$ {C}_{GG}\left(\mathbf{x},{\mathbf{x}}^{\hbox{'}}\right)={\sigma}_G^2R\left(\mathbf{x}-{\mathbf{x}}^{\hbox{'}}\right) $$

(A17)

where R(x − x ^') refers to the autocorrelation function, and the most commonly used one is the exponential one which reads:

$$ R\left(\mathbf{x}-{\mathbf{x}}^{\hbox{'}}\right)= \exp \left(-d\left(\mathbf{x},{\mathbf{x}}^{\hbox{'}}\right)\right) $$

(A18)

where d(x, x ^') indicates the distance between x and x ^', and it is commonly expressed as $d\left(\mathbf{x},{\mathbf{x}}^{\hbox{'}}\right)={\displaystyle \sum_{k=1}^n{a}_k{\left({x}_k-{x}_k^{\hbox{'}}\right)}^{b_k}}$ with a _k and b _k being the parameters of the Kriging model defining the strength of correlation of the Gaussian random field. Given a set of N training input samples {x ⁽¹⁾, …, x ^(N)} and the corresponding values z = {g(x ⁽¹⁾), …, g(x ^(N))}^T, the estimation of the limit state function at a new realization x of X turns out to be Gaussian random variable with mean value μ _Ĝ (x) and variance σ ² _Ĝ (x) estimated by (Echard et al. 2011; Dubourg et al. 2013):

$$ \left\{\begin{array}{l}{\mu}_{\widehat{G}}\left(\mathbf{x}\right)=\mathbf{f}{\left(\mathbf{x}\right)}^T\widehat{\boldsymbol{\upbeta}}+\mathbf{r}{\left(\mathbf{x}\right)}^T{\mathtt{R}}^{-1}\left(\mathbf{z}-\mathtt{F}\widehat{\boldsymbol{\upbeta}}\right)\\ {}{\sigma}_{\widehat{G}}^2\left(\mathbf{x}\right)={\sigma}_G^2\left(1-\mathbf{r}{\left(\mathbf{x}\right)}^T{\mathtt{R}}^{-1}\mathbf{r}\left(\mathbf{x}\right)+\mathbf{u}{\left(\mathbf{x}\right)}^T{\left({\mathtt{F}}^T{\mathtt{R}}^{-1}\mathtt{F}\right)}^{-1}\mathbf{u}\left(\mathbf{x}\right)\right)\end{array}\right. $$

(A19)

where $\mathtt{R}$ is the correlation matrix with the (i, j) th component R _ij  = R(x ⁽ⁱ⁾, x ^(j)) computed by (A18), r(x) is the vector of cross-correlations between the new realization x and the N training samples, i.e., r _j (x) = R(x, x ^(j)), $\mathtt{F}$ is the regression matrix defined by F _ij  = f _j (x ⁽ⁱ⁾) (i = 1, …, N, j = 1, …, d). The vector u(x) reads $\mathbf{u}\left(\mathbf{x}\right)={\mathtt{F}}^T{\mathtt{R}}^{-1}\mathbf{r}\left(\mathbf{x}\right)-\mathbf{f}\left(\mathbf{x}\right)$ and the least squares estimation $\widehat{\boldsymbol{\upbeta}}$ of the linear regression coefficients is $\widehat{\boldsymbol{\upbeta}}={\left({\mathtt{F}}^T{\mathtt{R}}^{-1}\mathtt{F}\right)}^{-1}{\mathtt{F}}^T{\mathtt{R}}^{-1}\mathbf{z}$. The surrogate model for the limit state function is then μ _Ĝ (x), and the Kriging variance σ ² _Ĝ (x) is the minimum of the mean square error between Ĝ(x) and G(x). Kriging is an interpolation method, which means that μ _Ĝ (x ^(j)) = z ^(j) and σ ² _Ĝ (x ^(j)) = 0 holds for j = 1, …, N.

Appendix D: Analytical expression of the limit state function of the ten-bar structure

With the virtual work principle, the vertical displacement 3 is derived as (Wei et al. 2014):

$$ {\varDelta}_3=\left({\displaystyle \sum_{i=1}^6{N}_i^0{N}_i+\sqrt{2}{\displaystyle \sum_{i=7}^{10}{N}_i^0{N}_i}}\right)\frac{L}{AE} $$

(A20)

where N _i is the axial internal force of the bar with number i, and N ⁰ _i is the axial internal force of the bar with number i when P ₁ = P ₃ = 0 and P ₂ = 1. The internal force N _i is derived as:

$$ \begin{array}{l}{N}_1={P}_2-\frac{\sqrt{2}}{2}{N}_8,\kern1em {N}_2=-\frac{\sqrt{2}}{2}{N}_{10},\kern0.5em {N}_3=-{P}_1-2{P}_2+{P}_3-\frac{\sqrt{2}}{2}{N}_8,\kern0.6em {N}_4=-{P}_2+{P}_3-\frac{\sqrt{2}}{2}{N}_{10}\\ {}{N}_5=-{P}_2-\frac{\sqrt{2}}{2}{N}_8-\frac{\sqrt{2}}{2}{N}_{10},\kern0.9em {N}_6=-\frac{\sqrt{2}}{2}{N}_{10},\kern1em {N}_7=\sqrt{2}\left({P}_1+{P}_2\right)+{N}_8,\kern0.8em {N}_9=\sqrt{2}{P}_2+{N}_{10}\\ {}{N}_8=\frac{a_{22}{b}_1-{a}_{12}{b}_2}{a_{11}{a}_{22}-{a}_{12}{a}_{21}},\kern1.8em {N}_{10}=\frac{a_{11}{b}_2-{a}_{21}{b}_1}{a_{11}{a}_{22}-{a}_{12}{a}_{21}}\end{array} $$

(A21)

where

$$ \begin{array}{l}{a}_{11}=\frac{\left(3+4\sqrt{2}\right)L}{2AE},\kern0.6em {a}_{22}=\frac{\left(2+2\sqrt{2}\right)L}{AE},\kern0.5em {a}_{12}={a}_{21}=\frac{L}{2AE},\kern1.4em \\ {}{b}_1=\frac{L\left[-\left(2\sqrt{2}+4\right){P}_2-\left(\sqrt{2}+4\right){P}_1+\sqrt{2}{P}_3\right]}{2AE},\kern0.8em {b}_2=\frac{L\left[\sqrt{2}{P}_3-\left(4+2\sqrt{2}\right){P}_2\right]}{2AE}\end{array} $$

(A22)