An efficient decoupled method for time-variant reliability-based design optimization

Abstract

Time-variant reliability-based design optimization (tRBDO) can rationally consider the time-variant uncertainties in engineering structures and find the optimal design that can keep reliable throughout its whole life cycle. However, solving the tRBDO involves a nested double-loop procedure and requires excessive computational cost. In this paper, a novel decoupled method called sequential approximate time-variant reliability analysis and optimization (SATO) is proposed to improve the efficiency of tRBDO. First, a two-step method is proposed to transform the original tRBDO problem into an equivalent deterministic optimization problem according to the results of time-variant reliability analysis (TRA). Second, a novel approximate TRA (ATRA) method based the least-square method is proposed to reduce the computational cost of TRA. Finally, the proposed SATO method decouples the original double-loop procedure in tRBDO into a sequential process of ATRA and deterministic optimization. Test results of a complicated welded beam problem verify that the proposed method can achieve similar accuracy and much higher efficiency than the compared methods. A rocket inter-stage structure problem demonstrates the capability of the proposed method in practical engineering applications.

Appendices

Appendix 1

1.1 AMPPT method for TRA and identifying critical time instants

The AMPPT method is based on the concept of MPP trajectory (Zhang et al. 2021). It can not only accurately calculate the time-variant reliability, but also identify the critical time instants within the time interval [0, T] via an adaptive sampling process.

For an arbitrary time instant t_a ∈ [0, T] , denote the MPP of the instantaneous performance function g(d, X, Y(t_a), t_a) as u_MPP(t_a). When t_a varies from 0 to T, the MPP u_MPP(t_a) will move from u_MPP(0) to u_MPP(T). If we connect all these MPPs, a curve u_MPP(t)(t ∈ [0, T]) in u-space can be obtained, which is defined as the MPP trajectory. Figure 14 shows a schematic diagram of the MPP trajectory, where the solid curves represent the limit-state boundaries at the critical time instants.

Fig. 14

Schematic diagram of the MPP trajectory and critical time instants

For a given TRA problem, the AMPPT method first approximates its MPP trajectory with the adaptive Kriging model ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$, and then calculates the time-variant reliability based on ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$, The solution process of AMPPT consists of three steps, which are briefly described as follows.

First, discretize [0, T] into N_init equidistant time instants, and perform MPP searches at these time instants to obtain the initial time-MPP samples. {(t_i, u_MPP(t_i))| i = 1, 2, .., N_init} Then, build a rough Kriging model ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$ with these samples. Afterwards, perform an adaptive sampling process to iteratively identify the critical time instants t^∗, at which the Kriging model ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$ has the largest prediction variance,

$$ {t}^{\ast }=\arg {\displaystyle \begin{array}{c}\max \\ {}t\in \left[{t}_s,{t}_e\right]\end{array}}\frac{\sum_{j=1}^{n+m}{\sigma}_j^2(t)}{n+m} $$

(34)

where ${\sigma}_j^2(t)\left(j=1,2,\cdots n+m\right)$ is the prediction variance of the j-th component of ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$. Then, perform MPP-search at t^∗ to obtain a new sample (t^∗, u_MPP(t^∗)) , and update ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$ accordingly. Repeat this process until the Kriging model ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$ is accurate enough.

Second, according to ${\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)$, transform the response of the time-variant performance function g(d, X, Y, (t), t) into an equivalent Gaussian process H(t) . The mean, standard deviation, and autocorrelation coefficient functions of H(t) are derived as:

$$ {\displaystyle \begin{array}{c}\mu (t)=-\beta (t)=-\left\Vert {\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)\right\Vert \\ {}\begin{array}{c}\sigma (t)=1\\ {}\rho \left({t}_1,{t}_2\right)=\frac{\mathrm{Cov}\left(H\left({t}_1\right),H\left({t}_2\right)\right)}{\sigma \left({t}_1\right)\sigma \left({t}_2\right)}\\ {}=\mathrm{Cov}\left(-\beta \left({t}_2\right)+{\left({\boldsymbol{a}}_{\mathrm{v}}\left({t}_1\right)\right)}^T\mathbf{V}+{\left({\boldsymbol{a}}_{\mathrm{w}}\left({t}_1\right)\right)}^T{\mathbf{W}}_{1,}-\beta \left({t}_2\right)+{\boldsymbol{a}}_{\mathrm{v}}\left({t}_2\right)\left){}^T\mathbf{V}+{\boldsymbol{a}}_{\mathrm{w}}\left({t}_1\right)\right){}^T{\mathbf{W}}_2\right)\end{array}\\ {}\begin{array}{c}=\mathrm{Cov}{\left(\Big({\boldsymbol{a}}_{\mathrm{v}}\left({t}_1\right)\right)}^T\mathbf{V}+{\left({\boldsymbol{a}}_{\mathrm{w}}\left({t}_1\right)\right)}^T{\mathbf{W}}_1,{\left({\boldsymbol{a}}_{\mathrm{v}}\left({t}_2\right)\right)}^T\mathbf{V}+{\left({\boldsymbol{a}}_{\mathrm{w}}\left({t}_2\right)\right)}^T{\mathrm{W}}_2\Big)\\ {}=\mathrm{Cov}\left(\Big({\boldsymbol{a}}_{\mathrm{v}}{\left({t}_1\right)}^T\mathbf{V},{\left({\boldsymbol{a}}_{\mathrm{v}}\Big({t}_2\right)}^T\mathbf{V}\right)+\mathrm{Cov}\left({\left({\boldsymbol{a}}_{\mathrm{w}}\left({t}_1\right)\right)}^T{\mathrm{W}}_1,{\left({\boldsymbol{a}}_w\left({t}_2\right)\right)}^T{\mathrm{W}}_2\right)\\ {}={\left({\boldsymbol{a}}_{\mathrm{v}}\left({t}_1\right)\right)}^T{\boldsymbol{a}}_{\mathrm{v}}\left({t}_2\right)+{\left({\boldsymbol{a}}_{\mathrm{w}}\left({t}_1\right)\right)}^T\mathbf{C}\left({t}_{1,}{t}_2\right){\boldsymbol{a}}_w\left({t}_2\right)\end{array}\end{array}} $$

(35)

where C(t₁, t₂) is a n_Y × n_Y correlation coefficient matrix. V and W are independent standard normal random variables transformed from X and Y(t), respectively. a_v(t) and a_W(t) are calculated by

$$ {\displaystyle \begin{array}{c}{\boldsymbol{\upalpha}}_{\mathrm{V}}(t)={\hat{\mathbf{u}}}_{\mathrm{MPP},\mathrm{V}}(t)/\beta (t)\\ {}{\boldsymbol{\upalpha}}_{\mathrm{W}}(t)={\hat{\mathbf{u}}}_{\mathrm{MPP},\mathrm{W}}(t)/\beta (t)\\ {}{\hat{\mathbf{u}}}_{\mathrm{MPP}}(t)=\left[{\hat{\mathbf{u}}}_{\mathrm{MPP},\mathrm{V}}(t),{\hat{\mathbf{u}}}_{\mathrm{MPP},\mathrm{W}}(t)\right]\end{array}} $$

(36)

Finally, calculate the time-variant reliability based on spectral decomposition (Sudret and Der Kiureghian 2002) and Monte Carlo Simulation (MCS). Discretize the time interval [0, T] into s equidistant time instants t_i(i = 1, 2, …, s), and construct a covariance matrix ∑ as

$$ \boldsymbol{\Sigma} ={\left[\begin{array}{cccc}\mathrm{Cov}\left({t}_1,{t}_1\right)& \mathrm{Cov}\left({t}_2,{t}_1\right)& \cdots & \mathrm{Cov}\left({t}_s,{t}_1\right)\\ {}\mathrm{Cov}\left({t}_1,{t}_2\right)& \mathrm{Cov}\left({t}_2,{t}_2\right)& \cdots & \mathrm{Cov}\left({t}_s,{t}_2\right)\\ {}\vdots & \vdots & \ddots & \vdots \\ {}\mathrm{Cov}\left({t}_1,{t}_s\right)& \mathrm{Cov}\left({t}_2,{t}_s\right)& \cdots & \mathrm{Cov}\left({t}_s,{t}_s\right)\end{array}\right]}_{s\times s} $$

(37)

where, Cov(t_i, t_j) = σ(t_i)σ(t_j)ρ(t_i, t_j), for i, j = 1, 2, ⋯, s. Then, H(t) can be decomposed as (Sudret and Der Kiureghian 2002)

$$ H(t)\approx \mu (t)+\sigma (t)\sum \limits_{k=1}^p\frac{\xi_k}{\sqrt{\lambda_k}}{\mathbf{Q}}_k^T\boldsymbol{\uprho} (t) $$

(38)

where ξ_k(k = 1, 2, …, p) are independent standard normal random variables;λ_k and Q_k are the eigenvalues and eigenvectors of ∑, respectively; ρ(t) = [Cov(t₁, t), Cov(t₂, t), .., Cov(t_p, t)]^T is a covariance vector.

Use (38) to generate N_MCS samples ${H}^{(j)}=\left[{h}_1^{(j)},{h}_2^{(j)},\cdots, {h}_s^{(j)}\right]$, (j = 1, 2, ⋯, N_MCS) of H(t), and the time-variant reliability index β_cur can be estimated by

$$ {\beta}_{\mathrm{cur}}={\Phi}^{-1}\left(P(T)\right)={\Phi}^{-1}\left(\frac{\sum \limits_{j=1}^{N_{\mathrm{MCS}}}I\left({H}^{(i)}\right)}{N_{\mathrm{MCS}}}\right) $$

(39)

where I(H^(j)) is an indicator function. If ${\max}_{i=l}^s\left({h}_i^{(j)}\right)<0$ , I(H^(j)) = 1; otherwise, I(H^(j)) = 0.

Appendix 2

1.1 Kriging model

In both the AMPPT method described above and the ATRA method proposed in Section 3.2, the Kriging model (Lophaven et al. 2002; Gano et al. 2006) is selected to approximate the MPP trajectory due to its advantage in providing the prediction variance and its successful applications in field of reliability analysis (Hawchar et al. 2018; Zhang et al. 2019; Li et al. 2020).

The Kriging model approximates the jth(j = 1, 2, …, n + m) component μ_{MPP, j}(t) of the MPP trajectory u_MPP(t) (see Fig. 14) as

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

(40)

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

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

(41)

where σ² is the variance of s(t), R(t_p, t_q) is the correlation coefficient, and θ is a parameter that can be determined by the maximum likelihood estimation (Giunta and Watson 1998).

Assume the number of “time-MPP” pairs {(t_i, u_MPP(t_i))⌊i = 1, 2, ⋯n} is n. Denote y = {u_{MPP, j}(t_i)}. The natural logarithm of the likelihood function is defined as

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

(42)

where R = [R(t_p, t_q)]_n × n is a n × n correlation matrix and A is a n × 1 unit vector. By setting the derivatives of (42) with respect to μ and σ² to zero, μ and σ²can be estimated as

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

(43)

Substituting (43) into (42),θ can be determined by maximizing the likelihood function

$$ \theta ={\displaystyle \begin{array}{c}\mathrm{argmax}\\ {}\theta \end{array}}\left(-\frac{nl\mathrm{n}{\hat{\sigma}}^2+\ln \mid \mathbf{R}\mid }{2}\right) $$

(44)

Once all hyper parameters are obtained, the Kriging model ${\hat{u}}_{\mathrm{MPP},j}(t)$ can be used to predict the jth (j = 1, 2, …, n + m) component of the MPP at an arbitrary time instant $\overset{\acute{\mkern6mu}}{t}$:

$$ {\hat{u}}_{\mathrm{MPP},j}\left(\overset{\acute{\mkern6mu}}{t}\right)=\hat{\mu}+{\mathbf{r}}^T{\mathbf{R}}^{-1}\left(\mathbf{y}-\mathbf{A}\hat{\mu}\right) $$

(45)

where r is a correlation vector defined by $\mathbf{r}={\left[R\left(\overset{\acute{\mkern6mu}}{t},{t}_1\right),R\left(\overset{\acute{\mkern6mu}}{t},{t}_2\right),\dots, R\left(\overset{\acute{\mkern6mu}}{t},{t}_n\right)\right]}^T$. The variance of the prediction in (45) is given by

$$ {\sigma}_j^2\left(\overset{\acute{\mkern6mu}}{t}\right)={\overset{\acute{\mkern6mu}}{\sigma}}^2\left[1-{\boldsymbol{r}}^T{\mathbf{R}}^{-1}\mathbf{r}+\frac{{\left(1-{\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{r}\right)}^2}{{\mathbf{A}}^T{\mathbf{R}}^{-1}\mathbf{A}}\right] $$

(46)