An Importance Sampling Framework for Time-Variant Reliability Analysis Involving Stochastic Processes

Abstract

In recent years, methods were proposed so as to efficiently perform time-variant reliability analysis. However, importance sampling (IS) for time-variant reliability analysis is barely studied in the literature. In this paper, an IS framework is proposed. A multi-dimensional integral is first derived to define the time-variant cumulative probability of failure, which has the similar expression to the classical definition of time-invariant failure probability. An IS framework is then developed according to the fact that time-invariant random variables are commonly involved in time-variant reliability analysis. The basic idea of the proposed framework is to simultaneously apply time-invariant IS and crude Monte Carlo simulation on time-invariant random variables and stochastic processes, respectively. Thus, the probability of acquiring failure trajectories of time-variant performance function is increased. Two auxiliary probability density functions are proposed to implement the IS framework. However, auxiliary PDFs available for the framework are not limited to the proposed two. Three examples are studied in order to validate the effectiveness of the proposed IS framework.

1. Introduction

Most mechanical structures and systems are vulnerable against uncertainties, such as loads and material properties. Reliability analysis provides an appropriate framework to evaluate the probability that a structure or system successfully performs its intended functions over a time interval of interest without failures, considering various forms of uncertainties [1]. Over the past decades, time-invariant reliability analysis methods have been well developed, e.g., analytical methods [2] and simulation-based methods [3,4]. Since dynamic or time-variant uncertainties are often involved, mechanical structures and systems are commonly related to performance degradation, and time-invariant methods cannot be directly used to perform time-variant reliability analysis. Time-variant reliability analysis has gained much attention as a method to deal with the issue. Methods for time-variant reliability analysis in the literature can be roughly summarized into two categories: first-crossing-based methods and extreme-value-based methods.

After the well-known Rice formula was proposed, first-crossing-based methods gained much attention [5,6,7,8]. This kind of method concentrates on calculating the probability that the out-crossing event occurs for the first time in a predefined time interval. The out-crossing event shows that the performance value of a structure exceeds a predefined threshold. With the assumption that out-crossing events are mutually independent when they occur, first-crossing-based methods approximate the cumulative probability of failure using an integration of the out-crossing rate which represents the mean number of out-crossing events per unit of time. Accordingly, calculating the out-crossing rate is vital to first-crossing-based methods. For instance, Renaud et al. [6] presented a PHI2 method which calculated the out-crossing rate by solving a two-component parallel system reliability problem. Sudret [9] proposed an improved version of PHI2 by providing analytical expressions of out-crossing rate. However, the independence assumption may lead to large errors in engineering when out-crossing events are strongly dependent. To address the deficiency, Hu and Du [10] relaxed the assumption using the concept of joint crossing rate, and Yu et al. [11] proposed an approximation of the first-crossing probability density function (PDF) method to avoid direct calculations of the out-crossing rate. The performance of a mechanical structure is commonly a stochastic process because of time-variant uncertainties and uncertainty propagation. Zhang et al. [12] first transformed the time-variant performance into an equivalent Gaussian process and then used crude Monte Carlo simulation (MCS) to calculate the cumulative probability of failure. In addition, composite limit state [13,14], failure process decomposition [15], total probability theorem [16] and stochastic process discretization [17,18] have also been proposed to evaluate the cumulative probability of failure.

It is acknowledged that a mechanical structure fails if the extreme value of its time-variant performance over the time interval of interest exceeds a predefined threshold. Therefore, the extreme value is the primary focus of extreme-value-based methods. Hu and Du [19] proposed a sampling method to estimate the distribution of the extreme value of a stochastic process. Ping et al. [20] proposed an extreme-value event evolution method to obtain the time-variant failure probability corresponding to arbitrary time interval and arbitrary failure threshold. Yu et al. [21] combined the extreme-value moment method and improved the maximum entropy method to address time-variant problems involving multiple failure modes and temporal parameters. In recent years, surrogate models have been adopted to improve the computational efficiency of extreme-value-based methods [22,23,24]. For instance, Hu et al. [25] proposed a single-loop Kriging (SILK) to reduce computational effort of the double-loop procedure [26,27]. In SILK, the Kriging model is adaptively refined based on the learning function U [28] that was originally developed for time-invariant reliability analysis. Considering that the acquisition of failure information may be difficult, Zafar and Wang [29] used transfer learning and the Kriging model to overcome this issue. Lara et al. [23] used polynomial chaos expansion to construct an approximate expression of a time-consuming model and then applied crude MCS on the surrogate model to obtain the cumulative probability of failure. Compared with other methods, surrogate-model-based methods remarkably reduce the number of evaluations of time-consuming models.

The random simulation-based method is also a representative extreme-value-based method. Crude MCS directly generates random trajectories of time-variant performance function. The computational cost presents a major challenge for the application of crude MCS even though it is easy to implement. Therefore, it is often combined with other methods to evaluate time-variant probability of failure, such as surrogate-model-based methods and a part of first-crossing-based methods. A great amount of the literature in regards to crude MCS as the benchmark compares efficiency and accuracy of other methods [6,24,25,30]. Subset simulations (SS), such as Markov Chain Monte Carlo (MCMC), SS with splitting method [31,32] and SS with splitting and partitioning in time [33], are efficient methods for precisely evaluating time-variant probability of failure. However, they are developed to deal with dynamical systems subject to stochastic excitation, and not applicable to problems involving general stochastic process (e.g., Gaussian processes). To address time-variant reliability analysis involving stochastic processes, Du et al. [34] applied parallel SS which was first used to evaluate the failure probabilities of multiple time-invariant limit states, and proposed a method to determine the so-called principal variable.

Importance sampling (IS) techniques for time-variant reliability analysis are barely studied in the literature, whereas IS for time-invariant reliability analysis have been well developed [35,36]. This paper intends to fill the gap and propose an importance sampling framework for time-variant reliability analysis. The input variables of a time-variant performance function of a mechanical structure commonly contain both time-invariant random variables and stochastic processes. Performance values of the structure at different time instants are correlated, because they share the same time-invariant variables and realizations of stochastic processes at different instants are correlated with each other. Considering that applying IS on stochastic processes is difficult, the IS framework proposed in this paper applies time-invariant IS on the time-invariant random variables and samples random trajectories of stochastic processes according to crude MCS in order to generate more failure trajectories of time-variant performance function and enhance the computational efficiency of time-variant reliability analysis.

The remainder of this paper is organized as follows. Section 2.1 provides the problem statement of time-variant reliability analysis and Section 2.2 briefly introduces the crude MCS. The proposed IS framework is developed in Section 3.1, which is followed by two auxiliary PDFs in Section 3.2. The proposed framework is validated in Section 4. Section 5 is the conclusion.

2. Time-Variant Reliability Analysis and Crude MCS

2.1. Time-Variant Reliability Analysis

In engineering, the reliability of a mechanical structure or system generally degrades over its service time t. The performance function G(X, Z(t), t), also called limit state function, is commonly used to determine whether a structure or system can fulfill its intended functions under intended conditions. $\mathit{X}={[X^{(1)},\dots ,X^{(M_1)}]}^T$ is the vector of random variables, collecting the time-invariant variables, e.g., geometric properties. $\mathit{Z}(t)={[X^{(1)}(t),\dots ,X^{(M_2)}(t)]}^{\mathrm{T}}$ represents the vector of stochastic processes, e.g., loadings and degradations in material properties. Failure occurs when the performance value is beyond a prescribed threshold. For simplicity, this paper defines the failure state as G(X,Z(t),t) ≤ 0. Therefore, the cumulative probability of failure associated with G(X, Z(t), t) can be expressed as:

$$P_{\mathrm{f},\mathrm{c}}\left(t_0,t_e\right)=\mathrm{Pr}\left\{G(\mathit{X},\mathit{Z}(t),t)\le 0,\exists t\in \left[t_0,t_e\right]\right\}$$

(1)

where [t₀, t_e] is the time interval of interest. $\mathrm{Pr}\{\cdot \}$ is the probability operator and “$\exists$” means “there exists”. It is worth noting that P_f,c(t₀,t_e) is relatively different from the instantaneous probability of failure P_f,i(t) at time $t\in [t_0,t_e]$ which has the following expression:

$$P_{\mathrm{f},\mathrm{i}}\left(t\right)=\mathrm{Pr}\left\{G(\mathit{X},\mathit{Z}(t),t)\le 0\right\}$$

(2)

As illustrated in Section 1, estimating the probability defined by Equation (1) has received wide attention in the literature, and two categories of methods, i.e., first-crossing-based methods and extreme-value-based methods, have been proposed.

The former category focuses on calculating the out-crossing rate,

$$\lambda (t)=\underset{\Delta t\to 0}{\lim }\frac{\mathrm{Pr}\{G(\mathit{X},\mathit{Z}(t),t)>0\cap G(\mathit{X},\mathit{Z}(t+\Delta t),t+\Delta t)\le 0\}}{\Delta t}$$

(3)

Sequentially, the cumulative probability of failure can be approximated by:

$$P_{\mathrm{f},\mathrm{c}}\left(t_0,t_e\right)\approx 1-\left(1-P_{\mathrm{f},\mathrm{i}}\left(t_0\right)\right)\exp -{{{\displaystyle \int }}^{}}_{t_0}^{t_e}\lambda (t)dt$$

(4)

where P_f,i(t₀) is the instantaneous probability of failure at the initial time instant t₀. Out-crossing rate methods are developed under the assumption that out-crossing events are mutually independent. They may provide inaccurate results, especially when out-crossing events are strongly dependent.

Extreme-value-based methods are based on the fact that a failure event occurs when the minimum of G(X,Z(t),t) over [t₀, t_e] is negative. The minimum of G(X,Z(t),t) relies on the realization of X and the trajectory of Z(t), so expansion techniques are essential to generate random trajectories of Z(t). Currently, the expansion optimal linear estimation method (EOLE) and Karhunen–Loève (KL) expansion are widely employed in the literature. In spite of their difference, both methods represent a stochastic process with a set of uncorrelated random variables. This section does not discuss which method is more suitable for discretizing a stochastic process and just assumes that the stochastic process Z^(m)(t) is modeled by the random vector ${\xi }^{(m)}=\left[{\xi }_1^{(m)},{\xi }_2^{(m)},\dots ,{\xi }_{p_m}^{(m)}\right]$ (m = 1, …, M₂). The notation W is introduced to collect all the uncorrelated variables ${\xi }_1^{(m)}$ (I = 1, …, p_m; m = 1, …, M₂). Consequentially, the trajectory of Z(t) fully relies on the realization of random vector W. From the perspective of extreme-value-based methods, the cumulative probability of failure can be expressed as a formulation of a multi-dimensional integral:

$$P_{\mathrm{f},c}\left(t_0,t_e\right)\approx {\displaystyle \int I(\mathit{x},\mathit{w})f_X(\mathit{x})f_W(\mathit{w})\mathrm{d}\mathit{x}\mathrm{d}\mathit{w}}$$

(5)

where f_X(x) and f_W(w) are the joint PDFs of X and W, respectively. I(x,w) is the cumulative failure indicator defined as follows:

$$I(\mathit{x},\mathit{w})=\{\begin{array}{cc}1& \mathrm{if}\underset{t_0\le t\le t_{\epsilon }}{\min }\{G(\mathit{x},\mathit{z}(t),t)\}\le 0\\ 0& \mathrm{otherwise}\end{array}$$

(6)

where z(t) (t₀ ≤ t ≤ t_e) is the trajectory of Z(t) determined by w.

2.2. Crude Monte Carlo Simulation

Crude MCS is obviously the most robust method to estimate the cumulative probability of failure defined by Equation (3). It estimates $P_{\mathrm{f},c}\left(t_0,t_e\right)$ using the expression below:

$${\widehat{P}}_{\mathrm{f},\mathrm{c}}^{MC}\left(t_0,t_e\right)\approx \frac{1}{N_{MC}}\underset{n=1}{\overset{N_{MC}}{{{\displaystyle \sum }}^{}}}I\left({\mathit{x}}_{MC,n},{\mathit{w}}_{MC,n}\right)$$

(7)

where ${\mathit{x}}_{\mathrm{MC},n}$ and ${\mathit{w}}_{\mathrm{MC},n}$ (n = 1, …, N_MC), i.i.d. realizations of X and W, are generated according to f_X(x) and f_W(w), respectively.

The coefficient of variation of ${\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{MC}}\left(t_0,t_e\right)$ is:

$${\delta }_{P\mathrm{f}}^{\mathrm{MC}}=\frac{\sqrt{{\mathrm{var}}_{P\mathrm{f}}^{\mathrm{MC}}}}{{\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{MC}}\left(t_0,t_e\right)}$$

(8)

where,

$${\mathrm{var}}_{P\mathrm{f}}^{\mathrm{MC}}=\frac{{\widehat{P}}_{\mathrm{f},c}^{\mathrm{MC}}\left(t_0,t_e\right)\left(1-{\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{MC}}\left(t_0,t_e\right)\right)}{N_{\mathrm{MC}}}$$

(9)

3. Importance Sampling for Time-Variant Reliability Analysis

For any x and w, there is a trajectory of G(X,Z(t),t) associated with them, and hundreds or even thousands of evaluations of the time-variant performance function are needed to determine the value of I(x,w). Moreover, crude MCS requires enormous trajectories of G(X,Z(t),t) to achieve a desired estimate of the cumulative failure probability. For example, to achieve a coefficient of variation close to 3%, about 10^p+3 random trajectories are required to access a probability of 10^−p. Therefore, regardless of whether crude MCS is used alone or is combined with a surrogate model, its computational cost is significantly high when $P_{\mathrm{f},\mathrm{c}}\left(t_0,t_e\right)$ is small. This section aims to propose a framework in order to reduce the size of random population.

3.1. The Importance Sampling Framework

IS methods to improve the efficiency of time-invariant reliability analysis have been extensively studied and adopted in the literature [37,38], e.g., those based on the most probable point (MPP) of failure and kernel-density-based IS methods. Their basic idea is to increase the probability that failure samples are generated by introducing an auxiliary density.

IS methods for time-variant reliability analysis are barely studied in the literature because stochastic processes are commonly involved in time-variant problems and IS is difficult to conduct on stochastic processes. According to Section 2.2, crude MCS for time-variant reliability analysis can be performed as follows: (1) generate random samples ${\mathit{x}}_{\mathrm{MC},n}$ (n = 1, …, N_MC) according to the joint PDF f_X(x); (2) generate ${\mathit{w}}_{\mathrm{MC},n}$ (n = 1, …, N_MC) according to f_W(w); (3) determine values of $I\left({\mathit{x}}_{\mathrm{MC},n},{\mathit{w}}_{\mathrm{MC},n}\right)$ (n = 1, …, N_MC) based on the combination $\left({\mathit{x}}_{\mathrm{MC},n},{\mathit{w}}_{\mathrm{MC},n}\right)$; (4) compute ${\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{MC}}\left(t_0,t_e\right)$. In other words, the random realizations of X and W are separately generated in the procedure of crude MCS. Therefore, this paper conducts IS on the random vector X by introducing the basic idea of IS methods for time-invariant reliability analysis into time-variant reliability analysis. Equation (3) can be rewritten as follows:

$$\begin{array}{c}{P}_{\mathrm{f},\mathrm{c}}\left(t_0,t_e\right)\approx {\displaystyle \int I(\mathit{x},\mathit{w})f_X(\mathit{x})f_W(\mathit{w})\mathrm{d}\mathit{x}\mathrm{d}\mathit{w}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={{\displaystyle \int }}^{}\left(I(\mathit{x},\mathit{w})\frac{f_{\mathit{X}}(\mathit{x})}{h_{\mathit{X}}(\mathit{x})}\right)h_{\mathit{X}}(\mathit{x})f_{\mathit{W}}(\mathit{w})\mathrm{d}\mathit{x}\mathrm{d}\mathit{w}\end{array}$$

(10)

where h_X(x) is the auxiliary PDF this paper introduces. Sequentially, $P_{\mathrm{f},\mathrm{c}}\left(t_0,t_e\right)$ can be estimated as:

$${\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{IS}}\left(t_0,t_e\right)\approx \frac{1}{N_{\mathrm{IS}}}\underset{n=1}{\overset{N_{\mathrm{IS}}}{{{\displaystyle \sum }}^{}}}\left(I\left({\mathit{x}}_{\mathrm{IS},n},{\mathit{w}}_{\mathrm{IS},n}\right)\frac{f_{\mathit{X}}\left({\mathit{x}}_{\mathrm{IS},n}\right)}{h_{\mathit{X}}\left({\mathit{x}}_{\mathrm{IS},n}\right)}\right)$$

(11)

where ${\mathit{x}}_{\mathrm{IS},n}$ and ${\mathit{w}}_{\mathrm{IS},n}$ (n = 1, …, N_IS) are generated according to h_X(x) and f_W(w), respectively. The coefficient of variation of ${\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{IS}}\left(t_0,t_e\right)$ is:

$${\delta }_{P\mathrm{f}}^{\mathrm{IS}}=\frac{\sqrt{{\mathrm{var}}_{P\mathrm{f}}^{\mathrm{IS}}}}{{\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{IS}}\left(t_0,t_e\right)}$$

(12)

where

$${\mathrm{var}}_{P\mathrm{f}}^{\mathrm{IS}}=\frac{1}{N_{\mathrm{IS}}}[\frac{1}{N_{\mathrm{IS}}}\underset{n=1}{\overset{N_{\mathrm{IS}}}{{{\displaystyle \sum }}^{}}}{\left(I_K\left({\mathit{x}}_{\mathrm{IS},n},{\mathit{w}}_{\mathrm{IS},n}\right)\cdot \frac{f_X\left({\mathit{x}}_{\mathrm{IS},n}\right)}{h_X\left({\mathit{x}}_{\mathrm{IS},n}\right)}\right)}^2-{\left({\widehat{P}}_{\mathrm{f},\mathrm{c}}^{\mathrm{Is}}\left(t_0,t_e\right)\right)}^2]$$

(13)

3.2. Auxiliary PDF

The auxiliary PDF h_X(x) plays an important role in the IS framework proposed for time-variant reliability analysis. This section proposes two strategies to construct h_X(x). However, it is worth noting that methods of constructing h_X(x) are not limited to the proposed two.

3.2.1. Single MPP-Based Auxiliary PDF

This section assumes that both X and Z(t_e) are subject to the standard Gaussian distribution. One can use isoprobabilistic transformation to make X and Z(t_e) satisfy the assumption. t_e is the end instant of the time interval of interest. It is chosen here because the instantaneous probability of failure at t_e is commonly closer to $P_{\mathrm{f},\mathrm{c}}\left(t_0,t_e\right)$ than any other instantaneous probability of failure. Shifting the sampling center from the origin to the MPP is widely employed in IS methods for time-invariant reliability analysis. Hence, this section shifts the sampling center of X and defines h_X(x) as:

$$h_X(\mathit{x})=f_X\left(\mathit{x}-{\mathit{\mu }}_{X,t_e}^{\ast }\right)$$

(14)

where ${\mathit{\mu }}_{X,t_e}^{\ast }$ is determined by the optimization problem Equation (13) with t = t_e.

$$\{\begin{array}{l}\left[{\mu }_{\mathit{X},t}^{\ast },{\mu }_{\mathit{Z}(t)}^{\ast }\right]=\arg {\min }_{[\mathit{X},\mathit{Z}(t)]}\left(\parallel \mathit{x}{\parallel }^2+\parallel \mathit{z}(t){\parallel }^2\right)\hfill \\ \mathrm{s}.\mathrm{t}. G(\mathit{x},\mathit{z}(t),t)\le 0\hfill \end{array}$$

(15)

It is obvious that $\left[{\mu }_{\mathit{X},t}^{\ast },{\mu }_{\mathit{Z}(t)}^{\ast }\right]$ is the MPP associated with the instantaneous performance function G(X,Z(t_e),t_e) and ${\mu }_{\mathit{X},t_e}^{\ast }$ is the component of the MPP corresponding to X. This section sets the standard deviation of h_X(x) equal to 1, but one can also choose a different value to tighten or spread out the samples ${\mathit{x}}_{\mathrm{IS},n}$ (n = 1, …, N_IS).

3.2.2. Multiple MPP-Based Auxiliary PDF

It is easily noted from Equation (4) that the structure whose performance function is G(X,Z(t),t) may fail at any time instant $t_i\in [t_0,t_e]$. The auxiliary PDF proposed above apparently pays more attention to the end time instant t_e than any other time instant. For any $t_i\in [t_0,t_e]$, the MPP $\left[{\mu }_{\mathit{X},t}^{\ast },{\mu }_{\mathit{Z}(t)}^{\ast }\right]$ associated with the instantaneous performance function G(X,Z(t_i),t_i) can be obtained according to Equation (13). ${\mu }_{\mathit{X},t_i}^{\ast }$ apparently varies with t_i. The auxiliary PDF in Section 3.2.1 is acceptable when the variation of ${\mu }_{\mathit{X},t_i}^{\ast }$ from t₀ to t_e is tiny. However, a part of the domain of importance may have very small value of h(·) determined by Equation (12) when the variation is significant. To overcome the deficiency, this section develops an auxiliary PDF based on multiple MPPs.

We first discrete the interval [t₀, t_e] into N_s + 1 uniform time instants t_n (n = 0, …, N_s). The auxiliary PDF based on multiple MPPs is then expressed as:

$$h_{\mathit{X}}(\mathit{x})=\frac{1}{N_s+1}\underset{n=0}{\overset{N_s}{{{\displaystyle \sum }}^{}}}{f}_X\left(\mathit{x}-{\mu }_{\mathit{X},t_n}^{\ast }\right)$$

(16)

where ${\mu }_{\mathit{X},t_n}^{\ast }$ (n = 0, …, N_s) are obtained by solving Equation (13) with t = t_n.

4. Validation of the Proposed Method

This section thoroughly investigates the performance of the IS proposed in Section 3. Three examples of varying complexities with respect to the dimension and the extent of nonlinearity are employed to conduct the investigation.

4.1. A Numerical Example

This example is modified from a numerical example involving two time-invariant variables and a stochastic process. The original version comes from References [34,39]. The performance function employed herein is:

$$\begin{array}{c}G(\mathit{X},Y(t),t)={\left(0.25X_1+3.5\right)}^2\left(0.25X_2+3.5\right)\\ -5\left(0.25X_1+3.5\right)(1+Y(t))t+\left(0.25X_2+4.5\right)t^2-20\end{array}$$

(17)

where X₁ and X₂ are mutually independent and subject to the standard normal distribution. Y(t) is a Gaussian process with zero mean and unit variance. The autocorrelation coefficient function is:

$$r\left(t_1,t_2\right)=\exp \left({\left(t_1-t_2\right)}^2\right)$$

(18)

The time interval of interest in the example is [0, 0.5]. EOLE is retained in this paper to discretize the stochastic process. The time interval [0, 0.5] is uniformly discretized into 100 instants. Crude MCS with 10⁹ simulation is regarded as the benchmark. The two auxiliary PDFs in Section 3.2 are respectively used to conduct the proposed IS (N_s = 4, Equation (14)). Some of the results are summarized in

Table 1. Results of example 1.

Method		${\widehat{\mathit{P}}}_{\mathbf{f},\mathbf{c}}$$\left({\mathit{\delta }}_{\mathit{P}\mathbf{f}}\right)$
		t = 0	[0, 0.1]	[0, 0.2]	[0, 0.3]	[0, 0.4]	[0, 0.5]
Benchmark	NMC = 109	1.77 × 10−5	1.1 × 10−4	1.21 × 10−3	8.46 × 10−3	3.09 × 10−2	7.02 × 10−2
MCS	NMC = 5 × 104	2 × 10−5 (100%)	10−4 (44.7%)	1.5 × 10−3 (11.5%)	8.24 × 10−3 (4.91%)	3.16 × 10−2 (2.47%)	7.15 × 10−2 (1.61%)
	NMC = 105	10−5 (100%)	8 × 10−5 (35.4%)	1.28 × 10−3 (8.83%)	8.31 × 10−3 (3.45%)	3.14 × 10−2 (1.76%)	7.04 × 10−2 (1.15%)
Single-MPP IS	NMC = 104	1.62 × 10−5 (58%)	1.37 × 10−4 (23.6%)	9.08 × 10−4 (13.3%)	7.17 × 10−3 (6.99%)	2.97 × 10−2 (4.27%)	6.69 × 10−2 (3.26%)
	NMC = 5 × 104	1.53 × 10−5 (25.7%)	1.14 × 10−4 (11.7%)	1.18 × 10−3 (5.34%)	8.26 × 10−3 (2.81%)	3.18 × 10−2 (1.82%)	7.02 × 10−2 (1.42%)
Multiple-MPP IS	NMC = 104	1.75 × 10−5 (3.91%)	1.04 × 10−4 (5.52%)	1.07 × 10−3 (7.28%)	7.01 × 10−3 (7.45%)	3.27 × 10−2 (5.95%)	7.33 × 10−2 (5.08%)
	NMC = 5 × 104	1.77 × 10−5 (1.74%)	1.09 × 10−4 (2.52%)	1.21 × 10−3 (3.48%)	8.28 × 10−3 (3.29%)	3.24 × 10−2 (2.62%)	7.12 × 10−2 (2.27%)

. In the table, single-MPP IS and multiple-MPP IS represent the proposed IS implemented with the single MPP-based auxiliary PDF (Equation (12)) and multiple MPP-based auxiliary PDF (Equation (14)), respectively.

Figure 1 shows lines of $P_{\mathrm{f},\mathrm{c}}(0,t)$ and the corresponding coefficient of variation ($t\in [0,0.5]$) obtained using crude MCS and the proposed IS. In each row of Figure 1, the size of the random population associated with the methods are the same in order to visualize the advantage of the proposed IS as well as the difference between single-MPP IS and multiple-MPP IS. According to Figure 1 and Table 1, the efficiency of crude MCS and the proposed IS are close in terms of estimating $P_{\mathrm{f},\mathrm{c}}(0,0.5)$. However, the proposed IS outperforms with respect to estimating the evolution of $P_{\mathrm{f},\mathrm{c}}(0,t)$ over the time interval of interest. Multiple-MPP IS is capable of efficiently providing an accurate evolution of $P_{\mathrm{f},\mathrm{c}}(0,t)$ ($t\in [0,0.5]$).

To have some insight into the performance of the methods, Figure 2 shows random points of X generated using crude MCS and the proposed IS and sampling centers of the methods. By comparing Figure 2a,b, it can be concluded that the single MPP-based auxiliary PDF generates a little more failure samples than crude MCS, so the single-MPP IS is just a little more efficient than crude MCS with respect to estimating $P_{\mathrm{f},\mathrm{c}}(0,0.5)$. Moreover, the single-MPP IS explores a part of domain of importance for $P_{\mathrm{f},\mathrm{c}}(0,t)$ $t\in [0,0.5]$, revealing the reason why it provides more accurate curve of $P_{\mathrm{f},\mathrm{c}}(0,t)$ than crude MCS. Figure 2c indicates that multiple MPP-based auxiliary PDF generates the most failure samples, and it explores most of domain of significance for the evolution of $P_{\mathrm{f},\mathrm{c}}(0,t)$. However, many of them are far from the origin and contribute little to $P_{\mathrm{f},\mathrm{c}}(0,0.5)$. The above explains why multiple-MPP IS provides the most accurate curve of ${\widehat{P}}_{\mathrm{f},\mathrm{c}}(0,t)$ but shows no advantage in estimating $P_{\mathrm{f},\mathrm{c}}(0,0.5)$.

4.2. A Corroded Beam

The second example deals with a steel bending beam whose cross section is linearly corroded [6,40]. Two loads are applied to the beam, i.e., dead load and a time-variant load. The dead load is equal to $p={\rho }_{st}{b}_0{h}_0(\mathrm{N}/\mathrm{m})$ where ${\rho }_{st}=78.5\mathrm{N}/{\mathrm{m}}^3$ is the force density of steel, and the pinpoint load F(t) applied at the mid span (Figure 3) is a Gaussian process. The performance function of the corroded beam is:

$$G(\mathit{X},F(t),t)=M_u(t)-M(t)$$

(19)

where,

X = [f_y,b₀,h₀]^T

$$M_u(t)=\frac{b(t)h^2(t)}{4}{f}_y$$

$$M(t)=\frac{F(t)L}{4}+\frac{{\rho }_{st}{b}_0{h}_0{L}^2}{8}$$

b(t) = b₀-2ωt

h(t) = h₀-2ωt

ω = 0.03 mm/year

Table 2. Distribution information of input variables (example 2).

Variable	Distribution	Mean	Standard Deviation	Autocorrelation Coefficient Function
fy	Lognormal	240 MPa	24 MPa	-
b0	Lognormal	0.2 m	0.01 m	-
h0	Lognormal	0.04 m	0.004 m	-
F(t)	Gaussian	3500 N	700 N	$\exp (-{(t_2-t_1)}^2)$

lists the distribution information of random variables and the stochastic process F(t).

The time interval of interest is [0, 20 year]. EOLE is still used to discretize the stochastic process F(t). To obtain enough accurate results, the interval [0, 20] is discretized into 1500 time instants. Crude MCS with 10⁹ random simulations is regarded as the benchmark. Crude MCS and the proposed auxiliary PDFs are compared by Figure 4 and Figure 5 and

Table 3. Results of example 2.

Method		${\widehat{\mathit{P}}}_{\mathbf{f},\mathbf{c}}$$\left({\mathit{\delta }}_{\mathit{P}\mathbf{f}}\right)$
		t = 0	[0, 5]	[0, 10]	[0, 15]	[0, 20]
Benchmark	NMC = 109	2.25 × 10−6	2.12 × 10−5	4.58 × 10−5	7.9 × 10−5	1.24 × 10−4
MCS	NMC = 106	7 × 10−6 (37.8%)	3.3 × 10−5 (17.4%)	6.3 × 10−3 12.6%)	1.0 × 10−4 (10%)	1.47 × 10−4 (8.25%)
	NMC = 5 × 106	2.6 × 10−6 (27.7%)	2.5 × 10−5 (8.94%)	5.66 × 10−5 (5.94%)	8.98 × 10−5 (4.72%)	1.31 × 10−4 (3.9%)
Single-MPP IS	NMC = 104	1.76 × 10−6 (15.4%)	1.93 × 10−5 (5.77%)	4.38 × 10−5 (5.47%)	7.35 × 10−5 (4.25%)	1.13 × 10−4 (3.72%)
	NMC = 5 × 104	2.31 × 10−6 (6.28%)	2.2 × 10−5 (2.91%)	4.66 × 10−5 (2.37%)	7.78 × 10−5 (1.97%)	1.2 × 10−4 (1.87%)
Multiple-MPP IS	NMC = 104	1.82 × 10−6 (12.4%)	2.16 × 10−5 (7.58%)	4.52 × 10−5 (5.66%)	7.84 × 10−5 (5.6%)	1.17 × 10−4 (4.42%)
	NMC = 5 × 104	2.35 × 10−6 (6.01%)	2.19 × 10−5 (3.07%)	4.7 × 10−5 (2.71%)	7.96 × 10−5 (2.32%)	1.23 × 10−4 (2.09%)

. According to the results, it is hard to identify which auxiliary PDF proposed in Section 3.2 is better, because the variation of ${\mu }_{\mathit{X},t}^{\ast }$ over the time interval [0, 20] is insignificant (

Table 4. Sampling centers of single-MPP IS and multiple-MPP IS (example 2). The sampling center of single-MPP IS is [−1.67, −0.69, −3.19] (Equation (12)), and all points listed in the table are adopted to construct the auxiliary PDF of multiple-MPP IS (Equation (14)).

Time Instant (t)	${\mathit{\mu }}_{\mathit{X},\mathit{t}}^{\ast }$
0	[−1.88, −0.776, −3.43]
5	[−1.83, −0.754, −3.37]
10	[−1.78, −0.733, −3.31]
15	[−1.72, −0.711, −3.25]
20	[−1.67, −0.69, −3.19]

). However, using the same size of random population, both of the proposed auxiliary PDFs are capable of providing more accurate results, e.g., $P_{\mathrm{f},\mathrm{c}}(0,20)$, the evolution of $P_{\mathrm{f},\mathrm{c}}(0,t)$ $t\in [0,20]$ (Figure 4) and instantaneous failure probabilities (Figure 5).

4.3. A Cantilever Tube Structure

A cantilever tube structure, modified from a previously published example [18,23], is studied in this section. It is adopted herein to investigate the performance of the proposed IS when involving multiple stochastic processes. As shown by Figure 6, four time-variant loads, i.e., F₁(t), F₂(t), P(t) and T(t), are applied to the cantilever tube. The yield stress is considered to linearly decrease over time resulting from the material degradation. The decreasing model is:

$${\sigma }_u\left(t\right)={\sigma }_0\left(1-0.01t\right)$$

(20)

where σ₀ is the initial yield stress.

The time interval of interest in this example is [0, 5 year]. The performance function associated with the cantilever tube is:

$$G(\mathit{X},\mathit{Z}(t),t)={\sigma }_u(t)-{\sigma }_{\max }(t)$$

(21)

where X = [d, h, σ₀]^T and Z(t) = [F₁(t), F₂(t), T(t), P(t)]^T. σ_max(t) is the maximum Von Mises stress having the following expression:

$${\sigma }_{\max }(t)=\sqrt{{\sigma }_x^2(t)+3{\tau }_{zx}^2(t)}$$

where,

$${\sigma }_x(t)=\frac{F_1(t)\sin \left({\theta }_1\right)+F_2(t)\sin \left({\theta }_2\right)+P(t)}{A}+\frac{M(t)d}{2I}$$

$${\tau }_{zx}(t)=\frac{T(t)d}{4I}$$

$$M(t)=F_1(t)\cos \left({\theta }_1\right)L_1+F_2(t)\cos \left({\theta }_2\right)L_2$$

$$A=\frac{\pi }{4}\left[d^2-{(d-2h)}^2\right]$$

$$I=\frac{\pi }{64}\left[d^4-{(d-2h)}^4\right]$$

Distribution parameters of X and Z(t) are listed in

Table 5. Distribution information of input variables (example 3).

Variable	Distribution	Mean	Standard Deviation	Autocorrelation Coefficient Function
F1(t) (N)	Gaussian process	1800	180	$\exp (-\left|t_2-t_1\right|/4)$
F2(t) (N)	Gaussian process	1800	180	$\exp (-\left|t_2-t_1\right|)$
T(t) (Nm)	Gaussian process	1900	190	$\exp (-4{\left|t_2-t_1\right|}^2)$
P(t) (N)	Gaussian process	1800	180	$\exp (-4\left|t_2-t_1\right|)$
d (mm)	Normal	42	0.5	-
h (mm)	Normal	5	0.1	-
σ0 (MPa)	Normal	560	56	-
L1 (mm)	Deterministic	60	-	-
L2 (mm)	Deterministic	120	-	-
θ1 (°)	Deterministic	10	-	-
θ2 (°)	Deterministic	5	-	-

. The time interval [0, 5 year] is uniformly discretized into 1000 time instants. Crude MCS with 10⁹ random simulations is still regarded as the benchmark. Figure 7 and Figure 8 and

Table 6. Results of example 3.

Method		${\widehat{\mathit{P}}}_{\mathbf{f},\mathbf{c}}$$\left({\mathit{\delta }}_{\mathit{P}\mathbf{f}}\right)$
		t = 0	[0, 1]	[0, 2]	[0, 3]	[0, 4]	[0, 5]
Benchmark	NMC = 109	6.83 × 10−4	2.25 × 10−3	3.92 × 10−3	5.73 × 10−3	7.76 × 10−3	9.94 × 10−3
MCS	NMC = 104	1.5 × 10−3 (25.8%)	3.7 × 10−3 (16.4%)	5.5 × 10−3 (13.4%)	7.2 × 10−3 (11.7%)	9.9 × 10−3 (10%)	1.13 × 10−2 (9.35%)
	NMC = 105	8.2 × 10−4 (11%)	2.43 × 10−3 (6.41%)	4.17 × 10−3 (4.89%)	6.02 × 10−3 (4.06%)	7.92 × 10−3 (3.54%)	1.03 × 10−2 (3.11%)
Single-MPP IS	NMC = 103	6.5 × 10−4 (15.3%)	2.06 × 10−3 (13.6%)	4.32 × 10−3 (10.6%)	6.11 × 10−3 (10.9%)	8.49 × 10−3 (10.4%)	1.08 × 10−2 (9.9%)
	NMC = 104	6.71 × 10−4 (5.46%)	2.16 × 10−3 (4.23%)	2.72 × 10−3 (3.67%)	5.42 × 10−3 (3.31%)	7.55 × 10−3 (2.99%)	9.82 × 10−3 (3.03%)
Multiple-MPP IS	NMC = 103	6.7 × 10−4 (16%)	2.35 × 10−3 (12.7%)	4.05 × 10−3 (9.69%)	5.6 × 10−3 (11.5%)	8.37 × 10−3 (10.3%)	1.08 × 10−2 (9.54%)
	NMC = 104	6.81 × 10−4 (5.5%)	2.16 × 10−3 (4.08%)	3.78 × 10−3 (4.18%)	5.42 × 10−3 (3.62%)	7.51 × 10−3 (3.1%)	9.85 × 10−3 (3.03%)

compare the proposed IS against crude MCS, and

Table 7. Sampling centers of single-MPP IS and multiple-MPP IS (example 3).

Time Instant (t)	${\mathit{\mu }}_{\mathit{X},\mathit{t}}^{\ast }$
0	[−0.554, −0.25, −2.67]
1.25	[−0.544, −0.245, −2.59]
2.5	[−0.534, −0.24, −2.52]
3.75	[−0.523, −0.236, −2.44]
5	[−0.511, −0.23, −2.37]

lists the sampling centers of auxiliary PDFs of the proposed IS. It can be found that ${\mu }_{\mathit{X},t}^{\ast }$ does not change much over the time interval [0, 5], and the performance of the auxiliary PDFs proposed in Section 3.2 seems evenly matched. Results indicate that the proposed IS requires significantly fewer simulations than crude MCS while maintaining the accuracy level of the estimate of P_f,c(0,5). Using the same size of random population, the proposed method achieves much more accurate results. Both auxiliary PDFs proposed in Section 3.2 have remarkable advantages over crude MCS.

5. Conclusions

This paper develops a framework of importance sampling for time-variant reliability analysis. Considering that time-invariant random variables are commonly involved in time-variant reliability analysis, the basic idea of the proposed framework is conducting IS on time-invariant variables to generate more failure trajectories of time-variant performance function. Two auxiliary PDFs based on MPP(s) are proposed in Section 3.2. Three examples are utilized to validate the efficiency of the proposed IS. The results show that the proposed IS remarkably reduces the size of the random population and saves computational cost when the cumulative failure probability over the time interval of interest is small (example 2 and 3). When the cumulative failure probability over the time interval of interest is large (example 1), the auxiliary PDFs proposed in Section 3.2 may show little advantage in terms of estimating P_f,c(t₀,t_e). However, the proposed IS is capable of efficiently providing accurate curves of cumulative failure probability P_f,c(t₀,t) and instantaneous failure probability P_f,i(t₀,t) ($t\in [t_0,t_e]$).

It is worth noting that auxiliary PDFs available for the proposed IS framework are not limited to the proposed two. More robust auxiliary PDF is in demand to address complex time-variant performance function. It is obvious that the proposed IS framework cannot deal with the time-variant problem involving no time-invariant random variables. An IS framework conducting both time-invariant random variables and stochastic processes may be able to address the limitation of the proposed framework and further increase the computational efficiency. In addition, when combining with surrogate models, e.g., adaptive Kriging model, the proposed IS framework can be used to perform time-variant reliability analysis involving time-consuming performance function. These ideas may be discussed in future work.

6. Replication of Results

The details of the proposed IS are provided in Section 3 and Section 4. The proposed IS is easy to use and authors are confident that all results of the proposed method can be replicated by readers with MATLAB. Readers are encouraged to contact the corresponding author by email if they have any questions about the proposed IS framework.