Extending the global reliability sensitivity analysis to the systems under double-stochastic uncertainty

Abstract

Systems with random variables and random excitations exist widely in various engineering problems. Extending the traditional global reliability sensitivity to this double-stochastic system has important guiding significance for its design optimization. However, because there is a certain coupling between the randomness of variables and the randomness of excitation, this coupling mechanism is difficult to determine in practical projects. Therefore, it is difficult to extend the traditional reliability sensitivity analysis method to this double-stochastic system. In this research, it is assumed that there is no correlation between variables and excitations. Then, combining the first-passage method–based dynamic strength formula and the variance-based sensitivity analysis method, an approximate global reliability sensitivity analysis method for this double-stochastic system is proposed. In order to improve the computational efficiency, a nested loop method based on seven-point estimation is proposed for reliability sensitivity analysis. In order to verify the accuracy and efficiency of the proposed method, a Monte Carlo simulation is given as a reference. Three examples are studied and discussed to illustrate the practicality and feasibility of the proposed method.

Keywords

double-stochastic global sensitivity index reliability seven-point estimation variance

Introduction

In practical engineering problems, systems are inevitably affected by stochastic excitation such as earthquakes, wind loads, and waves. The reliability analysis of these systems involving stochastic excitations is called dynamic reliability analysis (Chen and Li, 2005; Li and Chen, 2003; Zhang et al., 2008). The first-passage method–based dynamic strength formula (Asmussen, 1998; Gradišek et al., 2000; Wang et al., 2017) is one of the most widely used methods in dynamic reliability analysis. In general, if an engineer wants to improve the performance of these dynamic systems, dynamic optimization design (Du and Bode, 2004; Rufu et al., 2002) will be considered. However, if the system has too many variables, it is often difficult to conduct its reliability-based optimization (Wang et al., 2017). For this issue, it is generally necessary to first screen out those variables that have a significant effect on reliability, and second, ignore the remaining variables that have little or no effect on reliability to reduce the size of the optimization. Sensitivity analysis (Borgonovo, 2007; Kucherenko, 2009; Kucherenko et al., 2012; Melchers and Ahammed, 2004; Wu and Mohanty, 2006) is one of the effective ways to identify these important variables.

Reliability sensitivity can be classified into local reliability sensitivity (LRS) (Lu et al., 2009; Song et al., 2009) and global reliability sensitivity (GRS) (Liu and Homma, 2010; Wei et al., 2015; Zhou et al., 2013). The LRS index is usually defined as the partial derivative of the failure probability or reliability to the distribution parameter of input variable, which quantifies the effect of the distribution parameter of input variable on the failure probability or reliability at its nominal value. Relative to the LRS index, the GRS index reflects the contribution of the uncertainty of input variable over its entire distribution to failure probability or reliability. Dutta and Ramakrishnan (1998) improved the accuracy of design sensitivity of the structure under transient dynamic load by systematically achieving an adaptive mesh and a static correction method. Benfratello et al. (2000) obtained the sensitivity of order higher than two by using Kronecker algebra extensively. Chaudhuri and Chakraborty (2004) applied sensitivity analysis to seismic reliability analysis of structures. All of the above sensitivity analyses are partial derivatives of the dynamic response to the distribution parameters of input variables, that is, they belong to the local sensitivity analysis. Simultaneously, these indices do not involve reliability. Cao et al. (2013) extended the variance-based sensitivity index to the stochastic process and derived the sensitivity index of the structure under the Gaussian process, which mainly studied contribution of the uncertainty of variable to the output response rather than its reliability. Valdebenito et al. (2012) considered a local approximation of performance function and then estimated the sensitivity index by performing a finite difference. In this literature, the reliability sensitivity estimation of the linear system under stochastic excitation is classified as the LRS analysis because the author does not consider the uncertainty of the input variable over the entire distribution.

It is also worth noting that there is a lot of research on GRS using surrogate models. Palar et al. (2018) studied the global sensitivity analysis using multi-fidelity polynomial chaos expansion. Rohmer and Foerster (2011) studied the global sensitivity of large-scale numerical landslide models using Gaussian-process meta-modeling. Other literatures on surrogate model–based global sensitivity analysis can refer to Ciuffo et al. (2013), Crestaux et al. (2009), and Sudret (2008). Although this surrogate model–based method is widely applied to various engineering problems, their application objects are almost the general engineering problems that do not involve random excitation, rather than the engineering problems involving the double-stochastic uncertainties studied in our research.

As far as we know, all studies involving sensitivity analysis with double-stochastic uncertainties are local sensitivity analysis (Allen et al., 2001; Chen et al., 2009; Jensen et al., 2015) instead of global sensitivity analysis. In view of this, the article attempts to extend GRS analysis to engineering system with double-stochastic uncertainties. First, the first-passage method–based dynamic strength formula is employed to calculate the reliability of these structures. Then, the obtained reliability is assumed to be a “response” of the double-stochastic system. Finally, a variance-based GRS analysis method for the double-stochastic system is proposed under the assumption that there is no coupling between the randomness of the variables and the randomness of excitation. To improve the efficiency of the proposed method, a seven-point (SP) estimation-based nested loop method (Zhao and Ono, 2000) is also proposed in this research.

The outline of this article is organized as follows. Section “The responses of determinate structure under a single stationary stochastic excitation” reviews the structure response under a single stationary stochastic excitation. The first-passage-based strength dynamic formula is used to calculate the reliability of the dynamic system in section “The dynamic reliability based on the first-passage method.” In section “Calculate SP-based GRS index of the double-stochastic structure,” a GRS index considering double-stochastic uncertainties is proposed based on the variance decomposition. Simultaneously, based on the brief introduction of SP estimation method, the GRS index of double-stochastic dynamic systems is proposed. In section “Examples and discussions,” three examples are presented to illustrate the proposed method. Some conclusions are highlighted in section “Conclusion.”

The responses of determinate structure under a single stationary stochastic excitation

In this section, based on the frequency domain method, the stochastic vibration response analysis is briefly introduced from the following two aspects: single-degree-of-freedom and multi-degree-of-freedom.

Single-degree-of-freedom

Consider a single degree-of-freedom oscillator that is subject to a single stationary stochastic excitation. This dynamic system can be described by the following stochastic differential equation

m \overset{\cdot\cdot}{y} + c \overset{\cdot}{y} + k y = f (t)

(1)

where m, c, and k denote the mass, damping, and stiffness of dynamic system, respectively. $f (t)$ is a stationary stochastic processes with a constant power spectral density (PSD) value and assumes $S_{ff} (ω) = S_{0} = 1$ . $ζ = c / (2 \sqrt{mk})$ represents the damping ratio of the system.

For a single degree-of-freedom oscillator system, the PSD of the response y can be represented by the following analytical expression

S_{yy} (ω) = {| H (ω) |}^{2} S_{ff} (ω) = \frac{S_{0}}{{(k - m ω^{2})}^{2} + c^{2} ω^{2}}

(2)

Then, the standard deviation of displacement response $σ_{y}$ and the standard deviation of speed response $σ_{\overset{\cdot}{y}}$ can be derived and expressed as follows, respectively

σ_{y}^{2} = λ_{0} = \int_{- \infty}^{\infty} S_{yy} (ω) d ω = \frac{π S_{0}}{2 ζ k \sqrt{mk}} = \frac{π S_{0}}{ck}

(3)

σ_{\overset{\cdot}{y}}^{2} = λ_{2} = \int_{- \infty}^{\infty} ω^{2} S_{yy} (ω) d ω = \frac{π S_{0}}{2 ζ m \sqrt{mk}} = \frac{π S_{0}}{cm}

(4)

Multiple degrees-of-freedom case

Consider a multiple degrees-of-freedom system that is subjected to a single stationary stochastic excitation, the corresponding dynamic equation can be expressed as follows

M \overset{\cdot\cdot}{y} + C \overset{\cdot}{y} + K y = p x (t)

(5)

In equation (5), $M$ , $C$ , and K denote the mass matrix, damping matrix, and stiffness matrix of system, respectively. $\overset{\cdot\cdot}{y}$ , $\overset{\cdot}{y}$ , and $y$ represent the acceleration, velocity, and displacement column vector of system, respectively. $x (t)$ is a stationary stochastic processes with a known PSD, and p is given as a constant vector.

Then, according to the stochastic vibration theory, the PSD of response y can be represented as follows

S_{yy} (ω) = {| H (ω) |}^{2} S_{xx} (ω)

(6)

where $H (ω)$ is the frequency response function. Then, the spectral moments of the response y can be further obtained by the following equation (Lutes and Sarkani, 2009)

λ_{i} = \int_{- \infty}^{+ \infty} {| ω |}^{i} S_{yy} (ω) d ω = 2 \int_{0}^{+ \infty} ω^{i} S_{yy} (ω) d ω

(7)

Then, two responses $σ_{y}$ and $σ_{\overset{\cdot}{y}}$ can be calculated by the zero order moment and second order moment of y, respectively, and can be expressed as follows

σ_{y} = \sqrt{λ_{0}} = \sqrt{2 \int_{0}^{+ \infty} S_{yy} (ω) d ω}

(8)

σ_{\overset{\cdot}{y}} = \sqrt{λ_{2}} = \sqrt{2 \int_{0}^{+ \infty} ω^{2} S_{yy} (ω) d ω}

(9)

Obviously, three responses y , $\overset{\cdot}{y}$ , and $\overset{\cdot\cdot}{y}$ must be a stochastic process because the external excitation $x (t)$ is a stationary stochastic process.

As can be seen from the above analysis, the response of a stochastic vibration system is represented by variance. Based on the first-passage method proposed by Rice (1944), the following section is a brief introduction of how to use variance response to estimate the reliability of dynamic structures.

The dynamic reliability based on the first-passage method

Generally, for a given stochastic process $y (t)$ , consider that if the response y does not exceed a given threshold b over the entire time range, that is, $t \in [0, T]$ , the structure is safe, otherwise the structure is insecure. Therefore, the reliability can be expressed as follows

R^{+} (T) = Prob {y (t) \leq b, 0 \leq t \leq T}

(10)

Equation (10) is a unilateral reliability model. If the dynamic response y resides between the up threshold b and the lower threshold $- b$ over the entire time range, we consider this case to be a bilateral reliability model, and the corresponding dynamic reliability can be expressed as follows

R (T) = Prob {- b < y (t) < b, 0 \leq t \leq T}

(11)

It can be seen from the above analysis that as long as there is a crossing from security domain to the failure domain in the entire stochastic process, the failure will happen. This failure mode is the so-called first-passage failure criterion (Hu and Du, 2015).

Based on the Poisson hypothesis, the first-passage method–based unilateral and bilateral dynamic strength reliability formulae for dynamic systems subjected to a single stationary random excitation can be expressed as follows, respectively

R (T) = \exp [- \frac{T}{2 π} \frac{σ_{\overset{\cdot}{y}}}{σ_{y}} \exp (- \frac{b^{2}}{2 σ_{y}^{2}})]

(12)

R (T) = \exp [- \frac{T}{π} \frac{σ_{\overset{\cdot}{y}}}{σ_{y}} \exp (- \frac{b^{2}}{2 σ_{y}^{2}})]

(13)

It can be seen from the above analysis that the key to estimate the dynamic strength reliability is to solve $σ_{y}$ and $σ_{\overset{\cdot}{y}}$ of the dynamic systems. As can be seen from section “Multiple degrees-of-freedom case,” $σ_{y}$ and $σ_{\overset{\cdot}{y}}$ are the zero order spectral moment and second spectral moment of y, respectively. For a given single stationary stochastic excitation, the PSD of the response y can be obtained directly from the equation (6). Then, $σ_{y}$ and $σ_{\overset{\cdot}{y}}$ can be calculated by equation (7). Finally, the reliability of dynamic systems can be solved by equation (12) or equation (13).

From the above analysis, we can see that the dynamic reliability formulas (12) and (13) apply to deterministic structures that are subject to a single stationary stochastic excitation. For the uncertain structure affected by stochastic excitation, its dynamic reliability is also a random variable because of the randomness of its structure parameters. Next, based on the variance decomposition, an SP-GRS analysis method is proposed for the structures under the double-stochastic uncertainties.

Calculate SP-based GRS index of the double-stochastic structure

Variance-based GRS index with double-stochastic uncertainties

Review of the variance-based global sensitivity index

The analysis of variance (ANOVA) decomposition proposed by Sobol (2001) is one of the most commonly used methods in global sensitivity analysis. Based on the ANOVA decomposition, the variance of the performance function $Y = g (X)$ can be estimated by the sum of the individual decomposition terms, that is

V = \sum_{i = 1}^{n} V_{i} + \sum_{i \neq j} V_{ij} + \sum_{i \neq j \neq k} V_{ijk} + \dots + V_{12 \dots n}

(14)

where V is the unconditional variance of Y and can be expressed as follows

V = V (Y) = \int g^{2} (X) f_{X} (x) d x - g_{0}^{2}

(15)

The conditional variance of Y is given as follows

V_{i} = V (\int g (X) Π_{j \neq i}^{n} [f_{X_{j}} (x_{j}) d x_{j}]) = V [E (Y | X_{i})]

(16)

\begin{matrix} V_{i_{1}, i_{2}, \dots, i_{s}} = V (\int g (X) Π_{j \neq i_{1}, i_{2}, \dots, i_{s}}^{n} [f_{X_{j}} (x_{j}) d x_{j}]) \\ = V [E (Y | X_{i_{1}, i_{2}, \dots, i_{s}})] \end{matrix}

(17)

In equation (16) and equation (17), $V (\cdot)$ and $E (\cdot)$ are the variance operator and the expectation operator, respectively. $g_{0}$ is the mean value of the performance function Y. $f_{X_{j}} (X_{j})$ and $f (X)$ denote the marginal probability density function of the variable $X_{j}$ and the joint probability density function of the variables $X$ , respectively.

The variance-based global sensitivity index is defined as the ratio of partial variance to total variance and is expressed as follows

S_{ij \dots s} = \frac{V_{ij \dots s}}{V}

(18)

If only the individual contribution of one variable $X_{i}$ to the total variance V is considered, the above equation can be rewritten as follows

S_{i} = \frac{V_{i}}{V}

(19)

where $S_{i}$ is commonly termed as the main index of variable $X_{i}$ , it reflects the individual contribution of the input variable $X_{i}$ to the uncertainty of the output response Y within its “global” uncertain range.

In practical engineering problems, most variables interact with other variables. These interactions also contribute to the uncertainty of the response Y. The sum of these interactive contributions and the main contribution is the so-called total contribution, that is, total index. It can be expressed as follows

S_{Ti} = \frac{V_{Ti}}{V}

(20)

where $V_{Ti}$ is the average reduction of the variance of Y when fixing $X$ over their full distribution range except $X_{i}$ , and it can be represented as follows

V_{Ti} = V (Y) - V [E (Y | X_{- i})] = E [V (Y | X_{- i})]

(21)

Obviously, if the main index $S_{i}$ is close to its corresponding total index $S_{Ti}$ , indicating the interaction between variable $X_{i}$ and other variables has no effect on the uncertainty of its output response.

After the above discussion on the variance-based global sensitivity index, the variance-based GRS index in the scheme of double-stochastic process will be discussed and proposed in the following section.

Variance-based GRS index of the double-stochastic structure

As can be seen from the analysis in section “The dynamic reliability based on the first-passage method,” the dynamic reliability of the systems under the double-stochastic process is not a certain value but a random value, and it can be denoted as $R (t, X)$ . $X = (X_{1}, X_{2}, \dots, X_{n})$ represent the structure uncertainty parameters and are modeled by the random variables that obey a certain distribution. Due to the randomness of $R (t, X)$ , it is assumed to be the “output variable” of the dynamic systems. Then, the main GRS index and the total GRS index of the system under double-stochastic uncertainty can be proposed as follows, respectively

S_{i} = \frac{V [E (R (t, X) | X_{i})]}{V [R (t, X)]}

(22)

S_{Ti} = \frac{E [V (R (t, X) | X_{- i})]}{V [R (t, X)]}

(23)

Both of the above two indices are variance-based global dynamic reliability sensitivity indices and used to calculate the GRS index of the system under the double-stochastic uncertainties. It is worth noting that in this kind of “double-stochastic uncertainty,” there is a coupling phenomenon between the randomness of excitation and the randomness of system parameters, and this coupling can further affect the reliability of the structure. However, it is very difficult to determine the coupling mechanism, that is, the correlation, between excitation and system parameters in practical projects. Moreover, even if the correlation between the two types of parameters can be found, the correlation coefficient is difficult to determine. In fact, the influence of the coupling between the excitation and system parameters on the structural performance is of great research and application value, and it is also an intractable problem in the current kinetic analysis. In view of the above reasons, in this study, the reliability sensitivity analysis of the structure under random excitation is performed under the assumption that the randomness of the excitation is independent of the randomness of the system parameters. That is, the GRS indices in equations (22) and (23) are approximate GRS indices of the system under double-stochastic uncertainties.

As can be seen from the above analysis, the key to estimate the GRS index in equations (22) and (23) is to solve the unconditional variance $V [R (t, X)]$ , conditional variance $V [E (R (t, X) | X_{i})]$ , and conditional expectation $E [V (R (t, X) | X_{- i})]$ . In order to effectively solve the above three variables, a nested loop method based on SP estimation is proposed in this research.

SP estimation method

In recent years, the weight-point estimation method (Rosenblueth, 1981; Saltelli et al., 2010; Seo and Kwak, 2002), for estimating probability moments by using some feature points and weight points of performance function has been extensively studied. Among these methods, the SP estimation method (Zhao and Ono, 2000) is one of the most commonly used methods. Next, the SP estimation method is briefly introduced based on the univariate and multivariate cases.

Univariate case

For a univariate function $Y = g (X)$ , the expectation and variance can be approximated as follows, respectively

E (Y) = \sum_{l = 1}^{7} P_{l} \times g (T^{- 1} (u_{l}))

(24)

V (Y) = \sum_{l = 1}^{7} P_{l} \times {(g (T^{- 1} (u_{l})) - μ_{g})}^{2}

(25)

where $μ_{g}$ is the value of the function Y when the univariate variable X is fixed at its mean value $μ$ , $T^{- 1} (u_{l})$ is the inverse Rosenblatt transformation, $X_{l}$ is the corresponding value at point $u_{l}$ . If X is a normal random variable with expectation $μ$ and standard deviation $σ$ , then

X_{l} = T^{- 1} (u_{l}) = μ + u_{l} σ

(26)

where $u_{l}$ and $P_{l}$ represent the characteristic points and corresponding weights of random variables that follow the standard normal distribution and are represented as follows, respectively

\begin{matrix} u_{1} = 0, u_{2} = - u_{3} = 1.1544054, \\ u_{4} = - u_{5} = 2.3667594, u_{6} = - u_{7} = 3.7504397 \\ P_{1} = 16 / 35, P_{2} = P_{3} = 0.2401233, \\ P_{4} = P_{5} = 3.07571 \times 10^{- 2}, \\ P_{6} = P_{7} = 5.48269 \times 10^{- 4} \end{matrix}

Multivariate case

For a multivariable function $Y = g (X) = g (X_{1}, X_{2}, \dots, X_{n})$ , it can be approximated by the sum of a series of univariate functions (Zhao and Ono, 2000) and is expressed as follows

g \approx g' (X) = \sum_{i = 1}^{n} (g_{i} - g_{μ}) + g_{μ}

(27)

where

g_{μ} = g (μ_{1}, μ_{2}, \dots, μ_{n})

(28)

g_{i} = g (μ_{1}, \dots, μ_{i - 1}, X_{i}, μ_{i + 1}, \dots, μ_{n})

(29)

Obviously, $g_{i}$ is an univariate function of $X_{i}$ . Then, the expectation and variance of multivariate function can be expressed as follows

E (Y) = \sum_{i = 1}^{n} (μ_{g_{i}} - g_{μ}) + g_{μ}

(30)

V (Y) = \sum_{i = 1}^{n} σ_{g_{i}}^{2}

(31)

where $μ_{g_{i}}$ and $σ_{g_{i}}$ are the mean value and standard deviation of the univariate function $g_{i}$ , which can be calculated by equations (24) and (25), respectively.

Estimation of dynamic GRS index

In section “Variance-based GRS index with double-stochastic uncertainties,” two GRS indices of structures under the double-stochastic uncertainties are proposed. Then, a brief introduction of the SP method is given in section “SP estimation method.” Next, the SP method is used to estimate the two GRS indices in section “Variance-based GRS index with double-stochastic uncertainties.”

As can be seen from section “Variance-based GRS index with double-stochastic uncertainties,” the key to estimate $S_{i}$ and $S_{Ti}$ is to solve $Var [R (t, X)]$ , $Var [E (R (t, X) | X_{i})]$ , and $E [V (R (t, X) | X_{- i})]$ .

For the multivariate unconditional variance $V [R (t, X)]$ , it can be converted to the sum of univariate variance functions by equation (31) and expressed as follows

V [R (t, X)] = \sum_{i}^{n} σ_{R_{i}}^{2} = \sum_{i}^{n} V [R_{i} (t, X)]

(32)

where $R_{i} (t, X) = R_{i} (t, μ_{1}, \dots, μ_{i - 1}, X_{i}, μ_{i + 1}, \dots, μ_{n})$ . Obviously, $V [R_{i} (t, X)]$ is a univariate function that can be calculated using equation (25), that is

\begin{matrix} V [R_{i} (t, X)] = \sum_{l = 1}^{7} P_{l} \times {(R (t, μ_{1}, \dots, μ_{i - 1}, T^{- 1} (u_{l}), μ_{i + 1}, \dots, μ_{n}) - R (t, μ_{1}, \dots, μ_{i - 1}, μ_{i}, μ_{i + 1}, \dots, μ_{n}))}^{2} \end{matrix}

(33)

Then, the final expression of the unconditional variance $Var [R (t, X)]$ estimated by the SP method is given as follows

V [R (t, X)] = \sum_{i = 1}^{n} \sum_{l = 1}^{7} P_{l} \times {(R (t, μ_{1}, \dots, μ_{i - 1}, T^{- 1} (u_{l}), μ_{i + 1}, \dots, μ_{n}) - R (t, μ_{1}, \dots, μ_{i - 1}, μ_{i}, μ_{i + 1}, \dots, μ_{n}))}^{2}

(34)

For the conditional variance $V [E (R (t, X) | X_{i})]$ , $V [E (R (t, X) | X_{i})]$ is decomposed into internal and external two parts and is focused first on internal part $E (R (t, X) | X_{i})$ . Fixing $X_{i}$ at its realization $x_{i}^{*}$ and treat the remaining variables $X_{- i}$ as random variables. Then, we find that $R (t, X) | x_{i}^{*}$ can be regarded as a multivariate function. For simplicity, we use $ψ (X_{- i})$ to represent $R (t, X) | x_{i}^{*}$ , that is, $ψ (X_{- i}) = R (t, X) | x_{i}^{*} = ψ (X_{1}, \dots, x_{i}^{*}, \dots, X_{n})$ . Then, the expectation $E (ψ (X_{- i}))$ can be calculated by equation (30) and expressed as follows

E (ψ (X_{- i})) = \sum_{j = 1, j \neq i}^{n} (μ_{ψ_{j} (X_{- i})} - ψ {(X_{- i})}_{μ}) + ψ {(X_{- i})}_{μ}

(35)

where $μ_{ψ_{j} (X_{- i})}$ denotes the mean value of the univariate function $ψ_{j} (X_{- i})$ . It can be estimated by equation (24) and expressed as follows

\begin{matrix} μ_{ψ_{j} (X_{- i})} = E [ψ_{j} (X_{- i})] = E [ψ_{j} (μ_{1}, \dots, X_{j}, \dots, x_{i}^{*}, \dots, μ_{n})] \\ = \sum_{l = 1}^{7} P_{l} \times ψ_{j} (μ_{1}, \dots, T^{- 1} (u_{l}), \dots, x_{i}^{*}, \dots, μ_{n}) \end{matrix}

(36)

The expression of $ψ (X_{- i})_{μ}$ in equation (35) is given as follows

ψ {(X_{- i})}_{μ} = ψ (μ_{1}, \dots, x_{i}^{*}, \dots, μ_{n})

(37)

Substituting equations (36) and (37) into equation (35), we find that $E (ψ (X_{- i}))$ is a univariate function of $x_{i}^{*}$ . Replacing $E (ψ (X_{- i}))$ with $φ (x_{i}^{*})$ , the conditional variance $V [E (R (t, X) | X_{i})]$ can be finally estimated by equation (25), that is

\begin{matrix} V [E (R (t, X) | X_{i})] = V [φ (x_{i}^{*})] \\ = \sum_{l = 1}^{7} P_{l} \times [φ (T^{- 1} (u_{l})) - μ_{φ (x_{i}^{*})}] \end{matrix}

(38)

where $μ_{φ (x_{i}^{*})}$ is the mean value of the univariate function $φ (x_{i}^{*})$ .

For the conditional variance $E [V (R (t, X) | X_{- i})]$ , it can also be decomposed into internal and external two parts and focused first on the internal part $V (R (t, X) | X_{- i})$ . Fixing $X_{- i}$ at its realization $(x_{1}^{*}, \dots, X_{i}, \dots, x_{n}^{*})$ , we find $R (t, X) | X_{- i} = R (t, x_{1}^{*}, \dots, X_{i}, \dots, x_{n}^{*})$ is a univariate function of $X_{i}$ . Using $φ' (X_{- i})$ to represent $R (t, X) | X_{- i}$ , its variance can be calculated by equation (25) and expressed as follows

\begin{matrix} V [φ' (X_{- i})] = \sum_{l = 1}^{7} P_{l} \times {(φ' (x_{1}^{*}, \dots, T^{- 1} (u_{l}), \dots, x_{n}^{*}) - μ_{φ' (X_{- i})})}^{2} \end{matrix}

(39)

where $μ_{φ' (X_{- i})} = φ' (x_{1}^{*}, \dots, μ_{i}, \dots, x_{n}^{*})$ denotes the mean value of the univariate function $φ' (X_{- i})$ . Obviously, $V [φ' (X_{- i})]$ is a $n - 1$ dimension multivariate function of $X_{- i}$ . Replacing $V [φ' (X_{- i})]$ with $ψ' (X_{- i})$ , $E [V (R (t, X) | X_{- i})]$ can be calculated by equation (30) and expressed as follows

\begin{matrix} E [V (R (t, X) | X_{- i})] = E (V [φ' (X_{- i})]) = E (ψ' (X_{- i})) \\ = \sum_{j = 1, j \neq i}^{n} (μ_{{ψ'}_{j} (X_{- i})} - ψ' {(X_{- i})}_{μ}) - ψ' {(X_{- i})}_{μ} \end{matrix}

(40)

where $ψ'_{j} (X_{- i}) = ψ'_{j} (μ_{1}, \dots, x_{j}^{*}, \dots, μ_{i - 1}, T^{- 1} (u_{l}), \dots, μ_{n})$ is a univariate function of $x_{j}^{*} (j \neq i)$ . $ψ' (X_{- i})_{μ} = ψ' (μ_{1}, \dots, μ_{j}, \dots, μ_{i - 1}, \dots, T^{- 1} (u_{l}), \dots, μ_{n})$ , which is the mean value of the multivariate function $ψ' (X_{- i})$ . The mean value of univariate function $ψ'_{j} (X_{- i})$ , that is, $μ_{{ψ'}_{j} (X_{- i})}$ , can be calculated by equation (24) and expressed as follows

μ_{{ψ'}_{j} (X_{- i})} = E ({ψ'}_{j} (X_{- i})) = \sum_{l = 1}^{7} P_{l} \times ({{ψ'}_{j} (X_{- i}) |}_{x_{j}^{*} = T^{- 1} (u_{l})})

(41)

Then, $E [V (R (t, X) | X_{- i})]$ can be estimated by substituting equation (39) into equation (41).

Finally, the main index $S_{i}$ and the total index $S_{Ti}$ can be estimated by equations (22) and (23), respectively.

Monte Carlo simulation

The Monte Carlo simulation (MCS) is used as a reference to verify the accuracy and efficiency of the proposed SP method. The process is briefly described as follows (Saltelli et al., 2010).

Suppose that A and B are two sample matrices. $A_{B}^{(i)}$ is a sample matrix, where all columns are from A except the ith column is from B . Then, $V [E (Y | X_{i})]$ and $E [V (Y | X_{- i})]$ can be calculated by the following two formulas, respectively

V [E (Y | X_{i})] = V (Y) - \frac{1}{2 N} \sum_{j = 1}^{N} {(g {(B)}_{j} - g {(A_{B}^{(i)})}_{j})}^{2}

(42)

E [V (Y | X_{- i})] = \frac{1}{2 N} \sum_{j = 1}^{N} {(g {(A)}_{j} - g {(A_{B}^{(i)})}_{j})}^{2}

(43)

where $g (A)_{j}$ denotes the value of the performance function when the input variable is the jth column of the sample matrix A . $g (B)_{j}$ and $g (A_{B}^{(i)})_{j}$ have similar meanings to $g (A)_{j}$ . Then, the main index $S_{i}$ and the total index $S_{Ti}$ can be estimated by equations (19) and (20), respectively.

Examples and discussions

Single degree-of-freedom oscillator system

A single degree-of-freedom oscillator system shown in Figure 1 is studied in this example. This dynamic system can be described by equation (1). Assume that m, c, and k are normal distribution random variables with mean values of 15, 2.6, and 45, respectively. For the sake of simplicity, assume that the PSD value of $f (t)$ is a constant value, that is, $S_{ff} (ω) = S_{0} = 1$ .

Figure 1.

Single degree-of-freedom oscillator systems.

The bilateral reliability model, that is, equation (13), is employed in this example to calculate its reliability. In the case of T = 200(s), the reliability analysis result of the single degree-of-freedom oscillator system is shown in Figure 2. Obviously, reliability increases as the threshold b increases, and it is clear that the relationship between reliability and threshold is nonlinear rather than linear.

Figure 2.

Reliability at different threshold b when the random variables are fixed at their mean values.

When the variation coefficient $ζ$ and the threshold b are fixed at 0.001 and 0.7, respectively, the GRS index of this dynamic system is shown in Figure 3.

Figure 3.

Sensitivity indices of input variables when the variation coefficient $ζ$ and threshold b are fixed at 0.001 and 0.7, respectively: (a) Main sensitivity indices, and (b) Total sensitivity indices.

In this example, we use the results of the MCS method with 10⁵ samples as a reference, which indicates that the number of (2 + 3) × 10⁵ function evaluations are required to calculate the main indices and total indices for this example involving three random variables. However, 652 function evaluations are required by the proposed method. This illustrates the feasibility of the proposed method in practical engineering. Simultaneously, it can be clearly seen from Figure 3 that the results estimated by SP method match well with those obtained by the MCS method, which shows that the proposed method has a high accuracy.

Apparently, stiffness k and mass m have a significant contribution on the dynamic reliability of this example. The two GRS indices of damping c are very small and close to zero, indicating that damping has almost no effect on the reliability of this dynamic system. The importance ranking is given as $k > m > c$ .

In order to verify whether the variation coefficient $ζ$ of the input variables and the threshold b affect the importance ranking, the GRS indices at different values of variation coefficient $ζ$ are estimated (when the threshold b is fixed at 0.75) and is shown in Figure 4. At the same time, the GRS indices along with the change of the threshold b (when the variation coefficient $ζ$ is fixed at 0.001) is also studied and is shown in Figure 5.

Figure 4.

Sensitivity indices of input variables at different variation coefficient $ζ$ when the threshold b is fixed at 0.75: (a) Main sensitivity indices, and (b) Total sensitivity indices.

Figure 5.

Sensitivity indices of input variables at different threshold b when the variation coefficient $ζ$ is fixed at 0.001: (a) Main sensitivity indices, and (b) Total sensitivity indices.

From Figures 4 and 5, we can see that the importance ranking does not change with the change of the coefficient $ζ$ or the threshold b. We also find that the main and total importance indices change slightly with the change of the coefficient $ζ$ or the threshold b. However, it is worth noting that this conclusion applies only to this system, and different systems may draw different conclusions.

It can be seen from the results that the main indices are very close to the corresponding total indices, indicating that there is little or no interaction between the three input variables. Correctness of this conclusion can be confirmed by the motion equation in equation (1), where three input random variables m, c, and k obviously have no interaction with each other.

Simple beam

A beam subjected to stochastic excitation is shown in Figure 6. The beam is of uniform thickness $H = 0.02 m$ , width $W = 0.2 m$ , and length $L = 1 m$ . The elastic modulus, Poisson’s ratio, and density are 2e11 Pa, 0.3, and 7800 kg/m³, respectively. The random variables in this example are assumed to be H, W, and L. These three variables follow a normal distribution and have the same variation coefficient 0.001. An acceleration spectral density function of stationary stochastic excitation is shown in Figure 7.

Figure 6.

Simple beam sketch.

Figure 7.

Acceleration spectral density function of stationary stochastic excitation

According to the bilateral first-passage theory in section “The dynamic reliability based on the first-passage method,” the dynamic strength reliability formula can be expressed as follows

R_{s} (T) = \exp [- \frac{T}{π} \frac{σ_{\overset{\cdot}{σ}}}{σ_{σ}} \exp (- \frac{b^{2}}{2 σ_{σ}^{2}})]

(44)

where $σ_{σ}$ and $σ_{\overset{\cdot}{σ}}$ denote the standard deviation of the stress response and the derivative of $σ_{σ}$ . As can be seen from equation (44), the key to calculate dynamic strength reliability is to obtain $σ_{σ}$ and $σ_{\overset{\cdot}{σ}}$ . However, it is very difficult to obtain the $σ_{\overset{\cdot}{σ}}$ directly in practical engineering. Based on the approximate linear relationship between the stress and displacement of the elastic material, the following equation is established

\frac{σ_{\overset{\cdot}{σ}}}{σ_{σ}} = \frac{σ_{\overset{\cdot}{y}}}{σ_{y}}

(45)

Then, equation (44) can be rewritten as follows

R_{s} (T) = \exp [- \frac{T}{π} \frac{σ_{\overset{\cdot}{y}}}{σ_{y}} \exp (- \frac{b^{2}}{2 {(σ_{σ})}^{2}})]

(46)

According to equation (46), the three variables required to calculate reliability using equation (46) are stress response $σ_{σ}$ , displacement response $σ_{y}$ , and velocity response $σ_{\overset{\cdot}{y}}$ , respectively. These three quantities can be directly obtained by random vibration analysis, and the corresponding contour results are shown in Figures 8 to 10, respectively.

Figure 8.

Contour results of the stress response.

Figure 9.

Contour results of the displacement response.

Figure 10.

Contour results of the velocity response.

For simplicity, consider only the reliability of the node with the maximum response. Assume that the time T and the threshold b are $3.6 \times 10^{7}$ (s) and 8e9 (Pa), respectively. According to equation (46), the reliability of the node with the maximum response is R_s = 0.9909. With this reliability $R_{s}$ as an output response, the sensitivity indices of three random variables H, W, and L can be obtained according to the proposed SP method and the MCS method with 1×10⁵ simulations, and the results are shown in Figure 11. It is worth noting that the total number of function evaluations required by the MCS method is (2 + 3)×10⁵ instead of 1×10⁵.

Figure 11.

Sensitivity indices of three random variables: (a) Main sensitivity indices, and (b) Total sensitivity indices.

As can be seen from Figure 11, the sensitivity results obtained by the proposed SP method are in good agreement with the sensitivity results obtained by the MCS, indicating that the proposed method has high accuracy. The number of function evaluations required for the proposed method in this example is 652, which is approximately 0.13% of the number of function evaluations required by the MCS, indicating the proposed method has high computational efficiency.

It can also be seen from Figure 11 that the two sensitivity indices of the length L are the largest, followed by the height H and the width W, which means that the length L has the greatest effect on the reliability, while the width H has the least effect on the reliability. Simultaneously, the main indices of the three random variables are basically the same as their total indices, indicating that the interaction between the three random variables has almost no effect on reliability. Sensitivity information has important application value in engineering. According to the above sensitivity analysis, we can see that the length L and height H are the key variables affecting the reliability of this beam and should be given more attention in the actual engineering design.

An aeronautical hydraulic pipeline system

The above two examples verify the accuracy and efficiency of the proposed method. In this section, the proposed method is applied to the reliability sensitivity analysis of an aeronautical hydraulic pipeline system subjected to stochastic excitation to illustrate its practicality. Due to the flutter of the aircraft or the vibration of the pump source, the aeronautical hydraulic pipeline is usually in a random vibration environment (Wang et al., 2018). Therefore, it is very necessary to find out the important parameters that affect the reliability of the pipeline.

The failure modes of aeronautical hydraulic pipelines are generally divided into two categories: stress-based failure and strain-based failure. In this example, the first-passage method is first employed to calculate the stress-based dynamic reliability. Second, the sensitivity analysis is performed with the dynamic reliability as the objective function to find out the parameters that have significant effect on the dynamic reliability of the aeronautical hydraulic pipelines.

The finite element model of an aeronautical hydraulic pipeline system is shown in Figure 12. Figure 12(a) is the total pipeline system, the left and right ends are oil output, and the bottom end is the pump source. Experience shows that the part near the pump source is more vulnerable that other parts. Therefore, this part is taken out from the general model as the object model of this study, as shown in Figure 12(b). In Figure 12(b), point C is a fixed constraint. The spring element (stiffness k is given as 2.5 × 10⁶ N/m by experience) is used to constrain the three translational degrees of freedom of the point A and point B. The Z-direction and X-direction translational degrees of freedom of the points D, E, and F are constrained. For the sake of simplicity, assume that the random excitation in this example is the same to the excitation in case “Simple beam” and is shown in Figure 7.

Figure 12.

Finite element model of aeronautical hydraulic pipeline: (a) General pipeline model, and (b) Local pipeline model.

In this example, consider the elastic modulus E, material density $ρ$ , thickness t, density of the hydraulic fluid $ρ_{0}$ , and three support positions $l_{1}$ , $l_{2}$ , and $l_{3}$ in y-direction as the input variables. Assume all these input variables follow the normal distribution with mean values of $μ_{E} = 2.06 \times 10^{11} Pa$ , $μ_{ρ} = 7900 kg / m^{3}$ , $μ_{t} = 0.003 m$ , $μ_{ρ_{0}} = 900 kg / m^{3}$ , $μ_{l_{1}} = 0.125 m$ , $μ_{l_{2}} = 0.125 m$ , and $μ_{l_{3}} = 0.167 m$ , respectively. $ζ$ is the variation coefficient.

If the maximum stress of any node in the pipeline system exceeds the stress threshold b, the total piping system is considered invalid. Therefore, the pipeline system can be regarded as a series system in terms of reliability analysis. In view of this, it is first necessary to calculate the reliability $R_{i} (T)$ of each node. Then, the reliability of the pipeline system can be calculated by

R^{T} = Π_{i = 1}^{n} R_{i}

(47)

Due to space limitations, only the responses of some nodes of the pipeline system are listed in Table 1. The threshold b and the time T are 2×10⁹(Pa) and 3.6×10⁷(s) in this example, respectively.

Table 1.

Responses at certain nodes of the pipeline system.

Node number	$σ_{y} (10^{- 3})$	$σ_{\overset{\cdot}{y}}$	$σ_{σ} (10^{7})$	$R_{i}$
1	0.3584	0.1051	0.1671	1.0000
2	0.3722	0.2060	1.4067	1.0000
3	0.3972	0.3342	2.6677	0.9970
4	0.3967	0.3321	2.5230	0.9999
22	0.3582	0.0813	0.7300	1.0000
23	0.3982	0.3384	2.5365	0.9998
30	0.3584	0.0825	1.4466	1.0000

Then, the reliability of the pipeline system calculated by equation (47) is

R^{T} = Π_{i = 1}^{30} R_{i} = 0.9967

The contour results of the different output response of the hydraulic pipeline are shown in Figures 13 to 15, respectively. Then, using $R^{T}$ as the objective function, the global sensitivity analysis of the hydraulic pipeline is carried out, and the results are shown in Figure 16. The number of function evaluations required by the proposed method to estimate the GRS index for all input variables is 4165.

Figure 13.

Contour results of the displacement response.

Figure 14.

Contour results of the velocity response.

Figure 15.

Contour results of the stress response.

Figure 16.

Sensitivity indices of input variables (fix $ζ$ at 0.005): (a) Main sensitivity indices, and (b) Total sensitivity indices.

As can be seen from Figure 16, the two sensitivity indices of the support position $l_{3}$ are the largest, which indicates that it has the greatest impact on the reliability of the hydraulic pipeline system. Simultaneously, the two sensitivity indices of the hydraulic fluid $ρ_{0}$ are the smallest and close to zero, which means that it has the least impact on the reliability of the pipeline system. Apparently, the importance ranking of input variables to the reliability of the pipeline system is $l_{3} > ρ > E > l_{1} > l_{2} \approx t > ρ_{0}$ .

The above importance ranking is obtained by fixing the variation coefficient $ζ$ at 0.005, we also want to know whether the above importance ranking varies with the change of variation coefficient.

As can be clearly seen from Figure 17, the importance ranking does not change with the change of variation coefficient. It can also be found that the sensitivity index of each input variable increases as the variation coefficient $ζ$ increases, and the larger the index, the greater the changes. It is worth noting that in this example, the variation coefficient of all variables is consistent, that is, $ζ$ . If different variables have different variation coefficients, different conclusions may be drawn.

Figure 17.

Sensitivity indices of input variables with the changes of variation coefficient $ζ$ : (a) Main sensitivity indices, and (b) Total sensitivity indices.

It is also found that the total indices are slightly larger than their corresponding main indices, which means that there is a certain interaction between these input random variables. However, the interaction does not have a strong effect on reliability.

In this example, two position variables $l_{3}$ and $l_{1}$ and two pipeline material property variables, elastic modulus E and material density $ρ$ , are important “failure sources” for this dynamic pipeline system. If engineers want to improve the performance of this pipeline system, they should pay more attention to the above four variables, especially the position variable $l_{3}$ .

Conclusion

This article extends the variance-based GRS indices to a dynamic system in which stochastic input variables and stochastic process exist simultaneously. In order to improve the computational efficiency of the sensitivity index, a loop nesting method based on the SP estimation is proposed. These indices can quantify the contribution of input uncertainty on the reliability of the system under double-stochastic uncertainty and provide important references for improving the performance of engineering problems.

Our research mainly focuses on two aspects. One is to propose a loop nesting method based on the SP estimation method in Gaussian integral, and the other is to use the proposed method to perform the global sensitivity analysis of the structure under double-stochastic uncertainty. Three examples have demonstrated that the proposed method is of high efficiency and practicability compared with the MCS procedure.

Although the proposed method has high computational accuracy, its computational efficiency still needs to be further improved. Due to the rapid development of surrogate models in recent years, how to use the surrogate model to efficiently calculate the sensitivity of the structure under double random uncertainty is of great significance in practical engineering, and it is also an important aspect of our follow-up research.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The financial support by the Nature Science Foundation of China (NSFC51608446), the Natural Science Fundamental Research Plan of Shaan Xi Province (Grant No. 2017JQ1021), and the Fundamental Research Funds for the Central Universities (Grant No 3102018zy011).

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