A novel structural reliability analysis method via improved maximum entropy method based on nonlinear mapping and sparse grid numerical integration

Mechanical Systems and Signal Processing 133 (2019) 106247

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Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp

A novel structural reliability analysis method via improved maximum entropy method based on nonlinear mapping and sparse grid numerical integration Wanxin He, Yan Zeng, Gang Li ⇑ Department of Engineering Mechanics, State Key Laboratory of Structural Analysis for Industrial Equipment, Dalian University of Technology, Dalian 116024, China

a r t i c l e

i n f o

Article history: Received 31 January 2019 Received in revised form 13 June 2019 Accepted 16 July 2019

Keywords: Nonlinear transformation Cauchy distribution Maximum entropy method Structural reliability analysis Sparse grid numerical integration Statistical moment evaluation

a b s t r a c t This paper proposes an improved maximum entropy method for reliability analysis (i-MEM), in which the limit state function is transformed by a nonlinear mapping to predict the failure probability accurately. Through the nonlinear mapping, more statistical information can be obtained by the first-four statistical moments, and the truncation error originating from numerical integration is solved by the bounded limit state function after the nonlinear mapping, therefore the i-MEM can capture the tail information of the real probability distribution. In order to calculate the statistical moments in i-MEM with accuracy and efficiency, an improved sparse grid numerical integration method (i-SGNI) is developed on the basis of the normalized moment-based quadrature rule. Combining the i-MEM and i-SGNI, a novel reliability analysis method is proposed. To illustrate the accuracy, efficiency and numerical stability of the proposed method, six numerical examples and one engineering example are presented, compared with some common reliability analysis methods. The results show that the proposed method, with the combination of i-MEM and i-SGNI, can achieve a good balance between accuracy and efficiency for structural reliability analysis. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Uncertainty is the essential attribute in the objective world [1], which is a frequent occurrence in practical engineering, such as uncertainties in material properties, geometric dimensions and applied loads. These uncertainties can be propagated to the structural response and influence the structural reliability. Therefore, many methods related to structural reliability analysis have been developed recently [2–6], which can be divided into four categories generally. Firstly, direct sampling method, such as Monte Carlo simulation (MCS) method [7], is extensively used in reliability analysis. Though this method is quite simple and robust, it is always employed as the reference solution to validate other reliability analysis methods due to its unacceptable computational burden [8]. Some improved methods, such as the Importance Sampling [9–11], Subset Simulation [12] and Line Sampling [13,14], have been proposed to reduce computational cost, but the sampling-based methods are still confronted with the problem of inefficiency [15]. Secondly, approximate reliability analysis method is an alternative to improve the computational efficiency, such as the First-Order Reliability Method (FORM) [16,17] and Second Order Reliability Method (SORM). FORM is widely used in both ⇑ Corresponding author. E-mail address: [email protected] (G. Li). https://doi.org/10.1016/j.ymssp.2019.106247 0888-3270/Ó 2019 Elsevier Ltd. All rights reserved.

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W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

reliability analysis and reliability-based design optimization with probabilistic or non-probabilistic uncertainties [18,19]. However, because FORM or SORM requires the Taylor expansions of the limit state function at the most probable point (MPP), their accuracy, efficiency and convergence rely on the MPP and the complexity of the limit state function considerably. Thirdly, surrogate-based reliability analysis method draws more and more attention recently due to its efficiency, such as response surface model [20] and Kriging model [21]. Although surrogate-based method decreases the computational cost greatly, the prediction accuracy depends on the sample points and model parameters. Finally, moment-based method evaluates the failure probability based on the statistical moments of the limit state function, therefore the derivative and MPP are not required. Recently, various moment-based methods have been proposed for reliability analysis, such as Pearson and Johnson system, generalized Lambda distribution and maximum entropy method (MEM) [22–24]. The general guideline is developed by Xi el al. [25] to determine the optimal moment method for reliability analysis. The method for forward and inverse structural uncertainty propagations was proposed on the basis of the first-four statistical moments and lambda probability density function [26]. Though widely used, the moment-based methods still have some problems in practical applications. For example, the experiential distribution function in Pearson family, may lead to unstable results near the boundaries of different regions pertaining to different distribution types [27]. The performance of Johnson distribution is poor for calculation of the unknown parameters by the statistical moments of the sample data [28]. Saddlepoint approximations may fail to obtain the accurate result due to numerical singularity [29]. Though the generalized beta and lambda distribution is accurate for the problems with moderate to low reliability levels, it is nearly incompetent for the cases with the failure probability smaller than 0.01 [23]. As a kind of moment-based method, the MEM is widely employed in structural reliability analysis, whose main idea is to reconstruct the probability density function (PDF) via the statistical moments of the limit state function [30]. Jaynes [31] gives the maximum entropy principle to evaluate the most unbiased PDF, which accords with the available data and minimizes the spurious information [32]. Shore and Johnson [33] proved that no other probabilistic methods except the MEM can satisfy all consistency axioms when only moments constraints are available for the PDF prediction. Despite of the conceptual elegance, some shortcomings limit the application of MEM. Primarily, because low-order statistical moments may be insufficient to capture the tail information of the real PDF accurately, higher-order statistical moments with high precision are required for predicting the PDF of the limit state function for some cases. However, higher-order statistical moments are difficult to obtain accurately and may cause numerical instability [34–37]. Secondly, the numerical integration for the failure probability estimation should have been calculated on the infinite boundaries when the limit state function is defined on (
1, +1), and the truncation error is ineluctable in practice. Thirdly, the truncated form of the exponential polynomial function used in MEM cannot cover all types of probability density functions, although the infinite expansion of the exponential polynomial function approaches to the real one asymptotically [38]. Therefore, an improvement for MEM is required for the accuracy and numerical instability. To evaluate the statistical moments accurately is the key point in MEM, which, however, is a thorny problem because it involves the calculation of high dimensional integration. Generally speaking, three common methods are frequently used to calculate the integration [39]: analytical methods, simulation methods and numerical integration methods. Analytical methods can only solve simple problems, and simulation methods, such as Monte Carlo simulation and Importance Sampling, can be hardly accepted in practical engineering due to its computational cost. Numerical integration methods, such as Gauss– Hermite method and Gauss–Legendre method, are also confronted with the problem of inefficiency when the number of the input random variables is 4 or more. For the tradeoff between the accuracy and efficiency, Rahman and Xu [40] proposed a univariate dimension-reduction method (UDRM) for evaluating the statistical moments of structural response. Further, Xu and Rahman [39] developed a generalized multivariate dimension-reduction method (GDRM) for the statistical moment evaluation of complex limit state functions, which, however, is expensive in computational cost. Recently, sparse grid numerical integration (SGNI) [41,42] is used to evaluate the statistical moments of the limit state function due to its accuracy and efficiency. The main idea of SGNI is to calculate the high dimensional integration by Smolyak-type quadrature formula, which needs fewer integration points with good computational accuracy. Actually, decomposition method [43] can be regarded as a simplified version of SGNI [41]. As far as the authors’ knowledge, the integration points and weights used in SGNI are obtained by Gaussian integration in recent researches, which, however, is only applicable for some special probability distributions. And Rosenblatt or Nataf transformation is usually employed to transform the non-normal random variables into standard normal ones, which may cause extra error of calculation. In this paper, an improved MEM (i-MEM) based on nonlinear mapping is proposed to overcome the deficiencies of the traditional MEM. Through the improvement, more statistical information can be obtained by a few statistical moments for predicting the failure probability accurately, and the numerical instability due to high-order statistical moments can be avoided. Meanwhile, the truncation error originating from numerical integration is solved by the bounded limit state function after the nonlinear mapping. Furthermore, this paper develops an improved SGNI (i-SGNI) for the statistical moment evaluation of the limit state function, which is applicable to the problems with the input random variables following arbitrary probability distributions. Finally, a novel reliability analysis method is proposed through the combination of i-MEM and i-SGNI. Organization of the manuscript is as follows. In Section 2, the main ideas related to the traditional MEM are introduced. Section 3 presents the details about the proposed i-MEM. In Section 4, the proposed i-SGNI for evaluating the statistical moments is detailed. In order to demonstrate applicability of the proposed method for evaluating the statistical moments

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

3

and assessing the structural reliability, seven examples are presented in Section 5, followed by some discussions. The last section summarizes some concluding remarks. 2. Traditional maximum entropy method for reliability analysis The first-n statistical moments of a continuous random variable, Y, are defined as:

Z

E½Y i  ¼

yi f ðyÞdy; i ¼ 1; 2:::; n

ð1Þ

Y

where f(y) is the PDF of Y and E[∙] is the expectation operator. Recovering the PDF of Y from the statistical moment information in Eq. (1) is one of the classical moment problems [34], and one of the common approaches is the MEM, with the optimization formulation as:

8 f m ðyÞ > < find : R maximize : Hðf m ðyÞÞ ¼
Y f m ðyÞlnðf m ðyÞÞdy > R : s:t: : E½Y i  ¼ Y yi f m ðyÞdy; i ¼ 1; 2; :::; n

ð2Þ

where H is defined as the Shannon entropy of a continuous random variable. It has been proved that the optimal estimated result, fm(y), converges to the real PDF, f(y), as n ? +1 [30]. However, it is impractical to obtain massive high-order statistical moments accurately for industrial application. Moreover, the problem of numerical singularity due to high-order statistical moments are thorny when solving the optimal problem in Eq. (2). Therefore, n = 4 is in common use for recovering f(y). The result of Eq. (2) can be derived by the stationary value condition of the Lagrange function as follows:

f m ðyÞ ¼ expð
k0


4 X

ki yi Þ

ð3Þ

i¼1

where the Lagrange multiplier, k = (k1, k2, k3, k4) can be calculated by solving the constraint conditions in Eq. (2) and k0 is derived analytically based on the normalization axiom in probability theory:

"Z

k0 ¼ ln

expð


4 X

Y

#

ki yi Þdy

ð4Þ

i¼1

Then, the reliability analysis can be achieved via numerical integration over the failure domain, XY, which represents that the limit state function is less than 0 in this paper.

Z Pf ¼ PðY < 0Þ ¼

0


1

f m ðyÞdy

ð5Þ

3. Improved maximum entropy method based on nonlinear mapping Though the traditional maximum entropy method is widely used in reliability analysis, the performance of the method in terms of accuracy and stability still poses some challenges, such as numerical singularity caused by calculation of higherorder statistical moments, truncation error from numerical integration on (
1, +1), truncated form of the exponential polynomial function of PDFs, and numerical singularity caused by the magnitude of the limit state function. In this paper, we propose an improved maximum entropy method (i-MEM) based on a nonlinear mapping of the original limit state function to overcome the problems above. 3.1. Formulation of i-MEM Define the nonlinear mapping of Y 2 (
1, +1) as follows

T ¼ #ðYÞ

ð6Þ

which is of boundedness, monotonicity, differentiability and invertibility on (
1, +1). Thus, reconstruction of probability density function of Y is equivalent to reconstruction of probability density function of T and XY ? XT (the failure domain of T) is bijection. Based on the transformed limit state function, the optimal formulation of MEM can be written as follows:

8 um ðtÞ > < find : R maximize : Hðum ðtÞÞ ¼
T um ðtÞlnðum ðtÞÞdy > R : s:t: : E½T i  ¼ T ti um ðtÞdt; i ¼ 1; 2; :::; 4

ð7Þ

4

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Therefore, um(t) can be given similarly as

um ðtÞ ¼ exp
k0


4 X

!

ð8Þ

ki t i

i¼1

and

"Z k0 ¼ ln

exp


4 X

T

! # ki t i dt

ð9Þ

i¼1

Then, the failure probability can be calculated by the following numerical integration on XT

Z

Pf ¼ PðT < 0Þ ¼

um ðtÞdt

XT

ð10Þ

It can be inferred that the accuracy of the estimated failure probability relies on the form of the nonlinear mapping greatly. In this paper, an arctangent nonlinear transformation, Cauchy distribution function, is used in the i-MEM, which is expressed as follows

T¼

  0  1 2 Y arctan þ 1 2 ð0; 1Þ 2 p k

ð11Þ

where k is the scale parameter used to select the optimal nonlinear transformation and Y0 is the normalized limit state function to avoid the numerical problems caused by the magnitude of Y, defined as

Y0 ¼

Y YðlÞ

ð12Þ

in which Y(l) is the value of limit state function at the mean values of the input random variables. 3.2. Algorithm of i-MEM According to the derivation in Section 3.1, the unknown parameters in i-MEM are k and ki, (i = 1, 2, 3, 4). In order to bypass the highly nonlinear constraint conditions, an alternate formulation to calculate ki is developed on the basis of the minimization of the Kullback-Leibler (K-L) divergence [44] between the real PDF, u(t), and the estimated PDF, um(t, k), which is expressed as:

K½uðtÞ; um ðt; kÞ ¼

Z

T

uðtÞlog





uðtÞ dt um ðt; kÞ

ð13Þ

Substituting Eq. (8) into Eq. (13), the K-L divergence can be written as follows:

K½uðtÞ; um ðt; kÞ ¼
HðuÞ þ k0 þ

4 X

ki EðT i Þ

ð14Þ

i¼1

Because H is the entropy of the real PDF, minimization of the K-L divergence is equivalent to the following optimization problem [44]:

find :

k and k

minimize : Cðk; kÞ ¼ k0 þ

P4

i¼1 ki EðT

i

ð15Þ

Þ

Before the optimization problem in Eq. (15) is solved, two lemmas on correctness of the proposed algorithm must be presented. Lemma 1 (K-L divergence conservation law): If random variables, Y and T, conform to Eq. (6), the estimator of f(y), say fm(y), can be derived by the estimator of u(t), um(t). Thus, the following relationship is established:

K½f ðyÞ; f m ðyÞ ¼ K½uðtÞ; um ðtÞ

ð16Þ

Proof. According to Eq. (6), the following derivation can be given.

K½uðtÞ; um ðtÞ ¼

Z T

uðtÞlog



Z

f ðyÞlog

¼ Y





uðtÞ dt ¼ um ðtÞ 

"

Z

0

f ð#
1 ðtÞÞj½#
1 ðtÞ jlog T

f ðyÞ dy ¼ K½f ðyÞ; f m ðyÞ f m ðyÞ

which completes the proof of Lemma 1.

0

f ð#
1 ðtÞÞj½#
1 ðtÞ j 0

#

f m ð#
1 ðtÞÞj½#
1 ðtÞ j

Z dt ¼ T

    dy f ðyÞ f ðyÞ log dt dt f m ðyÞ

5

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Lemma 2 (consistency law between PDF and K-L divergence): Consider two nonlinear transformations of Eq. (6), T1 and T2. If the K-L divergences have the relationship of K½u1 ðt1 Þ; u0m ðt1 ; k1 Þ < K½u2 ðt 2 Þ; u00m ðt 2 ; k2 Þ, u0 m(t) is superior to u00 m(t) in the view of minimization of the K-L divergence, where u0 m(t, k1) and u00 m(t, k2) are the estimators of u1(t) and u2(t), the real PDFs of T1 and T2. Proof. Assume the two nonlinear transformations of Eq. (6) as follows:

T 1 ¼ #1 ðYÞ and T 2 ¼ #2 ðYÞ

ð17Þ

Thus, the relationship between T1 and T2 can be expressed by the following function

T1 ¼

-ðT 2 Þ ¼ #1 ð#
1 2 ðT 2 ÞÞ

Therefore, we have

ð18Þ

2  
1 0 3
1   u ðt 1 Þ  - ðt 1 Þ    2 0 5dt1 u2 -
1 ðt1 Þ  -
1 ðt1 Þ log4 u0m ðt1 ; k1 Þ T1  2  3 Z u2 ðt2 Þ -
1 ðt1 Þ 0  5dt 2 ¼ u2 ðt2 Þlog4 u0m ðt1 ; k1 Þ T2 

Z



u ðt Þ K½u1 ðt 1 Þ; u0m ðt 1 ; k1 Þ ¼ u1 ðt1 Þlog 0 1 1 dt1 ¼ u T1 m ðt 1 ; k1 Þ

Based on Lemma 1, if u0m ðt 1 ; k1 Þ – u00m

Z



  -
1 ðt1 Þ  -
1 ðt1 Þ 0 , K½u1 ðt1 Þ; u0m ðt1 ; k1 Þ – K½u2 ðt2 Þ; u00m ðt2 ; k2 Þ, therefore,



K½u1 ðt 1 Þ; u0m ðt 1 ; k1 Þ < K½u2 ðt2 Þ; u00m ðt2 ; k2 Þ can occur. Let f0 m(y) and f00 m(y) represent the estimators of Y derived by u0 m(t) and u0 0 m(t). According to Lemma 1, we have 0 00 K½u1 ðt 1 Þ; u0m ðt 1 ; k1 Þ = K½f ðyÞ; f m ðyÞ and K½u2 ðt2 Þ; u00m ðt2 ; k2 Þ = K½f ðyÞ; f m ðyÞ. Therefore, if K½u1 ðt1 Þ; u0m ðt1 ; k1 Þ < K½u2 ðt2 Þ; 0 00 00 0 um ðt2 ; k2 Þ, K½f ðyÞ; f m ðyÞ < K½f ðyÞ; f m ðyÞ can be obtained, namely, f m(y) is closer to f(y) than f00 m(y). Thus, it can be concluded that u0 m(t) is superior to u00 m(t) in the view of minimization of the K-L divergence. The proof of Lemma 2 is completed. On the basis of Lemmas 1 and 2, we can confirm that the proposed algorithm in Eq. (15) is reasonable, which is used for selecting the optimal form of the nonlinear mapping and the parameters in the estimated PDF. 3.3. An efficient strategy for calculating ki The optimization problem of Eq. (15) can be solved by a double-loop optimization algorithm, given as



min min fCðk; kÞg k

ð19Þ

k

where the calculation of k is the outer layer optimization and the calculation of k = (k1, k2, k3, k4) is the inner layer optimization. Though this algorithm is straightforward, it is so time-consuming [45]. Alternatively, a simplified method for solving k is employed on the basis of a linear equation. It has been proved that U(k, k) is convex about k [44], therefore KKT condition can be used to derive the global minimum [46] when k is given, as shown in Eq. (20)

@C ¼ 0 ! EðT i Þ; @ki

i ¼ 1; 2; 3; 4

ð20Þ

where E(Ti) can be written further as follows

Z

EðT i Þ ¼ T

t i um ðtÞdt;

i ¼ 1; 2; 3; 4

ð21Þ

Substituting Eq. (8) to Eq. (21) and performing integration by parts, we have

Z EðT i Þ ¼ T

t i um ðtÞdt ¼

1 1 1 ½tiþ1 um ðtÞ0
iþ1 iþ1

Z T

tiþ1 dum ðtÞ ¼

4 1 1 X u ð1Þ þ kk kEðT iþk Þ iþ1 m i þ 1 k¼1

ð22Þ

Replace i with i-1, and then Eq. (22) can be rewritten as a similar form

Z EðT i
1 Þ ¼ T

ti
1 um ðtÞdt ¼

1 i 1 1 ½t um ðtÞ0
i i

Z T

t i dum ðtÞ ¼

4 1 1X um ð1Þ þ kk kEðT i
1þk Þ i i k¼1

ð23Þ

According to Eqs. (22) and (23), k = (k1, k2, k3, k4) can be solved by the following linear equations [45,47]

ði þ 1ÞEðT i Þ
iEðT i
1 Þ ¼

4 X k¼1

kk k½EðT iþk Þ
EðT i
1þk Þ;

i ¼ 1; 2; 3; 4

ð24Þ

6

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Thus, the double-loop optimization in Eq. (19) can be simplified to a single-loop optimization expressed as follows

min fCðk; kÞg

ð25Þ

k

where k is solved by Eq. (24). 4. Improved sparse grid numerical integration In the algorithm of i-MEM, estimating the statistical moments E(Ti) is a crucial step. Recently, various approaches have been developed for calculating the statistical moments of limit state function. Recently, sparse grid numerical integration (SGNI) [41,42] is caused attention due to its accuracy and efficiency, in which the high dimensional integration is calculated by Smolyak-type quadrature formula, with fewer integration points. In this paper, an improved SGNI (i-SGNI) is proposed to estimate the statistical moments for the tradeoff between accuracy and efficiency, which is the combination of SGNI and normalized moment-based quadrature rule (NMBQR) [48] and applicable for the input random variables with arbitrary probability distributions. 4.1. Sparse grid numerical integration The estimation of the statistical moments in Eq. (21) can be achieved in the form of multivariate integration

Z EðT i Þ ¼ T

t i uðtÞdt ¼

Z X

t i ðx1 ; x2 ; :::; xN Þwðx1 Þwðx2 Þ:::wðxN Þdx1 dx2 :::dxN

ð26Þ

where N is the number of input random variables, x = (x1, x2, . . ., xN) are the realizations of the mutually independent input random variables X = (X1, X2, . . ., XN) and w(x1), w(x2) and w(xN) are their PDFs. The joint PDF of X can be regarded as the weight associated with kernel ti(x) [41]. From the Smolyak algorithm [49], the SGNI for the calculation of the statistical moments in Eq. (26) can be obtained as

Z Z  Z E Ti ¼ t i uðt Þdt ¼ ::: t i ðx1 ; x2 ; :::; xN Þwðx1 Þwðx2 Þ:::wðxN Þdx1 dx2 :::dxN X

T

X

¼

qþN
jjijj1

ð
1ÞqþN
jjijj1 C N
1

i2Hðq;NÞ

mi 1 X

:::

j1 ¼1

mi  N X t i xij11 ; :::; xijNN wij11 :::wijNN

ð27Þ

jN ¼1

where the nonnegative integer q is the precision parameter, the multi-index i = (i1, . . ., ik, . . ., iN) 2 N Nþ , the set H (q, N) is defined by

( Hðq; NÞ ¼

i2

N Nþ

:qþ16

N X

) ik 6 q þ N

ð28Þ

k¼1 i

i

xjkk is the jkth integration point of the kth input random variables, wjkk is the corresponding weight, and mik is the number of the integration points for the kth input random variables, given as

mik ¼ 1 for ik ¼ 1 and mik ¼ 2ik
1 þ 1 for ik > 1

ð29Þ

For the sake of succinctness, please refer to Ref. [41] for the further details about SGNI. 4.2. Normalized moment-based quadrature rule It is vital to obtain the integration points and weights for calculating the numerical integration in Eq. (27) accurately. Various quadrature rules have been developed, such as Gauss-Hermite, Gauss-Legendre and Gauss-Laguerre, which are applicable for some special probability distributions. The moment-based quadrature rule (MBQR) [39,40] was introduced to calculate the statistical moments of the limit state function of the mechanical system with the input random variables of arbitrary probability distributions, in which the integration points and weights can be obtained by solving a linear equation composed of the raw statistical moments of the input random variables. Readers can refer to Ref. [40] for details about MBQR. It was found that the linear equation in MBQR was ill-conditioned in many cases [27,50], causing considerable error of the higher-order statistical moments. Hence, NMBQR [48] was proposed to improve the stability of the linear equation, by replacing the raw statistical moments with the normalized statistical moments. Then the linear equation in NMBQR is expressed as follows:

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

2

l0j;n
1 6 6 l0 6 j;n 6 6 l0 6 j;nþ1 6 6 6 4


l0j;n
2
l0j;n
1
l0j;n

.. .

l0j;n
3    l0j;n
2    l0j;n
1   

.. .

.. .

ð
1Þn
1 l0j;0 ð
1Þn
1 l0j;1 ð
1Þn
1 l0j;2

.. .

.. .

l0j;2n
2
l0j;2n
3 l0j;2n
4    ð
1Þn
1 l0j;n
1

7

3

2 0 3 2 0 3 r j;1 lj;n 7 7 6 r 0 7 6 l0 7 7 6 j;2 7 6 j;nþ1 7 7 6 0 7 6 0 7 7 6 r j;3 7 6 lj;nþ2 7 76 7¼6 7 7 6 . 7 6 . 7 7 6 . 7 6 . 7 7 4 . 5 4 . 5 5 l0j;2n
1 r0j;n

ð30Þ

where

l0j;k ¼

Z

þ1

  xj
l j k

rj


1

f X j ðxj Þdxj ; k ¼ 1; 2; :::; n

ð31Þ

lj and rj are the mean value and standard deviation, respectively, and n is the number of integration points. The normalized integration points can be obtained by solving the following polynomial equation, whose coefficients are the solution of Eq. (30) znj
r 0j;1 zn
1 þ r 0j;2 zn
2
::: þ ð
1Þn r 0j;n ¼ 0 j j

ð32Þ

On the basis of the normalized integration points, the linear transformation in Eq. (33) is performed for the integration points in the original space

xj ¼ zj rj þ lj

ð33Þ

and the integration weights can be obtained as follows

R þ1 Qn wjk ¼


1

i¼1;i–k ðzj

Qn


zji Þf X j ðxj Þdxj

i¼1;i–k ðzjk


zji Þ

Pn
1 ¼

l

i 0 i¼0 ð
1Þ j;n
i
1 qj;k;i Qn i¼1;i–k ðzjk
zji Þ

ð34Þ

where qj,k,0 = 1, qj;k;i ¼ r 0j;i
zjk qj;k;ði
1Þ , zjk is the kth normalized integration point for the jkth input random variable and wjk is the weight at the kth integration point for the jkth random variable. Then, the integration in Eq. (26) can be calculated by the i-SGNI. In summary, we propose a novel approach to reconstruct the unknown PDF through the i-MEM based on i-SGNI, with the flow chart shown in Fig. 1. 5. Examples In order to illustrate the effectiveness of the proposed method, four numerical examples and one engineering example were tested. For the tradeoff between the accuracy and efficiency, q = 2 is provided as the precision parameter of the i-SGNI [42]. The results of the statistical moment estimation obtained from three methods are compared, say the proposed i-SGNI, UDRM with seven integration points and MCS. It should be noted that the relative errors of four important statistical parameters, mean value, standard deviation, skewness and kurtosis, are adopted as the evaluation criterion of the accuracy of statistical moment evaluation in this paper. The results of reliability analysis in each example are obtained by the proposed method (i-SGNI + i-MEM), i-SGNI + MEM, UDRM + i-MEM, UDRM + MEM and MCS. The relative error (e) is calculated by

e¼

jr
r mcs j r mcs

ð35Þ

where rmcs and r represent the results from MCS and other methods, respectively. 5.1. Example 1 A nonlinear numerical example [51] is first investigated, with the limit state function as

GðXÞ ¼ 1000X 1 X 2
7:51X 1 X 3 þ X 1 X 4 þ 40X 5
0:5X 23

ð36Þ

where Xi, i = 1, . . ., 5, are statistically independent and follow the normal distributions with different mean values, l1 = 1.2, l2 = 2.4, l3 = 50, l4 = 25, l5 = 10, and standard deviations, r1 = 0.36, r2 = 0.072, r3 = 3, r4 = 7.5, r5 = 5, respectively. 5.1.1. Statistical moment evaluation Some details of i-SGNI are illustrated in this example. Firstly, the set H (q, N) is obtained by Eq. (28), as shown in Table 1. Secondly, generate the integration grids and weights by NMBQR according to H (q, N). For part 1 in Table 1, the number of the integration points is one for each input random variable on the basis of Eq. (29), therefore the integration grid of part 1 is a point actually, as shown in Fig. 2. For parts 2 and 3, the integration grid of each component is in a line. For example, three

8

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Fig. 1. The flowchart of the proposed method.

integration points are required according to Eq. (29) for the first input random variable in component 1 of part 2 and one for others. Thus, the integration grid can be drawn in Fig. 3. By the same token, the integration grids of parts 4–7 are on a plane. For the sake of concision, the schematic view for integration grid of component 1 in part 4 is given, as shown in Fig. 4. Finally, substituting the integration grids and weights into Eq. (27), the raw statistical moments are obtained, from which standard deviation, skewness and kurtosis can be obtained incidentally by the following equations

9

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247 Table 1 Set H (q, N) in for the Example 1. Part Part Part Part Part Part Part Part

1 2 3 4 5 6 7

Component 1

Component 2

Component 3

Component 4

Component 5

(1,1,1,1,1) (2,1,1,1,1) (3,1,1,1,1) (2,2,1,1,1) (1,2,2,1,1) (1,1,2,2,1) (1,1,1,2,2)

– (1,2,1,1,1) (1,3,1,1,1) (2,1,2,1,1) (1,2,1,2,1) (1,1,2,1,2) –

– (1,1,2,1,1) (1,1,3,1,1) (2,1,1,2,1) (1,2,1,1,2) – –

– (1,1,1,2,1) (1,1,1,3,1) (2,1,1,1,2) – – –

– (1,1,1,1,2) (1,1,1,1,3) – – – –

Fig. 2. Part 1.

r¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi EðG2 Þ
EðGÞ 3

sk ¼ EðG ku ¼

Þ
3EðG2 ÞEðGÞþ2½EðGÞ3

ð37Þ

r3G

EðG4 Þ
4EðG3 ÞEðGÞþ6EðG2 Þ½EðGÞ2 þ3½EðGÞ4

r4G

where r, sk and ku represent standard deviation, skewness and kurtosis, respectively. Moreover, the function evaluations (FE) can be computed by 1 + (3–1)  5+(5–1)  5+(9–5)  C 25 = 71. From the results listed in Table 2, it can be seen that the UDRM needs only 31 FEs and provides the satisfactory evaluations for mean, standard deviation and kurtosis, but it gives an unreliable result for skewness with the relative error of

Fig. 3. Component 1 of part 2.

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W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Fig. 4. Component 1 of part 4.

Table 2 Comparisons of statistical moments for Example 1. Method

FE

Mean

Standard deviation

Skewness

Kurtosis

MCS UDRM Relative error i-SGNI Relative error

106 31 – 71 –

1604.0211 1604.9000 0.05% 1604.9000 0.05%

790.0054 789.4977 0.06% 789.9693 0.00%

0.0301
0.0017 105.72% 0.0315 4.83%

3.0084 3.0001 0.28% 3.0119 0.12%

105.72%. The proposed i-SGNI can give accurate statistical moment estimations without too much extra computational cost, whose maximum relative error is 4.83%. 5.1.2. Reliability analysis The estimated PDFs of the transformed and original limit state functions from the proposed method are compared with the reference solutions in Fig. 5, with close agreements with those of MCS. Parameters of the nonlinear transformation and the estimated PDF are given in Table 3. From the results of reliability analysis listed in Table 4, it can be concluded that the proposed method outperforms the other methods in terms of accuracy. Through further comparison between the UDRM + i-MEM and the proposed method, it can be seen that the accuracy of reliability analysis is improved obviously due to the accurate statistical moment evaluation by i-SGNI. From the results of the proposed method and i-SGNI + MEM, it can be concluded that i-MEM can be used to further promote the accuracy for reliability analysis.

(a) PDF of transformed limit state function

(b) PDF of original limit state function

Fig. 5. The PDFs for Example 1.

11

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247 Table 3 Parameters of estimated PDF in Example 1.

C

Scale parameter k

k0

k1

k2

k3

k4


2.4275

7.1442

688.86


4034.70

9310.00


10513.68

5029.97

Table 4 Prediction of failure probability for Example 1. Method

MCS

UDRM + MEM

i-SGNI + MEM

UDRM + i-MEM

Proposed method

Pf Relative error

0.0202 –

0.0211 4.46%

0.0204 0.99%

0.0210 3.96%

0.0203 0.50%

5.2. Example 2 This example considers a reinforced concrete beam under bending moment [15], with the limit state function as

gðXÞ ¼ X 1 X 2 X 3


X 21 X 22 X 4
X7 X5X6

ð38Þ

where the statistical information and descriptions of the input random variables are shown in Table 5 and Fig. 6. 5.2.1. Statistical moment evaluation Due to the complexity of the limit state function and input random variables, UDRM gives the statistical moment evaluation with large errors. As shown in Table 6, the errors of the skewness and kurtosis estimated by UDRM are 67.88% and 16.82%, which demonstrates that UDRM is incompetent for accurate evaluation of high-order statistical moments with complex limit state function. Thought i-SGNI pays more computational cost, the accuracy of the results is improved considerably. Therefore, i-SGNI is a good choice for statistical moment evaluation when considering the balance between accuracy and efficiency, especially for the cases with complex limit state function. 5.2.2. Reliability analysis The estimated PDFs by the proposed method are compared with those by MCS in Fig. 7, from which the proposed method provides accurate approximations for the unknown PDFs of the transformed and original limit state functions. The failure

Table 5 Statistical characteristics of the random variables for Example 2. Variable

Description

Distribution

Mean

C.O.V

X1 X2 X3 X4 X5 X6 X7

Area of reinforcement Yield stress of reinforcement Effective depth of reinforcement Stress–strain factor of concrete Compressive strength of concrete Width of section Applied bending moment

Lognormal Lognormal Lognormal Normal Weibull Normal Lognormal

1260 mm2 300 N/mm2 770 mm2 0.35 25 N/mm2 200 mm 100 kN∙m

0.15 0.15 0.10 0.25 0.20 0.10 0.20

Note: C.O.V is the abbreviation of coefficient of variation in this paper.

(a) Cross section of the beam

(b) Schematic view of ultimate stress distribution

Fig. 6. State of ultimate for reinforced concrete beam section.

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W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Table 6 Comparisons of statistical moments for Example 2. Method

FE

Mean

Standard deviation

Skewness

Kurtosis

MCS UDRM Relative error i-SGNI Relative error

106 43 – 127 –

180.0400 179.9981 0.02% 179.9616 0.04%

67.7877 67.4675 0.47% 67.7147 0.11%

0.5900 0.1895 67.88% 0.5878 0.38%

3.7147 3.0900 16.82% 3.5659 4.01%

(a) PDF of transformed limit state function

(b) PDF of original limit state function

Fig. 7. The PDFs for Example 2.

probability from different methods are listed in Table 7, from which it can be seen that the proposed method gives accurate result with the relative error of 0.87%. However, the results of UDRM + MEM and i-SGNI + MEM are nearly unauthentic, with the relative errors of 100% and 51.17%. The main reason for the inaccuracy is that the original PDF, shown in Fig. 7(b), has a heavy tail, which is difficultly captured by the MEM with enough accuracy. The advantage of the proposed i-MEM compared with traditional MEM is that the method can obtain more statistical information to approximate the tail of the unknown PDF accurately with the help of the nonlinear mapping, which will be discussed later. In this case, the UDRM + i-MEM fails to converge due to the numerical singularity. Therefore, the proposed method can provide accurate prediction for failure probability with benign numerical stability. The optimal parameters of the nonlinear mapping and the estimated PDF are listed in Table 8, which can assist the reader to verify the results. 5.3. Example 3 As shown in Fig. 8, the third example considers a column under the action of axial compressive load [52]. Because the column is sufficiently slender, it may fail due to buckling and the limit state function is written as

Table 7 Prediction of failure probability for Example 2. Method

MCS

UDRM + MEM

i-SGNI + MEM

UDRM + i-MEM

Proposed method

Pf Relative error

3.83  10
4 –

0.00 100%

1.87  10
4 51.17%

- (Without convergence) –

3.80  10
4 0.87%

Table 8 Parameters of estimated PDF in Example 2.

C

Scale parameter k

k0

k1

k2

k3

k4


1.5762

1.6998

201.174


973.210

1821.936


1642.594

620.229

13

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Fig. 8. Column with pinned ends.

LB ¼

p2 E n p h 2

L

64

ðD þ T Þ4
D4

io


P

ð39Þ

where the statistical information and descriptions of the input random variables are listed in Table 9. 5.3.1. Statistical moment evaluation Because the set H (q, N) of this example is the same with that of Example 1, the FEs are identical in the two cases. However, the computational accuracy of skewness and kurtosis from UDRM decreases sharply since the limit state function here is more complex than Eq. (36). As presented in Table 10, the errors of skewness and kurtosis calculated by UDRM reach 52.72% and 11.58%, respectively. Alternatively, i-SGNI can provide satisfactory results despite the limit state function is complex. It can be concluded that i-SGNI provides an appropriate choice for the statistical moment evaluation of complex limit state function. 5.3.2. Reliability analysis The comparisons of PDF approximations shown in Fig. 9 illustrate that the proposed method fits the reference solutions quite well though the original PDF has heavy and long tails. The failure probability from each method listed in Table 11

Table 9 Statistical characteristics of the random variables for Example 3. Variable

Description

Distribution

Mean

Standard deviation

E D T L P

Elastic modulus of material Dimension of section Dimension of section Height of the column Axial load

Lognormal Lognormal Lognormal Lognormal Gumbel

203 Gpa 23.5 mm 6 mm 2500 mm 4000 N

30.45 Gpa 0.5 mm 0.4 mm 50 mm 200 N

Table 10 Comparisons of statistical moments for Example 3. Method

FE

Mean

Standard deviation

Skewness

Kurtosis

MCS UDRM Relative error i-SGNI Relative error

106 31 – 71 –

3147.4946 3147.3334 0.01% 3147.3619 0.00%

1369.0002 1357.7839 0.82% 1368.5215 0.03%

0.5611 0.2653 52.72% 0.5582 0.52%

3.5655 3.1526 11.58% 3.4480 3.29%

14

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

(a) PDF of transformed limit state function

(b) PDF of original limit state function

Fig. 9. The PDFs for Example 3.

Table 11 Prediction of failure probability for Example 3. Method

MCS

UDRM + MEM

i-SGNI + MEM

UDRM + i-MEM

Proposed method

Pf Relative error

0.0019 –

0.0055 189.47%

0.0016 15.79%

0.0045 136.84%

0.0020 5.26%

shows that the proposed method outperforms the other methods in terms of accuracy, with the relative error of 5.26%. By contrast, the relative error of the result from i-SGNI + MEM is 15.79%. It can be concluded that the accuracy of failure probability prediction can be improved greatly by i-MEM since the proposed method can capture the information of the tail of the unknown PDF accurately. The results of UDRM + MEM and UDRM + i-MEM are poor with the relative errors of 189.47% and 136.84% due to the inaccurate statistical moment evaluation. Therefore, i-SGNI is required for MEM based reliability analysis when considering the tradeoff between the accuracy and efficiency. For verification, the involved parameters in this example are given in Table 12. 5.4. Example 4 This is a highly nonlinear example [53] with the limit state function expressed as:

Y ¼ 1
ðy
6Þ2
ðy
6Þ3 þ 0:6ðy
6Þ4
z

ð40Þ

where y ¼ 0:9063x1 þ 0:4226x2 and z ¼ 0:4226x1
0:9063x2 , in which x1 and x2 follow the normal distributions with different mean value m1 = 4.5580, m2 = 1.9645, and the same standard deviation, r1 = r2 = 0.3, respectively. 5.4.1. Statistical moment evaluation The first-four statistical moments of the limit state function from the proposed method and UDRM are presented in Table 13, from which it can be seen that the errors of UDRM are larger than those of i-SGNI, especially for skewness and kurtosis. In terms of efficiency, the FEs of the two methods are 13 and 17, respectively, therefore the i-SGNI can improve the computational accuracy without too much extra cost. 5.4.2. Reliability analysis From the reference solutions from MCS in Fig. 10, it can be seen that the PDF of the real limit state function has long and heavy tails, which is difficult to fit due to limited statistical information. Compared with the reference solutions, the

Table 12 Parameters of estimated PDF in Example 3.

C

Scale parameter k

k0

k1

k2

k3

k4


1.7244

2.7338

566.812


3227.17

6928.671


6733.654

2526.470

15

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247 Table 13 Comparisons of statistical moments for Example 4. Method

FE

Mean

Standard deviation

Skewness

Kurtosis

MCS UDRM Relative error i-SGNI Relative error

106 13 – 17 –

2.1508 2.1456 0.24% 2.1499 0.04%

1.6940 1.5658 7.56% 1.6913 0.16%

2.5121 2.1229 15.49% 2.4118 3.99%

12.8087 10.6588 16.78% 11.6922 8.71%

(a) PDF of transformed limit state function

(b) PDF of original limit state function

Fig. 10. The PDFs for Example 4.

estimated PDFs by the proposed method are of good agreement with the real ones. The failure probability from different methods are listed in Table 14, from which it can be concluded that the proposed method gives accurate result with the relative error of 2.71%. However, the results of the other methods are inaccurate, with the relative errors of 2940.24%, 50.52% and 33.53%, because the information of the tails is not captured accurately. It can be concluded that the proposed i-MEM can fit the tail of the unknown PDF better than traditional MEM owning to the nonlinear mapping, which will be discussed in detail in Section 6. For verification, the optimal parameters of the nonlinear mapping and the estimated PDF are given in Table 15. 5.5. Example 5 Finally, an aerospace engineering example is considered for demonstrating the applicability of the proposed method for the limit state function without the explicit expression. A typical curvilinearly stiffened panel with multiple cutouts was designed, manufactured and tested by NASA Langley Research Center [54]. Based on previous work, a hybrid descent mean value approach was proposed by Keshtegara and Hao [55,56] for optimizing the weight of the curvilinearly stiffened panel. As shown in Fig. 11, the dimensions of this panel are 609.6  711.2 mm, which is representative of a large wing engine pylon rib. There are four curvilinear stiffeners and two circular cutouts, and the cutouts are reinforced by the thick circular ring

Table 14 Prediction of failure probability for Example 4. Method

MCS

UDRM + MEM

i-SGNI + MEM

UDRM + i-MEM

Proposed method

Pf Relative error

6.71  10
4 –

2.04  10
2 2940.24%

1.01  10
3 50.52%

4.46  10
4 33.53%

6.86  10
4 2.71%

Table 15 Parameters of estimated PDF in Example 4.

C

Scale parameter k

k0

k1

k2

k3

k4

0.6169

1.9651


1.4155

0.7537

0.1901


0.3568

0.1195

16

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Fig. 11. Schematic view of aircraft curvilinearly stiffened shell problem.

Table 16 Statistical characteristics of the random variables for Example 5. Variable

Description

Distribution

Mean

Standard deviation

E

Young’s modulus Poisson’s ratio Skin thickness Stiffener height Stiffener thickness Equivalent external load

Normal Normal Normal Normal Normal Normal

72,504 Mpa 0.33 2.36 mm 18 mm 1.36 mm 35750 N

2100 Mpa 0.0099 0.118 mm 0.9 mm 0.068 mm 1787.5 N

t tsk hs tst P

areas. Specifically, the thicknesses of large and small reinforced rings are 5.9 mm and 4.7 mm, respectively. In this example, the equivalent external load, Young’s modulus, Poisson’s ratio, the stiffener height, thickness and the skin thickness are considered as random variables, whose uncertainty properties are listed in Table 16. The limit state function here can be expressed as

U ¼ CB
P

ð41Þ

where CB is the critical buckling capacity predicted by finite element analysis and P is the equivalent external load. Thus, if U < 0, the structural failure state is achieved.

5.5.1. Statistical moment evaluation The first-four statistical moments of the limit state function are calculated by UDRM and i-SGNI, as shown in Table 17. For the implicit limit state function, UDRM is still not capable of estimating the skewness accurately. Moreover, in terms of estimating the mean value, the performance of UDRM is also reduced, compared with the previous examples. Alternatively, the

Table 17 Comparisons of statistical moments for Example 5. Method

FE

Mean

Standard deviation

Skewness

Kurtosis

MCS UDRM Relative error i-SGNI Relative error

105 37 – 97 –

14705.2890 14358.6180 2.36% 14688.5897 0.11%

6151.4461 6064.5517 1.41% 6132.7952 0.30%

0.1629
0.4384 370.65% 0.1731 6.26%

3.0577 3.1266 2.25% 3.1454 2.87%

17

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247 Table 18 Parameters of estimated PDF in Example 5.

C

Scale parameter k

k0

k1

k2

k3

k4


1.7384

2.4632

8.785

174.920


748.658

855.381


240.552

(a) PDF of transformed limit state function

(b) PDF of original limit state function

Fig. 12. The PDFs for Example 5.

Table 19 Prediction of failure probability for Example 5. Method

MCS

UDRM + MEM

i-SGNI + MEM

UDRM + i-MEM

Proposed method

Pf Relative error

0.0063 –

0.0173 174.60%

0.0057 9.52%

0 100%

0.0065 3.18%

computational errors by i-SGNI are satisfactory. Hence, whether the function is explicit or not, it can be seen that i-SGNI is a promising method for statistical moment estimation.

5.5.2. Reliability analysis The optimal parameters are provided in Table 18, based on which the predictions of the PDFs by the proposed method are depicted in Fig. 12. Obviously, the PDF curves are considerably close to those by MCS. As shown in Table 19, the estimated failure probability by the proposed method is 0.0065, with the relative error of 3.18%, which is closest to the result of MCS. The relative errors from UDRM + MEM and UDRM + i-MEM are 174.60% and 100% due to the inaccurate statistical moment evaluation. Owing to the nonlinear mapping, the proposed method is more accurate than i-SGNI + MEM, with the relative errors of 3.18% and 9.52%, respectively. It can be concluded that the proposed method is a proper choice for the prediction of failure probability when the compromise between accuracy and efficiency is required.

6. Discussion The proposed method for reliability analysis is composed of i-MEM and i-SGNI, which are used to estimate the failure probability and the statistical moments, respectively. In the proposed i-MEM, the normalization in Eq. (12) and the scale parameter, k, of the nonlinear mapping are the key, by which i-MEM bypasses the shortcomings of traditional MEM. The normalization plays an important role in adjusting the magnitude of the limit state function and the shape of the estimated PDF curve, which is completed by the function value at the mean value of the input random variables. The scale parameter is an undetermined parameter solved by the optimal algorithm to obtain the accurate failure probability. The selection of k influences the accuracy of the predicted failure probability significantly. Herein, results of failure probability for Example 3 obtained by different k are given to illustrate the importance of this parameter. As show in Table 20, the result of the proposed method is closer to that of MCS with the scale parameter, k, approaching the optimum. The PDFs of different cases are

18

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Table 20 Predicted failure probability from different k for Example 3. Method

MCS

Proposed method

k Entropy Pf Relative error

– – 0.0019 –

1
1.2802 0.0009 52.63%

(a) PDF of transformed limit state function

2
1.5154 0.0021 10.53%

2.7338
1.7244 0.0020 5.26%

(b) PDF of original limit state function

Fig. 13. The PDFs obtained from k = 1 for Example 3.

(a) PDF of transformed limit state function

(b) PDF of original limit state function

Fig. 14. The PDFs obtained from k = 2 for Example 3.

also drawn in Figs. 13 and 14. It can be concluded that the accurate failure probability can be calculated only if the optimal scale parameter is obtained. From the comparisons between the results from the proposed method and i-SGNI + MEM, it can also be concluded that the proposed i-MEM outperforms traditional MEM in terms of accuracy. Herein, we give a further discussion about the advantage. Consider the Taylor series expansions of Eq. (11)

!      2n
1 ! n 1 2 Y 1 2X 1 Y n
1 þ1 ¼ þ1 T¼ arctan ð
1Þ 2 p kYðlÞ 2 p i¼1 2n
1 kYðlÞ

ð42Þ

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W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

Then, we have

" ! 2n
1 #! n 1 2X 1 Y n
1 E E½T ¼ ð
1Þ þ1 2 p i¼1 2n
1 kYðlÞ

ð43Þ

from which it can be seen that the expectation of T is the weighted summation of the massive statistical moments of Y. Therefore, a couple of statistical moments of T contain a large amount of statistical information about Y. In other words, the PDF of T or Y estimated by the proposed method represents the probability information of the real mechanical system better than that of Y by MEM. For structural reliability analysis, the high-order statistical moments influence the capture of the tail of real PDF, which observably determines the accuracy of failure probability prediction. Theoretically, since more information of high-order statistical moments is contained in the statistical moments of the transformed limit state function, the proposed i-MEM can capture the tail of the real PDF and estimate the failure probability better than the traditional MEM. For example, the reliability analysis results of Examples 3 and 4 considering higher-order statistical moments are listed in Tables 21 and 22, where MEM (4) and MEM (6) are the results obtained by the traditional MEM based on the first-four and the first-six statistical moments, respectively. In this part, the statistical moments are estimated by MCS in order to eliminate the influence of calculation error of statistical moment. From the results, traditional MEM requires more high-order statistical moments for accurate results in Example 3. However, the numerical singularity is a thorny problem due to high-order statistical moments [57], for example, the calculation in Example 2 failed in convergence because of the numerical singularity. In Example 4, the first-six statistical moments are employed, but the accuracy of the predicted failure probability is still unacceptable, with the error of 78.84%, because the PDF of the limit state function has long and heavy tails which have a great influence on the problems of low failure probability. Moreover, it is usually difficult to obtain the high-order statistical moments accurately. By contrast, owning to the nonlinear mapping, the proposed i-MEM estimates the failure probability accurately in the both cases. Estimating the statistical moments accurately is also indispensable in the proposed method. For complex limit state functions, especially implicit (Example 5), the estimation of their statistical moments is a thorny problem, since the tradeoff between the accuracy and efficiency should be considered. As shown in this paper, UDRM is incompetent, therefore iSGNI is proposed. Actually, there exist other methods for calculating the statistical moments accurately, but most of them are inefficient. For example, bivariate dimension reduction method (BDRM) is also a method for statistical moment evaluation. Herein, the results of statistical moment estimation for Example 3 by BDRM with five integration points is shown in Table 23. It can be seen that the BDRM pays more computational cost than i-SGNI, but the accuracy is not improved further. Therefore, i-SGNI is a better choice for estimating the statistical moments accurately and efficiently. In summary, the proposed method provides the approach to obtain more statistical information of limit state function and gives the feasible algorithm for the estimations of failure probability and statistical moment. All the results presented in this paper demonstrate that the proposed method, with the combination of i-SGNI and i-MEM, is accurate and stable for reliability analysis with acceptable efficiency.

Table 21 Prediction of failure probability with higher-order statistical moments for Example 3. Method

MCS

MEM (4)

MEM (6)

Proposed method

Pf Relative error

0.0019 –

0.0022 15.79%

0.0018 5.26%

0.0020 5.26%

Table 22 Prediction of failure probability with higher-order statistical moments for Example 4. Method

MCS

MEM (4)

MEM (6)

Proposed method

Pf Relative error

6.71  10
4 –

2.90  10
2 4221.91%

1.20  10
3 78.84%

6.86  10
4 2.71%

Table 23 Statistical moment evaluation by BDRM and i-SGNI for Example 3. Method

FE

Mean

Standard deviation

Skewness

Kurtosis

BDRM Relative error i-SGNI Relative error

261 – 71 –

3147.4113 0.00% 3147.3619 0.00%

1368.5266 0.03% 1368.5215 0.03%

0.5495 2.11% 0.5582 0.52%

3.4515 3.30% 3.4480 3.29%

20

W. He et al. / Mechanical Systems and Signal Processing 133 (2019) 106247

7. Concluding remarks This paper proposes a novel method for structural reliability analysis accurately and efficiently on the basis of the improved maximum entropy method (i-MEM) and the improved sparse grid numerical integration (i-SGNI). Primarily, the statistical moments are obtained by the proposed i-SGNI. Afterwards, the proposed i-MEM is used to approximate the PDF of the limit state function, based on which the failure probability is calculated by the numerical integration on the failure domain. Four numerical examples and one engineering example are presented to illustrate the validity of the proposed method, which are also calculated by some other methods for comparison. Some discussions are presented to confirm the advantages of the proposed method further, and the following conclusions can be obtained: (1) Through the nonlinear mapping, the truncation error due to numerical integration can be avoided and a couple of statistical moments of transformed limit state function can provide more information for the PDF prediction of the limit state function of the real mechanical system. Thus, the problems caused by high-order statistical moments can be avoided and information of the tail of the real PDF and structural failure probability can be captured accurately. (2) Owing to the normalization and optimal scale parameter in the nonlinear mapping, the proposed method overcomes the numerical problem caused by the magnitude of the limit state function, therefore it has good accuracy and stability. (3) The proposed i-SGNI can estimate the statistical moments of limit state function accurately with acceptable computational cost. The univariate dimension-reduction method (UDRM) may fail to provide accurate moment estimation when the limit state function is complex or implicit. Although the bivariate dimension reduction method (BDRM) may have the similar accuracy with i-SGNI, it pays more computational cost than i-SGNI. (4) Due to inaccurate statistical moment estimation, UDRM based MEM and UDRM based i-MEM is incompetent for accurate failure probability prediction. In summary, the proposed method is considerably promising for statistical moment estimation and reliability analysis when considering the balance between accuracy and efficiency. Acknowledgments The support of the National Basic Research Program of China (Grant No. 2014CB046506) and the National Natural Science Foundation of China (Grant No. 11872142) are greatly appreciated. References [1] H.N. Pollack, Uncertain science... uncertain world, Cambridge University Press, 2005. [2] J. Liu, H. Liu, C. Jiang, et al, A new measurement for structural uncertainty propagation based on pseudo-probability distribution, Appl. Math. Model. 63 (2018) 744–760. [3] D. Zhang, X. Han, C. Jiang, et al, The interval PHI2 analysis method for time-dependent reliability, Sci. Sinica Phys., Mech. Astron. 45 (5) (2015) 54601. [4] G. Li, B. Li, H. Hu, A novel first–order reliability method based on performance measure approach for highly nonlinear problems, Struct. Multidiscip. Optim. 57 (4) (2018) 1593–1610. [5] Z. Meng, H. Zhou, H. Hu, et al, Enhanced sequential approximate programming using second order reliability method for accurate and efficient structural reliability-based design optimization, Appl. Math. Model. (2018). [6] J. Xu, F. Kong, An adaptive cubature formula for efficient reliability assessment of nonlinear structural dynamic systems, Mech. Syst. Sig. Process. 104 (2018) 449–464. [7] R.Y. Rubinstein, D.P. Kroese, Simulation and the Monte Carlo method, John Wiley & Sons, 2016. [8] H.R. Maier, B.J. Lence, B.A. Tolson, et al, First-order reliability method for estimating reliability, vulnerability, and resilience, Water Resour. Res. 37 (3) (2001) 779–790. [9] S. Engelund, R. Rackwitz, A benchmark study on importance sampling techniques in structural reliability, Struct. Saf. 12 (4) (1993) 255–276. [10] R.E. Melchers, Importance sampling in structural systems, Struct. Saf. 6 (1) (1989) 3–10. [11] G.I. Schueller, Efficient Monte Carlo simulation procedures in structural uncertainty and reliability analysis-recent advances, Struct. Eng. Mech. 32 (1) (2009) 1–20. [12] S.K. Au, J.L. Beck, Estimation of small failure probabilities in high dimensions by subset simulation, Probab. Eng. Mech. 16 (4) (2001) 263–277. [13] P.S. Koutsourelakis, H.J. Pradlwarter, G.I. Schuëller, Reliability of structures in high dimensions, part I: algorithms and applications, Probab. Eng. Mech. 19 (4) (2004) 409–417. [14] P.S. Koutsourelakis, Reliability of structures in high dimensions. Part II. Theoretical validation, Probab. Eng. Mech. 19 (4) (2004) 419–423. [15] J. Xu, C. Dang, A new bivariate dimension reduction method for efficient structural reliability analysis, Mech. Syst. Sig. Process. 115 (2019) 281–300. [16] O. Ditlevsen, H.O. Madsen, Structural Reliability Methods, Wiley, New York, 1996. [17] Z. Meng, Y. Pu, H. Zhou, Adaptive stability transformation method of chaos control for first order reliability method, Eng. Comput. 34 (4) (2018) 671– 683. [18] Z. Meng, H. Zhou, New target performance approach for a super parametric convex model of non-probabilistic reliability-based design optimization, Comput. Methods Appl. Mech. Eng. 339 (2018) 644–662. [19] Z. Meng, H. Hu, H. Zhou, Super parametric convex model and its application for non-probabilistic reliability-based design optimization, Appl. Math. Model. 55 (2018) 354–370. [20] D. Zhang, X. Han, C. Jiang, et al, Time-dependent reliability analysis through response surface method, J. Mech. Des. 139 (4) (2017) 041404. [21] W. Jian, S. Zhili, Y. Qiang, et al, Two accuracy measures of the Kriging model for structural reliability analysis, Reliab. Eng. Syst. Saf. 167 (2017) 494– 505. [22] Y.G. Zhao, T. Ono, Moment methods for structural reliability, Struct. Saf. 23 (1) (2001) 47–75. [23] E. Acar, M. Rais-Rohani, C.D. Eamon, Reliability estimation using univariate dimension reduction and extended generalised lambda distribution, Int. J. Reliab. Saf. 4 (2–3) (2010) 166–187.

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[24] H.Y. Kang, B.M. Kwak, Application of maximum entropy principle for reliability-based design optimization, Struct. Multidiscip. Optim. 38 (4) (2009) 331–346. [25] Z. Xi, C. Hu, B.D. Youn, A comparative study of probability estimation methods for reliability analysis, Struct. Multidiscip. Optim. 45 (1) (2012) 33–52. [26] J. Liu, X. Meng, C. Xu, et al, Forward and inverse structural uncertainty propagations under stochastic variables with arbitrary probability distributions, Comput. Methods Appl. Mech. Eng. 342 (2018) 287–320. [27] B.D. Youn, Z. Xi, P. Wang, Eigenvector dimension reduction (EDR) method for sensitivity-free probability analysis, Struct. Multidiscip. Optim. 37 (1) (2008) 13–28. [28] N.L. Johnson, S. Kotz, N. Balakrishnan, Continuous Univariate Distributions, 2nd Edition., 1995. [29] B. Huang, X. Du, Uncertainty analysis by dimension reduction integration and saddlepoint approximations, J. Mech. Des. 128 (1) (2006) 26–33. [30] L.R. Mead, N. Papanicolaou, Maximum entropy in the problem of moments, J. Math. Phys. 25 (8) (1984) 2404–2417. [31] E.T. Jaynes, Information theory and statistical mechanics, Phys. Rev. 106 (4) (1957) 620. [32] X. Zhang, Efficient computational methods for structural reliability and global sensitivity analyses. 2013. [33] J. Shore, R. Johnson, Axiomatic derivation of the principle of maximum entropy and the principle of minimum cross-entropy, IEEE Trans. Inf. Theory 26 (1) (1980) 26–37. [34] R.M. Mnatsakanov, Hausdorff moment problem: reconstruction of distributions, Stat. Probab. Lett. 78 (12) (2008) 1612–1618. [35] R.M. Mnatsakanov, Hausdorff moment problem: reconstruction of probability density functions, Stat. Probab. Lett. 78 (13) (2008) 1869–1877. [36] A. Tagliani, Maximum entropy in the Hamburger moments problem, J. Math. Phys. 35 (9) (1994) 5087–5096. [37] A. Tagliani, Hausdorff moment problem and maximum entropy: a unified approach, Appl. Math. Comput. 105 (2–3) (1999) 291–305. [38] G. Li, C. Zhou, Y. Zeng, et al, New maximum entropy-based algorithm for structural design optimization, Appl. Math. Model. 66 (2019) 26–40. [39] H. Xu, S. Rahman, A generalized dimension-reduction method for multidimensional integration in stochastic mechanics, Int. J. Numer. Meth. Eng. 61 (12) (2004) 1992–2019. [40] S. Rahman, H. Xu, A univariate dimension-reduction method for multi-dimensional integration in stochastic mechanics, Probab. Eng. Mech. 19 (4) (2004) 393–408. [41] J. He, S. Gao, J. Gong, A sparse grid stochastic collocation method for structural reliability analysis, Struct. Saf. 51 (2014) 29–34. [42] S. Yu, Z. Wang, D. Meng, Time-variant reliability assessment for multiple failure modes and temporal parameters, Struct. Multidiscip. Optim. (2018) 1– 13. [43] H. Xu, S. Rahman, Decomposition methods for structural reliability analysis, Probab. Eng. Mech. 20 (3) (2005) 239–250. [44] H.K. Kevasan, J.N. Kapur, Entropy optimization principles with applications, Entropy Energy Dissipation Water Resour. (1992). [45] G. Li, W. He, Y. Zeng, An improved maximum entropy method via fractional moments with Laplace transform for reliability analysis, Struct. Multidiscip. Optim. (2018) 1–20. [46] D.P. Bertsekas, A. Scientific, Convex Optimization Algorithms, Athena Scientific, Belmont, 2015. [47] A. Tagliani, Hausdorff moment problem and fractional moments: a simplified procedure, Appl. Math. Comput. 218 (8) (2011) 4423–4432. [48] G. Li, K. Zhang, A combined reliability analysis approach with dimension reduction method and maximum entropy method, Struct. Multidiscip. Optim. 43 (1) (2011) 121–134. [49] S.A. Smolyak, Quadrature and interpolation formulas for tensor products of certain classes of functions, Doklady Akademii Nauk. Russ. Acad. Sci. 148 (5) (1963) 1042–1045. [50] B.H. Ju, B.C. Lee, Improved moment-based quadrature rule and its application to reliability-based design optimization, J. Mech. Sci. Technol. 21 (8) (2007) 1162–1171. [51] J. Xu, C. Dang, F. Kong, Efficient reliability analysis of structures with the rotational quasi-symmetric point-and the maximum entropy methods, Mech. Syst. Sig. Process. 95 (2017) 58–76. [52] X. Huang, Y. Zhang, Reliability–sensitivity analysis using dimension reduction methods and saddlepoint approximations, Int. J. Numer. Meth. Eng. 93 (8) (2013) 857–886. [53] Y.H. Sung, B.M. Kwak, Reliability bound based on the maximum entropy principle with respect to the first truncated moment, J. Mech. Sci. Technol. 24 (9) (2010) 1891–1900. [54] D. Havens, S. Shiyekar, A. Norris, et al. Design, optimization, and evaluation of integrally-stiffened al-2139 panel with curved stiffeners. 2011. [55] B. Keshtegar, P. Hao, A hybrid descent mean value for accurate and efficient performance measure approach of reliability-based design optimization, Comput. Methods Appl. Mech. Eng. 336 (2018) 237–259. [56] P. Hao, Y. Wang, C. Liu, et al, Hierarchical nondeterministic optimization of curvilinearly stiffened panel with multicutouts, AIAA Journal 56 (10) (2018) 4180–4194. [57] P.N. Inverardi, A. Petri, G. Pontuale, et al, Stieltjes moment problem via fractional moments, Appl. Math. Comput. 166 (3) (2005) 664–677.