AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability

Structural Safety 82 (2020) 101891

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Structural Safety journal homepage: www.elsevier.com/locate/strusafe

AK-ARBIS: An improved AK-MCS based on the adaptive radial-based importance sampling for small failure probability

T

Wanying Yuna,b, Zhenzhou Lua, , Xian Jiangc, Leigang Zhangd, Pengfei Heb, ⁎

⁎

a

School of Aeronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, China School of Aerospace Engineering and Applied Mechanics, Tongji University, Shanghai 200092, China c Aircraft Flight Test Technology Institute, Chinese Flight Test Establishment, Xi’an, Shaanxi 710089, China d China Academy of Launch Vehicle Technology, Beijing 100076, China b

ARTICLE INFO

ABSTRACT

Keywords: Adaptive radial-based importance sampling Kriging model Candidate sampling pool Training time Small failure probability

The pivotal problem in reliability analysis is how to use a smaller number of model evaluations to get more accurate failure probabilities. To achieve this aim, an iterative method based on the Monte Carlo simulation and the adaptive Kriging (AK) model (abbreviated as AK-MCS) has been proposed in 2011 by Echard et al. But for small failure probability, the number of the candidate points is extremely large for convergent solution. These points need to be evaluated by the current Kriging model to select the best next sample for updating the Kriging model in AK-MCS method, and the large candidate points will make the adaptive updating process of Kriging model much more time-consuming. Therefore, to improve the applicability of the AK-MCS method for small failure probability, the adaptive radial-based importance sampling (ARBIS) is employed to reduce the number of candidate points in the AK-MCS method, and an ARBIS combined with AK model method (abbreviated as AKARBIS) is proposed. The idea of the ARBIS is adaptively to find the optimal -sphere, i.e., the largest sphere of the safe domain, and then samples inside the optimal -sphere is directly recognized as safety and do not need to call the true limit state function to judge their states (safe or failed). During the adaptive process of finding the optimal -sphere, the Kriging model is ceaselessly updated layer after layer based on the U learning scheme in each sampling pool which only contains the samples between the current spherical rings. The updating process of Kriging model stops until the optimal -sphere is adaptively found and the convergent condition is satisfied. By finding the optimal -sphere, the total number of candidate samples is reduced which only includes the samples outside the optimal -sphere. Besides, the whole candidate sampling pool is partitioned into several sub-candidate sampling pool sequentially. The proposed method not only inherits the advantage of the AK-MCS but also reduces the reliability analysis time of the AK-MCS from two aspects. One is the size reduction of the candidate sampling pool, the other is the reduction of the actual limit state function evaluations because the sampling points locating inside the adaptively searched optimal -sphere do not need to participate in the training process. By analyzing a highly nonlinear numerical case, a non-linear oscillator system, a simplified wing box structural model, an aero-engine turbine disk and a planar ten-bar structure, the effectiveness and the accuracy of the proposed AK-ARBIS method for estimating the small failure probability are verified.

1. Introduction The key to reliability analysis is to solve the following integral,

Pf = Pr{g (X )

0} =

f g (X ) 0 X

( x ) dx

(1)

where Pr{·} is the probability operation, X is the vector of random input variables, g (X ) is the limit state function, fX (x ) is the joint probability density function (PDF) of the random input variables X and

F = {X : g (X ) 0} is the failure domain. At present, the reliability analysis methods can be mainly divided into four categories. The first category is the approximate analytical method such as the first-order reliability method (FORM) [1,2] and the second-order reliability method [3,4]. The second category is the moment based method such as the first-order third-moment based method [5], the fourth moment based method [6–9] and the fractional moment based moment [10,11]. The third category is the sampling-based

Corresponding authors. E-mail addresses: [email protected] (W. Yun), [email protected] (Z. Lu), [email protected] (X. Jiang), [email protected] (L. Zhang), [email protected] (P. He). ⁎

https://doi.org/10.1016/j.strusafe.2019.101891 Received 3 October 2018; Received in revised form 4 September 2019; Accepted 5 September 2019 0167-4730/ © 2019 Elsevier Ltd. All rights reserved.

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method such as the Monte Carlo simulation (MCS) method, importance sampling (IS) method [12–14], modified importance sampling (MIS) method [15], adaptive radial-based importance sampling (ARBIS) method [16], subset simulation (SS) method [17–19], line sampling (LS) method [20,21], directional simulation method [22], etc. The fourth category is the metamodel based method. The efficient global reliability Analysis method [23] and the active learning reliability method combining the adaptive Kriging (AK) and the MCS (abbreviated as AK-MCS) [24] were proposed earlier. Many research orientations of the AK based reliability analysis methods are studied in recent years. The research orientations are mainly divided into four categories. The first category aims at refining the learning function. Zhang et al. [25] proposed a learning function based on the probabilistic classification function and the most probable region. Lv et al. [26] proposed a new learning function based on information entropy. Wang et al. [27] proposed the maximum confidence enhancement based sequential sample scheme to refine the Kriging model. Zheng et al. [28] proposed a new active learning method based on the U learning function in AK-MCS. The second category aims at studying the stopping criterion. Hu et al. [29] used the maximum percentage error of the failure probability estimation less than one predefined limitation as a stopping criterion to construct a single-loop Kriging model for time-dependent reliability analysis. Lelievre et al. [30] proposed a more conservative stopping condition corresponding to the probability that all points are well classified. The third category mainly researches how to reduce the candidate sampling pool especially for the small failure probability because the candidate sampling pool in AK-MCS is so large that prohibits the usage of AK-MCS for estimating the small failure probability. Based on this research orientation, some AK model combined with the variance reduction techniques were proposed such as AK-IS [31], AKMIS [32], meta-IS [33], meta AK-IS2 [34], AK-SS [35,36], AK-LS [37], AK model combined with the SSIS method (abbreviated as the AK-SSIS) [38], etc. The fourth category aims at extending the AK based reliability analysis methods to the dynamic/time-variant reliability analysis, the system reliability analysis and the fuzzy reliability analysis. Wang et al. [39] proposed a double-loop adaptive sampling method for dynamic reliability analysis. Wang et al. [40] developed an AK based reliability analysis method for the time-variant reliability analysis by using the equivalent stochastic process transformation. Fauriat et al. [41] extended AK-MCS to system reliability analysis. Yun et al. [42] improved the AK-SYS method for system reliability analysis. Wei et al. [43] used multiple response Kriging model to assess the structural system reliability. Yun et al. [44] proposed a novel step-wise AK-MCS method to estimate the fuzzy failure probability. Fig. 1 summarizes the main research orientations of the AK based reliability analysis methods. The aim of this paper is to improve the application of the direct AKMCS in case of analyzing the small failure probability. For large failure probability Pf (Pf [10 1, 10 3]), AK-MCS is efficient. But for the problem with small failure probability (Pf [10 6, 10 3] or smaller), the large number of candidate points (10 2~10 4) Pf are required to find the best next sample to update the current Kriging model. The large number of candidate points limits the application of AK-MCS in analyzing the small failure probability. The large number of candidate points will influence the training time of Kriging model in AK-MCS and sometimes make the computer out of memory. This paper tries to reduce the size of the original candidate sampling pool in AK-MCS and without affecting the accuracy of analysis. By combining the ARBIS, the candidate samples in AK-MCS are divided into different non-overlapping subsets according to the in-process spherical rings. Then, the Kriging model is ceaselessly updated layer after layer until the optimal -sphere, i.e., the largest sphere in the safe domain, is found. The proposed AK combined with ARBIS method is abbreviated as the AK-ARBIS. The candidate sampling pool of the AK-ARBIS is a part candidate sampling pool of the AK-MCS because samples inside the optimal -sphere are directly regarded as the safe state so that the samples inside the optimal -sphere do not need to participate in the training process. Therefore, only

samples outside the optimal -sphere compose the candidate sampling pool in order to select the best next training point and update the Kriging model, which will reduce the number of total candidate samples and save the time to find the best next training point in comparison with the original AK-MCS. Besides, the proposed AK-ARBIS does not need any pre-processes such as finding the most probable point (MPP) in the failure domain or the IS direction, which makes it simpler to be implemented. The rest of this paper is organized as follows. Section 2 briefly reviews the original AK-MCS for reliability analysis. Section 3 introduces the ARBIS and the proposed AK-ARBIS in detail. A highly nonlinear limit state function, a non-linear oscillator system, a simplified wing box structural model, an aero-engine turbine disk and a planar ten-bar structure are analyzed in section 4, and the results of these case studies demonstrate the effectiveness of the proposed AK-ARBIS method in case of the nonlinear limit state function and the small failure probability. Finally, conclusions are summarized in Section 5. 2. Reviews of the structural reliability analysis and its AK-MCS estimation The important issue to analyze the structural reliability is to solve Eq. (1). Eq. (1) can be estimated by the following expectation, i.e.,

Pf =

g (X ) 0

fX (x ) dx = E (IF (X ))

1 N

N

IF (xi ) i=1

(2)

where E (·) is the operation of the expectation, N is the number of samples generated by the PDF fX (x ) , xi is the i th sample of inputs among these N samples and IF (X ) is the failure domain indicator function which is defined as follows,

IF (X ) =

0 g (X ) > 0 1 g (X ) 0

(3)

To identify the states (safe or failed) of the N samples, AK-MCS method is previously proposed which is based on the U learning scheme. In AK-MCS, the sign (negative or positive) of the limit state of each sample generated by the PDF fX (x ) is accurately identified by the AK model. To conveniently introduce the proposed AK-ARBIS method which is an improvement of the AK-MCS method, the computational steps of the original AK-MCS is briefly reviewed as follows. Step 1: Generate N -size samples of model input variables by the PDF fX (x ) and denote this sampling pool as the matrix x11 x21 x n1 x1 x12 x22 x n2 x2 S= = . In this step, the limit state value of each

x1N x2N xnN xN sample in matrix S does not need to be calculated. N ) initial samples from the Step 2: Randomly select N1-size (N1 matrix S and then calculate their limit state values by the actual limit state function g (X ) . Then, the initial training sample set T is constructed as T = {(x1, g (x1)), (x2 , g (x2)), ...,(x N1, g (x N1))} . Step 3: Construct/Update the Kriging model gK (X ) according to the current training sample set T by the toolbox DACE [45]. Kriging model is reviewed in the Appendix A Step 4: Identify the best next training sample in S and evaluate its limit state function value of the selected best next training sample. The identification of the best next sample is completed according to the learning function U , i.e.,

µ gK X ) U (X ) = gK

(X )

(4)

where µ gK (X ) is the Kriging mean and gK (X ) is the Kriging standard deviation. Then, the best next training sample denoted as x u is thus the sample 2

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Research orientations of AK based reliability analysis methods

Learning function

Stopping criterion

Reduction of candidate sampling pool

dynamic reliability system reliability fuzzy reliability

EFF [23]

maximum percentage error [29]

AK - IS [31]

Double - loop dynamic reliability [39]

probability of

AK - MIS [32]

correctness with whole population [30]

Single - loop dynamic reliability [40]

meta - IS [33]

U learning function [24]

Probabilistic classfication function [25]

meta AK - IS2 [34]

Information entropy based function [26]

AK - SS [35 36]

Maximum confidence enhancement function [27]

AK - LS [37]

Optimization based more uncertainty point [28]

AK - SSIS [38]

AK - SYS [41] system reliability AK - SYSi [42] system reliability multiple AK [43] system reliability

AK - MCS [44] fuzzy reliability

Fig. 1. Main research orientations of the AK based reliability analysis methods.

at which the function U is minimized among these N candidates, i.e., x u = arg min U (xi ) .

circle of AK-MCS is used to be the initial training sample set of the next circle of AK-MCS until the condition of C. O. V P^f < 0.05 is satisfied. MCS, a universal method, does not require knowing the moments of the limit state function, the partial derivatives of the limit state function and the MPP. The essence of the computational formula in AK-MCS is the same as that in the MCS and the only difference between AK-MCS and MCS is that the limit states of all inputs’ samples in AK-MCS are identified by the Kriging model gK (X ) while the limit states of all inputs’ samples in MCS are identified by the actual limit state function g (X ) . For large failure probability (Pf [10 1, 10 3]), AK-MCS is efficient and accurate enough because small number of candidate samples is involved in AK-MCS. For small failure probability Pf [10 6, 10 3], a large number of inputs’ samples need to be generated (usually 10 2 Pf ~ 10 4 Pf ) by the PDF fX (x ) in AK-MCS, which leads to a large number of candidate samples is involved in AK-MCS. The large scale of candidate samples makes the training process of Kriging model in AKMCS time-consuming. Next section, the proposed AK-ARBIS will be introduced which only selects a partial number of candidate samples in AK-MCS as final candidate samples by the strategy of finding the optimal -sphere adaptively.

i = 1,..., N

Due to the prior assumption of Kriging model, the response from Kriging model follow a normal distribution, i.e., gK (X )~N (µ gK (X ), g2K (X )) . Then, the probability of making a mistake on the sign is ( U ) and the probability of accurate identification of ( U ) = (U ) . The stopping criterion of updating the the sign is 1 AK model is that U (x u ) 2 , which means that the sign of every sample in N samples is correctly identified with a probability of (2) = 97.7% at least by its Kriging mean. If U (x u ) < 2 , calculate g (x u ) and add {x u , g (x u )} into the training sample set, i.e., T = T {(x u , g (xu )} and then return to the Step 3; Otherwise, execute the next step continuously. Step 5: Calculate the failure probability and its coefficient of variation (COV). By the current Kriging model gK (X ) which guarantees that the condition of min U (xi ) 2 is satisfied, the failure probability i = 1,..., N

and its COV are estimated by Pf and C. O. V P^f respectively shown in Eqs. (5) and (6), i.e.,

Pf =

1 N

N

IKF (xi )

(5)

i=1

C. O. V P^f =

IKF (x ) =

1

P^f NP^

1µ g K ( x )

3. The improved AK-MCS based on the ARBIS To reduce the candidate samples in AK-MCS especially for small failure probability but do not lost the universality of the AK-MCS, ARBIS is embedded into the AK-MCS method, in which the optimal -sphere, i.e., the largest sphere in the safe domain, is adaptively found. The limit states of all samples inside the optimal -sphere are safe due to the largest safe sphere. Therefore, only samples outside the optimal -sphere need to judge the limit states so that only samples outside the optimal -sphere need to participate in the training of AK model as candidates. Section 3.1 will briefly introduce the basic theory of the ARBIS method. Section 3.2 will introduce the proposed AK-ARBIS elaborately.

(6)

f

0

0µ gK (x ) > 0

If C. O. V P^f is larger than 0.05, S should be is increased with new samples coming from another MCS population and AK-MCS returns to the Step 4 until the stopping condition is met again. It is important to emphasize that no information about the previous evaluations of the limit state function is lost because the training sample set T in the last 3

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4

A

2

U2

B

i 1

0

number of failure samples between the outside of i -sphere and the inside of i 1-sphere, and denote it as Ni . Step 5: Repeat step 4 until there is no failure samples between the outside of the i -sphere and the inside of the i 1-sphere or use the stopping condition introduced in Ref. [16]. Then, the failure probability is estimated by Eq. (10)

Failure domain

MPP

Pf =

0

-2

Limit-state -4

-2

0

2

3.2. AK-ARBIS: An improved AK-MCS

4

From the review of the ARBIS method, it can be seen that only the samples outside the optimal -sphere are required to call the limit state function g (X ) , and the optimal -sphere is adaptively searched layer after layer until the convergent condition is satisfied. The next -sphere is found by the samples’ information outside the current -sphere. Combining the idea of the ARBIS, AK-ARBIS is proposed in this section. The proposed AK-ARBIS inherits the advantage of the fast convergence of the ARBIS and the strong prediction of the AK model. In the proposed AK-ARBIS, the Kriging model is adaptively updated layer after layer between the outside of the current i -sphere and the inside of the last i 1-sphere, and by the current updated Kriging model the next i + 1-sphere is searched. The above process is finished until the optimal -sphere is obtained. It can be seen the difference between the proposed AK-ARBIS and the AK-MCS is that the candidate samples of the AK-MCS are divided into different non-overlapping subsets to update the Kriging model sequentially, and the samples inside the optimal -sphere do not need to participate in the process of updating the Kriging model. The proposed AK-ARBIS directly reduces the whole training sampling pool and each circle’s training time. The processes of selecting and dividing the candidate sampling pool by the proposed AK-ARBIS are shown in Fig. 3. The flowchart of the proposed AK-ARBIS is shown in Fig. 4 and the concrete steps of the proposed AK-ARBIS are summarized as follows. Step 1: Generate the MCS sampling pool of model inputs. Generate MCS sampling pool (N samples) according to the joint standard normal PDF (u) and put these samples in the matrix Snorm . Step 2: Initialize the parameters. Set = 0 , SA = S norm and k = 1. Step 3: Select the samples outside the -sphere. Count how many samples in SA are located outside the k -sphere and put those satisfied samples in the matrix Aouter . If Aouter is empty, turn to the Step 7; Otherwise, execute the next step continuously. Step 4: Construct/Update the Kriging model. If k = 1, randomly choose N1 samples in Aouter to calculate their actual limit states by the limit state function H (u) , and construct the initial Kriging model by the training sample set Th which is composed by the N1 input–output samples, i.e., Th = {(u1, H (u1)), (u2, H (u2)), ...,(u N1, H (u N1))} and then use the U learning scheme to update the Kriging model constructed in the region of Aouter until the stopping criterion min U (ui ) 2 is sa-

U1 Fig. 2. The adaptive strategy to determine the optimal radial of sphere.

3.1. The brief introduction of the ARBIS The main idea of the ARBIS is to find the optimal -sphere step by step by using the samples' information outside the current -sphere and the line search procedure. Samples inside the optimal -sphere can be directly regarded as the safe samples and do not need to evaluate their limit states by actual limit state function g (X ) . The concrete steps of the ARBIS on reliability analysis are reviewed as follows and more details can be referred in Ref. [16]. The adaptive strategy to determine the optimal radial of sphere is shown in Fig. 2. Step 1: Set the initial radial of sphere 0 , i.e., 0

=

2 (1

Pf 0)

(7)

where is the inverse distribution function of the Chi-square distribution function 2 with n freedom degrees (n is the dimensionality of the input variables X ). Generally, Pf 0 = 10 6 . Step 2: Transform samples in the matrix S into the standard normal space u by the isoprobabilistic transformation u = T (x ) and denote as u1 u2 norm S = . Then denote g (T 1 (u )) as H (u) , and the failure prob2

uN ability is estimated as Pf =

H (u ) 0

(u ) du

1 N

N

I f (u i ) i=1

(8)

where

(u) is the joint standard normal PDF of u corresponding to 0 H (ui ) 0 . fX (x ) and IF (ui ) = 1 H (u i ) > 0 Step 3: Set i = 1, and select the samples outside the 0 -sphere and evaluate their limit state values by g (X ) to determine the number of failure samples outside the 0 -sphere, and denote it as N0 . Step 4: Set i = i + 1, and determine the new radial of i -sphere by line search procedure in the direction of the sample satisfying g (u ) 0 and u > i 1 and possessing maximum value of (u) . Denote the sample which has the maximum value of (u) and satisfies the conditions g (u ) 0 and u > i 1 as u max . Then, the new radial of i -sphere is found by the following equation, H

i

u max u max

Generally,

(10)

where Ni denotes the number of failure samples in matrix S . The essence of the computational formula for estimating the failure probability by the ARBIS is the same as the computational formula by the MCS. But ARBIS avoids a part evaluations of the limit state function values compared with the MCS by search the optimal -sphere. Therefore, compared with MCS, ARBIS can reduce a part of computational cost and guarantee the same accuracy as the MCS.

i

-4

Ni N

ui Aouter

tisfied; Otherwise, add the best next sample uu = arg min U (ui) and ui Aouter

its limit state value H (uu ) into the training sample set Th , i.e., Th = Th {(uu, H (uu )} and then use the current Th to update the Kriging model in succession until the stopping criterion min U (ui ) 2 is ui Aouter

=0 i

satisfied. Step 5: Count how many failed samples outside the current -sphere and calculate the next radius of sphere. Use the current Kriging model HK (u ) to pick out the failed samples in the region Aouter and record the

(9)

can be found by two-to-three iterations. Count the 4

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(b)

(a)

0

(c) 0

3

(d)

3 opt

1

opt 1

Fig. 3. The process of selecting the candidate sampling pool in AK-ARBIS method: (a) The samples of two-dimensional random variables generated by joint standard normal PDF and the samples are used as candidate samples in AK-MCS; (b)–(d) The stratified process of selecting candidate samples in AK-ARBIS.

number as NFk . If NFk = 0 , turn to the Step 7; Otherwise, choose the failure sample uF with the maximum value of (u) in the matrix Aouter , and then calculate the next denoted as new by solving the equation

HK

(

new

uF uF

the proposed AK-ARBIS can reduce the whole training sampling pool and save much more in-process time to update the Kriging model in comparison with the classical AK-MCS. It is important to point out that the optimal -sphere is not calculated before establishing the AK model. The optimal -sphere is searched by a series of iterations in which the samples outside the previous -sphere and the current Kriging model are employed to determine the next -sphere until the convergent condition is satisfied. The optimal -sphere, the final Kriging model and the failure probability estimation are obtained simultaneously. The basic computational formulas of the AK-MCS and the proposed AK-ARBIS are the same. The differences between the AK-MCS and the proposed AK-ARBIS and the advantages of the proposed AK-ARBIS are summarized as follows,

) = 0 (where H (·) means the current Kriging model) K

using the quadratic interpolation with three points. Step 6: Update the parameters. Set = new , SA = SA Aouter , and k = k + 1. Then, turn to the Step 3. Step 7: Estimate the failure probability. The failure probability and its COV are estimated by the two following equations:

Pf =

i=k i=1

NFi

(11)

N

C. O. V P^f =

1

P^f NP^ f

(1) The size of candidate samples of model inputs in AK-MCS is equal to the number of inputs’ samples in MCS, while the size of the candidate samples in the proposed AK-ARBIS equals to the number of inputs’ samples in MCS minus the samples inside the optimal -sphere. If the optimal -sphere is found, the states of samples inside the optimal -sphere can be directly recognized as safety and do not need to be determined by the Kriging model. Therefore, under the same level of accuracy and robustness, the size of candidate samples in the proposed AK-ARBIS is smaller than that in the AK-MCS, which may make the size of training sample set in the proposed AK-ARBIS is smaller than that in the AK-MCS and then save some evaluations of the actual limit state function.

(12)

Step 8: Update of the population. If C. O. V P^f is larger than 0.05, the algorithm starts all over again after enlarging N by using the current Kriging model as an initial Kriging model; Otherwise, output Pf and C. O. V P^f . From the above procedure, it can be seen that samples inside the optimal -sphere are never involved in the candidate sampling pool of the updating process. By interpolation technique and the strong prediction of the Kriging model, the optimal -sphere is gradually searched where the Kriging model is updated layer after layer between the outside of the current -sphere and the last -sphere, and the corresponding candidate samples are obtained layer after layer. Therefore, 5

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Fig. 4. Flowchart of the proposed AK-ARBIS.

(2) The total candidate samples in the proposed AK-ARBIS are divided into several small sets of candidate samples, and then the number of candidate samples in each circle of updating the Kriging model in the proposed AK-ARBIS is much smaller than that in each circle of updating the Kriging model in the AK-MCS. Therefore, the AKARBIS can save much more training time which is generated by taking all candidate samples into the Kriging model than AK-MCS especially for the small failure probability.

4.1. Case study I: A higher order limit state function A 2D higher order limit state function is used in this case study. The mathematical expression is

g (X ) = X14 + X24

5

(13)

where the two input variables mutually and independently follow the normal distribution with mean 2 and standard deviation 1. The classification of samples by the curve g (X ) = 0 in the standard normal space is shown in Fig. 5. From Fig. 5, it can be seen that the nonlinear degree of the limit state function is high. Therefore, FORM cannot be used to find the MPP which is illustrated in Ref. [46] so that some MPP-based method cannot be efficiently employed to solve this case. Table 1 shows the numerical results of the failure probabilities estimated by AK-MCS, the proposed AK-ARBIS, SS method and MCS. From Table 1, it can be seen that by the proposed AK-ARBIS, the 20,000 candidates in the original AK-MCS are reduced to 12,749, which means that there are 7251 samples inside the optimal -sphere and 12749 samples outside the optimal -sphere. Only the 12,749 samples actually participate in the adaptive updating process of the AK model. The limit states of the 7251 samples inside -sphere are directly regarded as safe samples and their signs do not need to use the Kriging model to identify. Table 1 also shows that the proposed AK-ARBIS uses fewer model evaluations and less training time than the original AK-MCS because of two reasons. One reason is that the candidate sampling pool in the proposed AK-ARBIS is smaller than that in AK-MCS. The other reason is

4. Case studies In this section, five case studies are analyzed. The first one is a high nonlinear limit state function whose MPP is difficult to be searched so that many AK model combined with other MPP-based methods cannot be easily adapted. The second one is a non-linear oscillator system, by adjusting the distribution parameters of one stochastic input, the magnitude of the failure probability can be changed flexibly. By different magnitude of the failure probability, some properties of the proposed AK-ARBIS are investigated. The third, the fourth and the fifth case studies are applications in a simplified wing box structural model, an aero-engine turbine disk and a planar ten-bar structure, respectively. In order to compare the efficiency of the proposed AK-ARBIS and the original AK-MCS, all analyses are carried out in this paper by using the same computer with a Inter (R) Xeon (R) CPU processor at 3.47 GHz 3.46 GHz with 48 GB RAM. 6

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Table 2 The distribution parameter of the input variables of Case study II.

X2

Input variable

Distribution

Mean

Standard deviation

M C1 C2 R T1 F1(Case 1) F1(Case 2) F1(Case 3)

Normal Normal Normal Normal Normal Normal Normal Normal

1 1 0.1 0.5 1 1 0.6 0.45

0.05 0.1 0.01 0.05 0.2 0.2 0.1 0.075

Table 3 Numerical results of the failure probability for Case 1. Method

X1

MCS*

AK-MCSi*

Fig. 5. The classification of the limit state function by the curve g (X ) = 0 in standard normal space.

AK-MCS* AK-MCS

AK-ARBIS FORM

Table 1 Numerical results of the failure probability of case study I.

SS

Method

Ncall

Time

Size of candidate sampling pool

Pf

C . O. V Pf (%)

AK-MCS AK-ARBIS SS MCS

34 30 10,000 20,000

3.5648 s 0.8952 s – –

20,000 12,794 – –

0.0785 0.0785 0.0732 0.0785

2.4 2.4 3.6 2.4

LS

Pf

–

–

7 × 10 4 85

2.834 × 10

2

2.832 × 10

2

2.834 × 10

2

2.831 × 10

2

2.831 × 10

2

3.318 × 10

2

2.894 × 10

2

2.880 × 10

2

55

7 × 10 4

435

7 × 10 4

35

7 × 10 4 45,688

6.5

–

–

–

–

–

–

62

65

63 30

10,000

30 + 600

C . O. V Pf (%) 2.2

2.14 2.2

2.21 2.21 –

4.78 1.07

1. The reliability analysis results of different cases obtained by the proposed AK-ARBIS are consistent with these obtained by AK-MCS and MCS. 2. The total analysis time and the whole size of the candidate sampling pool in the proposed AK-ARBIS are smaller than those in the original AK-MCS, respectively. 3. For failure probability smaller than 10 6 , AK-MCS cannot be implemented because large number of candidate samples makes the computational time of selecting the next best training point too long. The proposed AK-ARBIS is competent for the failure probability smaller than 10 6 . For case 3, the proposed AK-ARBIS reduces 99.998% candidate samples in AK-MCS.

4.2. Case study II: A non-linear oscillator system A non-linear undamped single degree of freedom system is shown in Fig. 6. This case study is also analyzed in Refs. [23,24,30,31]. The limit state function is 0 T1

2

Ncall

distribution and the distribution parameters are shown in Table 2 and (C1 + C2 ) M . 0 = To further investigate the effectiveness of the proposed AK-ARBIS for different orders of magnitude of failure probability, three cases corresponding to F1 with different distribution parameters are analyzed for comparison. Tables 3, 4 and 5 show the results of the three cases obtained by the proposed AK-ARBIS, AK-MCS, AK-IS, FORM, SS and LS, respectively. From Tables 3–5, three points can be obtained as follows:

that some candidate points in AK-MCS with U values less than 2 may locate inside the optimal -sphere, and these samples need not to be selected as the best next samples to update the Kriging model in AKARBIS. Thus, the number of training samples in the proposed AK-ARBIS may be smaller than that in AK-MCS.

2F1 sin M 02

CUP time(s)

(*results reproduced from Ref. [30])

Ncall : the number of actual limit state function evaluations.

G (C1, C2, M , R, T1, F1) = 3R

Size of candidate sampling pool

(14)

where the six random input variables follow the independent normal

Table 4 Numerical results of the failure probability for Case 2. Method

Size of candidate sampling pool

CUP time(s)

Ncall

Pf

MCS**

1.8 × 10 8 –

–

1.8 × 10 8 29

9.09 × 10

6

9.76 × 10

6

9.13 × 10

6

9.09 × 10

6

9.09 × 10

6

2.89 × 10

6

8.90 × 10

6

FORM** AK-IS**

AK-MCS

AK-ARBIS SS

LS

Fig. 6. Non-linear oscillator.

10 4

1.8 × 10 8 837,641 – –

–

–

125,875 341 – –

(**results reproduced from Ref. [31]) 7

29 + 38

77

71

80,000

30 + 1500

C . O. V Pf (%)

2.47 –

2.29 2.47 2.47

greater than 5 1.50

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Table 5 Numerical results of failure probability for Case 3. Method

Size of candidate sampling pool

CUP time(s)

Ncall

Pf

MCS**

9 × 1010 –

–

9 × 1010 29

1.55 × 10

8

1.56 × 10

8

1.53 × 10

8

1.44 × 10

8

1.56 × 10

8

FORM** AK-IS**

10 4

AK-ARBIS

5.73 × 1010 1,811,913

AK-MCSi*

–

–

29 + 38

–

6 h + 2283 s

77

76

C . O. V Pf (%)

2.68 –

2.70 3.4

2.67

*results reproduced from Ref. [30]; **results reproduced from Ref. [31]; 6 h: Time to generate 9 × 1010 samples; 2283 s: Time to construct/update the Kriging model

To further measure the efficiency of the proposed AK-ARBIS in comparison with the original AK-MCS, the candidate sample reduction ratio and the analysis time reduction ratio are respectively defined in Eqs. (15) and (16),

Candidate sample reduction ratio N (AK ARBIS) Nc (AK = c Nc (AK MCS)

MCS)

Analysis time reduction ratio Time(AK ARBIS) Time(AK = Time(AK MCS)

Fig. 7. Schematic diagram of the simplified wing box model.

the y , x and z directions are denoted by L x , L y and Lz respectively. The sectional area of all the bars in represented by A , the thickness of all the plates is denoted by TH , E is the elastic modulus of all bars and plates and P is the external load. The Poisson's ratio is 0.3. The seven input variables follow the independent normal distributions and the corresponding distribution parameters are shown in Table 7. Considering the maximum displacement of the structure in y direction not exceeding 0.018 m as the failure threshold, the limit state function is established y , where y = h (L x , L y , Lz , A, E , P , TH ) is an implicit as g = 0.018 function of stochastic inputs, and it is determined by the ANSYS software. The ANSYS model is shown in Fig. 8. The results of failure probability estimated by the proposed AKARBIS, AK-MCS, AK-IS and MCS are shown in Table 8. From Table 8, it can be seen that the failure probability of this simplified wing box structural model is 3.48 × 10 4 . For estimating this small failure probability, the original AK-MCS requires 2 × 106 candidate samples which is determined by the robustness level of MCS, while only 175,958 samples are finally picked out through the process of finding the optimal -sphere and only 175,958 samples finally make up the total candidate sampling pool of the proposed AK-ARBIS method. From the result Table, it can be seen that the proposed AK-ARBIS reduces 91.20% candidate samples and saves 95.79% analysis time compared to the original AK-MCS. Besides, the actual model evaluations of the proposed AK-ARBIS are smaller than those of the original AK-MCS. The analysis time and the size of candidate sampling pool in AK-IS are both smaller than those in AK-ARBIS. However, the application ranges of AK-IS are limited because the IS PDF is constructed by the MPP-based method. Besides, the number of actual limit state function evaluations in AK-IS is larger than that in AK-ARBIS because samples in AK-IS are generated by the IS PDF and then AK-IS may need to identify more failure surface than the proposed AK-ARBIS which is a direct improvement of the AKMCS.

(15)

MCS)

(16)

where Nc (AK ARBIS) and Nc (AK MCS) denote the number of candidate samples (or called the size of candidate sampling pool) in the proposed AK-ARBIS and the AK-MCS, respectively. Time(AK ARBIS) and Time(AK MCS) denote the total analysis time of estimating the failure probability by the proposed AK-ARBIS and the AK-MCS, respectively. Table 6 shows the candidate sample reduction ratio and analysis time reduction ratio of these three cases. From Table 6, it can be seen that as the magnitude of failure probability decreases, the analysis time reduction ratio and the candidate sample reduction ratio of the proposed AK-ARBIS increase, which means that the proposed AK-ARBIS is more suitable for estimation of the small failure probability. In other word, the proposed AK-ARBIS is a direct improvement of the original AK-MCS, and it makes the AK-MCS be able to deal with the problems with small failure probability. Compared with the AK-IS or other MPPbased Kriging analysis method for small failure probability estimation, the proposed AK-ARBIS does not require searching the MPP and the conditional samples. Therefore, the proposed AK-ARBIS is simpler and more flexibly to be implemented. 4.3. Application I: A simplified wing box structural model In this case study, a simplified wing box structural model is analyzed by the proposed AK-ARBIS to verify its engineering application. Fig. 7 shows the schematic diagram of the simplified wing box model. This structure consists of 64 bars and 42 plates. The 64 bars are divided into three groups based on their directions. The lengths of the bars in

4.4. Application II: An aero-engine turbine disk Turbine disk shown in Fig. 9 is a key component to aeroengine

Table 6 Comparison of the sample reduction ratio and the time reduction ration of the three cases. Case 1

Pf

Sample reduction ratio Time reduction ratio

2.834 × 10

34.73% 81.43%

Case 2 2

9.09 × 10

99.53% 99.73%

Table 7 The distribution parameter of the input variables of the wing box structure.

Case 3 6

1.56 × 10

Input variables

8

Mean

99.998% N/A

Standard deviation

N/A denotes the value is not available. 8

Lx (m) 0.4

0.04

L y (m) 0.2

0.02

Lz (m) 0.6

0.06

A (m2)

E (GPa)

1 × 10

4

1 × 10

5

71

7.1

P (N)

TH (m)

1.5 × 103

3 × 10

3

1.5 × 10 2

3 × 10

4

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Table 9 The distribution types and distribution parameters of input variables of Application II. Variable s

Distribution type

Mean

Normal

2 × 109 8240 5.67

Normal

1.22 × 10 200

Lognormal Lognormal Normal

C A

J n

6.2 × 10

Normal

Coefficient of variation 0.1

0.2 0.2 0.1

-3

0.1

-4

0.1

Table 10 Numerical results of failure probability of the simplified wing box structural model.

Fig. 8. The finite element model of the wing box structure.

Ncall

CUP time(s)

MCS

2 × 106 197

9002.72 s

AK-MCS

AK-ARBIS AK-IS

186 198

Size of candidate sampling pool

Pf

–

–

3.48 × 10

4

3.79

3.48 × 10

4

3.79

379.28 s

2 × 106 175,958

3.48 × 10

4

3.79

3.54 × 10

-4

3.42

52.26 s

3500

Ncall

CUP time(s)

MCS

5 × 106 94

2449.6 s

AK-MCS

Table 8 Numerical results of failure probability of the simplified wing box structural model. Method

Method

AK-ARBIS AK-IS

C . O. V P^ (%) f

86

110

–

37 s

7.15 s

Size of candidate sampling pool –

5,000,000 271,741 8000

C . O. V P^ (%)

Pf

f

2.86 × 10

-4

2. 6

2.86 × 10

-4

2. 6

2.86 × 10

-4

2. 6

2.96 × 10

-4

2.3

proposed AK-ARBIS, AK-MCS, AK-IS and MCS. From Table 10, it can be analyzed that under the same level of accuracy and robustness, the proposed AK-ARBIS reduces 94.57% candidate samples and save 98.49% analysis time in comparison with the AK-MCS. No matter that the analysis time and the size of candidate sampling pool in AK-IS are both smaller than those in the proposed AK-ARBIS, the number of actual limit state function evaluations in AK-IS is larger than that in the proposed AK-ARBIS. Besides, the application range of the proposed AKARBIS is wider than the AK-IS because the IS PDF is constructed by the MPP-based technique where the MPP is searching by the FORM. 4.5. Application III: A planar ten-bar structure The planar 10-bar structure is shown in Fig. 10, and the geometric dimension variables are the length of all the horizontal and the vertical bars, the node loads, the elastic modulus, and the section area of each bar, which are denoted as L , Pi (i = 1, 2, 3), E and Ai (i = 1, 2, ...,10) , respectively. About the connection behaviors of this planar 10-bar structure, the nodes 2, 3, 5 and 6 shown in the Fig. 10 are hinge joints. About the boundary conditions, the nodes 1 and 4 are fixed hinges on the support. The fifteen input variables are mutually independent and following the lognormal distribution. The distribution parameters are listed in Table 11. By use of the Ansys 11.0, the finite element model is constructed and is shown in Fig. 11. It is assumed that the limit state function is g = 0.0041 y , where y is the displacement of the node 3 in the vertical direction. When g < 0 , the planar 10-bar is failure,

Fig. 9. Diagram of crack of an aero-engine turbine disk.

rotating structure which endures the centrifugal force and the thermal stress in the processes of starting and accelerating. Combined with the complex shape, the stress concentration position may appear in the pin hole and the bottom of tongue-and-groove during the process of work. After working for a period of time, cracks may appear in these positions. The load applied on the aeroengine turbine disk is

F =

C 2

2

+2

2J

(17)

where , C , and J are mass density, coefficient, rotational speed, and cross sectional moment of inertia, respectively. = 2 n where n is the rotational frequency. Considering the safety and the applicability, the limit state function is established as follows,

g ( s, , C , A, J , n) =

sA

F

(18)

where s is the ultimate strength and A is the cross-sectional area. All random input variables s , , C , A , J and n are assumed as independent variables with the distribution types and parameters listed in Table 9. Table 10 shows the results of failure probability estimated by the

Fig. 10. Planar ten-bar structure. 9

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W. Yun, et al.

if the AK-IS can get accurate result in advance for its FORM based limitation, while the proposed AK-ARBIS has the same application scope as the AK-MCS, i.e., the universality. Besides, in most cases, the number of actual limit state function evaluations in the proposed AK-ARBIS is smaller than that in the AK-IS. That is to say, if the actual model is extremely complicated and time-consuming, the intermediate training time of Kriging model generated by taking all candidate samples into the Kriging model can be omitted in comparison with the time generated by running the actual complicated model.

Table 11 The distribution parameter of the input variables of the planar ten-bar structure. Input variable

Distribution

Mean

Coefficient of variation

P1 (kN) P2 (kN) P3 (kN) L (m)

Lognormal Lognormal Lognormal Lognormal Lognormal

80 10 10 1 0.001

0.05 0.05 0.05 0.05 0.05

Ai (m2) E (GPa)

Lognormal

100

0.05

5. Conclusion This paper aims at improving the efficiency of the original AK-MCS especially for the small failure probability (10 3~10 6 or smaller). The proposed AK-ARBIS inherits the simple and direct properties of the AKMCS and has the same basic computational formula as the AK-MCS. By employing the ARBIS technique and AK model, two aims are achieved. One is that the size of the candidate sampling pool in AK-MCS is reduced. The other is that the reduced candidate sampling pool is divided into several different non-overlapping subsets, on which the parallelized updating processes are achieved. Five case studies are analyzed and the conclusions are drawn on the basis of the results presented, 1. Only the samples outside the optimal -sphere are actually used as candidate samples to update the Kriging model based on the U learning scheme. 2. The optimal -sphere, the final updated Kriging model and the reliability analysis results are obtained simultaneously. 3. The efficiency of the proposed AK-ARBIS compared with AK-MCS is mainly measured by the analysis time and the number of candidate samples. The number of the candidate samples is extremely large for small failure probability in AK-MCS, and the Kriging model must be run at these candidate samples, which consumes large amount of computational time and significantly affects the applicability of the AK-MCS in engineering problem with small failure probability. From the results in Case studies Section, it can be seen that the proposed AK-ARBIS considerably reduces the analysis time and the number of the candidate samples compared with AK-MCS. 4. The actual model evaluations in the proposed AK-ARBIS may be smaller than those in the AK-MCS. The main reason is that some candidate points in AK-MCS with U values less than 2 may locate inside the optimal -sphere, and then the computational cost of updating in these candidate points can be saved by the proposed AKARBIS.

Fig. 11. The finite element model of the planar ten-bar structure.

otherwise, the planar 10-bar can normally work. Table 12 shows the numerical results of failure probability estimated by the proposed AK-ARBIS, AK-MCS, AK-IS and MCS. The failure probability is relatively small for this planar ten-bar structure. For this engineering application, the proposed AK-ARBIS reduces 84.51% candidate samples and save 98.50% analysis time compared to the AK-MCS. The size of candidate sampling pool and the analysis time in the AK-IS are smaller than those in the proposed AK-ARBIS, but the number of actual limit state function evaluations in the proposed AK-ARBIS is smaller than that in the AK-IS. Therefore, it can be concluded the advantages of the proposed AK-ARBIS from two viewpoints, i.e., (1) Compared with AK-MCS, the proposed AK-ARBIS not only reduces the training times and the size of candidate sampling pool but also further decreases the number of evaluations of the actual limit state function. Therefore, no matter from the perspective of the analysis time or from the perspective of the number of actual model evaluations, the proposed AK-ARBIS is more efficient than the original AK-MCS under the same accurate level and the same robustness. (2) Compared with the AK-IS, if the AK-IS can get accurate result, the training time and the size of candidate sampling pool in AK-IS are indeed smaller than those in AK-ARBIS. However, we cannot know

The proposed AK-ARBIS is efficient and suitable for estimation of small failure probability which generally exists in the ubiquitous reliable engineering systems. Acknowledgements: This work was supported by the National Natural Science Foundation of China (Grant 51775439), the National Postdoctoral Program for Innovative Talents (Grant No. BX20190244) and the National Science and Technology Major Project (Grant 2017-I-00090046).

Table 12 Numerical results of failure probability of the planar ten-bar structure. Method

Size of candidate sampling pool

CUP time(s)

Ncall

Pf

MCS

1.5 × 107

–

3.36 × 10

5

1.5 × 107 2,323,186

3106

3.36 × 10

5

AK-ARBIS

1.5 × 107 262

3.37 × 10

5

3.60 × 10

5

AK-MCS AK-IS

2000

Appendix A

C . O. V Pf (%)

Kriging model 206,951 149

215

246

4.45

The Kriging model is a semi-parametric interpolation technique based on the statistical theory [47] including the parametric linear regression part and the nonparametric stochastic process. For an unknown function g (X ) , the Kriging model is given as follows [48]:

4.45 4.45 4.77

10

Structural Safety 82 (2020) 101891

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gk (X ) =

B i (X )

+ Z (X ) = B T (X ) + Z (X )

i

(A1)

i=1

where B (X ) = [B1 (X ), B2 (X ), ...,Bp the base functions of vector X , = [ 1, 2, ..., is the regression coefficient vector, and p denotes the number of base functions. Z (X ) is a stationary Gaussian process [49] with the following statistic characteristics,

(X )]T are

T p]

(A2)

E (Z (X )) = 0

(A3)

2

Var (Z (X )) =

2R ((x

cov [Z (xi ), Z (xj )] =

i ),

(A4)

(xj ))

where 2 is the process variance, xi and xj are the i th and j th samples among the Nt training sample points, and R ((xi ), (xj )) is the correlation function about xi and xj with a correlation parameter vector . Gaussian correlative model is widely used and employed in this paper to describe the correlation function, i.e.,

R (xi , xj ) = exp

Nt

(k ) k (x i

[

x j(k ) ) ]

0, 0

k

2

(A5)

k=1

xi(k) ,

where Nt is the number of training sample set, the Kriging model and generally is set as 2. Besides,

R (xi , xj ) = exp

Nt

(k k (x i

[

x j(k ) 2]

x j(k )

and k are the k th components of xi , xj and respectively, connects with the smoothness of is usually substituted by a scalar . Therefore, Eq. (A5) can be simplified as,

0

(A6)

k=1

Define R =

R ((x1), (x1))

R ((x1), (x Nt ))

, B =

B1 (x1)

B2 (x1)

Bp (x1)

and g is corresponding vector of the limit state functions B1 (x Nt ) B2 (x Nt ) Bp (x Nt ) R ((x Nt ), (x1)) R ((x Nt ), (x Nt )) calculated at each experimental points (xi )(i = 1, 2, ...,Nt ) , i.e., g=[g (x1),g (x2 ), ...,g (x Nt )]T , and the unknown and 2 can be estimated as follows, (A7)

= (B TR 1B) 1BTR 1g 2

=

1 (g N0

T

B ) R 1 (g

B )

The correlation parameter 2

= arg min(Nt ln

(A8) can be optimized by the maximum likelihood estimation, i.e., (A9)

+ ln R )

Therefore, for any unknown point X , the Best Linear Unbiased Predictor of model gK (X ) is shown to be a Gaussian random variable

gK (X)~N µ gK (X ),

gK ( X )

where the mean and the variance of the Kriging model are given as follows:

µ gK (X ) = B T X ) ^ + r T (X ) R 1(g 2 gK

(X ) =

2 (1

r T (X ) R 1r (X )

+

B ^) [B TR 1r (X )

(A10)

B (X )]T (B TR 1B ) 1 [B TR 1r (X )

B (X )])

(A11)

where r T (X ) = [R ((X ), (x1)), ...,R ((X ), (x Nt ))]T .

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