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A sequential surrogate method for reliability analysis based on radial basis function
Structural Safety 73 (2018) 42–53
Contents lists available at ScienceDirect
Structural Safety journal homepage: www.elsevier.com/locate/strusafe
A sequential surrogate method for reliability analysis based on radial basis function Xu Li a, Chunlin Gong a, Liangxian Gu a, Wenkun Gao a, Zhao Jing b,⇑, Hua Su a a b
School of Astronautics, Northwestern Polytechnical University, Xi’an, PR China Department of Mechanics, Huazhong University of Science and Technology, Wuhan, PR China
a r t i c l e
i n f o
Article history: Received 27 December 2016 Received in revised form 18 October 2017 Accepted 19 February 2018
Keywords: Reliability analysis Radial basis function Surrogate model Limit state function
a b s t r a c t A radial basis function (RBF) based sequential surrogate reliability method (SSRM) is proposed, in which a special optimization problem is solved to update the surrogate model of the limit state function (LSF) iteratively. The objective of the optimization problem is to find a new point to maximize the probability density function (PDF), subject to the constraints that the new point is on the approximated LSF and the minimum distance to the existing points is greater than or equal to a given distance. By updating the surrogate model with the new points, the surrogate model of LSF becomes more and more accurate in the important region with a high failure probability and on the LSF boundary. Moreover, the accuracy of the unimportant region is further improved within the iteration due to the minimum distance constraint. SSRM takes advantage of the information of PDF and LSF to capture the failure features, which decrease the samples of implicit LSF defined by expensive finite element analysis. Several numerical examples show that SSRM improves the accuracy of the surrogate model in the important region around the failure boundary with a small number of samples and has a better adaptability to the nonlinear LSF, hence increases the accuracy and efficiency of the reliability analysis. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction In engineering design, when uncertainties are involved, the failure probability Pf of the limit state function (LSF) g(x) should be examined. Pf is essentially a multi-dimensional integral formulated as
Z Z Pf ¼
Z
pðxÞdx1 dx2 dxm
ð1Þ
D
where D is the failure region defined as D ¼ fxjgðxÞ 6 0; x 2 Rm g, and p(x) is the joint probability function. The numerical integral and direct Monte Carlo Simulation (MCS) are difficult for a complex system with implicit time-consuming analysis models. To calculate the integral with the original LSF faces enormous computational challenges [1–2]. Therefore, different approximations of the LSF are adopted to improve the computational efficiency of reliability analysis. The mean value method (MVM) [3–4] performs a first-order Taylor expansion of the LSF at the mean point (as shown in Fig. 1), in which the LSF is assumed to be normal distribution. ⇑ Corresponding author. E-mail address: [email protected] (Z. Jing). https://doi.org/10.1016/j.strusafe.2018.02.005 0167-4730/Ó 2018 Elsevier Ltd. All rights reserved.
The MVM is one of the most efficient reliability methods. However, it requires the independent input variables obey normal distribution, which is difficult to be satisfied in practical problems. Moreover, the approximation error increases with the increase of the nonlinearity. Therefore, this method is appropriate for fast estimation of the structural reliability with low nonlinearity. The first-order reliability method (FORM) [5–8] transforms random variables with different distributions into the same standard normal space U by Rosenblatt transformation [9] and then performs a first-order Taylor expansion at the most probable point (MPP) which has the maximum failure probability on the LSF (as shown in Fig. 1). Eventually, the normal distribution parameters (mean value and standard deviation) of LSF are figured out with the gradient of the approximation function, and then the failure probability is obtained analytically. Compared with the MVM, the FORM does not require the input variables to obey normal distribution and has a higher accuracy with the LSF expanded at the MPP. However, the optimization with an equality constraint to find the MPP increases the number of the LSF evaluations. Moreover, it increases the nonlinearity of the LSF g(x) during the Rosenblatt transformation, thus the approximation error is large when the nonlinearity of LSF is high [10–11].
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X. Li et al. / Structural Safety 73 (2018) 42–53
u2
x2
of SSRM with several numerical examples; conclusions are given in Section 5.
Failure Region g(x) 0 GSORM(u)=0 gMVM(x)=0 MPP u*
μ
2. Surrogate model GFORM(u)=0
2.1 Construction of surrogate model
||u*|| Safe Region g(x)>0
LSF G(u)=0
LSF g(x)=0 O
Original Design Space X
x1
O
Standard Normal Space U
u1
Fig. 1. Approximations of LSF in X and U space.
The second-order reliability method (SORM) [7,12–13] is similar to the FORM. First, the input random variables are transformed into the standard normal space U to get the LSF G(u). Second, G(u) is expanded with second-order Taylor expansion at the MPP to obtain a quadratic hypersurface. Finally, the reliability of the approximate model is analyzed with analytical methods [10]. The SORM has a higher nonlinear adaptability than the FORM, but it needs to perform the time-consuming second-order gradients. Moreover, the adaptability to nonlinear boundary is still limited [10–11]. The response surface is a different commonly used reliability analysis method [14–16]. By sampling in the neighborhood of the design point, the local approximation model of LSF is constructed. Since the response surface model evaluation time is far less than that of the original LSF, it is possible to use the approximate model for Mont Carlo Simulation (MCS) or Importance Sampling (IS) [17–19]. However, since the accuracy of the response surface method is poor with numbered samples, the complex structural behavior might not be captured. When considering factors such as the failure boundary and probability density, more samples are required to improve the local accuracy. Therefore, iterative response surface reliability analysis methods are proposed [20–30], which increases the samples in the important region near the MPP, and gradually improves the accuracy of the approximation model. This type of iterative method is also known adaptive or active learning method [31–32,45]. In general, the response surface approximation model can be replaced by other surrogate model (also known as meto-model) techniques such as radial basis function (RBF), Kriging, support vector regression (SVR), artificial neutral net (ANN), etc. [18,33–44]. The existing methods converge in a local region after increasing the sample density of the important region in some degree, but the accuracy does not increase in the less important region. In order to make full use of the information of the added samples to improve the approximate accuracy of the important and less important region, this paper proposes a sequential surrogate reliability method (SSRM) based on RBF. By adding points sequentially to the surrogate model, the failure features in the important region and on the boundary of the LSF are captured, and the failure probability is obtained with MCS by using the surrogate model. SSRM does not need to use the original LSF to search for MPP directly, but rather becomes close to the important area near MPP in the process of adding points, thus reducing the number of sample evaluations and avoiding the failure to find the optimal solution. Meanwhile, the SSRM method makes a trade-off between precision and computational cost. The remainder of this paper is structured as follows. Section 2 introduces the surrogate model technology used in SSRM; Section 3 describes the implementation process of SSRM, and analyzes its characteristics; Section 4 verifies the effectiveness and efficiency
The goal of the surrogate model is to construct a prediction model of a complex or unknown model with the observation samples. Instinctually, surrogate model is an interpolation or regression model, which is also a branch of machine learning [47]. The common surrogate models include polynomial response surface method (PRSM) [48–49], radial basis function (RBF) [48–49], Kriging [50–52], support vector regression (SVR) [50–52] and artificial neutral net (ANN) [53]. Since RBF has good nonlinear adaptability and is easy to implement, this paper constructs the sequential surrogate model with RBF. Assume that the observation samples are presented as
S ¼ fðxi ; yi Þji ¼ 1; 2; . . . ; ng
ð2Þ
denoted by a matrix form
X ¼ ½x1 ; x2 ; . . . ; xn T
ð3Þ
y ¼ ½y1 ; y2 ; . . . ; yn T
where n is the sample size, X denotes the input sample matrix, y is the output sample vector. x is an m-dimensional design variable. RBF uses a series of linear combinations of radial basis functions to approximate the expensive limit state function, which can be formulated as
b y ðxÞ ¼
n X
bi fðkx xi kÞ ¼ fðxÞT b
ð4Þ
i¼1
^ðxÞ denotes the predictive response at point x, bi is the ith where y component of the radial base coefficient vector b, and fðkx xi kÞ (see Table 1) is the ith component of the radial basis function vector fðxÞ. As shown in Table 1, r ¼ kx xi k is the Euclidian distance between two samples, and c is the shape parameter. Substitute the samples of Eq. (2) into Eq. (4),
3 32 3 2 b1 y1 fðr11 Þ fðr 12 Þ fðr1n Þ 6y 7 6 fðr Þ fðr Þ fðr Þ 76 b 7 22 2n 76 2 7 6 27 6 21 76 7 6 7 ¼ 6 .. .. .. 76 .. 7 6 .. 7 6 .. 4 . 5 4 . . . . 54 . 5 bn yn fðr n1 Þ fðr n2 Þ fðr nn Þ 2
ð5Þ
Rewritten as a matrix form
y ¼ Fb
ð6Þ
As the samples are different with each other, F 2 R
nn
is a non-
singular matrix, and Eq. (6) has a unique solution b ¼ F1 y. Thus the prediction model is given by
b y ðxÞ ¼ fðxÞT F1 y
ð7Þ
where f(x) is related to the prediction point x and sample input matrix X; F1 y is only related to X and y. For a new prediction ^ðxÞ. sample x, f(x) is calculated one time to get its predicted valuey
Table 1 Radial basis functions. Type
Function form fðrÞ
Gaussian Inverse Multiquadric
expðcr2 Þ
Thin plate spline
1=2
ð1 þ cr 2 Þ r 2 logð1 þ cr 2 Þ
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X. Li et al. / Structural Safety 73 (2018) 42–53
It should be pointed out that the shape parameter c, which has a great influence on the accuracy of the model, is included in F and is determined by experience or other optimization criteria. This paper uses the cross-validation criteria to optimize the shape parameter c. 2.2. Validation of surrogate model A common method to estimate the accuracy of the surrogate model is the root mean square errors (RMSE). However, since RBF requires the model strictly goes through sample points, RMSE is a constant zero, leads to the failure of the model estimation. To avoid the problem, the cross-validation method is adopted. The samples are divided into K roughly equal-sized parts. For the kth part (k = 1, 2, , K), the model is fitted with the other K 1 parts of the samples, and calculates the prediction error of the fitted model when predicting the kth part of the samples [47]. Crossvalidation can fully reflect the matching degree of the samples and the model. In particular, when K is equal to the sample size n, it is called leave-one-out cross-validation error (LOOCV) [47]. Thus, the RBF shape parameter c can be estimated by the following sub-optimization problem:
min c
n X 2 ½yi b y ðxi ; S fðxi ; yi ÞgÞ
ð8Þ
i¼1
In Eq. (8), it can be seen that the evaluation of LOOCV error requires n times construction of the surrogate model. However, the LOOCV error does not require additional verification points, which is capable of describing the matching degree between the samples and the prediction model. According to the surrogate model and obtained parameter c, a reliability analysis method based on the sequential surrogate model can be constructed. 3 Sequential surrogate reliability method This section presents the detailed procedure of the proposed sequential surrogate reliability method (SSRM). SSRM uses the sequential surrogate model of LSF to add samples to refine the surrogate model, which is also named active learning method in the community of machine learning. To get more meaningful failure information of LSF, the added samples are selected in the neighborhood of the most probable point (MPP) and on the LSF hypersurface. However, when the added sampled is too close to the existing samples, the added information is small. Therefore, a constraint of minimum distance is required. With the criteria of adding samples, SSRM reduces the samples in the unimportant region with small failure probability. In the iterative process, the failure boundary of the LSF is approached step by step, and eventually a surrogate model of LSF which captures most failure information is obtained. In order to facilitate the description of the algorithm, the independent random variables in original design space X are transformed into standard normal space U with Rosenblatt transformation [9], which are formulated as
ui ¼ F1 U i FX i ðxi Þ; ði ¼ 1; 2; ; mÞ
8 8 b > > Gðu; Sk1 Þ ¼ 0 > < > > > > uk ¼ argmin kuk s:t: dðu; Sk1 Þ P dmin ðSk1 Þ > > > u > : > > u L 6 u 6 uU ; u 2 R m > > > > < DSk ¼ fðDuk ; gðDuk ÞÞg
ð10Þ
where g(x) is the LSF in the X space and G(u) is the LSF in the U space.
ð11Þ
Sk ¼ Sk1 [ DSk > > > > b > > P k ¼ PMCS f Gðu; Sk Þ 6 0g; k ¼ 1; 2; 3; > > > > b > S0 Þ 6 0g Start : S0 ; P0 ¼ PMCS f Gðu; > > > > jP k Pk1 j : Stop :k P k 6 er Þ max or ðjP k P k1 j 6 ea & P k
where Duk is the sample added in the kth iteration, u denotes the Euclidean distance between u and the coordinate origin O. Since the joint probability density function (PDF) of the standard random variables is given by
! 2 m Y ui 1 kuk2 m=2 pffiffiffi exp : pðuÞ ¼ exp ¼2 2 2 2 i¼1
ð12Þ
when u gets the minimum value, p(u) has the maximum failure probability (see Fig. 2). Sk is the sample set in the kth iteration, ^ Gðu; Sk Þ represents the surrogate model of LSF based on Sk , ui is the ith input variable of the sample set Sk1 which totally has a number of jSk1 j samples. dðu; S k1 Þ is the minimum distance between the currently sample and the existing samples, and dmin ðS k1 Þ is the target minimum distance (for more details see Step 4). uL and uU are the lower and upper bounds of variables, m is the dimension of LSF, P k is the failure probability in the kth iteration, S0 is the initial sample set constructed with Latin hypercube sampling (LHS) [46], P0 is the failure probability based on the Monte Carlo simulation (MCS) with the initial surrogate model of LSF, kmax is the maximum number of samples allowed to add, ea and er denote absolute and relative errors, respectively. The procedure of SSRM is shown in Fig. 3, and more detailed steps of the algorithm are summarized as follows: Step 1: Selecting the initial sample points by LHS and calculating the responses of LSF to obtain the initial sample set S0 . Since the interval probability Pf5 6 U i 6 5g 0:99999943 of each dimension contains most information, here we set the lower and upper bounds of LHS to be 5 and 5 respectively. For the problem with small standard deviation, the initial sample number can be taken as jS0 j ¼ m þ 1, while for the large standard deviation and highnonlinear problems, the initial sample number can be set as jS0 j ¼ 2m þ 1, where m is the dimension of random variables. Step 2: Using the initial samples to construct the initial surro^ gate model of LSF, Gðu; S0 Þ , and optimizing the model shape u2
x2 Failure Region g(x) 0
G(u;S0)=0
ð9Þ
where X i is the ith random variable in the original space, U i is the ith standard normal random variables, xi and ui are the values of the random variables, FX i ðÞ and FUi ðÞ are cumulative distribution functions (CDF) of X i and U i , and m is the dimension of the variables. Therefore, the LSF in a vector form is formulated as
gðxÞ ¼ g½F1 X FU ðuÞ ¼ GðuÞ
The following algorithm is implemented in the U space. The key step of SSRM is an iteration process with special initial and terminal conditions, which can be briefly described as follows
μ
d(u;S0)
u
||u|| Safe Region g(x)>0 G(u)=0
g(x)=0 O
Original Design Space X
x1
O
Standard Normal Space U
u1
Fig. 2. Initial surrogate in U-Space. ‘‘ ” denotes the initial samples, and ‘‘ denotes the added sample based on the initial surrogate model.
”
45
X. Li et al. / Structural Safety 73 (2018) 42–53
differentiability is adopted here, and it is complemented with the MATLAB optimization toolbox. In addition, the minimum distance between the current sample u and the existing samples ui is given by
Start
Step 1
LHS
dðu; Sk1 Þ ¼ min fku ui kjðui ; gðui ÞÞ 2 Sk1 g ui
Initial LSF Evaluation 1
...
Step 2
Constructing Initial Surrogate Model
Step 3
Initial Reliability Analysis
Step 4
Surrogate Model Optimization
Step 5
Sample Set Update
Step 6
Surrogate Model Update
Step 7
MCS Reliability Analysis
Initial LSF Evaluation n
To give a reasonable value of the target minimum distance, the number of samples, the dimension of LSF and the interval width of the variables in each dimension should be taken into account. Such a target value can be defined as
dmin ðSk1 Þ ¼ kmaxminfkui uj kjðui ; gðui ÞÞ; ðuj ; gðuj ÞÞ 2 Sk1 ; i–jg ui
Step 8
where k is a scale factor in the range 0.1 to 0.5 (this paper takes k ¼ 0:2), which balances the local and global search. Moreover, the optimization problem contains an equality constraint. Therefore, the total constraints are very strong, which makes GA inefficient and difficult to converge in the high-dimensional problem. According to the idea of penalty function method, the optimization problem with strong constraints in Eq. (11) can be transformed into an approximate unconstrained optimization problem:
u
k=k+ 1 No
End
Fig. 3. Flow chart of SSRM.
parameter c. Here the surrogate model uses RBF which strictly goes through the sample points and has strong nonlinear adaptability (see Sections 2.1 and 2.2) and is built with the authors’ in-house MATLAB toolbox. ^ Step 3: Using the initial surrogate model Gðu; S0 Þ for reliability analysis with MCS. In order to eliminate the errors induced by different random samples, the MCS reliability analysis uses the same random samples which can be reproduced with the same random seed. Therefore, the initial failure probability ^ P 0 ¼ PMCS fGðu; S0 Þ 6 0g is presented as N 1X b b j ; S0 ÞÞ PMCS f Gðu; S0 Þ 6 0g Ið Gðu N j¼1
ð13Þ
ð14Þ
where uj is an m-dimensional sample generated by the standard normal distribution, N is the total number of samples for MCS, and IðÞ is the indicator function, defined as
IðxÞ ¼
1; if x 6 0 0; if x > 0
dðu;Sk1 Þ 1þexpð20ðdðu;Sk1 Þdmin ðSk1 ÞÞÞ
i
ð18Þ
u is too close to existing samples, say dðu; Sk1 Þ < dmin ðSk1 Þ, the objective function increase rapidly, which ensures the original inequality constraint to work. As an unconstrained optimization problem, the transformed problem is easier to solve for the highdimensional problems. However, the process of searching for new samples is not as accurate as the original optimization problem, which decreases the global accuracy of the surrogate model. Step 5: Estimating the sample found in Step 4 with practical LSF GðuÞ to obtain the sample set DSk ¼ fðDuk ; GðDuk ÞÞg and then updating the sample set Sk ¼ Sk1 [ DSk . ^ Step 6: Updating the surrogate model of LSF Gðu; Sk Þ and optimizing the shape parameter c. Step 7: Estimating the failure reliability with the updated surrogate model with MCS, which can be presented as ^ Pk ¼ PMCS fGðu; Sk Þ 6 0g and calculated with Eq. (13). Here the samples used in MCS are the same as Step 3. Step 8: Convergence check. If one of the termination conditions, (a) the maximum iteration condition k P kmax , (b) the relative convergence condition jP k Pk1 j=Pk 6 er ð¼ 1 104 Þ and the absolute
and the coefficient of variation (COV) for MCS is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 PMCS COVðPMCS Þ ¼ P MCS N
^ ð1 þ kukÞjGðu; Sk1 Þj
^ where the symbols are the same as Eq. (11). If Gðu; Sk1 Þ ¼ 0, the objective function gets the minimum value. However, if ^ Gðu; Sk1 Þ 0, the objective function value decreases with u. When
Yes Step 9
uj
ð17Þ
Duk ¼ argmin h Satisfying the stop criterion?
ð16Þ
ð15Þ
Step 4: Solving the optimization problem in Eq. (11) to search for the new sample. As the optimization problem has an inequality constraint of minimum distance which is not differentiable, gradient-based optimization algorithm, such as sequential quadratic programing (SQP), cannot be utilized here. Therefore, the genetic algorithm (GA) which is not restricted by the
convergence condition jPk Pk1 j 6 ea ð¼ 1 103 Þ, is satisfied, go to Step 9, otherwise set k = k + 1 and then go to Step 4. Step 9: End of SSRM. The main procedures of SSRM are the construction of surrogate model and the criterion to add points, which are also the core to guarantee the accuracy and efficiency in the iteration process. SSRM automatically increases the samples in the important region with large failure probability, which makes full use of the information of PDF and each sample. In addition, since the minimum distance constraint of the samples works, SSRM increases the number of the samples in the less important region when the important region includes enough points. Therefore, SSRM does not require many initial samples. After each surrogate model update, MCS reliability analysis is performed to achieve the estimation of failure probability. Since most of the failure features near the MPP and the LSF boundary are captured by the surrogate model
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X. Li et al. / Structural Safety 73 (2018) 42–53
Table 2 Distributions of random variables for the circular pipe structure. Variables
Mean
Standard deviation
Distribution
rf
301.079 0.503
14.78 0.049
Normal Normal
h
iteratively, the MCS based estimation has a good accuracy. Moreover, if the time of a single evaluation of the surrogate model is far less than that of the original LSF, the cost of model parameter optimization, MCS and GA based on the surrogate model is acceptable. Therefore, SSRM shows its superiority of accuracy and efficiency, when LSF is defined by time-consuming model, such as expensive finite element analyses. 4. Numerical examples In this section, seven examples are used to perform SSRM, and the results are compared with those of MCS and some other existing methods. Example 1–3 are of different nonlinearity, example 4 is the LSF of a speed reducer shaft with variables under different distributions, example 5 is a cantilever tube structure with highdimensional variable space, example 6 is a nonlinear oscillator with six random variables, and example 7 demonstrates the sensitivity of SSRM in high-dimensional problems. Assuming the transformed random variables in U space to be ui 2 ½5; 5; i ¼ 1; 2; ; m, the initial samples are selected by LHS, and the sample size is m + 1 or 2m + 1, where m is the number of the random variables. In these examples, FORM and SORM are realized by Isight5.6, a software for multidisciplinary design optimization (MDO), while SSRM and MCS are implemented with the authors’ in-house Matlab codes. In order to quantify the error, results of MCS are taken as the references, and random seeds are selected to obtain the same samples for MCS in each example.
Considering a circular pipe with circumferential through-wall crack subjected to a bending moment, the LSF is given by [4]:
ð19Þ
where rf , h, M, R and t are flow stress, half-crack angle, applied bending moment, radius of the pipe and thickness of the pipe, respectively. R = 0.3377 m, t = 0.03377 m, M = 3 106 Nm. rf and h are random variables. The statistical properties of the random variables involved in the problem are illustrated in Table 2. As shown in Fig. 4, three initial samples are selected and four extra samples are added iteratively. The approximate LSFs of FORM, SORM and SSRM are close to each other in the neighborhood of the MPP due to the low nonlinearity. In the region far away from the MPP, the approximate LSF of FORM has a poor accuracy, while SORM and SSRM have better accuracies with the approximate LSFs closer to the original LSF. As shown in Table 3, SSRM has the best accuracy with a relative error of 0.0175%. As the initial ^ SSRM ðuÞ in Fig. 4(a) is sample size is small, the initial surrogate LSF G far away from the original LSF, however, after 4 iterations, they are almost the same (See Table 4). 4.2. Hyper-sphere bound problem This example considers a hyper-sphere bound with higher nonlinearity, and the LSF is given by
gðXÞ ¼ 1 X 31 X 32
4.3. Cantilever beam This is also an example with higher nonlinearity, a cantilever beam with rectangular cross section subjected to uniformly distributed loading [20,53]. The LSF with respect to the maximum deflection at the free end being greater than l/325, is given by 4
g¼
l wbl 325 8EI
ð20Þ
ð21Þ
where I = bh3/12, where w, b, l, E, I and h are load per unit, width, span, modulus of elasticity, moment of inertia of the cross section and depth, respectively. The random variables are w and h, as shown in Table 6. Assuming E and l are 2.6 104 MPa and 6 m respectively, the LSF is reduced to
gðXÞ ¼ 18:46154 74769:23
4.1. Circular pipe structure
h 1 sinðhÞ M gðXÞ ¼ 4t rf R2 cos 2 2
As shown in Fig. 5, three initial samples are selected and nine extra samples are added iteratively. As expected, Table 5 illustrates that SSRM obtains more accurate result with fewer samples. Since the approximation of the LSF can be improved step by step in the process of adding points, the accuracy is significantly improved. FORM and SORM cannot capture most of the failure features of the important region due to the high nonlinearity, however SSRM updates the surrogate of LSF, thus a good approximation in the important region and its neighborhood are obtained. As shown in Fig. 5 (a), the initial surrogate model is not accurate, and the added points are far away from the original LSF at first, however, as the iteration goes on, the added points get closer to the LSF. Thus, each point makes contributions to the improvement of the accuracy of the LSF boundary, especially in the important region. Meanwhile, the region far away from the MPP and the LSF boundary has a few samples and the probability is very small, hence the region has little effect on the estimation. That is the reason why SSRM improves the accuracy with fewer samples.
X1 X 32
ð22Þ
As shown in Fig. 6, five initial samples are selected and 13 extra samples are added iteratively. Table 7 and Fig. 6 show that SSRM obtains a good accuracy with Pf = 0.009499 and 18 LSF evaluations. Meanwhile, Fig. 6(a) shows that the added samples are all near the LSF boundary, and keeps minimum distance to each other. It means that all the samples contribute a lot to obtain the failure probability with a good approximation to the LSF boundary. Moreover, it can be observed that the accuracy in the region near u = (4, 4) is relatively poor. However, since the region is far from the mean point u = (0, 0), the probability is very small, which means that the poor accuracy does not affect the estimation of the failure probability (See Table 8). 4.4. Speed reducer shaft In this section, considering the LSF of the reducer axis of multivariate random variables with different distributions [9]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðXÞ ¼ S 32=ðpD3 Þ F 2 L2 =16 þ T 2
ð23Þ
In this example, 6 initial samples are selected and 38 extra samples are added iteratively. Fig. 7 shows that the initial value of Pf is far from the MCS result, due to the small number of samples. However, with the increase of the samples during the iteration, the capture ability of the SSRM for the failure boundary is increased. Hence, Pf converges in a few iterations. As expected, the results in Table 9 show that SSRM has a good adaptability and accuracy for this multidimensional problem (See Fig. 8).
X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 4. Iterative process of SSRM for the circular pipe structure.
47
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X. Li et al. / Structural Safety 73 (2018) 42–53
Table 3 Comparison of different reliability methods for the circular pipe structure.
Table 5 Comparison of different reliability methods for the hyper-sphere problem.
Methods
Pf
Relative Error (%)
Number of LSF evaluations
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM SSRM MCS(COV = 0.14%)
0.033065 0.034211 0.034347 0.034353
3.7493 0.4134 0.0175 0.000
9 14 7 1 106
FORM SORM SSRM MCS(COV = 0.53%)
1.891 102 2.672 102 3.381 102 3.381 102
44.070 20.970 0.000 0.000
15 20 12 1 106
Table 6 Distributions of random variables for the cantilever beam.
Table 4 Distributions of random variables for the hyper-sphere problem. Variables
Mean
Standard deviation
Distribution
X1 X2
0.5 0.5
0.2 0.2
Normal Normal
Variables
Mean
Standard deviation
Distribution
X1 X2
1000 250.0
200 37.5
Normal Normal
4.5. Cantilever tube problem
I ¼ ðp=64Þ½d ðd 2tÞ J ¼ 2I 4
To further demonstrate the performance of SSRM in the case of different distributions in high dimensions, consider the following cantilever tube problem, the LSF is defined as follows [6]: (See Table 10).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðXÞ ¼ Sy r2x þ 3s2zx where
szx ¼ Td=ð2JÞ M ¼ F 1 L1 cosðh1 Þ þ F 2 L2 cosðh2 Þ A ¼ ðp=4Þ½d ðd 2tÞ h ¼ d=2 2
ð24Þ
4
2
rx ¼ ðP þ F 1 sinðh1 Þ þ F 2 sinðh1 ÞÞ=A þ Mh=I
Fig. 5. Iterative process of SSRM for the hyper-sphere problem.
ð25Þ
49
X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 6. Iterative process of SSRM for the cantilever beam.
Table 7 Comparison of different reliability methods for the cantilever beam.
a b c d e
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM DSa DS + RSb DS + SPc DS + NNd ISe IS + RS IS + SP IS + NN SSRM MCS(COV = 1.0%)
0.00988 0.00988 0.01000 0.00600 0.00700 0.00800 0.01000 0.00900 0.01000 0.01200 0.009499 0.009594
2.981 2.981 4.232 37.46 27.04 16.61 4.232 6.191 4.232 25.08 0.9902 0.0000
27 32 551 60 57 40 9312 2192 358 63 18 1 106
Directional Sampling. Response Surface. Splines. Neural networks. Important Sampling. These results are from Schueremans [53]. Fig. 7. Iterative process of SSRM for the speed reducer shaft.
Table 8 Distributions of random variables for the speed reducer shaft. Meana\ Lower Boundb
Variables
Diameter D(mm) Span L(mm) External Force F(N) Torques T(Nmm) Strength S(MPa) a b
Standard deviationa\ Upper Boundb
39 400 1500 250,000 70
0.1 0.1 350 35,000 80
Table 9 Comparison of different reliability methods for the speed reducer shaft.
Distribution
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM SSRM MCS(COV = 1.1%)
7.14392 104 7.12341 104 7.52 103 7.52 103
90.50 90.53 0.00 0.00
30 50 44 1 106
a
Normal Normala Grumble(Max) Normala Uniformb
a
The distribution parameters are mean and standard deviation respectively. The distribution parameters are lower and upper bound respectively.
Constants h1 ¼ 5 ; h2 ¼ 10 . In this example, 10 initial samples are selected and 8 extra samples are added iteratively. As a nine-dimensional problem, the
computational cost for FORM and SORM to solve the MPP increases dramatically but not improve the accuracy effectively, while SSRM converges to the MCS result with 8 iterations (see Fig. 9). Though the dimension of the variables is high, the uniform distribution interval and the coefficient of variation are small, thus the
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X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 8. Cantilever tube [6].
Table 10 Distributions of random variables for the cantilever tube. Variables
Meana\ Low Boundb
Standard deviationa\ Up Boundb
Distribution
t(mm) d(mm) L1(mm) L2(mm) F1(N) F2(N) P(N) T(Nmm) Sy(MPa)
5 42 119.75 59.75 3000 3000 12,000 90,000 220
0.1 0.5 120.25 60.25 300 300 1200 9000 22
Normala Normala Uniformb Uniformb Normala Normala Gumbela Normala Normala
Fig. 9. Iterative process of SSRM for the cantilever tube.
Table 11 Comparison of different reliability methods for the cantilever tube.
nonlinearity near the design point is not very high. Therefore, the failure features can be captured by SSRM with a few samples; accordingly, a good estimation of the failure probability (see Table 11) is achieved (See Fig. 10).
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM SSRM MCS(COV = 3.1%)
3.8644 104 3.8154 104 1.0520 103 1.0460 103
63.06 63.52 0.574 0.000
30 84 18 1 106
4.6. Dynamic response of a nonlinear oscillator This example consists of a nonlinear undamped single degree of freedom system with six random variables [20,34,53]. The LSF is given by
gðc1 ; c2 ; m; r; t 1 ; F 1 Þ ¼ 3r j
2F 1 x0 t 1 sinð Þj 2 mx20
4.7. High-dimensional problem This example which is from Bourinet [54] aims to evaluate the performance of SSRM to deal with high-dimensional problems. Failure probabilities with different numbers of random variables are discussed. The LSF is formulated as (See Table 14).
pffiffiffiffiffi gðXÞ ¼ m þ 3r m
Xi
i¼1
c1
ð27Þ
F(t)
m
ð26Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x0 ¼ ðc1 þ c2 Þ=m. The random variables are given in Table 12. In this problem, 13 initial samples are selected and 6 extra samples are added as shown in Fig. 11. As expected, the Pf of SSRM in Table 13 is also close to that of MCS with a relative error of 1.623%. Since the result of MCS is also with a coefficient of variation of 2.2%, the relative error of SSRM is acceptable. However, the total LSF evaluations of SSRM are 19, which is smaller than the existing methods (See Fig. 12).
m X
F(t)
z(t)
c2 t1
t
Fig. 10. Nonlinear oscillator [34].
Table 12 Distributions of random variables for the nonlinear oscillator. Variables
Mean
Standard deviation
Distribution
m c1 c2 r F1 t1
1 1 0.1 0.5 0.5 1
0.05 0.1 0.01 0.05 0.2 0.2
Normal Normal Normal Normal Normal Normal
where m is the number of random variables, X i ði ¼ 1; 2; ; mÞ are independent random variables which follow the lognormal distribution with mean value l = 1 and standard deviation r ¼ 0:2. The transformed LSF in U space is presented as
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X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 11. Iterative process of SSRM for the nonlinear oscillator.
Fig. 12. Iterative process of SSRM for the high-dimensional problem for m = 40, 100 and 250.
Table 13 Comparison of different reliability methods for the nonlinear oscillator.
a b c d e f g h
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM DSa DS + RSb DS + SPc DS + NNd ISe IS + RS IS + SP IS + NN AKf + MCS+Ug AK + MCS+EFFh SSRM MCS(COV = 2.2%)
3.108 102 2.840 102 3.500 102 3.400 102 3.400 102 2.800 102 2.700 102 2.500 102 2.700 102 3.100 102 2.834 102 2.851 102 2.880 102 2.834 102
9.668 0.212 23.500 19.972 19.972 1.200 4.728 11.785 4.728 9.386 0.000 0.600 1.623 0.000
35 48 1281 62 76 86 6144 109 67 68 58 45 19 7 104
m pffiffiffiffiffi X l2 gðUÞ ¼ m þ 3r m exp ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ U i l2 þ r2 i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! ln
r2 þ1 l2
ð28Þ
Directional Sampling. Response Surface. Splines. Neural networks. Important Sampling. These results are from Schueremans et. al. (2005) [53]. Active Learning. Learning Function U. Expected Feasibility Function. These results are from Echard et al. (2011) [34].
In this example, three cases (m = 40, 100 and 250) are discussed. The results are shown in Table 15, where results of FORM, SORM, 2 SMART and MCS are from reference [54]. As shown in Table 15, for m = 40, 81 initial samples are selected and 117 extra samples are added iteratively; for m = 100, 201 initial samples are selected and 147 extra samples are added iteratively; for m = 250, 501 initial samples are selected and 233 extra samples are added iteratively. Fig. 11 shows the iterative processes of different cases. As expected, the number of added samples increases with the dimension of LSF. Moreover, when the dimension is high, it is difficult to find the accurate most probable point, and the accuracy of Taylor approximation is poor, therefore FORM and SORM have more errors. The accuracy of SSRM is slightly lower than 2SMART, but the number of LSF evaluations is just about 5.3% to 6.9% of 2SMART. Thus, SSRM balances the accuracy and the number of sample points.
Table 14 Iteration process of SSRM for the nonlinear oscillator. Iteration No.
u1(m)
u2(c1)
u3(c2)
u4(r)
u5(F1)
u6(t1)
Pf
0
2.140 4.899 3.734 3.031 2.600 0.328 1.125 3.605 1.003 1.717 1.776 3.119 4.862 0.109 0.007 0.325 0.010 0.074 0.081
0.125 1.441 1.061 3.786 3.502 2.584 3.443 3.345 2.119 4.542 0.992 1.315 4.384 0.428 0.037 0.761 0.238 1.080 0.433
2.975 2.518 4.171 2.319 1.669 0.878 0.331 4.844 1.447 4.954 2.996 0.628 4.017 0.152 0.041 0.012 0.018 0.016 0.270
3.075 4.468 4.113 3.929 4.875 0.055 1.238 1.590 0.900 2.308 0.777 1.988 3.219 0.161 0.702 0.008 0.750 0.254 -0.817
2.505 0.549 0.357 4.085 1.794 3.379 4.870 2.749 2.168 1.326 3.906 0.608 4.269 1.307 0.806 1.546 1.742 1.600 1.157
1.525 2.988 1.517 0.482 2.596 4.064 0.467 4.786 3.782 2.092 4.729 0.191 2.870 1.136 1.641 1.947 0.759 1.105 1.154
0.04303
1 2 3 4 5 6
0.02921 0.02789 0.02680 0.02904 0.02881 0.02880
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X. Li et al. / Structural Safety 73 (2018) 42–53
Table 15 Comparison of different reliability methods for the high-dimensional problem. Methods FORM SORM SMART SSRM MCS(COV = 4.1%) FORM SORM 2 SMARTa SSRM MCS(COV = 4.4%) FORM SORM 2 SMART SSRM MCS(COV = 4.6%) 2
a
m 40 40 40 40 40 100 100 100 100 100 250 250 250 250 250
Pf 4
2.15 10 2.49 103 1.95 103 1.93 103 1.98 103 4.20 105 2.97 103 1.74 103 1.72 103 1.73 103 2.82 106 5.77 103 1.61 103 1.53 103 1.59 103
Relative Error (%)
Number of LSF evaluations
89.14 25.75 1.52 2.53 0.00 97.57 71.68 0.58 0.58 0.00 99.82 262.89 1.26 3.77 0.00
135 995 3729 198 3 105 315 5465 6036 348 3 105 765 32,390 10,707 734 3 105
Subset simulation by Support-vector Margin Algorithm for Reliability Estimation [54].
5. Conclusions In this paper, an efficient method for reliability analysis named SSRM is proposed. This method searches for new sample points, which involve the information about the joint probability density function and the limit state function, to refine the surrogate model of LSF iteratively. The sequential surrogate model captures more meaningful failure features in the important region, while reduces the samples in the region with low failure probability. In each iteration, MCS is used to evaluate the failure probability. Thus, a sequence of approximate failure probabilities is obtained. Several examples including numerical and engineering cases demonstrate that SSRM has a good adaptability to the LSFs with different nonlinearity and dimensions. For low-dimensional problems, SSRM has a significant advantage over existing methods in terms of accuracy and efficiency; for high-dimensional problems, SSRM has comparable precision to existing methods and higher efficiency. However, as the parameter optimization of the surrogate model, MCS, and GA take a number of surrogate evaluations, SSRM shows its superiority only when the LSFs are time-consuming implicit functions (e.g. finite element analysis). In the future study, the information of the computationally cheap low-fidelity model can be involved to construct a variable-fidelity surrogate model, so that the number of time-consuming high-fidelity sample points is expected to be further reduced.
Acknowledgements The research is supported by the Fundamental Research Funds for the Central Universities (No. G2016KY0302) and the National Natural Science Foundation of China (No. 51505385 and No. 11572134). The authors also thank Dr. Xueyu Li and the anonymous reviewers for the helpful work to improve the study.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.strusafe.2018.02. 005.
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Contents lists available at ScienceDirect
Structural Safety journal homepage: www.elsevier.com/locate/strusafe
A sequential surrogate method for reliability analysis based on radial basis function Xu Li a, Chunlin Gong a, Liangxian Gu a, Wenkun Gao a, Zhao Jing b,⇑, Hua Su a a b
School of Astronautics, Northwestern Polytechnical University, Xi’an, PR China Department of Mechanics, Huazhong University of Science and Technology, Wuhan, PR China
a r t i c l e
i n f o
Article history: Received 27 December 2016 Received in revised form 18 October 2017 Accepted 19 February 2018
Keywords: Reliability analysis Radial basis function Surrogate model Limit state function
a b s t r a c t A radial basis function (RBF) based sequential surrogate reliability method (SSRM) is proposed, in which a special optimization problem is solved to update the surrogate model of the limit state function (LSF) iteratively. The objective of the optimization problem is to find a new point to maximize the probability density function (PDF), subject to the constraints that the new point is on the approximated LSF and the minimum distance to the existing points is greater than or equal to a given distance. By updating the surrogate model with the new points, the surrogate model of LSF becomes more and more accurate in the important region with a high failure probability and on the LSF boundary. Moreover, the accuracy of the unimportant region is further improved within the iteration due to the minimum distance constraint. SSRM takes advantage of the information of PDF and LSF to capture the failure features, which decrease the samples of implicit LSF defined by expensive finite element analysis. Several numerical examples show that SSRM improves the accuracy of the surrogate model in the important region around the failure boundary with a small number of samples and has a better adaptability to the nonlinear LSF, hence increases the accuracy and efficiency of the reliability analysis. Ó 2018 Elsevier Ltd. All rights reserved.
1. Introduction In engineering design, when uncertainties are involved, the failure probability Pf of the limit state function (LSF) g(x) should be examined. Pf is essentially a multi-dimensional integral formulated as
Z Z Pf ¼
Z
pðxÞdx1 dx2 dxm
ð1Þ
D
where D is the failure region defined as D ¼ fxjgðxÞ 6 0; x 2 Rm g, and p(x) is the joint probability function. The numerical integral and direct Monte Carlo Simulation (MCS) are difficult for a complex system with implicit time-consuming analysis models. To calculate the integral with the original LSF faces enormous computational challenges [1–2]. Therefore, different approximations of the LSF are adopted to improve the computational efficiency of reliability analysis. The mean value method (MVM) [3–4] performs a first-order Taylor expansion of the LSF at the mean point (as shown in Fig. 1), in which the LSF is assumed to be normal distribution. ⇑ Corresponding author. E-mail address: [email protected] (Z. Jing). https://doi.org/10.1016/j.strusafe.2018.02.005 0167-4730/Ó 2018 Elsevier Ltd. All rights reserved.
The MVM is one of the most efficient reliability methods. However, it requires the independent input variables obey normal distribution, which is difficult to be satisfied in practical problems. Moreover, the approximation error increases with the increase of the nonlinearity. Therefore, this method is appropriate for fast estimation of the structural reliability with low nonlinearity. The first-order reliability method (FORM) [5–8] transforms random variables with different distributions into the same standard normal space U by Rosenblatt transformation [9] and then performs a first-order Taylor expansion at the most probable point (MPP) which has the maximum failure probability on the LSF (as shown in Fig. 1). Eventually, the normal distribution parameters (mean value and standard deviation) of LSF are figured out with the gradient of the approximation function, and then the failure probability is obtained analytically. Compared with the MVM, the FORM does not require the input variables to obey normal distribution and has a higher accuracy with the LSF expanded at the MPP. However, the optimization with an equality constraint to find the MPP increases the number of the LSF evaluations. Moreover, it increases the nonlinearity of the LSF g(x) during the Rosenblatt transformation, thus the approximation error is large when the nonlinearity of LSF is high [10–11].
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X. Li et al. / Structural Safety 73 (2018) 42–53
u2
x2
of SSRM with several numerical examples; conclusions are given in Section 5.
Failure Region g(x) 0 GSORM(u)=0 gMVM(x)=0 MPP u*
μ
2. Surrogate model GFORM(u)=0
2.1 Construction of surrogate model
||u*|| Safe Region g(x)>0
LSF G(u)=0
LSF g(x)=0 O
Original Design Space X
x1
O
Standard Normal Space U
u1
Fig. 1. Approximations of LSF in X and U space.
The second-order reliability method (SORM) [7,12–13] is similar to the FORM. First, the input random variables are transformed into the standard normal space U to get the LSF G(u). Second, G(u) is expanded with second-order Taylor expansion at the MPP to obtain a quadratic hypersurface. Finally, the reliability of the approximate model is analyzed with analytical methods [10]. The SORM has a higher nonlinear adaptability than the FORM, but it needs to perform the time-consuming second-order gradients. Moreover, the adaptability to nonlinear boundary is still limited [10–11]. The response surface is a different commonly used reliability analysis method [14–16]. By sampling in the neighborhood of the design point, the local approximation model of LSF is constructed. Since the response surface model evaluation time is far less than that of the original LSF, it is possible to use the approximate model for Mont Carlo Simulation (MCS) or Importance Sampling (IS) [17–19]. However, since the accuracy of the response surface method is poor with numbered samples, the complex structural behavior might not be captured. When considering factors such as the failure boundary and probability density, more samples are required to improve the local accuracy. Therefore, iterative response surface reliability analysis methods are proposed [20–30], which increases the samples in the important region near the MPP, and gradually improves the accuracy of the approximation model. This type of iterative method is also known adaptive or active learning method [31–32,45]. In general, the response surface approximation model can be replaced by other surrogate model (also known as meto-model) techniques such as radial basis function (RBF), Kriging, support vector regression (SVR), artificial neutral net (ANN), etc. [18,33–44]. The existing methods converge in a local region after increasing the sample density of the important region in some degree, but the accuracy does not increase in the less important region. In order to make full use of the information of the added samples to improve the approximate accuracy of the important and less important region, this paper proposes a sequential surrogate reliability method (SSRM) based on RBF. By adding points sequentially to the surrogate model, the failure features in the important region and on the boundary of the LSF are captured, and the failure probability is obtained with MCS by using the surrogate model. SSRM does not need to use the original LSF to search for MPP directly, but rather becomes close to the important area near MPP in the process of adding points, thus reducing the number of sample evaluations and avoiding the failure to find the optimal solution. Meanwhile, the SSRM method makes a trade-off between precision and computational cost. The remainder of this paper is structured as follows. Section 2 introduces the surrogate model technology used in SSRM; Section 3 describes the implementation process of SSRM, and analyzes its characteristics; Section 4 verifies the effectiveness and efficiency
The goal of the surrogate model is to construct a prediction model of a complex or unknown model with the observation samples. Instinctually, surrogate model is an interpolation or regression model, which is also a branch of machine learning [47]. The common surrogate models include polynomial response surface method (PRSM) [48–49], radial basis function (RBF) [48–49], Kriging [50–52], support vector regression (SVR) [50–52] and artificial neutral net (ANN) [53]. Since RBF has good nonlinear adaptability and is easy to implement, this paper constructs the sequential surrogate model with RBF. Assume that the observation samples are presented as
S ¼ fðxi ; yi Þji ¼ 1; 2; . . . ; ng
ð2Þ
denoted by a matrix form
X ¼ ½x1 ; x2 ; . . . ; xn T
ð3Þ
y ¼ ½y1 ; y2 ; . . . ; yn T
where n is the sample size, X denotes the input sample matrix, y is the output sample vector. x is an m-dimensional design variable. RBF uses a series of linear combinations of radial basis functions to approximate the expensive limit state function, which can be formulated as
b y ðxÞ ¼
n X
bi fðkx xi kÞ ¼ fðxÞT b
ð4Þ
i¼1
^ðxÞ denotes the predictive response at point x, bi is the ith where y component of the radial base coefficient vector b, and fðkx xi kÞ (see Table 1) is the ith component of the radial basis function vector fðxÞ. As shown in Table 1, r ¼ kx xi k is the Euclidian distance between two samples, and c is the shape parameter. Substitute the samples of Eq. (2) into Eq. (4),
3 32 3 2 b1 y1 fðr11 Þ fðr 12 Þ fðr1n Þ 6y 7 6 fðr Þ fðr Þ fðr Þ 76 b 7 22 2n 76 2 7 6 27 6 21 76 7 6 7 ¼ 6 .. .. .. 76 .. 7 6 .. 7 6 .. 4 . 5 4 . . . . 54 . 5 bn yn fðr n1 Þ fðr n2 Þ fðr nn Þ 2
ð5Þ
Rewritten as a matrix form
y ¼ Fb
ð6Þ
As the samples are different with each other, F 2 R
nn
is a non-
singular matrix, and Eq. (6) has a unique solution b ¼ F1 y. Thus the prediction model is given by
b y ðxÞ ¼ fðxÞT F1 y
ð7Þ
where f(x) is related to the prediction point x and sample input matrix X; F1 y is only related to X and y. For a new prediction ^ðxÞ. sample x, f(x) is calculated one time to get its predicted valuey
Table 1 Radial basis functions. Type
Function form fðrÞ
Gaussian Inverse Multiquadric
expðcr2 Þ
Thin plate spline
1=2
ð1 þ cr 2 Þ r 2 logð1 þ cr 2 Þ
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X. Li et al. / Structural Safety 73 (2018) 42–53
It should be pointed out that the shape parameter c, which has a great influence on the accuracy of the model, is included in F and is determined by experience or other optimization criteria. This paper uses the cross-validation criteria to optimize the shape parameter c. 2.2. Validation of surrogate model A common method to estimate the accuracy of the surrogate model is the root mean square errors (RMSE). However, since RBF requires the model strictly goes through sample points, RMSE is a constant zero, leads to the failure of the model estimation. To avoid the problem, the cross-validation method is adopted. The samples are divided into K roughly equal-sized parts. For the kth part (k = 1, 2, , K), the model is fitted with the other K 1 parts of the samples, and calculates the prediction error of the fitted model when predicting the kth part of the samples [47]. Crossvalidation can fully reflect the matching degree of the samples and the model. In particular, when K is equal to the sample size n, it is called leave-one-out cross-validation error (LOOCV) [47]. Thus, the RBF shape parameter c can be estimated by the following sub-optimization problem:
min c
n X 2 ½yi b y ðxi ; S fðxi ; yi ÞgÞ
ð8Þ
i¼1
In Eq. (8), it can be seen that the evaluation of LOOCV error requires n times construction of the surrogate model. However, the LOOCV error does not require additional verification points, which is capable of describing the matching degree between the samples and the prediction model. According to the surrogate model and obtained parameter c, a reliability analysis method based on the sequential surrogate model can be constructed. 3 Sequential surrogate reliability method This section presents the detailed procedure of the proposed sequential surrogate reliability method (SSRM). SSRM uses the sequential surrogate model of LSF to add samples to refine the surrogate model, which is also named active learning method in the community of machine learning. To get more meaningful failure information of LSF, the added samples are selected in the neighborhood of the most probable point (MPP) and on the LSF hypersurface. However, when the added sampled is too close to the existing samples, the added information is small. Therefore, a constraint of minimum distance is required. With the criteria of adding samples, SSRM reduces the samples in the unimportant region with small failure probability. In the iterative process, the failure boundary of the LSF is approached step by step, and eventually a surrogate model of LSF which captures most failure information is obtained. In order to facilitate the description of the algorithm, the independent random variables in original design space X are transformed into standard normal space U with Rosenblatt transformation [9], which are formulated as
ui ¼ F1 U i FX i ðxi Þ; ði ¼ 1; 2; ; mÞ
8 8 b > > Gðu; Sk1 Þ ¼ 0 > < > > > > uk ¼ argmin kuk s:t: dðu; Sk1 Þ P dmin ðSk1 Þ > > > u > : > > u L 6 u 6 uU ; u 2 R m > > > > < DSk ¼ fðDuk ; gðDuk ÞÞg
ð10Þ
where g(x) is the LSF in the X space and G(u) is the LSF in the U space.
ð11Þ
Sk ¼ Sk1 [ DSk > > > > b > > P k ¼ PMCS f Gðu; Sk Þ 6 0g; k ¼ 1; 2; 3; > > > > b > S0 Þ 6 0g Start : S0 ; P0 ¼ PMCS f Gðu; > > > > jP k Pk1 j : Stop :k P k 6 er Þ max or ðjP k P k1 j 6 ea & P k
where Duk is the sample added in the kth iteration, u denotes the Euclidean distance between u and the coordinate origin O. Since the joint probability density function (PDF) of the standard random variables is given by
! 2 m Y ui 1 kuk2 m=2 pffiffiffi exp : pðuÞ ¼ exp ¼2 2 2 2 i¼1
ð12Þ
when u gets the minimum value, p(u) has the maximum failure probability (see Fig. 2). Sk is the sample set in the kth iteration, ^ Gðu; Sk Þ represents the surrogate model of LSF based on Sk , ui is the ith input variable of the sample set Sk1 which totally has a number of jSk1 j samples. dðu; S k1 Þ is the minimum distance between the currently sample and the existing samples, and dmin ðS k1 Þ is the target minimum distance (for more details see Step 4). uL and uU are the lower and upper bounds of variables, m is the dimension of LSF, P k is the failure probability in the kth iteration, S0 is the initial sample set constructed with Latin hypercube sampling (LHS) [46], P0 is the failure probability based on the Monte Carlo simulation (MCS) with the initial surrogate model of LSF, kmax is the maximum number of samples allowed to add, ea and er denote absolute and relative errors, respectively. The procedure of SSRM is shown in Fig. 3, and more detailed steps of the algorithm are summarized as follows: Step 1: Selecting the initial sample points by LHS and calculating the responses of LSF to obtain the initial sample set S0 . Since the interval probability Pf5 6 U i 6 5g 0:99999943 of each dimension contains most information, here we set the lower and upper bounds of LHS to be 5 and 5 respectively. For the problem with small standard deviation, the initial sample number can be taken as jS0 j ¼ m þ 1, while for the large standard deviation and highnonlinear problems, the initial sample number can be set as jS0 j ¼ 2m þ 1, where m is the dimension of random variables. Step 2: Using the initial samples to construct the initial surro^ gate model of LSF, Gðu; S0 Þ , and optimizing the model shape u2
x2 Failure Region g(x) 0
G(u;S0)=0
ð9Þ
where X i is the ith random variable in the original space, U i is the ith standard normal random variables, xi and ui are the values of the random variables, FX i ðÞ and FUi ðÞ are cumulative distribution functions (CDF) of X i and U i , and m is the dimension of the variables. Therefore, the LSF in a vector form is formulated as
gðxÞ ¼ g½F1 X FU ðuÞ ¼ GðuÞ
The following algorithm is implemented in the U space. The key step of SSRM is an iteration process with special initial and terminal conditions, which can be briefly described as follows
μ
d(u;S0)
u
||u|| Safe Region g(x)>0 G(u)=0
g(x)=0 O
Original Design Space X
x1
O
Standard Normal Space U
u1
Fig. 2. Initial surrogate in U-Space. ‘‘ ” denotes the initial samples, and ‘‘ denotes the added sample based on the initial surrogate model.
”
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X. Li et al. / Structural Safety 73 (2018) 42–53
differentiability is adopted here, and it is complemented with the MATLAB optimization toolbox. In addition, the minimum distance between the current sample u and the existing samples ui is given by
Start
Step 1
LHS
dðu; Sk1 Þ ¼ min fku ui kjðui ; gðui ÞÞ 2 Sk1 g ui
Initial LSF Evaluation 1
...
Step 2
Constructing Initial Surrogate Model
Step 3
Initial Reliability Analysis
Step 4
Surrogate Model Optimization
Step 5
Sample Set Update
Step 6
Surrogate Model Update
Step 7
MCS Reliability Analysis
Initial LSF Evaluation n
To give a reasonable value of the target minimum distance, the number of samples, the dimension of LSF and the interval width of the variables in each dimension should be taken into account. Such a target value can be defined as
dmin ðSk1 Þ ¼ kmaxminfkui uj kjðui ; gðui ÞÞ; ðuj ; gðuj ÞÞ 2 Sk1 ; i–jg ui
Step 8
where k is a scale factor in the range 0.1 to 0.5 (this paper takes k ¼ 0:2), which balances the local and global search. Moreover, the optimization problem contains an equality constraint. Therefore, the total constraints are very strong, which makes GA inefficient and difficult to converge in the high-dimensional problem. According to the idea of penalty function method, the optimization problem with strong constraints in Eq. (11) can be transformed into an approximate unconstrained optimization problem:
u
k=k+ 1 No
End
Fig. 3. Flow chart of SSRM.
parameter c. Here the surrogate model uses RBF which strictly goes through the sample points and has strong nonlinear adaptability (see Sections 2.1 and 2.2) and is built with the authors’ in-house MATLAB toolbox. ^ Step 3: Using the initial surrogate model Gðu; S0 Þ for reliability analysis with MCS. In order to eliminate the errors induced by different random samples, the MCS reliability analysis uses the same random samples which can be reproduced with the same random seed. Therefore, the initial failure probability ^ P 0 ¼ PMCS fGðu; S0 Þ 6 0g is presented as N 1X b b j ; S0 ÞÞ PMCS f Gðu; S0 Þ 6 0g Ið Gðu N j¼1
ð13Þ
ð14Þ
where uj is an m-dimensional sample generated by the standard normal distribution, N is the total number of samples for MCS, and IðÞ is the indicator function, defined as
IðxÞ ¼
1; if x 6 0 0; if x > 0
dðu;Sk1 Þ 1þexpð20ðdðu;Sk1 Þdmin ðSk1 ÞÞÞ
i
ð18Þ
u is too close to existing samples, say dðu; Sk1 Þ < dmin ðSk1 Þ, the objective function increase rapidly, which ensures the original inequality constraint to work. As an unconstrained optimization problem, the transformed problem is easier to solve for the highdimensional problems. However, the process of searching for new samples is not as accurate as the original optimization problem, which decreases the global accuracy of the surrogate model. Step 5: Estimating the sample found in Step 4 with practical LSF GðuÞ to obtain the sample set DSk ¼ fðDuk ; GðDuk ÞÞg and then updating the sample set Sk ¼ Sk1 [ DSk . ^ Step 6: Updating the surrogate model of LSF Gðu; Sk Þ and optimizing the shape parameter c. Step 7: Estimating the failure reliability with the updated surrogate model with MCS, which can be presented as ^ Pk ¼ PMCS fGðu; Sk Þ 6 0g and calculated with Eq. (13). Here the samples used in MCS are the same as Step 3. Step 8: Convergence check. If one of the termination conditions, (a) the maximum iteration condition k P kmax , (b) the relative convergence condition jP k Pk1 j=Pk 6 er ð¼ 1 104 Þ and the absolute
and the coefficient of variation (COV) for MCS is given by
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 PMCS COVðPMCS Þ ¼ P MCS N
^ ð1 þ kukÞjGðu; Sk1 Þj
^ where the symbols are the same as Eq. (11). If Gðu; Sk1 Þ ¼ 0, the objective function gets the minimum value. However, if ^ Gðu; Sk1 Þ 0, the objective function value decreases with u. When
Yes Step 9
uj
ð17Þ
Duk ¼ argmin h Satisfying the stop criterion?
ð16Þ
ð15Þ
Step 4: Solving the optimization problem in Eq. (11) to search for the new sample. As the optimization problem has an inequality constraint of minimum distance which is not differentiable, gradient-based optimization algorithm, such as sequential quadratic programing (SQP), cannot be utilized here. Therefore, the genetic algorithm (GA) which is not restricted by the
convergence condition jPk Pk1 j 6 ea ð¼ 1 103 Þ, is satisfied, go to Step 9, otherwise set k = k + 1 and then go to Step 4. Step 9: End of SSRM. The main procedures of SSRM are the construction of surrogate model and the criterion to add points, which are also the core to guarantee the accuracy and efficiency in the iteration process. SSRM automatically increases the samples in the important region with large failure probability, which makes full use of the information of PDF and each sample. In addition, since the minimum distance constraint of the samples works, SSRM increases the number of the samples in the less important region when the important region includes enough points. Therefore, SSRM does not require many initial samples. After each surrogate model update, MCS reliability analysis is performed to achieve the estimation of failure probability. Since most of the failure features near the MPP and the LSF boundary are captured by the surrogate model
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X. Li et al. / Structural Safety 73 (2018) 42–53
Table 2 Distributions of random variables for the circular pipe structure. Variables
Mean
Standard deviation
Distribution
rf
301.079 0.503
14.78 0.049
Normal Normal
h
iteratively, the MCS based estimation has a good accuracy. Moreover, if the time of a single evaluation of the surrogate model is far less than that of the original LSF, the cost of model parameter optimization, MCS and GA based on the surrogate model is acceptable. Therefore, SSRM shows its superiority of accuracy and efficiency, when LSF is defined by time-consuming model, such as expensive finite element analyses. 4. Numerical examples In this section, seven examples are used to perform SSRM, and the results are compared with those of MCS and some other existing methods. Example 1–3 are of different nonlinearity, example 4 is the LSF of a speed reducer shaft with variables under different distributions, example 5 is a cantilever tube structure with highdimensional variable space, example 6 is a nonlinear oscillator with six random variables, and example 7 demonstrates the sensitivity of SSRM in high-dimensional problems. Assuming the transformed random variables in U space to be ui 2 ½5; 5; i ¼ 1; 2; ; m, the initial samples are selected by LHS, and the sample size is m + 1 or 2m + 1, where m is the number of the random variables. In these examples, FORM and SORM are realized by Isight5.6, a software for multidisciplinary design optimization (MDO), while SSRM and MCS are implemented with the authors’ in-house Matlab codes. In order to quantify the error, results of MCS are taken as the references, and random seeds are selected to obtain the same samples for MCS in each example.
Considering a circular pipe with circumferential through-wall crack subjected to a bending moment, the LSF is given by [4]:
ð19Þ
where rf , h, M, R and t are flow stress, half-crack angle, applied bending moment, radius of the pipe and thickness of the pipe, respectively. R = 0.3377 m, t = 0.03377 m, M = 3 106 Nm. rf and h are random variables. The statistical properties of the random variables involved in the problem are illustrated in Table 2. As shown in Fig. 4, three initial samples are selected and four extra samples are added iteratively. The approximate LSFs of FORM, SORM and SSRM are close to each other in the neighborhood of the MPP due to the low nonlinearity. In the region far away from the MPP, the approximate LSF of FORM has a poor accuracy, while SORM and SSRM have better accuracies with the approximate LSFs closer to the original LSF. As shown in Table 3, SSRM has the best accuracy with a relative error of 0.0175%. As the initial ^ SSRM ðuÞ in Fig. 4(a) is sample size is small, the initial surrogate LSF G far away from the original LSF, however, after 4 iterations, they are almost the same (See Table 4). 4.2. Hyper-sphere bound problem This example considers a hyper-sphere bound with higher nonlinearity, and the LSF is given by
gðXÞ ¼ 1 X 31 X 32
4.3. Cantilever beam This is also an example with higher nonlinearity, a cantilever beam with rectangular cross section subjected to uniformly distributed loading [20,53]. The LSF with respect to the maximum deflection at the free end being greater than l/325, is given by 4
g¼
l wbl 325 8EI
ð20Þ
ð21Þ
where I = bh3/12, where w, b, l, E, I and h are load per unit, width, span, modulus of elasticity, moment of inertia of the cross section and depth, respectively. The random variables are w and h, as shown in Table 6. Assuming E and l are 2.6 104 MPa and 6 m respectively, the LSF is reduced to
gðXÞ ¼ 18:46154 74769:23
4.1. Circular pipe structure
h 1 sinðhÞ M gðXÞ ¼ 4t rf R2 cos 2 2
As shown in Fig. 5, three initial samples are selected and nine extra samples are added iteratively. As expected, Table 5 illustrates that SSRM obtains more accurate result with fewer samples. Since the approximation of the LSF can be improved step by step in the process of adding points, the accuracy is significantly improved. FORM and SORM cannot capture most of the failure features of the important region due to the high nonlinearity, however SSRM updates the surrogate of LSF, thus a good approximation in the important region and its neighborhood are obtained. As shown in Fig. 5 (a), the initial surrogate model is not accurate, and the added points are far away from the original LSF at first, however, as the iteration goes on, the added points get closer to the LSF. Thus, each point makes contributions to the improvement of the accuracy of the LSF boundary, especially in the important region. Meanwhile, the region far away from the MPP and the LSF boundary has a few samples and the probability is very small, hence the region has little effect on the estimation. That is the reason why SSRM improves the accuracy with fewer samples.
X1 X 32
ð22Þ
As shown in Fig. 6, five initial samples are selected and 13 extra samples are added iteratively. Table 7 and Fig. 6 show that SSRM obtains a good accuracy with Pf = 0.009499 and 18 LSF evaluations. Meanwhile, Fig. 6(a) shows that the added samples are all near the LSF boundary, and keeps minimum distance to each other. It means that all the samples contribute a lot to obtain the failure probability with a good approximation to the LSF boundary. Moreover, it can be observed that the accuracy in the region near u = (4, 4) is relatively poor. However, since the region is far from the mean point u = (0, 0), the probability is very small, which means that the poor accuracy does not affect the estimation of the failure probability (See Table 8). 4.4. Speed reducer shaft In this section, considering the LSF of the reducer axis of multivariate random variables with different distributions [9]
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðXÞ ¼ S 32=ðpD3 Þ F 2 L2 =16 þ T 2
ð23Þ
In this example, 6 initial samples are selected and 38 extra samples are added iteratively. Fig. 7 shows that the initial value of Pf is far from the MCS result, due to the small number of samples. However, with the increase of the samples during the iteration, the capture ability of the SSRM for the failure boundary is increased. Hence, Pf converges in a few iterations. As expected, the results in Table 9 show that SSRM has a good adaptability and accuracy for this multidimensional problem (See Fig. 8).
X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 4. Iterative process of SSRM for the circular pipe structure.
47
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X. Li et al. / Structural Safety 73 (2018) 42–53
Table 3 Comparison of different reliability methods for the circular pipe structure.
Table 5 Comparison of different reliability methods for the hyper-sphere problem.
Methods
Pf
Relative Error (%)
Number of LSF evaluations
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM SSRM MCS(COV = 0.14%)
0.033065 0.034211 0.034347 0.034353
3.7493 0.4134 0.0175 0.000
9 14 7 1 106
FORM SORM SSRM MCS(COV = 0.53%)
1.891 102 2.672 102 3.381 102 3.381 102
44.070 20.970 0.000 0.000
15 20 12 1 106
Table 6 Distributions of random variables for the cantilever beam.
Table 4 Distributions of random variables for the hyper-sphere problem. Variables
Mean
Standard deviation
Distribution
X1 X2
0.5 0.5
0.2 0.2
Normal Normal
Variables
Mean
Standard deviation
Distribution
X1 X2
1000 250.0
200 37.5
Normal Normal
4.5. Cantilever tube problem
I ¼ ðp=64Þ½d ðd 2tÞ J ¼ 2I 4
To further demonstrate the performance of SSRM in the case of different distributions in high dimensions, consider the following cantilever tube problem, the LSF is defined as follows [6]: (See Table 10).
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gðXÞ ¼ Sy r2x þ 3s2zx where
szx ¼ Td=ð2JÞ M ¼ F 1 L1 cosðh1 Þ þ F 2 L2 cosðh2 Þ A ¼ ðp=4Þ½d ðd 2tÞ h ¼ d=2 2
ð24Þ
4
2
rx ¼ ðP þ F 1 sinðh1 Þ þ F 2 sinðh1 ÞÞ=A þ Mh=I
Fig. 5. Iterative process of SSRM for the hyper-sphere problem.
ð25Þ
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X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 6. Iterative process of SSRM for the cantilever beam.
Table 7 Comparison of different reliability methods for the cantilever beam.
a b c d e
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM DSa DS + RSb DS + SPc DS + NNd ISe IS + RS IS + SP IS + NN SSRM MCS(COV = 1.0%)
0.00988 0.00988 0.01000 0.00600 0.00700 0.00800 0.01000 0.00900 0.01000 0.01200 0.009499 0.009594
2.981 2.981 4.232 37.46 27.04 16.61 4.232 6.191 4.232 25.08 0.9902 0.0000
27 32 551 60 57 40 9312 2192 358 63 18 1 106
Directional Sampling. Response Surface. Splines. Neural networks. Important Sampling. These results are from Schueremans [53]. Fig. 7. Iterative process of SSRM for the speed reducer shaft.
Table 8 Distributions of random variables for the speed reducer shaft. Meana\ Lower Boundb
Variables
Diameter D(mm) Span L(mm) External Force F(N) Torques T(Nmm) Strength S(MPa) a b
Standard deviationa\ Upper Boundb
39 400 1500 250,000 70
0.1 0.1 350 35,000 80
Table 9 Comparison of different reliability methods for the speed reducer shaft.
Distribution
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM SSRM MCS(COV = 1.1%)
7.14392 104 7.12341 104 7.52 103 7.52 103
90.50 90.53 0.00 0.00
30 50 44 1 106
a
Normal Normala Grumble(Max) Normala Uniformb
a
The distribution parameters are mean and standard deviation respectively. The distribution parameters are lower and upper bound respectively.
Constants h1 ¼ 5 ; h2 ¼ 10 . In this example, 10 initial samples are selected and 8 extra samples are added iteratively. As a nine-dimensional problem, the
computational cost for FORM and SORM to solve the MPP increases dramatically but not improve the accuracy effectively, while SSRM converges to the MCS result with 8 iterations (see Fig. 9). Though the dimension of the variables is high, the uniform distribution interval and the coefficient of variation are small, thus the
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X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 8. Cantilever tube [6].
Table 10 Distributions of random variables for the cantilever tube. Variables
Meana\ Low Boundb
Standard deviationa\ Up Boundb
Distribution
t(mm) d(mm) L1(mm) L2(mm) F1(N) F2(N) P(N) T(Nmm) Sy(MPa)
5 42 119.75 59.75 3000 3000 12,000 90,000 220
0.1 0.5 120.25 60.25 300 300 1200 9000 22
Normala Normala Uniformb Uniformb Normala Normala Gumbela Normala Normala
Fig. 9. Iterative process of SSRM for the cantilever tube.
Table 11 Comparison of different reliability methods for the cantilever tube.
nonlinearity near the design point is not very high. Therefore, the failure features can be captured by SSRM with a few samples; accordingly, a good estimation of the failure probability (see Table 11) is achieved (See Fig. 10).
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM SSRM MCS(COV = 3.1%)
3.8644 104 3.8154 104 1.0520 103 1.0460 103
63.06 63.52 0.574 0.000
30 84 18 1 106
4.6. Dynamic response of a nonlinear oscillator This example consists of a nonlinear undamped single degree of freedom system with six random variables [20,34,53]. The LSF is given by
gðc1 ; c2 ; m; r; t 1 ; F 1 Þ ¼ 3r j
2F 1 x0 t 1 sinð Þj 2 mx20
4.7. High-dimensional problem This example which is from Bourinet [54] aims to evaluate the performance of SSRM to deal with high-dimensional problems. Failure probabilities with different numbers of random variables are discussed. The LSF is formulated as (See Table 14).
pffiffiffiffiffi gðXÞ ¼ m þ 3r m
Xi
i¼1
c1
ð27Þ
F(t)
m
ð26Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where x0 ¼ ðc1 þ c2 Þ=m. The random variables are given in Table 12. In this problem, 13 initial samples are selected and 6 extra samples are added as shown in Fig. 11. As expected, the Pf of SSRM in Table 13 is also close to that of MCS with a relative error of 1.623%. Since the result of MCS is also with a coefficient of variation of 2.2%, the relative error of SSRM is acceptable. However, the total LSF evaluations of SSRM are 19, which is smaller than the existing methods (See Fig. 12).
m X
F(t)
z(t)
c2 t1
t
Fig. 10. Nonlinear oscillator [34].
Table 12 Distributions of random variables for the nonlinear oscillator. Variables
Mean
Standard deviation
Distribution
m c1 c2 r F1 t1
1 1 0.1 0.5 0.5 1
0.05 0.1 0.01 0.05 0.2 0.2
Normal Normal Normal Normal Normal Normal
where m is the number of random variables, X i ði ¼ 1; 2; ; mÞ are independent random variables which follow the lognormal distribution with mean value l = 1 and standard deviation r ¼ 0:2. The transformed LSF in U space is presented as
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X. Li et al. / Structural Safety 73 (2018) 42–53
Fig. 11. Iterative process of SSRM for the nonlinear oscillator.
Fig. 12. Iterative process of SSRM for the high-dimensional problem for m = 40, 100 and 250.
Table 13 Comparison of different reliability methods for the nonlinear oscillator.
a b c d e f g h
Methods
Pf
Relative Error (%)
Number of LSF evaluations
FORM SORM DSa DS + RSb DS + SPc DS + NNd ISe IS + RS IS + SP IS + NN AKf + MCS+Ug AK + MCS+EFFh SSRM MCS(COV = 2.2%)
3.108 102 2.840 102 3.500 102 3.400 102 3.400 102 2.800 102 2.700 102 2.500 102 2.700 102 3.100 102 2.834 102 2.851 102 2.880 102 2.834 102
9.668 0.212 23.500 19.972 19.972 1.200 4.728 11.785 4.728 9.386 0.000 0.600 1.623 0.000
35 48 1281 62 76 86 6144 109 67 68 58 45 19 7 104
m pffiffiffiffiffi X l2 gðUÞ ¼ m þ 3r m exp ln pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ U i l2 þ r2 i¼1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi! ln
r2 þ1 l2
ð28Þ
Directional Sampling. Response Surface. Splines. Neural networks. Important Sampling. These results are from Schueremans et. al. (2005) [53]. Active Learning. Learning Function U. Expected Feasibility Function. These results are from Echard et al. (2011) [34].
In this example, three cases (m = 40, 100 and 250) are discussed. The results are shown in Table 15, where results of FORM, SORM, 2 SMART and MCS are from reference [54]. As shown in Table 15, for m = 40, 81 initial samples are selected and 117 extra samples are added iteratively; for m = 100, 201 initial samples are selected and 147 extra samples are added iteratively; for m = 250, 501 initial samples are selected and 233 extra samples are added iteratively. Fig. 11 shows the iterative processes of different cases. As expected, the number of added samples increases with the dimension of LSF. Moreover, when the dimension is high, it is difficult to find the accurate most probable point, and the accuracy of Taylor approximation is poor, therefore FORM and SORM have more errors. The accuracy of SSRM is slightly lower than 2SMART, but the number of LSF evaluations is just about 5.3% to 6.9% of 2SMART. Thus, SSRM balances the accuracy and the number of sample points.
Table 14 Iteration process of SSRM for the nonlinear oscillator. Iteration No.
u1(m)
u2(c1)
u3(c2)
u4(r)
u5(F1)
u6(t1)
Pf
0
2.140 4.899 3.734 3.031 2.600 0.328 1.125 3.605 1.003 1.717 1.776 3.119 4.862 0.109 0.007 0.325 0.010 0.074 0.081
0.125 1.441 1.061 3.786 3.502 2.584 3.443 3.345 2.119 4.542 0.992 1.315 4.384 0.428 0.037 0.761 0.238 1.080 0.433
2.975 2.518 4.171 2.319 1.669 0.878 0.331 4.844 1.447 4.954 2.996 0.628 4.017 0.152 0.041 0.012 0.018 0.016 0.270
3.075 4.468 4.113 3.929 4.875 0.055 1.238 1.590 0.900 2.308 0.777 1.988 3.219 0.161 0.702 0.008 0.750 0.254 -0.817
2.505 0.549 0.357 4.085 1.794 3.379 4.870 2.749 2.168 1.326 3.906 0.608 4.269 1.307 0.806 1.546 1.742 1.600 1.157
1.525 2.988 1.517 0.482 2.596 4.064 0.467 4.786 3.782 2.092 4.729 0.191 2.870 1.136 1.641 1.947 0.759 1.105 1.154
0.04303
1 2 3 4 5 6
0.02921 0.02789 0.02680 0.02904 0.02881 0.02880
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Table 15 Comparison of different reliability methods for the high-dimensional problem. Methods FORM SORM SMART SSRM MCS(COV = 4.1%) FORM SORM 2 SMARTa SSRM MCS(COV = 4.4%) FORM SORM 2 SMART SSRM MCS(COV = 4.6%) 2
a
m 40 40 40 40 40 100 100 100 100 100 250 250 250 250 250
Pf 4
2.15 10 2.49 103 1.95 103 1.93 103 1.98 103 4.20 105 2.97 103 1.74 103 1.72 103 1.73 103 2.82 106 5.77 103 1.61 103 1.53 103 1.59 103
Relative Error (%)
Number of LSF evaluations
89.14 25.75 1.52 2.53 0.00 97.57 71.68 0.58 0.58 0.00 99.82 262.89 1.26 3.77 0.00
135 995 3729 198 3 105 315 5465 6036 348 3 105 765 32,390 10,707 734 3 105
Subset simulation by Support-vector Margin Algorithm for Reliability Estimation [54].
5. Conclusions In this paper, an efficient method for reliability analysis named SSRM is proposed. This method searches for new sample points, which involve the information about the joint probability density function and the limit state function, to refine the surrogate model of LSF iteratively. The sequential surrogate model captures more meaningful failure features in the important region, while reduces the samples in the region with low failure probability. In each iteration, MCS is used to evaluate the failure probability. Thus, a sequence of approximate failure probabilities is obtained. Several examples including numerical and engineering cases demonstrate that SSRM has a good adaptability to the LSFs with different nonlinearity and dimensions. For low-dimensional problems, SSRM has a significant advantage over existing methods in terms of accuracy and efficiency; for high-dimensional problems, SSRM has comparable precision to existing methods and higher efficiency. However, as the parameter optimization of the surrogate model, MCS, and GA take a number of surrogate evaluations, SSRM shows its superiority only when the LSFs are time-consuming implicit functions (e.g. finite element analysis). In the future study, the information of the computationally cheap low-fidelity model can be involved to construct a variable-fidelity surrogate model, so that the number of time-consuming high-fidelity sample points is expected to be further reduced.
Acknowledgements The research is supported by the Fundamental Research Funds for the Central Universities (No. G2016KY0302) and the National Natural Science Foundation of China (No. 51505385 and No. 11572134). The authors also thank Dr. Xueyu Li and the anonymous reviewers for the helpful work to improve the study.
Appendix A. Supplementary data Supplementary data associated with this article can be found, in the online version, at https://doi.org/10.1016/j.strusafe.2018.02. 005.
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