Reliability Engineering and System Safety 142 (2015) 346–356
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Reliability Engineering and System Safety journal homepage: www.elsevier.com/locate/ress
A double-loop adaptive sampling approach for sensitivity-free dynamic reliability analysis Zequn Wang, Pingfeng Wang n Department of Industrial and Manufacturing Engineering, Wichita State University, Wichita, KS 67260, USA
art ic l e i nf o
a b s t r a c t
Article history: Received 2 November 2013 Received in revised form 24 April 2015 Accepted 15 May 2015 Available online 6 June 2015
Dynamic reliability measures reliability of an engineered system considering time-variant operation condition and component deterioration. Due to high computational costs, conducting dynamic reliability analysis at an early system design stage remains challenging. This paper presents a confidence-based meta-modeling approach, referred to as double-loop adaptive sampling (DLAS), for efficient sensitivityfree dynamic reliability analysis. The DLAS builds a Gaussian process (GP) model sequentially to approximate extreme system responses over time, so that Monte Carlo simulation (MCS) can be employed directly to estimate dynamic reliability. A generic confidence measure is developed to evaluate the accuracy of dynamic reliability estimation while using the MCS approach based on developed GP models. A double-loop adaptive sampling scheme is developed to efficiently update the GP model in a sequential manner, by considering system input variables and time concurrently in two sampling loops. The model updating process using the developed sampling scheme can be terminated once the user defined confidence target is satisfied. The developed DLAS approach eliminates computationally expensive sensitivity analysis process, thus substantially improves the efficiency of dynamic reliability analysis. Three case studies are used to demonstrate the efficacy of DLAS for dynamic reliability analysis. & 2015 Elsevier Ltd. All rights reserved.
Keywords: Reliability analysis Surrogate model Sequential sampling Dynamic Sensitivity free
1. Introduction Engineered systems generally degrade over time and could fail due to time-variant operational conditions and component deterioration, which may lead to catastrophic consequences such as substantial economic and societal losses. To measure the performance of engineered systems against potential system failures, reliability is defined as the probability that the system or component will perform the required function for a given period of time under inherent uncertainties and certain operation conditions. In the literature, two types of reliability analysis have been conducted, referred to as static reliability analysis and dynamic reliability analysis, depending on whether time-variant characteristics are considered in reliability analysis processes. To conduct static reliability analysis, various numerical methods, including both analytical and simulation-based approaches, have been developed, such as most probable point (MPP) based methods [1–3], dimension reduction method (DRM) [4–6], polynomial chaos expansion (PCE) [7–10] and Kriging-based methods
n
Corresponding author. E-mail addresses:
[email protected] (Z. Wang),
[email protected] (P. Wang). http://dx.doi.org/10.1016/j.ress.2015.05.007 0951-8320/& 2015 Elsevier Ltd. All rights reserved.
[11–14]. In addition, studies have also been done to handle multiply limit states [15,16] and epidemic uncertainty [17–19] in static reliability analysis. In MPP-based methods such as the first order reliability method (FORM), reliability index is calculated as the distance between the MPP and the origin in the U-space by iteratively locating MPP on the limit state function. Due to an iterative MPP searching process, sensitivity information of performance functions with respect to random variables is required in order to pinpoint the next potential MPP point and carry forward the searching process. However, accurate sensitivity information of the performance function is usually not readily available in practical engineering applications. The DRM simplifies a single multi-dimensional integration for reliability analysis to multiple one-dimensional integrations using an additive decomposition formula, and then estimates reliability based on statistical moments of system performance functions. Although it is a sensitivity-free approach for reliability analysis, the DRM may introduce significant error for limit state functions with high nonlinearity. The PCE method constructs a stochastic response surface with multi-dimensional polynomials over the sample space of random variables, updates the stochastic response surface by incorporating more samples and then approximates reliability directly using Monte Carlo simulation (MCS) based on the developed stochastic response surface. The accuracy of the PCE can be
Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
Nomenclature R
Φ βt EI
reliability standard Gaussian cumulative distribution function target reliability index expected improvement
improved by increasing the order of stochastic polynomial terms; however, the computational cost can be prohibitively high for problems with a large number of random input variables. Surrogate models have also been employed for reliability analysis to replace original computationally expensive simulation models, so that reliability can be approximated less expensively. For this type of approaches, major challenges include proposing appropriate metrics to quantify the accuracy of reliability estimation and developing efficient sampling schemes for surrogate models. Compared with static reliability analysis, performing dynamic reliability analysis is even more computationally expansive in practical engineering applications, because of the time-dependency of system failure events. In the literature, two categories of methods: extreme performance based approaches [20–22,26] and first-passage based approaches [23–25], have been developed for dynamic reliability analysis. The extreme performance approaches define the failure event according to extreme value of the performance function, and then quantify uncertainty of the extreme performances in order to approximate dynamic reliability. Instead of extreme performances, first-passage based approaches focus on out-crossing events, when the performance function exceeds the upper bound or falls below the lower bound of the safety threshold, and estimate dynamic reliability by computing an out-crossing rate measure. In the first category of dynamic reliability methods, the composite limit state (CLS) approach [26] has also been developed to tackle the timedependency issue and calculate the cumulative probability of failure based on MCS. As the CLS converts the continuous time to discrete time intervals and constructs a composite limit state by combining all instantaneous limit states of discretized time intervals in a series manner, it is extremely expensive to perform the dynamic reliability analysis using the CLS, as illustrated by reported case studies [26]. Recently, the nested extreme response surface method (NERS) [22] utilized the Kriging technique to efficiently identify extreme time responses corresponding to extreme performances, so that dynamic reliability can be performed by only focusing extreme events using existing static reliability tools such as the FORM and MCS. Although NERS can tackle the time-dependent issue efficiently, error can also be induced by using FORM as a static reliability analysis tool. As a representative of the first-passage methods, the PHI2 approach [27] was developed for dynamic reliability estimation, in which the FORM was also utilized to calculate out-crossing rates. Although static reliability analysis tools such as FORM can be integrated with the PHI2 method, the error of dynamic reliability estimation could be very significant for two reasons: high nonlinearity of the limit states and improper time step while discretizing the time variable. Another limitation of PHI2 is that it requires accurate sensitivity information of the performance function with respect to random input variables, which is usually not available in practical engineering applications. To handle time-dependency of system failure events and reduce extremely high computational costs in dynamic reliability analysis, this paper presents a confidence-based meta-modeling approach, referred to as double-loop adaptive sampling (DLAS), for efficient sensitivityfree dynamic reliability analysis. In order to evaluate dynamic reliability directly by MCS, Gaussian Process (GP) regression is adopted to construct a meta-model for extreme performance function over time while the DLAS technique is developed to enhance the fidelity of metamodel sequentially by considering the model input variables and time
fx(x) f( | ) If Pf vþ
347
probability density function conditional probability density function or likelihood function indicator function probability of failure out-crossing rate
concurrently in two sampling loops. The rest of paper is organized as follows. Section 2 introduces dynamic reliability analysis and existing methods. Section 3 details the developed DLAS approach for dynamic reliability analysis. Three case studies are used to demonstrate the effectiveness of the developed methodology in Section 4.
2. Review of dynamic reliability analysis For engineered systems, system failure events occur if system performance function goes beyond its failure thresholds. Consequently, a limit state function, denoted as G(x)¼ 0, can be defined which separates the safe and failure events in the random input space. For static reliability analysis, the probability of failure is defined as Z Z P f ¼ PrðGðxÞ o 0Þ ¼ ⋯ f x ðxÞdx ð1Þ GðxÞ o 0
where fx(x) is the joint probability density function. However, the performance function is also governed by the time-variant uncertainties such as loading conditions and component deterioration. Time parameter can be implicitly involved in the limit state function when input random processes are taken into account. In this work, we assume that the limit state function is an explicit function of the random variable x and time parameter t. Thus, a time-variant limit state function can be generally derived as G(x, t)¼ 0 by taking the time parameter t into account in reliability analysis. Let tl be the designed system life time of interest, the probability of failure within [0, tl] can be described as P f ð0; t l Þ ¼ Prð ( t A ½0; t l ; Gðx; t Þ o 0Þ
ð2Þ
Thus, the task of dynamic reliability analysis is to estimate the Pf in an efficient and accurate manner. The rest of this section provides a brief review of three representative dynamic reliability analysis approaches: the composite limit state (CLS) approach, the nested extreme response surface (NERS) approach, and the outcrossing rate approach. 2.1. Composite limit state approach If the CLS is used, the time interval [0, tl] will be discretized to NT time nodes with a fixed time step Δt. Let G(x, tn) ¼ 0 (n ¼1,…, NT) denotes the instantaneous limit state at the nth time node tn, the composite limit state is defined as the union event of all instantaneous limit states. The cumulative probability of failure can then be described as P f ð0; t l Þ ¼ Pr [ N ð3Þ n ¼ 0 Gðx; t n Þ o0 where failure occurs if any of the instantaneous limit states is violated. With the development of the composite limit state, dynamic reliability can be estimated using existing static reliability analysis tools; however, identifying composite limit states is computationally very expensive because it requires evaluations of all instantaneous performances for each design point. While using the CLS method, time variable is discretized to simplify the dynamic reliability
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Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
analysis, which may in turn induce error if time-dependent limit state functions are highly nonlinear within different time intervals. 2.2. Nested extreme response surface approach While employing the nested extreme response surface (NERS) method, dynamic reliability analysis is carried out by only focusing on instant time scenarios at each system design point that lead to extreme response of the system performance function by developing a nested extreme response surface of time. Let T(x) denotes the response surface of time corresponding to the extreme value of the limit state function for input x, expressed as TðxÞ ¼ tj min Gðx; tÞ; t A ½0; t l ð4Þ t
Therefore, the probability of failure can be derived as P f ð0; t l Þ ¼ PrðGðx; TðxÞÞ o0Þ
ð5Þ
To efficiently identify T(x) in NERS, a Kriging model is constructed and an adaptive response prediction and model maturation mechanism was developed based on the mean square error measure to concurrently improve the accuracy and computational efficiency of the NERS approach. With the response surface of time T(x), dynamic reliability analysis is converted into a static reliability analysis problem and performed using existing reliability analysis tools such as FORM and MCS. NERS is able to reduce computational costs significantly; however, the accuracy of dynamic reliability approximation could be sacrificed because of the use of the FORM for static reliability analysis in NERS.
For the out-crossing rate based approaches, probability of failure is approximated based on the out-crossing rate defined as Pr½Gðx; t Þ 4 0 \ G x; t þ Δt o0 v þ ðtÞ ¼ lim ð6Þ Δt Δt-0 Let N(0, tl) denotes the number of out-crossing events within the time interval [0, tl], the probability of failure can be approximated by
The bounds of Pf (0, tl) is expressed as max P f ðτÞ rP f ð0; t l Þ r P f ð0Þ þ E½N ð0; t l Þ
0 r τ r tl
3.1. Gaussian process based meta-modeling for dynamic reliability analysis For dynamic reliability analysis, the limit state function can be defined by extreme responses of performance function G(x,t) and expressed as Ge ðxÞ ¼ min Gðx; tÞ; t A ½0; t l ð10Þ Thus, the probability of failure is defined as Z Z P f ¼ PðGe ðxÞ o 0Þ ¼ ::: f x ðxÞdx
ð11Þ
Ge ðxÞ o 0
Due to generally very high computational costs for calculating the probability of failure using the MCS approach, Gaussian process (GP) based meta-modeling technique [28–34] can be generally employed to develop a surrogate model so that the MCS can be implemented efficiently based on the developed surrogate model. Let us consider k random input variables, denoted as x ¼[x1,…,xk]. Based on the GP model, extreme performances of a system are assumed to be Ge ðxÞ ¼ f ðxÞα þ SðxÞ þ ε T
ð12Þ
where f (x)¼ [f1(x),…,fb(x)] is the basis function vector, α ¼[α1,…, αb] is the regression coefficient vector, S(x) is a Gaussian stochastic process with zero mean and certain covariance matrix, and ε is the uncorrelated noise variable that follows standard Gaussian distribution. In this study, it is assumed that ε ¼ 0, since system responses are assumed to be obtained from the same simulation code instead of true experiments. In addition, the term fT(x) α is assumed to be a constant global mean μ in this study for two reasons. First, it is more reasonable to assume a global mean if there is no prior information about the global trend of the extreme performance function. Second, a global mean GP model generally owns adequate abilities in modeling complex engineered systems. With the above assumptions, the GP model for the extreme performance is derived as T
2.3. Out-crossing rate-based approach
P f ð0; t l Þ ¼ PrðfGðx; 0Þ o 0g [ fNð0; t l Þ 4 0gÞ
for sensitivity-free dynamic reliability analysis. Section 3.1 presents the GP meta-modeling technique for dynamic reliability analysis and the confidence level of reliability approximation using GP models. Section 3.2 details the developed DLAS approach for efficient updating of GP models while Section 3.3 summarizes the numerical procedure of employing DLAS for dynamic reliability analysis.
ð7Þ
ð8Þ
where the mean number of out-crossing events is computed as Z tl v þ ðtÞdt ð9Þ E½N ð0; t l Þ ¼ 0
Out-crossing rate based approach provides a rigorous mathematical derivation of the dynamic reliability while it also suffers the difficulty of evaluating the out-crossing rates for the general stochastic processes. Given the limitations of existing dynamic reliability methods, this paper presents a new double loop adaptive sampling (DLAS) approach for dynamic reliability analysis based on surrogate models, in which the accuracy of dynamic reliability approximation is quantified by a new confidence measure developed in this study and the efficiency can then be guaranteed by a novel double loop adaptive sampling mechanism.
3. DLAS approach for dynamic reliability analysis This section presents a confidence-based meta-modeling approach, referred to as double-loop adaptive sampling (DLAS),
Ge ðxÞ ¼ μ þ SðxÞ:
ð13Þ
Note that a variety of covariance functions could be used for GP models [28], and in this study a commonly used Gaussian covariance function is adopted, in which the covariance function between two input variables xi and xj is expressed as Covði;jÞ ¼ σ 2 Rði;jÞ
ð14Þ
where R represents the correlation matrix. The (i, j) entry of R is described as " # k X p p bp Rði;jÞ ¼ Corrðxi ; xj Þ ¼ exp ap j xi xj j ð15Þ p¼1
where Corr is the correlation function, and ap and bp are parameters of the GP model to be determined. With n number of observations (x, Ge(x)) in which x¼ [x1,…,xn] and Ge(x)¼ [Ge(x1),…, Ge(xn)] for i¼1,…,n, the log likelihood function of the GP model can be provided as 1 1 LGP ¼ n lnð2π Þ þ n ln σ 2 þ lnjRj þ 2 ðGe AμÞT R 1 ðGe AμÞ 2 2σ ð16Þ
Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
where A is an n 1 unit vector. Then μ and maximizing the likelihood function as h i1 μ ¼ A T R 1 A A T R 1 Ge
σ2 ¼
ðGe AμÞT R 1 ðGe AμÞ n
σ2 can be obtained by ð17Þ ð18Þ
With the GP model being developed, the response for any given new input point x’ can be estimated as ^ e ðx'Þ ¼ μ þ rT R 1 ðGe AμÞ G
ð19Þ
where r is the correlation vector between xʹ and the samples x ¼ [x1,…,xn], of which the ith element of r is given by r(i)¼ Corr(xʹ,xi). The mean square error e(xʹ) can be estimated by " # ð1 AT R 1 rÞ2 e^ ðx'Þ ¼ σ 2 1 rT R 1 r þ ð20Þ AT R 1 A Let Ω¼{x, | Ge(x) o0} denotes the failure region, thus the probability of failure can be expressed as Z P f ¼ Prðx A ΩÞ ¼ IðxÞf x ðxÞdx ¼ E½IðxÞ ð21Þ Rk
where E[ ] denotes the expectation operator; Rk is a real number space; I(x) is an indicator function and defined as ( ^ e ðxÞ o0 1; G IðxÞ ¼ ð22Þ 0; otherwise To employ the MCS for dynamic reliability analysis, N random samples X ¼[x1,…,xN] are generated according to the randomness of input variables. The dynamic reliability can be calculated by R ¼ 1 Pf ¼ 1
N 1X Iðx Þ Ni¼1 i
ð23Þ
For the ith MCS sample X ¼[x1,…,xN], xi can be simply classified as failure or safe according Eq. (22). The probability of the correct classification for xi is accordingly obtained as
^
Ge ðxi Þ
Prc ðxi Þ ¼ Φð pffiffiffiffiffiffiffiffiffiffi Þ ð24Þ e^ ðxi Þ where | | is the absolute operator. Thus the confidence of reliability approximation using MCS based on the GP model is obtained as C ðGP;MCSÞ ¼ E½Prc ¼
N 1X Prc ðxi Þ Ni¼1
ð25Þ
Note that C(GP,MCS) is a positive value within (0.5, 1]; a big value of C(GP,MCS) represents high accuracy of dynamic reliability approximation. 3.2. Double-loop adaptive sampling scheme Constructed GP models should be updated sequentially by adding new samples if the confidence of reliability approximation is lower than the confidence target. In this study, a double-loop adaptive sampling scheme is developed for the sequential sampling and updating of GP models. The developed adaptive sampling scheme contains updating mechanisms in two loops for input variables and time, respectively, as explained below. In the outer loop, a sample should be selected within the N MCS samples X ¼[x1,. ,xN] as candidates to update the current GP model if the confidence of the dynamic reliability approximation C(GP,MCS) is lower than the confidence target. It is straightforward that a sample that can improve C(GP,MCS) to the most degree should be selected to update the GP model. However, it is impossible to
349
compute the confidence improvement of C(GP,MCS) by adding new sample xi if the true performance G(xi) is not available. Thus, a sampling criterion is needed for the selection of new samples that could provide the largest confidence improvement without actually evaluating the true performances. In this study, a new sampling criterion for the selection of samples to update the GP model in the top level is developed, which is expressed as qffiffiffiffiffiffiffiffiffiffi C A ðxi Þ ¼ ð1 Prc ðxi ÞÞ f x ðxi Þ e^ ðxi Þ ð26Þ where Prc(xi) is the probability of correct classification for xi; fx(xi) is the probability density function value at xi; e^ (xi) is the estimated mean square error of the prediction using the GP model. By the developed sampling criterion, a specific sample x^ will be selected for updating the GP model through maximizing CA as x^ ¼ arg maxC A ðxi Þ; i ¼ 1; :::; N
ð27Þ
xi
By maximizing the CA, the first term in Eq. (26) tries to locate a new sample which currently has a low probability of correct classification; the second term pushes the new sample to the place where relative big probability density function values exist in the probabilistic space; and the third term favors the sample with big estimation uncertainty of the GP model. Once the new sample x^ is selected in the outer sampling loop, the inner sampling loop is activated to extract the extreme response value of the selected sample iteratively. In order to find the extreme response Ge(x^ ) at the sample point x^ , a onedimensional ordinary Kriging model is established based on initial time samples [t1,…, tm] evenly distributed over the time interval of interest. The ordinary Kriging method is adopted in this study since it is capable of searching the global optimum efficiently while dealing with nonlinear performance functions. As this paper is mainly focused on the adaptive sampling scheme, the detail of ordinary Kriging method, which can be found from the references [32–34], is omitted. Given the selected updating point x^ in the probabilistic space from the outer loop updating scheme, the responses of the performance function are evaluated as G(x^ , ti), i ¼1,…, m. A one-dimensional response surface of performance function can then be built using the ordinary Kriging method for G (x^ , t) where t is within the time interval [0, tl], expressed as Gðx^ ; tÞ ¼ μðx^ Þ þ eðtÞ
ð28Þ
where μ(x^ ) is the global mean of the ordinary model and e(t) is error term which follows a zero mean Gaussian distribution. The developed one-dimensional ordinary Kriging model is updated iteratively through continuously searching the most useful sample point till the extreme performance is convergence. Similar with efficient global optimization algorithm [35], the expected improvement metric is adopted to quantify the potential contribution of a new sample point and the sample point that gives the largest expected improvement will be selected for updating of the ordinary Kriging model. In the ordinary Kriging model, the unknown response at time t can be approximated by a normal distribution with the mean by the approximated response GK(x^ , t) using the ordinary Kriging model and the variance by the approximated mean squared error mse(x^ , t). Thus, the improvement at time t can be defined as IðtÞ ¼ max GK ðx^ Þmin GK ðx^ ; tÞ; 0 ð29Þ where GK(x^ )min is approximated global minimum using the Kriging model. Thus, the expected maximum improvement at t can be expressed as E½IðtÞ ¼ E max GK ðx^ Þmin GK ðx^ ; tÞ; 0 ! GK ðx^ Þmin GK ðx^ ; tÞ ¼ GK ðx^ Þmin GK ðx^ ; tÞ Φ mseðx^ ; tÞ0:5
350
Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
þ mseðx^ ; tÞ0:5 ϕ
! GK ðx^ Þmin GK ðx^ ; tÞ mseðx^ ; tÞ0:5
ð30Þ
where ϕ( ) is the probability density function of the standard Gaussian distribution. The new time sample t^ can be determined by maximizing the expected improvement, evaluated and used to update the Kriging model sequentially. Thus the global minimum performance GK(x^ )min is updated iteratively till the maximum expected improvement is less than a critical value. Once the minimum performance GK(x^ )min is obtained in the bottom level scheme, the GP model in the up level is updated by incorporating new data [x^ , GK(x^ )min]. The procedure of updating the GP model using the double loop adaptive sampling scheme is outlined in Table 1.
3.3. Procedure of dynamic reliability analysis using the DLAS approach
paper. Once the extreme response is obtained for the point x^ , the GP model in the top level is updated by adding a new data [x^ , GK(x^ )min]. The top-level updating process stops if the confidence level target is satisfied. The procedure of employing the developed double-loop adaptive sampling approach for dynamic reliability analysis is summarized in Fig. 1.
4. Case studies In this section, three examples, a mathematical problem, a fourbar function generator problem, and an aircraft tubing case study are used to demonstrate the developed DLAS approach for dynamic reliability analysis. 4.1. Case study I: a mathematical problem A time-variant limit state function G(x, t) is given by
The dynamic reliability analysis algorithm starts with generating n initial samples x ¼[x1,…,xn] in the probabilistic space using Latin hypercube sampling or grid sampling methods. For each sample point in x, m (m Z 3) number of different time samples [t1,. tm] are generated evenly over time interval [0, tl]. The performance function will be elevated at all the time samples for a given sample point in x and then used to build ordinary Kriging models. The extreme response of each sample is extracted according to the bottom level updating scheme, and then used to build a GP model for global extreme performance Ge(x). For dynamic reliably analysis, N samples are generated according to the randomness of input for MCS. The extreme responses are approximated by Eq. (19), and the estimated mean square error of the approximations is given by Eq. (20). The probability of failure and dynamic reliability are then computed by Eqs. (21) and (23). In order to measure the accuracy of the dynamic reliability approximation, the accuracy level C(GP,MCS) can be calculated by Eq. (25). If the current C(GP,MCS) is greater than the confidence target, the algorithm is terminated and dynamic reliability is readily computed; otherwise, the twolevel sequential DOE scheme is activated to update GP model efficiently. In the top level, the alternative criterion CA in Eq. (26) is computed for all the N samples in MCS, and the sample x^ with the biggest value of CA is selected for updating purpose as shown in Eq. (27). In the bottom-level, m time samples [t1, …, tm] are generated evenly over time domain for the sample x^ , and then evaluated for performances. An ordinary Kriging model is built for the performance G(x^ , t) where t is the only variable. To improve the accuracy of extreme performance approximation, a new time sample t^ is selected by maximizing the expected improvement in Eq. (30). The updating process in the bottomlevel is terminated once the maximum expected improvement is less than a threshold CB, which is set as CB ¼1%n|GK(x^ )min| in this
Gðx; t Þ ¼ x1 2 x2 5x1 t þ ðx2 þ1Þt 2 20
ð31Þ
where t represents the time variable varying within [0, 5]. Random variables x1 and x2 are normally distributed: x1 Normal (3.5, 0.32) and x2 Normal (3.5, 0.32). Thus the probabilistic space can be determined as [3.5–5n0.3, 3.5 þ5n0.3] that is [2.0, 5.0] for both x1 and x2 in this case. Fig. 2 shows instantaneous limit states within the time interval [0, 5] in which the black line is the true limit state for dynamic reliability analysis. As a reference, brute MCS is employed to calculate the dynamic reliability in which 1,000,000 samples are generated in the probabilistic space according to the randomness of the input and then evaluated at 100 time nodes evenly distributed within the time interval [0, 5]. Following the procedure detailed in Section 3.3, five samples x are generated in the probabilistic space {(x1, x2) | 2.0 ox1 o 5.0; 2.0 ox2 o 5.0} using Latin hypercube sampling. To extract the Initialize x and Ge(x); Set CR; Generate X; Build GP model for Extreme Performance
Compute Extreme Performance Gmin(x,t)
Dynamic Reliability Analysis Using MCS Compute Confidence Level C(GP,MCS) C(GP,MCS)>CR Yes Dynamic Reliability by MCS
Update x and Ge(x) Data Set
Evaluate G(x,t ) No Yes
Max(EI) < CB
Compute EI and Locate Sample t
Locate New Sample x with Maximum CA
Build Ordinary Kriging for G(x,t)
Compute CA for all MCS Samples
ti and Response G(x,ti) (i=1,...m)
Outer Loop Sampling
Inner Loop Sampling
No
Fig. 1. Flowchart of dynamic reliability analysis using DLAS.
Table 1 Procedure of double-loop adaptive sample for dynamic reliability analysis. Steps
Procedure
Step Identify initial data set x ¼[x1,…,xn] and Ge(x) ¼ [ Ge(x1),…,Ge(xn)] for GP model development; set the confidence target CR and generate N MCS samples X ¼[x1,. , xN]; 1: Step Built a GP model using data sets [x, Ge(x)], compute the dynamic reliability R using MCS based on the developed GP model and calculate the confidence level C(GP, 2: MCS); Step If C(GP,MCS) o CR: go to Step 4 3: Otherwise, stop. Step Compute CA(xi) i¼ 1,…,N for MCS samples and locate the new sample x^ in probability space with maximum CA in the top level; 4: Step 5 Built an ordinary Kriging for G(x^ , t), locate new sample t^ iteratively in time space with biggest expected improvement in the bottom level; approximate extreme performance GK(x^ )min; Step 6 Update date x and Ge(x) by adding (x^ , GK(x^ )min); go to Step 2;
Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
Fig. 2. Limit states functions within time interval [0, 5].
Table 2 Samples Using for the GP Model in Case Study I.
Fig. 3. Sequential updating process during dynamic reliability analysis.
Table 3 Comparison results of PHI2, CLS and the proposed approach.
x1
x2
Initial samples
4.8791 3.9321 2.1274 3.6441 2.7066
4.6109 63.3077 2.7369 3.5427 4.1722 6.4117 3.5768 9.3623 2.0278 20.2460
5 6 4 5 5
Sequential updating samples
3.4037 3.4197 3.5013 3.1517 3.1201 3.5271
3.2347 0.1807 0.8976
5 5 5
Estimated extreme performance
351
No. of evaluations
extreme performance, three time samples are generated for this case study and evaluated to construct initial ordinary Kriging models. Thus a GP model can be constructed for the extreme performance Ge(x), and N ¼ 100,000 samples X is generated according to the randomness of input for dynamic reliability analysis using MCS. Then the developed double-loop adaptive sampling scheme is employed to enhance the accuracy of dynamic reliability approximation till the confidence level C(GP,MCS) is greater than the target CR ¼ 0.99. Table 2 shows the initial and updating samples, their corresponding extreme performances and number of evaluations. The sequential updating process is shown in Fig. 3. As shown in the figure, the limit state obtained by the GP model is almost identical to the true one at place near the mean point after iterative updating process; the confidence level of dynamic reliability approximation is increased significantly and the reliability is converged very soon. To do a comparison study, the mathematic example is also solved by PHI2 and CLS methods. For both of the methods, the step size of the time has a significant impact in terms of accuracy and efficiency. For CLS, the dynamic reliability estimation is 0.8312 by setting the time step as 0.5 in which more than 1000 performances are evaluated. In PHI2, 438 evaluations are required to obtain dynamic reliability 0.8334 with time step 0.14. In Table 3, the comparison results clearly demonstrate that the proposed approach is able to obtain more accurate dynamic reliability with less computational cost.
Approach
Reliability
Error
Num. of Evaluations
PHI2 CLS DLAS MCS
0. 8334 0.8312 0.8169 0. 8163
2.09% 1.82% 0.07% –
438 41000 40 100,000,000
Fig. 4. Four-bar function generator mechanism.
shows a four-bar function generator mechanism which is designed to generate the movement function y(x)¼601þ601nsin [0.75 (x 971)]. The angle between crank AB and AD is the input x within the range [971, 2171] whereas the angle between CD and DE is the output y. L1, L2, L3 and L4 are the length of four bars which are random variables following normal distributions with parameters given in Table 4. By treating the input x as time variable t in dynamic reliability analysis, the real motion output y can be obtained [36] as 0 yðtÞ ¼ 2arctan@
P 2
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 P 22 þ P 21 P 23 A P3 P1
ð32Þ
where P1, P2 and P3 are given by P 1 ¼ 2L4 ðL1 L2 cos ðt 97ÞÞ P 2 ¼ 2L2 L4 sin t P 3 ¼ L1 2 þ L2 2 þ L4 2 L3 2 2L1 L2 cos t
ð33Þ
4.2. Case study II: four-bar function generator A function generator mechanism is commonly used to realize a certain function between its motion input and the output. Fig. 4
In order to perform the desired functionality, the real motion output y(t) should match the expected output within certain allowable tolerance. Thus, the instantaneous limit state function
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Table 4 Random variables for the function generator mechanism. Variables
Distribution
Mean (mm)
Standard deviation (mm)
L1 L2 L3 L4
Normal Normal Normal Normal
100 55.5 144.1 72.5
0.05 0.05 0.05 0.05
of the four-bar function generator can be defined as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 P 2 P 22 þ P 21 P 23 A GðL1 ; L2 ; L3 ; L4 ; tÞ ¼ C absð2arctan@ P3 P1 ð60 þ 60 sin ½0:75ðt 97ÞÞÞ
ð34Þ
where C is the allowable threshold and t is the time variable within the interval [971, 2171]. Brute MCS is also employed for dynamic reliability analysis to provide a reference, in which 1,000,000 samples are generated according to the randomness of the input and then evaluated at 1000 time nodes evenly distributed within the time interval [97,217]. Thus dynamic reliability can be approximated readily marked as RMCS. Then, the developed double-loop adaptive sampling approach is employed to compute dynamic reliability, denoted by R(GP,MCS), and the percent error can be calculated by
RðGP;MCSÞ RMCS
100% ð35Þ Error ¼ RMCS All the dynamic reliability analysis including MCS and the proposed approach is carried out at two different levels of the allowable threshold C as 0.9 and 0.98. The confidence target for the two different levels of the threseshold C is set as 0.99. According the randomness of input, the lower and upper bound of probabilistic space can be determined as BL ¼[100 5n0.05, 55.5 5n0.05, 144.1 5n0.05, 72.5 5n0.05] and BU ¼[100 þ 5n0.05, 55.5 þ5n0.05, 144.1þ 5n0.05, 72.5 þ 5n0.05]. Following the procedure detailed in Section 3.3, 20 samples x are generated in the probabilistic space using Latin hypercube sampling and then evaluated for the extreme performances. To extract the extreme performance for each initial sample, five time samples are generated evenly distributed on the time interval [92,217] and evaluated for the performances. Then ordinary Kriging models are constructed and updated sequentially to approximate extreme performances, in which the time sample with the biggest expected improvement is selected for updating. The initial samples x, the corresponding extreme performances and number of evaluations are detailed in Table 5. Due to the different allowable threshold C, the corresponding extreme performances for two levels of the allowable threshold are different. In Table 5, the vaules within parentheses are for the case that the allowable threshold C is 0.98 while the values outside the parentheses are for the case that the allowable threshold C is 0.90. Based on the observations in Table 5, a GP model can be built for the extreme performance Ge(L1,L2,L3,L4). The input of GP mode are a realization of random variables (L1,L2,L3,L4) in the probabilistic space while the output of GP model is the approximated extreme performance. Then N ¼1,000,000 MCS samples X are generated according to the randomness of input for dynamic reliability analysis. By employing the developed double-loop adaptive sampling scheme, the confidence level C(GP,MCS) is generally increased by gradually evaluating new samples and updating the GP model. The iterative updating process stops after 27 and 4 iterations for two different tolerant levels C ¼0.9 and 0.98 respectively. Fig. 5 displays the iterative dynamic reliability estimation during updating process while the history of confidence of
dynamic reliability approximations is shown in Fig. 6 for the first case. The sequential updating samples, the corresponding estimated extreme responses and the number of evaluations for the two scenarios are detailed in Tables 6 and 7. All the reliability approximations using DLAS method are compared with the results from MCS in Table 8. To perform dynamic reliability analysis with the threshold 0.90, a total number of 269 function evaluations are required and 47 samples in probablistic space are used to built the GP model, then the dynamic reliability is approxiamted as 0.8248 with 0.3022% error compared with MCS. For the scenario with threshold C ¼0.98, 125 performance evluations and 24 samples in probabilistic space are required to approxiamte the dyamic reliabilty with 0.3777% error compared with MCS. It is observed that more evaluations are required for the second scenario even though the confidence target is the same. The major reason is that the reliability of the first scenario is lower than the second one. Generally more failure surface will be involved in reliability analysis for the case with lower reliability, and more computational efforts are required to identify these important areas in the double-loop adaptive sampling approach. 4.3. Case study III: aircraft tubing In aerospace industry, potential failures of tubing assemblies may lead to catastrophic system failures as they are widely integrated in many subsystems of aircraft such as hydraulic system, fuel system, and environmental control system etc. It is of vital importance to accurately and efficiently approximate the probability of tubing failure in aircraft design stage, which is still technically challenging due to the existence of time-dependent uncertainties. This work employs the proposed DLAS approach for the reliability analysis of an aircraft tubing subsystem. The geometry of tubing system is shown in Fig. 7 with the all the parameters detailed in Table 9. The thickness T and inner diameter D are assumed as random variables while all the other parameters are deterministic. The tube will experience timevariant inner pressure P(t) during each operation cycle, which is given by P(t)¼10n(2t t2) MPa while t is the time parameter within a range [0, 1]. In addition, Young's modulus M(t) is a function of time due to time-variant thermal condition as M(t) ¼ 200n(1 tþ t2) GPa. In this study, a finite element model is developed for the tube to obtain the maximum displacement for different inputs during the dynamic reliability analysis process. A failure is defined as the maximum displacement during one hour operation cycle is larger than a critical threshold of 0.045 cm. Fig. 8 shows a displacement contour of tubing system in ANSYS 14, in which the maximum displacement of the design is 0.044967 cm. Follow the numerical procedure in Section 3.3, the proposed DLAS approach is employed to perform dynamic reliability analysis for the tubing system. According the randomness of input, the lower and upper bound of probabilistic space can be determined as BL ¼[10 5n0.05, 2.25 5n0.01] and BU ¼[10 þ5n0.05, 2.25 þ 5n0.01]. To build an initial GP model for extreme response, ten sample points x are generated in the probabilistic space using Latin hypercube sampling and then evaluated for the extreme performances. For each sample point in x, four time sample points are generated evenly distributed on the time interval [0, 1] and evaluated for the performances. Then the double loop adaptive sampling scheme will be triggered if the confidence level of dynamic reliability approximation by the GP model does not satisfy a confidence target 0.9999. Fig. 9 demonstrates the inner loop sampling process for extracting the extreme performance of a specific tubing system design. As shown in Fig. 9, an initial Kriging model was constructed using four time sample points [0, 0.33, 0.66, 1] and updated based on the inner sampling rule. In this
Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
353
Table 5 Initial samples and extreme performances. Random variables
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
L1
L2
L3
L4
99.7794 99.8047 100.1193 99.8742 100.0532 99.9954 99.8487 100.1933 99.9634 100.2469 100.1699 100.0055 99.8976 100.0794 99.7540 100.1319 100.0459 99.9417 100.2030 99.9052
55.3621 55.3949 55.5952 55.6315 55.3141 55.4658 55.6976 55.7279 55.4284 55.2525 55.5070 55.4154 55.6687 55.3398 55.7047 55.6178 55.5276 55.4795 55.5659 55.2810
144.1024 144.2685 143.9024 144.2154 144.0079 144.0279 144.1846 143.9719 144.2372 144.3046 144.0767 143.8945 143.9253 144.1432 144.3498 144.1736 143.9788 143.8695 144.0590 144.2954
72.4980 72.6971 72.3715 72.6335 72.7360 72.2769 72.5489 72.3487 72.2522 72.4392 72.4623 72.5522 72.3757 72.5213 72.5965 72.6609 72.7067 72.3084 72.4137 72.6186
Extreme value of G
No. of evaluation
0.1544( 0.0744) 0.1804( 0.1004) 0.0629(0.0171) 0.0813( 0.0013) 0.0897(0.1697) 0.0122(0.0922) 0.1168( 0.0368) 0.1082( 0.0282) 0.2336( 0.1536) 0.059(0.1390) 0.05(0.1300) 0.0436(0.1236) 0.0831(0.1631) 0.1026(0.1826) 0.3375( 0.2575) 0.0721(0.1521) 0.0021(0.0821) 0.1102(0.1902) 0.001(0.0790) 0.1599( 0.0799)
5(5) 5(5) 5(5) 5(8) 5(5) 5(5) 5(5) 5(5) 5(5) 5(5) 5(5) 5(5) 6(6) 5(5) 5(5) 5(5) 5(5) 6(6) 5(5) 5(5)
Table 6 Updated samples and extreme performances for C¼ 0.90. Random variables
Fig. 5. Iterative dynamic reliability history.
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27
Extreme of G No. of Evaluation
L1
L2
L3
L4
100.0111 99.9393 99.9684 99.9602 99.9468 99.9585 99.9848 99.9917 99.9994 99.9151 100.0251 99.9423 99.9387 100.0146 99.9499 99.9719 99.9717 99.9104 99.9794 99.9932 99.9282 100.0426 100.0016 100.0429 99.8947 99.9472 100.0137
55.5069 55.4851 55.4434 55.5162 55.5268 55.5083 55.5555 55.4883 55.4946 55.5045 55.5244 55.4732 55.4592 55.4616 55.565 55.4753 55.5464 55.4489 55.4478 55.42 55.5499 55.4818 55.4969 55.4889 55.5157 55.4104 55.5741
144.0973 144.1366 144.1165 144.1503 144.1154 144.1094 144.1432 144.1595 144.1837 144.0748 144.1774 144.1222 144.092 144.167 144.0999 144.1641 144.1606 144.0846 144.1101 144.152 144.119 144.2017 144.1277 144.1658 144.0784 144.1088 144.1829
72.5046 72.4862 72.4958 72.4821 72.513 72.4503 72.4599 72.4562 72.5066 72.4944 72.4531 72.5278 72.4805 72.4607 72.4633 72.5358 72.5264 72.5177 72.4345 72.4961 72.5262 72.4873 72.4141 72.4213 72.5372 72.498 72.4965
0.084 0.0326 0.0163 0.0259 0.0116 0.0059 0.0048 0.0204 0.009 0.0087 0.0047 0.005 0.0043 0.0051 0.0058 0.0023 0.0008 0.0034 0.0007 0.0045 0.0009 0.0039 0.0024 0.004 0.0093 0.0024 0.0056
5 5 5 5 5 5 8 5 5 5 8 5 5 5 8 8 9 5 9 5 8 5 8 8 5 5 8
Table 7 Updated Samples and Extreme Performances for C¼ 0.98. Random variables
1 2 3 4 Fig. 6. Confidence level of dynamic reliability approximation.
Extreme value of G
L1
L2
L3
L4
99.9682 99.9086 99.9476 99.9313
55.4885 55.4584 55.4212 55.4931
144.1395 144.1381 144.1689 144.1832
72.4949 0.0769 72.4814 0.0124 72.4763 0.0149 72.4509 0.0227
No. of Evaluation
5 5 5 5
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0.04
Table 8 Dynamic reliability analysis results.
0.90 0.98
Dynamic reliability DLAS
MCS
0.8248 0.9834
0.8273 0.9797
Error
0.3022% 0.3777%
0.035
No. of evaluations
269 125
Maximum Displacement
C
0.03 0.025 0.02 0.015 Kriging Prediction in EGO Sample Points for Kriging Actural FEA Response
0.01 0.005 0
0
0.2
0.4
0.6
0.8
1
Time Fig. 9. Inner loop sampling process.
Fig. 7. Geometry of tubing.
2.6
0.03
2.5 Table 9 Parameters for the aircraft tubing.
0.03
0.03
Distribution
Mean
Standard deviation
Inner diameter Di Thickness T Length L Young's modulus M(t) Inner pressure P(t)
Normal Normal – – –
10 cm 2.25 cm 100 cm 200n(1 tþ t2) GPa 10n(2t t2) MPa
0.5 cm 0.1 cm
Thickness
2.4
Variables
2.3 2.2 2.1
0.04
0.04
0.04
2 1.9
0.05 8
0.05 9
10
0.05 11
12
Inner Diameter Fig. 10. Contour of extreme performance.
Fig. 8. Finite element analysis of tubing system.
study, the stopping criteria of the inner sampling scheme CB is set as 10 10 in order to maintain high level accuracy. With only one updating sample point t¼ 0.7390, the extreme performance can be obtained with the stopping criteria satisfied. In Fig. 9, the response prediction from the updated Kriging model is almost identical with the actual time series response obtained directly from FEA tools. With all the extreme performance of initial sample points identified, an initial GP model was constructed. Fig. 10 shows the
contour of extreme performance in the uncertain space. By employing the developed double-loop adaptive sampling scheme, the confidence level C(GP,MCS) is enhanced by iteratively identifying important sample points for updating the GP model. In this study, the iterative updating process stops after six iterations to reach 0.9999 confidence target. Totally, 16 sample points are required while 80 finite element analysis are evaluated for dynamic reliability in DLAS. Fig. 11 displays six identified important sample points at different sampling iterations in the DLAS for dynamic reliability analysis and the approximated limit state function by the updated GP model. Clearly, all the identified sample points are very close to the failure surface, by which the accuracy of reliability approximation can be improved efficiently. Fig. 12 shows the iterative dynamic reliability estimation during updating process while the history of confidence of dynamic reliability approximations is shown in Fig. 13.
5. Conclusion This paper presented a confidence-based meta-modeling approach, referred to double-loop adaptive sampling, for
Z. Wang, P. Wang / Reliability Engineering and System Safety 142 (2015) 346–356
generic confidence level measure is developed to evaluate the accuracy of dynamic reliability estimation based on the GP model. With the confidence measure, a double-loop adaptive sampling scheme is developed to update the GP model sequentially by concurrently exacting new samples for random input variables and time concurrently in two sampling loops. Three case studies are implemented to demonstrate effectiveness of the proposed approach. The case study results showed that the developed approach can substantially reduce the computational costs for dynamic reliability analysis while maintaining good accuracy. In addition, the developed approach is a completely sensitivity-free method, thus, can be easily integrated into dynamic reliabilitybased design optimization for product design with time-variant probabilistic constraints.
2.8
Thickness
2.6
2.4
2.2
2
1.8
8
9
10 Inner Diameter
11
12
0.9908 0.9908
References
Reliability
0.9907 0.9907 0.9906 0.9906 0.9905
1
2
3 4 Iterations
5
6
Fig. 12. Iterative dynamic reliability history in DLAS.
1 0.9999 0.9998 0.9997 0.9996 0.9995 0.9994
Acknowledgement This research is partially supported by National Science Foundation through Faculty Early Career Development (CAREER) award (CMMI- 1351414) and the Award (CMMI-1200597).
Fig. 11. Approximated limit state and the updating samples.
Confidence Level
355
1
2
3 4 Iterations
5
6
Fig. 13. Iterative CONFIDENCE HISTORY in DLAS.
sensitivity-free dynamic reliability. The developed approach constructs a GP model for extreme responses of the performance function in the probabilistic space, so that Monte Carlo simulation can be readily employed for dynamic reliability estimation. A
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