Yingshi HU, Zhenzhou LU, Ning WEI, Xia JIANG, Changcong ZHOU

School of Aeronautics, Northwestern Polytechnical University, Xi’an 710072, China

KEYWORDS Kriging;Learning function;Lifetime;System reliability;Time-dependent reliability analysis

Abstract It is important to determine the safety lifetime of Multi-mode Time-Dependent Structural System (MTDSS). However, there is still a lack of corresponding analysis methods.Therefore,this paper establishes MTDSS safety lifetime model firstly,and then proposes a Kriging surrogate model based method to estimate safety lifetime. The first step of proposed method is to construct the Kriging model of MTDSS performance function by using extremum learning function. By identifying possible extremum mode of MTDSS, the performance function of MTDSS can be equivalently transformed into the one of Single-mode Time-Dependent Structure (STDS).The second step is to use the Advanced First Failure Instant Learning Function(AFFILF)to train the Kriging model constructed in the first step, so that the convergent Kriging model can identify the possible First Failure Instant (FFI) of STDS. Then safety lifetime can be searched quickly by dichotomy search.By using AFFILF,the minimum instant that the state is not accurately identified by the current Kriging model is selected as the training point, which avoids the unnecessary calculation which may be introduced into the existing First Failure Instant Learning Function(FFILF).In addition, the Candidate Sample Pool (CSP) reduction strategy is also adopted. By adaptively deleting the random candidate sample points whose FFI have been accurately identified by the current Kriging model,the training efficiency is further improved.Three cases show that the proposed method is accurate and efficient.

The single-mode time-dependent reliability analysis method has been widely investigated during the past decades.1–3At present,the crossing rate method4–8,extremum transformation method9–11and the surrogate model method12–15have been proposed.Based on the assumption that crossings are independent and obey the Poisson distribution,7the crossing rate method estimate the Time-Dependent Failure Probability(TDFP) by integrating the crossing rate in the Time Interval of Interest (TIOI). The extremum transformation method

1. Introduction

focuses on the extremum in the TIOI. After transforming the time-dependent problem into the time-independent one by searching the extremum of each random input variable,10a lot of efficient time-independent reliability analysis methods such as the moment based method16–19, the sampling based method20–22and the surrogate model method23–28can be adopted. The surrogate model method uses the surrogate model of the original time-dependent performance function to estimate the TDFP.Due to the ability of giving the predicting mean and variance of the output, the Kriging model has been the most commonly used surrogate model.12–15

For multi-mode time-independent structural systems,Hohenbichler and Rackwitz29estimated the reliability index of each mode using the First-Order Reliability Method(FORM), and then estimate the failure probability of the system by the multivariate normal joint distribution function φpof system response. Since FORM is a linear approximation method, the error of the results obtained will be large when the system is high nonlinearity. Thus, Pandey30and Du31introduced moment estimation method and saddle point estimation method to obtain φprespectively to improve the accuracy. Gong32and Li et al.33respectively introduced important sampling method and subset simulation method into the multimode structural system reliability analysis to improve the efficiency of estimating the failure probability. Xing et al.34proposed an extended sequential compound method, which can calculate the correlation coefficient between the failure modes.Ref.35–37 introduced Kriging surrogate model into the multimode time-independent structural system. By using different learning functions, the accuracy and efficiency of estimating the failure probability of multi-mode time-independent structural system are further improved.

Multi-mode Time-Dependent Structural System (MTDSS)is composed of multiple time-dependent components through series, parallel and other connections. However, the reliability analysis methods mentioned above are all for single-mode time-dependent structure or multi-mode time-independent system, when the analysis object changes to the MTDSS, the above reliability analysis methods cannot be directly used.Therefore, in recent years, the reliability analysis methods for MTDSS have been paid much attention.38–40Son and Savage41used set theoretic formulation to estimate the TDFP of the multi-mode time-dependent series system. Jiang et al.42proposed a reliability analysis method for the MTDSS based on the discretization of stochastic process. Yu et al.43combined the extreme value moment method and improved maximum entropy method, and proposed a reliability analysis method for the MTDSS with temporal parameters. Jiang et al.44extended the crossing rate method into the MTDSS reliability analysis, and proposed an efficient reliability analysis method.Qian et al.45proposed a single-loop Kriging surrogate model method for the MTDSS combined the extremum transformation method with the single-loop Kriging surrogate model method in the single-mode time-dependent reliability analysis.12

The safety lifetime of the time-dependent structure can be expressed as the maximum service time under the constraint of a given TDFP. At present, a lot of researches have concerned the safety lifetime analysis method for the singlemode time-dependent structure. Yun et al.46proposed a Single-Loop Adaptive Sampling (SLAS) surrogate model method. Hu et al.15pointed out the importance of accurately identifying the First Failure Instant (FFI) to estimate the TDFP in any time subinterval, and proposed a single-loop Kriging surrogate model method combined with the First Failure Instant Learning Function (FFILF), which further improved the efficiency of constructing the Kriging model.For the time-dependent structure with fuzzy input,Fan et al.47proposed an equivalent constraint transformation method that replaces the constraint of the actual time-dependent failure possibility with that of the lower boundary of the minimum output response, so as to greatly improve the efficiency of the safety lifetime analysis. However, these methods are proposed for the single-mode time-dependent structure,when analyzing the MTDSS, these methods cannot be used directly.Therefore, this paper will propose a method to analyze the safety lifetime of the MTDSS.

The core contribution of this paper is to establish the safety lifetime analysis model of the MTDSS according to the definition of the TDFP of the MTDSS firstly, and then propose a safety lifetime analysis method based on the Kriging surrogate model combining the extremum learning function with the Advanced First Failure Instant Learning Function(AFFILF).In the proposed method, the extremum learning function is used firstly to select the most likely extremum mode identified by the current Kriging model at each random input realization and time instant, and then the performance function of the MTDSS can be equivalently transformed into the one of the Single-mode Time-Dependent extremum Structure (STDS).Then,for the converted STDS, the AFFILF is used to extract the most probable FFI identified by the current Kriging model,and the sample point of the most probable FFI will be added to the corresponding training set,so that the convergent Kriging model can accurately identify the FFI of each random input sample point and accurately estimate the TDFP of the MTDSS in any subinterval of the TIOI. Then the safety lifetime can be obtained by applying the dichotomy search into the convergent Kriging model. The existing FFILF selects the instant with the most inaccurately judged state as the new training point.If the real FFI is less than the FFI obtained by the current Kriging model, the unnecessary calculation of adding training point may be introduced by the FFILF.Therefore, the AFFILF is established in this paper. The AFFILF selects the minimum instant whose state is not accurately judged as the new training point, so as to avoid the unnecessary calculation that may be introduced by the existing FFILF and improve the efficiency.At the meanwhile,in the process of training the Kriging model,the sample points whose FFI have been accurately identified by the current Kriging model in the random input Candidate Sample Pool (CSP) will be deleted,then the size of the CSP can be continuously reduced, so as to further improve the efficiency of training the Kriging model.In order to ensure that the convergent Kriging model can accurately identify the FFI of all random input sample points, this paper also proposes a calibration strategy, which checks and calibrates the convergent Kriging model based on the CSP reduction strategy until the FFI of all sample points in the unreduced CSP are accurately identified. The results of two examples illustrate the accuracy and efficiency of the proposed method.

The safety lifetime analysis model of the MTDSS is established in Section 2. In Section 3, an efficient safety lifetime analysis method based on the Kriging surrogate model for the MTDSS is proposed on the basic of the model established in Section 2, and the specific implementation steps of the proposed method will also be given. The analysis of three cases is given in Section 4 and Section 5 gives the conclusion.

2. Basic solution of the MTDSS safety lifetime

The safety lifetime analysis model of the MTDSS is established in this section, and the corresponding basic solution method for the MTDSS safety lifetime will be given from the point of view of the FFI of the MTDSS.

2.1. Safety lifetime analysis model of MTDSS

The MTDSS performance function {gc（X，Y（t)，t)，c=1，2，...，m} with m modes generally includes random vector X,random process vector Y（t) and time variable t. Since Y（t)can be converted into a series of independent combination of random variables and time variable,48–50this paper will only study the performance function {gc（X，t)，c=1，2，...，m}with random input vector. For the problems containing stochastic process, the stochastic process can be transformed first, and then the proposed method can be applied to analyze the safety lifetime.

For the MTDSS performance function {gc（X，t)，c=1，2，...，m} with m modes and n dimension random input vectors, the TDFP Pf（0，te) in the TIOI t ?[0，te] can be given as follows,

Since Pf（0，ts)is a non-decreasing function of ts,the dichotomy search can be applied to solve Eq. (5).

2.2. Relationship between the FFI and the TDFP estimation of MTDSS

From Eq.(5),it can be seen that the key to solve the safety lifetime accurately by dichotomy search is to accurately estimate the TDFP Pf（0，tm) of MTDSS in any subinterval[0，tm]?[0，te] of TIOI [0，te], that is, if Pf（0，tm) can be estimated accurately, tscan be obtained accurately.

According to Ref. 15, for the single-mode time-dependent structure, the key to accurately estimate is to accurately estimate the FFI tf（x)of each random input sample point x.Since the STDS performance function ge（X，t)obtained by Eq.(2)is equivalent to the MTDSS {gc（X，t)，c=1，2，...，m}, the FFI of x in the ge（X，t) is the one in the {gc（X，t)，c=1，2，...，m}.Therefore, if tf（x) in ge（X，t) can be accurately estimated, the TDFP of MTDSS in any subinterval [0，tm] of TIOI [0，te]can be accurately estimated.

For the STDS performance function ge（X，t), its timedependent failure domain F（0，te) in TIOI [0，te] is given as follows,

It can be seen from Eq. (9) and Eq. (10) that if the FFI of the STDS, which is equivalent to the MTDSS{gc（X，t)，c=1，2，...，m}, can be accurately estimated, the TDFP Pf（0，tm) of the MTDSS in any subinterval [0，tm] of the TIOI[0，te]can be accurately estimated, so as to efficiently solve the safety lifetime of the MTDSS by using dichotomy search according to Eq.(5).Therefore,in Section 3,this paper will propose a safety lifetime analysis method based on the Kriging surrogate model combining the extremum learning function with the AFFILF.

3. Safety lifetime analysis method based on Kriging surrogate model for MTDSS

3.1. Basic theory

In this section, a safety lifetime analysis method based on the Kriging surrogate model for the MTDSS is established by combining the extremum learning function and the AFFILF.Firstly, the extremum learning function is used to identify the most possible extremum mode of the MTDSS predicted by the current Kriging model, and then the MTDSS performance function can be equivalently transformed to the STDS performance function. Then the AFFILF is used to help the convergent Kriging model accurately identify the FFI of the STDS. The FFI of the MTDSS can be accurately estimated by the convergent Kriging model constructed by the above adaptive learning process, and the TDFP of the MTDSS in any subinterval of TIOI can be estimated accurately, so that the safety lifetime of the MTDSS can be efficiently obtained by the dichotomy search.

In order to improve the efficiency of constructing the Kriging model, the problems of the existing FFILF are discussed,and a more efficient AFFILF is proposed in this section. In addition,an adaptive CSP reduction strategy is also proposed.According to the adaptive CSP reduction strategy,the random input sample point whose FFI can be accurately identified by the current Kriging model will be deleted from the CSP, so as to reduce the calculation amount of predicting in the learning process. In order to ensure that the final convergent Kriging model can accurately identify the FFI of all random input sample points, the corresponding calibration strategy is also proposed. The proposed calibration strategy checks whether the convergent Kriging model obtained by the adaptive CSP reduction strategy can accurately identify the FFI of all random input sample points in the CSP. If there has any sample points not accurately identified, the CSP reduction strategy will be performed again in the complete CSP based on the current Kriging model, until the convergent Kriging model can accurately identify the FFI of all random input sample points in the complete CSP. The CSP reduction strategy can greatly reduce the amount of prediction in the process of updating the Kriging model, and greatly improve the convergence efficiency, while the calibration strategy can avoid the misjudgment that may be introduced in the process of CSP reduction.

3.2. Equivalent transformation from the MTDSS to STDS

3.2.1. Basic theory

It can be seen from Eq. (14) that the MTDSS performance function can be equivalently transformed into the STDS performance function by only constructing a reasonable extremum learning function to select the correct extremum mode.

3.2.2. Extremum learning function

Firstly,the Sobol set sampling method52–53is applied to generate Nx-size random input variables CSP Sx={x1，x2，...，xNx}according to the joint PDF of the n dimensional random input variables x, and the Nt-size time CSP St={t1，t2，...，tNt} by uniformly discretizing the TIOI [0，te]. For each mode, the Nin-size initial training set Tc={（x1，t1)，（x2，t2)，...，（xNin，tNin)} and the corresponding output set Yc={gc（x1，t1)，gc（x2，t2)，...，gc（xNin，tNin)} (c=1，2，...，m) are generated uniformly distributed in the random sample space. Then the Kriging surrogate model ^gc（x，t) of the c-th mode can be constructed by Tcand Yc. According to the Kriging model theory,54the U learning function Uc（x，t) of ^gc（x，t) at （x，t)can be given as follows,

3.3. FFILF for STDS

3.3.1. Discussion of existing FFILF

For the Kriging model ^ge（x，t) of the STDS with predicted mean μ^ge（x，t) and U learning function U^ge（x，t) determined by Eq.(18) and Eq. (19), Ref. 15 gives the FFI ^tf（x)predicted by the ^ge（x，t) at the random input sample point x as follows,

where ueis the threshold value of the advanced U learning function to accurately identify the states at the sample point.

Then the safety lifetime can be obtained by combining the Kriging model ^ge（x，t) of the STDS performance function ge（x，t) which is equivalent to the MTDSS performance function with the dichotomy search given in Eq. (5).

This existing FFILF improves the learning efficiency by avoiding training the sample points after the FFI. However,in some cases, the efficiency of this learning function can still be improved.

Table 1 shows the prediction at 5 time instants in St={1，2，3，4，5} by the current Kriging model ^ge（x，t) of the STDS performance function ge（x，t) = 0.5（t-x1)（t-x2)when take x1=1.5, x2=4.5 and the training CSP T={0.5，0.9，4.3，4.6}.U^ge（x，t)and μ^ge（x，t)are the U learning function and the mean predicted by the ^ge（x，t)at（x，t),respectively. ge（x，t) is the actual output value of the STDS at （x，t).

It can be seen from Table 1 that actual FFI in Stis t=2,but the FFI determined by the current Kriging model ^ge（x，t)is t=4. If ue=2, there still are some instants when the U learning function is less than ueafter the transformation of the U learning function by Eq. (21). Therefore, according toEq. (22), t=4 will be added to the training set, that is,Tnew=T ∪{4}={0.5，0.9，4.3，4.6，4}. Table 2 gives the prediction of the Kriging model by Tnew.

Table 1 Prediction of the Kriging model by ST.

According to Table 2 we can see that the FFI is not still accurate identified by the Kriging model and t=2 will be added to the training set to promise the convergent Kriging model to accurately estimate the FFI.This means that for this case, the FFILF needs at least 2 new training points to accurately identify the FFI. However, if t=2 but not t=4 is selected as the first new training instant to update the Kriging model, the updated Kriging model can accurately identify the FFI. Table 3 gives the prediction of the Kriging model by Tanother=T ∪{2}={0.5，0.9，4.3，4.6，2}.

It can be seen from Table 3 that the Kriging model is convergent to accurately identify the FFI after one update and avoid the learning at t=4, so as to improve the efficiency. It should be pointed out that the conclusion from this single case is universal. For the cases with multiple crossings, the current Kriging model may misidentify the last few crossing instants as the FFI.For this kind of cases,the FFILF may firstly focus on the accurate identification of the sign around the wrong FFI,and then the true FFI.This way will lead to a lot of useless calculation in constructing the Kriging model.If we directly focus on the accurate identification of the sign around the true FFI at the beginning, the efficiency will be greatly improved.

Therefore, an AFFILF is proposed in Section 3.3.2 to further improve the accuracy of training the Kriging model.

3.3.2. AFFILF

Eq.(25)shows that if ω1（x)≠φ,there are the instants when the state is not accurately identified at the random input sample point x by the current Kriging model, then the minimum instant when the state cannot be accurately identified by the current Kriging model will be taken as the training instant

Table 2 Prediction of the Kriging model by Tnew.

Table 3 Prediction of the Kriging model by Tanother.

tadd（x).Otherwise,the FFI^tf（x)estimated by the current Kriging model is accurate,then the^tf（x)can be used as the training instant at x. After the training instant tadd（x) of all x is obtained, the new training random input sample point x*should be selected as the one with the minimum U learning function value , that is,

After obtaining the new training points according to Eqs.(26)–(28), let Tc*= Tc*∪{（x*，t*)} and Yc*= Yc*∪{gc*（x*，t*)}, and then update the Kriging model^gc*（x，t) of the c*-th mode. Then the TDFP Pf（0，tm) of the MTDSS in any subinterval [0，tm] of TIOI [0，te] can be estimated by the convergent Kriging model according to Eq.(24).As we can see that for the case given in Table 1, according to the AFFILF given by Eqs. (26)–(28), t=2 will be selected as the new training point, then according to Table 3 we can see that the Kriging model after once update can accurately identify the FFI, so as to avoid the learning at t=4 when using the FFILF in Eq. (22), and further improve the efficiency.

3.4. Adaptive CSP reduction strategy for Kriging surrogate model

According to the basic theory of the Kriging model,55with the increase of the prediction points and the training set points,the prediction efficiency of the Kriging model will decrease significantly.The learning function introduced in this paper needs to predict the mean and the standard deviation of all sample points in the whole CSP by the current Kriging model when selecting a new training sample point. And for MTDSS, the number of points need to be predicted in each update is Nx×Nt×m,usually on the order of 108.If we adopt the usual method, the efficiency of the program will be low. Therefore,this section will introduce an adaptive CSP reduction strategy and the corresponding calibration strategy.

3.4.1. Adaptive CSP reduction strategy

3.5. The procedure and flowchart of estimating the safety lifetime of the MTDSS

After obtaining the convergent Kriging model according to Section 3.4, the dichotomy search can be applied to solve the safety life of the MTDSS.The specific procedure of estimating the safety lifetime of the MTDSS based on the Kriging model is shown as below,and the corresponding flowchart is given in Fig. 1.

Step 1 Construct the Kriging model

Step 1.1 Generate the random input variable CSP Sx={x1，x2，...，xNx} according to the joint PDF of the random input variables, and the time variable CSP St={t1，t2，...，tNt} by regarding the time as a uniformly distributed variable in the TIOI [0，te]. For each mode, generate the corresponding initial training set Tc={（x1，t1)，（x2，t2)，...，（xNin，tNin)} and the corresponding output set Yc={gc（x1，t1)，gc（x2，t2)，...，gc（xNin，tNin)} (c=1，2，...，m).

4. Case study

Fig. 1 Flowchart of estimating the safety lifetime of the MTDSS.

Three examples are given in this section. According to Section 2, we can see that the first step to analyze the safety lifetime of the MTDSS is to equivalently transform the MTDSS performance function to the STDS performance function.For the Kriging based method, the extremum learning function mentioned in Section 3.2.2 can help us to complete this equivalent transformation. After the transformation, we can use the FFILF or AFFILF to estimate the safety lifetime of the equivalent STDS performance function. In addition, if we continue to use the extremum learning function but not the FFILF or AFFILF, we can estimate the TDFP of the equivalent STDS accurately. In order to simplify the expression, we use EX + EX to represent the method that uses the extremum learning function to transform the MTDSS performance function and the extremum learning function to estimate the safety lifetime of the equivalently STDS performance function, and the specific procedure of the EX + EX method is given in Appendix A. The EX + FF method uses the extremum learning function to transform the MTDSS performance function and estimate the safety lifetime by using the FFILF given in Section 3.3.1.The EX+AF method uses the extremum learning function to transform the MTDSS performance function and estimate the safety lifetime by using the AFFILF given in Section 3.3.2.At the meanwhile,the EX+EX method does not adopt the adaptive CSP reduction strategy given in Section 3.4, while the EX+FF method and EX + AF method adopt the adaptive CSP reduction strategy. In this paper, the DACE toolbox56is used to construct the Kriging model.

4.1. Numerical case

Consider a 4-mode time-dependent parallel system without random process input as follows,

where the random input variables X1and X2are independent normal variables,and the means are μX1=2,μX2=3,the standard deviations are σX1=σX2=0.6. t is the time variable and t ?[0，3].

In order to illustrate the efficiency and accuracy of the proposed method,Table 4 gives the results of the estimated TDFP in the TIOI by using Monte Carlo Simulation(MCS)method,EX + EX method, EX + FF method and the proposed EX+AF method,respectively.The number of initial training samples of each mode is 10, the number of the random input samples is 105, and the number of the time samples is 300.The accuracy level of each method is measured by the relative error between the results of these methods and the results of MCS. Where Ncallis the times of calling the MTDSS performance function, Nloopis the times of executing calibration strategy in adaptive CSP reduction strategy shown in Section 3.4, and Time represents the time used to construct the convergent Kriging model.

It can be seen from Table 4 that the EX + EX method,EX+FF method and EX+AF method all have good accuracy in estimating Pf（0，3),and the times of calling the MTDSS performance function are far less than MCS method, which fully shows the efficiency and accuracy of these three methods in estimating the TDFP of the MTDSS in the TIOI. At the same time, through the comparison of these three methods,we can see that the EX + EX method calls the MTDSS performance function the least, while the EX + FF and EX + AF call the MTDSS performance function the same and much more than the EX+EX method.The reason is that the EX + EX method only needs to accurately identify one failure instant of the random input sample point, while EX + FF and EX + AF need to accurately identify the FFI of the random input sample point.Ref.15 points out that the difficulty of identifying one failure instant is far less than that of identifying the FFI,so the EX+EX method needs less calls of the MTDSS performance function. Although the EX + EX method calls the MTDSS performance function the least, the time needed to constructed the convergent Kriging model by the EX+EX method is not the least.The reason is that the EX+EX method does not adopt the adaptive CSP reduction strategy shown in Section 3.4, so it needs to predict all the random input sample points in the whole random input CSP every time it selects a new training point, which leads to low prediction efficiency,as shown that although only 45 training points are added, it takes 40657 s to construct the convergent Kriging model. The EX + FF method uses the adaptive CSP reduction method, so that even if 252 training points are added, it only takes 62459 s to construct the convergent Kriging model. The EX + AF method takes the least time(23852 s) to construct the convergent Kriging model with 251 added training points.The reason is that the times of executing the calibration strategy by the EX + AF method is far less than that by the EX+FF method.Since in the calibration strategy, all the random input sample points need to be predicted,and in the first few steps of the adaptive CSP reduction strategy,the number of sample points in the prediction CSP is still large, the time costed to construct the convergent Kriging model will increase with the increase of the number of times of executing the calibration strategy.Thus the time costed by the EX + FF method will be much longer than that of the EX+AF method.The reason why EX+FF method spends much more times of executing the calibration strategy is that the FFI selected by the EX+AF method is the minimum time instant whose state is not accurately identified in the time CSP,which makes it easier for the updated Kriging model to accurately identify the FFI of all random input sample points, so that the EX + AF method can converge after three times of executing the calibration strategy.

In order to illustrate the efficiency and accuracy of the proposed method in estimating the safety lifetime of the MTDSS,Table 5 gives the results of the safety lifetime tsestimated by the above four methods under different TDFP constraints P*f.It can be seen from Table 5 that the error of safety lifetime estimated by the EX + EX method is large, while the safety lifetime obtained by the EX + FF method and the EX + AF method is completely consistent with the MCS method. The main reason is that the EX + EX method only focuses on accurately estimating one failure instant,which will lead to the EX+EX method unable to accurately estimate the TDFP of the MTDSS in any subinterval of TIOI,thus leading to the large error of the safety lifetime estimated by the EX + EX method. The safety lifetime obtained by the EX + FF method and the EX + AF method can accurately identify the FFI of each random input sample point, so that these two methods can accurately estimate the TDFP of the MTDSS in any subinterval of TIOI, and then can accurately estimate the safety lifetime. In order to further illustrate this point, Figs. 2–4 show the change of the TDFP Pf（0，tm) in any subinterval [0，tm] of TIOI [0，te] with tmand the relative error ε by the Kriging model constructed by the EX + EX,EX + FF and EX + AF, respectively. It can be seen from Fig. 2 that the EX + EX method will produce large error inestimating Pf（0，tm), while we can see from Fig. 3 and Fig. 4 that the EX + FF method and the EX + AF method can accurately estimate Pf（0，tm).

Table 4 TDFP of Section 4.1 in the time interval t ?[0，3].

Table 5 ts under different given TDFP constraint P*f.

In order to further illustrate the high efficiency and accuracy of the proposed method, Section 4.2 will analyze a corroded bending beam with random process input.

4.2. Corroded bending beam

A corroded bending beam57is given in Fig.5,The force on it is P（t)=（1-0.05sin（2πt))F（t), where F（t)is a random process,and the length （a1，a2) and the width （b1，b2) of the cross section of the longitudinal bar are affected by corrosion, which are reduced k1=2×10-2inch/year and k2=1.5×10-2inch/year, respectively. When both longitudinal bars yield, the system fails, so the system is a parallel system, and its corresponding MTDSS performance function is,

Fig. 2 Pf（0，tm) and ε vs tm by EX + EX.

Fig. 3 Pf（0，tm) and ε vs tm by EX + FF.

Fig. 4 Pf（0，tm) and ε vs tm by EX + AF.

Table 7 shows the TDFP of the MTDSS in the TIOI estimated by the MCS method, the EX + EX method, the EX + FF method and the EX + AF method, respectively.The number of initial training samples in each mode is 20,the number of random input samples is 105, and the number of time sample points is 800.

Fig. 5 Corroded bending beam.

Table 6 Parameters of the random input.

It can be seen from Table 7 that for the example with random process input,the Pf（0，te)estimated by these three methods are all accurate. Since the EX + EX only focuses on accurately identifying one failure instant, it has the least number of times to call the MTDSS performance function. The EX + FF method and the EX + AF method need to accurately identify the FFI of each random input sample point, so that these two methods need more times to call the MTDSS performance function than EX + EX. At the same time, we can see that compared with the EX + FF method, the EX + AF method needs less calls and running time, which fully illustrates the efficiency of the AFFILF proposed in this paper.

Fig. 6 Pf（0，tm) vs tm by different methods.

4.3. A rotate vector reducer of industrial robot

Fig. 7 gives the model sketch of industrial robot and Rotate Vector (RV) reducer.45The surface wear of the cycloidal gear and the wear of the roller bearing have the greatest impact on the reliability of the RV reducer. The time-dependent performance functions of these two modes are given as,correlation coefficient function of random process P（t) is ρ（t1，t2)=exp（-（t2-t1)2/4) and the detailed information of the random variables is given in Table 9.

Table 7 TDFP of Section 4.2 in the time interval t ?[0，4].

Table 8 ts under different given TDFP constraint P*f of Section 4.2.

Fig. 7 Model sketch of industrial robot and RV reducer.45

Table 9 Detailed information of the random inputs.

Table 10ts under different given TDFP constraint P*f under t heTIOI[0,10].

The RV reducer is a series system since either failure of the two failure modes can cause the system failure.Table 10 shows the results obtained under the TIOI [0，10]. According to Table 10, the same conclusion in the parallel system can be obtained for the series system.

5. Conclusions

The core contribution of this paper is to establish the safety lifetime analysis model of the Multi-mode Time-Dependent Structural System (MTDSS) for the first time, and according to the established model, a Kriging surrogate model method combining the extremum learning function and the Advanced First Failure Instant (AFFI) learning function is proposed. In the first step of constructing the Kriging model, the extremum learning function is applied to equivalently transform the MTDSS performance function into the Single-mode Time-Dependent extremum Structural (STDS) performance function.In the second step,the AFFI learning function is applied for the converted STDS performance function to select the most likely first failure instant sample point identified by the current Kriging model, and add the selected sample point to the training set of the corresponding extremum mode. The convergent Kriging model constructed by these two steps can accurately identify the first failure instant of each random input sample point, and then the convergent Kriging model can be applied to accurately estimate the Time-Dependent Failure Probability (TDFP) of the MTDSS in any subinterval of the time interval of interest,so that the constructed convergent Kriging model can be used to estimate the safety lifetime of the MTDSS by dichotomy search.This paper also proposes the AFFI learning function, which selects the first failure instant as the smallest time sample point whose state is not identified accurately in the time Candidate Sample Pool(CSP), so as to avoid the unnecessary calculation of the existing first failure instant learning function. At the same time, in order to avoid the inefficiency of the Kriging model in predicting large samples, this paper also proposes an adaptive CSP reduction strategy, which will delete the random input sample points whose first failure instant can be accurately identified by the current Kriging model from the prediction sample pool after each update, so as to improve the efficiency of constructing the Kriging model.The results of two cases verify the efficiency of the proposed method.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (No.52075442); the National Science and Technology Major Project (2017-IV-0009-0046) and the National Natural Science Foundation of China(No.51975476).

Appendix A. Procedure of EX + EX method

where ω3= {t｜U^ge（x，t)≤ue，t ?St} is the set of instant when the states are not accurately identified by the current Kriging model ^ge（x，t), ω4= {t｜U^ge（x，t)>ue，t ?St}=ω-3(ω-3is the complementary set of ω3) is the set of instant when the states are accurately identified by the^ge（x，t),ueis the threshold value of the U learning function, φ is the empty set.

After obtaining the tmin（x),the STDS performance function can be converted into the single-mode time-independent performance function according to the extremum transformation method of the time-dependent reliability analysis.The U learning function UX（x)and the failure domain indication function^IF（0，te)（x) of the single-mode time-independent performance function at the random input sample point x are,