Reliability analysis is time consuming, and high efficiency could be maintained through the integration of the Kriging method and Monte Carlo simulation (MCS). This Kriging-based MCS reduces the computational cost by building a surrogate model to replace the original limit-state function through MCS. The objective of this research is to further improve the efficiency of reliability analysis with a new strategy for building the surrogate model. The major approach used in this research is to refine (update) the surrogate model by accounting for the full information available from the Kriging method. The existing Kriging-based MCS uses only partial information. Higher efficiency is achieved by the following strategies: (1) a new formulation defined by the expectation of the probability of failure at all the MCS sample points, (2) the use of a new learning function to choose training points (TPs). The learning function accounts for dependencies between Kriging predictions at all the MCS samples, thereby resulting in more effective TPs, and (3) the employment of a new convergence criterion. The new method is suitable for highly nonlinear limit-state functions for which the traditional first- and second-order reliability methods (FORM and SORM) are not accurate. Its performance is compared with that of existing Kriging-based MCS method through five examples.
Reliability analysis evaluates the likelihood of failure for components or systems in the presence of randomness . Seeking for a good balance between accuracy and efficiency is always the focus on the methodology development of reliability analysis. MCS [2,3] is commonly used for reliability analysis. MCS can produce high accuracy given a sufficiently large sample size. But it is computationally expensive if the sample size is too large. On the other hand, approximation methods, such as the FORM and SORM [2,4–6], are in general much more efficient, but may not be accurate for highly nonlinear limit-state functions. To this end, design of experiments (DoE) based MCS methods have been developed for high accuracy and efficiency.
DoE methods are used to generate initial TPs, and a surrogate model is built for a limit-state function, then MCS is performed based on the surrogate model. The DoE methods for reliability analysis include response surface modeling [7–10], artificial neural networks [7,11], support vector machines , polynomial chaos expansions , and Kriging [7,14–17]. Most of these methods evaluate the limit-state function at a number of predefined points and then create a surrogate model to replace the limit-state function in the subsequent MCS.
The Kriging-based active learning reliability method is used to create the surrogate model in a sequential manner . After the initial surrogate model is built with a small number of initial TPs, more TPs are added one-by-one until the surrogate model accurately represents the original limit-state function. A learning function is employed in the model building process to select the best TPs intelligently and refine the surrogate model in a most efficient fashion.
A learning function is a function that defines the criteria of selecting a best TP so that the surrogate model can be refined with improved accuracy. Different Kriging-based reliability methods use different learning functions. Based on the efficient global optimization , the efficient global reliability analysis (EGRA)  uses the expected feasibility function to determine TPs, while the active learning reliability method combining Kriging and Monte Carlo Simulation (AK-MCS)  uses the probability of predicting the correct sign of the limit-state function as its learning function. Other leaning functions are also available [20,21], and discussions on learning functions can be found in Refs.  and .
EGRA and AK-MCS have also been applied to system reliability [24,25] and time-dependent reliability analyses [26,27], and other improvements have been made [21,22,28–30]. The Kriging-based reliability methods can be further improved with respect to accuracy and efficiency because of the following reason: even though the responses predicted by Kriging are realizations of a Gaussian process and are therefore dependent on one another, the above methods do not account for the dependencies between the responses.
To further improve the efficiency of Kriging-based reliability methods, this work proposes a new Kriging-based reliability method. Its general process is similar to that of AK-MCS. An initial surrogate model is created with a small number of TPs. Then, the surrogate model is refined with more TPs. Once the surrogate model becomes accurate, MCS is used to estimate the probability of failure. The contributions of this research include the following new components that help select the new TPs more efficiently.
The method selects a new TP using the complete Gaussian process output of the surrogate model that is available from the Kriging method. It can therefore fully account for the correlations between output variables at all the MCS sample points.
The new method calculates both the mean and variance of the estimated probability of failure with new formulas that involve the mean and covariance functions of the above Gaussian process. This makes it more effective to select new TPs.
Instead of focusing on the accuracy of the limit-state function, the new method focuses directly on the accuracy of the reliability estimate with a new convergence criterion. This improves the efficiency of the reliability analysis without jeopardizing the accuracy.
With the above new components, the new method is in general more effective than the methods that use independent Kriging predictions.
Background and Literature Review
In this section, we review the definition of reliability and Kriging-based reliability methods.
As we discussed previously, or can be estimated by MCS, surrogate MCS, FORM, and SORM.
Independent Kriging Methods (IKM).
The output of the surrogate model from the Kriging method follows a Gaussian process. As a result, two output variables predicted by the surrogate model are two realizations of the Gaussian process and are likely dependent. The IKM ignore such dependence. In other words, the output variables are assumed independent. This assumption simplifies the process of building the surrogate model, but may adversely affect the efficiency of the model building process.
The other assumption of IKM is that the surrogate model at the limit state will produce an accurate reliability estimate if the surrogate model is accurate. Although this assumption is valid, it emphasizes the accuracy of the surrogate model, instead of devoting effort directly for improving the accuracy of the reliability estimate. This may also affect the efficiency.
A sequential process is involved as the surrogate model is built iteratively with the help of MCS. After obtaining the accurate surrogate model, MCS is performed on it, and the Kriging predictions of the surrogate model are treated independently. The general idea of IKM is as follows:
Generate a small number of TPs for input random variables , denoted by . And build an initial surrogate model based on these TPs.
Generate a large number of Monte Carlo sample points for . These points serve as candidates for the TPs and are called candidate points (CPs), denoted by .
If the surrogate model is not accurate, a learning function is used to select the best TP to refine the surrogate model in a most efficient fashion.
Add the new TP to the existing TPs and then refine the surrogate model.
Steps (3) and (4) are repeated until convergence is attained. The flowchart of the process is provided in Fig. 1. Note that the size of CPs may change during the process if the error of reliability analysis from MCS is large. The error can be found in Ref. .
The methods differ from one another by their learning functions. While the AK-MCS  uses a probability measure in its learning function, the EGRA method  uses the limit-state function value directly. Since the two methods have the similar performance , we will compare the proposed method with only AK-MCS, which is reviewed briefly next.
where and are the mean and standard deviation of the Kriging prediction at , respectively.
The smaller is , and the higher is . Hence, a new TP is identified with the minimum among CPs. When is sufficiently large, the surrogate model will be accurate at the limit state , and the process will then converge. is used in AK-MCS , and it is equivalent to .
where is the size of CPs. If , will be increased .
The above equation shows that the estimate of only uses the sign information of the predictions, and correlations between the predictions are not considered. The surrogate model built with Kriging, however, is a Gaussian process, and its output variables (the Kriging predictions) are dependent. This work develops a new method that accounts for such prediction dependency in order to improve the performance of IKM.
Dependent Kriging Method (DKM)
We now discuss the new method that accounts for the dependencies between Kriging predictions. The method is referred to as the DKM for brevity.
The key points of DKM are as follows:
DKM considers correlations between Kriging predictions at all CPs.
As a result, DKM uses complete statistical characteristics of the Gaussian process, including not only the mean and standard deviation functions of the Gaussian process but also its correlation function.
DKM focuses directly on the accuracy of the estimate of reliability, instead of that of the limit-state function considered by IKM. DKM is therefore probability oriented, instead of function value oriented.
With the above considerations, new components of the proposed method are developed, including a new way to estimate , a new learning function, a new convergence criterion, and a new procedure.
We now discuss the aforementioned new components.
A New Way to Calculate pf.
A New Learning Function.
where is the kth point of CPs .
This indicates that the contribution to at the newly added TP becomes zero. Recall that this point has the highest contribution to before it is added to TPs. Keep on adding new TPs this way will provide the highest effectiveness way to reach convergence.
The learning function of DKM uses all the information of the Gaussian process, including its mean, variance, and correlation functions, while the IKM uses only the mean and variance functions of the Gaussian process. DKM is also more direct because it focuses on the probability of failure itself while IDM employs two steps—create an accurate surrogate model first and then calculate the probability of failure. As a result, the former method will be in general more effective than the latter method.
A New Convergence Criterion.
The convergence criterion is therefore given by the ratio of to . This convergence criterion is directly linked to the error of the estimate of the probability of failure, and such a direct link does not exist in IKM.
With the use of the full information of the Gaussian process, DKM will be in general more effective than IKM. Directly using the strategy of the new learning function, however, will be computationally intensive because of calculating the bivariate joint probabilities for all the CPs. The number of such calculations is . If the size of the CPs is 105, the number of calculations will be . (But note that does not require calling the original limit-state function.)
To avoid using all the CPs, we select a small portion of the CPs. The points in this small portion are called selected candidate points (SCPs). SCPs are selected based on two criteria: small errors in the estimate of and high potential contributions to .
The second criterion requires high potential contributions to . Recall that the contribution of a CP is given by . To avoid calculating the bivariate probabilities , we only consider the first term , which is the variance of the indicator function at . As a result, the CPs that have the highest variances of indicator functions will be added to the set of SCPs, and the number of these points is . The SCPs therefore consist of all the points in the failure region and other points with the highest indicator function variances in the safe region.
After the set of SCPs is formed, the learning function at each point of SCPs is calculated, and the SCP with the highest learning function value will be chosen as a new TP. Recall that evaluating the learning function needs to calculate bivariate probabilities. With the use of SCPs, the total number of bivariate probability calculations will be reduced to . If 200 SCPs are used, the total number of bivariate probability calculations will be , which is much less than the number when all the CPs are used.
The symbols in the above equation are the same as those in Appendix A.
Figure 2 shows the domains of CPs, SCPs, and the failure region, denoted by , , and , respectively. From the figure, we have
This means that we can just use the SCPs to determine the convergence criterion. After the convergence criterion is satisfied, MCS is performed on the final surrogate model to evaluate the probability of failure. The flowchart of the DKM is shown in Fig. 3.
where is from MCS with a large sample size and the original limit-state function. is therefore regarded as an accurate solution for accuracy comparison. is from a non-MCS method, namely, DKM, IKM, or AK-MCS. Since both DKM and IKM are based on random sampling, their results are also random. We therefore run DKM and IKM 20 times independently and then report their average results.
To have a fair comparison between DKM and IKM, ideally, we should incorporate the same convergence criteria. The direct equivalency of the convergence criteria between the two methods, however, does not exist. Thus, we implement the following strategy for the comparison.
For DKM, set the confidence in Eq. (31) to be 98%, or , and the allowable error to be . The number of SCPs is 200.
Run DKM until convergence and record the number of function calls .
Use the same value of and initial TPs to run IKM. This means that if the total number of TPs reaches , IKM terminates.
Repeat the above steps 20 times and report the average , , , and the standard deviation of . With the above strategy, the accuracy of the two methods is compared with the same number of TPs or function calls.
As discussed previously, it is not easy to estimate the error of the estimated probability of failure if we use the existing Kriging-based reliability methods. DKM can easily overcome this drawback because the process of model training terminates once the estimated error of the probability of failure is small enough. To show this advantage, we also perform AK-MCS with its original procedure  20 times using the same CPs as those of DKM. The parameters we use for AK-MCS are those in Ref. , and they are and . Then, the results from MCS, DKM, IKM, and AK-MCS are put together in a table for an easy comparison.
The process of building the surrogate model actually takes place in the space of independent random variables that follow standard normal distributions. This means that all the random variables are transformed into standard normal variables during the analysis. This transformation makes programing the process more convenient, and it does not affect the performance, such as the accuracy and efficiency, of the reliability analysis.
where and are independently and normally distributed with and .
The contour of the limit-state function is plotted in Fig. 4, which shows the high nonlinearity of the limit-state function. This figure also shows the initial TPs, added TPs, CPs, and SCPs in the last iteration for one of the 20 DKM runs. The procedure is as follows:
Generate 12 initial TPs, indicated by pentagrams in Fig. 4, and use them to build an initial surrogate model.
Generate sample points as CPs, denoted by solid dots in Fig. 4. is determined by Eq. (34), where remains the same for all the examples. Then, all the new SCPs and TPs, indicated by stars and circles, respectively, in Fig. 4, will be selected from the CPs.
Select SCPs from CPs based on the state of each CP (either in the safe or failure region) and the variance of the indicator function.
Select a new TP from SCPs if the point has the highest contribution.
Add the new TP and its response to the existing set of TPs; update the surrogate model.
Steps (2)–(5) are repeated until convergence. The contour of the final surrogate model at the limit state is plotted in Fig. 5, which shows that the surrogate model is accurate in the region where the random variables have high probability density and is less accurate in the region where the random variables have low probability density. This feature keeps the number of TPs minimal. Then, Eq. (19) is used to calculate the probability of failure using the final surrogate model and the same CPs.
After DKM is completed, the total number of limit-state function calls , which is also the total number of TPs, is recorded. Then, with the same initial TPs, IKM is performed, and its TPs are added iteratively until the total number of TPs reaches the recoded number . Then, Eq. (12) is used to calculate the probability of failure using the final surrogate model and the same CPs.
For comparison, the initial TPs, added TPs, and CPs of IKM are also plotted in Fig. 6. By comparing Figs. 4 and 6, we see that patterns of the added TPs of IKM and DKM are similar even though the two methods generate different TPs. The TPs of IKM are generated to minimize the error of the wrong sign of the limit-state function while the TPs of DKM are generated to minimize the error in the estimate of the probability of failure. As discussed previously, both the two TP updating strategies help increase the accuracy, and this is the reason that the two patterns are similar; the two strategies also have different foci, and this is the reason that the individual TPs from the two strategies are different. As indicated in the results, the strategy of DKM makes the updating process more efficient.
After DKM and IKM are performed 20 times, the average results are calculated and are shown in the row of DKM and IKM, respectively, in Table 1. With the same average 26.3 function calls or the same efficiency, DKM is more accurate than IKM. The results also show that DKM is more robust since the standard deviation of is smaller than that of IKM and AK-MCS.
The original AK-MCS is also performed 20 times with the same initial TPs as that of DKM or IKM. The results are given in the last row of Table 1. Both its average errors and number of function calls are larger than those of DKM and IKM. AK-MCS is less accurate for this problem because it requires a sample size smaller than that of DKM. AK-MCS is also less efficient because it requires a minimum value (or the minimum probability of wrong sign = 0.0228). This requirement does not have a direct link to the error of probability of failure, and it is satisfied with more function calls than that of DKM.
where , and . The distributions are given in Table 2.
The results are shown in Table 3, which indicate that DKM has higher accuracy than IKM, and both DKM and IKM outperform AK-MCS.
A roof truss structure [38,39] is shown in Fig. 8. Assume the truss bars bear a uniformly distributed load , which can be transformed into nodal load . The perpendicular deflection of the truss peak node is calculated by
where and are the cross-sectional areas of the reinforced concrete and steel bars, respectively; and are their corresponding elastic modulus; and is the length of the truss.
where . All the random variables are independent, and their distributions are given in Table 4.
Table 5 shows the average results from 20 runs, which indicate that DKM is more accurate than IKM and is also more accurate and efficient than AK-MCS.
The cantilever tube  shown in Fig. 9 is subjected to external forces , , , and torsion . The performance function is defined as the difference between the yield strength and the maximum stress
The cross-sectional area of the tube is , and the moment of inertia of the tube is . The torsional stress at the origin is , where . The distributions of the independent random variables are given in Table 6.
The results from Table 7 also show the higher accuracy of DKM.
The maximum motion error is shown in Fig. 11, and the failure region is shown in Fig. 12. The two figures indicate the irregularity and nonlinearity of the performance function and the failure region, for which traditional reliability methods, such as the FORM and SORM, will not be accurate. The DKM works quite well for this problem, as indicated by the results in Table 8. The results show that DKM is more accurate than IKM.
The efficiency of the reliability analysis is critical because it calls the associated limit-state function repeatedly. Kriging-based reliability methods are computationally efficient. As a result, they have increasingly been researched and applied, especially for highly nonlinear limit-state functions, for which the traditional FORM and SORM are not applicable. This study clearly demonstrates that the efficiency can be further improved by accounting for the dependencies between Kriging predictions.
The new DKM in this work improves the efficiency with its three new components. The first component is the new formula of calculating the probability of failure. The formula uses the average probability of failure at all the Monte Carlo samples, as well as both means and standard deviations of the Kriging predictions. The second component is the new learning function for selecting TPs. For a single sample point, the learning function considers not only the contribution of the point itself to the error of the probability of failure but also those of the dependencies from all the other points. The third component is the new stopping criterion that guarantees a good balance between accuracy and efficiency. The five examples indicate that DKM is more accurate than the Kriging-based methods that use only independent Kriging predictions.
The future work includes the following directions: (1) Improve the performance of DKM for problems with an extremely low probability of failure. (2) Extend DKM for system reliability analysis with at least two limit-state functions. (3) Incorporate DKM in reliability-based design, and (4) develop new DKM for time-dependent reliability analysis.
This material is based upon the work supported by the National Science Foundation through Grant Nos. CMMI 1234855 and CMMI 1300870. The support from the Intelligent Systems Center (ISC) at the Missouri University of Science and Technology is also acknowledged.
Appendix A: Kriging Method
Appendix B: IKM as a Special Case of DKM
is monotonic with respect to as shown in Fig. 13. This indicates that maximizing in DKM without considering correlations is equivalent to minimizing in IKM. The learning function of IKM is therefore a special case of that of DKM.