Complex implicit performance functions widely exist in many engineering problems. The reliability analysis of these problems has always been a challenge. Using surrogate model instead of real performance function is one of the methods to solve this kind of problem. Kriging is one of the surrogate models with precise interpolation technique. In order to make the kriging model achieve higher accuracy using a small number of samples, i.e., improve its practicability and feasibility in practical engineering problems, some active learning equations are wildly studied. Expected improvement function (EIF) is one of them. However, the EIF has a great disadvantage in selecting the added sample point. Therefore, a joint active learning mechanism, J-EIF, is proposed to obtain the ideal added point. The J-EIF active learning mechanism combines the two active learning mechanisms and makes full use of the characters of kriging model. It overcomes the shortcoming of EIF active learning mechanism in the selection of added sample points. Then, using Monte Carlo Simulation (MCS) results as a reference, the reliability of two examples is estimated. The results are discussed showing that the learning efficiency and accuracy of the improved EIF are both higher than those of the traditional EIF.
Reliability analysis plays an increasingly important role in practical engineering problems [
Several surrogate models, for instance, neural network [
For improving the approximating ability of the kriging model, some elaborate active learning functions are constructed and employed to update the kriging model in an iterative way. An expected improvement function (EIF) [
For the EIF active learning mechanism, firstly it is needed to use the current best solution, usually the minimum value of the real system at current training sample points, to construct a learning function. Secondly, find the sample point that maximizes the EIF in a large number of samples; notice that this process is performed with the kriging model rather than real objective function. Finally, add this sample and its corresponding real function value to training samples to update the kriging model. Meanwhile, the above active learning will terminate when the iteration process satisfies a given convergence criterion. This paper studied the EIF active learning mechanism and found some shortcomings in the process of screening the added sample point. The improvement is made for EIF to optimize its learning process and then improve the prediction ability of the kriging model. In particular, a joint active learning mechanism, J-EIF, is proposed to obtain the ideal added point. The J-EIF active learning mechanism combines the two active learning mechanisms which have the same principle but different integral area. It overcomes the shortcoming of EIF active learning mechanism in the selection of added sample points.
It is well known that three key points should be considered in a surrogate model with high accuracy and efficiency. The first point is the accuracy of the metamodel itself, the second point is the performance of the active learning mechanisms, and the third is the selection of appropriate experimental design points. In this paper, the target of the research is to improve the performance of the active learning mechanisms to make the accuracy and efficiency of the surrogate model better, rather than the metamodel or experimental design points.
The remaining of this paper is organized as follows. A brief introduction of MCS procedure for reliability analysis and kriging metamodel is given in Section
Suppose that
It can be seen from (
Use
Given
It can be seen from (
It can be seen clearly that the prediction of the kriging model at a new sample vector
It can be seen from the above analysis that a good kriging prediction model is a guarantee of reliability estimations. The EIF active learning mechanism is a function which can improve the prediction ability of the kriging model by adding new sample to training samples. Next, a brief introduction is given on the EIF. Then, some discussion on the EIF is given and a J-EIF active learning mechanism is proposed.
The EIF is an active learning function used to select the location at which the value of EIF is the largest. Then, the corresponding sample point and its objective function value should be added to the training sample points to improve the prediction ability of the kriging model. Its learning mechanism is given as follows [
This indicates that
For simplicity, a single variable function
Kriging predictor and its standard error.
It can be seen from (
Let us suppose that the minimum value of objective function
It can be seen from (
Similar to EIF active learning function, a new EIF active learning function is introduced and expressed as follows:
Comparing (
Suppose that
In view of the above problems, a joint active learning mechanism, J-EIF, is proposed to obtain the ideal added point. It is expressed as follows:
which indicates that the ideal added point
In the following sections, a numerical example is studied to illustrate the accuracy of the proposed J-EIF. The result of the MCS procedure with 106 samples is used as a reference. Then, a certain system in example 2 demonstrates the practicality and feasibility of kriging model which combined the J-EIF active learning mechanisms. The computational efficiency and accuracy of the J-EIF and the EIF active learning equation are also compared in example 2.
In order to illustrate the accuracy of the proposed J-EIF, a numerical example is studied in this section with the reference results of MCS. Assume the performance function is
In this example, variables
It can be seen from Tables
In order to intuitively compare the results of the two methods, the reliability listed in Tables
Reliability changes when variation coefficient
| 0.3 | 0.32 | 0.34 | 0.36 | 0.38 |
---|---|---|---|---|---|
MCS | 0.9992 | 0.9986 | 0.9977 | 0.9967 | 0.9953 |
J-EIF | 0.9992 | 0.9986 | 0.9977 | 0.9966 | 0.9952 |
Error | 0.00% | 0.00% | 0.00% | 0.01% | 0.01% |
Reliability changes when threshold
| 4.0 | 4.2 | 4.4 | 4.6 | 4.8 |
---|---|---|---|---|---|
MCS | 0.9300 | 0.9478 | 0.9619 | 0.9727 | 0.9809 |
J-EIF | 0.9308 | 0.9484 | 0.9622 | 0.9729 | 0.9811 |
Error | 0.09% | 0.06% | 0.03% | 0.02% | 0.02% |
Reliability changes when variation coefficient varies.
Reliability changes when threshold varies.
It can be seen from Figure
Numerical example 1 fully demonstrates the high accuracy of the proposed J-EIF. Next, a certain system example is studied and used to verify the superiority of the proposed J-EIF method.
Assume the limit state function of a certain system is
According the above information, the reliability results of this example at different variation coefficient
The computational efficiency and accuracy of improved method J-EIF are estimated by two indices, i.e., the number of the function calls
In order to observe how reliability changes when variation coefficient
It can be seen from the above analysis that the improved J-EIF active learning mechanism performs well with the variety of variation coefficient. Its performance at different thresholds is listed in Table
It can be seen from Table
It can be intuitively seen from Figure
The accuracy and the number of the function calls are two important indicators that reflect the performance of the proposed method. In this paper, the first example just reflects the accuracy of the proposed method, and the second example not only reflects the accuracy of the proposed method, but also illustrates the two important indicators in detail. It is worth noting that example 2 verifies the accuracy and the number of the function calls of the proposed method from different angles; i.e., the two important indicators of the proposed method are illustrated by the different variation coefficients or different thresholds. Although there are only two examples, the sufficient calculations have already illustrated the accuracy and the number of the function calls of the proposed method in detail.
Random variables and their distribution parameters of a certain system.
Variables name | | | | | | | |
---|---|---|---|---|---|---|---|
Mean value | 50 | 43 | 35 | 32 | 1000 | 1000 | 2E11 |
Reliability results at different levels of variation coefficient
| 0.003 | 0.0035 | 0.004 | 0.0045 | 0.005 |
---|---|---|---|---|---|
MCS | 0.9994 | 0.9972 | 0.9924 | 0.9844 | 0.9739 |
| |||||
J-EIF | 0.9995 | 0.9974 | 0.9926 | 0.9843 | 0.9733 |
| 0.01% | 0.02% | 0.02% | 0.01% | 0.06% |
| 12+5 | 12+5 | 12+5 | 12+4 | 12+4 |
| |||||
EIF | 0.9997 | 0.9982 | 0.9941 | 0.9875 | 0.9792 |
| 0.03% | 0.10% | 0.17% | 0.31% | 0.54% |
| 12+8 | 12+8 | 12+7 | 12+7 | 12+7 |
Reliability results with the changes of threshold
| 0.0121 | 0.0122 | 0.0123 | 0.0124 | 0.0125 |
---|---|---|---|---|---|
MCS | 0.9904 | 0.9942 | 0.9965 | 0.9980 | 0.9989 |
| |||||
J-EIF | 0.9904 | 0.9943 | 0.9967 | 0.9982 | 0.9990 |
| 0.00% | 0.01% | 0.02% | 0.02% | 0.01% |
| |||||
EIF | 0.9931 | 0.9962 | 0.9980 | 0.9990 | 0.9995 |
| 0.27% | 0.20% | 0.15% | 0.10% | 0.06% |
The reliability at different levels of variation coefficient
The reliability at different levels of threshold
The selection of a new added sample point by the EIF has some shortcomings; i.e., the added sample point selected by the EIF may not be the best solution. In view of the above problems, this paper proposes a J-EIF active learning function, which extends the unilateral EIF active learning mechanism to a bilateral active learning mechanism and makes full use of the characters of kriging model. The high accuracy of the proposed J-EIF can be seen from numerical example 1. In addition, the results show that the relationship between the reliability and variation coefficient or threshold is nonlinear rather than linear. To fully verify the superiority of the proposed method, a certain system is studied by the proposed J-EIF and original EIF method. The results show that the proposed J-EIF reaches the convergence criterion by using fewer samples than EIF, indicating that the J-EIF active learning function has a higher convergence speed than the EIF. Moreover, using MCS results as a reference, it can be found that the kriging model based on the J-EIF has higher accuracy than that based on the EIF.
The data used to support the findings of this study are available from the corresponding author upon request.
The authors declare that they have no conflicts of interest.