In structural reliability analysis, simulation methods are widely used. The statistical characteristics of failure probability estimate of these methods have been well investigated. In this study, the sensitivities of the failure probability estimate and its statistical characteristics with regard to sample, called ‘contribution indexes’, are proposed to measure the contribution of sample. The contribution indexes in four widely simulation methods, i.e., Monte Carlo simulation (MCS), importance sampling (IS), line sampling (LS) and subset simulation (SS) are derived and analyzed. The proposed contribution indexes of sample can provide valuable information understanding the methods deeply, and enlighten potential improvement of methods. It is found that the main differences between these investigated methods lie in the contribution indexes of the safety samples, which are the main factors to the efficiency of the methods. Moreover, numerical examples are used to validate these findings.

Reliability analysis plays an important role in the structural design. In reliability analysis, the evaluation of the failure probability is a basic problem. In the past decades many methods have been presented to address this issue. There are analytical methods, e.g., first-order reliability method (FORM) [

Though FORM and SORM are two elementary approaches and often very efficient, neither of them is robust in handling the case with a complex limit state function, such as a highly non-linear limit state function, or multiple failure states. Due to their inherent assumptions, both of them may not produce accurate estimates. MCS is a universal method, and is robust to the type and dimension of the problem. But it is inefficient when handing problem with small failure probability. Lots of variance reduction methods have been proposed, i.e., IS, LS and SS. IS is one of the most effective variance reduction method, and it generates samples according to an auxiliary distribution instead of the original distribution. LS employs lines instead of random points to explore the failure region of the problem. SS expresses the small failure probability as a product of large conditional failure probabilities, thus the small failure probability can be obtained by computing a series of big conditional failure probabilities with smaller sample size. On the other hand, MCS, IS, LS and SS are very inefficient compared with FORM or SORM, and the convergence to the exact solution is guaranteed for an increasing number of simulations, and confidence bounds on the solution are available in the case of a finite number of simulations. Furthermore, these methods are very robust in the sense that they can handle complex limit states. The surrogate model method is an important approach which owns highly efficient, however, the design of experiment (DoE) is the key of the accuracy of methods. Recently, the joint use of simulation methods and surrogate model methods, which is called ‘active learning method’, has been proposed, i.e., Kriging model with MCS [

Also, the sensitivity analysis for variables has been attracted more and more attentions [

It can be seen from above, the simulation-based reliability analysis and sensitivity analysis have been extensively studied [

In this work, three sample contribution indexes are proposed to quantify the effect of the samples in reliability analysis and four widely used reliability analysis methods are investigated by the proposed indexes. This work is motivated by empirical observations on the calculation of failure probability by using simulation method. Three contribution indexes have been proposed, which are associated with the sensitivity of the failure probability estimate with respect to sample, i.e., the contribution (sensitivity) of sample to the estimate of failure probability and its statistical characteristics in simulation-based method. And these indexes in four widely used simulation-based methods are derived and examined, i.e., Monte Carlo simulation (MCS), importance sampling (IS), line sampling (LS) and subset simulation (SS). It is of practical interest to ascertain the effectiveness and the role of the samples in different methods. Analysis of the contribution of each sample can show the tendency of the estimate and statistical characteristics as samples are generated, and also can compare the efficiency of each method for the reliability analysis. Meanwhile, it can be properly used in the active learning method in improving the efficiency of the method.

The paper is outlined as follows. First, in Section 2 the definitions of three sample contribution indexes are given which will be investigated through the paper. In the following section, the contribution indexes are derived for three different methods, i.e., MCS, IS, LS, and SS. Then in Section 4, some numerical examples are given to illustrate the contribution indexes. At last, some conclusions are drawn.

In order to study the contribution of sample, three indexes are proposed here to quantify and assess the contribution of the samples.

Suppose in a simulation-based reliability analysis, the

where _{j}_{j}

Similarly, the contribution of the sample to the variance and the coefficient of variation (c.o.v.) of the estimate can be also defined as:

where _{j}

In this section, the contribution of sample in four simulation-based methods are analyzed here. The four methods are widely used in the present practice engineering, which are MCS, IS, LS and SS, respectively.

Suppose that a set of samples

where _{f}_{s}

For the given set of samples, the contribution index of a certain sample, say

There are two kinds of samples in the variable space, the failure samples and the safety samples, respectively. Then the contribution index can be also simplified according to these two cases.

➀ When

where

In the context of small failure probability, the contribution of a failure sample is approximately reciprocal of the number of failure samples. The meaning of the index can be interpreted as that excluding this point will result in 100/_{f}_{f}

The total contribution of all the failure samples can also be obtained

➁ For another case which the sample falls in the safety region, that is

It can be seen that all the samples in the safety region are equally contributed to the failure probability estimate, which is approximately reciprocal of the number of safety samples.

The total contribution of all the safety samples can also be obtained

Based on

Comparing case ➀ and ➁, we can obtain that

_{f}

As it is well-known that, the estimate of failure probability in MCS is unbiased, that is

where

Substitute

Note that _{j}

However, in the calculation process, the item

In this context, suppose _{f}_{f}

In the other case that _{f,−j} = _{f}

It is seen from

In importance sampling method [

where

In this context, the contribution of a certain sample,

Similarly, we discuss the calculation of _{j}

➀ When

It seems that the failure samples in importance sampling method are not equally contributed to the failure probability estimate comparing to MCS. The contribution of the failure sample is nearly proportional to its weighted function value

And all the contribution of the failure samples nearly adds up to 1, that is

Especially, for problem with only normal variables, some properties can be obtained. As normal variables can be easily transformed to standard normal ones, we discuss in standard normal space for simplicity. In the standard normal space, the basic random variables

where

In order to see the characteristic of contour line of contribution index, let

where

Meanwhile, for points in different contour lines, the values of contribution indexes are exponentially decreasing/increasing, which are shown as

➁ For another case that

It can be seen that the expression for contribution index of safety sample is exactly the same as the one in MCS as shown in

And all the contribution of the safety sample is given by

Thus it seems that

As well-known that, the estimate of failure probability in importance sampling method is unbiased. The variance and c.o.v. of the estimate can be given by

Similarly, the theoretical contribution indexes of the sample,

Surprisingly, the expression is exactly the same as those of MCS given in

Similarly, in the computational process, all the expectation items in the variance and c.o.v. of the estimate are usually calculated by the samples, i.e.,

Thus, when

And when

Finally, the contribution indexes to the variance and c.o.v. can be computed by

In line sampling simulation [_{f}

with the conditional failure probabilities [

where

The contribution of a certain sample,

Approximately, it can be rewritten as:

It seems that the contribution of sample is proportional to its corresponding failure probability component.

In line sampling, the estimate of failure probability is also unbiased, and the variance and c.o.v. of the failure probability estimate can be given by

Theoretically,

Similarly, in computational process, _{f}

In subset simulation [_{f}

where

For a given set of samples generated in certain level, namely

Similarly, we still discuss it in two cases, safety sample and failure sample, respectively.

➀ Clearly, the samples in the 1th to

When

When

From _{i}

➁ when

For the final m-th level, when _{m}_{m}_{0} or

The statistical properties of the

First, the variance and c.o.v. of

where

where _{i}_{i}

It should be noted that although the

And hence the variance can be approximated by

In theory, the contribution indexes can be given by

In computational process, _{i+1} is estimated by

In order to compute the actual contribution index, suppose the sample,

where

Thus the estimated contribution indexes

_{j, s}, _{j, s}, _{j}_{j, s}, and _{j, s} quantify the contribution of samples to the failure probability estimate in reliability analysis simulation from two aspect, failure event (sample) and safety event (sample), respectively; while _{j}

MCS | IS | LS | SS | |
---|---|---|---|---|

_{f} |
||||

_{j, s} |
||||

_{j, f} |
||||

_{j} |
||||

Numerical examples are given here to calculate the contribution indexes of the four methods given above. Note that these examples are quite simple reliability problems by themselves, as they are selected to illustrate the findings given above with figures, and the simulation-based method is dependent of the number of dimensions and less affected by the complication of the problem. The first example is a normal linear case that the failure probability is varied from 10^{−3} to 10^{−5}. The second example is a highly nonlinear case.

The limit state function for the first example, which was also studied in [

where

The purpose is to investigate the performance of the contribution indexes for different probability levels and different methods.

MCS | IS | SS | |
---|---|---|---|

10^{4} |
10^{3} |
||

_{f} |
228 | 501 | 215 |

_{f} |
|||

_{j, s} |
|||

_{j, f} |
|||

_{j} |
– | ||

– | |||

It can be seen from _{f}

For IS method, some findings can also be seen from _{j, f} is approximately an exponent function from a certain viewpoint in the three-dimension plot, this is also pointed out in Section 3. And it is can be seen that the computed contribution index

For SS method, the following findings can be addressed from

In order to investigate the performance of indexes for different levels of failure probability, _{j, f} values of different methods are approximately in the same order of magnitude, however, the contribution _{j, s} values are different from each other considerably. In comparison, the contribution of safety sample of SS method is the biggest in the absolute value among the three methods, and that of MCS method is the smallest. Similar conclusions can be also drawn for the contribution to the statistical characteristics of estimate. This demonstrates that the high efficiency of reliability method is gained from the high contribution of safety sample. For the different levels of failure probability, this becomes more obvious. As in

MCS | IS | SS | |
---|---|---|---|

10^{5} |
10^{3} |
||

_{f} |
128 | 519 | 112 |

_{f} |
|||

_{j, s} |
|||

_{j, f} |
|||

_{j} |
– | ||

– | |||

MCS | IS | SS | |
---|---|---|---|

10^{3} |
|||

_{f} |
14 | 492 | 291 |

_{f} |
|||

_{j, s} |
|||

_{j, f} |
|||

_{j} |
– | ||

– | |||

The limit state function is given by

where _{1}, and _{2} are independent normally distributed random variables,

The contribution indexes of samples of four methods, i.e., MCS, IS, LS and SS, are computed and the results are given in _{j, f} values are approximately in the same order of magnitude as those of different methods.

MCS | IS | LS | SS | |
---|---|---|---|---|

10^{5} |
1000 | 500 | ||

_{f} |
582 | 383 | – | 583 |

_{f} |
||||

_{j, s} |
||||

_{j, f} |
– | |||

_{j} |
– | – | ||

– | – | |||

– | ||||

– | ||||

_{j, s}, and to c.o.v. _{j, s} is exponent, and

_{j}

_{j, s} for the samples in each level is shown in

The reliability of turbine disk is the key to the safety of the aeronautical engine. The fatigue life of an aeronautical engine turbine disk structure (See

According to the well-known Mason-Coffin law which is consider the effect of mean stress and mean strain on the fatigue life, the fatigue life can be computed as

where

Considering the actual life under of 0-takeoff-0 load cycle must exceeding the required fatigue life, the limit state function can be expressed as

where _{f0} is the required minimum service life and it is set as a const _{f0} = 10^{6} (cyc); _{f}

No. | Random variable | Mean | C.o.v. | Distribution |
---|---|---|---|---|

1 | 2029.0 | 0.1 | Normal | |

2 | 0.0196 | 0.1 | Normal | |

3 | 536.6 | 0.1 | Normal | |

4 | 0.0002225 | 0.1 | Normal | |

5 | −0.096 | 0.05 | Normal | |

6 | −0.41 | 0.05 | Normal |

The contribution indexes of samples of three methods, i.e., MCS, IS, and LS, are computed and the results are given in _{j}_{j, f} of IS method is about 100 times of the one of MCS, but for failure samples, the contribution indexes, _{j, f}, for both of these two methods are in the same order of magnitude. For LS method, _{j}^{−6} to 10^{−2}.

MCS | IS | LS | |
---|---|---|---|

10^{5} |
500 | 500 | |

_{f} |
190 | 261 | – |

_{f} |
|||

_{j, s} |
|||

_{j, f} |
– | ||

_{j} |
– | ||

– | |||

– | |||

– |

Meanwhile it is found that the contribution

In this paper, three contribution indexes have been proposed, which are the relative changes when a sample is included in reliability calculation or not. The indexes in three simulation-based methods are examined, i.e., Monte Carlo simulation, importance sampling and subset simulation. These indexes are proposed to quantify the sample contribution to the failure probability estimate and its statistical characteristics, thus investigate the efficiency of widely used reliability analysis methods from the contribution of the sample aspect.

Summarizing the arguments, the following findings can be concluded:

For Monte Carlo simulation, results show that the contribution of the failure sample to the estimate is bigger than that of safety sample in failure probability estimation.

For Importance sampling, the contribution index of the failure sample to the estimate is approximately proportional to its weighted function value while the safety samples contribute equally.

For line sampling method, the contribution indexes of the failure sample and safety sample are nearly the same.

For subset simulation, the sample contribution index is related with the conditional probability, but not the target probability to be computed. This can be a good explanation of why SS method owns high efficiency.

The further work should be the implementation of the proposed finding into the active learning in the DoE of surrogate methods, such as the combining of Kriging model with MCS [

The authors would like to acknowledge financial support from NSAF (Grant No. U1530122), the Aeronautical Science Foundation of China (Grant No. ASFC-20170968002) and the Fundamental Research Funds for the Central Universities of China (XMU, 20720180072).