The reliability and sensitivity analyses of stator blade regulator usually involve complex characteristics like high-nonlinearity, multi-failure regions, and small failure probability, which brings in unacceptable computing efficiency and accuracy of the current analysis methods. In this case, by fitting the implicit limit state function (LSF) with active Kriging (AK) model and reducing candidate sample pool with adaptive importance sampling (AIS), a novel AK-AIS method is proposed. Herein, the AK model and Markov chain Monte Carlo (MCMC) are first established to identify the most probable failure region(s) (MPFRs), and the adaptive kernel density estimation (AKDE) importance sampling function is constructed to select the candidate samples. With the best samples sequentially attained in the reduced candidate samples and employed to update the Kriging-fitted LSF, the failure probability and sensitivity indices are acquired at a lower cost. The proposed method is verified by two multi-failure numerical examples, and then applied to the reliability and sensitivity analyses of a typical stator blade regulator. With methods comparison, the proposed AK-AIS is proven to hold the computing advantages on accuracy and efficiency in complex reliability and sensitivity analysis problems.
During the air compression process of aeroengine compressor, stator blade regulator plays a key role in increasing airflow and preventing surge phenomenon [
For probability analysis problems involving multiple random input variables, the key point is to calculate the following multivariate integral, i.e.,
To obtain the failure probability by solving
In recent years, by shifting the sampling center from the origin to the most probable point (MPP), AK combined with importance sampling (AK-IS) can produce satisfactory results as long as the MPP can be well identified. Moreover, it has become hot solution paths in addressing small probability problems [
Under such circumstances, by fitting the implicit LSF with AK model and reducing candidate sample pool with adaptive importance sampling (AIS), a novel AK-AIS method is proposed to address the high-nonlinearity, multiple failure regions, and small failure probability problems. In the proposed method, AK model is adopted to determine the initial state of Markov chain and identify the MPFRs, and the adaptive kernel density estimation (AKDE) function is constructed to generate the candidate importance samples in adaptive reduced sample pool. By sequentially importing samples to update the Kriging model, the reliability and sensitivity indices can be obtained through few real LSF calls. Compared with the current relevant methods [
The organization of this paper is summarized as follows: The AK-AIS method is expounded in
In this section, to reduce the calls of real LSF and improve the computing accuracy of complex reliability and sensitivity analysis problems, a novel AK-AIS method is presented, which includes the identification of the MPFRs, adaptive importance sampling and the corresponding procedures. The basic principle of the proposed AK-AIS method is illustrated in
By determining the initial state of Markov chain with AK model and calibrate the boundaries by M-H algorithm, the accurate identification of the MPFRs is realized as follows.
To acquire the precise steady state distribution (i.e., optimal importance sampling density function) in complex reliability and sensitivity problems, the AK model instead of subjective experience [
Considering Latin hypercube sampling (LHS) technique, the initial samples
Considering the large-scale randomly generated samples (i.e., 10^{4} samples) as candidate sampling pool, based on active learning function (i.e., U [
Based on the updated Kriging model, the failure region samples with
Moreover, it should be noted that compared with the direct subjective given initial state of Markov chain, although the additional large candidate sampling pool (i.e., 10^{4} samples) is used to update Kriging model, only few LSF calls are required during the determination of the MCMC initial state.
During the transition of the Markov chain from one state to another, by introducing an acceptance mechanism, M-H algorithm is adopted to accept the new transition state with a certain probability. With the combination of the state transition probability and acceptance probability, the failure samples can be generated quickly in Markov chain simulation. Considering the distributions of failure samples, the multiple failure boundaries can be accurately calibrated. The main steps are introduced as follows.
Considering the obtained initial state of Markov chain {
Regarding the (
According to the improved M-H criterion [
Repeat Step (2) until
To address the large tail error problem caused by the fixed KDE function and low modeling efficiency problem caused by the large candidate sampling pool in IS method, an adaptive kernel density estimation (AKDE) function is constructed based on the identified MPFRs, and an adaptive reduced sampling pool varying with the fitted Kriging model is established. By combining the AKDE to generate important samples and the adaptive reduction sample pool to reduce candidate important samples, the adaptive importance sampling (AIS) method is proposed to guarantee a small tail error and few calls of real LSF.
Considering the generated samples in identified MFPRs, by combing Gaussian kernel function with high robustness and adaptive kernel window width parameter, a width-varying KDE function (i.e., optimal important sampling density function) is built. This function can be regarded as the optimal important sampling density function, which can generate a set of important samples based on discrete random integer interpolation. Since the AKDE function fitted by the samples in identified MFPRs is closer to the optimal sampling density function, the sampling accuracy is promising to be elevated.
Based on the generated samples {
Assuming that a discrete random integers
The window width parameter controls the smoothness of the AKDE function. By adaptively adjusting the kernel window width in the KDE process, the constructed AKDE function is promising to improve the smoothness degree and accelerate the convergence of the optimal sampling density function, then the desired important samples can be generated precisely.
Regarding the large candidate importance sampling pool will reduce the finding efficiency of the desired samples, by only selecting the importance samples close to the Kriging-fitted LSF as the candidate sample pool [
With the reduced candidate sampling pool, the number of candidate samples is significantly reduced. Moreover, the reduced candidate sampling pool moves adaptively as the Kriging-fitted LSF updated ceaselessly, which ensures that the desired samples can be precisely found. In addition, since the Kriging update process is based on AK model described in
The estimated variance
With the samples of input random parameters gained by massive samplings, the local sensitivity index of the failure probability
To assess the effect of
Therefore, the sensitivity index is transformed into a variance-based index of the indicator function, i.e.,
Based on the total variance law
The normalized version of the main effect index and total effect index are given as
The flowchart of the proposed algorithm is depicted in
Noticeably, by precisely identifying the most probable failure regions with AK model and efficiently re-updating the Kriging-fitted LSF by adaptive importance sampling, the proposed AK-AIS method holds the potential to improve the computational efficiency and accuracy for complex reliability and sensitivity analysis problems.
To demonstrate the accuracy and efficiency of the proposed method, two numerical examples are selected to compare the proposed method with the existing methods. All computations are performed on an Inter(R) Core (TM) Desktop Computer (3 GHz CPU and 16 GB RAM).
The first example is taken from references [
To identify the MPFRs using the proposed AK-AIS method, the real LSF functions are called 13 and 25 times to build and update the Kriging model, respectively; in the AIS process, the real LSF functions are further called 13 times to approximate the LSF curves. With 51 (i.e., 13 + 25 + 13) real LSF calls, the failure probability is achieved as 3.39 × 10^{−3} and the sensitivity indices are listed in
Indices | Mean PRS |
Standard deviation PRS |
Main effect index |
Total effect index |
---|---|---|---|---|
−0.1976 | 2.6395 | 0.0165 | 0.5438 | |
1.4447 | 7.0030 | 0.4609 | 0.9845 |
To validate the computational advantages of the proposed method, MCS, FORM, Subset, Meta-IS, MetaAK-IS^{2}, and KAIS are also performed the example. From
Methods | Computing error | |||
---|---|---|---|---|
Crude MCS [ |
120000 | 3.35 × 10^{−3} | – | <5% |
FORM [ |
7 | 1.35 × 10^{−3} | 59.7% | – |
Subset [ |
300000 | 3.48 × 10^{−3} | 3.88% | <3% |
Au and Beck [ |
100 + 500 | 2.47 × 10^{−3} | 26.27% | 8% |
Meta-IS [ |
44 + 600 | 3.54 × 10^{−3} | 5.67% | <5% |
MetaAK-IS^{2} [ |
48 + 69 | 3.47 × 10^{−3} | 3.58% | <5% |
KAIS [ |
35 + 65 + 63 | 2.69 × 10^{−3} | 19.7% | 5% |
AK-AIS+EFF | 13 + 44 = 57 | 3.43 × 10^{−3} | 2.39% | 1.27% |
AK-AIS+REIF | 13 + 54 = 67 | 3.36 × 10^{−3} | 0.29% | 0.76% |
AK-AIS+REIF2 | 13 + 72 = 85 | 3.41 × 10^{−3} | 1.79% | 1.00% |
AK-AIS+U | 13 + 38 = 51 | 3.39 × 10^{−3} | 1.19% | 1.05% |
Note: Assuming that the
The second example [
Using the proposed method, the real LSF are called 10, 19, and 23 times to build, update, and re-update the Kriging model, four failure regions are identified and the adaptive importance sampling is performed, respectively. With 52 (i.e., 10 + 19 + 23) times of real LSF calls, the failure probability is achieved as 2.22 × 10^{−3} and the sensitivity indices are acquired in
Indices | Mean PRS |
Standard deviation PRS |
Main effect index |
Total effect index |
---|---|---|---|---|
−0.0340 | 5.1936 | 0.0424 | 0.9255 | |
−0.0394 | 5.1628 | 0.0408 | 0.9290 |
To validate the computational advantages of the proposed method,
Methods | Computing error | |||
---|---|---|---|---|
Crude MCS [ |
781016 | 2.24 × 10^{−3} | – | 2.3% |
FORM [ |
7 | 1.35 × 10^{−3} | 39.73% | – |
DS [ |
1800 | 2.22 × 10^{−3} | 0.89% | – |
Subset [ |
600000 | 2.22 × 10^{−3} | 0.89% | 1.5% |
SMART [ |
1035 | 2.21 × 10^{−3} | 1.34% | – |
MetaAK-IS^{2} [ |
48+90 | 2.22 × 10^{−3} | 0.89% | 1.7% |
AK-AIS+EFF | 10+43=53 | 2.19 × 10^{−3} | 2.23% | 1.16% |
AK-AIS+REIF | 10+46=56 | 2.22 × 10^{−3} | 0.89% | 1.75% |
AK-AIS+REIF2 | 10+43=53 | 2.21 × 10^{−3} | 1.34% | 0.87% |
AK-AIS+U | 10+42=52 | 2.22 × 10^{−3} | 0.89% | 0.83% |
Note: Assuming that the
Under the multiple uncertain parameters of flexible deformation, joint friction and multi-physics loads, the HCF life of stator blade regulator exhibits large dispersion, which seriously affects the fatigue reliability and security, and prone to HCF failure. A typical stator blade regulator with TC4 titanium alloy is illustrated in
According to the operating loads, material properties and model parameters pose important influence to the HCF life dispersion of stator blade regulator, the gas temperature
According to Latin hypercube sampling (LHS) technique and the distributed traits of input variables, 65 groups of input variables are extracted, and the corresponding real outputs (fatigue life
Based on the LHS technique, 10,000 samples are generated and imported into the Kriging-fitted LSF curves instead of the complex rigid-flexible coupling model. Then, the marginal probability distributions of mean stress
Indices | ||||
---|---|---|---|---|
0.1580 | −0.2157 | 0.0001 | 0.2424 | |
0.2766 | −0.0637 | 0.0001 | 0.1933 | |
−0.0013 | −0.1571 | −0.0004 | 0.1624 | |
−0.1831 | −0.0232 | −0.0002 | 0.1266 | |
0.0771 | 0.0290 | −0.0001 | 0.0753 | |
−0.0350 | −0.0074 | −0.0001 | 0.0347 | |
−0.1321 | −0.0495 | 0.0001 | 0.0948 | |
0.5097 | −0.4415 | 0.0021 | 0.4953 | |
2.4230 | 5.2505 | 0.1472 | 0.9386 | |
−1.3340 | 1.5346 | 0.0113 | 0.6886 | |
0.3230 | 0.0450 | 0.0004 | 0.2020 |
To verify the advantages of the proposed AK-AIS method, we compare the failure probability analysis results with MCS, FORM and AK-MCS. As shown in
Methods | |||
---|---|---|---|
Crude MCS | 10000 | 0.0011 | – |
FORM | 60 | 0.0778 | – |
AK-MCS | 498 | 0.00085 | 0.077 |
AK-AIS | 484 | 0.0010 | 0.085 |
To address the high-nonlinearity, multi-failure regions, and small failure probability problems in reliability and sensitivity analyses of stator blade regulator, by combining the advantages of active Kriging (AK), Markov chain Monte Carlo (MCMC), adaptive kernel density estimation (AKDE) and importance sampling (IS), a novel active Kriging-based adaptive importance sampling (AK-AIS) method is developed. The proposed method is validated by two numerical examples with multiple failure regions and then is applied to a typical reliability and sensitivity analysis of aero-engine stator blade regulator. Some conclusions are summarized as follows:
(1) Numerical examples verify that the proposed AK-AIS method holds high accuracy and efficiency in MPFRs identification and the Markov chain initial state determination.
(2) Application case indicates that the proposed method can efficiently and accurately accomplish the reliability and sensitivity analysis of stator blade regulator. Moreover, it has been found that the material constant
(3) The proposed method is suitable for the complex reliability and sensitivity analyses with high-nonlinearity, multiple failure regions, and small failure probability, which provides theoretical guidance for determining the initial state of Markov chain in complex engineering.
Although the study provides a feasible and efficient approach for the reliability and sensitivity analyses of stator blade regulators, limitations do exist. Most deviations from expected solution can be attributed to incomplete factors considered in this study. According to the present study and the questions raised, the following problems require to be addressed for further application of the proposed approach in future.
(1) To improve the computing quality of complex mechanism reliability analysis, more additional factors (design tolerance, performance degradation, vibration mechanics, and so forth) should be analyzed and investigated.
(2) To further improve the computational efficiency of reliability analysis with high-dimensional, small failure probability problem, a more effective method to determine the initial state of Markov chain needs to be further studied.
(3) An advanced kernel function should be adopted in high precision kernel sampling density function, such as Gaussian linear mixture kernel function, ANOVA Kernel, and Sigmoid Kernel.
Active Kriging
Adaptive Importance Sampling
Adaptive Kernel Density Estimation
AK combined with Importance Sampling
AK combined with MCS
Active Kriging-based Adaptive Importance Sampling
Coefficient of Variation
Directional Simulation
First-Order Reliability Method
High Cycle Fatigue
Instrumental PDF
Important Sampling
Kriging-based Adaptive Importance Sampling
Limit State Function
Metropolis Hastings
Monte Carlo Simulation
Most Probable Point
Markov chain Monte Carlo
Most Probable Failure Region(s)
Meta-model-based Importance Sampling
Probability Density Function
Second-Order Reliability Method