Turbine blisks are assembled using blades, disks and casings. They can endure complex loads at a high temperature, high pressure and high speed. The safe operation of assembled structures depends on the reliability of each component. Monte Carlo (MC) simulation is commonly used to analyze structural reliability, but this method needs to run thousands of computations. In order to assess the clearance reliability of assembled structures in an efficient and precise manner, the novel Kriging-based decomposed-coordinated (DC) (DCNK) approach is proposed by integrating the DC strategy, the Kriging model and the importance sampling-based Markov chain (MCIS) technique. In this method, the DC strategy is used to decompose a multi-objective problem into many single-objective problems. The relationships between these many single-objectives and the overall objective are then coordinated. The Kriging model is applied to establish the limit state functions of the single-objectives and multi-objective problems, while the MCIS method is used to assess the structural assembled clearance reliability. Moreover, a highly nonlinear complex compound function is first utilized to verify the DCNK model from a mathematical perspective. Then, the reliability of an aeroengine high-pressure turbine (HPT) blade-tip radial running clearance (BTRRC) is analyzed to validate the DCNK approach by considering thermo-structural interaction. The analytical results show that the reliability is 0.9976 when the allowance value of the BTTRC is 1.7650 × 10−3 m. Compared with different methodologies (including direct simulation, the classical Kriging model, and the weighted response surface method (WRSM)), the proposed method holds obvious advantages in computing time and precision, as well as simulation efficiency and precision. The efforts of this paper provide a useful approach to analyzing assembled clearance reliability and contribute to the development of structural reliability theory.
Assembled structures usually involve many components in accordance with specific principles. These structures typically suffer from interaction loads between multi-physical fields. For instance, an aeroengine high-pressure turbine is assembled by blisks and casings, which enables it to endure thermal loads and structural loads during operation. Moreover, the overall safety of assembled structures is determined by the reliability of its components. As for assembled structures, if a component is not sufficiently reliable, the function of the entire system is affected, and a catastrophic accident can occur. Therefore, the clearance reliability of assembled structures must be analyzed by considering the randomness of influencing factors.
A number of direct methods have been proposed to analyze structural reliability. Rezaei et al. [
In order to overcome the shortcomings of direct methods, indirect approaches (surrogate models) have been proposed to analyze the reliability of complex structures. Tandjiria et al. [
In order to address the aforementioned issues, we develop a surrogate modeling strategy, namely the novel Kriging-based decomposed-coordinated (DCNK) approach, for the purpose of assessing structural assembled clearance reliability. In this case, the decomposed-coordinated (DC) strategy is used to decompose the multi-objective problem into many single-objective problems. Then, this strategy is used to coordinate the relationships between single-objectives and the overall objective. Furthermore, the Kriging model is applied to establish the limit state functions of the single-objective and multi-objective problems, while the importance sampling-based Markov chain (MCIS) technique is employed to assess structural assembled clearance reliability. In addition, we select a highly nonlinear compound function and an aeroengine high-pressure turbine (HPT) to validate the proposed method.
The rest of this paper is outlined as follows. In Section 2, the basic theory of structural assembled clearance reliability assessment with the DCNK approach is discussed. In Section 3, a highly nonlinear compound function is used to verify the DCNK approach in terms of predictive performance. In Section 4, the reliability analysis for the HPT blade-tip radial running clearance (BTRRC) is derived, so as to validate the analytical precision and computing efficiency of the DCNK method. Finally, in Section 5, the main conclusions are summarized.
In order to analyze the structural reliability of multiple objectives, the DCNK approach is developed by integrating the Kriging model, the DC strategy, the importance sampling principle and the Markov chain method. For the DCNK approach, the DC strategy is employed to decompose a “big” problem into many “small” problems, and to coordinate the “small” problems to process a “big” problem. Moreover, the Kriging model is used to derive the limit state functions of the related objectives, while the MCIS technique is adopted to generate samples and assess structural assembled clearance reliability. The structural assembled clearance reliability assessment process (including the DCNK approach) is shown in
As seen in
(1) Deterministic analysis—it is necessary to establish finite element (FE) models of objective structures, set constraints and boundary conditions, and execute the deterministic analysis based on the established FE models.
(2) Sample generation—it is necessary to determine the study time point depending on the variation in output response; ensure input parameters and their numerical features; obtain samples of input parameters using the Latin hypercube sampling (LHS) method [
(3) DCNK modeling—it is necessary to establish the DCNK model based on the training samples and validate the established DCNK model using the testing samples. If the DCNK model prediction accuracy cannot satisfy the precision requirement, return to Step (2). Otherwise, move to Step (4).
(4) Reliability assessment—it is necessary to extract the candidate sampling pool using the MCIS method; obtain these values for the relevant output responses; and determine the structural assembled clearance reliability and output analysis results.
In this section, we elaborate on the basic theory underpinning the DCNK approach, including the Kriging model.
The Kriging model was first introduced into geostatistics by Krige [
where
where
where
where
The estimated variance
where
The stochastic component
where
Although the traditional Kriging model is suitable for general reliability problems, it is unable to effectively assess the clearance reliability of assembled structures, because this method requires multiple models to be established. In addition, the DC strategy proposed by Adomian has been utilized to approximate the complex compound function and performs well [
As shown in
The assembled structure clearance function is given by:
where
where
where
Based on
where
According to the Kriging model, the
where
Similarly,
The overall model layer is derived in the form of the Kriging model,
where
Furthermore, the limit state function of the assembled structure clearance is given by:
where
Through the above analysis, the limit state function of the assembled structure clearance is decomposed into many subfunctions. Each subfunction is established based on the Kriging model, and then the overall model of the assembled structure clearance function is derived based on the relationship between the whole object and its sub-objects.
In this section, the probabilistic analysis is conducted with the limit state function of the assembled structure clearance. The candidate sample pool is determined
The Markov chain based on the Metropolis-Hasting algorithm is adopted in order to simulate the conditional samples of the failure domain, and to improve their efficiency [
(1) Define the stationary distribution of the Markov Chain. According to the analysis results, the failure probability of the assembled structure clearance is given by:
where the vector
where
(2) Select the proper proposal distribution. The proposal distribution
where
(3) Select the initial state of the Markov Chain
(4) Determine the
According to the Metropolis-Hasting algorithm,
where
(5) Generate
In summary, the conditional samples
In
The importance sampling PDF is introduced to solve the failure probability [
where
As the MPP is the point with the largest joint PDF value in the failure domain, the importance sampling PDF is constructed at the MPP,
The MCIS can be used to simulate the failure domain samples without information concerning the true samples, such that it may determine the MPP with the largest joint PDF value and thus construct the importance sampling PDF. Therefore, the MCIS is suitable for the highly nonlinear and high-dimensional limit state function.
In order to analyze the reliability of the assembled structure clearance efficiently, the MCIS and DCNK models are combined. The MCIS is used to construct the importance sampling PDF
The flowchart for analyzing the assembled structure clearance reliability is summarized in
(1) Construct the importance sampling PDF
(2) Generate the candidate sampling pool
(3) Compute the DCNK model
(4) Calculate the failure probability
In order to validate the proposed DCNK model by comparing the Kriging model and the WRSM, a highly nonlinear complex compound function is considered as the case study. All computations are completed on a 64-bit desk computer with Intel Core i5–10400 of 2.9 GHz CPU and 32 GB RAM.
In the numerical example,
Variable | Mean | Standard deviation |
---|---|---|
3 | 0.1 | |
4 | 0.1 | |
2.5 | 0.1 | |
5 | 0.1 | |
0.5 | 0.1 | |
−0.5 | 0.1 |
The output response relationship between the complex compound function and subfunctions is expressed by
In order to establish the DCNK model, 50 samples are obtained using the LHS method in accordance with the input variables’ distribution parameters. 30 samples are used as training samples to establish the DCNK model, while the remaining 20 samples are taken as the testing samples to verify the performance of the DCNK model. The decomposed and coordinated models of the subfunctions are given by:
Then,
Using these 30 samples as the training samples, a Kriging model
Similarly, the WRSM model
The accuracy of the DCNK model is tested using the remaining 20 samples. Using the true response for the compound function as a reference, the prediction error of the DCNK model is calculated by comparing the Kriging model and the WRSM. The prediction errors, including the absolute error (
where
The input variable
3.3728 | 3.8552 | 2.5696 | 5.0774 | 0.1710 | −0.7201 | |
3.0441 | 3.8383 | 2.7297 | 4.6712 | 0.1355 | −0.6781 | |
3.1817 | 4.1805 | 2.3721 | 5.0593 | 0.6995 | −0.5954 | |
2.8818 | 4.0135 | 2.3790 | 5.3479 | 0.6187 | −0.4723 | |
2.9538 | 4.1479 | 2.4430 | 4.7604 | 0.5560 | −0.1794 | |
2.6504 | 4.0794 | 2.6390 | 4.8726 | 0.6934 | −0.4245 | |
3.0140 | 4.1872 | 2.5213 | 4.7966 | 0.5135 | −0.4188 | |
3.1854 | 3.5486 | 2.7251 | 5.2258 | 0.5280 | −0.3913 | |
2.7884 | 3.7956 | 2.3456 | 4.8887 | 0.3250 | −0.2593 | |
2.9555 | 3.9656 | 2.5822 | 5.1503 | 0.5019 | −0.3839 |
Input variables | True |
DCNK | Kriging | WRSM | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|
value | value | value | ||||||||||
6.6019 | 6.6248 | 0.0229 | 0.0069 | 6.2636 | 0.3382 | 0.2155 | 6.7557 | 0.1539 | 0.0643 | |||
7.1396 | 7.1367 | 0.0029 | 7.5982 | 0.4586 | 7.9760 | 0.8364 | ||||||
4.5041 | 4.4269 | 0.0773 | 5.2464 | 0.7422 | 4.5895 | 0.0853 | ||||||
4.5135 | 4.4821 | 0.0314 | 4.2592 | 0.2543 | 4.7365 | 0.2230 | ||||||
4.8819 | 4.8751 | 0.0068 | 4.4979 | 0.3840 | 4.9731 | 0.0912 | ||||||
6.9529 | 6.9824 | 0.0295 | 6.4689 | 0.4840 | 7.1697 | 0.2169 | ||||||
5.6904 | 5.6848 | 0.0056 | 5.3006 | 0.3898 | 5.7460 | 0.0556 | ||||||
8.5013 | 8.4855 | 0.0158 | 10.0864 | 1.5851 | 8.5572 | 0.0559 | ||||||
4.1692 | 4.1670 | 0.0022 | 5.7383 | 1.5692 | 4.2888 | 0.1196 | ||||||
6.7974 | 6.8102 | 0.0128 | 6.5370 | 0.2604 | 6.7070 | 0.0904 |
As seen in
Methods | Modeling and prediction times |
---|---|
DCNK | 0.0829 s |
Kriging | 0.1077 s |
WRSM | 15.2695 s |
For different test samples, the absolute error curves of the DCNK model, the Kriging model and the WRSM are shown in
In summary, the DCNK model performs better with regard to the complex compound function with large-scale parameters for a highly nonlinear problem. It is indicated that the DCNK model is more robust and stable as a model. Therefore, the DCNK method is used to estimate the assembled clearance reliability of an aeroengine HTP BTRRC.
The BTRRC refers to a critical assembly relationship in an aeroengine high-pressure turbine. Its running clearance seriously affects the performance and reliability of an aeroengine. In this section, we consider the BTRRC of an aeroengine as the study object for the purpose of verifying the proposed DCNK approach.
Aeroengine turbines endure complex loads during operation, such as thermal loads, and structural loads. In order to analyze the thermo-structural coupling, we simplify the HPT model by ignoring tenons, pin holes, and cooling holes [
The radial deformations of the blisks and casings are the largest in the climb phase of aeroengines [
The key parameters are selected as random variables in BTRRC analysis [
Object | Variables | Mean | Standard deviation |
---|---|---|---|
Blisk | 540 | 16.2 | |
210 | 6.3 | ||
200 | 6.0 | ||
245 | 7.35 | ||
320 | 9.6 | ||
1527 | 45.81 | ||
1082 | 32.86 | ||
864 | 25.92 | ||
1030 | 31 | ||
980 | 29.4 | ||
820 | 24.6 | ||
11756 | 352.68 | ||
8253 | 247.59 | ||
6547 | 196.41 | ||
1168 | 35.04 | ||
8210 | 0.123 | ||
Casing | 1050 | 31.5 | |
320 | 9.6 | ||
6000 | 180 | ||
5400 | 162 | ||
4800 | 144 | ||
4200 | 126 | ||
2600 | 78 | ||
8400 | 252 |
Based on the characteristics of the random variables in
The BTRRC coordinated model
The limit state function of the BTRRC is denoted as:
where
The convergence analysis of the BTRRC limit state function is performed using different DCNK approach simulations. The results of the convergence analysis are shown in
As shown in
The simulation history and BTRRC deformation distribution histogram are shown in
As demonstrated in
In order to verify the DCNK approach in the BTRRC reliability assessment, the BTRRC limit state equation is simulated for different times using the four methods (MC simulations, the Kriging model, the WRSM, and the DCNK model). The failure probabilities and reliability are listed in
Sampling |
Failure probability | Reliability | Precision (%) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
MC | WRSM | Kriging | DCNK | MC | WRSM | Kriging | DCNK | WRSM | Kriging | DCNK | |
100 | 0 | 0.01 | 0.01 | 0.0035 | 1 | 0.99 | 0.99 | 0.9965 | 99 | 99 | 99.65 |
1000 | 0.002 | 0.004 | 0.001 | 0.0026 | 0.998 | 0.996 | 0.999 | 0.9974 | 99.70 | 99.92 | 99.96 |
2000 | 0.0022 | 0.003 | 0.0028 | 0.0024 | 0.9978 | 0.997 | 0.9972 | 0.9976 | 99.92 | 99.94 | 99.98 |
10000 | – | 0.0031 | 0.0027 | 0.0024 | – | 0.9969 | 0.9973 | 0.9976 | – | – | – |
As illustrated in
In this paper, a new surrogate model method is proposed for the probabilistic analysis involving assembled structure clearance with high nonlinearity and hyperparameters. In order to establish the relationship between the input parameters and output response of the assembled clearance, the DCNK approach is proposed by combining the Kriging model and the DC strategy. The MCIS method and the DCNK model are used to assess the reliability of the assembled structure clearance. The effectiveness and applicability of the DCNK approach are verified numerically and by assessing aeroengine HTP BTRRC reliability. Some conclusions are summarized as follows:
(1) The average absolute error of the DCNK model is 0.0069, which is far less than that of the Kriging model and the WRSM (
(2) The prediction accuracy of the DCNK model is 99.3%, which is closer to the true value than that of the Kriging model and the WRSM (78.5% and 93.6%, respectively).
(3) The modeling and prediction time for the DCNK model (0.0829 s) are smaller than those of the Kriging model (0.1077 s) and the WRSM (15.2695 s).
(4) The failure probability is 0.0024 and reliability is 0.9976 when the allowance value of the BTTRC is 1.7650 × 10−3 m. The failure probability and reliability of the DCNK approach converges when the number of samples reaches 2000.
(5) The precision of the DCNK approach (99.98%) is higher than that of the Kriging model and the WRSM (99.92% and 99.94%).
In summary, compared to the Kriging model and the WRSM, the DCNK approach performs better for the complex compound function with large-scale parameters for a highly nonlinear problem, when modeling approximation accuracy (modeling accuracy) and simulation performance (computational efficiency and precision). The present study offers an effective approach to highly nonlinear assembled structures, and promising insights for the probabilistic optimal design of the HPT BTRRC.