Fatigue reliability-based design optimization of aeroengine structures involves multiple repeated calculations of reliability degree and large-scale calls of implicit high-nonlinearity limit state function, leading to the traditional direct Monte Claro and surrogate methods prone to unacceptable computing efficiency and accuracy. In this case, by fusing the random subspace strategy and weight allocation technology into bagging ensemble theory, a random forest (RF) model is presented to enhance the computing efficiency of reliability degree; moreover, by embedding the RF model into multilevel optimization model, an efficient RF-assisted fatigue reliability-based design optimization framework is developed. Regarding the low-cycle fatigue reliability-based design optimization of aeroengine turbine disc as a case, the effectiveness of the presented framework is validated. The reliability-based design optimization results exhibit that the proposed framework holds high computing accuracy and computing efficiency. The current efforts shed a light on the theory/method development of reliability-based design optimization of complex engineering structures.

Aeroengine structures like turbine discs usually operate in harsh multi-physics environments for long amounts of time and must fulfill a number of strict requirements like long service life and high fatigue reliability [

Currently, extensive efforts have been carried out to boost the resist fatigue performance of aeroengine structures [

Apart from the consideration of multiple uncertain factors, the modeling methods and/or solving algorithms are other key problems for performing reliability-based design optimization. Up to now, plenty of progress in this field has been succeed [

For the reliability-based design optimization of aerospace structures, the mapping relationships between structural responses and random variables always show complex high-nonlinearity traits, leading to the traditional Monte Carlo methods or moment-based methods incur the issues of insufficient computing efficiency or computing accuracy. As one valuable computing method, surrogate modeling methods have emerged and attracted much attention in reliability design fields [

In this paper, to perform high-accuracy and high-efficiency fatigue reliability-based design optimization of aeroengine structures, by fusing the random subspace strategy and weight allocation technology into bagging ensemble theory [

Due to the reliability degree, regression always involves the complex computations of high-nonlinearity and high-complexity, and establishing one individual model is frequently insufficient to achieve satisfactory regression accuracy in the fatigue reliability-based design optimization of aeroengine structures. In this case, by integrating several individual basis models into the integrated framework, ensemble models like random forests have emerged and are widely applied in probabilistic analysis [

In this case, by merging random subspace strategy and weight allocation technology, an assigning weight technique-based RF approach is presented to map the input variables to output response accurately. The basic thought of the presented method is: firstly, multiple sets (_{1}, _{2}, …, _{M}) of independent sample subsets are established using bootstrap sampling techniques; then, regarding the random subspace strategy-based parallel bagging framework, the basic architecture of RF is established; moreover, instead of simply averaging of all decision trees, the weight allocation technology is employed to assign different weights for each decision tree, the RF model is accurately mathematical modeled, to enhance the generalization accuracy and regression performance. The schematic diagram of the presented assigning weight technique-based RF model is illustrated in

Considering the extracted sample _{ij}, _{ij}( _{ij})}, the sample space _{1}, _{2}, …, _{M} by random subspace strategy, the basis decision tree

During the training process of the above decision trees, it is necessary to consider how to select and measure segmentation features and points. To find the best segmentation variables and points, the quality of segmentation features and points are measured by the weighted sum of impure degrees _{i}, _{ij}) of each sub node, i.e.,
_{i} indicates the segmentation feature of a node; _{ij} the segmentation value for segmentation features; _{left}, _{right} the sample number in the left sub node, and the right sub node, respectively; _{s} and the current node; _{left}, _{right} the training samples in the left, right sub node, respectively; _{s} represents the samples in the current node;

By substituting the

Regarding the dividing method, assuming the _{l} example (_{i}, _{i}) falls into the _{l}, i.e., _{l}), the corresponding output

Furthermore, assuming the RF consists of ^{t} across all of the decision trees, the output _{RF} is acquired as

By assigning the weight _{t} to the _{1}, …, _{T}) indicates the weight vector, ^{1}, …, ^{T})^{T} the output vector corresponding to example

By calculating the mean values of the output results in each decision tree, the distributed output response of the _{t} indicates the independent random variables; _{ij}, _{t}) is the output response of the decision tree based on _{ij} and _{t}.

For fatigue reliability-based design optimization problems of aeroengine structures, each optimization cycle involves multiple repeated calculations of reliability degree, and each calculation of reliability degree also contains large-scale calls of implicit high-nonlinearity limit state function, resulting in expensive computing cost and unsatisfactory optimization accuracy when using the direct Monte Claro Simulation and traditional surrogate methods [

By employing the presented random forest modeling method, the constitutive responses like mean stress and strain range of aeroengine structures can be mapped by

For the widespread low-cycle fatigue problems in aeroengine structures, fatigue life is usually evaluated using the improved Manson-Coffin model [_{m} indicates the mean stress; ^{L} the low-cycle fatigue life; _{f} the fatigue strength coefficient; _{f} the fatigue ductility coefficient; _{0} as the actual number of operating cycles, the fatigue damage

By regarding the

According to the Monte Carlo simulation thought [

By setting the high-sensitivity variables including operating loads _{L}, material parameters _{M} and life model parameters _{P} as the design variables, fatigue life _{f} as optimization objective, reliability degree

Therefore, by fusing the presented random forest model into the reliability optimization model, the large-scale of the real high-nonlinearity reliability degree calculations can be avoided effectively, which is conducive to reducing huge computational tasks and enhancing the optimal accuracy for the fatigue reliability-based design optimization of aeroengine structures. The RF-based fatigue reliability optimization workflow is constructed in

During the operation of aircraft engines, the high-pressure turbine disc is subjected to complex environmental loads such as high-temperature, high-pressure and high-speed, resulting in uneven temperature gradients, large stress loads, and low cycle fatigue damage. To achieve the fatigue reliability-based design optimization and improve the fatigue reliability performance of aeroengine turbine disc, the presented RF method is employed, where the RF-I represents the RF method with random subspace strategy, and RF-II presents the RF method with random subspace strategy and weight allocation technology. The sketch of aeroengine turbine disc is shown in

In view of the multisource uncertain factors co-determine the fatigue life of the turbine disc, we select the physical uncertain parameters as random variables [_{max} and strain range Δ_{t} appear at the center of the disc. In this study, the mean stress _{m} = (_{max} + _{min})/2 and the strain range ∆_{t} _{max } − _{min} at the risk section of the turbine disc are considered as the first layer output responses [

Variables | Rotating speed | Temperature | Elastic modulus | Material density | Thermal conductivity | Thermal expansion |
---|---|---|---|---|---|---|

^{o}C |
^{−9} (t/mm^{3}) |
^{o}C)) |
^{−6} (^{o}C) |
|||

Mean | 1198 | 650 | 182 | 8.21 | 18.3 | 13.9 |

Standard | 23.96 | 13 | 3.64 | 0.164 | 0.366 | 0.278 |

Distribution | Normal | Normal | Normal | Normal | Normal | Normal |

Temperature ^{o}C |
100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 |
---|---|---|---|---|---|---|---|---|---|

204 | 193 | 182 | 173 | 163 | 163 | 159 | 141 | 132 | |

^{o}C)) |
12.1 | 14.2 | 16.7 | 18.8 | 21.4 | 23.7 | 26.2 | 27.1 | 28.5 |

^{−6 o}C |
11.6 | 12.3 | 12.4 | 13.3 | 13.8 | 14.4 | 15.1 | 15.7 | 16.5 |

Variables | Fatigue strength index | Fatigue ductility index | Fatigue strength coefficient | Fatigue ductility coefficient |
---|---|---|---|---|

_{f} |
_{f} |
|||

Mean | −0.08 | −0.94 | 1318 | 0.976 |

Standard | 0.0016 | 0.0188 | 26.36 | 0.01952 |

Distribution | Normal | Normal | Normal | Lognormal |

To reduce the dimensionality of design variables and downsize the computing scale of fatigue reliability-based design optimization, high sensitivity parameters (_{f}_{m} and Δ_{t}) as objectives, and the value span of _{m}, Δ_{t}, _{f}_{f} as objective, and the fatigue reliability and value span as the constrained conditions, the second layer reliability optimization model is established. The built optimal model is shown in

Variables | ^{−9} (t/mm^{3}) |
_{f} |
||||
---|---|---|---|---|---|---|

Upper limit | 1186.02 | 643.5 | 180.18 | 8.128 | 1304.82 | −0.0792 |

Lower limit | 1209.98 | 656.5 | 183.82 | 8.374 | 1331.18 | −0.0808 |

Mean | 1198 | 650 | 182 | 8.21 | 1318 | −0.08 |

Standard | 23.96 | 13 | 3.64 | 0.164 | 26.36 | 0.0016 |

Distribution | Normal | Normal | Normal | Normal | Normal | Normal |

Based on the linkage sampling technique [^{−3} m/m) and standard deviations (7.7 MPa, 1.103 × 10^{−4} m/m), respectively.

According to the stress/strain responses obtained in

Based on the built reliability-based design optimization model, the fatigue reliability-based design optimization of the turbine disc is completed by using the presented RF methods (i.e., RF-I, RF-II). The probabilistic distributions of maximum stress, strain range and fatigue life before and after optimized by RF-I, RF-II are revealed. In

Methods | LCF life/cycle | Modeling time/s | Iterations | Optimization time/s | ||
---|---|---|---|---|---|---|

Before optimization | After optimization | Increase | ||||

QP | 6304 | 6803 | 499 | 35880 | 156 | 1.11 × 10^{7} |

SVR | 6304 | 7067 | 763 | 12920 | 76 | 4.02 × 10^{5} |

ANN | 6304 | 7526 | 1222 | 8450 | 56 | 2.62 × 10^{5} |

RF-I | 6304 | 8069 | 1765 | 2990 | 25 | 8.67 × 10^{4} |

RF-II | 6304 | 9258 | 2954 | 2530 | 23 | 7.59 × 10^{4} |

From

As shown in the iteration times and optimization time in

Therefore, the presented RF-II method can greatly reduce optimal iterations and save computational time while keeping the solving accuracy in fatigue reliability-based design optimization problems for aeroengine structures.

To perform high-efficiency and high-accuracy fatigue reliability-based design optimization for aeroengine structures, a random forest (RF) surrogate model is first presented by fusing the random subspace strategy and weight allocation technology into bagging ensemble theory; by embedding the RF surrogate into the multilevel optimal model, the RF-based fatigue reliability optimization framework is further developed. The presented framework is verified by the low-cycle fatigue reliability-based design optimization of aeroengine turbine disc considering the multisource uncertainties. Some conclusions are summarized as follows:

(1) From the fatigue reliability-based design optimization of turbine disc, the presented RF model is validated to be efficiently converged to the accurate reliability results, which can greatly accelerate the solving process in optimal models.

(2) Through the method comparisons, it has been confirmed that the RF-based reliability optimization framework that has been created has the computational benefits of high accuracy and high efficiency when it comes to designing aeroengine structures.

(3) The current work offers a brand-new RF-assisted fatigue reliability-based design optimization framework for aeroengine structures and thereby promoting the development of modeling and methodology for fatigue optimization design of aeroengine structures.

The work in this study proposed a feasible and effective method for fatigue reliability-based design optimization of aeroengine structures (i.e., Random Forest (RF) method). However, this method still has limitations regarding accuracy deviating from actual engineering. This deviation is mainly attributed to the incomplete factors considered in this study. According to current research, to further apply the proposed method framework in the future, the following issues need to be addressed:

(1) In the current reliability design of aeroengine structures, further improvement is needed to determine the random variable coefficients of model uncertainty and material randomness parameters.

(2) More sensitivity parameters that have a significant impact on the fatigue life should be further considered, the more accurate fatigue reliability optimization design should be performed.

(3) More accurate fatigue mechanics, such as creep-fatigue, and high-low cycle composite fatigue, etc., should be considered to improve the engineering application value of aerospace structures.

Monte Carlo simulation

Random forest

RF with random subspace strategy

RF with random subspace strategy and weight allocation technology

Quadratic polynomial

Support vector regression

Artificial neural network

rotational speed

temperature

elastic modulus

material density

thermal conductivity coefficient

coefficient of thermal expansion

fatigue strength index

fatigue ductility index

_{f}′

fatigue strength coefficient

_{f}

fatigue ductility coefficient

_{max}

maximum stress

_{t}

strain range

_{m}

mean stress

The authors express their appreciation to the National Natural Science Foundation of China. The authors also express gratitude for the support of the Hong Kong Polytechnic University and Beihang University.

This work was supported by the National Natural Science Foundation of China under Grant (Number: 52105136), the Hong Kong Scholar program under Grant (Number: XJ2022013) China Postdoctoral Science Foundation under Grant (Number: 2021M690290), and Academic Excellence Foundation of BUAA under Grant (Number: BY2004103). The authors would like to thank them.

The authors confirm contribution to the paper as follows: study conception and design: Lu-Kai Song; data collection: Xue-Qin Li; analysis and interpretation of results: Lu-Kai Song; draft manuscript preparation: Xue-Qin Li. All authors reviewed the results and approved the final version of the manuscript.

The data that support the findings of this study will be made available on request.

The authors declare that they have no conflicts of interest to report regarding the present study.