# Decomposed Collaborative Modeling Approach for Probabilistic Fatigue Life Evaluation of Turbine Rotor

^{1}

^{2}

^{*}

## Abstract

**:**

_{p}and fatigue strength coefficient σ

_{f}′ are the main affecting factors to fatigue life, whose effect probability are 28% and 22%, respectively. By comparing with direct Monte Carlo method, KM method, IKM method and DC response surface method, the presented DCIKM is validated to hold high efficiency and accuracy in probabilistic fatigue life evaluation.

## 1. Introduction

## 2. Decomposed Collaborative Modeling Approach

#### 2.1. Intelligent Kriging Modeling

#### 2.1.1. Kriging Model Overview

_{1}, X

_{2}, …, X

_{m})

^{T}and their output responses Y = (y

_{1}, y

_{2}, …, y

_{m})

^{T}, the nonlinear relationship between input points and output responses can be mapped by an interpolation Kriging model, i.e.,

_{ε}

^{2}is the process variance, R(∙) the spatial correlation function, θ=(θ

_{1}, θ

_{2}, …, θ

_{n}) the correlation parameter vector.

_{d}, its response value Ŷ(X

_{d}) can be estimated by product of weight w and known response Y, that is Ŷ(X

_{d}) = w

^{T}Y. The prediction error can be obtained as:

_{d}and sampling points,

**R**(∙) the correlation matrix.

#### 2.1.2. Intelligent Algorithm

_{i}is the i-th correlation parameters; n the number of input variables; f(θ) the fitness function.

_{ave}/f

_{max}) is greater than some certain value, the high-quality individuals are difficult to generate due to the lack of the species diversity, then the dynamic mutation operation would be set with a high mutation rate to enhance the mutation rate; otherwise the dynamic crossover operation would be set with a high crossover rate to enhance the crossover rate [38]. The arcsin(f

_{ave}/f

_{max}) is adopted as the judgment function of species diversity, since it can change faster with the increase of average fitness f

_{ave}. The dynamic crossover rate p

_{c}and dynamic mutation rate p

_{m}are designed as

_{ave}indicates average fitness; f

_{max}the maximum fitness; arcsin(∙) the arc sine function. In this paper, k

_{1}and k

_{2}are chosen as 1 and 0.005, respectively. Moreover, π/6 is adopted as a guideline because arcsin(f

_{ave}/f

_{max}) ≥ π/ 6 equals to f

_{ave}/f

_{max}≥ 1/2, which reflects the diversity of species. Furthermore, the reason for dividing by π/2 is to ensure that arcsin(f

_{ave}/f

_{max})/(π/2) ≤ 1.

#### 2.1.3. Intelligent Algorithm with Kriging Model, IKM

_{p}) are obtained:

**R*** are the regression coefficient, process variance and correlation parameters matrix corresponding to the optimal correlation parameter θ*, respectively.

#### 2.2. Decomposed Collaborative IKM, DCIKM

#### 2.2.1. Basic thought of DCIKM

- Regarding the evaluation layer and response traits, the complex model with all input variables and total output response is divided into multiple simple submodels, each of which contains fewer input variables and one output response. It is assumed that the submodels are independent of each other.
- Considering the plasticity of materials, the thermal-structure coupling deterministic analysis is accomplished through FE simulation.
- The output responses of sub-models are obtained by importing several input variables into FE calculation, and the input variables and output responses are treated as training and testing data.
- With the extracted samples, the decomposed IKM of sub-models are constructed by the proposed IKM thought.
- Massive sampling for input variables is performed by Latin hypercube sampling (LHS) technique, and the statistical characteristics of output responses are obtained by decomposed IKM simulation.
- Taking the output responses of decomposed IKM models as the input variables, the collaborative IKM is established. By employing the simple DCIKM approach instead of time-consuming direct MC simulation, the probabilistic fatigue life evaluation is accomplished.

#### 2.2.2. Mathematical Modeling of DCIKM

^{(p)}represents the input variable of the p-th layer, the corresponding output response Y

^{(p)}is:

^{(p)}of surrogate model at the p-layer can be described as:

^{(1)}, Y

^{(2)}, …, Y

^{(r)}} as input variables $\overline{X}$, $\overline{Y}$ represents the output response of overall surrogate model, then the collaborative IKM model is constructed as:

## 3. Probabilistic Fatigue Life Evaluation Theory

^{(p)}], the limit state function G

^{(p)}(x) based on decomposed IKM model can be expressed as:

^{(p)}(x) can be determined by the indicator function of failure domain:

_{l}indicates the l-th data set; l = 1, 2, …, s.

_{i}means the i-th input vector of all sample variables; Mean(·) the mean function; Var(·) the variance function.

_{r}[G(x

_{l})] is the indicator function of secure domain; N

_{r}the sample number in secure domain; N the total sample number.

_{f}[G(x

_{l})] the indicator function of total failure domain; N

_{b}the sample number of total failure domain.

## 4. Case Study

#### 4.1. Material Preparations

#### 4.1.1. Finite Element Model

^{−3}m and “minimum edge length” as 1.9995 × 10

^{−3}m, and refined the local mesh of blade root by setting “element size” as 3.6 × 10

^{−3}m. From the convergence effects and simulation accuracy in reference [3,30], with the set meshing procedures, the meshing effects is guaranteed effectively. As shown in Figure 6, the FE model of simplified turbine rotor involves 18,454 quadrilateral elements and 30,911 element nodes. Moreover, an appropriate symmetric boundary constraint is imposed on the sector disc, and axial and circumferential constraints are loaded on the inner diameter arc. Furthermore, for facilitating the fatigue life calculation, we simplify the actual complex load spectrum of turbine rotor into the trapezoidal load spectrum [32].

#### 4.1.2. Variable Selection

_{f}′ and fatigue ductility coefficient ε

_{f}′, are considered as the second part of input random variables and its distribution characteristics are shown in Table 2.

_{*}σ, μ + f

_{*}σ] (where μ is the variable mean, σ the standard deviation, f the positive constant). The parameter f determines the sampling domain range, f is set to 4 in this study, since this sampling domain contains 99.99% variable fluctuation information [44] and the corresponding failure possibility.

#### 4.2. Deterministic Fatigue Life Evaluation

_{max}-0, the mean stress σ

_{m}is obtained as 0.5 × (σ

_{min}+σ

_{max}). With the calculated mean stress σ

_{m}, elastic strain range ∆ε

_{e}, plastic strain range ∆ε

_{p}and Manson–Coffin model, the low-cycle fatigue life of turbine rotor is obtained as 3606 cycles. Manson–Coffin model is shown in Equation (27).

_{f}′ the fatigue strength coefficient; ε

_{f}′ the fatigue ductility coefficient; b the fatigue strength index; c the fatigue ductility index; and N

_{f}the failure cycle number.

#### 4.3. Decomposed Stress-Strain Prediction

#### 4.3.1. Decomposed IKM Modeling

#### 4.3.2. Stress–Strain Prediction with Decomposed IKM Model

^{−4}m/m, standard deviations of 4.46 MPa and 1.64 × 10

^{−4}m/m, respectively. The elastic strain range nearly obeys standard normal distribution with mean value 4.82 × 10

^{−3}m/m and standard deviation 1.15 × 10

^{−4}m/m.

#### 4.3.3. Sensitivity Analysis with Decomposed IKM Model

_{m}, elastic strain range

**∆**ε

_{e}and plastic strain range

**∆**ε

_{p}. To further reflect the correlation of output response and prominent input invariables, the scatter sketches are drawn in Figure 14, Figure 15 and Figure 16, respectively. Note that the scatter points in Figure 16 are close to the X-axis because some extracted plastic strain range responses are close to zero.

#### 4.4. Collaborative Fatigue Life Evaluation

#### 4.4.1. Collaborative IKM Modeling

_{f}′, fatigue ductility coefficient ε

_{f}′) and decomposed output responses (mean stress σ

_{m}, elastic strain range ∆ε

_{e}plastic strain range ∆ε

_{p}) are taken as input variables of collaborative IKM, and the fatigue life of turbine rotor is taken as total output response. Similar to the establishment of decomposed IKM, 107 groups of samples are extracted, where 30 samples for testing samples and the remaining for training samples. The modeling process of collaborative IKM with DMIGA is shown in Figure 17. Through the comparison of real test outputs with the estimated outputs, the model performance of collaborative IKM is validated in Figure 18. Note that the data in Figure 18 comes from the testing dataset.

#### 4.4.2. Fatigue Life Evaluation with Collaborative IKM Model

_{f}are obtained by simulating the collaborative IKM with 10,000 simulations. The simulation history and distribution feature of fatigue life N

_{f}are shown in Figure 19. Obviously, the fatigue life N

_{f}nearly obeys log-normal distribution. In view of the reliability analysis theory in Equation (24), the probabilistic fatigue life under reliability 99.87% is 3296 cycles. It should be noted that 3296 cycles are the probability fatigue life corresponding to 10,000 simulations, and the operating hours corresponding to 3296 cycles based on the flight load spectrum [32] are 1207 h.

#### 4.4.3. Sensitivity Analysis with Collaborative IKM Model

_{p}and fatigue strength coefficient σ

_{f}′ are the main factors affecting the fatigue life, accounting for 28% and 22%, respectively. Therefore, ∆ε

_{p}and σ

_{f}′ should be preferentially regarded in fatigue reliability design of turbine rotor. Moreover, we also find that the decrease of ∆ε

_{p}will result in the increase of fatigue life, while the increase of σ

_{f}′ will lead to the increase of fatigue life. The correlation between fatigue life and main influencing parameters are depicted in Figure 21. Note that the scatter points in Figure 21a are close to the Y-axis is owing to some plastic strain range responses are near to zero.

#### 4.5. Method Validations

_{f}′, c, b, ε

_{f}′] and output response N

_{f}directly, while DCRSM and DCIKM adopt parallel calculation by simulating elastic strain range Δε

_{e}, plastic strain range Δε

_{p}, mean stress σ

_{m}, and fatigue life N

_{f}in several computer devices. The computing costs and computing precision of different evaluation methods are compared in Table 4 and Table 5, respectively.

## 5. Conclusions and Outlooks

- The simulation history and distribution characteristics of fatigue life are obtained and the reliability-based fatigue life N
_{f}= 3296 cycles is recommended for the turbine rotor fatigue life design, which is conducive to greatly enhance the safety performance of turbine rotor. - The sensitivity analysis results show that rotor speed and gas temperature are the main factors on mean stress, elastic strain range and plastic strain range, while plastic strain range and fatigue strength coefficient are the major factors on fatigue life, which provides a valuable guidance for further optimization of turbine rotor.
- Methods comparison (MCM, KM, IKM, and DCRSM, DCIKM) illustrates that the proposed DCIKM holds superiority in computing efficiency and accuracy. Accordingly, it is proved that the intelligent algorithm searching for optimal Kriging parameters is promising to build a higher-precision Kriging model. Moreover, the decomposed collaborative strategy is suitable to decrease the nonlinearity of probabilistic design of turbine rotor.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 5.**Modeling procedure of the presented decomposed collaborative intelligent Kriging model (DCIKM).

**Figure 7.**Nephogram of the stress-strain and temperature for turbine rotor: (

**a**) stress, (

**b**) temperature, (

**c**) elastic strain range, (

**d**) plastic strain range.

**Figure 9.**Prediction results of decomposed IKMs: (

**a**) decomposed IKM-1, (

**b**) decomposed IKM-2, (

**c**) decomposed IKM-3.

**Figure 10.**Output responses of decomposed IKM-1: (

**a**) simulation history, (

**b**) probabilistic distribution.

**Figure 11.**Output responses of decomposed IKM-2: (

**a**) simulation history and, (

**b**) probabilistic distribution.

**Figure 12.**Output responses of decomposed IKM-3: (

**a**) simulation history, (

**b**) probabilistic distribution.

**Figure 13.**Sensitivities and effect probabilities with different decomposed output responses: (

**a**) mean stress σ

_{m}, (

**b**) elastic strain range

**∆**ε

_{e}, (

**c**) plastic strain range ∆ε

_{p}.

**Figure 21.**Scatter sketches of fatigue life: (

**a**) plastic strain range, (

**b**) fatigue strength coefficient.

Random Variables | Mean | Standard Deviation | Distribution |
---|---|---|---|

Rotate speed ω, rad/s | 922 | 18.4 | Normal |

Gas temperature T, k | 773.2 | 15.5 | Normal |

Density ρ, 10^{−9} t/mm^{3} | 8.21 | 0.164 | Normal |

Modulus of elasticity E, GPa | 163 | 3.26 | Normal |

Heat conductivity λ, W/(m °C) | 21.4 | 0.428 | Normal |

Thermal expansion coefficient α, 10^{−6} °C | 13.8 | 0.276 | Normal |

Random Variables | Mean | Standard Deviation | Distribution |
---|---|---|---|

Fatigue strength index b | −0.1 | 0.002 | Normal |

Fatigue ductility index c | −0.84 | 0.0168 | Normal |

Fatigue strength coefficient σ_{f}′ | 1419 | 28.38 | Normal |

Fatigue ductility coefficient ε_{f}′ | 0.505 | 0.0101 | Lognormal |

Temperature (°C) | 100 | 200 | 300 | 400 | 500 | 600 | 700 | 800 | 900 |
---|---|---|---|---|---|---|---|---|---|

E, GPa | 205 | 196 | 182 | 173 | 163 | 163 | 159 | 141 | 134 |

λ, W/m °C | 12.1 | 14.2 | 16.7 | 18.8 | 21.4 | 23.7 | 26.2 | 27.6 | 28.9 |

α, 10^{−6} °C | 11.6 | 12.3 | 12.4 | 13.3 | 13.8 | 14.4 | 15.1 | 15.7 | 16.5 |

n | MCM | KM | IKM | DCRSM | DCIKM | |||
---|---|---|---|---|---|---|---|---|

Time, s | Time, s | Time, s | Improved Efficiency, % | Time, s | Improved Efficiency, % | Time, s | Improved Efficiency, % | |

10^{2} | 5754 | 45.7 | 40.1 | 12.25 | 31.9 | 30.19 | 22.7 | 50.33 |

10^{3} | 60,890 | 47.1 | 41.2 | 12.53 | 32.8 | 30.36 | 23.2 | 50.74 |

10^{4} | 798,954 | 49.8 | 43.1 | 13.65 | 34.6 | 30.52 | 24.5 | 50.80 |

10^{5} | — | 58.7 | 50.4 | 14.14 | 39.7 | 32.37 | 28.3 | 51.79 |

_{c}− T

_{KM})/T

_{KM}, where T

_{c}is the calculation time of the compared method, T

_{KM}the calculation time of KM method.

**Table 5.**Reliability analysis results of five methods for turbine rotor fatigue life (N = 3296 cycles).

n | MCM | KM | IKM | DCRSM | DCIKM | ||||
---|---|---|---|---|---|---|---|---|---|

Reliability | Reliability | Precision, % | Reliability | Precision, % | Reliability | Precision, % | Reliability | Precision, % | |

10^{2} | 0.92 | 0.81 | 88.04 | 0.90 | 97.83 | 0.87 | 94.57 | 0.91 | 98.91 |

10^{3} | 0.984 | 0.915 | 92.99 | 0.971 | 98.68 | 0.942 | 95.73 | 0.975 | 99.09 |

10^{4} | 0.9977 | 0.9521 | 95.43 | 0.9969 | 99.92 | 0.9731 | 97.53 | 0.9972 | 99.95 |

10^{5} | — | 0.9579 | 96.01 | 0.9971 | 99.94 | 0.9739 | 97.61 | 0.9970 | 99.93 |

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## Share and Cite

**MDPI and ACS Style**

Huang, Y.; Bai, G.-C.; Song, L.-K.; Wang, B.-W.
Decomposed Collaborative Modeling Approach for Probabilistic Fatigue Life Evaluation of Turbine Rotor. *Materials* **2020**, *13*, 3239.
https://doi.org/10.3390/ma13143239

**AMA Style**

Huang Y, Bai G-C, Song L-K, Wang B-W.
Decomposed Collaborative Modeling Approach for Probabilistic Fatigue Life Evaluation of Turbine Rotor. *Materials*. 2020; 13(14):3239.
https://doi.org/10.3390/ma13143239

**Chicago/Turabian Style**

Huang, Ying, Guang-Chen Bai, Lu-Kai Song, and Bo-Wei Wang.
2020. "Decomposed Collaborative Modeling Approach for Probabilistic Fatigue Life Evaluation of Turbine Rotor" *Materials* 13, no. 14: 3239.
https://doi.org/10.3390/ma13143239