# Fuzzy Multi-SVR Learning Model for Reliability-Based Design Optimization of Turbine Blades

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## Abstract

**:**

## 1. Introduction

## 2. Theory and Method

#### 2.1. Fuzzy Multi-SVR Learning Method

#### 2.1.1. Fuzzy Multi-SVR Learning Model

**X**

^{(j)}, and the corresponding output response is y

^{(j)}(

**X**

^{(j)}), the response surface curve was constructed by the sample set{y

^{(j)}(

**X**

^{(j)}): j∈Z

^{+}}. The relationship between

**X**

^{(j)}and y is expressed by:

_{ji}is ith sample in jth SVR model, x

_{j}

^{`}is the center point of sample set

**x**

_{j}of jth SVR model, σ is the width of Gauss Kernel function

**k**

^{(j)}.

**θ**should be optimized to improve the modeling precision of SVR model in Equation (2). Search method is the traditional methods in seeking for the parameters

**θ**and has some blindness as it largely depends on the experience of researchers. Thus, more efficient algorithm should adopt to find the parameters

**θ**in SVR modeling.

#### 2.1.2. Multi-Objective Genetic Algorithm

**θ**. Comparing with GA, MOGA holds more flexible and adaptive design space exploration, and potentially avoid the influence of the plateau-like function profile [34]. Besides, MOGA breaks the limitation of a single population evolution of GA and uses multiple populations with different control parameters for optimization iterations [24,25]. Substantially, MOGA originates from GA and inherits natural selection and genetic characteristics, and the optimal solution of the objective function can be gained via successive iterations with selection, crossover, and mutation. The basic principle of the MOGA is summarized below. We first generate N initial populations (i.e., blade density, rotor speed, temperature, aerodynamic pressure, and gravity) with binary encoding, then the N new populations are obtained by the procedures of selection operator, crossover operator, and mutation. We further select the optimal individuals of each excellent population via the artificial selection operator, which are applied to the structure elite population to search for the optimal value of the objective function. Obviously, MOGA is essentially a combination of multiple GAs by a certain relationship. However, in MOGA, different control parameters (i.e., crossover probability p

_{c}

_{,l}and mutation probability p

_{m}

_{,l}) are used to complete the collaborative evaluation of multiple populations (l = 1, 2 …, N). Thus, the MOGA has both global and local search abilities by introducing immigrant operator to exchange messages among populations and avoid the destruction and loss of optimal individual information. In this process, the elite population does not participate as selection, crossover, or mutation operators. The minimum reserved generation is usually regarded as the terminal condition of optimization iterations. The procedure of MOGA is shown in Figure 1.

#### 2.2. RBDO Model with Fuzzy Multi-SVR Learning Method

_{j}of jth failure mode, and performance constraints allowable value [σ

_{io}] are considered as constraints. In this case, the RBDO model of multi-failure structures with fuzzy multi-SVR learning method is expressed by:

_{j}is the reliability degree under jth failure mode, [R

_{j0}] denotes the allowable reliability under jth failure mode, R stands for the comprehensive reliability of multi-failure modes, [R

_{0}] the allowable comprehensive reliability of multi-failure modes, ${\underset{\_}{x}}_{i},{\overline{x}}_{i}$ indicate upper and bottom boundary of ith fuzzy variables. The upper and bottom bounds of the transitional interval are determined by introducing amplification coefficient, i.e., ${\underset{\_}{x}}^{n}=\underset{\_}{x}$ ${\underset{\_}{x}}^{l}=\underset{\_}{\beta}\times \underset{\_}{x}$; ${\overline{x}}^{u}=\overline{\beta}\times \overline{x},{\overline{x}}^{l}=\overline{x}$; $\left\{\sigma \right\}=\left[{\sigma}_{x},{\sigma}_{y},{\sigma}_{z},{\tau}_{xy},{\tau}_{yz},{\tau}_{zx}\right]$, $\overline{\beta}=1.05~1.3$; ${\lambda}^{*}$ is the optimal level Scut set.

#### 2.3. Flowchart of RBDO with Fuzzy Multi-SVR Learning Method

**Step 1**: Build the FE model of turbine blades, and select blade density, rotor speed, temperature, aerodynamic pressure, and gravity as input variables, and regard the deformation and stress of turbine blades as two failure modes (i.e., optimized objects), and consider the fuzziness of rotor speed, gas temperature, and boundary conditions.

**Step 2**: Perform the static deterministic analysis of blades based on FE model and thermal-structural interaction, by regarding the means of random variables, to find the maximum points of blade stress and deformation as the object of turbine blade optimization.

**Step 3**: Regard the randomness and fuzziness of design parameters to extract handful samples of random variables by Latin hypercube sampling (LHS) technique which have been validated to be a highly effective sampling approach [35], and to calculate the output response of stress and deformation as output samples by FE simulations with the extracted samples of input variables.

**Step 4**: Normalize the samples comprising of input samples and output samples as the training samples to find the optimal parameters of SVR model by using an artificial bee colony (ABC) algorithm, which was verified to be an efficient parameter optimization approach [25], and then to build fuzzy multi-SVR learning models for the deformation and stress of turbine blades.

**Step 5**: Acquire enough samples gained by the linkage LHS technique [35], to conduct the probabilistic simulations of multi-failure structure based on the developed fuzzy multi-SVR learning models.

**Step 6**: Establish the RBDO model with the developed fuzzy multi-SVR learning method by employing the MOGA to find the optimal parameters in the RBDO, in which we regard load parameters (i.e., rotor speed, gas temperature, aerodynamic pressure, etc.) and material parameters as design variables, and stress and deformation as the objective functions (design objectives), as well as reliability and fuzzy boundary conditions as constraint functions.

**Step 7**: Implement the RBDO of a multi-failure turbine blade with the fuzzy multi-SVR learning method to search for the optimal design parameters subject to design objectives and constraints.

## 3. Fuzzy Reliability-Based Design of Multi-Failure Turbine Blade

#### 3.1. Deterministic Analysis of Turbine Blade

#### 3.2. Modeling for Fuzzy Multi-SVR Learning Method

**θ**= (c, σ, ε) of SVR model were optimized by an artificial bee colony (ABC) algorithm [25]. The fitness function of the ABC algorithm was defined as the mean square error (MSE) in the training process of the SVR model. Two hundred iterations were generally performed to gain the fitting curves of blade stress’s SVR (SVR-1) and deformation’s SVR (SVR-2). The two curves are drawn in Figure 6. SVR models are determined in respect of the optimal parameters (c, σ, ε) searched by the ABC algorithm. The coefficients of fuzzy multi-SVR learning models involving blade stress and deformation are shown in Equations (5) and (6). In coming work, we employ the two fuzzy multi-SVR learning models to carry out the RBDO of the multi-failure turbine blade.

#### 3.3. RBDO of Multi-Failure Turbine Blade

_{max}and maximum deformation δ

_{max}of the blade are minimized with regard to the RBDO model with the fuzzy multi-SVR learning method. We consider input random variables (ω, T) as design variables, and both blade stress σ and deformation δ as design objective function, as well as reliability index and boundary loads as constraint conditions, thus the RBDO model of turbine blade are shown in Equation (7). Herein, the upper and lower limits of fuzzy constraints are listed in Table 2.

#### 3.4. Fuzzy Multi-SVR Learning Method Validation

^{4}, because MC method cannot perform the calculation for a too-large computational burden for probabilistic analysis of blade FE models. Thus, it is inefficient for the MC method to conduct the design analysis of complex structure with large-scale simulations, (2) time-cost for blade probabilistic analysis increases with the increase of MC simulations for three methods, (3) the time consumption of the fuzzy multi-SVR learning method is far less than those of the MC method and the SVM method for the same number of simulations. For instance, the fuzzy multi-SVR learning method only spends 0.156 s for 1000 simulations, which is only about 1/2.2 × 10

^{6}that of the MC method and 47.5% that of the SVM method. Meanwhile, the strength of fuzzy multi-SVR learning method in time computation presents more obvious with increasing simulations. It is thus demonstrated that the efficiency of fuzzy multi-SVR learning method is far higher than the MC method and SVM method in calculation, and the fuzzy multi-SVR learning method is an efficient approach replacing FE models and SVM models, for the probabilistic analysis of multi-failure structures. (4) For the same simulations, the reliability degrees of blade coupling failure with the fuzzy multi-SVR learning method are almost consistent with these with the MC method, and are higher than the traditional SVM method. Moreover, the reliability degree of the blade increases and becomes higher with the rise of simulations. It is illustrated that more precise results such as reliability degree, can be gained by increasing the number of MC simulations against the response surface models, for structure design analysis from a probabilistic perspective.

## 4. Conclusions

- (1)
- From the RBDO of a turbine blade with deformation and stress failures with the presented fuzzy multi-SVR learning method, we gain that the stress and deformation of the blade under operation reduced by 92.38 MPa and 0.09838 mm, in the promise of acceptable computational precision and efficiency, which is promising to improve the reliability of turbine blade.
- (2)
- With regard to the probabilistic failure analysis of the bladed disk, we find that the developed fuzzy multi-SVR learning method does not only costs a small amount of analytical time and high computational efficiency relative to the Monte Carlo (MC) method and the SVM method, but also has an acceptable computational precision in the reliability degree as its optimization results are almost consistent with that of the FE method based on MC simulations. Moreover, the strengths of the proposed fuzzy multi-SVR learning method in modeling and simulation become more obvious with the increasing simulations.
- (3)
- In terms of the fuzzy RBDO of the multi-failure blade, it is illustrated that the developed fuzzy multi-SVR learning method is more workable than the MC method and SVM method. The reason is that the optimal parameters including design parameters and optimization objects are preferable as larger reductions and higher reliability degree.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Reliability-based design optimization (RBDO) procedure with fuzzy multi-support vector machine of regression (SVR) learning method.

**Figure 7.**Membership functions of blade allowable failures. (

**a**) Allowable stress membership function, (

**b**) allowable deformation membership function.

**Figure 9.**Distributions of blade stress and deformation before and after optimization. (

**a**) Stress distributions, (

**b**) deformation distributions.

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

Density ρ, kg/m^{3} | 8210 | 414.1934 | Normal |

Rotor speed ω, rad/s | 1168 | 104.7138 | Normal |

Temperature T, K | 1173.2 | 105.18 | Normal |

Aerodynamic pressure P, MPa | 0.5 | 0.0448 | Normal |

Gravity g, m/s^{2} | 9.8 | 0.294 | Normal |

Upper and Lower Limit | [σ], MPa | [δ], mm | ω, rad/s | T, K | |
---|---|---|---|---|---|

Upper bound | Upper limit | 604.75 | 2.01195 | 1349.0 | 1355.0 |

Lower limit | 574.75 | 0.00195 | 1284.8 | 1290.5 | |

Lower bound | Upper limit | - | - | 1051.2 | 1055.9 |

Lower limit | - | - | 735.84 | 739.13 |

Design Variables | Original Data | Optimization Results |
---|---|---|

ω, rad/s | 1168 | 1200.1 |

T, K | 1173.2 | 1110.9 |

**Table 4.**Computing time and reliability degrees of blade probabilistic analysis with different methods.

Number of Samples | Computing Time, s | Reliability Degree, % | ||||
---|---|---|---|---|---|---|

MC Method | Traditional SVM | Fuzzy Multi-SVR Learning Method | MC Method | Traditional SVM | Fuzzy Multi-SVR Learning Method | |

10^{2} | 54,000 | 0.0108 | 0.0062 | 99 | 97 | 98 |

10^{3} | 339,200 | 0.329 | 0.156 | 99.5 | 98.3 | 99.2 |

10^{4} | - | 0.789 | 0.468 | 99.34 | 98.65 | 99.29 |

10^{5} | - | 2.013 | 1.232 | - | 98.791 | 99.782 |

Objective Functions | Before Optimization | MC Method | Traditional SVM | Fuzzy Multi-SVR Learning Method | |||
---|---|---|---|---|---|---|---|

After Optimization | Reduction | After Optimization | Reduction | After Optimization | Reduction | ||

σ, MPa | 583.75 | 552.59 | 31.16 | 530.23 | 53.52 | 491.37 | 92.38 |

δ, mm | 1.0195 | 0.98814 | 0.03136 | 1.0001 | 0.0194 | 0.92112 | 0.09838 |

R | 95.40 | 96.9 | - | 97.83 | - | 98.85 | - |

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## Share and Cite

**MDPI and ACS Style**

Zhang, C.-Y.; Wang, Z.; Fei, C.-W.; Yuan, Z.-S.; Wei, J.-S.; Tang, W.-Z.
Fuzzy Multi-SVR Learning Model for Reliability-Based Design Optimization of Turbine Blades. *Materials* **2019**, *12*, 2341.
https://doi.org/10.3390/ma12152341

**AMA Style**

Zhang C-Y, Wang Z, Fei C-W, Yuan Z-S, Wei J-S, Tang W-Z.
Fuzzy Multi-SVR Learning Model for Reliability-Based Design Optimization of Turbine Blades. *Materials*. 2019; 12(15):2341.
https://doi.org/10.3390/ma12152341

**Chicago/Turabian Style**

Zhang, Chun-Yi, Ze Wang, Cheng-Wei Fei, Zhe-Shan Yuan, Jing-Shan Wei, and Wen-Zhong Tang.
2019. "Fuzzy Multi-SVR Learning Model for Reliability-Based Design Optimization of Turbine Blades" *Materials* 12, no. 15: 2341.
https://doi.org/10.3390/ma12152341