# Weighted Regression-Based Extremum Response Surface Method for Structural Dynamic Fuzzy Reliability Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Basic Theory on Dynamic Fuzzy Reliability Analysis

#### 2.1. Weighted Regression Extremum Response Surface Method (WR-ERSM) Modeling

**x**) denotes the extremum of output response within the time domain [0, T], corresponding to the input variables

**x**= [x

_{1}, x

_{2}, …, x

_{k}]

^{T}, where k is the number of inputs, the ERSM model can be expressed as:

**B**and

**C**indicate constant term, linear term and quadratic term.

**B**and

**C**are denoted as:

**E**

_{l}(l = 1, 2, …, s) is the l-th sampling category, namely experimental condition, which is the rule of generated sample set of random variable with respect to both the mean μ and standard deviation σ; the subscripts i, j indicate the i-th and j-th random variables; the subscript s expresses the number of sampling types; f denotes the empirical coefficient which is usually selected from 1 to 3.

**v**is the n × (k + 1) matrix of input variables, in which n is the number of samples;

**d**is the vector of undetermined coefficients in the ERSM model.

_{WR}(

**x**) is structured as:

_{WR},

**B**

_{WR}and

**C**

_{WR}are the weighted constant, weighted linear vector and weighted quadratic matrix, respectively.

**B**

_{WR}and

**C**

_{WR}are denoted as

_{WR}, b

_{WR,i}and c

_{WR,i}are the weighted coefficient of A,

**B**and

**C**, respectively.

_{obj}(

**x**) with respect to n samples, the weighted values are then obtained by dividing the minimum value y

_{obj}(

**x**) from all the output responses y

_{true}(

**x**), which are ensured by dynamic deterministic analysis with the FE model. The m efficient samples with larger weights are selected from this pool of n samples, and the weighted matrix

**w**is restructured. The related formulas are:

_{true}

_{, j}(

**x**) indicates the true value of the j-th output response; w

_{j}is the weighted value of the j-th sample; m (m ≥ (2k + 1)) is the number of efficient samples.

**d**

_{WR}denotes the vector of undetermined coefficients in the WR-ERSM model;

**v**

_{WR}is the matrix of efficient samples;

**y**

_{WR}is the output response corresponding to the efficient samples.

**d**

_{WR},

**v**

_{WR}and

**y**

_{WR}are structured as:

#### 2.2. Safety Criterion Transformation

_{eq}and σ

_{eq}indicates the mean and standard deviation of equivalent random parameter; ${\mu}_{\tilde{y}}\left(x\right)$ is the membership function of fuzzy safety criterion; G

_{x}is defined as:

_{u}and u

_{l}are the upper bound and lower bound of fuzzy variable interval, respectively.

#### 2.3. Structural Dynamic Fuzzy Reliability Analysis

_{z}and σ

_{z}are both mean value and standard deviation, respectively.

_{z}and σ

_{z}present the mean value and standard deviation of output response; μ

_{eq}and σ

_{eq}indicate the mean value and standard deviation of the safety criterion.

## 3. Example Analysis

#### 3.1. Deterministic Analysis for Turbine Blisk

#### 3.2. The WR-ERSM Model of Turbine Blisk

**x**) in Equation (18) obey a normal distribution, the dynamic fuzzy reliability analysis of turbine blisk is accomplished with the MC method. The simulation histories and stress histograms of turbine blisk are drawn in Figure 10 and Figure 11, respectively.

_{z}= 9.669 × 10

^{8}Pa and standard deviation σ

_{z}= 5.743 × 10

^{6}Pa. Moreover, in light of Equation (16), the built model in Equation (18) is rewritten as the PDF, i.e.,

#### 3.3. Turbine Blisk Reliability Evaluation

_{eq}and standard deviation σ

_{eq}are consequently achieved. The PDF f(z

_{eq}) can be expressed by:

_{eq}) of stress and the safety criterion of the turbine blisk are drawn in Figure 13, respectively.

^{8}Pa and 9.862 × 10

^{8}Pa) and standard deviations (5.743 × 10

^{6}Pa and 4.980 × 10

^{6}Pa) of the two methods are acquired. In line with Equation (17), the structural reliability index and reliability degree are β = 2.751 and Pr = 0.9970, respectively.

## 4. WR-ERSM Verification Procedure

#### 4.1. Model-Fitting Properties

^{2}and maximum absolute error r

_{max}to test the fitting accuracy for the WR-ERSM and ERSM. The r

^{2}and r

_{max}are illustrated as follows,

_{i}denotes the authentic output responses; ${\widehat{y}}_{i}$ is the output responses gained by the mathematical models; $\overline{y}$ indicates the mean of the experimental data; S expresses the standard deviation of experimental data. If the square-error r

^{2}is close to 1 and the relative maximum absolute error r

_{max}is close to 0, the fitting accuracy is high.

^{2}and r

_{max}. The results are listed in Table 3.

#### 4.2. Simulation Performances for Dynamic Fuzzy Reliability Analysis of Turbine Blisk

## 5. Conclusions

- (1)
- The WR-ERSM is highly precise and efficient in structural dynamic reliability evaluation, since ERSM has the capacity of processing the transient problem;
- (2)
- The WR approach can improve modeling accuracy so that the proposed WR-ERSM possesses high fitting efficiency and accuracy, due to the requirement of small samples;
- (3)
- WR-ERSM possesses good simulation performance in structural dynamic fuzzy reliability evaluation, as the fuzzy safety criterion is considered to improve the precision;
- (4)
- The change rule of turbine blisk structural stress from start to cruise for an aircraft is acquired with the maximum value of structural stress at t = 165 s and the reliability degree (Pr = 0.997) of the turbine blisk.
- (5)
- The efforts of this study provide a promising method for the dynamic reliability analysis and evaluation of complex structures with respect to the working process.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Flow chart of structural dynamic fuzzy reliability analysis with weighted regression extremum response surface method (WR-ERSM).

Parameters | Variable | Distribution | Mean, μ | St.Dev., δ |
---|---|---|---|---|

Inlet velocity (m·s^{−1}) | v | Normal | 168 | 5.04 |

Inlet pressure (Pa) | P | Normal | 600,000 | 12,000 |

Material density (kg·m^{−3}) | ρ | Normal | 8210 | 246 |

Angular speed (rad·s^{−1}) | w | Normal | 1168 | 35 |

Parameters and Weighted Coefficient | Parameters and Weighted Coefficient | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

v m·s^{−1} | P, ×10^{5} Pa | ρ kg·m^{−3} | w rad·s^{−1} | σ × 10^{8} Pa | W | v, m·s^{−1} | P, ×10^{5} Pa | ρ, kg·m^{−3} | w, rad·s^{−1} | σ, ×10^{8} Pa | W |

168.00 | 6.00 | 8210 | 1168 | 9.687 | 0.9105 | 173.04 | 6.00 | 8210 | 1133 | 9.098 | 0.9694 |

162.96 | 6.00 | 8210 | 1168 | 9.693 | 0.9099 | 168.00 | 6.12 | 8210 | 1133 | 9.105 | 0.9687 |

168.00 | 5.88 | 8210 | 1168 | 9.686 | 0.9106 | 168.00 | 6.00 | 7964 | 1133 | 8.827 | 0.9992 |

168.00 | 6.00 | 7964 | 1168 | 9.392 | 0.9391 | 173.04 | 5.88 | 7964 | 1168 | 9.385 | 0.9398 |

168.00 | 6.00 | 8210 | 1133 | 9.105 | 0.9687 | 173.04 | 5.88 | 8210 | 1133 | 9.098 | 0.9694 |

173.04 | 6.00 | 8210 | 1168 | 9.686 | 0.9391 | 162.96 | 6.12 | 7964 | 1168 | 9.398 | 0.9385 |

168.00 | 6.12 | 8210 | 1168 | 9.687 | 0.9105 | 162.96 | 6.12 | 8210 | 1133 | 9.111 | 0.9681 |

168.00 | 6.12 | 8210 | 1203 | 10.29 | 0.8576 | 168.00 | 6.00 | 8456 | 1203 | 10.59 | 0.8575 |

162.96 | 5.88 | 8210 | 1168 | 9.687 | 0.9105 | 168.00 | 6.12 | 7964 | 1133 | 8.829 | 0.9989 |

162.96 | 6.00 | 7964 | 1168 | 9.391 | 0.9392 | 162.96 | 6.00 | 8456 | 1133 | 9.389 | 0.9394 |

162.96 | 6.00 | 8210 | 1133 | 9.105 | 0.9687 | 168.00 | 5.88 | 8456 | 1133 | 9.383 | 0.9400 |

168.00 | 5.88 | 7964 | 1168 | 9.391 | 0.9392 | 173.04 | 6.12 | 7964 | 1168 | 9.385 | 0.9398 |

168.00 | 5.88 | 8210 | 1133 | 9.105 | 0.9687 | 173.04 | 6.12 | 8210 | 1133 | 9.098 | 0.9694 |

168.00 | 6.00 | 7964 | 1133 | 8.827 | 0.9992 | 173.04 | 6.00 | 8456 | 1133 | 9.376 | 0.9407 |

173.04 | 6.12 | 8210 | 1168 | 9.687 | 0.9105 | 168.00 | 6.12 | 8456 | 1133 | 9.383 | 0.9400 |

162.96 | 6.12 | 8210 | 1168 | 9.693 | 0.9099 | 162.96 | 5.88 | 7964 | 1168 | 9.398 | 0.9385 |

168.00 | 6.00 | 8210 | 1203 | 10.28 | 0.8576 | 173.04 | 6.00 | 8210 | 1203 | 10.28 | 0.8576 |

173.04 | 5.88 | 8210 | 1168 | 9.681 | 0.9111 | 162.96 | 5.88 | 8210 | 1133 | 9.111 | 0.9681 |

173.04 | 6.00 | 7964 | 1168 | 9.385 | 0.9398 | 162.96 | 6.00 | 7964 | 1133 | 8.827 | 0.9992 |

168.00 | 6.12 | 7964 | 1168 | 9.391 | 0.9392 | 168.00 | 5.88 | 7964 | 1133 | 8.826 | 0.9993 |

Method | Fitting ERSM Model | Fitting Accuracy | ||
---|---|---|---|---|

Sample Number | Fitting Time, h | r^{2} | r_{max} | |

WR-ERSM | 9 | 7.05 | 0.9984 | 0.0535 |

ERSM | 29 | 22.39 | 0.9742 | 0.0834 |

Methods | P_{r} | Errors | Precision, % |
---|---|---|---|

MC method | 0.9981 | - | - |

ESTM | 0.9962 | 0.0019 | 99.81 |

ERSM | 0.9937 | 0.0044 | 99.56 |

WR-ERSM | 0.9970 | 0.0011 | 99.89 |

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## Share and Cite

**MDPI and ACS Style**

Lu, C.; Feng, Y.-W.; Fei, C.-W.
Weighted Regression-Based Extremum Response Surface Method for Structural Dynamic Fuzzy Reliability Analysis. *Energies* **2019**, *12*, 1588.
https://doi.org/10.3390/en12091588

**AMA Style**

Lu C, Feng Y-W, Fei C-W.
Weighted Regression-Based Extremum Response Surface Method for Structural Dynamic Fuzzy Reliability Analysis. *Energies*. 2019; 12(9):1588.
https://doi.org/10.3390/en12091588

**Chicago/Turabian Style**

Lu, Cheng, Yun-Wen Feng, and Cheng-Wei Fei.
2019. "Weighted Regression-Based Extremum Response Surface Method for Structural Dynamic Fuzzy Reliability Analysis" *Energies* 12, no. 9: 1588.
https://doi.org/10.3390/en12091588