# Multi-Physics and Multi-Objective Optimization for Fixing Cubic Fabry–Pérot Cavities Based on Data Learning

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## Abstract

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## 1. Introduction

## 2. Multi-Physics Coupling Theory and Finite Element Method for FP Cavity

#### 2.1. Multi-Physics Coupling Theory

#### 2.2. Finite Element Method and Model Establishment

## 3. Results and Analysis of Multi-Physics Coupling

#### 3.1. Displacement Distribution of FP Cavity under the Fixing Force

#### 3.2. Analysis of Cavity Length Change under the Random Vibration Experiment

## 4. Mechanical Optimization for Fixing a Cubic FP Cavity

#### 4.1. Determination of Design Spaces and Performance Indexes

#### 4.2. Orthogonal Experiments: Design and Implementation

#### 4.3. Establishment and Comparison of Data Learning Models

#### 4.4. Evolutionary Algorithm Optimization and Performance Verification

## 5. Conclusions

- Performance indices acquired by multi-physics coupling simulation and key design variables determination: a multi-physics model for the cubic FP cavity is established and the performances under fixing force and random vibrations are obtained via finite element analysis, then the key performance indices (V, ${w}_{F}$, ${w}_{F}$) and key design variables (d, l, F) are determined considering the features of the FP cavity.
- Training data are obtained by orthogonal experiment and a fitting model is established based on a neural network: 49 sets of data obtained from the orthogonal experiment are used to establish and compare different data learning models (NN, RSF, KRG), with the results indicating that the neural network has the best performance.
- Finally, an optimized combination of key design variables is obtained using an evolutionary algorithm: NSGA-II is adopted as the optimization algorithm, the Pareto-optimal front is obtained, and the optimal combination of design variables is finally determined as $\{5,32,250\}$. The performance after optimization demonstrates a great improvement, with the displacement under the fixing force and vibration test both decreasing by more than 60%.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 9.**EEMD results for the cavity length change along the $45\xb0$ and −45° directions in the vibration test.

**Figure 10.**Kurtogram of the cavity length change along the $45\xb0$ and $-45\xb0$ directions under the vibration test.

Frequency Range/Hz | Power Spectral Density |
---|---|

10~50 | 3 db/oct (rising slope) |

50~300 | 0.25 g${}^{2}$/Hz (holding value) |

300~2000 | −12 db/oct (falling slope) |

Number | d/mm | l/mm | F/N | V/m${}^{3}$ ($\times {10}^{-4}$) | ${\mathit{w}}_{\mathit{F}}/$mm ($\times {10}^{-7}$) | ${\mathit{w}}_{\mathbf{vib}}$/mm ($\times {10}^{-21}$) |
---|---|---|---|---|---|---|

1 | 5 | 20 | 100 | 9.854 | 7.842 | 4.145 |

2 | 5 | 25 | 200 | 9.752 | 14.827 | 2.064 |

3 | 5 | 30 | 300 | 9.601 | 17.132 | 5.376 |

4 | 5 | 35 | 400 | 9.389 | 14.815 | 4.837 |

5 | 5 | 40 | 150 | 9.107 | 3.606 | 0.132 |

6 | 5 | 45 | 250 | 8.746 | 14.391 | 1.073 |

7 | 5 | 20 | 350 | 9.854 | 27.447 | 6.374 |

8 | 5 | 20 | 200 | 9.854 | 15.684 | 6.743 |

9 | 5 | 25 | 300 | 9.752 | 22.240 | 0.914 |

10 | 5 | 30 | 400 | 9.601 | 22.843 | 0.523 |

11 | 7 | 20 | 400 | 9.800 | 40.118 | 17.790 |

12 | 7 | 25 | 150 | 9.699 | 13.495 | 0.635 |

13 | 7 | 30 | 250 | 9.547 | 19.568 | 11.998 |

14 | 7 | 35 | 350 | 9.335 | 17.763 | 2.962 |

15 | 7 | 40 | 100 | 9.054 | 3.706 | 2.251 |

16 | 7 | 45 | 200 | 8.692 | 14.024 | 5.684 |

17 | 7 | 25 | 300 | 9.699 | 26.989 | 1.270 |

18 | 7 | 35 | 150 | 9.335 | 7.613 | 3.432 |

19 | 7 | 40 | 250 | 9.054 | 9.265 | 1.705 |

20 | 7 | 45 | 350 | 8.692 | 24.541 | 1.662 |

21 | 9 | 20 | 350 | 9.730 | 46.076 | 10.823 |

22 | 9 | 25 | 100 | 9.629 | 12.392 | 3.465 |

23 | 9 | 30 | 200 | 9.477 | 21.960 | 0.461 |

24 | 9 | 35 | 300 | 9.265 | 25.628 | 8.813 |

25 | 9 | 40 | 400 | 8.984 | 24.200 | 9.371 |

26 | 9 | 45 | 150 | 8.622 | 13.379 | 4.828 |

27 | 9 | 30 | 250 | 9.477 | 27.450 | 4.631 |

28 | 9 | 45 | 100 | 8.622 | 8.919 | 0.530 |

29 | 9 | 20 | 150 | 9.730 | 19.747 | 1.841 |

30 | 9 | 25 | 250 | 9.629 | 30.980 | 3.798 |

31 | 11 | 20 | 300 | 9.645 | 55.853 | 4.380 |

32 | 11 | 25 | 400 | 9.543 | 70.869 | 16.213 |

33 | 11 | 30 | 150 | 9.391 | 24.309 | 5.896 |

34 | 11 | 35 | 250 | 9.180 | 33.908 | 15.272 |

35 | 11 | 40 | 350 | 8.898 | 37.615 | 0.556 |

36 | 11 | 45 | 100 | 8.536 | 11.767 | 0.636 |

37 | 11 | 35 | 200 | 9.180 | 27.126 | 0.652 |

38 | 11 | 30 | 350 | 9.391 | 56.720 | 9.568 |

39 | 11 | 35 | 100 | 9.180 | 13.563 | 4.416 |

40 | 11 | 40 | 200 | 8.898 | 21.494 | 5.854 |

41 | 13 | 20 | 250 | 9.544 | 62.855 | 3.054 |

42 | 13 | 25 | 350 | 9.442 | 85.326 | 21.218 |

43 | 13 | 30 | 100 | 9.291 | 22.917 | 3.813 |

44 | 13 | 35 | 200 | 9.079 | 42.097 | 14.350 |

45 | 13 | 40 | 300 | 8.797 | 54.660 | 0.338 |

46 | 13 | 45 | 400 | 8.436 | 59.989 | 17.794 |

47 | 13 | 20 | 400 | 9.544 | 100.568 | 30.596 |

48 | 13 | 40 | 150 | 8.797 | 27.330 | 0.169 |

49 | 13 | 45 | 300 | 8.436 | 44.992 | 7.038 |

Model | $\mathit{V}/\%$ | ${\mathit{w}}_{\mathit{F}}/\%$ | ${\mathit{w}}_{\mathbf{vib}}/\%$ |
---|---|---|---|

NN | 0.12 | 0.79 | 2.49 |

RSF | 0.10 | 1.43 | 4.76 |

KRG | 0.27 | 1.56 | 4.63 |

d/mm | l/mm | F/N | V/m${}^{3}$$(\times {10}^{-4})$ | ${\mathit{w}}_{\mathit{F}}/$mm $(\times {10}^{-7})$ | ${\mathit{w}}_{\mathbf{vib}}/$mm $(\times {10}^{-21})$ | |
---|---|---|---|---|---|---|

Before | 10 | 25 | 200 | 9.588 | 14.858 | 7.626 |

After | 5 | 32 | 250 | 9.524 | 5.658 | 2.852 |

$\Delta $ | $-50\%$ | $+28\%$ | $+25\%$ | $-1\%$ | $-62\%$ | $-63\%$ |

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**MDPI and ACS Style**

Zhao, H.; Meng, F.; Wang, Z.; Yin, X.; Meng, L.; Jia, J.
Multi-Physics and Multi-Objective Optimization for Fixing Cubic Fabry–Pérot Cavities Based on Data Learning. *Appl. Sci.* **2023**, *13*, 13115.
https://doi.org/10.3390/app132413115

**AMA Style**

Zhao H, Meng F, Wang Z, Yin X, Meng L, Jia J.
Multi-Physics and Multi-Objective Optimization for Fixing Cubic Fabry–Pérot Cavities Based on Data Learning. *Applied Sciences*. 2023; 13(24):13115.
https://doi.org/10.3390/app132413115

**Chicago/Turabian Style**

Zhao, Hang, Fanchao Meng, Zhongge Wang, Xiongfei Yin, Lingqiang Meng, and Jianjun Jia.
2023. "Multi-Physics and Multi-Objective Optimization for Fixing Cubic Fabry–Pérot Cavities Based on Data Learning" *Applied Sciences* 13, no. 24: 13115.
https://doi.org/10.3390/app132413115