# Fast Optimization Design of the Flexure for a Space Mirror Based on Mesh Deformation

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## Abstract

**:**

## 1. Introduction

## 2. The Design of the Space Mirror Assembly

#### 2.1. The Design Index of the Space Mirror Assembly

#### 2.2. Topology Optimization of the Space Mirror

_{0}= 0.2) to find the distribution of the lightweight ribs. After 20 iterations, the best material distribution of the ribs is clear. The topology optimization result of the solid mirror is shown in Figure 2b. From this result, we can see that the lightweight structure of the mirror can use eight lightweight ribs. The closer it is to the central hole, the more important the material is. The material far away from the central hole can be removed. Considering the manufacturing difficulty of the mirror, the mirror is designed with eight lightweight ribs, and we add ring ribs in the middle to increase its stiffness, so the initial mirror structure is shown in Figure 3.

#### 2.3. The Integrated Optimization Design

_{1,}t

_{2,}t

_{3,}t

_{4}and t

_{5}indicate the thickness of the respective lightweight ribs. Because there are only five parameters, which are used as variables for integrated optimization design, the pointer algorithm is used to optimize the space mirror in this paper. Four algorithms comprise the pointer algorithm, which has strong adaptability and is suitable for complex optimization problems. The mathematical model Equation (4) in size optimization is used to optimize the ribs of the mirror.

#### 2.4. The Design of the Support Structure

## 3. The Fast Optimization of the Space Mirror Assembly

## 4. Experimental Verification

#### 4.1. Optical Test

#### 4.2. The Stability Test of the Space Mirror Assembly

#### 4.3. Mechanical Test

#### 4.4. The Wave Aberration of the Telescope

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**The flexure of the space mirror assembly (

**a**) the initial structure of flexure; (

**b**) the improved flexure.

**Figure 9.**The final result under the four load cases: (

**a**) axial gravity; (

**b**) lateral gravity; (

**c**) 4 °C temperature change; (

**d**) 0.01 mm assembly error.

**Figure 14.**The test of mechanical vibration: (

**a**) Z-direction vibration (axial) (

**b**) Y-direction vibration (lateral).

Component | Material | Elastic Modlus (Gpa) | Poisson Ratio | Desity (g/cm^{3}) | Thermal Expansion Coefficient (10 ^{−6}/°C) |
---|---|---|---|---|---|

Mirror | SiC | 330 | 0.25 | 3.05 | 2.5 |

Sleeve | Invar | 141 | 0.25 | 8.9 | 2.5 |

Flexure | Titanium | 109 | 0.34 | 4.44 | 8.9 |

Variable | Domain | Initial Value | Optimal Value |
---|---|---|---|

t_{1} (mm) | (3, 6) | 4 | 3.6 |

t_{2} (mm) | (3, 6) | 4 | 5 |

t_{3} (mm) | (3, 6) | 4 | 4.7 |

t_{4} (mm) | (3, 8) | 4 | 4.8 |

t_{5} (mm) | (4, 8) | 6 | 4.8 |

Mass (kg) | - | 1.234 | 1.347 |

RMSz (nm) | - | 0.886 | 0.691 |

Design Variable | Value Range (mm) | Initial Value (mm) | Optimal Value (mm) | Design Value (mm) |
---|---|---|---|---|

H | (2.5–3.5) | 3.5 | 3.346 | 3.3 |

T | (3–3.5) | 3 | 3.038 | 3 |

R | (0.5–1) | 1 | 0.514 | 0.5 |

OPI | RMSt (nm) | RMSy (nm) | RMSz (nm) | RMS0.01 (nm) | f1 (Hz) | |
---|---|---|---|---|---|---|

Initial design | 4.67 | 1.29 | 7.98 | 1 | 1.5 | 220 |

Optimal design | 3.89 | 1.1 | 4.96 | 1 | 5.38 | 268.6 |

Finial design | 3.13 | 1.48 | 3.44 | 1 | 4.84 | 274.4 |

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**MDPI and ACS Style**

Liu, F.; Li, W.; Zhao, W.; Wang, X.; Wang, X. Fast Optimization Design of the Flexure for a Space Mirror Based on Mesh Deformation. *Photonics* **2021**, *8*, 567.
https://doi.org/10.3390/photonics8120567

**AMA Style**

Liu F, Li W, Zhao W, Wang X, Wang X. Fast Optimization Design of the Flexure for a Space Mirror Based on Mesh Deformation. *Photonics*. 2021; 8(12):567.
https://doi.org/10.3390/photonics8120567

**Chicago/Turabian Style**

Liu, Fengchang, Wei Li, Weiguo Zhao, Xiaodong Wang, and Xiaoyu Wang. 2021. "Fast Optimization Design of the Flexure for a Space Mirror Based on Mesh Deformation" *Photonics* 8, no. 12: 567.
https://doi.org/10.3390/photonics8120567