复合材料星载天线反射面热变形有限元模拟及拓扑优化

谭昆林; 曹静; 黄宏才; 魏培君; 张从发

复合材料星载天线反射面热变形有限元模拟及拓扑优化

1.
北京科技大学数理学院, 北京 100083
2.
北京空间飞行器总体设计部, 北京 100086

详细信息

通讯作者:
魏培君，教授，博导，研究方向为弹性波传播、声子晶体及超材料　E-mail: weipj@ustb.edu.cn

中图分类号: TB332
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- 被引次数: 0
出版历程
- 收稿日期: 2023-12-26
- 修回日期: 2024-02-21
- 录用日期: 2024-03-11
- 网络出版日期: 2024-04-16

Finite element simulation and topology optimization of thermal deformation of reflector of composite spaceborne antenna

1.
School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, China
2.
Beijing Space Vehicle General Design Department, Beijing 100086, China

摘要

摘要: 为控制星载天线反射面的热变形，本文以一口径为1200 mm的正方形格栅反射面为研究对象，首先，确定了预测反射面热变形的有限元建模策略，计算了反射面在三种不同温度荷载下的型面热变形均方根(RMS)。在此基础上提供了一种新的有限元建模策略——层合板等效方式，求出层合板的等效弹性模量和等效热膨胀系数。其次，以正方形格栅反射面为原型对格栅芯子进行拓扑优化，并与其他三种几何形状的芯子进行了比较，发现正方形格栅反射面为最优的结构形式。然后，对正方形格栅反射面做参数优化，找出了最优的正方形格栅单胞尺寸、蒙皮铺层方式、正方形格栅铺层方式和胶层厚度。最后，对正方形格栅反射面做热稳定性指标置信度分析，得到RMS的概率密度函数分布图和各个设计参数的贡献率Pareto图，找出了影响反射面型面热变形的关键因素。
- 反射面 /
- RMS /
- 层合板等效 /
- 拓扑优化 /
- 参数优化 /
- 置信度分析
Abstract: In order to control the thermal deformation of satellite-borne antenna reflector, a square grille reflector with a diameter of 1200 mm was taken as the research object. Firstly, the finite element modeling strategy for predicting the thermal deformation of the reflector was determined, and the thermal deformation RMS of the reflector under three different temperature loads was calculated. On this basis, a new finite element modeling strategy, the equivalent method of laminates, was provided to obtain the equivalent elastic modulus and equivalent thermal expansion coefficient of laminates. Secondly, the grid core was optimized topologically using the square grid reflector as prototype, and compared with the other three geometries, it is found that the square grid reflector is the best structure. Then, the parameters of the square grating reflector were optimized, and the optimal square grating cell size, skin laying mode, square grating laying mode and adhesive layer thickness were found out. Finally, the probability density function distribution of RMS and the contribution rate Pareto diagram of each design parameter were obtained by the confidence analysis of the thermal stability index of the square grating reflector, and the key factors affecting the thermal deformation of the reflector profile were found out.
- reflecting surface /
- RMS /
- laminated board equivalent /
- topology optimization /
- parameter optimization /
- confidence analysis

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图 1 反射面结构：(a)整体结构；(b)瓜瓣单元结构

Figure 1. Reflector structure: (a) Overall structure; (b) Melon petal unit structure

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图 2 反射面瓜瓣单元的有限元模型

Figure 2. Finite element model of reflecting surface melon petal element

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图 3 反射面热变形型面误差计算示意图

Figure 3. Schematic diagram of error calculation of thermal deformation profile of reflector

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图 4 反射面在三种温度荷载下的z向位移：(a)均匀温升80℃；(b)面内温度梯度0~100℃；(c)面外温度梯度0~2℃

Figure 4. z displacement of the reflector under three temperature loads: (a) Uniform temperature rise of 80℃; (b) In-plane temperature gradient 0-100℃; (c) Out-of-plane temperature gradient 0-2℃

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图 5 整体坐标系与局部坐标系的关系

Figure 5. Relation between global coordinate system and local coordinate system

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图 6 四种拓扑结构：(a)正方形格栅单胞；(b)蜂窝单胞；(c)三角形格栅单胞；(d)圆管单胞

Figure 6. Four topological structures: (a) Square grid cell; (b) Honeycomb monocytes; (c) Triangular grid cell; (d) Cylindrids

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图 7 $ {a}_{1} $与RMS和等效密度

Figure 7. $ {a}_{1} $ to RMS and equivalent density

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图 8 蒙特卡洛模拟：(a)RMS的概率密度函数分布图；(b)各个设计参数的贡献率Pareto图

Figure 8. Monte Carlo simulation: (a) RMS probability density function distribution; (b) Pareto diagram of contribution rate of each design parameter

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表 1 M55 J型CFRP单层板性能参数

Table 1. Performance parameters of M55 J CFRP single-layer plate

Elasticity modulus /GPa		Poisson's ratio	Shear elasticity /GPa			CTE/(×10⁻⁶·⁻¹)
$ {E}_{1} $	$ {E}_{2} $	$ \nu $	$ {G}_{12} $	$ {G}_{13} $	$ {G}_{23} $	$ {\alpha }_{1} $	$ {\alpha }_{2} $	$ {\alpha }_{3} $
290	10	0.27	4.5	4.5	2.1	−1	35	35
Notes: CTE—coefficient of thermal expansion

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表 2 胶的性能参数

Table 2. Performance parameters of adhesive

Elasticity modulus /GPa	Poisson's ratio	CTE/(×10⁻⁶·⁻¹)
2.5	0.3	50

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表 3 三种温度荷载下的RMS

Table 3. RMS under three temperature loads

Temperature load	Uniform temperature rise of 80℃	0-100℃ inside the surface	0-2℃ outside the surface
RMS/$ \mathrm{\mu }\mathrm{m} $	90.59	116.3	0.92

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表 4 蒙皮和格栅的等效材料参数

Table 4. Equivalent material parameters of skin and grille

Structure	Elasticity modulus /GPa		Poisson's ratio	shear elasticity /GPa			CTE/(×10⁻⁶·⁻¹)
Structure	$ {E}_{x} $	$ {E}_{y} $	$ {\nu }_{xy} $	$ {G}_{xy} $	$ {G}_{x{\textit{z}}} $	$ {G}_{y{\textit{z}}} $	$ {\alpha }_{x} $	$ {\alpha }_{y} $	$ {\alpha }_{{\textit{z}}} $
Stressed skin [0,90,45,−45]	48.77	48.77	0.762	13.8	3.3	3.3	−2.9	−2.9	222.6
Grid[0,90,45,−45,0,90,45,−45,0,90]	59.81	59.81	0.708	10.17	3.3	3.3	5.56	38.98	2400

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表 5 单层板铺层方式和层合板等效方式下的RMS

Table 5. RMS of single layer and laminated board equivalent mode

Structure	Uniform temperature rise of 80℃RMS/$ \mathrm{\mu }\mathrm{m} $	0-100℃ inside the surface RMS/$ \mathrm{\mu }\mathrm{m} $	0-2℃ outside the surface RMS/$ \mathrm{\mu }\mathrm{m} $
Single ply	90.59	116.3	0.92
Laminate equivalent	82.34	125.81	0.73

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表 6 四种拓扑结构下的RMS

Table 6. RMS in four topologies

Load structural style	Temperature rise 80℃ $ \mathrm{R}\mathrm{M}\mathrm{S}/\mathrm{\mu }\mathrm{m} $	0-100℃ inside the surface $ \mathrm{R}\mathrm{M}\mathrm{S}/\mathrm{\mu }\mathrm{m} $	0-2℃ outside the surface $ \mathrm{R}\mathrm{M}\mathrm{S}/\mathrm{\mu }\mathrm{m} $
Square grating	90.59	116.3	0.92
Honeycomb	108	148.5	1.18
Triangular grating	103.5	128.3	1.04
circular tube	116.8	176.7	1.32

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表 7 $ {a}_{1} $与RMS和等效密度的关系

Table 7. Relation of $ {a}_{1} $ to RMS and equivalent density

$ {a}_{1}/\mathrm{m}\mathrm{m} $	Temperature rise 80℃ $ \mathrm{R}\mathrm{M}\mathrm{S}/\mathrm{\mu }\mathrm{m} $	0-100℃ inside the surface $ \mathrm{R}\mathrm{M}\mathrm{S}/\mathrm{\mu }\mathrm{m} $	0-2℃ outside the surface $ \mathrm{R}\mathrm{M}\mathrm{S}/\mathrm{\mu }\mathrm{m} $	Equivalent density$ \left(\mathrm{k}\mathrm{g}/{\mathrm{m}}^{3}\right) $
20	66.7	92.2	0.4	1700$ \times $0.19=323
25	74.6	100.4	0.63	1700$ \times $0.15=255
30	90.59	116.3	0.92	1700$ \times $0.13=221
40	103.1	125.9	1.13	1700$ \times $0.10=170
Notes: $ {a}_{1} $-Square grille cell outer length

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表 8 正交实验因素水平

Table 8. Orthogonal experimental factor levels

Factor	Skin laying method	Grid laying method
Level 1	$ \left[{\left(0，90\right)}_{5}\right] $	$ \left[{\left(0，90\right)}_{5}\right] $
Level 2	$ {\left(0，90，0，90，0\right)}_{s} $	$ {\left(0，90，0，90，0\right)}_{s} $
Level 3	$ \left[{\left(\pm 45\right)}_{5}\right] $	$ \left[{\left(\pm 45\right)}_{5}\right] $
Level 4	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $
Level 5	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $

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表 9 正交实验结果

Table 9. Orthogonal experiment results

Serial number	Skin laying method	Grid laying method	RMS/$ \mathrm\;{\mu }\mathrm{m} $
1	$ \left[{\left(0，90\right)}_{5}\right] $	$ \left[{\left(0，90\right)}_{5}\right] $	36.4
2	$ \left[{\left(0，90\right)}_{5}\right] $	$ {\left(0，90，0，90，0\right)}_{s} $	30.8
3	$ \left[{\left(0，90\right)}_{5}\right] $	$ \left[{\left(\pm 45\right)}_{5}\right] $	68.6
4	$ \left[{\left(0，90\right)}_{5}\right] $	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	68.8
5	$ \left[{\left(0，90\right)}_{5}\right] $	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	27.4
6	$ {\left(0，90，0，90，0\right)}_{s} $	$ \left[{\left(0，90\right)}_{5}\right] $	40
7	$ {\left(0，90，0，90，0\right)}_{s} $	$ {\left(0，90，0，90，0\right)}_{s} $	34.1
8	$ {\left(0，90，0，90，0\right)}_{s} $	$ \left[{\left(\pm 45\right)}_{5}\right] $	71.5
9	$ {\left(0，90，0，90，0\right)}_{s} $	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	71.6
10	$ {\left(0，90，0，90，0\right)}_{s} $	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	29.9
11	$ \left[{\left(\pm 45\right)}_{5}\right] $	$ \left[{\left(0，90\right)}_{5}\right] $	40.4
12	$ \left[{\left(\pm 45\right)}_{5}\right] $	$ {\left(0，90，0，90，0\right)}_{s} $	36.8
13	$ \left[{\left(\pm 45\right)}_{5}\right] $	$ \left[{\left(\pm 45\right)}_{5}\right] $	74.4
14	$ \left[{\left(\pm 45\right)}_{5}\right] $	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	93.9
15	$ \left[{\left(\pm 45\right)}_{5}\right] $	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	36.9
16	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	$ \left[{\left(0，90\right)}_{5}\right] $	39.6
17	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	$ {\left(0，90，0，90，0\right)}_{s} $	36.1
18	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	$ \left[{\left(\pm 45\right)}_{5}\right] $	73.6
19	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	73.1
20	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	36.2
21	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	$ \left[{\left(0，90\right)}_{5}\right] $	38.9
22	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	$ {\left(0，90，0，90，0\right)}_{s} $	33.4
23	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	$ \left[{\left(\pm 45\right)}_{5}\right] $	69.7
24	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	$ \left[{\left(45,-\mathrm{45,45},-\mathrm{45,45}\right)}_{s}\right] $	69.8
25	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	$ \left[\left(\begin{array}{c}\mathrm{0,45,90},-\mathrm{45,0},\\ \mathrm{45,90},-\mathrm{45,0},45\end{array}\right)\right] $	29.3
$ {K}_{1} $	46．4	39.06	-
$ {K}_{2} $	49.42	34.24	-
$ {K}_{3} $	56.48	71.56	-
$ {K}_{4} $	51.72	75.44	-
$ {K}_{5} $	48.22	31.88	-
R	10.08	43.56	-
Notes:$ {K}_{i}(i=\mathrm{1,2},\mathrm{3,4},5) $ represents the mean of the RMS when the factors are at each level, and R represents the range of the factors.

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表 10 胶层厚度与RMS的关系

Table 10. Relation between adhesive layer thickness and RMS

Bondline thickness /mm	0.1	0.2	0.3	0.4	0.5
Temperature rise 80℃RMS/$ \mathrm{\mu }\mathrm{m} $	27.4	28.2	29.4	30.4	31.4

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表 11 M55 J平纹布/氰酸酯和T300碳纤维/氰酸酯复合材料单层板性能参数

Table 11. Performance parameters of M55 J plain cloth/cyanate and T300 carbon fiber/cyanate composite monolayers

Material	Elasticity modulus /GPa		Poisson's ratio	Shear elasticity /GPa			CTE/(×10⁻⁶·⁻¹)
Material	$ {E}_{1} $	$ {E}_{2} $	$ \upsilon $	$ {G}_{12} $	$ {G}_{13} $	$ {G}_{23} $	$ {\alpha }_{1} $	$ {\alpha }_{2} $	$ {\alpha }_{3} $
M55 J	113.6	113.6	0.11	8	4	4	0.14	0.14	30
T300	60	60	0.13	8.41	4	4	0.14	0.14	30

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表 12 不同复合材料下的RMS

Table 12. RMS under different composite materials

Material	M55 J type carbon fiber	M55 J plain cloth/cyanate ester	T300 Carbon fiber/cyanate
RMS/$ \mathrm\;{\mu }\mathrm{m} $	27.4	0.2	0.8

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