载荷不确定的周期性结构稳健拓扑优化

付志方; 王春洁

doi:10.13700/j.bh.1001-5965.2016.0822

载荷不确定的周期性结构稳健拓扑优化

doi: 10.13700/j.bh.1001-5965.2016.0822

付志方¹,
王春洁^{1, 2, ,}

1.
北京航空航天大学机械工程及自动化学院, 北京 100083
2.
北京航空航天大学虚拟现实技术与系统国家重点实验室, 北京 100083

详细信息

作者简介:
付志方, 男, 博士研究生。主要研究方向:结构优化、仿真

王春洁, 女, 博士, 教授, 博士生导师。主要研究方向:机械设计, 结构优化、仿真, 结构动力学

通讯作者:
王春洁, E-mail:wangcj@buaa.edu.cn

中图分类号: TH122;O327
计量
- 文章访问数: 1018
- HTML全文浏览量: 190
- PDF下载量: 552
- 被引次数: 0
出版历程
- 收稿日期: 2016-10-24
- 录用日期: 2016-11-25
- 网络出版日期: 2017-04-20

Robust topology optimization of periodic structures under uncertain loading

FU Zhifang¹,
WANG Chunjie^{1, 2
, ,}

1.
School of Mechanical Engineering and Automation, Beijing University of Aeronautics and Astronautics, Beijing 100083, China
2.
State Key Laboratory of Virtual Reality Technology and Systems, Beijing University of Aeronautics and Astronautics, Beijing 100083, China

More Information

Corresponding author: WANG Chunjie, E-mail:wangcj@buaa.edu.cn

摘要

摘要:
提出了一种考虑周期性几何约束的结构稳健拓扑优化设计方法。针对线弹性体结构，提出了载荷不确定性条件下周期性结构稳健拓扑优化模型；推导了周期性结构柔度均值与方差的敏度数表达式，并基于软删双向渐进结构优化法提出了载荷不确定性条件下的周期性结构稳健拓扑优化设计方法。2个不同约束条件下的实例表明：本文提出的优化模型稳定性较好；考虑载荷不确定性得到的结构与确定性载荷下得到的结构有很大的区别，且稳健性优化设计得到的结构更加稳定。
- 周期性 /
- 稳健性 /
- 拓扑优化 /
- 载荷不确定性 /
- 渐进结构优化法
Abstract:
This paper proposes a method of structural robust topology optimization under the periodicity constraint. A robust topology optimization model for periodic structures is proposed for linear elastic structures. Then, the formula of the sensitivity number is developed, and the robust topology optimal design of periodic structure under uncertain loadings is performed using soft-kill bi-directional evolutionary structural optimization method. Two numerical examples under different constraints demonstrate the stability of the proposed method. There are significant differences between the optimal structures under deterministic loads and the optimal structures under uncertain loads, and the robust design is more stable than the deterministic design.
- periodicity /
- robustness /
- topology optimization /
- loading uncertainty /
- evolutionary structural optimization method

HTML全文

图 1 2D设计区域被分割成6个子域

Figure 1. 2D design domain with 6 unit cells

下载: 全尺寸图片幻灯片

图 2 双侧固定梁结构的设计域

Figure 2. Design domain for a bilateral clamped beam structure

下载: 全尺寸图片幻灯片

图 3 双侧固定梁拓扑优化得到的构型

Figure 3. Layouts obtained from topology optimization of bilateral clamped beam

下载: 全尺寸图片幻灯片

图 4 双侧固定梁拓扑优化迭代历史

Figure 4. Iteration history of topology optimization of bilateral clamped beam

下载: 全尺寸图片幻灯片

图 5 初始设计区域

Figure 5. Primary design domain

下载: 全尺寸图片幻灯片

图 6 米歇尔结构拓扑优化得到的构型

Figure 6. Layouts obtained from topology optimization of Michelle structure

下载: 全尺寸图片幻灯片

图 7 米歇尔结构迭代历史

Figure 7. Iteration history of Michelle structure

下载: 全尺寸图片幻灯片

表 1 双侧固定梁拓扑优化结果

Table 1. Topology optimization result of bilateral clamped beam

J
周期数	稳健性优化			确定性优化
周期数	均值	标准差	目标函数	均值	标准差	目标函数
M=1×1	525.62	11.98	268.80	529.72	7.84	268.78
M=1×2	601.58	1.78	301.68	615.17	17.70	316.44
M=2×4	862.01	35.40	448.71	884.58	33.04	458.81

下载: 导出CSV

表 2 米歇尔结构拓扑优化结果

Table 2. Topology optimization result of Michelle structure

J
周期数	稳健性优化			确定性优化
周期数	均值	标准差	目标函数	均值	标准差	目标函数
M=1×1	92.43	10.46	51.45	141.66	67.41	104.54
M=1×2	120.52	14.87	67.70	359.52	235.67	297.60
M=1×4	139.89	20.44	80.17	265.13	132.34	198.74

下载: 导出CSV

参考文献(20)

[1]	WADLEY H N G, FLECK N A, EVAN A G.Fabrication and structural performance of periodic cellular metal sandwich structures[J].Composites Science and Technology, 2003, 63(16):2331-2343. doi: 10.1016/S0266-3538(03)00266-5
[2]	HUANG, X D, XIE Y M.Optimal design of periodic structures using evolutionary topology optimization[J].Structural and Multidisciplinary Optimization, 2008, 36(6):597-606. doi: 10.1007/s00158-007-0196-1
[3]	焦洪宇, 周奇才, 李文军, 等.基于变密度法的周期性拓扑优化[J].机械工程学报, 2013, 49(13):132-138. http://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201313020.htm JIAO H Y, ZHOU Q C, LI W J, et al.Periodic topology optimization using variable density method[J].Journal of Mechanical Engineering, 2013, 49(13):132-138(in Chinese). http://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201313020.htm
[4]	荣见华, 廖莺, 赵志军, 等.基于位移约束的类周期性连续体结构拓扑优化设计[J].应用力学学报, 2013, 30(6):876-881. doi: 10.11776/cjam.30.06.B131 RONG J H, LIAO Y, ZHAO Z J, et al.Topology optimization of periodic-like structures with displacement constrains[J].Chinese Journal of Applied Mechanics, 2013, 30(6):876-881(in Chinese). doi: 10.11776/cjam.30.06.B131
[5]	焦洪宇, 周奇才, 吴青龙, 等.桥式起重机箱型主梁周期性拓扑优化设计[J].机械工程学报, 2014, 50(23):134-139. http://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201423021.htm JIAO H Y, ZHOU Q C, WU Q L, et al.Periodic topology optimization of the box-type girder of bridge crane[J].Journal of Mechanical Engineering, 2014, 50(23):134-139(in Chinese). http://www.cnki.com.cn/Article/CJFDTOTAL-JXXB201423021.htm
[6]	ZUO Z H, HUANG X D, YANG X Y.Comparing optimal material microstructures with optimal periodic structures[J].Computational Materials Science, 2013, 69(1):137-147.
[7]	LUO Z, TONG L, KANG Z.A level set method for structural shape and topology optimization using radial basis functions[J].Computers and Structures, 2009, 87(7):425-434.
[8]	GUEST J K, IGUSA T.Structural optimization under uncertain loads and nodal locations[J].Computer Methods in Applied Mechanics and Engineering, 2008, 198(1):116-124. doi: 10.1016/j.cma.2008.04.009
[9]	DUNNING P D, KIM H A, MULLINEUX G.Introducing loading uncertainty in topology optimization[J].AIAA Journal, 2011, 49(4):760-768. doi: 10.2514/1.J050670
[10]	CHEN S, CHEN W, LEE S.Level set based robust shape and topology optimization under random field uncertainties[J].Structural Multidisciplinary Optimization, 2010, 41(4):507-524. doi: 10.1007/s00158-009-0449-2
[11]	CHEN S, CHEN W.A new level-set based approach to shape and topology optimization under geometric uncertainty[J].Structural Multidisciplinary Optimization, 2011, 44(1):1-18. doi: 10.1007/s00158-011-0660-9
[12]	DUNNING P D, KIM H A.Robust topology optimization:Minimization of expected and variance of compliance[J].AIAA Journal, 2013, 51(11):2656-2664. doi: 10.2514/1.J052183
[13]	KANG Z, BAI S.On robust design optimization of truss structures with bounded uncertainties[J].Structural and Multidisciplinary Optimization, 2013, 47(5):699-714. doi: 10.1007/s00158-012-0868-3
[14]	赵军鹏, 王春洁.载荷不确定条件下的结构拓扑优化算法[J].北京航空航天大学学报, 2014, 40(7):959-964. http://bhxb.buaa.edu.cn/CN/abstract/abstract12979.shtml ZHAO J P, WANG C J.Algorithm of structural topology optimization under loading uncertainty[J].Journal of Beijing University of Aeronautics and Astronautics, 2014, 40(7):959-964(in Chinese). http://bhxb.buaa.edu.cn/CN/abstract/abstract12979.shtml
[15]	ZHAO J P, WANG C J.Robust topology optimization under loading uncertainty based on linear elastic theory and orthogonal diagonalization of symmetric matrices[J].Computer Methods in Applied Mechanics and Engineering, 2014, 273(2):204-218.
[16]	罗阳军, 亢战, 邓子辰.多工况下结构鲁棒性拓扑优化设计[J].力学学报, 2011, 43(1):227-234. doi: 10.6052/0459-1879-2011-1-lxxb2010-147 LUO Y J, KANG Z, DENG Z C.Robust topology optimization design of structures with multiple load cases[J].Chinese Journal of Theoretical and Applied Mechanics, 2011, 43(1):227-234(in Chinese). doi: 10.6052/0459-1879-2011-1-lxxb2010-147
[17]	付志方, 赵军鹏, 王春洁.多工况线性结构稳健拓扑优化设计[J].力学学报, 2015, 47(4):642-650. doi: 10.6052/0459-1879-15-072 FU Z F, ZHAO J P, WANG C J.Robust topology optimization design of structures with multiple load cases[J].Chinese Journal of Theoretical and Applied Mechanics, 2015, 47(4):642-650(in Chinese). doi: 10.6052/0459-1879-15-072
[18]	XIE Y M, ZUO Z H, HUANG X D, et al.Application of topological optimization technology to bridge design[J].Structural Engineering International, 2014, 24(2):185-191. doi: 10.2749/101686614X13830790993366
[19]	刘丰睿, 赵丽滨, 韩邦成, 等.磁悬浮控制力矩陀螺框架结构的拓扑优化设计[J].北京航空航天大学学报, 2010, 36(4):455-458. http://bhxb.buaa.edu.cn/CN/abstract/abstract8479.shtml LIU F R, ZHAO L B, HAN B C, et al.Topology optimization design of frame structure for magnetic suspension control moment gyroscope[J].Journal of Beijing University of Aeronautics and Astronautics, 2010, 36(4):455-458(in Chinese). http://bhxb.buaa.edu.cn/CN/abstract/abstract8479.shtml
[20]	HUANG X D, XIE Y M.Evolutionary topology optimization of continuum structures:Methods and applications[M].Chichester:John Wiley & Sons, 2010:53-64.