基于特征值分析的多尺度结构优化设计方法

孙国民; 张效忠; 孙延华

doi:10.21656/1000-0887.390207

基于特征值分析的多尺度结构优化设计方法

doi: 10.21656/1000-0887.390207

贵州工程应用技术学院土木建筑工程学院, 贵州毕节 551700

基金项目: 贵州省教育厅自然科学研究青年项目（黔教合KY字[2015]466）；贵州省科技计划项目（黔科合基础[2017]1062）

详细信息

作者简介:
孙国民（1985—），男，讲师，硕士(通讯作者. E-mail: sunguomin815@163.com).

中图分类号: O343.7
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出版历程
- 收稿日期: 2018-07-24
- 修回日期: 2019-04-18
- 刊出日期: 2019-06-01

Multi-Scale Structure Optimization Design Based on Eigenvalue Analysis

School of Civil and Architectural Engineering, Guizhou University of Engineering Science, Bijie, Guizhou 551700, P.R.China

摘要

摘要: 基于特征值分析，提出了多尺度结构优化设计方法.该方法被用于分析宏观结构上作用有最不利荷载时，使宏观结构刚度最大的宏观拓扑结构和微观材料分布.引入约束条件为最不利荷载的Euclid范数等于1，根据Rayleigh-Ritz定理，可以将结构的柔顺度转换为一个与局部荷载向量维数相同的对称矩阵，这样就将作用有最不利荷载的柔顺度最小问题转换为求解对称矩阵的最大特征值最小问题，同时最不利荷载可以通过最大特征值矩阵的特征向量求得.最后通过算例验证所提多尺度结构优化设计方法的有效性，并说明宏观拓扑结构和微观材料分布的合理性.所提出的多尺度优化方法具有迭代稳定、收敛迅速等特点.该文拓扑优化中密度函数的更新是基于灵敏度分析和移动渐近线方法（method of moving asymptotes，MMA）.
- 结构优化设计 /
- 多尺度 /
- 宏观结构 /
- 微观材料 /
- 特征值分析
Abstract: A multi-scale structure optimization method was proposed based on eigenvlue analysis, to find the macrostructure and microstructure of maximum macro stiffness under the worst load. The constraint that the Euclidian norm of the uncertain load is 1 was introduced, the structural compliance was calculated according to the Rayleigh-Ritz theorem, and the compliance was transformed to a symmetric matrix with the same dimensions as the local load vector. In this way, the compliance minimization problem under the worst load was transformed to the minimum problem of the maximum eigenvalue of the symmetric matrix. Moreover, the worst load case was determined with the eigenvector corresponding to the maximum eigenvalue of the matrix. Several numerical experiments demonstrated the validity of the proposed method, and illustrated the reasonability of the macro topological structure and the micro material distribution. The proposed multi-scale optimization method has virtues of iterative stability and rapid convergence. The update of the density function in the topological optimization was performed based on sensitivity analysis and the method of moving asymptotes (MMA).
- optimum structural design /
- multi-scale /
- macrostructure /
- micro material /
- eigenvalue analysis

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参考文献(14)

[1]	SANCHEZ-PALENCIA E. Non-Homogeneous Media and Vibration Theory [M]. Berlin: Springer-Verlag, 1980.
[2]	BENSSOUSAN A, LIONS J L, PAPANICOULAU G. Asymptotic Analysis for Periodic Structures [M]. Amesterdam: North Holland Publishing Company, 1978.
[3]	CIORANESCU D, PAULIN J S J. Homogenization in open sets with holes[J]. Journal of Mathematical Analysis and Applications,1979,71(2): 590-607.
[4]	BENDSE M P, KIKUCHI N. Generating optimal topologies in structural design using a homogenization method[J]. Computer Methods in Applied Mechanics and Engineering,1988,71(2): 197-224.
[5]	SIGMUND O. Materials with prescribed constitutive parameters: an inverse homogenization problem[J]. Journals & Books Create Account Sign in International Journal of Solids and Structures,1994,31(17): 2313-2329.
[6]	RODRIGUES H, GUEDES J M, BENDSOE M P. Hierarchical optimization of material and structure[J]. Structural and Multidisciplinary Optimization,2002,24(1): 1-10.
[7]	阎军. 超轻金属结构与材料性能多尺度分析与协同优化设计[D]. 博士学位论文. 大连: 大连理工大学, 2007.(YAN Jun. Multiscale analysis and concurrent optimization fo ultra-light metal structures and materials[D]. PhD Thesis. Dalian: Dalian University of Technology, 2007.(in Chinese))
[8]	LIU L, YAN J, CHENG G D. Optimum structure with homogeneous optimum truss-like material[J]. Computers and Structures,2008,86(13/14): 1417-1425.
[9]	COELHO P G, FERNANDES P R, GUEDES J M, et al. A hierarchical model for concurrent material and topology optimisation of three-dimensional structures[J]. Structural and Multidisciplinary Optimization,2008,35(2): 107-115.
[10]	TAKEZAWA A, NII S, KITAMURA M, et al. Topology optimization for worst load conditions based on the eigenvalue analysis of an aggregated linear system[J]. Computer Methods in Applied Mechanics and Engineering,2011,200(25/28): 2268-2281.
[11]	GUO X, ZHAO X F, ZHANG W S, et al. Multi-scale robust design and optimization considering load uncertainties[J]. Computer Methods in Applied Mechanics and Engineering,2015,〖STHZ〗 283: 994-1009.
[12]	ZHANG W S, ZHONG W L, GUO X. An explicit length scale control approach in SIMP-based topology optimization[J]. Computer Methods in Applied Mechanics and Engineering,2014,282: 71-86.
[13]	阎军, 邓佳东, 程耿东. 基于柔顺性与热变形双目标的多孔材料与结构几何多尺度优化设计[J]. 固体力学学报, 2011,32(2): 119-132.(YAN Jun, DENG Jiadong, CHENG Gengdong. Multi geometrical scale optimization for porous structure and material with multi-objective of structural compliance and thermal deformation[J]. Acta Mechanica Solida Sinica,2011,32(2): 119-132.(in Chiniese))
[14]	YAN J, DUAN Z Y, ERIK L, et al. Concurrent multi-scale design optimization of composite frame structures using the Heaviside penalization of discrete material model[J]. Acta Mechanica Sinica,2016,32(3): 430-441.