﻿ 桁架拓扑优化几何稳定性判定法和约束方案比较<sup>*</sup>
 文章快速检索 高级检索

Comparison of determining methods and constraint schemes for geometric stability in truss topology optimization
HAO Baoxin, ZHOU Zhicheng, QU Guangji, LI Dongze
China Academy of Space Technology, Beijing 100094, China
Received: 2018-10-29; Accepted: 2019-04-15; Published online: 2019-04-23 16:12
Foundation item: National Natural Science Foundation of China (11402281)
Corresponding author. ZHOU Zhicheng, E-mail: zhouzhicheng@cast.cn
Abstract: To improve the accuracy of determining truss geometric stability and the practicability of truss topology optimization results, several ways of determining truss geometric stability were compared, and the validity of three schemes for guaranteeing truss geometric stability of topology optimization results were discussed. First, by comparing several ways to identify truss geometric stability through some illustrative tiny trusses, a simple procedure was outlined to evaluate truss geometric stability. Second, a unified semidefinite programming (SDP) formulation of the truss topology optimization problem was established for three kinds of constraints to address the geometric stability issue. Finally, three truss structures were optimized with the SDP formulation, and the geometric stabilities of the resultant trusses were evaluated by the given simple scheme to reveal the validity of the three kinds of constraints to guarantee geometric stability. The results show that considering additional loads or the global stability constraint cannot guarantee the geometric stability of the optimized trusses while the fundamental frequency constraint can do when the constraint values are reasonably chosen.
Keywords: truss topology optimization     geometric stability     semidefinite programming (SDP)     additional loads     fundamental frequency constraint     global stability constraint

1 桁架结构几何稳定性的判定 1.1 桁架结构几何稳定性的判定方法

1) 检查节点处连接杆件的情况

2) 检查是否满足Maxwell准则

3) 检查刚度矩阵K的正定性

4) 检查结构平衡矩阵A是否行满秩

1.2 对几何稳定性判定方法的讨论

 编号 拓扑 方法1) 方法2) 方法3) 方法4) nDOF kDOF 1 × × × × 2 2 2 √ × × × 1 1 3 √ × × × 1 1 4 × √ × × 0 1

1.3 评估桁架结构几何稳定性的一种简单流程

 图 1 桁架结构几何稳定性的判定流程 Fig. 1 Flowchart for determining truss geometric stability

2 桁架结构几何稳定约束方案对比

2.1 基于半定规划的优化问题建模

2.2 桁架结构拓扑优化算例

2.2.1 算例1——考虑附加载荷

 图 2 10杆桁架基结构及不同工况下优化后的拓扑 Fig. 2 10-bar truss ground structure and optimized topologies under different load combinations

2.2.2 算例2——考虑基频约束

 图 3 33杆桁架基结构及不同基频约束下优化后的拓扑 Fig. 3 33-bar truss ground structure and optimized topologies under different fundamental frequency constraints

 图 4 33杆桁架不合理基频约束下的拓扑优化结果 Fig. 4 Topology optimization results of 33-bar truss under unreasonable fundamental frequency constraints

2.2.3 算例3——考虑全局稳定约束

 图 5 4杆桁架基结构及不同约束和载荷值下的拓扑优化结果 Fig. 5 4-bar truss ground structure and topology optimization results under different constraints and loads

2.2.4 算例4——不同约束方案对比

 图 6 88杆桁架基结构及不同设定下优化后的拓扑 Fig. 6 88-bar truss ground structure and optimized topologies under different settings

C=0.03，在不考虑任何约束方案的情况下求解问题A，优化后拓扑见图 6(b)；根据图 6(b)结果，在受压杆件连接的节点上施加附加载荷，幅值取为主载荷的1%，取C=0.03求解问题A，优化后拓扑见图 6(c)；考虑基频约束，取C=0.03、f=0.080 9求解问题B，优化后拓扑见图 6(d)；考虑全局稳定约束，取C=0.03、λcr=1求解问题C，优化后拓扑见图 6(e)。结构特性对比见表 2

 结构 杆数 体积 几何稳定 柔度 基频 临界屈曲载荷因子 基结构 88 157 是 0.03 0.080 9 64.244* 图 6(b) 22 66.879 否(N20) 0.03 307.39 图 6(c) 42 67.704 否(N3) 0.03 1.133 8 图 6(d) 55 67.392 是 0.03 0.079 5 0.548 9 图 6(e) 41 66.927 否(N8) 0.03 0.869 1 注：*表示本文分析所得基结构的临界屈曲载荷因子为64.244，与Nastran分析结果一致；文献[14]中该数据值为1.074，疑有误。

2.3 对几何稳定约束方案的讨论

3 结论

1) 通过对4种判定方法的对比，确定了桁架结构几何稳定的准确定义，可避免无效判定方法的盲目使用；结合不同判定方法的特点给出一种简单流程，可用于桁架结构几何稳定性的快速准确判定。

2) 使用统一的SDP模型对3种几何稳定约束方案的对比表明，在优化模型中考虑附加载荷或全局稳定约束均不能确保优化后拓扑的几何稳定性，但在约束合理设置的情况下，考虑基频约束则可以保证。

3) 传统的基结构法框架存在删除过细杆件的后处理方式。由于删除细杆对结构特性的影响，约束上下限设置的合理性会影响计算结果的合理性甚至优化问题数值求解的正确性。为避免对细杆的处理，下一步可考虑在基于独立拓扑变量的基结构法框架下进行对比研究。

4) 指出桁架结构的几何稳定性是一种仅与拓扑构型相关的属性，与载荷无关，这是桁架几何稳定性与局部稳定性、全局稳定性的本质区别。桁架结构全局稳定并不能保证其几何稳定。

5) 仅在全局稳定约束下进行桁架结构体积最小化设计，所得结果的应力分布水平可能极不合理，不满足实际工程要求。可见，随着结构优化问题建模和求解能力的不断提高，对不同约束组合下优化问题的适定性、约束之间的相互作用以及约束本身特性的研究也应引起足够注意。

① 对任意矩阵ACrm×n，有

 (A1)

② 若矩阵PCm×mQCn×n均可逆，则对任意矩阵ACrm×n，有

 (A2)

 (A3)

 [1] RULE W K. Automatic truss design by optimized growth[J]. Journal of Structural Engineering, 1994, 120(10): 3063-3070. DOI:10.1061/(ASCE)0733-9445(1994)120:10(3063) [2] MCKEOWN J J. Growing optimal pin-jointed frames[J]. Structural Optimization, 1998, 15(2): 92-100. [3] MARTÍNEZ P, MARTÍ P, QUERIN O M. Growth method for size, topology, and geometry optimization of trusss structures[J]. Structural and Multidisciplinary Optimization, 2007, 33(1): 13-26. [4] HAGISHITA H, OHSAKI M. Topology optimization of trusses by growing ground structure method[J]. Structural and Multidisciplinary Optimization, 2009, 37(4): 377-393. DOI:10.1007/s00158-008-0237-4 [5] HOOSHMAND A, CAMPBELL M I. Truss layout design and optimization using a generative synthesis[J]. Computers and Structures, 2016, 163: 1-28. DOI:10.1016/j.compstruc.2015.09.010 [6] DORN W, GOMORY R, GREENBERG M. Automatic design of optimal structures[J]. Journal de Mechanique, 1964, 3: 25-52. [7] TYAS A, GILBERT M, PRITCHARD T. Practical plastic layout optimization of trusses incorporating stability considerations[J]. Computers and Structures, 2006, 84: 115-126. DOI:10.1016/j.compstruc.2005.09.032 [8] DESCAMPS B, COELHO R F. The nominal force method for truss geometry and topology optimization incorporating stability considerations[J]. International Journal of Solids and Structures, 2014, 51: 2390-2399. DOI:10.1016/j.ijsolstr.2014.03.003 [9] OHSAKI M, KATOH N. Topology optimization of trusses with stress and local constraints on nodal stability and member intersection[J]. Structural and Multidisciplinary Optimization, 2005, 29(3): 190-197. [10] CERVEIRA A, AGRA A, BASTOS F, et al. A new branch and bound method for a discrete truss topology design problem[J]. Computational Optimization and Applications, 2013, 54(1): 163-187. DOI:10.1007/s10589-012-9487-6 [11] MELA K. Resolving issues with member buckling in truss topology optimization using a mixed variable approach[J]. Structural and Multidisciplinary Optimization, 2014, 50(6): 1037-1049. DOI:10.1007/s00158-014-1095-x [12] 冷国俊, 张卓, 保宏, 等. 考虑重叠过滤及稳定性约束的桁架拓扑优化方法[J]. 工程力学, 2013, 30(2): 8-12. LENG G J, ZHANG Z, BAO H, et al. Topology optimization of truss structure based on overlapping-filter and stability constraints[J]. Engineering Mechanics, 2013, 30(2): 8-12. (in Chinese) [13] GUO X, CHENG G D, OLHOFF N. Optimum design of truss topology under buckling constraints[J]. Structural and Multidisciplinary Optimization, 2005, 30(3): 169-180. DOI:10.1007/s00158-004-0511-z [14] KOČVARA M. On the modelling and solving of the truss design problem with global stability constraints[J]. Structural and Multidisciplinary Optimization, 2002, 23(3): 189-203. [15] DEB K, GULATI S. Design of truss-structures for minimum weight using genetic algorithms[J]. Finite Elements in Analysis and Design, 2001, 37(5): 447-465. DOI:10.1016/S0168-874X(00)00057-3 [16] SAVSANI V J, TEJANI G G, PATEL V K, et al. Modified meta-heuristics using random mutation for truss topology optimization with static and dynamic constraints[J]. Journal of Computational Design and Engineering, 2017, 4(2): 106-130. DOI:10.1016/j.jcde.2016.10.002 [17] RICHARDSON J N, ADRIAENSSENS S, BOUILLARD P, et al. Multiobjective topology optimization of truss structures with kinematic stability repair[J]. Structural and Multidisciplinary Optimization, 2012, 46(4): 513-532. DOI:10.1007/s00158-012-0777-5 [18] AHRARI A, DEB K. An improved fully stressed design evolution strategy for layout optimization of truss structures[J]. Computers and Structures, 2016, 164: 127-144. DOI:10.1016/j.compstruc.2015.11.009 [19] PELLEGRINO S, CALLADINE C R. Matrix analysis of statically and kinematically indeterminate frameworks[J]. International Journal of Solids and Structures, 1986, 22(4): 409-428. DOI:10.1016/0020-7683(86)90014-4 [20] PELLEGRINO S. Structural computations with the singular value decomposition of the equilibrium matrix[J]. International Journal of Solids and Structures, 1993, 30(21): 3025-3035. DOI:10.1016/0020-7683(93)90210-X [21] 阎军, 杨春秋. 计算结构力学[M]. 北京: 科学出版社, 2014: 1-3. YAN J, YANG C Q. Computational structural mechanics[M]. Beijing: Science Press, 2014: 1-3. (in Chinese) [22] 修乃华, 罗自炎. 半定规划[M]. 北京: 北京交通大学出版社, 2014: 1-79. XIU N H, LUO Z Y. Semidefinite programming[M]. Beijing: Beijing Jiaotong University Press, 2014: 1-79. (in Chinese) [23] BEN-TAL A, NEMIROVSKI A. Robust truss topology design via semidefinite programming[J]. SIAM Journal on Optimization, 1997, 7(4): 991-1016. DOI:10.1137/S1052623495291951 [24] OHSAKI M, FUJISAWA K, KATOH N, et al. Semi-definite programming for topology optimization of trusses under multiple eigenvalue constraints[J]. Computer Methods in Applied Mechanics and Engineering, 1999, 180(1-2): 203-217. DOI:10.1016/S0045-7825(99)00056-0 [25] BEN-TAL A, JARRE F, KOČVARA M, et al. Optimal design of trusses under a nonconvex global buckling constraint[J]. Optimization and Engineering, 2000, 1(2): 189-213. DOI:10.1023/A:1010091831812 [26] ACHTZIGER W, KOČVARA M. On the maximization of the fundamental eigenvalue in topology optimization[J]. Structural and Multidisciplinary Optimization, 2007, 34(3): 181-195. DOI:10.1007/s00158-007-0117-3 [27] STURM J F. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones[J]. Optimization Methods and Software, 1999, 11(1-4): 625-653. DOI:10.1080/10556789908805766 [28] TÜTÜNCÜ R H, TOH K C, TODD M J. Solving semidefinite-quadratic-linear programs using SDPT3[J]. Mathematical Programming, 2003, 95(2): 189-217. DOI:10.1007/s10107-002-0347-5 [29] FIALA J, KOČVARA M, STINGL M.PENLAB: A MATLAB solver for nonlinear semidefinite optimization[J/OL].(2013-11-20)[2018-08-20].http://arxiv.org/abs/1311.5240. [30] KANNO Y, OHSAKI M, KATOH N. Sequential semidefinite programming for optimization of framed structures under multimodal buckling constraints[J]. International Journal of Structural Stability and Dynamics, 2001, 1(4): 585-602. DOI:10.1142/S0219455401000305 [31] 张贤达. 矩阵分析与应用[M]. 2版. 北京: 清华大学出版社, 2013: 61-67. ZHANG X D. Matrix analysis and applications[M]. 2nd ed. Beijing: Tsinghua University Press, 2013: 61-67. (in Chinese)

#### 文章信息

HAO Baoxin, ZHOU Zhicheng, QU Guangji, LI Dongze

Comparison of determining methods and constraint schemes for geometric stability in truss topology optimization

Journal of Beijing University of Aeronautics and Astronsutics, 2019, 45(8): 1663-1673
http://dx.doi.org/10.13700/j.bh.1001-5965.2018.0624