Please wait a minute...
浙江大学学报(工学版)
Journal of ZheJiang University (Engineering Science)  2020, Vol. 54 Issue (11): 2169-2178    DOI: 10.3785/j.issn.1008-973X.2020.11.012
    
Stress-based topology optimization based on global measure of distort energy density
Yun-kai GAO(),Chao MA,Zhe LIU,Ya-nan XU
School of Automotive Studies, Tongji University, Shanghai 201804, China
Download: HTML     PDF(1773KB) HTML
Export: BibTeX | EndNote (RIS)      

Abstract  

A modified bi-directional evolutionary structural optimization (BESO) method for stress minimization topology optimization of continuum structures was proposed. A global measure was formulated by Kreisselmeier-Steihauser aggregation function to reduce the computational cost. The sensitivity numbers were derived by the computationally efficient adjoint variable method. The optimization process was stabilized by a sensitivity filtering and correction scheme. Design variables were updated by BESO with its material addition and removal scheme that drove the initial structure gradually evolved to the optimal design. The effectiveness of the proposed method was verified by three representative numerical examples. The efficiency of the topology optimization process was significantly improved by the proposed method. Compared with the compliance minimization design, the proposed method with appropriate stress norm parameter can effectively alleviate stress concentration. The maximum stress values of the optimal designs showed various degrees of decrease, thus enhancing the strength of structures. BESO method using discrete variables avoids the stress singularity and obtains the black-and-white design.

Key words： topology optimization      distort energy density      aggregation function      bi-directional evolutionary structural optimization      structural design     
Received: 17 December 2019      Published: 15 December 2020
CLC:  O 342  
  TH 122  
Service
E-mail this article
Add to my bookshelf
Add to citation manager
E-mail Alert
RSS
Articles by authors
Yun-kai GAO
Chao MA
Zhe LIU
Ya-nan XU
Cite this article:

Yun-kai GAO,Chao MA,Zhe LIU,Ya-nan XU. Stress-based topology optimization based on global measure of distort energy density. Journal of ZheJiang University (Engineering Science), 2020, 54(11): 2169-2178.

URL:

http://www.zjujournals.com/eng/10.3785/j.issn.1008-973X.2020.11.012     OR     http://www.zjujournals.com/eng/Y2020/V54/I11/2169


基于畸变比能全局化策略的应力拓扑优化

针对受体积约束的应力最小化连续体结构拓扑优化，提出一种改进的双向渐进结构优化方法. 使用Kreisselmeier-Steihauser函数建立畸变比能的全局化函数，以克服应力优化计算量大的难题. 利用伴随变量法求解单元灵敏度，并引入灵敏度过滤及修正方法稳定优化过程. 双向渐进结构优化方法通过逐渐增加和删除单元，使结构进化至最优构型. 3个典型的拓扑优化算例结果表明：提出的方法可有效提升应力拓扑优化的计算效率；与柔度最小化拓扑优化相比，使用适当的凝聚函数参数消除了结构中的应力集中效应，最优设计中的最大应力值较原设计有不同程度下降，提高了结构的强度. 双向渐进结构优化方法使用离散设计变量避免了应力奇异现象，拓扑优化结果的边界清晰.


关键词： 拓扑优化,  畸变比能,  凝聚函数,  双向渐进结构优化,  结构设计 
Fig.1 Dimension and boundary conditions of L-bracket
Fig.2 Optimal designs and von Mises stress contours of L-bracket
Fig.3 Evolution history of volume fraction and maximum von Mises stress of L-bracket
Fig.4 Optimal designs and maximal von Mises stress contours of L-bracket with different volume fractions
Fig.5 Optimization process of L-bracket with μ=10
Fig.6 Dimension and boundary conditions of MBB beam
Fig.7 Optimal designs and von Mises stress contours of MBB beam
Fig.8 Optimization process of MBB beam with μ=10
Fig.9 Dimension and boundary conditions of 3D cantilever
Fig.10 Optimal designs and von Mises stress contours of 3D cantilever
Fig.11 Optimization process of 3D cantilever with μ of 10
[1]   BENDSOE M P, KIKUCHI N Generating optimal topologies in structural design using a homogenization method[J]. Applied Mechanics and Engineering, 1988, 71 (2): 197- 224
doi: 10.1016/0045-7825(88)90086-2
[2]   张焕宇, 郝志勇 飞轮壳结构刚度对机体NVH性能的影响[J]. 浙江大学学报: 工学版, 2013, 47 (2): 261- 266
ZHANG Huan-yu, HAO Zhi-yong Influence of flywheel cover structural stiffness on engine body NVH performance[J]. Journal of Zhejiang University: Engineering Science, 2013, 47 (2): 261- 266
[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
JIAO Hong-yu, ZHOU Qi-cai, LI Wen-jun, et al Periodic topology optimization using variable density method[J]. Journal of Mechanical Engineering, 2013, 49 (13): 132- 138
doi: 10.3901/JME.2013.13.132
[4]   ALLAIRE G, JOUVE F, TOADER A, et al A level-set method for shape optimization[J]. Comptes Rendus Mathematique, 2002, 334 (12): 1125- 1130
doi: 10.1016/S1631-073X(02)02412-3
[5]   PICELLI R, TOWNSEND S, BRAMPTON C J, et al Stress-based shape and topology optimization with the level set method[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 329: 1- 23
doi: 10.1016/j.cma.2017.09.001
[6]   隋允康, 任旭春, 龙连春, 等 基于ICM方法的刚架拓扑优化[J]. 计算力学学报, 2003, 20 (3): 286- 289
SUI Yun-kang, REN Xu-chun, LONG Lian-chun, et al Topological optimization of frame based on ICM method[J]. Chinese Journal of Computational Mechanics, 2003, 20 (3): 286- 289
[7]   隋允康, 彭细荣 结构拓扑优化ICM方法的改善[J]. 力学学报, 2005, 37 (2): 190- 198
SUI Yun-kang, PENG Xi-rong The improvement for the ICM method of structural topology optimization[J]. Acta Mechanica Sinica, 2005, 37 (2): 190- 198
[8]   隋允康, 叶红玲. 连续体结构拓扑优化的ICM方法[M]. 北京: 科学出版社, 2013: 80-82.
[9]   XIE Y M, STEVEN G P A simple evolutionary procedure for structural optimization[J]. Computers and Structures, 1993, 49 (5): 885- 896
doi: 10.1016/0045-7949(93)90035-C
[10]   YOUNG V, QUERIN O M, STEVEN G P, et al 3D and multiple load case bi-directional evolutionary structural optimization (BESO)[J]. Structural Optimization, 1999, 18 (2): 183- 192
doi: 10.1007/s001580050119
[11]   WU J, CLAUSEN A, SIGMUND O, et al Minimum compliance topology optimization of shell-infill composites for additive manufacturing[J]. Computer Methods in Applied Mechanics and Engineering, 2017, 326: 358- 375
doi: 10.1016/j.cma.2017.08.018
[12]   PICELLI R, VICENTE W M, PAVANELLO R, et al Evolutionary topology optimization for structural compliance minimization considering design-dependent FSI loads[J]. Finite Elements in Analysis and Design, 2017, 135: 44- 55
doi: 10.1016/j.finel.2017.07.005
[13]   占金青, 卢清华, 张宪民 多相材料的连续体结构拓扑优化设计[J]. 中国机械工程, 2013, 24 (20): 2764- 2768
ZHAN Jin-qing, LU Qing-hua, ZHANG Xian-min Topology optimization of continuum structure with multiple materials[J]. China Mechanical Engineering, 2013, 24 (20): 2764- 2768
[14]   LE C H, NORATO J A, BRUNS T E, et al Stress-based topology optimization for continua[J]. Structural and Multidisciplinary Optimization, 2010, 41 (4): 605- 620
doi: 10.1007/s00158-009-0440-y
[15]   LIU B, GUO D, JIANG C, et al Stress optimization of smooth continuum structures based on the distortion strain energy density[J]. Computer Methods in Applied Mechanics and Engineering, 2019, 343: 276- 296
doi: 10.1016/j.cma.2018.08.031
[16]   CHENG G, GUO X ε-relaxed approach in structural topology optimization [J]. Structural Optimization, 1997, 13 (4): 258- 266
doi: 10.1007/BF01197454
[17]   BRUGGI M On an alternative approach to stress constraints relaxation in topology optimization[J]. Structural and Multidisciplinary Optimization, 2008, 36 (2): 125- 141
doi: 10.1007/s00158-007-0203-6
[18]   HUANG X, XIE Y M Convergent and mesh-independent solutions for the bi-directional evolutionary structural optimization method[J]. Finite Elements in Analysis and Design, 2007, 43 (14): 1039- 1049
doi: 10.1016/j.finel.2007.06.006
[19]   YANG X Y, XIE Y M, STEVEN G P, et al Bidirectional evolutionary method for stiffness optimization[J]. AIAA Journal, 1999, 37 (11): 1483- 1488
doi: 10.2514/2.626
[20]   王选, 刘宏亮, 龙凯, 等 基于改进的双向渐进结构优化法的应力约束拓扑优化[J]. 力学学报, 2018, 50 (2): 385- 394
WANG Xuan, LIU Hong-liang, LONG Kai, et al Stress-constrained topology optimization based on improved bi-directional evolutionary optimization method[J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50 (2): 385- 394
[21]   LUO Y, WANG M Y, KANG Z, et al An enhanced aggregation method for topology optimization with local stress constraints[J]. Computer Methods in Applied Mechanics and Engineering, 2013, 254: 31- 41
doi: 10.1016/j.cma.2012.10.019
[22]   XIA L, ZHANG L, XIA Q, et al Stress-based topology optimization using bi-directional evolutionary structural optimization method[J]. Computer Methods in Applied Mechanics and Engineering, 2018, 333: 356- 370
doi: 10.1016/j.cma.2018.01.035
[23]   隋允康, 铁军 结构拓扑优化ICM显式化与抛物型凝聚函数对于应力约束的集成化[J]. 工程力学, 2010, 27 (Suppl.2): 124- 134
SUI Yun-kang, TIE Jun The ICM explicitation approach to the structural topology optimization and the integrating approach to stress constraints based on the parabolic aggregation function[J]. Engineering Mechanics, 2010, 27 (Suppl.2): 124- 134
[24]   隋允康, 叶红玲, 彭细荣, 等 连续体结构拓扑优化应力约束凝聚化的ICM方法[J]. 力学学报, 2007, 39 (4): 554- 563
SUI Yun-kang, YE Hong-ling, PENG Xi-rong, et al Stress-constrained topology optimization based on improved bi-directional evolutionary optimization method[J]. Chinese Journal of Theoretical and Applied Mechanics, 2007, 39 (4): 554- 563
[25]   隋允康, 张学胜, 龙连春 应力约束处理为应变能集成的连续体结构拓扑优化[J]. 计算力学学报, 2007, 24 (5): 602- 608
SUI Yun-kang, ZHANG Xue-sheng, LONG Lian-chun ICM method of the topology optimization for continuum structures with stress constraints approached by the integration of strain energies[J]. Chinese Journal of Computational Mechanics, 2007, 24 (5): 602- 608
[26]   宣东海, 隋允康, 铁军, 等 结构畸变比能处理的应力约束全局化的连续体结构拓扑优化[J]. 工程力学, 2011, 28 (10): 1- 8
XUAN Dong-hai, SUI Yun-kang, TIE Jun, et al Continuum structural topology optimization with globalized stress constraint treated by structural distortional strain energy density[J]. Engineering Mechanics, 2011, 28 (10): 1- 8
[1] Xiao-lei DENG,Ze-feng SHENG,Jiang-lin ZHANG,Xiao-wen LV,Zhong HE,Jian-chen WANG,Jian-zhong FU. Thermal topology optimization design method of spindle under temperature-structure field coupling condition based on irregular cell[J]. Journal of ZheJiang University (Engineering Science), 2020, 54(1): 23-32.
[2] XIONG Fei, HAO Zhi yong, ZHENG Xu, LIU Rui jun. Design of low loudness timing cover based on multiobjective topology optimization[J]. Journal of ZheJiang University (Engineering Science), 2016, 50(5): 970-977.
[3] ZHANG Huan-yu, HAO Zhi-yong. Influence of flywheel cover structural stiffness on engine
body NVH performance[J]. Journal of ZheJiang University (Engineering Science), 2013, 47(2): 261-266.
[4] SHAN Yan-ling, GAO Bo-qing. Analysis of latticed shell structure’s robust configurations based on continuum topology optimization[J]. Journal of ZheJiang University (Engineering Science), 2013, 47(12): 2118-2124.
[5] JIANG Jun-Xia, FAN Zi-Chun, GUO Zhi-Min, KE Yang-Lin. Design and optimization of flexible assembly platform used for
three sections joint[J]. Journal of ZheJiang University (Engineering Science), 2010, 44(9): 1798-1804.
[6] ZHONG Mei-Feng, ZHENG Shui-Yang, BO Xiao-Hong. Finite element analysis and topology optimization of direct-drive compressor valve[J]. Journal of ZheJiang University (Engineering Science), 2009, 43(12): 2305-2308.