Periodic Topology Optimization under Stiffness Constraint Condition
-
摘要: 板条状结构的设计域具有较大的长宽比,常规的拓扑优化方法无法获得清晰的、易于加工的拓扑形式或求解困难。提出了一种刚度约束条件下基于变密度理论固体各向同性微结构材料惩罚模型(SIMP)的周期性拓扑优化的方法。建立了以结构的体积作为目标函数,单元相对密度为设计变量的周期性拓扑优化问题的数学模型。为了保证优化结构可以获得周期性的拓扑形式,在数学模型中设置额外的约束条件。通过优化准则法推导出虚拟子域设计变量的迭代公式,利用刚度约束计算出拉格朗日乘子。引入过滤函数解决拓扑优化容易出现数值计算不稳定,导致棋盘格、网格依赖性等问题。利用所提出的方法,对平面矩形悬臂梁结构进行拓扑优化研究,获得清晰的、易于加工的周期性拓扑形式。结果表明:当子域数目取值不同时,均可获得清晰的、易于加工的周期性拓扑形式,且具有良好的一致性。通过该典型算例验证了利用变密度理论SIMP插值模型实现周期性拓扑优化的可行性和有效性。Abstract: The design domain of lath-shaped structure have a large length-width ratio,so it is difficult to obtain a clear and periodic topology configuration using conventional topology optimization algorithms. Using variable density theory and solid isotropic microstructures with penalization(SIMP) model,this paper presents a periodic topology optimization method under stiffness constraint condition. The mathematical models for periodic topology optimization is built,in which the volume of structure is taken as objective function and relative densities of elements are taken as design variables. In order to obtain a topology structure which possesses periodicity,an additional constraint condition is introduced into the mathematical model. The iterative formula of virtual sub-domain design variables is deduced by taking the advantage of optimality criteria method and Lagrange multiplier is calculated using stiffness constraint. A filtering function is imported in order to solve the checkerboard and mesh-dependent problems. A clear and periodic topology configuration of a cantilever beam is obtained by the proposed method. The results show that the periodic topology configuration with good consistency cab be achieved when the number of sub-domain is different. This classical example proves that it is feasible and effective for applying variable density method to achieve periodic topology optimization.
-
Key words:
- algorithms /
- design /
- efficiency /
- Lagrange multipliers /
- mathematical models
-
[1] Bledsoe M P,Kikuchi N. Generating optimal topologiesin structural design using a homogenization method[J].Comput Method,1988,71:197-224 [2] Su-Young Chang,Sung-Kie Youn. Material cloud method for topology optimization[J]. InternationalJournal for Numerical Methods in Engineering,2006,65:585-1607 [3] Burke D R,Moslemi-Tabrizi S,Smy T J. Simulation ofinhomogeneous models using the finite cloud method[J]. Materialwiss Werkstofftech,2010,41(5):336-340 [4] 罗阳军,亢战,邓子晨. 多工况下结构鲁棒性拓扑优化研究[J]. 力学学报,2011,43(1),227-234Luo Y J,Kang Z,Deng Z C. Robust topology optimizationdesign of structures with multiple load cases [J]. ChineseJournal of Theoretical and Applied Mechanics,2011,43(1):227-234 (in Chinese) [5] 张桥,张卫红,朱继宏. 动态响应约束下的结构拓扑优化设计[J]. 机械工程学报,2010,46(15):45-51Zhang Q,Zhang W H,Zhu J H. Topology optimizationof structures under dynamic response constrains[J].Journal of Mechanical Engineering,2010,46(15):45-51(in Chinese) [6] 舒兵,龚宪生,廉超. 爆破试验下离心压缩机齿轮箱体的结构拓扑优化[J]. 机械科学与术,2012,31(9):1516-1521Shu B,Gong X S,Lian C. Topology optimization for theGear-box of the centrifugal compressor on the conditionof blasting experiment[J]. Mechanical Science andTechnology for Aerospace Engineering,2012,31 (9):1516-1521 (in Chinese) [7] 薄纪康,吴敏. 大型单闸板防喷阀阀体结构拓扑优化方法研 究[J]. 机 械 科 学 与 技 术,2012,31 (8):1373-1376Bo J K,Wu M. Optimizing structural topology ofblowout prevention valve of large-scale singe ram[J].Mechanical Science and Technology for AerospaceEngineering,2012,31(8):1373-1376 (in Chinese) [8] 刘书田,贾海朋,王德伦. 狭长结构拓扑优化[J]. 计算力学学报,2004,21(6):653-657Liu S T,Jia H P,Wang D L. Topology optimization ofthin-and-long structures [J]. Chinese Journal ofComputational Mechanics,2004,21 (6):653-657 (in Chinese) [9] 王庆,徐斌,何佳琦. 基于拓扑优化的结构动力学边界条件优化设计[J]. 机械科学与技术,2012,31(11):1845-1850Wang Q,Xu B,He J Q. Optimal design of structuraldynamic boundary conditions based on topologyoptimization method [J]. Mechanical Science andTechnology for Aerospace Engineering,2012,31 (11):1845-1850 (in Chinese) [10] 郑娟,龙述尧,熊渊博. 基于无网格数值求解技术的二维连续体结构拓扑优化设计[J]. 计算力学学报,2010,27(2):232-237Zheng J,Long S Y,Xiong Y B. The topologyoptimization design of the continuum structure based meshless numerical technique[J]. Chinese Journalof Computational Mechanics,2010,27(2):232-237 (in Chinese) [11] 刘国梁,陈建军,马洪波. 一种基于非概率可靠性的结构水平集拓扑优化[J]. 工程力学,2012,29(6):58-62Liu G L,Cheng J J,Ma H B. Structural topologicaloptimization for non-probability reliability in level setmethod[J]. Engineering Mechanics,2012,29(6):58-62 (in Chinese) [12] 罗震,陈立平,黄玉盈,等. 基于 RAMP 密度-刚度差值格式的结构拓扑优化[J]. 计算力学学报,2005,22(5):585-591Luo Z,Cheng L P,Huang Y Y,et al. Topologicaloptimization using ramp interpolation scheme [J].Chinese Journal of Computational Mechanics,2005,22(5):585-591 (in Chinese) [13] Huang X,Xie Y M. Optimal design of periodic structuresusing evolutionary topology optimization [J]. StructMultidisc Optim.,2008,36:597-606 -
![WeChat](/fileJXKXYJS/journal/article/jxkxyjs/2014/9/PIC/wechat_cn_4664.jpg)
计量
- 文章访问数: 219
- HTML全文浏览量: 29
- PDF下载量: 3
- 被引次数: 0