动力学自然单元法的谐波激励下的连续体结构拓扑优化

PDF(1240 KB)

振动与冲击 ›› 2019, Vol. 38 ›› Issue (21) : 252-258.

论文

徐家琪1，马永其1, 2

作者信息 +

Topology optimization of continuum structures under frequency excitation load based on dynamic natural element method

XU Jiaqi1MA Yongqi1,2

Author information +

文章历史 +

摘要

自然单元法是一种基于Voronoi图构造形函数的无网格方法，根据自然单元法的优点，提出了动力学自然单元法频率激励载荷下连续体的结构拓扑优化计算。采用各向同性固体微结构惩罚（SIMP）模型，将节点相对密度作为设计变量，建立以动柔度最小为目标函数，频率激励载荷作用下的拓扑优化模型。采用伴随分析法进行灵敏度分析并利用优化准则法对优化模型进行求解。通过数值算例计算，不仅得到了无棋盘格现象的优化结果，而且相比其它无网格方法提高了计算效率，说明该方法具有可行性和优越性。

Abstract

The natural element method is a meshless one with shape function formed using Voronoi diagram.According to advantages of the natural element method, topology optimization method of continuum structures under frequency excitation load based on dynamic natural element method was proposed.Using isotropic solid micro-structure penalty (ISMP) model, taking relative density of nodes as design variables and the dynamic flexibility being the minimum as the objective, a topology optimization model was built under frequency excitation load.The adjoint analysis method was used to perform sensitivity analysis, and the optimization criteria method was used to solve the optimization model.The numerical example computation results showed that this method can be used not only to obtain the optimization results with chess-free phenomenon, but also has a higher computational efficiency compared with other meshless methods to verify its feasibility and effectiveness.

导出引用

徐家琪1，马永其1, 2. 动力学自然单元法的谐波激励下的连续体结构拓扑优化[J]. 振动与冲击, 2019, 38(21): 252-258

XU Jiaqi1MA Yongqi1,2 . Topology optimization of continuum structures under frequency excitation load based on dynamic natural element method[J]. Journal of Vibration and Shock, 2019, 38(21): 252-258

参考文献

[1] A.G.M.Michell M.C.E. LVIII. The limits of economy of material in frame-structures [J]. Philosophical Magazine, 1904, 8(47):589-597.
[2] Bendsøe M P, Kikuchi N. Generating optimal topologies in structural design using a homogenization method [J]. Computer Methods in Applied Mechanics & Engineering, 1988, 71(2):197-224.
[3] Bendsøe M P, Sigmund O. Material interpolation schemes in topology optimization [J]. Archive of Applied Mechanics, 1999, 69(9-10):635-654.
[4] Sui Y, Yang D. A new method for structural topological optimization based on the concept of independent continuous variables and smooth model [J]. Acta Mechanica Sinica, 1998, 14(2):179-185.
[5] Xie Y M, Steven G P. A simple evolutionary procedure for structural optimization [J]. Computers & Structures, 1993, 49(5):885-896.
[6] Xie Y M, Steven G P. Evolutionary structural optimization for dynamic problems [J]. Computers & Structures, 1996, 58(6):1067-1073.
[7] 邱海, 隋允康, 叶红玲. 频率约束板结构拓扑优化[J]. 固体力学学报, 2012, 33(2):189-198.
[8] Díaaz A R, Kikuchi N. Solutions to shape and topology eigenvalue optimization problems using a homogenization method [J]. International Journal for Numerical Methods in Engineering, 2010, 35(7):1487-1502.
[9] Pedersen N L. Maximization of eigenvalues using topology optimization [J]. Structural & Multidisciplinary Optimization, 2000, 20(1):2-11.
[10] 魏星, 陈莘莘, 童谷生. 基于自然单元法的功能梯度板固有频率优化[J]. 力学季刊, 2017(1):135-143.
[11] Ma Z D, Kikuchi N, Cheng H C. Topological design for vibrating structures[J]. Comp.meth.in Appl.mech. & Engrg, 1995, 121(s 1–4):259-280.
[12] Jog C S. Topology Design of Structures Subjected to Periodic Loading [J]. Journal of Sound & Vibration, 2002, 253(3):687-709.
[13] 徐斌, 管欣, 荣见华. 谐和激励下的连续体结构拓扑优化[J]. 西北工业大学学报, 2004, 22(3):313-316. (Xu Bin, Guan Xin, Rong Jianhua. On topology Optimization of Continuous structures under Harmonic Excitation, Journal of North western Polytechnical University, 2004, 22(3):313-316.)
[14] 刘虎, 张卫红, 朱继宏. 简谐力激励下结构拓扑优化与频率影响分析[J]. 力学学报, 2013, 45(4):588-597.
[15] 左孔天,王书亭,陈立平,张云清,钟毅芳.拓扑优化中去除数值不稳定性的算法研究[J].机械科学与技术,2005(01):86-89+93.(Zuo Kongtian, Wang Shuting, Chen Liping, Zhang Yunqing, Zhong Yifang. Research on algorithms to eliminate numerical instabilities in topology optimization, Mechanical Science and Technology, 2005(01):86-89+93.)
[16] 程耿东. 结构优化新方法及其计算机实现[J]. 力学与实践,1992,(01):1-6.(Cheng Gengdong. New method of structural optimization and its realization. Mechanics in Engineering, 1992,(01):1-6.)
[17] 隋允康,叶红玲,杜家政.结构拓扑优化的发展及其模型转化为独立层次的迫切性[J].工程力学,2005(S1):107-118.(Sui Yunkang, Ye Hongling, Du Jiazheng. Development of structure topology optimization and imminency of its model transformation into independent level. Engineering Mechanics, 2005(S1):107-118.)
[18] 夏天翔, 姚卫星. 连续体结构拓扑优化方法评述[J]. 航空工程进展, 2011, 02(1):1-11.(Xia Tianxiang, Yao Weixing. A survey of topology optimization of continuum structure. Advances in aeronautical science and engineering, 2011, 02(1):1-11.)
[19] Lucy L B. A numerical approach to the testing of the fission hypothesis[J]. Astronomical Journal, 1977, 82(82):1013-1024.Lancaster P, Salkauskas K. Surfaces Generated by Moving Least Squares Methods[J]. Mathematics of Computation, 1981, 37(155):141-158.
[20] Belytschko T, Lu Y Y, Gu L. Element-Free Galerkin methods[J]. International Journal for Numerical Methods in Engineering, 1994, 37(2):229-256.
[21] Liu W K, Jun S J S, Zhang Y F. Reproducing kernel particle methods. Int J Numer Meth Eng[J]. International Journal for Numerical Methods in Fluids, 1995, 20(8‐9):1081-1106.
[22] Liu G R, Gu Y T. A point interpolation method for two-dimensional solids[J]. International Journal for Numerical Methods in Engineering, 2015, 50(4):937-951.
[23] Braun J, Sambridge M. A numerical method for solving partial differential equations on highly irregular evolving grids[J]. Nature, 1995, 376(6542):655-660.
[24] Zhou J X, Zou W. Meshless approximation combined with implicit topology description for optimization of continua[J]. Structural & Multidisciplinary Optimization, 2008, 36(4):347-353.
[25] Zheng J, Long S, Li G. The topology optimization design for continuum structures based on the element free Galerkin method[J]. Engineering Analysis with Boundary Elements, 2010, 34(7):666-672.
[26] 郑娟, 龙述尧, 熊渊博,等. 无网格径向点插值法在拓扑优化中的应用[J]. 固体力学学报, 2010, 31(4):427-432. (Zheng Juan, Long Shurao, Xiong Yuanbo. An application of the meshless radial point interpolation method to the structural topology optimization design, C hinese journal of solid mechanics, 2010, 31(4):427-432).
[27] Wu Y, Ma Y, Feng W. Topology optimization using the improved element-free Galerkin method for elasticity[J]. Chinese Physics B, 2017, 26(8):32-39.
[28] 郑娟, 龙述尧, 李光耀,等. 基于无网格径向点插值方法的简谐激励下的连续体结构拓扑优化[J]. 计算机辅助工程, 2011, 20(1):50-55. (Zheng Juan, Long Shurao, Li Guangyao. Topology optimization of continuum structures subject to harmonic excitation based on meshless radial point interpolation method, Computer Aided Engineering, 2011, 20(1):50-55.)
[29] 郑娟, 龙述尧, 李光耀. 基于无单元Galerkin方法的受迫振动下的连续体结构拓扑优化[J]. 固体力学学报, 2011, 32(5):527-533. (Zheng Juan, Long Shurao, Li Guangyao. Topology optimization of continuum structures subjected to forced vibration based on the element-free Galerkin method, Chinese Journal of solid mechanics, 2011, 32(5):527-533.).