基于面光滑有限元的复杂三维结构拓扑优化

中国机械工程 ›› 2015, Vol. 26 ›› Issue (7): 864-870.

基于面光滑有限元的复杂三维结构拓扑优化

何智成1;陈少伟1;李光耀1;张桂勇2

1.湖南大学汽车车身先进设计制造国家重点实验室，长沙，410082
2.大连理工大学工业装备结构分析国家重点实验室，大连，116023

出版日期:2015-04-10 发布日期:2015-04-24
基金资助:
国家自然科学基金资助项目（11202074）；湖南大学汽车车身先进设计制造国家重点实验室自主研究课题(51375001)；工业装备结构分析国家重点实验室开放基金资助项目(GZ1403)

Topology Optimization Using FS-FEM for Complex Three-dimensional Models

He Zhicheng1;Chen Shaowei1;Li Guangyao1;Zhang Guiyong2

1.State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body,Hunan University,Changsha,410082
2.State Key Laboratory of Structural Analysis for Industrial Equipment,Dalian University of Technology,Dalian,Liaoning,116023

Online:2015-04-10 Published:2015-04-24
Supported by:
National Natural Science Foundation of China（No. 11202074）

摘要/Abstract

摘要：

为了增强拓扑优化计算对任意复杂模型的适应性，改进基于线性四面体有限元的拓扑优化结果，引入了一种新型高精度的基于面光滑有限元模型(FS-FEM)来进行拓扑优化，通过每次迭代时提供很好的梯度解及位移解，从而达到改善拓扑优化结果的目的。在基于面光滑有限元模型的拓扑优化中，以柔度最小作为目标函数，建立了基于固体各向同性材料惩罚插值(SIMP)的拓扑优化数学模型，该数学模型通过最优准则进行求解。多个不同载荷的拓扑优化数值算例说明，采用基于面光滑有限元进行拓扑优化，结果都能够单调收敛，且采用该方法建立的拓扑优化模型能抑制棋盘格现象。与商业软件OptiStruct的计算比较表明，该方法相比有限元方法能得到更合理的拓扑结构。

关键词: 拓扑优化, 光滑伽辽金, 面光滑有限元, 四面体单元

Abstract:

The FS-FEM was proposed recently based on tetrahedral mesh, and showed great property in solid mechanics, such as providing very good gradient solutions. The topology optimization design of the continuum structures under static load was formulated in the scheme of FS-FEM. As the face-based smoothing domains were the sub-units of assembling stiffness matrix in the discretized system of smoothed Galerkin weak form, thus the relative density of face-based smoothing domains were served as design variables. In this formulation, the minimization compliance was considered as an objective function, and the mathematic of the topology optimization model was developed using the solid isotropic material with penalization (SIMP) interpolation scheme. The topology optimization problem was then solved by the optimality criteria method. Finally, the feasibility and efficiency of the proposed method were illustrated with several 3D examples that were widely used in the topology optimization design.

Key words: topology optimization, smoothed Galerkin weak form, face-based smoothed finite element method (FS-FEM);tetrahedral mesh

中图分类号:

TH122

何智成, 陈少伟, 李光耀, 张桂勇. 基于面光滑有限元的复杂三维结构拓扑优化[J]. 中国机械工程, 2015, 26(7): 864-870.

He Zhicheng, Chen Shaowei, Li Guangyao, Zhang Guiyong. Topology Optimization Using FS-FEM for Complex Three-dimensional Models[J]. China Mechanical Engineering, 2015, 26(7): 864-870.

参考文献

[1]Bendsoe M P,Kikuchi N.Generating Optimal Topologies in Structural Design Using a Homogenization Method[J]. Comput. Methods Appl. Mech. Eng.，1988，71(2):197-224.
[2]Bendsoe M P, Sigmund O.Material Interpolation Schemes in Topology Optimization[J].Arch. Appl. Mech., 1999,69(9/10):635-654.
[3]罗震.基于变密度法的连续体结构拓扑优化设计技术研究[D].武汉:华中科技大学, 2005.
[4]Challis J V.A Discrete Level-set Topology Optimization Code Written in MATLAB[J].Struct. Multidisc. Optim.,2010,41(3):453-464.
[5]隋允康, 杨德庆.多工况应力和位移约束下连续体结构拓扑优化[J].力学学报, 2000, 32(2): 171-179.
Sui Yunkang,Yang Deqing,Wang Bei.Topological Optimization of Continuum Structure with Stress and Displacement Constraints under Multiple Loading Cases[J]. Acta Mech. Sin.,2000,32(2): 171-179.
[6]Xie Y M, Steven G P.A Simple Evolutionary Procedure for Structural Optimization[J]. Comput. & Struct., 1993,49(5):885-896.
[7]Rozvany G I N.A Critical Review of Established Methods of Structural Topology Optimization[J].Struct. Multidisc. Optim., 2010, 37(3):217-237.
[8]Bruggi M.On the Solution of the Checkerboard Problem in Mixed-FEM Topology Optimization[J]. Comput. & Struct. , 2008, 86(19/20): 1819-1829.
[9]Zheng J, Long S Y, Xiong Y B,et al.A Topology Optimization Design for the Continuum Structure Based on the Meshless Numerical Technique[J]. CMES:Comput. Model. Eng.& Sci., 2008,34(2):137-154.
[10]Zhou J X, Zou W.Meshless Approximation Combined with Implicit Topology Description for Optimization of Continua[J]. Struct. Multidisc. Optim., 2008,36(4):347-353.
[11]Bendse M P, Sigmund O.Topology Optimization Theory, Methods and Applications[M].New York: Springer, 2003.
[12]Nguyen-Thoi T, Liu G R, Lam K Y,et al.A Face-based Smoothed Finite Element Method (FS-FEM) for 3D Linear and Geometrically Nonlinear Solid Mechanics Problems Using 4-node Tetrahedral Elements[J].Int. J. Numer. Mech. Engrg., 2009, 78(3):324-353.
[13]Liu G R.A Generalized Gradient Smoothing Technique and the Smoothed Bilinear Form for Galerkin Formulation of a Wide Class of Computational Methods[J].Int. J. Computational Methods, 2008,5(2): 199-236.
[14]Fujii D, Kikuchi N.Improvement of Numerical Instabilities in Topology Optimization Using SLP Method[J].Struct. Multidisc. Optim., 2000,19(2):113-121.
[15]Svanberg K.The Method of Moving Asymptotes-a New Method for Structural Optimization[J].Int. J. Numer. Meth. Engrg., 1987,24(2):359-373.
[16]Olhoff N, Ronholt E, Scheel J.Topology Optimization of Three Dimensional Structures Using Optimum Microstructures[J]. Struct. Optim., 1998,16(1):1-18.