基于水平集方法及COMSOL的弹性结构拓扑优化
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摘要
针对现有弹性结构拓扑优化方法计算效率差、结构边界不光滑等缺点,提出了一种基于水平集方法及COMSOL的弹性结构拓扑优化方法。优化方法利用水平集函数作为设计变量,结构的整体柔度最小为目标函数,实体材料所占的体积比为约束条件,结合COMSOL偏微分方程(PDE)模块中反应扩散方程有限元求解弹性结构拓扑优化问题。该方法与密度惩罚法相比,得到了光滑的结构边界;与传统的水平集法相比,不用有限差分求解复杂的Hamilton-Jacobi方程,摆脱了柯朗-弗里德里希斯-列维(CFL)条件的限制,提高了计算效率。数值算例结果表明该方法能够提高计算效率并获得光滑的结构边界。
In view of the existing elastic structural topology optimization method for computing efficiency, structure difference boundary is not smooth and other shortcomings, proposes a topology optimization method for elastic structure based on level set method and COMSOL. Optimization method using the level set function as design variables, the overall structure of the minimum compliance as the objective function, solid material volume ratio as the constraint condition, the combination of COMSOL partial differential equation(PDE) reaction diffusion equation of the finite element solution of elastic structure topology optimization module. Compared with the method of density of penalty method, obtains the structure boundary smooth; compared with the traditional level set method, not the Hamilton-Jacobi finite difference equation for solving complex, get rid of the Courant Friedrichs Levi(CFL) conditions, improving the calculation efficiency. The numerical results show that the method can improve calculation efficiency and obtain the structure boundary smooth.
In view of the existing elastic structural topology optimization method for computing efficiency, structure difference boundary is not smooth and other shortcomings, proposes a topology optimization method for elastic structure based on level set method and COMSOL. Optimization method using the level set function as design variables, the overall structure of the minimum compliance as the objective function, solid material volume ratio as the constraint condition, the combination of COMSOL partial differential equation(PDE) reaction diffusion equation of the finite element solution of elastic structure topology optimization module. Compared with the method of density of penalty method, obtains the structure boundary smooth; compared with the traditional level set method, not the Hamilton-Jacobi finite difference equation for solving complex, get rid of the Courant Friedrichs Levi(CFL) conditions, improving the calculation efficiency. The numerical results show that the method can improve calculation efficiency and obtain the structure boundary smooth.
引文
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[2]Bendsoe M.P.,Sigmund O.Material interpolations in topology optimization[J].Arch Appl Mech,1999(69):635-654.
[3]Osher S,Santosa F.Level set methods for optimization problems involving geometry and constraints I.Frequencies of a two density inhomogeneous drum[J].J Comput Phys,2001(171):272-288.
[4]Wang M.Y.,Wang X,Guo D.A level set method for structure topology optimization[J].Comput Methods Appl Mech Eng,2003(192):227-246.
[5]Allaire G,Jouv F,Toader A.Structure optimization using sensitivity analysis and a level-set method[J].J Comput Phys,2004(194):363-393.
[6]Jia Haipeng,Beom H.G.,Wang Yuxin.,et al.Evolutionary level set method for structural topology optimization[J].Computers and Structures,2011(89):445-454.
[7]Shojaee S,Mohammadian M.Structural topology optimization using an enhanced level set method[J].Scientia Iranica A,2012,19(5):1157-1167.
[8]Osher S,Sethian J A.Front propagating with curvature dependent speed:Algorithms based on Hamilton-Jacobi Formulation[J].Journal of Computational Physics,1988(79):12-49.
[9]Sethian J A,Wiegmann A.Structural boundary design via level set and immersed interface methods[J].Journal of Computational Physics,2000,163(2):489-528.
[10]Courant R,Friedrichs K,Lewyt H.On the Partial Difference Equations of Mathematical Physics[J].Mathematische Annalen,1928(100):32-74.