基于位移约束的类周期性连续体结构拓扑优化设计  

Topology optimization of periodic-like structures with displacement constraints

作　　者：荣见华[1,2] 廖莺[1,2] 赵志军[3] 谢亿民 易继军[1,2] 

机构地区：[1]长沙理工大学工程车辆轻量化与可靠性技术湖南省高校重点实验室,长沙410004 [2]长沙理工大学工程车辆安全性设计与可靠性技术湖南省重点实验室,长沙410004 [3]长沙学院土木系,长沙410004 [4]澳大利亚皇家墨尔本理工大学土木、环境与化学工程学院BOX2476墨尔本

出　　处：《应用力学学报》2013年第6期876-881,953-954,共6页Chinese Journal of Applied Mechanics

基　　金：国家自然科学基金(10872036;51228801);国家863项目(2008AA04Z118)

摘　　要：提出了一种基于位移约束的类周期性连续体结构拓扑优化设计的方法。为了获得类周期性结构的最优拓扑,将优化的结构区域划分成若干个子区域;为了解决目标函数振荡问题,在每一迭代步形成并引入变位移约束限,以单元相对密度指数幂的倒数作为设计变量,建立了位移约束的显式近似式,并形成了以结构质量作为目标函数、以位移作为约束条件的类周期性结构拓扑优化近似模型;本文通过改进对偶求解方法,建立了拉格朗日乘子迭代求解公式;引入虚拟子域设计变量,建立了类周期性结构各子区域单元设计变量之间的联系,满足了指定的类周期性约束条件,并推导出了设计变量迭代公式;最后给出了梁结构拓扑设计和双坡梯形屋钢屋架设计的算例。结果表明:随着周期数的增加,子结构尺寸对最优拓扑的影响减弱;优化迭代过程中没有目标函数振荡现象。以上结果验证了本文方法的可行性和有效性。A method for topology optimization of periodic-like structures with structural mass being the objective function and structural displacements being the constraint functions is proposed so that the optimum topology for any periodic-like structures can be obtained. Firstly, the optimization domain is divided into some sub-domains. Secondly, varying displacement constraint limits are formed and introduced to the optimization model at each iteration step to deal with the objective oscillation. Then, the reciprocal density exponents of structural elements are taken as design variables, explicit functions for displacement constraints are constructed, and a topological optimization model of a periodic-like continuum structure is formed. An improving dual solving method is given and a set of iteration formula for Lagrange multipliers is built. Virtual sub-domain design variables are introduced to establish the relation of corresponding variables between all the sub-domains of the periodic-like continuum structure in order to enforce structurally periodic-like requirement, and a set of iteration formula for design variables is established. Two examples of a beam structural design and a roof frame topology design are given, and the sub-domain size effect on the optimal topology decreases with structural periodicity increasing under the same displacement constraints. Example results show that there is not any objective oscillation phenomenon in optimization iterations, and the proposed method is of validity and effectiveness.

关 键 词：类周期性结构 拓扑优化 位移约束 二次规划法 拉格朗日乘子 

分 类 号：O342[理学—固体力学]