刚度约束条件的周期性拓扑优化研究  

Periodic Topology Optimization under Stiffness Constraint Condition

作　　者：焦洪宇[1,2] 周奇才[1] 李文军[1] 李英[2] 

机构地区：[1]同济大学机械与能源工程学院,上海201804 [2]常熟理工学院机械工程学院,苏州215500

出　　处：《机械科学与技术》2014年第9期1281-1286,共6页Mechanical Science and Technology for Aerospace Engineering

基　　金：国家自然科学基金项目(51375345);"十二五"国家科技支撑计划项目(2011BAF11B02)资助

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。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.

关 键 词：周期性 拓扑优化 刚度约束 SIMP插值模型 虚拟子域 

分 类 号：TH11[机械工程—机械设计及理论]