连续体结构的拓扑优化设计

对基于有限元数值求解技术的连续体结构的拓扑优化设计技术进行了综述. 利用密度-刚度插值格式和优化准则方法, 以结构的柔度最小化作为优化的目标函数, 论述并建立线弹性结构的静力学拓扑优化设计的数学模型和设计变量显示的迭代格式; 基于数学规划方法中的一种凸规划方法----移动渐近线方法和密度方法, 以结构的频率最大化作为优化的目标函数, 论述并建立了特征值问题拓扑优化设计的数学模型和设计变量隐式的更新方法. 对多目标拓扑优化问题、柔性机构的拓扑优化问题以及多物理场拓扑优化设计问题进行了讨论. 对优化结构中出现的棋盘格式和网格依赖性等数值计算问题进行剖析和讨论, 介绍和分析了目前解决数值计算问题常见的方法, 在此基础上对边界扩散现象进行了讨论. 给出了连续体结构拓扑优化设计的程序流程, 并用Matlab程序实现了算法, 通过几个典型的算例证明所综述方法的有效性.

TOPOLOGICAL OPTIMIZATION DESIGN FOR CONTINUUM STRUCTURES

Structural topology optimization techniques are briefly discussed based on finite element approach. The mathematical model, with the objective to minimize compliance, as well as the explicit updating scheme for design variables are both worked out based on density-stiffness interpolation schemes and Optimality Criteria method. The computational model to maximize the fundamental eigenvalue, and the implicit retrieving pattern of design variables for vibrating structures, are also worked out using density method and a convex approach the method of moving asymptotes. Compliant mechanism designs employing topology optimization method, Multi-objective topology optimization problems considering static and vibrating structures, and multi-physics topology optimization problems are all discussed. Numerical instabilities, such as checkerboards and mesh dependencies, are introduced and investigated. The flowchart for topology optimization design is given. The efficiency and validity of methodologies in this paper are demonstrated by several typical examples. The future perspectives of topology optimization designs are outlined.