拓扑优化设计中刚度与密度关系的一种力学模型

A Mechanical Model about the Relation Between Stiffness and Density Variable in Topology Optimal Design

为了探讨变密度法中伪密度的变化对材料刚度的影响,并建立两者之间的力学关系,本文采用有限元法研究这一问题。提出利用"实心"与"空心"两种八节点体单元来构建"微元体"模型,通过各种比例的"实心"和"空心"体单元的均匀组合,来模拟伪密度数值的变化过程。理论分析与推导表明:确定"微元体"的刚度时,本模型不仅与加载和约束形式无关,而且与"微元体"的形状亦无关。通过数值试验,可以给出这种模型的刚度与伪密度的离散数值关系;最后,利用幂指函数获得其解析关系。计算结果发现一个十分有趣的现象:刚度与伪密度不遵循人为给出的幂函数关系。

In order to explore the change of pseudo-density to the effect of material rigidity and set up the relation between them for density variable method, we presented FEM to deal with the problem. We proposed that the micro-body model was constructed by 'solid' and 'void' 8-node volume elements. Through even combinations of variant proportions with those two elements, we simulated the process of pseudo-density change. Theoretical analysis and deduction expressed that the posed model had nothing with not only loads and constraints but also the shape of micro-body. Then, by means of numerical test, we got the discrete data relation between rigidity and pseudo-density. Finally, by way of power and exponential function, we obtained their analytical relation. Results reveal that the obtained relation differs from the artificial power law.