狭长结构拓扑优化

通常的拓扑优化是在给定区域内，通过设计材料分布实现结构拓扑形式优化。对于设计区域的长和宽相近的平面问题，现行的方法可得到清晰的拓扑。但是，狭长结构的设计域具有大的长宽比。为了保证基结构包含足够多的拓扑形式，宽度方向要求有一定量的有限元分割，从而导致整体网格数和设计变量多、问题求解困难。本文提出了通过基本结构拼装的狭长结构拓扑优化方法，建立了以最小平均柔顺性密度为目标、同时设计材料分布和设计域几何尺度的基本结构的拓扑优化问题的数学提法和求解方法。利用所提出的问题提法和求解方法，设计了狭长悬臂梁的拓扑形式，讨论了危险截面的弯矩与剪力的相对值以及材料体积约束对拓扑形式的影响。数值结果表明，不同的弯矩与剪力的相对数值对应不同的拓扑形式，随着相对数值的增加，梁的拓扑形式由类桁架结构逐渐变成竖直立板加强的框架式结构。

Topology optimization of thin-and-long structures

Generally, an optimal topology is obtained by optimizing the material distribution in a given design domain. A distinctive topology configuration can be obtained when the design domain has similar size along the width and length for a plane problem. However, it is often difficult to solve the topology optimization problem for a long-and-thin structure using conventional algorithm. To ensure that the ground structure includes topology configurations as more as possible, enough divisions along the width should be made, which leads to the addition of the elements and/or design variables and makes the problem difficult to solve. In this paper, a method of topology optimization of long-and-thin structures through repeating the base structure is proposed. The base structure can be obtained by a minimum averaged compliance density (ACD) based algorithm, in which the ACD is taken as the objective function, and the topology or the material distribution and the domain dimensions of the structure are optimized simultaneously. As an illustrated example, a cantilever beam with large aspect ratio is optimized, and the effects of the relative value of the moment to shear force on the dangerous section and the weight limit on the optimal topology configurations are discussed in detail. Results show that different relative value of the moment may correspond to different optimal topology, and the optimal topology varies from truss-like structure to vertical stiffened box-like structure with increasing the moment relative to shear force.