桁架结构拓扑优化设计可行域研究

本文提出了桁架结构拓扑优化设计的k维可行子域、相邻可行子域、k维连通可行子域、k维奇异可行子域的定义，在这些定义的基础上，采用集合描述的方法，对Rozvany关于结构拓扑优化设计奇异最优解的定义进行了重新描述。基于以上定义，本文研究了桁架结构拓扑优化设计的可行域，证明了对于截面尺寸下限为零，且无尺寸上限的桁架结构受应力约束的拓扑优化设计问题，其设计空间不同拓扑的可行子域总是连通的，同时也给出了对于具有尺寸下限约束、具有局部稳定性约束的桁架结构拓扑优化设计的可行子域不连通的实例。

A Study on Feasible Domain in Topology Optimization of Trusses

Some definitions, suck as k dimensional feasible sub-domain, neighbor feasible sub-domain, k dimensional connecting feasible sub-domain and k dimensional singular feasible sub-domain, were proposed. Based on these definitions, the definition of singular optima in topology optimization of structure, which was given by Rozvany, was modified. Furthermore, the feasible domain in topology optimization of trusses was analyzed. The sufficient and necessary condition for the connectivity of sub-domain was given, and it is proved that the sub-domains of different topologies in design space are always connected for the topology optimization of trusses with stress constraints, zero sectional area lower limit constraints and without the upper limit constraints of sectional area. Simultaneously, some examples with non-connected sub-domains were presented, which related to the topology optimizations of trusses with lower bound constraints and local buckling constraints.