连续体结构破损-安全拓扑优化预估失效区域的理性准则

本文指出，连续体结构拓扑优化研究方向，从2016年Zhou的研究开始，进入了一个新阶段：离散结构的设计继续从连续体拓扑优化的结果获益，同时连续体拓扑优化的理论可以从离散结构的拓扑优化中受益。借此新阶段，本文针对连续体结构破损-安全的拓扑优化问题，通过几何分析途径，建立了预估破损区域的理性准则，即给出了结构局部破损模式尺寸上、下限及相邻局部破损区域的间距上、下限。以此理性准则，分析了Janson及Zhou给出的破损区域预估分布策略的各自优缺点，并通过算例对于相关策略进行了验证。结果表明：Janson的策略过于保守而导致不必要的极大计算量；Zhou的无缝平铺的策略不能保证所有拓扑生成离散元件通过破损测试，但在多数情况下仍可以得到具有足够冗余的最优拓扑；本文提出的以满足理性准则的方式布置破损区域，可以保证所有拓扑生成离散元件通过破损测试，并保证得到更多冗余的最优拓扑。本文的研究表明，预估破损区域的理性准则条件，为连续体结构破损-安全拓扑优化问题，提供了表述局部破损区域的理论进展。

Rational Criterion of Pre-estimation Distribution of Failure Regions for Fail-safe Topology Optimization of Continuum Structures

In this paper，it is pointed out that the topology optimization of continuum structures had entered a new stage since Zhou’s research in 2016. The new stage includes two points: (1) Designs of discrete structures have been benefitted from the results of the topology optimization of continuum structures. (2)At the same time, the theory of topology optimization of continuum structures can benefit from the topology optimization of discrete structures. The rational criterion of the pre-estimation distribution of failure regions is established based on geometrical analysis, that is, the upper limit and lower limit of the size of the local failure mode and the upper limit and lower limit of the space of adjacent local failure regions are given. The advantages and disadvantages of the pre-estimation distribution strategies of failure regions proposed by Janson and Zhou are analyzed according to the proposed rational criterion. Numerical examples are given to verify the relevant strategies. The results show that Janson’s strategy is too conservative to lead to unnecessary computation. Zhou's strategy with the gapless fill of failure regions can’t guarantee that all discrete members yielded by topology optimization pass the failure test, but in most cases, the optimal topology with sufficient redundancy can be obtained. While the pre-estimation distribution of failure regions satisfies the proposed rational criterion, all discrete members yielded by topology optimization can be ensured to pass the failure test and optimal topology with more redundant can be achieved. The study shows that the rational criterion for the pre-estimation distribution of failure regions provides theoretical progress in describing local failure regions for the failure-safety topology optimization continuum structures.