Three-scale concurrent topology optimization for the design of the hierarchical cellular structure

Natural structures and some researches about artificial three-level structures have demonstrated that these structures have good performance in many aspects. For obtaining such structures, the traditional two-scale design method needs very finer meshes, which lead to expensive computational costs. Therefore, our purpose is to propose an efficient topology optimization method for designing the three-level structure with excellent performance. The proposed method that is also called the three-scale design method in this paper divides the design domains into three scales, which are connected by the homogenization method. At each scale, the optimal material layout can be found by using the SIMP (Solid Isotropic Material with Penalization) method. Then, the proposed three-scale method is integrated into the topology optimization, two optimization strategies are provided to design the three-level structure. The first design strategy considers structural compliance as an optimization objective, which is usually common in multi-scale design. The decoupled sensitivity analysis method is used to improve the computational efficiency of this algorithm. Another effective strategy is to take buckling performance as the optimization objective, it can build a direct link between the good structural performance of the multi-level structure and optimization formula. Several numerical examples are provided to verify the effectiveness of the two design strategies. Meanwhile, the results of performance analysis show that adding a third scale does improve the performance of the structure in some aspects, such as buckling performance, robustness and ultra-light.