Topology optimization of hierarchical structures based on floating projection

Representative of the advanced engineering designs, hierarchical structures have attracted particular attention in academia and engineering. This study proposed a novel design strategy to obtain hierarchical structures with three layers. The macroscopic layer contained the structures to be optimized, which were periodically fashioned by multi-phase and multi-configuration mesoscopic structures. The configurations of the mesoscopic structures were determined by several classical types of material micro-structures at the microlayer. The micro-structures and mesoscopic structures were independent of optimization, while the equivalent elastic properties of the mesoscopic structures were evaluated using the homogenization method. The floating projection technique was employed to search for the final macro-structural topologies, where the minimum volume was used as the objective function when subjected to one or more displacement constraints. All constraints were added to the objective function through Lagrange multipliers, which were iteratively computed using the Newton-Raphson method. Moreover, a heuristic optimality criterion was established to stably update the design variables. Numerical examples were provided to reveal the effect of material properties and microscopic and mesoscopic structures on the design of hierarchical structures.