Topology optimization design of quasi-periodic cellular structures based on erode–dilate operators

Abstract

In this paper, a novel density-based topology optimization method for cellular structures with quasi-periodic microstructures is proposed. Here, ‘quasi-periodic’ is developed from periodic microstructures composed with a base unit cell, through gradually changing one or multiple alterable microstructural sizing/shape parameters. However, it is a challenge and key issue to identify the explicit alterable parameters in the density-based topology frame, since the microstructural topology is described by voxel densities. For solving this problem, an erode–dilate based method for describing the quasi-periodic microstructures is proposed. A family of quasi-periodic microstructures is formed by executing the erode–dilate operators on the base unit cell with different operator parameters. The topology optimization of quasi-periodic cellular structures is formulated as simultaneously optimizing the topology of the base unit cell and the volume fractions of macro-elements that correspond to the candidate microstructures. Furthermore, to avoid the small structural members being eliminated in the eroded process, the structural skeleton is extracted to modify the eroded process. Sensitivities of the structural compliance with respect to the two types of design variables are derived, and the gradient-based optimization method is applied to update the design variables. In the obtained results, the neighboring microstructures have similar topologies and varying volume fractions. The design space has been expanded compared with the periodic cellular structures, and the connectivity issue of the conventional heterogeneous microstructures has been resolved. The numerical examples validate the effectiveness of the proposed method, and the results show that the quasi-periodic cellular structures have better performances than single-scale structures and periodic cellular structures.