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.

本文提出了一种基于密度的准周期微结构蜂窝结构拓扑优化方法。在这里，“准周期”是通过逐渐改变一个或多个可变的微结构尺寸/形状参数，由由基本单位单元组成的周期性微结构发展而来的。然而，由于基于体素的微观结构拓扑结构是识别基于密度的拓扑结构框架中显式可变参数的一个挑战和关键问题。密度。为了解决这个问题，提出了一种基于腐蚀-扩张的方法来描述准周期的微观结构。通过在具有不同算子参数的基本单位单元上执行腐蚀-扩张算子，形成了准周期微结构族。拟定准周期细胞结构的拓扑优化，是为了同时优化基本单位单元的拓扑和对应于候选微结构的宏元素的体积分数。此外，为了避免在腐蚀过程中消除小的结构构件，提取结构骨架以修改腐蚀过程。得出关于两种类型的设计变量的结构顺应性的敏感性，采用基于梯度的优化方法更新设计变量。在获得的结果中，相邻的微结构具有相似的拓扑结构和变化的体积分数。与周期性的蜂窝结构相比，设计空间得到了扩展，并且解决了常规异质微结构的连通性问题。数值算例验证了所提方法的有效性，结果表明准周期胞状结构的性能优于单尺度结构和周期性胞状结构。常规异质组织的连通性问题已经解决。数值算例验证了所提方法的有效性，结果表明准周期胞结构比单尺度结构和周期性胞结构具有更好的性能。常规异质组织的连通性问题已经解决。数值算例验证了所提方法的有效性，结果表明准周期胞状结构的性能优于单尺度结构和周期性胞状结构。