Periodic topology optimization has been suggested as an effective means to design efficient structures which address a range of practical constraints, such as manufacturability, transportability, replaceability and ease of assembly. This study proposes a new approach for design of finite periodic structures by allowing variable orientation states of individual unit-cells. In some design instances of periodic structures, the unit-cell may exhibit certain geometric features allowing multiple possible assembly orientations (e.g. facing up or down). For the first time, this work incorporates such assembly flexibility within the periodic topology optimization, which enables to greatly expand the conventional periodic design space and take more advantage of structural periodicity. Given its broad applications, a methodology for the design of more efficient periodic structures while maintaining the same degree of periodic constraint may be of significant benefit to engineering practice. In this study, several numerical examples are presented to demonstrate the effectiveness of this new approach for both static and vibratory criteria. Brute force analysis is also utilized to compare all possible assembly configurations for several periodic structures with a small number of unit-cells. A heuristic approach is suggested for selecting more beneficially oriented configurations in periodic structures with a large number of unit-cells for which an exhaustive search may be computationally infeasible. It is found that in all the presented cases, the oriented periodic structures outperform the conventional non-oriented (or namely translational) periodic counterparts. Finally, an educational MATLAB code is provided for replication of the design results in this paper.

周期性拓扑优化已被建议作为设计高效结构的有效手段，解决一系列实际约束，例如可制造性、可运输性、可替换性和易于组装。这项研究提出了一种通过允许单个晶胞的可变取向状态来设计有限周期结构的新方法。在周期性结构的一些设计实例中，单元格可能表现出某些几何特征，允许多个可能的组装方向（例如面朝上或朝下）。这项工作首次将这种装配灵活性纳入周期性拓扑优化中，从而大大扩展了传统的周期性设计空间，并更多地利用了结构周期性。鉴于其广泛的应用，在保持相同程度的周期约束的同时设计更有效的周期结构的方法可能对工程实践有很大的好处。在这项研究中，提出了几个数值例子来证明这种新方法对于静态和振动标准的有效性。蛮力分析也用于比较具有少量单元格的几个周期性结构的所有可能的装配配置。建议使用启发式方法在具有大量晶胞的周期性结构中选择更有利的定向配置，对于这些结构，穷举搜索可能在计算上不可行。发现在所有提交的案例中，定向周期结构优于传统的非定向（或平移）周期结构。最后，提供了用于复制本文设计结果的教学 MATLAB 代码。