Design optimization of beam structures is a significant topic as beams are an efficient load-carrying component in engineering applications. Most of the earlier researches concentrate on the structural optimization of beams with invariant cross-section topology, and design optimization of periodic beam structures, which cross-section topology varies along the axial direction, has not been extensively investigated. This work presents a novel approach based on the relaxed Saint-Venant solution to conduct microstructural topology optimization of periodic beam structures for minimum structural compliance. One benefit of adopting Saint-Venant solution based compliance formulation is that the strain energy induced by transverse shear loading is incorporated in the structural compliance, which is not reflected in that based on first-order homogenization of the asymptotic homogenization (AH) theory, so that a more reasonable objective function is adopted for the optimization problem. In addition, a material connectivity constraint, which is constructed by restraining the ratio of the strain energy calculated from relaxed Saint-Venant solution to that obtained from first-order homogenization of the AH theory, is further proposed to prevent material separation or to strengthen material connection through the thickness direction. The detailed sensitivity analysis of the objective function and the constraints are carried out with the adjoint method, and several numerical examples are given to show the validity of the relaxed Saint-Venant solution based compliance formulation and the effectiveness of the proposed material connectivity constraint.

梁的设计优化是一个重要的主题，因为梁是工程应用中的有效承载组件。早期的研究大多集中在具有不变截面拓扑的梁的结构优化上，而对截面拓扑沿轴向变化的周期性梁结构的设计优化，还没有进行广泛的研究。这项工作提出了一种基于松弛Saint-Venant解决方案的新颖方法，可以对周期性梁结构进行微结构拓扑优化，以最小化结构的柔度。采用基于Saint-Vantant解决方案的顺应性公式的一个好处是，在结构顺应性中纳入了横向剪切载荷引起的应变能，这一点没有体现在基于渐近均质化（AH）理论的一阶均质化的问题上，因此对优化问题采用了更合理的目标函数。此外，还提出了一种材料连通性约束，该约束通过限制从松弛的Saint-Venant解计算得出的应变能与从AH理论的一阶均化获得的应变能之比构成，以防止材料分离或增强材料沿厚度方向连接。使用伴随方法对目标函数和约束条件进行了详细的灵敏度分析，