三周期极小曲面（triply periodic minimal surface，TPMS）点阵结构因其优异的综合性能受到中外学者的广泛关注。在点阵结构实际应用过程中，常常需要对其进行优化设计以兼顾轻量化与承载性能两方面的要求。目前，对TPMS点阵结构的优化设计主要集中于密度梯度层面，未综合考虑载荷方向对其力学性能的影响。为此，首先研究了TPMS点阵结构的各向异性特征。基于平均场均匀化方法求解了不同类型TPMS点阵结构的等效弹性矩阵，通过Matlab插值计算，绘制了其在三维空间范围内的杨氏模量图。发现不同类型的TPMS点阵结构呈现出不同的各向异性特征，其中W点阵结构在[100]等轴线方向上性能较强，在[111]等斜向对角方向上性能较弱，而P点阵结构则刚好相反。根据TPMS点阵结构的各向异性，同时考虑主应力方向以及相对密度分布对其性能的影响，提出了TPMS点阵结构的密度梯度杂交优化设计方法。以悬臂梁模型为基础，基于载荷边界条件对其进行拓扑优化设计，并将拓扑优化密度云映射为点阵结构的相对密度分布，从而实现密度梯度设计。根据TPMS点阵结构的各向异性特征以及单元主应力方向分别选择W和P点阵单胞填充悬臂梁，使主应力方向位于点阵结构性能较强的方向，避免点阵结构在性能薄弱的方向承受较大的应力。将不同类型的TPMS点阵单元合理分布后，利用激活函数将它们进行杂交连接，实现结构梯度设计。综合相对密度分布和单元结构分布，生成密度梯度杂交点阵结构。采用有限元仿真方法对比分析优化设计前后点阵结构的承载性能，结果表明密度梯度W和P点阵结构的刚度与对应的均质点阵结构相比都有明显提高，而由W和P两种点阵单胞组成的密度梯度杂交点阵结构刚度最大，比密度梯度W和P点阵结构分别提高4.63%和33.63%。该结果表明在密度优化的基础上，根据承载时单元主应力方向将不同类型的点阵结构进行合理分布以及混合杂交设计能够进一步提高结构的整体刚度。建立的TPMS点阵结构密度梯度杂交优化方法为其在轻量化设计等方面的应用提供了一定的指导。

The triply periodic minimal surface (TPMS) lattice structures have attracted extensive attention from scholars worldwide. In practical applications, these lattice structures are typically designed optimally to meet the requirements of both lightweight and load-bearing capacity. However, current optimal designs for TPMS lattice structures are limited to density gradients, and the influence of loading directions on their mechanical properties has not been comprehensively considered. To address this gap, the anisotropic characteristics of TPMS lattice structures were investigated. Their equivalent elastic matrixes were calculated by using the homogenization method, and three-dimensional Young’s modulus diagrams were generated with Matlab. The results showed distinct anisotropy characteristics for different types of TPMS lattice structures. For instance, the W structure exhibited higher strength in the axial direction [100] and weaker strength in the diagonal direction [111]; whereas the P structure showed the opposite trend. Subsequently, an optimization design method was proposed, combining density gradient with hybridization, considering both density distribution and principal stress directions. The optimization process involved topology optimization of a cantilever beam structure, and mapping the obtained density cloud to the relative density distribution of the lattice structure. Based on the anisotropic characteristics of TPMS lattice structures, W and P lattice cells were selected to fill the cantilever beam, aligning the principal stress directions with the strong mechanical properties of the lattice cells. After reasonable distribution of TPMS lattice cells of different types, they were smoothly connected by an activation function. Finally, the relative density and lattice cell type distributions were combined to obtain a density-graded hybrid lattice structure. The load-bearing performances of lattice structures before and after optimization designs were compared through finite element analysis. The results showed that the stiffness of density gradient W and P lattice structures was significantly improved compared with uniform structures. Moreover, the stiffness of the graded hybrid lattice structure was the highest, surpassing the density gradient W and P lattice structures by 4.63% and 33.63%, respectively. This demonstrates that hybridization design, achieved through a reasonable distribution of different lattice cells according to principal stress directions, can further improve overall stiffness. The established optimization method, combining density gradient with hybridization for TPMS lattice structures, provides a guidance for their application in lightweight designs.

TH164

重庆市自然科学基金重点项目资助（cstc2020jcyj-zdxmX0021）。