Aligning lattices based on local stress distribution is crucial for achieving exceptional structural stiffness. However, this aspect has primarily been investigated under a single load condition, where stress in 2D can be described by two orthogonal principal stress directions. In this paper, we introduce a novel approach for designing and optimizing triangular lattice structures to accommodate multiple loading conditions, which means multiple stress fields. Our method comprises two main steps: homogenization-based topology optimization and geometry-based de-homogenization. To ensure the geometric regularity of triangular lattices, we propose a simplified version of the general rank-$3$ laminate and parameterize the design domain using equilateral triangles with unique thickness per edge. During optimization, the thicknesses and orientation of each equilateral triangle are adjusted based on the homogenized properties of triangular lattices. Our numerical findings demonstrate that this proposed simplification results in only a slight decrease in stiffness, while achieving triangular lattice structures with a compelling geometric regularity. In geometry-based de-homogenization, we adopt a field-aligned triangulation approach to generate a globally consistent triangle mesh, with each triangle oriented according to the optimized orientation field. Our approach for handling multiple loading conditions, akin to de-homogenization techniques for single loading conditions, yields highly detailed, optimized, spatially varying lattice structures. The method is computationally efficient, as simulations and optimizations are conducted at a low-resolution discretization of the design domain. Furthermore, since our approach is geometry-based, obtained structures are encoded into a compact geometric format that facilitates downstream operations such as editing and fabrication.

翻译：基于局部应力分布对齐晶格结构对实现优异结构刚度至关重要。然而，这一特性目前主要针对单一载荷条件（此时二维应力可由两个正交主应力方向描述）开展研究。本文提出了一种新颖的三角晶格结构设计与优化方法，可适用于包含多个应力场的多载荷条件。该方法包含两个主要步骤：基于均匀化的拓扑优化与基于几何的去均匀化。为确保三角晶格的几何规则性，我们提出了广义秩-3层压结构的简化版本，并通过每条边具有独立厚度的等边三角形对设计域进行参数化。在优化过程中，基于三角晶格的均匀化特性调整每个等边三角形的厚度与取向。数值结果表明，该简化方案仅导致刚度轻微下降，同时获得了几何规则性显著的三角晶格结构。在基于几何的去均匀化阶段，我们采用场对齐三角剖分方法生成全局一致的三角网格，其中每个三角形依据优化后的取向场定向。与单一载荷条件下的去均匀化技术类似，本方法在处理多载荷条件时，能够生成高度精细化且具有空间变异性优化的晶格结构。由于仿真与优化均在低分辨率的设计域离散网格上进行，该方法具有高效的计算性能。此外，基于几何的特性使所得结构可编码为紧凑的几何格式，便于后续编辑与制造等下游操作。