  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.
