Design and Optimization of Functionally-graded Triangular Lattices for Multiple Loading Conditions

TL;DR

This work addresses designing functionally graded lattice infill that remains stiff under multiple loading conditions. It introduces a two-step pipeline: homogenization-based topology optimization over a simplified rank-3 laminate expressed as equilateral triangles with tunable edge thicknesses and a single orientation per cell, followed by a geometry-based de-homogenization using field-aligned triangulation to produce a globally consistent triangular lattice. The method achieves high stiffness and strong geometric regularity while reducing computational cost by operating at low-resolution design domains and encoding the outcome into a compact geometric format for editing and fabrication. The proposed approach outperforms uniform lattices and porous infill in stiffness and demonstrates robust performance across various boundary conditions, with potential extensions to 3D using tetrahedral analogs.

Abstract

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.

Figures (12)

• Figure 1: Method overview. (a) The optimization problem: Two loading conditions $F_1$ (orange) and $F_2$ (green) are applied, and the bottom is fixed. (b) The optimized density layout (background) and orientations (streamlines) generated by homogenization-based topology optimization. (c) The triangular parametrization with edges aligned to the tangents of the optimized rank-3 layers. The entire design domain and the optimized density layout are partitioned into sub-domains by this field-aligned mesh. (d) The triangular lattice design obtained via de-homogenization.
• Figure 2: Schematic composition of a rank-3 laminate and its geometric interpretation. (a) The design domain and boundary conditions for optimization, where each simulation cell is depicted by a set of specific specifications of a rank-3 laminate. (b) The multi-scale rank-3 laminate in a selected simulation cell. The material is composed of three differently oriented layers, shown in (c), (d), and (e), respectively. $\alpha$ is the relative width of each layer. $\bm{n}$ and $\bm{t}$ refer to the normal and tangent direction of a layer, respectively. (f) The proposed single-scale geometric interpretation of the rank-3 laminate as an equilateral triangle, with a unique thickness per edge.
• Figure 3: (a) Streamlines generated from the initial orientation field. (b) Optimization history of compliance and orientation regularization over the entire optimization process.
• Figure 4: (a) The domain discretization (background) and streamlines in the three different orientation fields of an optimized 6-RoSy field. (b) The randomly distributed positions (and respective directions) used for position optimization. The distribution is generated by considering a pre-defined edge length of the output mesh and the resolution of the input grid. (c) The position field after optimization and the representative positions (green dots) used to construct the output mesh. (d) The output mesh whose edges align with the optimized orientation field. Orientations are indicated by arrows, and edge colors distinguish corresponding layer orientations. To reduce visual clutter, only three of the six pairwise bidirectional 6-RoSy field directions are illustrated.
• Figure 5: Diagrammatic view of per-lattice de-homogenization. (a) The field-aligned triangular element with the covered cells. $\tilde{\theta}^j_i$ refer to the tangents of layers $i\in {1,2,3}$ in element $j$. Three arrow glyphs on each cell indicate the tangents. Corresponding width variables $\alpha^j_i$ are encoded into the arrow lengths. (b) The representative values of widths and orientations of the cells in the triangular element. (c) Relation of the representative values of orientations of three layers to the corresponding edges of the triangular element by detecting their orientation deviations. (d) The de-homogenized triangular lattice element whose unique edge thicknesses reflect the representative values of widths. The solid regions correspond to the deposition ratio indicated in (a).