Star-Shaped Distance Voronoi Diagrams for 3D Metamaterial Design

Abstract

3D cellular metamaterials are valued for many unique and useful mechanical properties. They enable lightweight, high-strength structures, with a wide range of directional stiffness profiles and possible auxetic behaviour. Infill patterns based on triply-periodic minimal surfaces (TPMS) are commonly used in additive manufacturing due to their high strength-to-weight ratio and near-isotropic mechanical behaviour. While existing work provides a wide range of cellular metamaterials to choose from, optimization of these patterns remains a significant challenge due to the diverse space of possible surface topologies and the lack of a unified parameterization. As a promising alternative, Voronoi diagrams with star-shaped distance metrics have been shown to provide a continuous parameterization of 2D cellular metamaterials, opening a rich space of possible designs. Extending the work of Zhou et al. 2025, we provide a novel, differentiable construction of 3D volumetric Voronoi diagrams with star-shaped metrics. We integrate our formulation into a complete pipeline for mechanical metamaterial optimization, demonstrating the flexibility of star-shaped metric Voronoi diagrams to create periodic structures with a diverse range of directional stiffness profiles and stress-strain curves. Furthermore, we demonstrate the applicability of this framework to heterogeneous, smoothly graded cellular structures.