Graph neural networks (GNNs) have emerged as a versatile and efficient option for modeling the dynamic behavior of deformable materials. While GNNs generalize readily to arbitrary shapes, mesh topologies, and material parameters, existing architectures struggle to correctly predict the temporal evolution of key physical quantities such as linear and angular momentum. In this work, we propose MomentumGNN—a novel architecture designed to accurately track momentum by construction. Unlike existing GNNs that output unconstrained nodal accelerations, our model predicts per-edge stretching and bending impulses which guarantee the preservation of linear and angular momentum. We train our network in an unsupervised fashion using a physics-based loss, and we show that our method outperforms baselines in a number of common scenarios where momentum plays a pivotal role.

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.

Cellular patterns, from planar ornaments to architectural surfaces and mechanical metamaterials, blend aesthetics with functionality. Homogeneous patterns like isohedral tilings offer simplicity and symmetry but lack flexibility, particularly for heterogeneous designs. They cannot smoothly interpolate between tilings or adapt to double-curved surfaces without distortion. Voronoi diagrams provide a more adaptable patterning solution. They can be generalized to star-shaped metrics, enabling diverse cell shapes and continuous grading by interpolating metric parameters. Martínez et al. [2019] explored this idea in 2D using a rasterization-based algorithm to create compelling patterns. However, this discrete approach precludes gradient-based optimization, limiting control over pattern quality. We introduce a novel, closed-form, fully differentiable formulation for Voronoi diagrams with piecewise linear star-shaped metrics, enabling optimization of site positions and metric parameters to meet aesthetic and functional goals. It naturally extends to arbitrary dimensions, including curved 3D surfaces. For improved on-surface patterning, we propose a per-sector parameterization of star-shaped metrics, ensuring uniform cell shapes in non-regular neighborhoods. We demonstrate our approach by generating diverse patterns, from homogeneous to continuously graded designs, with applications in decorative surfaces and metamaterials.

Computing intrinsic distances on discrete surfaces is at the heart of many minimization problems in geometry processing and beyond. Solving these problems is extremely challenging as it demands the computation of on-surface distances along with their derivatives. We present a novel approach for intrinsic minimization of distance-based objectives defined on triangle meshes. Using a variational formulation of shortest-path geodesics, we compute first and second-order distance derivatives based on the implicit function theorem, thus opening the door to efficient Newton-type minimization solvers. We demonstrate our differentiable geodesic distance framework on a wide range of examples, including geodesic networks and membranes on surfaces of arbitrary genus, two-way coupling between hosting surface and embedded system, differentiable geodesic Voronoi diagrams, and efficient computation of Karcher means on complex shapes. Our analysis shows that second-order descent methods based on our differentiable geodesics outperform existing first-order and quasi-Newton methods by large margins.

Navigating topological transitions in cellular mechanical systems is a significant challenge for existing simulation methods. While abstract models lack predictive capabilities at the cellular level, explicit network representations struggle with topology changes, and per-cell representations are computationally too demanding for large-scale simulations. To address these challenges, we propose a novel cell-centered approach based on differentiable Voronoi diagrams. Representing each cell with a Voronoi site, our method defines shape and topology of the interface network implicitly. In this way, we substantially reduce the number of problem variables, eliminate the need for explicit contact handling, and ensure continuous geometry changes during topological transitions. Closed-form derivatives of network positions facilitate simulation with Newton-type methods for a wide range of per-cell energies. Finally, we extend our differentiable Voronoi diagrams to enable coupling with arbitrary rigid and deformable boundaries. We apply our approach to a diverse set of examples, highlighting splitting and merging of cells as well as neighborhood changes. We illustrate applications to inverse problems by matching soap foam simulations to real-world images. Comparative analysis with explicit cell models reveals that our method achieves qualitatively comparable results at significantly faster computation times.