DDPM-Polycube: a denoising diffusion probabilistic model for polycube-based hexahedral mesh generation and volumetric spline construction

Abstract In this paper, we propose DDPM-Polycube, a generative polycube method based on denoising diffusion probabilistic models (DDPM) for generating hexahedral (hex) meshes and constructing volumetric splines. Unlike DL-Polycube methods that depend on large libraries of predefined polycube structure templates and on learning explicit mappings from input geometries to those templates, DDPM-Polycube learns the deformation from the input geometry to its corresponding polycube structures as a denoising task. The model is trained on a template set consisting of two geometric primitives, a cube and a cube with a hole, together with a single combination type. By learning the deformation characteristics from this template set, the DDPM-Polycube model progressively generates polycube structures from the input geometry by removing Gaussian noise. Once valid polycube structures are generated, they are used for parametric mapping to generate hex meshes. Truncated hierarchical B-splines are then applied to construct volumetric splines that satisfy the requirements of isogeometric analysis (IGA). Experiments on models up to genus 2 demonstrate that DDPM-Polycube model can directly generate polycube structures from input geometries, even when the topology of these geometries falls outside its trained range. Overall, this research shows the potential of diffusion models in advancing mesh generation and IGA applications.