DDPM-Polycube: A Denoising Diffusion Probabilistic Model for Polycube-Based Hexahedral Mesh Generation and Volumetric Spline Construction

TL;DR

DDPM-Polycube introduces a diffusion-based framework to deform input geometries into polycube structures for hex-mesh generation and volumetric spline construction. By learning deformations from simple primitives and using a non-standard Gaussian noise term, the method directly predicts valid polycube topologies without reliance on predefined templates, enabling robust generalization to complex engineering geometries. The pipeline couples surface segmentation with parametric mapping to produce high-quality all-hex meshes, followed by TH-spline3D-based volumetric splines that support isogeometric analysis and BEzier extraction for solver integration. Experimental results show successful generation across multiple genus levels, high-quality hex meshes with favorable Jacobian metrics, and compatible IGA outputs, underscoring the approach’s potential to streamline CAD-to-IGA workflows and broaden diffusion-model applications in computational geometry.

Abstract

In this paper, we propose DDPM-Polycube, a generative polycube creation approach based on denoising diffusion probabilistic models (DDPM) for generating high-quality hexahedral (hex) meshes and constructing volumetric splines. Unlike DL-Polycube methods that rely on predefined polycube structure templates, DDPM-Polycube models the deformation from input geometry to its corresponding polycube structures as a denoising task. By learning the deformation characteristics of simple geometric primitives (a cube and a cube with a hole), the DDPM-Polycube model progressively reconstructs polycube structures from input geometry by removing non-standard Gaussian noise. Once valid polycube structures are generated, they are used for surface segmentation and parametric mapping to generate high-quality hex meshes. Truncated hierarchical B-splines are then applied to construct volumetric splines that satisfy the requirements of isogeometric analysis (IGA). Experimental results 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. This provides greater generalization and adaptability for diverse engineering geometries. Overall, this research shows the potential of diffusion models in advancing mesh generation and IGA applications.

Figures (7)

• Figure 1: The DDPM-Polycube pipeline. (a) Converting CAD geometries into triangular meshes and point clouds; (b) the DDPM-Polycube model; (c) the polycube structure generated by the DDPM-Polycube model; (d) an all-hex control mesh generated through octree subdivision, parametric mapping and quality improvement techniques, some elements are removed to show the interior; and (e) volumetric spline with IGA simulation results using ANSYS-DYNA. The diffusion model relies on two substeps: (1) a forward diffusion process that adds non-standard Gaussian noise to the point cloud data of the polycube structure, and (2) a reverse diffusion process that removes the noise to reconstruct the polycube structure.
• Figure 2: 9 geometric configuration types for dataset generation. The first 8 columns contain one geometric primitive, either a cube or a cube with a hole oriented along the $Z$, $X$, or $Y$ axis, placed in different positions. The 9th column represents composite structures formed by two primitives. These configurations, along with the one-hot encoding, provide the starting point for the forward diffusion process.
• Figure 3: Overview of the DDPM-Polycube model architecture. (a) Input point cloud features are preprocessed into a 3-channel format. (b) The ResNet module within the U-Net extracts and refines geometric features through residual connections. (c) U-Net employs skip connections between downsampling and upsampling modules for detailed feature preservation. (d) Timestep and context embeddings are fused with convolutional features to enhance noise modeling. (e) The output layer predicts the non-standard Gaussian noise. (f) The denoising process iteratively removes non-standard Gaussian noise from the input geometry to generate the polycube structure.
• Figure 4: Loss of the DDPM-Polycube model as a function of epochs.
• Figure 5: The DDPM-Polycube algorithm's capability to generate polycube structures for various geometric models with different genus levels, demonstrating its ability to adapt to complex topologies and generate polycube structures beyond its training set. Examples include genus-0 models (Dice, Old man), genus-1 models (Ring, Rod, Lantern), and genus-2 models (Eight). The reverse diffusion process starts from the input geometry ($x'_{500}$), progresses through latent variable models ($x'_{499}$, $x'_{479}$, $x'_{19}$), and gradually removes the noise to reconstruct the final polycube structure ($x'_{0}$).

Theorems & Definitions (6)

• Remark 1: Training data design
• Remark 2: Grid configuration design
• Remark 3: Non-standard Gaussian noise
• Remark 4: Computation of $\mathbf{q}$
• Remark 5: Reparameterization trick
• Remark 6: Reasoning behind the model's effectiveness