NeurFrame: Learning Continuous Frame Fields for Structured Mesh Generation

Abstract

Structured meshes, composed of quadrilateral elements in 2D and hexahedral elements in 3D, are widely used in industrial applications and engineering simulations due to their regularity and superior accuracy in finite element analysis. Generating high-quality structured meshes, however, remains challenging, especially for complex geometries and singularities. Field-guided approaches, which construct cross fields in 2D and frame fields in 3D to encode element orientation, are promising but are typically defined on discrete meshes, limiting continuity and computational efficiency. To address these challenges, we introduce \emph{NeurFrame}, a neural framework that represents frame fields continuously over the domain, supporting infinite-resolution evaluation. Trained in a self-supervised manner on discrete mesh samples, NeurFrame produces smooth, high-quality frame fields without relying on dense tetrahedral discretizations. The resulting fields simultaneously guide high-quality quadrilateral surface meshes and hexahedral volumetric meshes, with fewer and better-distributed singularities. By using a single network, NeurFrame also achieves lower computational cost compared to prior self-supervised neural methods that jointly optimize multiple fields.

Figures (13)

• Figure 1: The field-guided structured mesh generation pipeline begins with an input triangular or tetrahedral mesh. A cross or frame field is optimized on the surface or volume to guide parameterization, aligning the mesh elements with the field directions. The structured mesh is then extracted from this parameterization. Compared to other quadrilateral or hexahedral mesh generation methods, such as the triangle-merge Remacle-2012 and octree-based Gao-2019 approaches, field-guided methods yield higher-quality meshes with fewer singularities (red and blue) and a more balanced singularity distribution.
• Figure 2: Spherical harmonic (SH) representation of a frame. The leftmost image shows a frame $\mathbf{V}$ together with its associated spherical function $F_{\mathbf{V}}$ on the unit sphere, where the function minima indicate the frame directions. The remaining plots show the projection of $F_{\mathbf{V}}$ onto the SH basis functions $Y_{0,0}$ in band 0 and $Y_{4,i}$ in band 4, with $i=-4,\dots,4$. The color scale encodes function values, from minimum (dark) to maximum (bright).
• Figure 3: Scaling the input surface or volumetric mesh changes the number of quadrilateral or hexahedral elements while preserving the singularity structure defined by the cross or frame field.
• Figure 4: Comparison of quadrilateral meshes generated by feature-aligned methods. Feature edges are identified as mesh edges whose adjacent face normals differ by more than $\pi/4$ and are marked in red. Singular points, i.e., vertices with valence not equal to four, are shown in red (higher valence) and blue (lower valence).
• Figure 5: Comparison of quadrilateral meshes guided by cross fields generated using different methods. In challenging saddle-shaped regions, such as the cat’s back, only NeurCross achieves alignment comparable to NeurFrame. For fine-scale features, including the cat’s ears, only the polyvector field and Ginzburg-Landau field produce cross fields with alignment quality similar to ours.