Hex Machina

Python implementation of the SRF approach to hexahedral meshing. (Work in Progress)

TODO

• The final volume parametrization step is not working (linear mixed-integer CG).
• Edge collapse adjustments need to be implemented.
• Move numerical optimizations to CUDA.
• Back-propagation to compute gradients in L-BFGS, hopefully improve runtime.
• Maybe run additional smoothing on surface cross-field.

Algorithm

Inputs : Triangle mesh as .stl (binary)

Given an input triangle mesh, a tetrahedral mesh is generated (using TetGen) and its boundary surface is extracted. The vertex-based curvatures and normals of the boundary surface are computed, as well as other topological information. A 3D frame is initialized at the centroid of each tetrahedron. The frame field smoothness is optimized by minimizing a non-linear energy function, solved using the efficient L-BFGS method. Moreover, some case-by-case adjustments are made to ensure the field is singularity-restricted. After the optimization, a volume parametrization is fit to the framefield as an atlas of linear maps defined on the vertices, using a CG (Conjuate Gradient) method. The hexahedral mesh is extracted based on the integer isosurface intersections.

Outputs: Saved to disk as .vtk files, which can be viewed in Paraview. This includes tetrahedral mesh, curvature cross-field, optimized 3D frame field, singular graph and the hexahedral mesh.

State of the Art

A non-exhaustive list of papers that inspired this implementation.

Volume parametrization

• All-Hex Meshing using Singularity-Restricted Field
• CubeCover - Parameterization of 3D Volumes
• Boundary Aligned Smooth 3D Cross-Frame Field

Boundary parametrization

• General Planar Quadrilateral Mesh Design Using Conjugate Direction Field
• QuadCover - Surface Parameterization using Branched Coverings
• Estimating Curvatures and Their Derivatives on Triangle Meshes
• Trivial Connections on Discrete Surfaces

Dependencies

Python 3, and set up other dependencies with:

pip install requirements.txt