What a polyhedral cell is
A polyhedral mesh fills the volume with irregular many-faced cells — a typical polyhedron has on the order of 12–14 faces, versus 4 for a tetrahedron and 6 for a hexahedron. Most poly meshers build these cells as the dual of an underlying tetrahedral mesh, agglomerating the tets around each node into a single polyhedron.
Why more faces help
Each face is where the solver exchanges fluxes with a neighbor, so more faces per cell means more neighbors — and that has real consequences:
- Better gradient accuracy: gradients reconstructed from ~14 neighbors are more robust than from a tet’s 4, so results are less sensitive to cell orientation.
- Fewer cells: as an industry rule of thumb, poly meshes often need roughly 3–5× fewer cells than tets for comparable accuracy.
- Better convergence: the extra connectivity smooths the discretization and often lowers iteration counts.
- Less skewness sensitivity: polyhedra tolerate irregular geometry more gracefully than stretched tets.
The trade-offs
Polyhedra are not free:
- Each cell stores more faces and neighbors, so memory and per-cell cost are higher.
- Construction is more complex than tet meshing, and not every solver or post-processor handles arbitrary polyhedra equally well.
- Building good anisotropic prism layers into a poly workflow is its own challenge.
Where poly shines
Polyhedral meshes are strongest on complex, dirty, real-world geometry where a clean hex mesh is impractical and a tet mesh would be enormous — external aerodynamics, manifolds and ducting, porous and subsurface domains, and multi-part assemblies.
How AutoMesh-Geo helps
AutoMesh-Geo produces polyhedral (Voronoi) cells that conform to faults and features, aiming to keep the gradient-accuracy and cell-count advantages of poly meshing while respecting the geometry that usually forces compromises. See Voronoi vs Delaunay for the underlying method.