Combinatorial Problems that Make
Hex-Meshing Hard

Kilian Verhetsel

Why do we care?

  • Subdividing the above into hex would be nice
    • Requires mesh with a very specific boundary Schneiders 1995

Conditions for Topological Meshes

  • Degenerate example
    • can still be used for topological invariant computation
  • Not useful for numerical computations
    • Need rules to exclude these cases

Rule Ⅰ: Distinct faces

We refer to elements by their vertices:

  • Edges connect 2 distinct vertices
  • Quads bounded by 4 distinct edges
  • Cubes bounded by 6 distinct quads

Rule Ⅱ: Intersections

Two distinct cubes can share:

  • Nothing
  • A vertex
  • An edge
  • A quadrangular facet

Rule Ⅲ: Interior & Boundary Faces

  • Boundary facet: in exactly one hex
  • All other facets: in exactly two hex

Topological Rules

Which mesh respects our rules?

Homology-Based Rules

  • For disks: every cycle bounds a set of quads
  • For balls: every closed surface bounds a set of hexahedra
  • Useful rule, though still insufficient

Is it even possible?

Yes if:

  1. Even number of quads
    Thurston 1993, Mitchell 1995
  2. No interior surface bounded by
    odd number of edges Erickson 2014

How do we build a solution?

  • Subdivide a tet-mesh into hex

Constructing a Solution

  • The mesh is almost complete…
  • Only issue: boundary was refined

Finishing the Construction:
Buffer Cells

  • Each buffer cell must have even number of quads
    • Need two variants

Meshing the Buffer Cells

It is an amusing exercise to fill out these cases by hand Eppstein 1999
It is not difficult to construct explicit hex meshes for these subdivided cubes by hand Erickson 2014

First explicit solution:
76 881 hex Carbonera 2010, Weill 2016

Meshing the Buffer Cells, Practically

  • $\le 40$ hex, but too hard for humans to find

Attempt Ⅰ: Try Everything

Verhetsel 2018

  1. Pick a boundary quad $Q$
  2. Try every hex containing $Q$

Problems with a Simple Approach

  • Non-manifold boundary after hex insertion
  • Difficult to maintain topological invariants

Attempt Ⅱ: A nicer construction

  • Boundary always remains a sphere
    Müller-Hanneman 1999, Xiang 2018, Verhetsel 2019
  • Equivalence between boundary transformation and hex insertion

Right Before a Solution…

  • The end of the search is always the same

Remembering Small Meshes Helps

  1. Precompute all meshes with up to $n$ hexahedra
  2. Compare them with unmeshed region
    • Skips the last $n$ levels of the search tree

Remembering Small Meshes Helps a Lot

  • Precomputed solutions don't always fit
  • We skip more than $n$ levels!

Meshing Small Inputs

  • Topological mesh computed for all small quadrangulations ($n \le 20$)
    • Most are harder than the above

Hex-Dominant Meshing

  • Subdividing tets and pyramids generates bad elements
  • Can compute mesh of larger cavity instead

Combinatorial Constraints in
Block Decompositions

  • Compute block structure from boundary decomposition
  • Some boundaries are much harder than others

Thank You!