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

# 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:

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