Overcoming the e-sampling requirement for minimal-element, non-uniformal Volume Meshing, with applications in Medical Finite Element Analysis

1. 1Challenge the future
Topology-conform segmented volume
meshing of volume images

2. 2Challenge the future
A Review – the basic challenge

3. 3Challenge the future
A Review – the basic challenge
topologically wrong (deformed)
mesh
Unordinary mesh elements

4. 4Challenge the future
A Review – the basic challenge
   0,,  xxBESx 
3D Delaunay Triangulation
– ε-sample requirement:
Problem:
Sharp features
Corners
Edges
Medial Axis touches edges & corners
λ(x) = 0 => unlimited samples at
features
How can we mesh sharp features ?
?

5. 5Challenge the future
A Review – the basic challenge
Assumption:
Any conformal surface mesh
can be converted into a
volume mesh via 3D Delaunay
Tetrahedralization without
Steiner-Point insertion on the
surface
Local Triangulation of the
surface in 2D-tangent plane

6. 6Challenge the future
First Result ...

7. 7Challenge the future
Further concept
Reason for whole: Neighbourhood approximation (k-Nearest
Neighbours)
• nearest neighbours ≠ real neighbours (1-Ring
Neighbourhood) in non-uniformal samples
N(x) ϵ S, d(x, s) < μ • d(x, s1) kNN (k=6)

8. 8Challenge the future
Further concept
• Solution: Natural Neighbours
• reflecting 1-Ring Neighbourhood
• Centres of neighbouring Voronoi Cells  Delaunay Vertices
• Problem: Neighbours = 3D Delaunay Triangulation Vertices
 ε-sampling requirement ?!
No: only candidates needed, not exclusively Neighbours
 N(x) ϵ {NN1, NN2, ... NNn, p1, p2 ...} ≠ S

9. 9Challenge the future
Guideline for Proof of Concept
 practical proof of concept:
CH-LDT with subsequent 3D Delaunay Tetrahedralization can
generate volumes meshes with relaxed, sparse sample
requirement independent from ε-sample requirement
Local Delaunay Triangulation (LDT)
with Convex Hull-constraint