1. Localized Delaunay Refinement for Volumes
Tamal K Dey and Andrew G Slatton
The Ohio State University

2. Problem Input: Volume O bounded by smooth 2-manifold ∂O
Output: Tetrahedral mesh approximates O
Constraints: Use a localized framework

3. Restricted Delaunay
Del S|M: Collection of Delaunay simplices t where Vt intersects M

4. Limitations
Traditional refinement maintains Delaunay triangulation in memory
This does not scale well
Causes memory thrashing
May be aborted by OS

5. Our Contribution
A simple algorithm that avoids the scaling issues of the Delaunay triangulation
Avoids memory thrashing
Topological and geometric guarantees
Guarantee of termination
Potentially parallelizable

6. Basic Approach
Divide sample in octree and refine each node individually
[DLS10]
Applying to volumes
[DLS10] and [ORY05]
New challenges

7. Difficulties: Consistency
Without some additional processing, meshes will not fit consistently across boundaries
Addressed in [DLS10]

8. Difficulties: Termination
Arbitrarily close insertions
Addressed in [DLS10]

9. New Difficulties Sample points must lie in bounded domain
Not a problem in [DLS10]
Outside vs inside

10. New Difficulties All vertices of restricted triangles must lie on ∂O
May lead to arbitrarily dense refinement

11. Algorithm Parameters: λ κ Sizing Sample density Approximation quality
Points per node

12. Algorithm: Overview Add octree root to processing queue
Process node at head of queue
May split into new nodes or re-enqueue some existing nodes
Repeat this step until queue is empty

13. Node Processing Split Do when |P| > κ, where P = P ∩ 
Divide P among children of 

14. Node Processing Refine
Do while |P| ≤ κ
Initialize node  with Del(R) = Del(P U N)
When a node  is not being refined, keep only P and UpϵPTp

15. Refinement Criteria Restricted triangle size, rf < λ
Vertices of restricted triangles lie on ∂O
Topological disk
Voronoi edge intersects at most once
Tetrahedron size, rt < λ
Radius-edge ratio, ratio < 2

16. Point Insertion Strategy is similar to that in [DLS10]
Termination
Key difference: We may delete some points after an insertion
Topological guarantees
Does not prevent termination

17. Reprocessing Re-enqueue ’ if  ≠ ’ inserts new point q in P’ or N’
Necessary for consistency

18. Output
Output UpϵPTp
Union of all UpϵPTp over all nodes  in octree

19. Termination Theorem 1: The algorithm terminates.
Use a packing argument to prove this

20. Termination Each refinement criterion implies some LB
(C1)→d’ss ≥ min{dss,λ} and d’sv ≥ min{dsv,λ}
(C2)→d’ss ≥ min{dss,dsv/2,dvv/2} and d’sv ≥ min{dsv,λ}
(C3)→d’ss ≥ min{dss,λ∂O} and d’sv ≥ min{dsv,λ}
(C4)→d’ss ≥ min{dss,λ*} and d’sv ≥ min{dsv,λ}
(C5)→d’sv ≥ min{dsv,λ} and d’vv ≥ min{dvv,λ}.
(C6)→d’sv,d’vv ≥ min{2dss,dsv,dvv} or d’ss ≥ min{dss,dsv,dvv}.
Apply results from [CDRR07], [BO05], [Dey06], [AB99]

21. Topology & Geometry Theorem 2: T=UpTp is subcomplex of Del P|O
∂T is a 2-manifold without boundary
Output is no more then λ distance from O
For small λ:
T is isotopic to O
Hausdorff distance O(λ2)

22. Results

23. Results

24. Results

25. Closing Remarks Key observations Shortcomings Future work
Localized beats non-localized
We are faster than CGAL
Shortcomings
Slivers
Future work
Sliver elimination
Piecewise-smooth complexes

26. Thank You!