localized delaunay refinement for volumes

1/27Department of Computer Science and Engineering

Localized Delaunay Refinement for Volumes

Tamal K Dey and Andrew G Slatton

The Ohio State University


Problem

• Input: Volume O bounded by smooth 2-manifold ∂O

• Output: Tetrahedral mesh approximates O

• Constraints: Use a localized framework


Restricted Delaunay

• Del S|M: Collection of Delaunay simplices t where Vt intersects M


• Traditional refinement maintains Delaunay triangulation in memory

• This does not scale well• Causes memory thrashing

• May be aborted by OS

Limitations


• A simple algorithm that avoids the scaling issues of the Delaunay triangulation• Avoids memory thrashing

• Topological and geometric guarantees

• Guarantee of termination

• Potentially parallelizable

Our Contribution


Basic Approach

• Divide sample in octree and refine each node individually• [DLS10]

• Applying to volumes• [DLS10] and [ORY05]

• New challenges


Difficulties: Consistency

• Without some additional processing, meshes will not fit consistently across boundaries• Addressed in [DLS10]


Difficulties: Termination

• Arbitrarily close insertions• Addressed in [DLS10]


• Sample points must lie in bounded domain• Not a problem in

[DLS10]

• Outside vs inside

New Difficulties


New Difficulties

• All vertices of restricted triangles must lie on ∂O

• May lead to arbitrarily dense refinement


Algorithm

• Parameters:• λ

• Sizing • Sample density• Approximation quality

• κ• Points per node


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


Node Processing

• Split • Do when |P| > κ, where P = P ∩

• Divide P among children of


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 UpϵPTp


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


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


Reprocessing

• Re-enqueue ’ if ≠ ’ inserts new point q in P’ or N’

• Necessary for consistency


Output

• Output UpϵPTp

• Union of all UpϵPTp over all nodes in octree


Termination

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


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]


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)


Results


Closing Remarks

• Key observations• Localized beats non-localized

• We are faster than CGAL

• Shortcomings• Slivers

• Future work• Sliver elimination

• Piecewise-smooth complexes


Thank You!

localized delaunay refinement for volumes

Documents