Interleaving Delaunay Refinement and Optimization for 2D Triangle Mesh Generation

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Interleaving Delaunay Refinement and Optimization
for 2D Triangle Mesh Generation

• Jane Tournois, Pierre Alliez, Olivier Devillers.

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Outline

• Introduction
• State of the Art
• Overview
• Algorithm
• Refinement
• Optimization
• Results
• Discussion future work

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The problem

• 2D Triangle mesh
• Smallest triangulation with good elements
• Respect the constraints

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State of the Art DELAUNAY REFINEMENT

• Rupperts algorithm Ruppert, 1995

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State of the Art MESH OPTIMIZATION

• Eppstein, 2001
• Well spaced points
• Bounded aspect ratios (? angles)
• Low number of points

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State of the Art MESH OPTIMIZATION

• Why using Lloyds method?

(but)Du et al.,1999 CVT is a local minimum.
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State of the Art MESH OPTIMIZATION

• Why using Lloyds method?
• Centroidal Voronoi Tessellation Du et al., 1999
• 1D and 2D simultaneous optimization

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Overview
Input PSLG
Refinement
Compute new sizing
Smoothing
Output final mesh
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Outline

• Introduction
• State of the Art
• Overview
• Algorithm
• Refinement
• Optimization
• Results
• Discussion future work

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Algorithm

• Input A PSLG domain O
• repeat
• Batch refinement in CDT(O)
• (Compute (new) sizing criterion)
• repeat
• Smoothing of CDT(O) by Lloyds iterations
• until a stopping criterion S
• until no new insertion
• repeat Smoothing of CDT(O) by Lloyds iterations
• until a stronger stopping criterion S
• Output The final triangle mesh

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Algorithm
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REFINEMENT Steiner points

• Inserts Steiner points
• 1D constrained edges midpoints
• 2D faces circumcenters

Refine BIG simplices wrt to current target sizing
ratio
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REFINEMENT Sizing function
k 5
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REFINEMENT Sizing ratio
1D simplex
2D simplex
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REFINEMENT Algorithm

• 1. Compute the

2. Choose the simplices to be refined? list of
Steiner points
3. Insert the Steiner points in batch
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OPTIMIZATION
Lloyd iteration (in 1D 2D)
Over a

• Bounded Voronoi Diagram
• its Utility
• its Construction

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OPTIMIZATION

• Lloyd iteration

Move the generators to the centroids of the
Voronoi cells
In 1D(? uniform)
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OPTIMIZATION

• Lloyd iteration

Move the generators to the centroids of the
Voronoi cells
In 2D(? uniform)
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OPTIMIZATION

• Lloyd iteration

Move the generators to the centroids of the
Voronoi cells
Simultaneous 1D 2D (? uniform)
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OPTIMIZATION

• Lloyd iteration

Move the generators to the centroids of the
Voronoi cells
Simultaneous 1D 2D (? uniform)
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OPTIMIZATION

• Lloyd iteration

Move the generators to the centroids of the
Voronoi cells
Simultaneous 1D 2D (? uniform)
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Why?
Bounded Voronoi Diagram
 Ordinary  Voronoi Diagram
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each constrained edge in the triangulation1.
Tag the  blind  triangles (predicate)
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each vertex in the triangulation2.
Construction of its bounded cell, circulating on
its incident faces.
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each vertex in the triangulation2.
Construction of its bounded cell, circulating on
its incident faces.
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each vertex in the triangulation2.
Construction of its bounded cell, circulating on
its incident faces.
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each vertex in the triangulation2.
Construction of its bounded cell, circulating on
its incident faces.
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each vertex in the triangulation2.
Construction of its bounded cell, circulating on
its incident faces.
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OPTIMIZATION
Bounded Voronoi Diagram Seidel, 1988
Construction algorithm
For each vertex in the triangulation2.
Construction of its bounded cell, circulating on
its incident faces.
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Outline

• Introduction
• State of the Art
• Overview
• Algorithm
• Refinement
• Optimization
• Results
• Discussion future work

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Comparing DR and our algorithm(1)
sizing ?0.05
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Comparing DR and our algorithm(1)
sizing ?0.05
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Comparing DR and our algorithm(1)
sizing ?0.05
-20 Steiner points
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Comparing DR and our algorithm(1)
sizing ?0.05
Our method
angles
Delaunay Refinement
0
60
120
180
-20 Steiner points
angles
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Comparing DR and our algorithm(2)
sizing ?0.01
CDT(Input PSLG) 166 vertices
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Comparing DR and our algorithm(2)
sizing ?0.01
Delaunay Refinement 8006 vertices
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Comparing DR and our algorithm(2)
sizing ?0.01
Our method 6076 vertices
-25 Steiner points
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Comparing DR and our algorithm(2)
sizing ?0.01
Our method
angles
Delaunay Refinement
0
60
120
180
-25 Steiner points
angles
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With an adaptive sizing function
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With an adaptive sizing function
k 0.4
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Outline

• Introduction
• State of the Art
• Overview
• Algorithm
• Refinement
• Optimization
• Results
• Discussion future work

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Contributions

• 2D triangle mesh
• Isotropic
• Respects constraints
• Interleaving

• Batch refinement
• Lloyds smoothing method (1D 2D)
• Bounded Voronoi Diagram
• Quadratures (1D 2D)

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Added value (compared to DR or DR Lloyd)

• Increased quality of triangles
• Fewer Steiner points
• ? uniform -28
• ? non-uniform -23
• Slowly decreasing sizing criterion

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Added value (compared to DR Lloyd)
Delaunay Refinement 1988 vertices
Delaunay Refinement Lloyd 1988 vertices
Our method 1519 vertices
-24 Steiner points
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Added value (compared to DR Lloyd)
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Added value (compared to DR Lloyd)
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Limitations

• Slower (100 times)
• ? needed (for refinement smoothing)
• Refinement ? Size of elementsSmoothing ? Shape
  of elements
• may need to run Delaunay Refinement
  once (without a sizing criterion)

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Limitations
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Future work
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Future work
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Thank you.