delaunay triangulation and mesh generationstaff.ustc.edu.cn/~lgliu/courses/dgp_2012_spring... ·...
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Delaunay Triangulationand Mesh Generation
Ligang LiuGraphics&Geometric Computing Lab
USTChttp://staff.ustc.edu.cn/~lgliu
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Outline
• Concepts
• Delaunay Triangulation
• Optimal Delaunay Triangulation (ODT)
• Centroidal Vonoroi Tessellation (CVT)
• 3D mesh generation
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Boris N. Delaunay
• Russian mathematician
• March 15, 1890 ‐ July 17, 1980
• Introduce Delaunay triangulation in 1934
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Georgy F. Voronoi
• Russian mathematician
• April 28, 1868 ‐ November 20, 1908
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Properties of DT (1)
• Empty sphere property: no points inside the circum‐sphere of any simplex– Delaunay edge
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Properties of DT (2)
• DT maximizes the smallest angle– [Lawson 1977] and [Sibson 1978]
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Properties of DT (3)
• Convex hull: union of all triangles
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Properties of DT (4)
• DT maximizes the arithmetic mean of the radius of inscribed circles of the triangles. – [Lambert 1994]
• DT minimizes roughness (the Dirichlet energy of any piecewise‐linear scalar function)– [Rippa 1990]
• DT minimizes the maximum containing radius (the radius of the smallest sphere containing the simplex)– [Azevedo and Simpson 1989], [Rajan 1991]
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Properties of DT (5)
• The DT in d‐dimensional spaces is the projection of the points of convex hull onto a (d+1)‐dimensional paraboloid.– [Brown 1979]
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Properties of DT (6)
• DT minimizes the spectrum of the geometric Laplacian (spectral characterization)– [Chen et al. 2010]
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Edge Swapping/Flipping [Sibson 1978]
• Start with any triangulation– 1. find any two adjacent triangles that form a convex quadrilateral that does not satisfy empty sphere condition
– 2. swap the diagonal of the quadrilateral to be a Deluany triangulation of that four points
– 3. repeat step 1,2 until stuck.
Convergence? Is it possible to end with an infinite loop?
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Mesh Generation
• Given a fixed point set, Delaunay triangulation will try to make the triangulation more shape regular and thus is considered as a “good” unstructured mesh.
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Mesh Quality
• What do we mean a “good” mesh/simplex (triangle)? – Minimal angle
– Mean ratio
– Aspect/radius ratio
– …• It is not easy to define a universal mesh quality acceptable by
everyone. But everyone agrees on the "best" simplex: equilateral triangle and tetrahedra.
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DT is not necessary a good mesh
DT only optimize the connectivity when points are fixed. The distribution of points is more important for a good mesh.
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What is the usage of a mesh?
• One Answer: Mesh is used to approximate functions.
On the other hand, by choosing appropriate functions, we can get meshes with good geometric shape.
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Optimality of Delaunay Triangulation
[Chen and Xu 2004]
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Graph of the Linear Interpolation
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Convergence of Edge Swapping Algorithm
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Lifting Trick
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Qhull: a Global Algorithm to Construct DT
• Lift points to the parabolid
• Form the lowest convex hull in Rn+1
• Project the lowest convex hull to Rn
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Optimal Delaunay Triangulation (ODT)
• [Chen and Xu 2004] proved the existence of ODT. In general, it is not unique.
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However…
• ODT energy function is only C0 (piecewise smooth)– Both the point positions and its connectivity are the variables in the function
• Can not be solved using general numeric optimization methods
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Mesh Optimization based on ODT
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Optimize the Connectivity
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Optimize the Location
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Generalization: non‐uniform density
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Generalization: anisotropic metric
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Centroidal Voronoi Tessellation (CVT)
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• Definition: The VT is a centroidal Voronoi tessellation (CVT) , if each seed coincides with the centroid of its Voronoi cell
Centroidal Voronoi Tessellation
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• Definition: The VT is a centroidal Voronoi tessellation (CVT) , if each seed coincides with the centroid of its Voronoi cell
• CVT energy function:
• CVT is a critical point of , an optimalCVT is a global minimizer of
Centroidal Voronoi Tessellation
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• Theorem: The CVT energy with identical to 1, is the volume difference between the circumscribed polytope and the paraboloid.– An optimal CVT is a best circumscribed piecewise linear approximant to the paraboloid
Geometric Interpretation
i∗x
ix
j∗x
jx
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Lloyd Algorithm
• Construct the VT associated with the points
• Compute the centroids of the Voronoi regions
• Move the points to the centroids
• Iterate until convergent
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CVT Energy
• CVT energy is C2 smooth!– [Liu et al. 2009]
• Could be solved much faster than Lloyd algorithm
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Compare ODT and CVT
ODT
CVT
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Mesh Generation — 3DRegular tetrahedra can't tile the spaceTetrahedra classified by bad angles
well‐spaced points generate only round or sliver Delaunay tetrahedra [Eppstein 01]
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Comparison
40 slivers < 15 deg26 slivers < 10 deg7 slivers < 5 deg
0 sliver < 5 deg 4 slivers < 10 deg 16 slivers < 15 deg
ODT:4000 seeds21726 tets
CVT:4000 seeds20179 tets
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Tetrahedron Qualities• Dihedral Angle distribution • Radius ratio distribution
20o 140o 0.3 1.060o
CVTODT
65o90o
Radius ratio=3 in
cir
rR⋅
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3D Tetrahedral Mesh
• [Alliez et al. 2005, Tournois et al. 2009]
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Summary
• CVT– best circumscribed PL approximant
– compact Voronoi cells
– more regular triangles in 2D meshing
• ODT– best inscribed PL interpolant– compact simplices– Less slivers in 3D meshing
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Q&A