anisotropic voronoi diagrams and guaranteed-quality anisotropic mesh generation françois labelle...
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Anisotropic Voronoi Diagrams and Guaranteed-Quality Anisotropic
Mesh Generation
François LabelleJonathan Richard Shewchuk
Computer Science Division University of California at Berkeley
Berkeley, California
Presented by Jessica Schoen
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Outline
Anisotropic meshes
Anisotropic Voronoi diagrams
Algorithm for anisotropic mesh generation
Current research
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I. Anisotropic Meshes
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What Are Anisotropic Meshes?Meshes with long, skinny triangles (in the right places).
Why are they important?•Often provide better interpolation of multivariate functions with fewer triangles.
•Used in finite element methods to resolve boundary layers and shocks. Source: “Grid Generation by the Delaunay
Triangulation,” Nigel P. Weatherill, 1994.
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Distance MeasuresMetric tensor Mp: distances & angles measured by p.
Deformation tensor Fp: maps physical to rectified space.
Mp = FpT
Fp.Physical Space
Fp Fq
FqFp-1
pq
p qFpFq
-1
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Distance MeasuresMetric tensor Mp: distances & angles measured by p.
Deformation tensor Fp: maps physical to rectified space.
Mp = FpT
Fp.Physical Space
Fp Fq
FqFp-1
Every point wants to be in a “nice” triangle in rectified space.
pq
p qFpFq
-1
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The Anisotropic Mesh Generation Problem
Given polygonal domain and metric tensor field M,
generate anisotropic mesh.
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A Hard Problem (Especially in Theory)
• Quadtree-based methods can be adapted to horizontal and vertical stretching, but not to diagonal stretching.
Common approaches to guaranteed-quality mesh generation do not adapt well to anisotropy.
• Delaunay triangulations lose their global optimality properties when adapted to anisotropy. No “empty circumellipse” property.
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Heuristic Algorithms forGenerating Anisotropic Meshes
Bossen-Heckbert [1996] George-Borouchaki [1998]
Li-Teng-Üngör [1999]Shimada-Yamada-Itoh [1997]
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II. Anisotropic Voronoi Diagrams
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Voronoi Diagram: DefinitionGiven a set V of sites in Ed, decompose Ed into cells. The cell Vor(v) is the set of points “closer” to v than to any other site in V.
Mathematically:
Vor(v) = {p in Ed: dv(p)≤ dw(p) for every w in V.}
distance from v to p as measured by v
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Distance Function Examples
1. Standard Voronoi diagram
dv(p) = || p – v ||2
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Distance Function Examples
2. Multiplicatively weighted Voronoi diagram
dv(p) = cv|| p – v ||2
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Distance Function Examples
3. Anisotropic Voronoi diagram
dv(p) = [(p – v)TMv(p – v)]1/2
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Anisotropic Voronoi Diagram
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Duality
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Two Sites Define a Wedge
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Dual Triangulation Theorem
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III. Anisotropic Mesh Generation
by Voronoi Refinement
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Easy Case: M = constant
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Easy Case: M = constant
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Voronoi Refinement Algorithm
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Voronoi Refinement Algorithm
Insert new sites on unwedged portions of arcs.
Islands
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Voronoi Refinement Algorithm
Insert new sites on unwedged portions of arcs.
Orphan
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Voronoi Refinement Algorithm
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Encroachment
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Special Rules for the Boundary
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Special Rules for the Boundary
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Main Result
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Why Does It Work?
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Why Does It Work?
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Numerical Problem
Red Voronoi vertex is intersection of conic sections
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Numerical Problem
Intersection is computed numerically
?
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Numerical ProblemWhich side of the red line is the vertex on?
?
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Numerical ProblemWhich side of the red line is the vertex on?
Geometric predicates are not always truthfuland the program crashes.
?
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IV. My Current Research
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Star of a Vertex: Definition
The star of a vertex v is the set of all simplices having v for a face.
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Star Based Anisotropic Meshing
Each vertex computes its own star independently
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Inconsistent StarsIf the arcs and vertices of the corresponding anisotropic Voronoi diagram are not all wedged,
the diagram may not dualize to a triangulation, and the independently constructed stars may not form a consistent triangulation.
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Equivalence TheoremIf the arcs and vertices of the anisotropic Voronoi diagram are all wedged, then
the independently constructed star of v
contains the same sites as star(v) in the dual of the anisotropic Voronoi
diagram.
v v
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