Anisotropic Voronoi Diagrams and GuaranteedQuality Anisotropic Mesh Generation

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Anisotropic Voronoi Diagrams and
Guaranteed-Quality Anisotropic Mesh Generation

• François Labelle

Jonathan 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 Measures

• Metric tensor Mp distances angles measured by
  p.
• Deformation tensor Fp maps physical to rectified
  space.
• Mp FpT Fp.

Physical Space
q
Fp
Fq
p
FqFp-1
p
q
FpFq-1
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Distance Measures

• Metric tensor Mp distances angles measured by
  p.
• Deformation tensor Fp maps physical to rectified
  space.
• Mp FpT Fp.

Physical Space
q
Fp
Fq
p
FqFp-1
p
q
FpFq-1
Every point wants to be in a nice triangle in
rectified space.
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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)
Common approaches to guaranteed-quality mesh
generation do not adapt well to anisotropy.

• Quadtree-based methods can be adapted to
  horizontal and vertical stretching, but not to
  diagonal stretching.

• 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
Shimada-Yamada-Itoh 1997
Li-Teng-Üngör 1999
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II. Anisotropic Voronoi Diagrams
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Voronoi Diagram Definition

• Given 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

• 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
Islands
Insert new sites on unwedged portions of arcs.
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Voronoi Refinement Algorithm
Orphan
Insert new sites on unwedged portions of arcs.
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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 Problem
Which side of the red line is the vertex on?
?
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Numerical Problem
Which side of the red line is the vertex on?
Geometric predicates are not always truthful
and 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 Stars

• If 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 Theorem

• If the arcs and vertices of the anisotropic
  Voronoi diagram are all wedged, then

v
v
contains the same sites as star(v) in the dual of
the anisotropic Voronoi diagram.
the independently constructed star of v
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