The document discusses mesh generation, which involves decomposing a domain into simple elements like triangles or tetrahedra. An optimal mesh has good element quality, conforms to the input domain, and uses the minimum number of points needed to make all Voronoi cells sufficiently "fat" or well-shaped according to metrics like radius-edge ratios. The talk presents analysis showing that the optimal mesh size is determined by the "feature size measure" of the input points, which involves the distance to each point's second nearest neighbor.

1. New Bounds
on the
Size of Optimal Meshes
Don Sheehy
Geometrica
INRIA

2. Mesh Generation

3. Mesh Generation
1 Decompose a volume
into simplices.

4. Mesh Generation
1 Decompose a volume
into simplices.
2 Simplices should be
quality.

5. Mesh Generation
1 Decompose a volume
into simplices.
2 Simplices should be
quality.

6. Mesh Generation
1 Decompose a volume
into simplices.
2 Simplices should be
quality.
3 Output should
conform to input.

7. Mesh Generation
1 Decompose a volume
into simplices.
2 Simplices should be
quality.
3 Output should
conform to input.

8. Mesh Generation

9. Mesh Generation
Uses:

10. Mesh Generation
Uses:
PDEs via FEM

11. Mesh Generation
Uses:
PDEs via FEM
Data Analysis

12. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:

13. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:
Triangle

14. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:
Triangle
CGAL

15. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:
Triangle
CGAL
TetGen

16. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:
Triangle
CGAL
TetGen
Theoretical Guarantees:

17. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:
Triangle
CGAL
TetGen
Theoretical Guarantees:
Sliver Removal

18. Mesh Generation
Uses:
PDEs via FEM
Data Analysis
Good Codes:
Triangle
CGAL
TetGen
Theoretical Guarantees:
Sliver Removal
Surface Reconstruction

19. Local Refinement Algorithms

20. Local Refinement Algorithms

21. Local Refinement Algorithms

22. Local Refinement Algorithms

23. Local Refinement Algorithms

24. Local Refinement Algorithms

25. Local Refinement Algorithms

26. Local Refinement Algorithms

27. Local Refinement Algorithms

28. Local Refinement Algorithms

29. Local Refinement Algorithms

30. Local Refinement Algorithms

31. Local Refinement Algorithms

32. Local Refinement Algorithms

33. Local Refinement Algorithms

34. Local Refinement Algorithms

35. Local Refinement Algorithms

36. Local Refinement Algorithms

37. Local Refinement Algorithms

38. Local Refinement Algorithms

39. Local Refinement Algorithms

40. Local Refinement Algorithms
Pros:

41. Local Refinement Algorithms
Pros:
Easy to implement

42. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel

43. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:

44. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination?

45. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination?
Accumulations?

46. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination?
Accumulations?

47. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination?
Accumulations?

48. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination?
Accumulations?
How many points?

49. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination?
Accumulations?
How many points?

50. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination? Yes.
Accumulations?
How many points?

51. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination? Yes.
Accumulations? No.
How many points?

52. Local Refinement Algorithms
Pros:
Easy to implement
Often Parallel
Cons:
Termination? Yes.
Accumulations? No.
How many points?
This is what we’ll answer.

53. The size of an optimal mesh is given by
the feature size measure.

54. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .

55. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .

56. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .

57. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
x

58. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
s( x)
x lf

59. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
x

60. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
x lfs(
x)

61. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
dx
Optimal Mesh Size = Θ Ω lfs(x)d
x lfs(
x)

62. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
number of vertices
dx
Optimal Mesh Size = Θ Ω lfs(x)d
x lfs(
x)

63. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
number of vertices
dx
Optimal Mesh Size = Θ Ω lfs(x)d
hides simple exponential in d
x lfs(
x)

64. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
number of vertices
dx
Optimal Mesh Size = Θ Ω lfs(x)d
hides simple exponential in d
dx
The Feature Size Measure: µP (Ω) = lfsP (x)d
Ω
x lfs(
x)

65. The size of an optimal mesh is given by
the feature size measure.
lfsP (x) := Distance to second nearest neighbor in P .
number of vertices
dx
Optimal Mesh Size = Θ Ω lfs(x)d
hides simple exponential in d
dx
The Feature Size Measure: µP (Ω) = lfsP (x)d
Ω
When is µP (Ω) = O(n)?

66. A canonical bad case for meshing is two
points in a big empty space.

67. A canonical bad case for meshing is two
points in a big empty space.

68. A canonical bad case for meshing is two
points in a big empty space.

69. A canonical bad case for meshing is two
points in a big empty space.

70. A canonical bad case for meshing is two
points in a big empty space.

71. A canonical bad case for meshing is two
points in a big empty space.

72. A canonical bad case for meshing is two
points in a big empty space.

73. A canonical bad case for meshing is two
points in a big empty space.

74. A canonical bad case for meshing is two
points in a big empty space.

75. A canonical bad case for meshing is two
points in a big empty space.

76. The feature size measure can be
bounded in terms of the pacing.

77. The feature size measure can be
bounded in terms of the pacing.
Order the points.

78. The feature size measure can be
bounded in terms of the pacing.
Order the points.

79. The feature size measure can be
bounded in terms of the pacing.
Order the points.

80. The feature size measure can be
bounded in terms of the pacing.
Order the points.

81. The feature size measure can be
bounded in terms of the pacing.
Order the points.

82. The feature size measure can be
bounded in terms of the pacing.
Order the points.

83. The feature size measure can be
bounded in terms of the pacing.
pi
Order the points.

84. The feature size measure can be
bounded in terms of the pacing.
pi
Order the points.
a = pi − NN(pi )

85. The feature size measure can be
bounded in terms of the pacing.
pi
Order the points.
a = pi − NN(pi )
b = pi − 2NN(pi )

86. The feature size measure can be
bounded in terms of the pacing.
pi
Order the points.
a = pi − NN(pi )
b = pi − 2NN(pi )
b
The pacing of the ith point is φi = a .

87. The feature size measure can be
bounded in terms of the pacing.
pi
Order the points.
a = pi − NN(pi )
b = pi − 2NN(pi )
b
The pacing of the ith point is φi = a .
Let φ be the geometric mean, so log φi = n log φ.

88. The feature size measure can be
bounded in terms of the pacing.
pi
Order the points.
a = pi − NN(pi )
b = pi − 2NN(pi )
b
The pacing of the ith point is φi = a .
Let φ be the geometric mean, so log φi = n log φ.
φ is the pacing of the ordering.

89. The trick is to write the feature size
measure as a telescoping sum.

90. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }

91. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }
n
µ P = µ P2 + µPi − µPi−1
i=3

92. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }
n
µ P = µ P2 + µPi − µPi−1
i=3
effect of adding the ith point.

93. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }
n
µ P = µ P2 + µPi − µPi−1
i=3
effect of adding the ith point.
µPi (Ω) − µPi−1 (Ω) = Θ(1 + log φi )

94. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }
n
µ P = µ P2 + µPi − µPi−1
i=3
effect of adding the ith point.
µPi (Ω) − µPi−1 (Ω) = Θ(1 + log φi )
n
log φi = n log φ
i=3

95. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }
n
µ P = µ P2 + µPi − µPi−1
i=3
effect of adding the ith point.
µPi (Ω) − µPi−1 (Ω) = Θ(1 + log φi )
n
log φi = n log φ Θ(n + n log φ)
i=3

96. The trick is to write the feature size
measure as a telescoping sum.
Pi = {p1 , . . . , pi }
n
µ P = µ P2 + µPi − µPi−1
i=3
effect of adding the ith point.
µPi (Ω) − µPi−1 (Ω) = Θ(1 + log φi )
n
log φi = n log φ Θ(n + n log φ)
i=3
d
Previous bound: O(n + φ n).

97. Pacing analysis has already led to
new results.

98. Pacing analysis has already led to
new results.
The Scaffold Theorem (SODA 2009)
Given n points well-spaced on
a surface, the volume mesh
has size O(n).

99. Pacing analysis has already led to
new results.
The Scaffold Theorem (SODA 2009)
Given n points well-spaced on
a surface, the volume mesh
has size O(n).
Time-Optimal Point Meshing (SoCG 2011)
Build a mesh in O(n log n + m) time.
Algorithm explicitly computes the pacing for each insertion.

100. Some takeaway messages:

101. Some takeaway messages:
1 The amortized change in the number of vertices in a mesh
as a result of adding one new point is determined by the
pacing of that point.

102. Some takeaway messages:
1 The amortized change in the number of vertices in a mesh
as a result of adding one new point is determined by the
pacing of that point.
2 Point sets that admit linear size meshes are exactly those
with constant pacing.

103. Some takeaway messages:
1 The amortized change in the number of vertices in a mesh
as a result of adding one new point is determined by the
pacing of that point.
2 Point sets that admit linear size meshes are exactly those
with constant pacing.
Thank you.

106. Mesh Generation 13

107. Mesh Generation 13
Decompose a domain
into simple elements.

108. Mesh Generation 13
Decompose a domain
into simple elements.

109. Mesh Generation 13
Decompose a domain
Mesh Quality
into simple elements.
Radius/Edge < const

110. Mesh Generation 13
Decompose a domain
Mesh Quality
into simple elements.
X ✓ X
Radius/Edge < const

111. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const

112. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const

113. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const

114. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const

115. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const
Voronoi Diagram

116. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const
Voronoi Diagram OutRadius/InRadius < const

117. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const
X ✓
Voronoi Diagram OutRadius/InRadius < const

118. Mesh Generation 13
Decompose a domain
Mesh Quality Conforming to Input
into simple elements.
X ✓ X
Radius/Edge < const
X ✓
Voronoi Diagram OutRadius/InRadius < const

119. Optimal meshing adds the fewest points
to make all Voronoi cells fat.*
* Equivalent to radius-edge condition on Delaunay simplices.

120. Optimal meshing adds the fewest points
to make all Voronoi cells fat.*
* Equivalent to radius-edge condition on Delaunay simplices.

121. Optimal meshing adds the fewest points
to make all Voronoi cells fat.*
* Equivalent to radius-edge condition on Delaunay simplices.

122. Optimal meshing adds the fewest points
to make all Voronoi cells fat.*
* Equivalent to radius-edge condition on Delaunay simplices.

123. Meshing Points 15
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
n = |P |, m = |M |

124. Meshing Points 15
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
n = |P |, m = |M |

125. Meshing Points 15
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
n = |P |, m = |M |

126. Meshing Points 15
Input: P ⊂ Rd
Output: M ⊃ P with a “nice” Voronoi diagram
n = |P |, m = |M |

127. How to prove a meshing algorithm is optimal.16

128. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P

129. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P

130. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
x

131. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
x
fP (x)

132. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
x

133. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
( x)
x fP

134. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P

135. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
dx
For all v ∈ M, fM (v) ≥ KfP (v) m=Θ
Ω fP (x)d

136. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
dx
For all v ∈ M, fM (v) ≥ KfP (v) m=Θ
Ω fP (x)d
“No 2 points too close together” “Optimal Size Output”

137. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
dx
For all v ∈ M, fM (v) ≥ KfP (v) m=Θ
Ω fP (x)d
“No 2 points too close together” “Optimal Size Output”
v

138. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
dx
For all v ∈ M, fM (v) ≥ KfP (v) m=Θ
Ω fP (x)d
“No 2 points too close together” “Optimal Size Output”
v
fM (v)

139. How to prove a meshing algorithm is optimal.16
The Ruppert Feature Size: fP (x) := distance to 2nd NN of x in P
dx
For all v ∈ M, fM (v) ≥ KfP (v) m=Θ
Ω fP (x)d
“No 2 points too close together” “Optimal Size Output”
v
( v) fM (v)
fP