1. Generating Well-Shaped Delaunay Mesh
Xiang-Yang Li
Department of Computer Science
Illinois Institute of Technology

2. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Outline
Preliminaries
Considered questions and previous works
Our Solutions:
perturbation
refinement
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

3. BioGeometry: Geometry & Biology
Representation of molecular structures and the simulation of biochemical processes
ligand-to-protein docking, ab initio structure prediction, and protein folding
Drug and protein design
Focus on geometric and topological representations
“geometric methods should be an essential component of any attempt to understand and simulate biological systems”
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

4. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Delaunay & Biology
Delaunay tessellations and Voronoi diagrams capture proximity relationships among sets of points.
When applied to points representing protein atoms or residue positions, they are used
to compute molecular surfaces and protein volumes,
to define cavities and pockets,
to analyze and score packing interactions, and
to find structural motifs
Almost Delaunay Tessellation (ACM SODA’04)
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

5. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Meshes
Modeling and Scientific Computing
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

6. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Meshes
Biology, by Herbert Edelsbrunner et al.
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

7. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Quality: Aspect Ratio R
R: circumradius; r
r: inradius
For numerical simulation, the accuracy and convergence depends on the quality of the mesh elements, I.e., the aspect ratio of the element. Generally, we can define the aspect ratio as the radius of the smallest sphere containing the element over the radius of the largest sphere contained in the element. Notice this definition also applies to non-simplex.
But in our work, we use the radius-edge ratio to measure the quality of the element. Which is defined as the ratio of the radius of the circumsphere over the length of the shortest edge of the simplex.
Notice that if an element has good aspect ratio, then it always has good radius edge ratio. But not vice versa--- sliver is the only element that has good radius-edge ratio, but its aspect ratio can be as large as infinite.
Aspect ratio: R/r
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

8. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Badly Shaped Element
Wedge
Spade
For numerical simulation, the accuracy and convergency depends on the quality of the mesh elements, I.e., the aspect ratio of the element. Generally, we can define the aspect ratio as the radius of the smallest sphere containing the element over the radius of the largest sphere contained in the element. Notice this definition also applies to non-simplex.
But in our work, we use the radius-edge ratio to measure the quality of the element. Which is defined as the ratio of the radius of the circumsphere over the length of the shortest edge of the simplex.
Notice that if an element has good aspect ratio, then it always has good radius edge ratio. But not vice versa--- sliver is the only element that has good radius-edge ratio, but its aspect ratio can be as large as infinite.
Spire
Spear
Spindle
Spike
Splinter
Cap
Sliver
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

9. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Radius-Edge Ratio
aspect ratio R/r R/L ratio R r R L
For numerical simulation, the accuracy and convergence depends on the quality of the mesh elements, I.e., the aspect ratio of the element. Generally, we can define the aspect ratio as the radius of the smallest sphere containing the element over the radius of the largest sphere contained in the element. Notice this definition also applies to non-simplex.
But in our work, we use the radius-edge ratio to measure the quality of the element. Which is defined as the ratio of the radius of the circumsphere over the length of the shortest edge of the simplex.
Notice that if an element has good aspect ratio, then it always has good radius edge ratio. But not vice versa--- sliver is the only element that has good radius-edge ratio, but its aspect ratio can be as large as infinite.
L: shortest edge length
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

10. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Sliver R c L q r s p
Small R/L
Large aspect ratio bad numerical accuracy
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

11. Delaunay Triangulationc
Delaunay triangulation is widely used in the mesh generation area.
In 3D, a mesh is Delaunay, if for any tetrahedron, its circumsphere does not contain any mesh vertex inside.
Notice that, in 2D, Delaunay triangulation always maximize the minimal angle among all possible triangulations.
But in 3D, it does not maximize the minimal dihedral angle among all triangulations.
For every simplex (triangle in 2D, tetrahedron in 3D),
circumsphere B(c, R) is empty.
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

12. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Examples p q r s s r p q
Delaunay triangulation is widely used in the mesh generation area.
In 3D, a mesh is Delaunay, if for any tetrahedron, its circumsphere does not contain any mesh vertex inside.
Notice that, in 2D, Delaunay triangulation always maximize the minimal angle among all possible triangulations.
But in 3D, it does not maximize the minimal dihedral angle among all triangulations. No
Yes
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

13. Generate Small R/L Delaunay Mesh
Delaunay Refinement (Chew, Ruppert, Shewchuk)
Generate mesh using input points
add circumcenter of simplex with large R/l >2;
enforce boundary protection
Repeat until no large R/L
Generate mesh with bounded R/L
Sphere-packing (Miller, Teng et al.)
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

14. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Encroachment x y r v c x y r v c z
C encroaches subfacet if it is inside equatorial sphere of xyz
C encroaches subsegment if it is inside diametral sphere of xy
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

15. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Boundary treatment
Encroachment (Ruppert, Shewchuk)
Bad simplex
Encroached subfacet
Encroached segment
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

16. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Considered Questions
Given a 3D PLC domain, generate
Sliver-free (with small aspect ratio)
Delaunay mesh
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

17. Priori Art: Sliver Exudation
Weighted Delaunay (Cheng-Dey-Edelsbrunner-Facello-Teng [SCG99])
Generate Delaunay mesh with small R/L
assign weights to mesh points
mesh with weighted Delaunay triangulation p2 q2 u v
Dist(u,v)2=uv2-(p+q)2
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

18. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Pros and Cons
No slivers inside the domain
No boundary treatments!
Impossible to guarantee sliver-free totally
Need implement weighted Delaunay
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

19. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Our solutions
Perturbation based method
First eliminate elements with large R/L
Then perturb vertices
Refinement based method add points inside the bad elements
Then eliminate slivers
Eliminate all created elements with large R/L first
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

20. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Observation
For qrs, small region for p to form sliver pqrs
Select p out of this region! r R
R < r L e
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

21. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Sliver region
Parameterizing sliver
Sliver region Rqrs R c L q r s p q r s f Y
Torus region
volume < C(r) s2N(p)3
Sliver: V/L3 < s
( from Sliver Exudation)
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

22. Smoothing and Cleaning-up Slivers
The topic of my Ph.D thesis proposal is
Generating High quality mesh and its applications.

23. Approach: Point Perturbation
For qrs, the region of p to form sliver pqrs is small
Move p out of this region! q r s f Y
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

24. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
w-Perturbation
S’: a w-perturbation of point set S.
for v of S, v’ is in B(v, wN(v)) where w <0.5 depends on R/l.
wN(v) v
N(v)
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

25. R/L for perturbed points
Assume S is a periodic point set
Del(S’) has bounded R/L, if w is small
Proof omitted
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

26. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Circular Dependency?
Perturbing p may create new slivers nonincident to p! p’ p
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

27. Break circular dependency
union graph G=(V,E):
V is mesh vertices set.
if exist perturbation so (p’q’) in Del(S’), then (p,q) in E.
constant degree:
G has constant degree D depends on R/L
constant C tetrahedra incident in all Del(s’)
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

28. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Subregion Existence
All tori regions can not cover the perturbation ball! q r s f Y
wN(p) p
N(p)
Volume =4p(wN(p))3/3
volume < C s2N(p)3
Need: D *C s2N(p)3 < 4p(wN(p))3/3
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

29. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Algorithm
Input: an almost good mesh M S is vertex set of M.
Smooth_Away_Sliver(M)
compute the union complex K of S;
perturb points in S;
construct the Delaunay triangulation of S’.
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

30. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Pros and Cons
Consider periodic point set (as exudation)
S=S0+Z3, where S0 is points in half-open unit cube.
No boundary treatment
w is too small!
However, get Delaunay meshes!
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

31. Removing Slivers by Refinement
The topic of my Ph.D thesis proposal is
Generating High quality mesh and its applications.

32. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Algorithm Outline
For any tet (with R/L>r)
add its circumcenter c
For any sliver
Find a point p near its circumcenter c, s.t. inserting p avoids small slivers
Insert point p and update the Del. triangulation
If c encroaches boundary
apply boundary protection instead
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

33. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Need What?
Need:
What is near?
What is small sliver?
Existence of such p?
Termination guarantee?
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

34. Near Center: Picking regiondR
sphere (c, dR), where d < 1
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

35. Small Slivers: Need To Avoidc
d R Rt p
Tet t is small sliver if: Rt < b R
Where b>1 is a constant.
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

36. Existence: Constant Small Slivers
Constant triangles form slivers with points in picking region
Almost good mesh: constant lengths variation. e E p v
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

37. Termination: Flow of Bad Tet
Original Slivers
Insert point
Created Slivers
Tet with R/L > r
High priority
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

38. Shortest Edge Is Not Short!
Parent element f(t)
Tet t (with R/L > r, or slivers)
Created tet m (good or bad)
Lf(t) Lt Lm
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

39. Termination Guarantee!
Step 1: eliminating R/L > r
If r>2, then the shortest edge of the mesh does not decrease! R c
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

40. Termination Guarantee!
Step 2: eliminating sliver t
If t sliver f(t)
Shortest edge > (1-d)b Lf(t) /4
If t f(t) with R/L>r
Shortest edge > (1-d)r Lf(t) /2 R c
d R Rt p
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

41. Split Original Sliversc l q r s p
R>L/2
L’=(1-d)R > L (1-d)/2
Decrease bounded!
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

42. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Split Created Slivers
Rt > b R
R > L/2 R c
d R Rt p
L’ > (1-d)bR
> L (1-d)b/2
Split this sliver
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

43. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Split Large R/L R c
R>r L
L’ >(1-d) r L
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

44. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Split Subfacets x
bf(y) R v
d R c r
c inside (v,R)
L’ > (1-d) r /2
R>r/2
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

45. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Split Subsegments x
bf(y) R v
d R c r
c inside (v,R)
L’ > (1-d) r /2
R>r/2
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

46. Termination Guarantee!
Guaranteed to terminate, if
(1-d)r > 8 and
(1-d)b > 4
Shortest edge length is at least
(1-d)/4 of original
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

47. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Good Grading
N(v) ~ lfs(v)
Define ev: shortest edge incident v
Define parent p of v
Show ev >c ep
Relate ev with lfs(v) for input v
Relate ev with N(v) for all v.
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

48. Local Feature Size lfs(x)
The radius of the smallest sphere at x intersects or contains two non-incident input features. u v y p x
In previous slide, we say the a spacing function specifies the desired element size. So what is the element size?
Usually, we use the nearest neighbor function, or edge length function to define the element size of point x.
The nearest neighbor function of a point x is the distance to its second nearest mesh vertex. ( Notice here, if x is mesh vertex, then x is the nearest vertex of itself.)
Edge length of a point x is the length of the longest edge of all elements contain x. q
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

49. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Nearest neighbor N(x)
distance to the nearest mesh vertex, if x is vertex distance to the second nearest mesh vertex, otherwise x y z y x
In previous slide, we say the a spacing function specifies the desired element size. So what is the element size?
Usually, we use the nearest neighbor function, or edge length function to define the element size of point x.
The nearest neighbor function of a point x is the distance to its second nearest mesh vertex. ( Notice here, if x is mesh vertex, then x is the nearest vertex of itself.)
Edge length of a point x is the length of the longest edge of all elements contain x.
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

50. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Mesh Size
Size at constant factor of well-shaped meshes.
N(v) ~lfs(v) for well-shaped mesh!
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

51. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Merits
R/L as good as Delaunay Refinement
Theoretic Dihedral angle much better
Complete boundary treatment
Good grading guarantee
Easy implementation!
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

52. Weighted Delaunay and Refinement
Cheng and Dey (2002)
Combine weighted Delaunay triangulation and our Delaunay refinement method
It is deterministic
It can also handle domain boundary
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

53. XiangYang Li, Well-Shaped Delaunay Mesh, IIT
Acknowledgment
H. Edelsbrunner, G. Miller, A. Stathopoulos, D. Talmor, S.-H. Teng, A. Ungor, N. Walkington
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT

54. Questions and Comments
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XiangYang Li, Well-Shaped Delaunay Mesh, IIT