Generating Well-Shaped Delaunay Mesh

1
Generating Well-Shaped Delaunay Mesh

• Xiang-Yang Li
• Department of Computer Science
• Illinois Institute of Technology

2
Outline

• Preliminaries
• Considered questions and previous works
• Our Solutions
• perturbation
• refinement

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

4
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 SODA04)

5
Meshes
Modeling and Scientific Computing
6
Meshes
Biology, by Herbert Edelsbrunner et al.
7
Quality Aspect Ratio

Aspect ratio R/r
8
Badly Shaped Element
Wedge
Spade
Spire
Spear
Spindle
Spike
Splinter
Cap
Sliver
9
Radius-Edge Ratio

• aspect ratio R/r R/L ratio

L
L shortest edge length
10
Sliver

• Small R/L
  
• Large aspect ratio
  bad numerical accuracy

11
Delaunay Triangulation
For every simplex (triangle in 2D, tetrahedron in
3D), circumsphere B(c, R) is empty.
12
Examples
s
r
p
q
No
Yes
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 gt2
• enforce boundary protection
• Repeat until no large R/L

• Generate mesh with bounded R/L
• Sphere-packing (Miller, Teng et al.)

14
Encroachment
C encroaches subfacet if it is inside equatorial
sphere of xyz
C encroaches subsegment if it is inside diametral
sphere of xy
15
Boundary treatment

• Encroachment (Ruppert, Shewchuk)

Encroached segment
Encroached subfacet
Bad simplex
16
Considered Questions

• Given a 3D PLC domain, generate
• Sliver-free (with small aspect ratio)
• Delaunay mesh

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

Dist(u,v)2uv2-(pq)2
18
Pros and Cons

• No slivers inside the domain
• No boundary treatments!
• Impossible to guarantee sliver-free totally
• Need implement weighted Delaunay

19
Our solutions

• Perturbation based method
• First eliminate elements with large R/L
• Then perturb vertices
• Refinement based method
  add points inside the bad elements
• First eliminate elements with large R/L
• Then eliminate slivers
• Eliminate all created elements with large R/L
  first

20
Observation

• For qrs, small region for p to form sliver pqrs
• Select p out of this region!

R lt r L
e
21
Sliver region
Parameterizing sliver
Sliver region Rqrs
Torus region volume lt C(r) s2N(p)3
Sliver V/L3 lt s ( from Sliver Exudation)
22
Smoothing and Cleaning-up Slivers
23
Approach Point Perturbation

• For qrs, the region of p to form sliver pqrs is
  small
• Move p out of this region!

24
w-Perturbation

• S a w-perturbation of point set S.
• for v of S, v is in B(v, wN(v))
  where w lt0.5 depends on R/l.

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

26
Circular Dependency?

• Perturbing p may create new slivers nonincident
  to p!

27
Break circular dependency

• union graph G(V,E)
• V is mesh vertices set.
• if exist perturbation so (pq) 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)

28
Subregion Existence
All tori regions can not cover the perturbation
ball!
Volume 4p(wN(p))3/3
volume lt C s2N(p)3
Need D C s2N(p)3 lt 4p(wN(p))3/3
29
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.

30
Pros and Cons

• Consider periodic point set (as exudation)
• SS0Z3, where S0 is points in half-open unit
  cube.
• No boundary treatment
• w is too small!
• However, get Delaunay meshes!

31
Removing Slivers by Refinement
32
Algorithm Outline

• For any tet (with R/Lgtr)
• 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

33
Need What?

• Need
• What is near?
• What is small sliver?
• Existence of such p?
• Termination guarantee?

34
Near Center Picking region
sphere (c, dR), where d lt 1
35
Small Slivers Need To Avoid
Tet t is small sliver if Rt lt b R Where bgt1 is
a constant.
36
Existence Constant Small Slivers

• Constant triangles form slivers with points in
  picking region
• Almost good mesh constant lengths variation.

37
Termination Flow of Bad Tet
Original Slivers
Insert point
Created Slivers
Tet with R/L gt r
High priority
38
Shortest Edge Is Not Short!
39
Termination Guarantee!

• Step 1 eliminating R/L gt r
• If rgt2, then the shortest edge of the mesh does
  not decrease!

40
Termination Guarantee!

• Step 2 eliminating sliver t
• If t sliver f(t)
• Shortest edge gt (1-d)b Lf(t) /4
• If t f(t) with R/Lgtr
• Shortest edge gt (1-d)r Lf(t) /2

41
Split Original Slivers
RgtL/2
L(1-d)R gt L (1-d)/2
Decrease bounded!
42
Split Created Slivers
Rt gt b R R gt L/2
L gt (1-d)bR gt L (1-d)b/2
Split this sliver
43
Split Large R/L
Rgtr L
L gt(1-d) r L
44
Split Subfacets
45
Split Subsegments
46
Termination Guarantee!

• Guaranteed to terminate, if
• (1-d)r gt 8 and
• (1-d)b gt 4
• Shortest edge length is at least
• (1-d)/4 of original

47
Good Grading

• N(v) lfs(v)
• Define ev shortest edge incident v
• Define parent p of v
• Show ev gtc ep
• Relate ev with lfs(v) for input v
• Relate ev with N(v) for all v.

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
q
49
Nearest neighbor N(x)

• distance to the nearest mesh vertex, if x is
  vertex distance to the second nearest mesh
  vertex, otherwise

50
Mesh Size

• Size at constant factor of well-shaped meshes.
• N(v) lfs(v) for well-shaped mesh!

51
Merits

• R/L as good as Delaunay Refinement
• Theoretic Dihedral angle much better
• Complete boundary treatment
• Good grading guarantee
• Easy implementation!

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

53
Acknowledgment

• H. Edelsbrunner, G. Miller, A. Stathopoulos, D.
  Talmor, S.-H. Teng, A. Ungor, N. Walkington

54
Questions and Comments