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Stability Aware Delaunay Refinement
Bishal Acharya
Department of Computer Science UNLV
July 11 2013
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Main Contributions
Development of efficient algorithms for stable mesh refinement
Proposing a new technique for reliable mesh refinement
Propose two algorithms for reliable mesh refinement
User friendly implementations of refinement algorithms in Java
Results of experimental investigations
Original mesh refined mesh
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Purpose of Mesh Refinement
Obtain better quality solution in finite element analysisTo remove bad triangles in the mesh
Offer control over the size of the triangles in the mesh
Make the mesh more stable
Initial mesh refined mesh
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Planar Graph
A graph that can be drawn in the plane without any intersecting edges A planar straight line graph PSLG is a planar graph with all straight edge
planar Non planar PSLG
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What is a Delaunay Triangulation ?
Triangulation proposed by Borish Delaunay in 1934
If an empty circle passes through two nodes then that forms Delaunay edge
The circumcircle of any triangle is empty
It is a type of planar graph which tries to maximize the minimum angle
Research on Delaunay triangulation are still actively conducted. [ Fortune 1988Yvinec, Mariett 2010 ]
Circumscribing circle should be empty Circumcircle of any triangle i
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Relationship with Voronoi DiagramDelaunay triangulation is the dual of Voronoi diagram
Overlay of Voronoi Diagram & Delaunay Tr
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Empty Circle Test
CircumscribingCircumscribing circle is empty
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Data Structure for Storing 2D - Mesh(Doubly Connected Edge List)
Each edge is represented by a pair of half-edgesRecord for half- edge
Record for nodes
v1 v2
v4
v3
Planar Graph G Representation in DC
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Face
F1
F2
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Review of Mesh Refinement
Refinement by quadrangulation [ 1998 ]Ruppert's Refinement [ 1994 ]
Chew's Refinement [ 1987 ]
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Characterizing Skinny Triangles
Height of a skinny triangle is much longer than its base
Skinny triangle can either be a Cap or a Needle .One of the goals of mesh refinement is to eliminate the skinny triangles
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Mesh Refinement By Quadrangulation
r
S
S
St
q
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Center of Gravity Refinement
Refine by inserting a new point at CG of the triangles in the mesh.
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Circumcircle Refinement
Refine by inserting a new point at CC of the triangles in the mesh.
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Ruppert's Mesh Refinement
Produces refined mesh with quality triangles
Input to the algorithm is PSLG
Two operations for refinement are split an encroached segment at isplit a skinny triangle at its circumcircle
Terminates for poor quality threshold of over 30 degrees
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Encroached Segment
Encroached
point lies insid
Encroached segment 2
point lies inside the circle
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Segment Split
Segment to split Mesh after spli
new point inserted
at midpoint of theold segment
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Ruppert's Algorithm
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Nontermination of Ruppert's Algorithm1 st ab is encroached by point csplit ab into half by inserting d
2 nd newly inserted point d encroachesupon segment ac . so, segment ac issplit into half by inserting e
e
encroachment and splitcontinues forever.
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Small Angle Non-termination Solution
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Ruppert's Mesh Refinement
refined mesh with angle : 3 area : 0.34
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Stable Mesh Refinement
Ruppert's algorithm only focuses on the quality aspect of a mesh
Insertion of new vertex into the mesh can possibly change the connectivity of edges in the triangulation.
A stable mesh requires that the insertion of a new node brings about least chanin the connectivity of existing edges of the triangulation.
We propose two criteria for inserting a new node into the triangulation.
Two algorithms are presented for stable mesh refinement
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Stable Node Position
Initial pos
Final pos
Dashed edges are edges before point is moved
Dotted edges are newly formed edges
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Limit Edge & Radial Triangles
Limit
radia
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Radial Roaming Region RR(i)
radi
radialregio
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Lateral Triangles
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Lateral Roaming Region LR(i)L
lateral roaming
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Free Roaming Region R(i)
radial roamin
g
free roaming
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CG of a Regular Polygon
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Proposed Algorithm 1
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Largest Empty Circle
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Proposed Algorithm 2
2
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Snapshots 1 : Random Points
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Results & Snapshots 2
quadra
1. resua resulskinnyangle itwo.
2. runs3. origtrianglhas 6k
4. doesaccoun
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Results & Snapshots 3
CG ref
1. resua resulskinnyregion
2. runs
3. If ork trianmesh h
4. doesaccoun
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Results & Snapshots 4
CC ref
1. skinnot locregion
2. runs
3. If ork trianmesh h
4. doesaccoun
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Results & Snapshots 5
Rupperesults
1. skinremovmesh.
2. runstime
3. Numtrianglthe siz
4. doesaccoun
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Snapshots 6 : Radial discs & RadialRoaming Region
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Snapshots 7 : lateral discs & lateralroaming region
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Snapshots 8 : Free roaming region
faor
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Tabulated Results 1
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Tabulated Results 2
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Tabulated Results 3
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Conclusion
Introduced the concept of free roaming region in mesh refinement
Proposed polynomial time algorithms for reliable mesh refinement
Investigated stability in existing mesh refinement algorithms
Presented results of experimental investigations
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Future work & extensions
Generalization of proposed algorithm for exterior nodes
Reduce time complexity of proposed algorithms
Get more insight into free region for stable refinement
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Demo
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Thank You !
Questions ?