Stability Aware Delaunay Refinement

7/28/2019 Stability Aware Delaunay Refinement

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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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CG ref

1. resua resulskinnyregion

2. runs

3. If ork trianmesh h

4. doesaccoun


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CC ref

1. skinnot locregion

2. runs

3. If ork trianmesh h

4. doesaccoun


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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 ?

Documents

Stability Aware Delaunay Refinement