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Longest edge n-section refinement- overview and open problems
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The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Longest-‐Edge n-‐sec7on Renement: overview and open
problems Jose Pablo Suárez
University of Las Palmas de Gran Canaria
@josepablosuarez
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Outline 1. Short intro
The Team and Division 2. Why meshes? 3. Where meshes? 4. Methods 5. Longest Edge Refinement
2D and 3D 6. Theoretical support for LE n-secting 7. Final remarks
Smart CiBes and FI-‐WARE: innovaBon to ciBzenship service, Wednesday the 16th. Santander, Cantabria, España
2
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
MAGiC Division University of Las Palmas
de Gran Canaria
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October Sociedad de la Información en Canarias -‐ 11 de junio de 2013 4
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Our team
Sociedad de la Información en Canarias -‐ 11 de junio de 2013 5
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Why meshes?
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Applied MathemaBcs & CompuBng
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes as a part of the body
PDE’s FEM CDF Computer Graf
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes: a part of the body
• Meshes
The skin Body
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes: EVERYWHERE
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes .. where?
…
NumericalMethods
Com. Aid. Geom.
Comp.Graphics
Comp. Gemetry
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Benefits of Edge Refinement
15
FEM / MulBgrid
Terrain Modelling
Geometry Streaming
Surfaces CAGD
VisualizaBon Games, GIS .. LOD
Comp. Geomet.
Compression
Medical Imaging
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Live & outdoor
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes .. Let’s move ..
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The problem: mesh refinement
Surface subdivision
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
IntroducBon to mesh refinement
19
Image Processing: border detection
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The problem: mesh refinement
• FEM in velociBes field
Plaza et Al. CNME, 1994
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The problem: mesh refinement
Dynamic meshes to solve a nonlinear fire propagation problem
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
AdapBve Mesh Refinement
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE refinement for terrain modelling
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Le n-‐secBon for terrain modelling
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
The meshing/solver process
3
2 2 kN
1. Build CAD Model 2. Meshing 3. Apply Loads and Boundary Condi7ons
4. Computa7onal Analysis
7. VisualizaBon
5. Error Es7ma7on 6. Remesh/Refine/Improve
Error < ε
7. Visualiza7on
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes and CAD systems
Hard the study of complex set of parts There is a challenge
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes and CAD systems
Mature Study of simplex parts: simulaBon
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshes Methods
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Meshing Methods
S. Owen, An IntroducBon to Unstructured Mesh GeneraBon, SANDIA Labs
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Tet & Hex
Pro’s: Structured solvers are very efficient. Good control over hex cell quality including stretching. Math behind. Con’s: Consuming time to generate meshes Complex in treating Topology Limitted Application: CFD ..
Pro’s: High degree of automation Tets suit very well to geometry Simplicity Con’s: Hard to assure good shapes Extensive Application: linear CSM, CEM, inviscid CFD, Computer graphics, CAD, etc.
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Some ways to subdivide in 2D
31
Baricentric
G.F. Carey, 1976 Similar
Rosenberg, 1975 Simplex Bisection
Rivara, 1984 4T-LE
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
… passion for Tri/Tet meshes
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Necesity of Quality/Degeneracy study
33
• Solution of Finite Element and Volume Element
• Poorly shaped, distorted or inverted elements
• Numerical difficulties
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Our contribu7on to meshing
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Longest edge based refinement
1. Pick up triangle t0 2. Find Longest Edge 3. Insert point on LE 4. Subdivison
t1 t2
t0 -‐> t1 and t2 t0
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
t
LE BisecBon
Longest-‐Edge bisecBon is good –> minimum angle does not vanish
Conforming meshes (Korotov et al. )
PropagaBng refinement Rivara / Plaza / Suárez
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
3T-‐LE Refinement Algorithm
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE TRISECTION c = 6.7052025350 Plaza et al. 2011
Longest edge in 2D
LE BISECTION
c = 2 Rosenberg, Stenger 1975
α min angle k ref step
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
… a more evidence to respect LE refinement
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Motivation of this work
• When triyng to triangulate this 6 points in 2D:
49
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
Classical Delaunay
Routines
FAIL!!"
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
What does fail?
• TRIANGLE: Jonathan Richard Shewchuk (7/1/2005)
Non-terminated infinite Loop No output produced
50
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
1 2
3
45
6
What does fail? • CGAL in Matlab R2009b / QHULL Matlab R14
51
Nice!!""
But ...
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
• Inspecting with more detail:
52
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
1 2
3
45
6
5 triangles generated"
3 6 4" 5 6 3"
6 2 4" 1 6 5" 4 2 3"
"
ü"ü"ü"ü" x"Collinearity here Ü
CGAL in Matlab R2009b / QHULL Matlab R14
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
A more trial
• A huge pain !
53
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.6
• Aeasily solved by an LE 3-secting method • (7 LE refinement method)
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
benefits of LE
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Benefits of LE
• They are cheap (linear) • TheoreBcally managed (geometry and math)
55
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Go ahead with LE n-‐sec7on in R^m
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Why LE n-‐secBon?
Facts: – No explored yet … – Cheap … – AffecBon for the LE – High subdivision raBo – High locality refinement
Open ques7ons: – Interior angles ? – Shapes axonomy ?
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE n-section in R^m
LE= Longest Edge –> Tri and Tets
number of equal parts Bisection n=2 / Trisection n=3 / Quatersection n=4 …
Dimension Tris, m=2 Tets, m=3 4-simplex m=4 …
n m
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE n-‐secBon at a glance
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Le n-‐secBon (n>=4) in r2
A limiBng property:
Interior angles vanish
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE 4-‐secBon (quatersecBon) 2d
Proof that Interior angles vanish
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Bolzano-‐Weirstrass 1817
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Championship in 2D LE n-‐sec7on face to face
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE n-‐secBon in 2D
c=2
c=6.7
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Limitng property beyond 2D
What about any other Dimension? The 3D case
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Similar ParBBon
66
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Baricentric ParBBon
67
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
8T-‐LE parBBon
68
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Angles in LE n-‐secBon in RM, n>=4 m>=3 vanish -‐> simplices degeneraBon
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Championship in 3D LE n-‐sec7on face to face
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE n-‐secBon in 3d (m=3)
Only Numerical support
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Championship in nD LE n-‐sec7on face to face
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
LE n-‐secBon in RM m>3
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Promising research areas by
• Empowering the pracBcal use LE n-‐secBon
• Combining with other compeBBve methods:
– Conforming face-‐to-‐face refinement
– Delaunay
– Edge swapping
• Feasible exploring in N dimension
• TRI/TET MESHES ARE FRIENDLY AND POWERFULL
TOOLS IN AMC
The Second BCAM Workshop on Computa7onal Mathema7cs 17-‐18 October
Final remarks
1. Longest Edge subdivision (LE n-section) suits well in triangle mesh
refinement
2. We prove a limitting property of vanishing angles in R^m
3. However, LE n-section methods seem to be a promising subdivision
method
4. Some Applications and numerical behaviour have been showed