longest edge n-section refinement- overview and open problems

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

[email protected]

@josepablosuarez


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


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


Our team

Sociedad de la Información en Canarias -‐ 11 de junio de 2013 5


Why meshes?


Applied MathemaBcs & CompuBng


Meshes as a part of the body

PDE’s FEM CDF Computer Graf


Meshes: a part of the body

•  Meshes

The skin Body


Meshes: EVERYWHERE


Meshes .. where?

…

NumericalMethods

Com. Aid. Geom.

Comp.Graphics

Comp. Gemetry


Benefits of Edge Refinement

15

FEM / MulBgrid

Terrain Modelling

Geometry Streaming

Surfaces CAGD

VisualizaBon Games, GIS .. LOD

Comp. Geomet.

Compression

Medical Imaging


Live & outdoor


Meshes .. Let’s move ..


The problem: mesh refinement

Surface subdivision


IntroducBon to mesh refinement

19

Image Processing: border detection



•  FEM in velociBes field

Plaza et Al. CNME, 1994



Dynamic meshes to solve a nonlinear fire propagation problem


AdapBve Mesh Refinement


LE refinement for terrain modelling


Le n-‐secBon for terrain modelling


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


Meshes and CAD systems

Hard the study of complex set of parts There is a challenge


Meshes and CAD systems

Mature Study of simplex parts: simulaBon


Meshes Methods


Meshing Methods

S. Owen, An IntroducBon to Unstructured Mesh GeneraBon, SANDIA Labs


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.


Some ways to subdivide in 2D

31

Baricentric

G.F. Carey, 1976 Similar

Rosenberg, 1975 Simplex Bisection

Rivara, 1984 4T-LE


… passion for Tri/Tet meshes


Necesity of Quality/Degeneracy study

33

• Solution of Finite Element and Volume Element

• Poorly shaped, distorted or inverted elements

• Numerical difficulties


Our contribu7on to meshing


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


t

LE BisecBon

Longest-‐Edge bisecBon is good –> minimum angle does not vanish

Conforming meshes (Korotov et al. )

PropagaBng refinement Rivara / Plaza / Suárez


3T-‐LE Refinement Algorithm


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


… a more evidence to respect LE refinement


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!!"


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


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 ...


•  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


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)


benefits of LE


Benefits of LE

• They are cheap (linear) • TheoreBcally managed (geometry and math)

55


Go ahead with LE n-‐sec7on in R^m


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 ?


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


LE n-‐secBon at a glance


Le n-‐secBon (n>=4) in r2

A limiBng property:

Interior angles vanish


LE 4-‐secBon (quatersecBon) 2d

Proof that Interior angles vanish


Bolzano-‐Weirstrass 1817


Championship in 2D LE n-‐sec7on face to face


LE n-‐secBon in 2D

c=2

c=6.7


Limitng property beyond 2D

What about any other Dimension? The 3D case


Similar ParBBon

66


Baricentric ParBBon

67


8T-‐LE parBBon

68


Angles in LE n-‐secBon in RM, n>=4 m>=3 vanish -‐> simplices degeneraBon


Championship in 3D LE n-‐sec7on face to face


LE n-‐secBon in 3d (m=3)

Only Numerical support


Championship in nD LE n-‐sec7on face to face


LE n-‐secBon in RM m>3


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


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

longest edge n-section refinement- overview and open problems

Education