an introduction to mesh generation algorithms part 2...an introduction to mesh generation algorithms...

An Introduction to Mesh Generation An Introduction to Mesh Generation AlgorithmsAlgorithms

Part 2Part 2

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 1

Part 2Part 2

Fathi El-YafiProject and Software Development Manager

Engineering Simulation

Overview

� Remeshing

� Decimation

Adaptive Meshing:

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 2

� Decimation

� Optimization

Finite Element Context

Objective:

2D Adaptive Mesh according to the specifications sizes imposed by the user

Adaptive Meshing

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 3

Method:� Calculating the density of the mesh from a tree Quaternary� Nodal generation within each cell of the tree� Meshing by advancing front method according to a criterion of Delaunay

Quaternary Tree

Finite Element Context

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 4

In computer science, a B+ tree (also known as a Quaternary Tree ) is a type of tree, which represents sorted data in a way that allows for efficient insertion, retrieval and removal of records, each of which is identified by a key. It is a dynamic, multilevel index, with maximum and minimum bounds on the number of keys in each index segment (usually called a block' or node'). In a B+ tree, in contrast to a B+ tree, all records are stored at the lowest level of the tree; only keys are stored in interior blocks.

Remeshing a Polygon

n points

?

Finite Element Context

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 5

Possible Remeshing?

?

Principle

=

Finite Element Context

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 6

=

=

Principle

n(n − 1)(n − 2)

n nodes lead to n-2 triangles

=

Finite Element Context

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 7

Tn = n(n − 1)(n − 2)

6It shows that we can create Tn

different triangles

120845635201041

109876543

N=10, 1430 possible triangulations!

Adaptive Mesh

State of the Art

� Methods that use the surface parameterization:

� Appropriate Plane + transport functions

� Appropriate Plane + Metric: INRIA

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 8

� Discrete Models:

Very Refined Initial Mesh + Simplifications

Reconstruction:

• Direct Method: Frontal Triangulation (R. Löhner)

• Remeshing: HDI (A. Rassineux)

Typical Examples

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 9

Adaptive Meshing using Diffuse Interpolation

Objective:

Building an adaptive mesh of a 3D surface according to size map, using the initial mesh as only data

Initial Mesh

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 10

Initial Mesh

Local geometric model

Diffuse Interpolation

Hermite type

Adaptive mesh

Remeshing Procedure

Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 11

Some Remeshing Tools

Edge splitting

Edge collapsing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 12

Swapping

Nodal Shifting

Node removing

Edge collapsing and nodal insertion

Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 13

Remeshing according to a size map

Simplification of a mesh (70%)

Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 14

Initial Mesh Intermediate Mesh Optimized Mesh

Surface Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 15

Initial Mesh

Optimized Mesh

Surface Remeshing

Reduction: 60%Reduction: 75%

Initial model

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 16

Adapted to curvature

Surface Meshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 17

Adapted to curvature

Simplified model

Simplified Curvature

In a plane 360iθ = °∑

ααααιιιι

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 18

Θ1

Θ2

Of all the triangles 360iα ≥ °∑default angle = f (Gaussian Curvature)( ) =∆=−∑ θθα ii

Feature Lines Detection

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 19

Feature Lines Detection

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 20

� Curvature using Diffuse Interpolation Model

(a) (b)

� Identification of geometrical parameters (radius, axis)

� Detecting neighborhood

Principal Curvatures

� Segmentation into subsets

� Surface Identification

� Detection of Feature lines

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 21

� Meshing

Example� Segmentation into subsets

� Surface Identification

� Detection of Feature lines

� Meshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 22…

� Meshing

Main Steps

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 23…

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 24

Diffuse Model

z

y

αα TPyxyxyxyxfz =>=<= 22 ,,,,,1),(Local model

MLS Technique : Diffuse approximation

Minimisation of an interpolation criterion

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 25

x

( ) ( )( )21

1 ),( iiT

ni

iix zxxpxxwJ −−×=∑

=

=αα

Moving Least Squares

( ) ( )2 21, , , , , ,T Tx y x xy y p x y p x< >= =� Polynomial Base

( )i n=

n = Total number of points ⇒ «standard» Least Squares

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 26

( )( )2

1

( ) ( )i n

Tx i i

i

J p x x f x xα α=

=

= − − −∑

X

Z

R

� Select a neighbor V(x) around a point(x,y)

( )( )2

( )

( ) ( )Tx i i

i V x

J p x x f x xα α∈

= − − −∑

⇒ The approximation is local

Weighted Moving Least Squares

( ) ( )( )2( ) ( )TJ w x x p x x f x xα α= − − − −∑

Weight Function

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 27

( ) ( )( )2

( )

( ) ( )Tx i i i

i V x

J w x x p x x f x xα α∈

= − − − −∑

/R d l

)l()-l(,x)w(x

w,x)w(x

ii

iii

xR

xxirefi

=+×=

=

−

avec

211 2

)(

Moving Least Squares Interpolation

( ) ( )( )2

( )

( ) ( )Tx i i i

i V x

J w x x p x x f x xα α∈

= − − − −∑

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 28

[ ]

−→

),(1

),(),(

xxw

xxwxxw

i

ii

If w(x,xi)→→→→ ∞∞∞∞ ⇒⇒⇒⇒ interpolation at point xi

CAD-STL Face Per Face

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 29

Finite Element Context

STL/Mesh/Adaptive Mesh

STL Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 30

Mesh Face per Face Adaptive Mesh

STL/Adaptive Mesh

Adaptive Mesh

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 31

STL Mesh

CAD/Adaptive Meshes

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 32

Mesh Details

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 33

Feature Lines

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 34

Surface Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 35

Initial model Simplified model

Surface Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 36

Surface Remeshing

Adaptive Mesh

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 37

Initial Mesh

Adaptive Mesh

Surface Remeshing

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 38

Repairing Surface Mesh

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 39

Demonstrations

Adaptive Mesh

Meshing Generation TutorialInstituto Tecnológico de Veracruz 21-25 April 2008 40