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:


� Decimation

� Optimization

Finite Element Context

Objective:

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

Adaptive Meshing


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



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

?



Possible Remeshing?

?

Principle

=



=

=

Principle

n(n − 1)(n − 2)

n nodes lead to n-2 triangles

=



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


� Discrete Models:

Very Refined Initial Mesh + Simplifications

Reconstruction:

• Direct Method: Frontal Triangulation (R. Löhner)

• Remeshing: HDI (A. Rassineux)

Typical Examples

Adaptive Mesh


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


Initial Mesh

Local geometric model

Diffuse Interpolation

Hermite type

Adaptive mesh

Remeshing Procedure

Remeshing

Adaptive Mesh


Some Remeshing Tools

Edge splitting

Edge collapsing

Adaptive Mesh


Swapping

Nodal Shifting

Node removing

Edge collapsing and nodal insertion

Remeshing

Adaptive Mesh


Remeshing according to a size map

Simplification of a mesh (70%)

Remeshing

Adaptive Mesh


Initial Mesh Intermediate Mesh Optimized Mesh

Surface Remeshing

Adaptive Mesh


Initial Mesh

Optimized Mesh

Surface Remeshing

Reduction: 60%Reduction: 75%

Initial model

Adaptive Mesh


Adapted to curvature

Surface Meshing

Adaptive Mesh


Adapted to curvature

Simplified model

Simplified Curvature

In a plane 360iθ = °∑

ααααιιιι

Adaptive Mesh


Θ1

Θ2

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

Feature Lines Detection

Adaptive Mesh


Feature Lines Detection

Adaptive Mesh


� 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

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


Diffuse Model

z

y

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

MLS Technique : Diffuse approximation

Minimisation of an interpolation criterion

Adaptive Mesh


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


( )( )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


( ) ( )( )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


[ ]

−→

),(1

),(),(

xxw

xxwxxw

i

ii

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

CAD-STL Face Per Face

Adaptive Mesh



STL/Mesh/Adaptive Mesh

STL Mesh


Mesh Face per Face Adaptive Mesh

STL/Adaptive Mesh

Adaptive Mesh

Adaptive Mesh


STL Mesh

CAD/Adaptive Meshes

Adaptive Mesh


Mesh Details

Adaptive Mesh


Feature Lines

Adaptive Mesh


Surface Remeshing

Adaptive Mesh


Initial model Simplified model

Surface Remeshing

Adaptive Mesh


Surface Remeshing

Adaptive Mesh

Adaptive Mesh


Initial Mesh

Adaptive Mesh

Surface Remeshing

Adaptive Mesh


Repairing Surface Mesh

Adaptive Mesh


Demonstrations

Adaptive Mesh


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

Documents