An Introduction to Mesh Generation Algorithms Part 2dynadata.com/ITVER/Elementos Finitos/Diapositivas/4-ReMeshing-Tu… · Instituto Tecnológico de Veracruz 21-25 January 2008Mesh

Mesh Generation Tutorial 1Instituto Tecnológico de Veracruz 21-25 January 2008

Fathi El-YafiSenior Product Engineer

Product Department of EXA Corporation

An Introduction to Mesh Generation An Introduction to Mesh Generation AlgorithmsAlgorithms

Part 2Part 2


Overview

Remeshing

Decimation

Optimization

Adaptive Meshing:


Finite Element Context

Method:Calculating the density of the mesh from a tree QuaternaryNodal generation within each cell of the treeMeshing by advancing front method according to a

criterion of Delaunay

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

Adaptive Meshing



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.

Quaternary Tree



Remeshing a Polygon

n points

Possible Remeshing?

?



Principle

=

=

=



Principle

Tn =

n(n − 1)(n − 2)6

It shows that we can create Tndifferent triangles

n nodes lead to n-2 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)


Adaptive Mesh

Typical Examples


Adaptive Mesh

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

Initial Mesh

Local geometric model

Diffuse Interpolation

Hermite type

Adaptive mesh

Remeshing Procedure


Adaptive Mesh

Remeshing


Adaptive Mesh

Some Remeshing Tools

Edge splitting

Swapping

Nodal Shifting

Node removing

Edge collapsing

Edge collapsing and nodal insertion


Adaptive Mesh

Remeshing

Remeshing according to a size map

Simplification of a mesh (70%)


Adaptive Mesh

Remeshing

Initial Mesh Intermediate Mesh Optimized Mesh


Adaptive Mesh

Surface Remeshing

Initial Mesh

Optimized Mesh


Adaptive Mesh

Surface Remeshing

Reduction: 60%Reduction: 75%

Initial model

Adapted to curvature


Adaptive Mesh

Surface Meshing

Adapted to curvature

Simplified model


Adaptive Mesh

Simplified Curvature

In a plane 360iθ = °∑

Θ1

Θ2

Of all the triangles 360iα ≥ °∑

αι

default angle = f (Gaussian Curvature)( ) =∆=−∑ θθα ii


Adaptive Mesh

Feature Lines Detection


Adaptive Mesh

Feature Lines Detection

Curvature using Diffuse Interpolation Model

(a) (b)

Identification of geometrical parameters (radius, axis)

Detecting neighborhood


Adaptive Mesh

Principal Curvatures

Segmentation into subsets

Surface Identification

Detection of Feature lines

Meshing

Mesh Generation Tutorial 22Instituto Tecnológico de Veracruz 21-25 January 2008…

Adaptive Mesh

ExampleSegmentation into subsets

Surface Identification

Detection of Feature lines

Meshing


Adaptive Mesh

Main Steps

…


Adaptive Mesh


Adaptive Mesh

Diffuse Model

xz

y

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

MLS Technique : Diffuse approximationMinimisation of an interpolation criterion

( ) ( )( )21

1 ),( iiT

ni

iix zxxpxxwJ −−×= ∑

=

=αα


Adaptive Mesh

Moving Least Squares

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

( )( )2

1( ) ( )

i nT

x i ii

J p x x f x xα α=

=

= − − −∑

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

X

Z

R

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

( )( )2

( )( ) ( )T

x i ii V x

J p x x f x xα α∈

= − − −∑

⇒ The approximation is local


Adaptive Mesh

Weighted Moving Least Squares

( ) ( )( )2

( )( ) ( )T

x i i ii V x

J w x x p x x f x xα α∈

= − − − −∑

Weight Function

/R d l)l()-l(,x)w(x

w,x)w(x

ii

iii

xRxxi

refi

=+×=

=

−

avec 211 2

)(


Adaptive Mesh

Moving Least Squares Interpolation

[ ]

−

→),(1

),(),(xxw

xxwxxwi

ii

( ) ( )( )2

( )( ) ( )T

x i i ii V x

J w x x p x x f x xα α∈

= − − − −∑

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


Adaptive Mesh

CAD-STL Face Per Face



STL/Mesh/Adaptive Mesh

STL Mesh

Mesh Face per Face Adaptive Mesh


STL/Adaptive Mesh

STL Mesh

Adaptive Mesh

Adaptive Mesh


CAD/Adaptive Meshes

Adaptive Mesh


Adaptive Mesh

Mesh Details


Adaptive Mesh

Feature Lines


Adaptive Mesh

Surface Remeshing

Initial model Simplified model


Adaptive Mesh

Surface Remeshing


Adaptive Mesh

Surface Remeshing

Initial Mesh

Adaptive Mesh


Adaptive Mesh

Surface Remeshing


Adaptive Mesh

Repairing Surface Mesh


Adaptive Mesh

Demonstrations

Documents

An Introduction to Mesh Generation Algorithms Part 2dynadata.com/ITVER/Elementos Finitos/Diapositivas/4-ReMeshing-Tu… · Instituto Tecnológico de Veracruz 21-25 January 2008Mesh