An Introduction to Mesh Generation Algorithms Part 2...An Introduction to Mesh Generation Algorithms...
Transcript of 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
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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
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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
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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
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� Discrete Models:
Very Refined Initial Mesh + Simplifications
Reconstruction:
• Direct Method: Frontal Triangulation (R. Löhner)
• Remeshing: HDI (A. Rassineux)
Typical Examples
Adaptive Mesh
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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
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Initial Mesh
Local geometric model
Diffuse Interpolation
Hermite type
Adaptive mesh
Remeshing Procedure
Remeshing
Adaptive Mesh
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Some Remeshing Tools
Edge splitting
Edge collapsing
Adaptive Mesh
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Swapping
Nodal Shifting
Node removing
Edge collapsing and nodal insertion
Remeshing
Adaptive Mesh
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Remeshing according to a size map
Simplification of a mesh (70%)
Remeshing
Adaptive Mesh
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Initial Mesh Intermediate Mesh Optimized Mesh
Surface Remeshing
Adaptive Mesh
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Initial Mesh
Optimized Mesh
Surface Remeshing
Reduction: 60%Reduction: 75%
Initial model
Adaptive Mesh
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Adapted to curvature
Surface Meshing
Adaptive Mesh
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Adapted to curvature
Simplified model
Simplified Curvature
In a plane 360iθ = °∑
ααααιιιι
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Θ1
Θ2
Of all the triangles 360iα ≥ °∑default angle = f (Gaussian Curvature)( ) =∆=−∑ θθα ii
Feature Lines Detection
Adaptive Mesh
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Feature Lines Detection
Adaptive Mesh
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� 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
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� Meshing
Example� Segmentation into subsets
� Surface Identification
� Detection of Feature lines
� Meshing
Adaptive Mesh
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� Meshing
Main Steps
Adaptive Mesh
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Adaptive Mesh
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Diffuse Model
z
y
αα TPyxyxyxyxfz =>=<= 22 ,,,,,1),(Local model
MLS Technique : Diffuse approximation
Minimisation of an interpolation criterion
Adaptive Mesh
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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
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( )( )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
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( ) ( )( )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
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[ ]
−→
),(1
),(),(
xxw
xxwxxw
i
ii
If w(x,xi)→→→→ ∞∞∞∞ ⇒⇒⇒⇒ interpolation at point xi
CAD-STL Face Per Face
Adaptive Mesh
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Finite Element Context
STL/Mesh/Adaptive Mesh
STL Mesh
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Mesh Face per Face Adaptive Mesh
STL/Adaptive Mesh
Adaptive Mesh
Adaptive Mesh
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STL Mesh
CAD/Adaptive Meshes
Adaptive Mesh
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Mesh Details
Adaptive Mesh
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Feature Lines
Adaptive Mesh
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Surface Remeshing
Adaptive Mesh
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Initial model Simplified model
Surface Remeshing
Adaptive Mesh
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Surface Remeshing
Adaptive Mesh
Adaptive Mesh
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Initial Mesh
Adaptive Mesh
Surface Remeshing
Adaptive Mesh
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Repairing Surface Mesh
Adaptive Mesh
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Demonstrations
Adaptive Mesh
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