Perspectives de l'adaptation de maillages dans la pratique de l'ingénieur
Julien Dompierre, juin 2003
Les sciences de l'ingénieur utilisent traditionnellement deux approches
complémentaires pour appréhender le monde: l'analyse théorique et l'étude
expérimentale. Depuis l'avènement des ordinateurs, la simulation numérique
représente une possible troisième voie. Elle permet d'analyser des systèmes
plus complexes que l'analyse théorique et d'étudier des systèmes
inaccessibles à l'étude expérimentale. Cependant, la simulation numérique
étant récente, le recul manque pour évaluer la qualité des résultats. Par
ailleurs, le principal coût des simulations numériques est le temps que
passe l'ingénieur à construire le modèle géométrique avec un système de CAO,
à construire un maillage avec un mailleur, à analyser la solution et à
rétroagir jusqu'à obtenir une solution satisfaisante. La confiance dans les
résultats et le coût humain sont deux obstacles majeurs à une plus grande
pénétration de la simulation numérique dans la pratique de l'ingénieur.
La recherche que je mène depuis une dizaine d'années porte sur la génération
et l'adaptation de maillages. Elle vise à accroître la fiabilité et à
réduire le coût des simulations numériques en en augmentant
l'automatisation. L'automatisation consiste à développer des algorithmes
numériques fiables et robustes qui réduisent les interventions de l'usager.
Grâce à ces recherches sur de nouvelles méthodes numériques, le processus de
simulation numérique deviendra plus fiable et devrait aboutir à une réponse
indépendante de l'utilisateur et des outils de modélisation utilisés.
Adaptation de maillages
La recherche en adaptation de maillages recouvre trois sujets
complémentaires: l'estimation d'erreur, les techniques de maillage et les
méthodes de couplage avec le résoluteur. Ce sont aussi les trois axes de
recherche que je compte mener: améliorer et étendre les estimateurs
d'erreurs, rendre le mailleur tridimensionnel plus robuste et rapide, et
diversifier les applications de simulation numérique.
J'ai développé une approche qui consiste à découpler l'estimation de
l'erreur des techniques de maillages par l'introduction d'une carte de
taille, isotrope ou anisotrope, qui transmet les spécifications de
l'estimateur d'erreur vers l'adapteur de maillages. Le logiciel OORT
(Object-Oriented Remeshing Toolkit) est basé sur cette approche.
L'adapteur de maillages construit un maillage qui satisfait aux
spécifications de la carte de taille. Il procède en modifiant de manière
itérative un maillage initial par un algorithme d'optimisation. Cet
algorithme optimise simultanément des variables discrètes (le nombre de
sommets et la connectivité entre les sommets) et des variables continues
(les coordonnées des sommets). Il converge vers un minimum et peut être
rendu plus efficace en accélérant la convergence. La construction d'un
maillage tétraédrique anisotrope est à la pointe de la recherche.
Intégration de la technologie
La génération et l'adaptation de maillages est une discipline en soi,
cependant, nous avons toujours voulu qu'elle soit applicable et intégrée
dans un processus de simulation numérique. Un volet important de la
recherche concerne donc l'intégration de la génération de maillages avec un
modèle issu de la CAO, et le couplage de l'adaptation de maillages avec des
résoluteurs éléments finis ou volumes finis. Cette recherche trouve son
sens dans les collaborations avec des équipes de génie qui développent ou
utilisent un processus de simulation numérique.
Au cours des cinq dernières années, des collaborations ont été mises en
oeuvre, tant avec des universitaires qu'avec des industriels. En
particulier, je collabore actuellement avec Général Électrique du Canada
pour coupler OORT avec CFX-5 et avec Steven Dufour, du Département de
mathématiques et de génie industr
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
1. Outline
Delaunay Mesh and its Generalization
Control of Error in Numerical Simulation
Conclusions
Generalization of Delaunay Meshes
for the Error Control
in Numerical Simulations
Julien Dompierre
Department of Mathematics and Computer Science
Laurentian University
Sudbury, October 2, 2009
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 1
2. Outline
Delaunay Mesh and its Generalization
General Framework
Control of Error in Numerical Simulation
Conclusions
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 2
3. Outline
Delaunay Mesh and its Generalization
General Framework
Control of Error in Numerical Simulation
Conclusions
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 3
4. Outline
Delaunay Mesh and its Generalization
General Framework
Control of Error in Numerical Simulation
Conclusions
General Framework of Numerical Simulation
CAD System Mesh Generator Solver
CAD Model Mesh Solution
Adaptor
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 4
5. Outline
Delaunay Mesh and its Generalization
General Framework
Control of Error in Numerical Simulation
Conclusions
General Framework with Feedback
CAD System Mesh Generator Solver
CAD Model Mesh Solution
Adaptor
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 5
6. Outline
Delaunay Mesh and its Generalization
General Framework
Control of Error in Numerical Simulation
Conclusions
Mesh Adaptation Loop
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 6
7. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 7
8. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Lesson on Voronoi Diagram
The Voronoi diagrams are partitions of space based on the
notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 8
9. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Voronoi Diagram
Georgy Fedoseevich Vorono¨ April 28,
ı.
1868, Ukraine – November 20, 1908,
Warsaw. Nouvelles applications des
param`tres continus ` la th´orie des
e a e
formes quadratiques. Recherches sur les
parall´llo`des primitifs. Journal Reine
e e
Angew. Math, Vol 134, 1908.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 9
10. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Perpendicular Bisector
Let S1 and S2 be two ver-
P tices in I 2 .
R The perpendi-
d(P, S1 )
S1 cular bisector M(S1 , S2 ) is the
d(P, S2 ) locus of points equidistant to
S1 and S2 . M(S1 , S2 ) =
2
S2 {P ∈ I | d(P, S1 ) = d(P, S2 )},
R
where d(·, ·) is the Euclidean
M distance between two points of
space.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 10
11. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
A Set of Vertices
Let S = {Si }i=1,...,N be a set of N vertices.
S2 S11
S9 S10
S5 S6 S4 S8
S1
S7 S12 S3
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 11
12. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Voronoi Cell
Definition: The Voronoi cell C(Si ) associated to the vertex Si is
the locus of points of space which are closer to Si than any other
vertex:
C(Si ) = {P ∈ I 2 | d(P, Si ) ≤ d(P, Sj ), ∀j = i}.
R
C(Si )
Si
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 12
13. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Voronoi Diagram
The set of Voronoi cells associated with all the vertices of the set
of vertices is called the Voronoi diagram.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 13
14. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Properties of the Voronoi Diagram
The Voronoi cells are polygons in 2D, polyhedra in 3D and
n-polytopes in nD.
The Voronoi cells are convex.
The Voronoi cells cover space without overlapping.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 14
15. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
What to Retain
The Voronoi diagrams are partitions of space into cells based
on the notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 15
16. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Lesson on Delaunay Triangulation
A Delaunay triangulation of a set of vertices is a
triangulation also based on the notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 16
17. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Delaunay Triangulation
Boris Nikolaevich Delone or Delau-
nay. 15 mars 1890, Saint Petersbourg
— 1980. Sur la sph`re vide. A la
e `
m´moire de Georges Voronoi, Bulletin of
e
the Academy of Sciences of the USSR,
Vol. 7, pp. 793–800, 1934.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 17
18. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
A Set of Vertices
S2 S11
S9 S10
S5 S6 S4 S8
S1
S7 S12 S3
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 18
19. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Triangulation of a Set of Vertices
The same set of vertices can be triangulated in many different
fashions.
...
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 19
20. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Triangulation of a Set of Vertices
...
...
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 20
21. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Triangulation of a Set of Vertices
...
...
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 21
22. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Delaunay Triangulation
Among all these fashions, there is one (or maybe many)
triangulation of the convex hull of the set of vertices that is said to
be a Delaunay triangulation.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 22
23. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Empty Sphere Criterion of Delaunay
Empty sphere criterion: A simplex K satisfies the empty sphere
criterion if the open circumscribed ball of the simplex K is empty
(ie, does not contain any other vertex of the triangulation).
K
K
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 23
24. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Violation of the Empty Sphere Criterion
A simplex K does not satisfy the empty sphere criterion if the
opened circumscribed ball of simplex K is not empty (ie, it
contains at least one vertex of the triangulation).
K
K
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 24
25. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Delaunay Triangulation
Delaunay Triangulation: If all the simplices K of a triangulation
T satisfy the empty sphere criterion, then the triangulation is said
to be a Delaunay triangulation.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 25
26. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Delaunay Algorithm
The circumscribed S3
sphere of a simplex has
to be computed. S2
ρout
This amounts to
computing the center of C
a simplex.
The center is the point
at equal distance to all
d
the vertices of the
simplex.
S1
P
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 26
27. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Delaunay Algorithm
How can we know if a
point P violates the
empty sphere criterion S3
for a simplex K ?
S2
The distance d
ρout
between the point P
and the center C has to C
be computed.
If the distance d is
greater than the radius d
ρ, the point P is not in
the circumscribed
sphere of the simplex S1
P
K.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 27
28. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Duality Delaunay-Vorono¨
ı
The Vorono¨ diagram is the dual of the Delaunay triangulation and
ı
vice versa.
Delaunay triangulations have many regularity properties.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 28
29. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
What to Retain
The Voronoi diagram of a set of vertices is a partition of
space into cells based on the notion of distance.
A Delaunay triangulation of a set of vertices is a
triangulation also based on the notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 29
30. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 30
31. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Vorono¨ and Delaunay in Nature
ı
Vorono¨ diagrams and Delaunay triangulations are not just a
ı
mathematician’s whim, they represent structures that can be found
in nature.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 31
32. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Giraffe Hair Coat
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 32
33. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
A Turtle
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 33
34. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
A Pineapple
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 34
35. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Devil’s Tower
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 35
36. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Dry Mud
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 36
37. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Bee Cells
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 37
38. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Dragonfly Wings
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 38
39. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Fly Eyes
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 39
40. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Pop Corn
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 40
41. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Carbon Nanotubes
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 41
42. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Soap Bubbles
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 42
43. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
A Geodesic Dome
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 43
44. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Biosph`re de Montr´al
e e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 44
45. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Streets of Paris
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 45
46. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Roads in France
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 46
47. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Roads in France
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 47
48. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 48
49. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Key Point of this Lecture
For a given set of vertices, the Vorono¨ diagram and the
ı
Delaunay triangulation are partitions of space based on the
notion of distance.
The notion of distance can be generalized.
And so, the notions of Vorono¨ diagram and Delaunay
ı
triangulation can be generalized.
J. Dompierre, M.-G. Vallet, P. Labb´ and F. Guibault. “An Analysis of Simplex
e
Shape Measures for Anisotropic Meshes”. Computer Methods in Applied
Mechanics and Engineering. vol. 194, p. 4895–4914, 2005
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 49
50. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Key Point of this Lecture
For a given set of vertices, the Vorono¨ diagram and the
ı
Delaunay triangulation are partitions of space based on the
notion of distance.
The notion of distance can be generalized.
And so, the notions of Vorono¨ diagram and Delaunay
ı
triangulation can be generalized.
J. Dompierre, M.-G. Vallet, P. Labb´ and F. Guibault. “An Analysis of Simplex
e
Shape Measures for Anisotropic Meshes”. Computer Methods in Applied
Mechanics and Engineering. vol. 194, p. 4895–4914, 2005
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 49
51. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Key Point of this Lecture
For a given set of vertices, the Vorono¨ diagram and the
ı
Delaunay triangulation are partitions of space based on the
notion of distance.
The notion of distance can be generalized.
And so, the notions of Vorono¨ diagram and Delaunay
ı
triangulation can be generalized.
J. Dompierre, M.-G. Vallet, P. Labb´ and F. Guibault. “An Analysis of Simplex
e
Shape Measures for Anisotropic Meshes”. Computer Methods in Applied
Mechanics and Engineering. vol. 194, p. 4895–4914, 2005
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 49
52. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich
LOBACHEVSKY, 1 d´cembre e
1792, Nizhny Novgorod — 24
f´vrier 1856, Kazan.
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 50
53. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
J´nos Bolyai
a
´
Janos BOLYAI, 15 d´cembre 1802
e
` Kolozsv´r, Empire Austrichien
a a
(Cluj, Roumanie) — 27 janvier 1860
` Marosv´s´rhely, Empire Austrichien
a aa
(Tirgu-Mures, Roumanie).
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 51
54. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Bernhard RIEMANN
Georg Friedrich Bernhard RIE-
MANN, 7 septembre 1826, Hanovre
¨
— 20 juillet 1866, Selasca. Uber die
Hypothesen welche der Geometrie zu
Grunde liegen. 10 juin 1854.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 52
55. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Non Euclidean Geometry
Riemann has generalized Euclidean geometry in the plane to
Riemannian geometry on a surface.
He has defined the distance between two points on a surface as the
length of the shortest path between these two points (geodesic).
He has introduced the Riemannian metric that defines the
curvature of space.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 53
56. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Definition of a Metric
If S is any set, then the function
d : S×S → I
R
is called a metric on S if it satisfies
(i) d(A, B) ≥ 0 for all A, B in S;
(ii) d(A, B) = 0 if and only if A = B;
(iii) d(A, B) = d(B, A) for all A, B in S;
(iv) d(A, B) ≤ d(A, C ) + d(C , B) for all A, B, C in S.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 54
57. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Euclidean Distance is a Metric
In the previous definition of a metric, let the set S be I 2 , the
R
function
d : I 2 ×I 2 → I
R R R
xA x
× B → (xB − xA )2 + (yB − yA )2
yA yB
is a metric on I 2 .
R
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 55
58. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Scalar Product is a Metric
Let a vectorial space with its scalar product ·, · . Then the norm
of the scalar product of the difference of two elements of the
vectorial space is a metric.
d(A, B) = B −A ,
1/2
= B − A, B − A ,
− − 1/2
→ →
= AB, AB ,
− T−
→ →
= AB AB.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 56
59. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
The Scalar Product is a Metric
If the vectorial space is I 2 , then the norm of the scalar product of
R
− →
the vector AB is the Euclidean distance.
1/2 − T−
→ →
d(A, B) = B − A, B − A = AB AB,
T
xB − x A xB − x A
= ,
yB − y A yB − y A
= (xB − xA )2 + (yB − yA )2 .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 57
60. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Metric Tensor
A metric tensor M is a symmetric positive definite matrix
m11 m12
M= in 2D,
m12 m22
m11 m12 m13
M = m12 m22 m23 in 3D.
m13 m23 m33
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 58
61. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Metric Length
−→
The length LM (AB) of an edge between vertices A and B in the
metric M is given by
−→ − − 1/2
→ →
LM (AB) = AB, AB M ,
−→ −→
= AB, M AB 1/2 ,
− T
→ −→
= AB M AB.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 59
62. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Euclidean Length with M = I
−→ −→ −→ 1/2 − T
→ −→
LM (AB) = AB, M AB = AB M AB,
T
xB − x A 1 0 xB − x A
= ,
yB − y A 0 1 yB − y A
−→
LE (AB) = (xB − xA )2 + (yB − yA )2 .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 60
63. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
αβ
Metric Length with M = βγ
−→ −→ −→ 1/2 − T
→ −→
LM (AB) = AB, M AB = AB M AB,
T
xB − x A α β xB − x A
= ,
yB − y A β γ yB − y A
−→
LM (AB) = α(xB − xA )2 + 2β(xB − xA )(yB − yA )
1/2
+γ(yB − yA )2 .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 61
64. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Length in a Variable Metric
In the general sense, the metric tensor M is not constant but
varies continuously for every point of space. The length of a
parameterized curve γ(t) = {(x(t), y (t), z(t)) , t ∈ [0, 1]} is
evaluated in the metric
1
LM (γ) = (γ ′ (t))T M (γ(t)) γ ′ (t) dt,
0
where γ(t) is a point of the curve and γ ′ (t) is the tangent vector
of the curve at that point. LM (γ) is always bigger or equal to the
geodesic between the end points of the curve.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 62
65. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Area and Volume in a Metric
Area of the triangle K in a metric M:
AM (K ) = det(M) dA.
K
Volume of the tetrahedron K in a metric M:
VM (K ) = det(M) dV .
K
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 63
66. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Example of a Metric Tensor Field
This analytical test case is defined in George and Borouchaki
(1997).
The domain is a [0, 7] × [0, 9] rectangle.
This test case has an anisotropic Riemannian metric defined by :
−2
h1 (x, y ) 0
M= −2 ,...
0 h2 (x, y )
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 64
67. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Example of a Metric Tensor Field
. . . where h1 (x, y ) is given by:
1 − 19x/40
if x ∈ [0, 2],
(2x−7)/3
20 if x ∈ ]2, 3.5],
h1 (x, y ) =
5(7−2x)/3
if x ∈ ]3.5, 5],
1 4 x−5 4
5 + 5 2 if x ∈ ]5, 7], . . .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 65
68. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Example of a Metric Tensor Field
. . . and h2 (x, y ) is given by:
1 − 19y /40
if y ∈ [0, 2],
(2y −9)/5
20
if y ∈ ]2, 4.5],
h2 (x, y ) = (9−2y )/5
5
if y ∈ ]4.5, 7],
1 4 y −7
4
+ if y ∈ ]7, 9].
5 5 2
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 66
69. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Metric and Delaunay Mesh
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 67
70. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
What to Retain
What appears to everybody to be a skewed triangle
could be an equilateral triangle in the corresponding
skewed space.
An adpated mesh is a only a regular uniform (probably
Delaunay) mesh in a skewed space.
Question 1: From where the Riemannian metric tensor
come from?
Question 2: How to build a regular uniform mesh in a
skewed space?
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 68
71. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 69
72. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Lesson on Mesh Adaptation
Mesh adaptation is an optimisation problem.
The optimal mesh usually does not exist.
Our algorithm is a metaheuristic closed to simulated
annealing that converges iteratively towards a better mesh.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 70
73. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Le crit`re de Delaunay n’est pas un g´n´rateur de maillage
e e e
Le crit`re de Delaunay permet de relier des sommets pour former
e
une triangulation.
Le crit`re de Delaunay peut “assez facilement” se g´n´raliser ` une
e e e a
m´trique riemannienne.
e
Mais, le crit`re n’indique pas combien de sommets il faut g´n´rer
e e e
ni o` il faut les g´n´rer.
u e e
Associer un g´n´rateur de sommets ` un algorithme de Delaunay
e e a
est une approche constructive de la g´n´ration de maillage
e e
(approche gloutonne, sans retour arri`re).
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 71
74. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Maillage unitaire
Un maillage de Delaunay dans la m´trique n’est pas
e
n´cessairement de la bonne taille.
e
On veut plus qu’un maillage de Delaunay dans la m´trique, on en
e
veut un de la bonne taille, ie, dont les arˆtes ont une longueur
e
unitaire avec la m´trique riemannienne.
e
On ne peut pas y arriver de fa¸on directe, mais par des
c
modifications successives.
Dans la boucle d’adaptation, pour que ca marche bien, le solveur
¸
doit converger, le mailleur doit converger, et la boucle compl`te
e
solveur-mailleur doit converger.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 72
75. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
La g´n´ration d’un maillage unitaire est un probl`me
e e e
d’optimisation
Les degr´s de libert´ sont le nombre et la position des sommets,
e e
ainsi que la connectivit´ entre eux.
e
Le probl`me a une partie continue (la position des sommets) et
e
une partie combinatoire (le nombre de sommets et la connectivit´).
e
On consid`re que c’est probablement un probl`me NP-Complet.
e e
On approche le maillage optimal avec une m´taheuristique qui
e
s’apparente ` du recuit-simul´ qui explore l’espace des maillages
a e
possibles.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 73
76. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
M´thode des voisinages
e
Soit M l’ensemble des maillages conformes et simpliciaux qui
discr´tisent un domaine. On veut construire une suite de maillages
e
mi ∈ M telle que mi+1 est un maillage dans le voisinage de mi et
telle que la suite converge vers un maillage optimal.
Un maillage mi+1 est voisin du maillage mi si mi+1 peut-ˆtre
e
obtenu de mi ` l’aide d’une transformation ´l´mentaire et locale.
a ee
Les op´rateurs de voisinage sont l’ajout ou la suppression d’un
e
sommet, la reconnection entre les sommets avec le retournement
d’un arˆte ou d’une face triangulaire, ou encore le d´placement
e e
d’un sommet.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 74
77. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Ajout d’un sommet
Le raffinement consiste ` ajouter un sommet au milieu d’une arˆte
a e
trop longue et ` couper en deux les faces et les t´tra`dres
a e e
adjacents.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 75
78. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Omission d’un sommet
Le maillage peut ˆtre d´raffin´ en enlevant les arˆtes trop courtes.
e e e e
Les ´l´ments autour de l’arˆte sont d´truits et les deux sommets de
ee e e
l’arˆte ne font plus qu’un.
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 76
79. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Retournement de faces
Chaque face interne est entour´e de deux t´tra`dres. Cette face
e e e
peut ˆtre retourn´e en une arˆte entour´e de trois t´tra`dres.
e e e e e e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 77
80. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Retournement d’arˆtes
e
S4 S3 S4 S3
S5 S5
A A
B B
S2 S2
S1 S1
Une arˆte AB entour´e de n t´tra`dres peut ˆtre retourn´e en n − 2
e e e e e e
triangles qui donnent 2(n − 2) t´tra`dres avec les sommets A et B.
e e
Quand n augmente, le nombre de configurations retourn´ese
augmente exponentiellement.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 78
81. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
D´placement d’un sommet
e
x4 x3
k3
k4
x k2
x5 k5
x2
k1
k6
x6 x1
Les sommets sont d´plac´s au “centre” de leurs voisins.
e e
Le “centre” doit ˆtre ´valu´e avec la m´trique riemannienne.
e e e e
C’est la seule m´thode disponible pour adapter des maillages
e
structur´s.
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 79
82. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Fonction coˆt
u
Pour piloter le processus d’optimisation, il faut d´finir une fonction
e
coˆt. Pour un simplexe donn´, cette fonction mesure la conformit´
u e e
en taille et en forme entre le simplexe et la m´trique riemannienne.
e
P. Labb´, J. Dompierre, M.-G. Vallet, F. Guibault et J.-Y. Tr´panier. “A
e e
Universal Measure of the Conformity of a Mesh with Respect to an Anisotropic
Metric Field”. International Journal for Numerical Methods in Engineering.
vol 61, p. 2675–2695, 2004.
Y. Sirois, J. Dompierre, M.-G. Vallet et F. Guibault. “Measuring the conformity
of non-simplicial elements to an anisotropic metric field”, International Journal
for Numerical Methods in Engineering. vol 64, p. 1944–1958, 2005.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 80
83. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Georg Friedrich Bernhard RIEMANN
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 81
84. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Best Grid, 8 ICNGG
“Best Grid” ` la session poster
a
de la 8th International Confer-
ence on Numerical Grid Gen-
eration in Computational Field
Simulations, juin 2002, Hon-
olulu, Hawa¨I.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 82
85. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Meshing Mæstro, 11 IMR
“Meshing Mæstro” ` la session
a
poster de la 11th International
Meshing Roundtable, septem-
bre 2002, Ithaca, New York.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 83
86. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
Adaptation de maillages anisotropes
En 3D, il reste du travail.
L’espace n’est pas pavable par des t´tra`dres r´guliers.
e e e
L’int´gration ` la CAO est cruciale.
e a
L’algorithme doit ˆtre robuste.
e
Le temps de calcul devient contraignant.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 84
87. Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes
What to Retain
We want more than just a Delaunay mesh in the
Riemannian metric. We want a Delaunay UNIT mesh in
the Riemannian metric.
Mesh adaptation is a optimisation problem with a
discrete part and a continuous part.
Our algorithm is a metaheuristic that converges
iteratively towards a better mesh by succesive local
modifications.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 85
88. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 86
89. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Lesson on Interpolation Error
For piecewise linear functions, the interpolation error is
controlled by second order derivatives.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 87
90. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
L’erreur d’interpolation
u
a b
Soit u la solution exacte d’un probl`me dans l’intervalle [a, b].
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 88
91. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Discr´tisation du domaine
e
u
a Th b
Soit Th une triangulation du domaine.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 89
92. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
La solution interpol´e Πh u
e
u
Πh u
a Th b
Soit Πh u, la solution u interpol´e sur l’ensemble des fonctions de
e
base lin´aires d´finies sur la triangulation Th .
e e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 90
93. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
L’erreur d’interpolation u − Πh u
u
Πh u
a Th b
L’erreur d’interpolation u − Πh u est la diff´rence entre la
e
solution exacte u et la solution interpol´e Πh u.
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 91
94. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
L’erreur d’interpolation u − Πh u
u
Πh u
a Th b
L’erreur d’interpolation u − Πh u pour des fonctions de base
lin´aires est domin´e par la d´riv´e seconde.
e e e e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 92
95. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Maillage optimal
u
Πh u
a Th b
Pour un nombre donn´ de sommets, le maillage qui minimise
e
l’erreur d’interpolation u − Πh u est celui qui concentre les
sommets l` o` la courbure est forte.
a u
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 93
96. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Erreur d’interpolation en 2D et 3D
En 2D, les d´riv´es secondes de la solution u forment une matrice
e e
hessienne
∂ 2 u/∂x 2 ∂ 2 u/∂x∂y
.
∂ 2 u/∂y ∂x ∂ 2 u/∂y 2
Si on rend la matrice hessienne d´finie positive, elle devient un
e
tenseur m´trique.
e
On d´finit ainsi un estimateur d’erreur anisotrope, qui ouvre la voie
e
` l’adaptation de maillage anisotrope.
a
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 94
97. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Exemple analytique
Le domaine Ω est le carr´ [0, 1]×[0, 1]. Le probl`me est d´fini
e e e
comme suit:
−∆u + k 2 u = 0 dans Ω
u = g sur ∂Ω,
o` la condition de Dirichlet g est d´finie de telle sorte que la
u e
solution analytique est donn´e par
e
u = e −kx + e −ky .
Cette solution a des couches limites pour de grandes valeurs de k.
F. Guibault, P. Labb´ et J. Dompierre. “Adaptivity Works! Controling the
e
Interpolation Error in 3D”. Fifth World Congress on Computational Mechanics,
Vienna University of Technology, 2002.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 95
98. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Solution analytique
u = e −kx + e −ky , k = 100.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 96
99. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Maillages adapt´s
e
Gauche: Maillage uniforme de 268 sommets.
Centre: Maillage adapt´ isotrope de 268 sommets.
e
Droite: Maillage adapt´ anisotrope de 260 sommets.
e
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 97
100. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Erreur d’interpolation
1
0.1
0.01
Total L2 error
0.001
0.0001
1e-05
1e-06
0.01 0.1
1/sqrt(N)
L’erreur d’interpolation en norme L2 converge en O(h2 ).
Pour obtenir une erreur de 0.001, il faudrait
200 ´l´ments avec un maillage adapt´ anisotrope,
ee e
2000 ´l´ments avec un maillage adapt´ isotrope,
ee e
20000 ´l´ments avec un maillage uniforme.
ee
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 98
101. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
What to Retain
For piecewise linear functions, the interpolation error of a
function u is dominated by second order derivatives.
The hessian matrix is used to defined the metric tensor for
mesh adaptation.
Adapted anisotropic meshes minimize the interpolation error
for a given number of nodes.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 99
102. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 100
103. Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control
Lesson on Approximation Error
The approximation error is bounded by the interpolation
error.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 101