Outline  Delaunay Mesh and its GeneralizationControl of Error in Numerical Simulation                             Conclusi...
Outline            Delaunay Mesh and its Generalization                                                     General Framew...
Outline            Delaunay Mesh and its Generalization                                                     General Framew...
Outline          Delaunay Mesh and its Generalization                                                   General Framework ...
Outline         Delaunay Mesh and its Generalization                                                  General Framework   ...
Outline         Delaunay Mesh and its Generalization                                                  General Framework   ...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
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Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
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Outline         Vorono¨ Diagrams and Delaunay Meshes                                                               ı      ...
Outline         Vorono¨ Diagrams and Delaunay Meshes                                                               ı      ...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                       ı        Delaunay Mes...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                       ı        Delaunay Mes...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                       ı        Delaunay Mes...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                       ı        Delaunay Mes...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                       ı        Delaunay Mes...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline      Vorono¨ Diagrams and Delaunay Meshes                                                             ı           ...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline    Vorono¨ Diagrams and Delaunay Meshes                                                          ı          Delaun...
Outline         Vorono¨ Diagrams and Delaunay Meshes                                                              ı       ...
Outline         Vorono¨ Diagrams and Delaunay Meshes                                                              ı       ...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                         ı          Delaunay...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline        Vorono¨ Diagrams and Delaunay Meshes                                                               ı       ...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                           ı            Dela...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                          ı           Delaun...
Outline   Vorono¨ Diagrams and Delaunay Meshes                                                        ı         Delaunay M...
Outline   Interpolation Error            Delaunay Mesh and its Generalization     Approximation Error          Control of ...
Outline   Interpolation Error          Delaunay Mesh and its Generalization     Approximation Error        Control of Erro...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error           Delaunay Mesh and its Generalization     Approximation Error         Control of Er...
Outline   Interpolation Error            Delaunay Mesh and its Generalization     Approximation Error          Control of ...
Outline   Interpolation Error          Delaunay Mesh and its Generalization     Approximation Error        Control of Erro...
Outline   Interpolation Error          Delaunay Mesh and its Generalization     Approximation Error        Control of Erro...
Outline                  Interpolation Error          Delaunay Mesh and its Generalization                    Approximatio...
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
Generalization of Delaunay Meshes for the Error Control in Numerical Simulations
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Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

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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

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Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

  1. 1. Outline Delaunay Mesh and its GeneralizationControl 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. 2. Outline Delaunay Mesh and its Generalization General Framework Control of Error in Numerical Simulation ConclusionsOutline 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. 3. Outline Delaunay Mesh and its Generalization General Framework Control of Error in Numerical Simulation ConclusionsOutline 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. 4. Outline Delaunay Mesh and its Generalization General Framework Control of Error in Numerical Simulation ConclusionsGeneral 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. 5. Outline Delaunay Mesh and its Generalization General Framework Control of Error in Numerical Simulation ConclusionsGeneral Framework with Feedback CAD System Mesh Generator Solver CAD Model Mesh Solution Adaptor Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 5
  6. 6. Outline Delaunay Mesh and its Generalization General Framework Control of Error in Numerical Simulation ConclusionsMesh Adaptation Loop Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 6
  7. 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 MeshesOutline 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. 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 MeshesLesson 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. 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 MeshesVoronoi 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. 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 MeshesThe 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. 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 MeshesA 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. 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 MeshesThe 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. 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 MeshesThe 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. 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 MeshesProperties 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. 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 MeshesWhat 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. 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 MeshesLesson 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. 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 MeshesDelaunay 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. 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 MeshesA 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. 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 MeshesTriangulation 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. 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 MeshesTriangulation of a Set of Vertices ... ... Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 20
  21. 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 MeshesTriangulation of a Set of Vertices ... ... Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 21
  22. 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 MeshesDelaunay 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. 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 MeshesEmpty 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. 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 MeshesViolation 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. 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 MeshesDelaunay 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. 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 MeshesDelaunay 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. 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 MeshesDelaunay 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. 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 MeshesDuality 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. 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 MeshesWhat 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. 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 MeshesOutline 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. 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 MeshesVorono¨ 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. 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 MeshesGiraffe Hair Coat Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 32
  33. 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 MeshesA Turtle Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 33
  34. 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 MeshesA Pineapple Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 34
  35. 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 MeshesThe Devil’s Tower Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 35
  36. 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 MeshesDry Mud Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 36
  37. 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 MeshesBee Cells Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 37
  38. 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 MeshesDragonfly Wings Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 38
  39. 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 MeshesFly Eyes Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 39
  40. 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 MeshesPop Corn Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 40
  41. 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 MeshesCarbon Nanotubes Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 41
  42. 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 MeshesSoap Bubbles Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 42
  43. 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 MeshesA Geodesic Dome Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 43
  44. 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 MeshesBiosph`re de Montr´al e e Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 44
  45. 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 MeshesStreets of Paris Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 45
  46. 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 MeshesRoads in France Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 46
  47. 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 MeshesRoads in France Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 47
  48. 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 MeshesOutline 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. 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 MeshesThe 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. 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 MeshesThe 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. 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 MeshesThe 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. 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 MeshesNikolai 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. 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 MeshesJ´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. 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 MeshesBernhard 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. 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 MeshesNon 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. 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 MeshesDefinition 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. 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 MeshesThe 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. 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 MeshesThe 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. 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 MeshesThe 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. 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 MeshesMetric 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. 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 MeshesMetric 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. 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 MeshesEuclidean 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. 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 αβMetric 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. 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 MeshesLength 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. 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 MeshesArea 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. 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 MeshesExample 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. 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 MeshesExample 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. 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 MeshesExample 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. 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 MeshesMetric and Delaunay Mesh Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 67
  70. 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 MeshesWhat 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. 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 MeshesOutline 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. 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 MeshesLesson 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. 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 MeshesLe 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. 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 MeshesMaillage 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. 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 MeshesLa g´n´ration d’un maillage unitaire est un probl`me e e ed’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. 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 MeshesM´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. 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 MeshesAjout 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. 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 MeshesOmission 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. 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 MeshesRetournement 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. 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 MeshesRetournement 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. 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 MeshesD´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. 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 MeshesFonction 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. 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 MeshesGeorg Friedrich Bernhard RIEMANN Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 81
  84. 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 MeshesBest 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. 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 MeshesMeshing 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. 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 MeshesAdaptation 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. 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 MeshesWhat 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. 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 ControlOutline 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. 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 ControlLesson 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. 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 ControlL’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. 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 ControlDiscr´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. 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 ControlLa 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. 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 ControlL’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. 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 ControlL’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. 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 ControlMaillage 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. 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 ControlErreur 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. 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 ControlExemple 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. 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 ControlSolution analytique u = e −kx + e −ky , k = 100. Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 96
  99. 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 ControlMaillages 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. 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 ControlErreur 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

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