generalization of delaunay meshes for the error control in numerical simulations
DESCRIPTION
Perspectives de l'adaptation de maillages dans la pratique de l'ingénieurJulien Dompierre, juin 2003Les sciences de l'ingénieur utilisent traditionnellement deux approchescomplémentaires pour appréhender le monde: l'analyse théorique et l'étudeexpérimentale. Depuis l'avènement des ordinateurs, la simulation numériquereprésente une possible troisième voie. Elle permet d'analyser des systèmesplus complexes que l'analyse théorique et d'étudier des systèmesinaccessibles à l'étude expérimentale. Cependant, la simulation numériqueétant récente, le recul manque pour évaluer la qualité des résultats. Parailleurs, le principal coût des simulations numériques est le temps quepasse l'ingénieur à construire le modèle géométrique avec un système de CAO,à construire un maillage avec un mailleur, à analyser la solution et àrétroagir jusqu'à obtenir une solution satisfaisante. La confiance dans lesrésultats et le coût humain sont deux obstacles majeurs à une plus grandepénétration de la simulation numérique dans la pratique de l'ingénieur.La recherche que je mène depuis une dizaine d'années porte sur la générationet l'adaptation de maillages. Elle vise à accroître la fiabilité et àréduire le coût des simulations numériques en en augmentantl'automatisation. L'automatisation consiste à développer des algorithmesnumériques fiables et robustes qui réduisent les interventions de l'usager.Grâce à ces recherches sur de nouvelles méthodes numériques, le processus desimulation numérique deviendra plus fiable et devrait aboutir à une réponseindépendante de l'utilisateur et des outils de modélisation utilisés.Adaptation de maillagesLa recherche en adaptation de maillages recouvre trois sujetscomplémentaires: l'estimation d'erreur, les techniques de maillage et lesméthodes de couplage avec le résoluteur. Ce sont aussi les trois axes derecherche que je compte mener: améliorer et étendre les estimateursd'erreurs, rendre le mailleur tridimensionnel plus robuste et rapide, etdiversifier les applications de simulation numérique.J'ai développé une approche qui consiste à découpler l'estimation del'erreur des techniques de maillages par l'introduction d'une carte detaille, isotrope ou anisotrope, qui transmet les spécifications del'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 auxspécifications de la carte de taille. Il procède en modifiant de manièreitérative un maillage initial par un algorithme d'optimisation. Cetalgorithme optimise simultanément des variables discrètes (le nombre desommets et la connectivité entre les sommets) et des variables continues(les coordonnées des sommets). Il converge vers un minimum et peut êtrerendu plus efficace en accélérant la convergence. La construction d'unmaillage tétraédrique anisotrope est à la pointe de la recherche.Intégration de la technologieLa génération et l'adaptation de maillages est une discipline en soi,cependant, nous avons toujours voulu qu'elle soit applicable et intégréedans un processus de simulation numérique. Un volet important de larecherche concerne donc l'intégration de la génération de maillages avec unmodèle issu de la CAO, et le couplage de l'adaptation de maillages avec desrésoluteurs éléments finis ou volumes finis. Cette recherche trouve sonsens dans les collaborations avec des équipes de génie qui développent ouutilisent un processus de simulation numérique.Au cours des cinq dernières années, des collaborations ont été mises enoeuvre, tant avec des universitaires qu'avec des industriels. Enparticulier, je collabore actuellement avec Général Électrique du Canadapour coupler OORT avec CFX-5 et avec Steven Dufour, du Département demathématiques et de génie industrTRANSCRIPT
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Generalization of Delaunay Meshesfor the Error Control
in Numerical Simulations
Julien Dompierre
Department of Mathematics and Computer ScienceLaurentian University
Sudbury, October 2, 2009
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 1
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
General Framework
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 2
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
General Framework
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 3
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
General Framework
General Framework of Numerical Simulation
Mesh Generator
Solution
CAD System
Mesh
Solver
Adaptor
CAD Model
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 4
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
General Framework
General Framework with Feedback
Mesh Generator
Solution
CAD System
Mesh
Solver
Adaptor
CAD Model
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 5
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
General Framework
Mesh Adaptation Loop
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 6
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 7
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Lesson on Voronoi Diagram
The Voronoi diagrams are partitions of space based on thenotion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 8
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Voronoi Diagram
Georgy Fedoseevich Voronoı. April 28,1868, Ukraine – November 20, 1908,Warsaw. Nouvelles applications desparametres continus a la theorie desformes quadratiques. Recherches sur lesparallelloedes primitifs. Journal ReineAngew. Math, Vol 134, 1908.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 9
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Perpendicular Bisector
S1
S2
d(P, S2)
d(P, S1)
M
P
Let S1 and S2 be two ver-tices in IR
2. The perpendi-cular bisector M(S1, S2) is thelocus of points equidistant toS1 and S2. M(S1, S2) =P ∈ IR
2 | d(P, S1) = d(P, S2),where d(·, ·) is the Euclideandistance between two points ofspace.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 10
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
A Set of Vertices
Let S = Sii=1,...,N be a set of N vertices.
S6
S11S2S10
S4
S3S12S7
S9
S8
S5
S1
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 11
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Voronoi Cell
Definition: The Voronoi cell C(Si ) associated to the vertex Si isthe locus of points of space which are closer to Si than any othervertex:
C(Si ) = P ∈ IR2 | d(P, Si ) ≤ d(P, Sj),∀j 6= i.
Si
C(Si )
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 12
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Voronoi Diagram
The set of Voronoi cells associated with all the vertices of the setof vertices is called the Voronoi diagram.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 13
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Properties of the Voronoi Diagram
The Voronoi cells are polygons in 2D, polyhedra in 3D andn-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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
What to Retain
The Voronoi diagrams are partitions of space into cells basedon the notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 15
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Lesson on Delaunay Triangulation
A Delaunay triangulation of a set of vertices is atriangulation also based on the notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 16
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Delaunay Triangulation
Boris Nikolaevich Delone or Delau-
nay. 15 mars 1890, Saint Petersbourg— 1980. Sur la sphere vide. A lamemoire de Georges Voronoi, Bulletin ofthe Academy of Sciences of the USSR,Vol. 7, pp. 793–800, 1934.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 17
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
A Set of Vertices
S6
S11S2S10
S4
S3S12S7
S9
S8
S5
S1
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 18
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Triangulation of a Set of Vertices
The same set of vertices can be triangulated in many differentfashions.
. . .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 19
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Triangulation of a Set of Vertices
. . .
. . .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 20
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Triangulation of a Set of Vertices
. . .
. . .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 21
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction 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 tobe a Delaunay triangulation.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 22
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Empty Sphere Criterion of Delaunay
Empty sphere criterion: A simplex K satisfies the empty spherecriterion 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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Violation of the Empty Sphere Criterion
A simplex K does not satisfy the empty sphere criterion if theopened circumscribed ball of simplex K is not empty (ie, itcontains at least one vertex of the triangulation).
K
K
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 24
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Delaunay Triangulation
Delaunay Triangulation: If all the simplices K of a triangulationT satisfy the empty sphere criterion, then the triangulation is saidto be a Delaunay triangulation.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 25
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Delaunay Algorithm
The circumscribedsphere of a simplex hasto be computed.
This amounts tocomputing the center ofa simplex.
The center is the pointat equal distance to allthe vertices of thesimplex.
S2
S1
S3
C
ρout
P
d
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 26
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Delaunay Algorithm
How can we know if apoint P violates theempty sphere criterionfor a simplex K?
The distance dbetween the point Pand the center C has tobe computed.
If the distance d isgreater than the radiusρ, the point P is not inthe circumscribedsphere of the simplexK .
S2
S1
S3
C
ρout
P
d
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 27
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Duality Delaunay-Voronoı
The Voronoı diagram is the dual of the Delaunay triangulation andvice versa.
Delaunay triangulations have many regularity properties.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 28
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
What to Retain
The Voronoi diagram of a set of vertices is a partition ofspace into cells based on the notion of distance.
A Delaunay triangulation of a set of vertices is atriangulation also based on the notion of distance.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 29
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 30
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Voronoı and Delaunay in Nature
Voronoı diagrams and Delaunay triangulations are not just amathematician’s whim, they represent structures that can be foundin nature.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 31
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Giraffe Hair Coat
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 32
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
A Turtle
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 33
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
A Pineapple
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 34
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Devil’s Tower
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 35
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Dry Mud
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 36
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Bee Cells
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 37
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Dragonfly Wings
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 38
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Fly Eyes
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 39
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Pop Corn
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 40
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Carbon Nanotubes
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 41
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Soap Bubbles
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 42
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
A Geodesic Dome
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 43
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Biosphere de Montreal
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 44
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Streets of Paris
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 45
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Roads in France
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 46
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Roads in France
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 47
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 48
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Key Point of this Lecture
For a given set of vertices, the Voronoı diagram and theDelaunay triangulation are partitions of space based on thenotion of distance.
The notion of distance can be generalized.
And so, the notions of Voronoı diagram and Delaunaytriangulation can be generalized.
J. Dompierre, M.-G. Vallet, P. Labbe and F. Guibault. “An Analysis of Simplex
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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Key Point of this Lecture
For a given set of vertices, the Voronoı diagram and theDelaunay triangulation are partitions of space based on thenotion of distance.
The notion of distance can be generalized.
And so, the notions of Voronoı diagram and Delaunaytriangulation can be generalized.
J. Dompierre, M.-G. Vallet, P. Labbe and F. Guibault. “An Analysis of Simplex
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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Key Point of this Lecture
For a given set of vertices, the Voronoı diagram and theDelaunay triangulation are partitions of space based on thenotion of distance.
The notion of distance can be generalized.
And so, the notions of Voronoı diagram and Delaunaytriangulation can be generalized.
J. Dompierre, M.-G. Vallet, P. Labbe and F. Guibault. “An Analysis of Simplex
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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Nikolai Ivanovich Lobachevsky
Nikolai Ivanovich
LOBACHEVSKY, 1 decembre1792, Nizhny Novgorod — 24fevrier 1856, Kazan.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 50
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Janos Bolyai
Janos BOLYAI, 15 decembre 1802a Kolozsvar, Empire Austrichien(Cluj, Roumanie) — 27 janvier 1860a Marosvasarhely, Empire Austrichien(Tirgu-Mures, Roumanie).
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 51
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Bernhard RIEMANN
Georg Friedrich Bernhard RIE-
MANN, 7 septembre 1826, Hanovre— 20 juillet 1866, Selasca. Uber dieHypothesen welche der Geometrie zuGrunde liegen. 10 juin 1854.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 52
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Non Euclidean Geometry
Riemann has generalized Euclidean geometry in the plane toRiemannian geometry on a surface.
He has defined the distance between two points on a surface as thelength of the shortest path between these two points (geodesic).
He has introduced the Riemannian metric that defines thecurvature of space.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 53
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Definition of a Metric
If S is any set, then the function
d : S×S → IR
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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Euclidean Distance is a Metric
In the previous definition of a metric, let the set S be IR2, the
function
d : IR2×IR
2 → IR(
xA
yA
)
×
(
xB
yB
)
→√
(xB − xA)2 + (yB − yA)2
is a metric on IR2.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 55
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Scalar Product is a Metric
Let a vectorial space with its scalar product 〈·, ·〉. Then the normof the scalar product of the difference of two elements of thevectorial space is a metric.
d(A, B) = ‖B − A‖,
= 〈B − A, B − A〉1/2,
= 〈−→AB,
−→AB〉1/2,
=
√
−→ABT
−→AB.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 56
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
The Scalar Product is a Metric
If the vectorial space is IR2, then the norm of the scalar product of
the vector−→AB is the Euclidean distance.
d(A, B) = 〈B − A, B − A〉1/2 =
√
−→ABT
−→AB,
=
√
(
xB − xA
yB − yA
)T (
xB − xA
yB − yA
)
,
=√
(xB − xA)2 + (yB − yA)2.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 57
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Metric Tensor
A metric tensor M is a symmetric positive definite matrix
M =
(
m11 m12
m12 m22
)
in 2D,
M =
m11 m12 m13
m12 m22 m23
m13 m23 m33
in 3D.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 58
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Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction 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
LM(−→AB) = 〈
−→AB ,
−→AB〉
1/2M
,
= 〈−→AB ,M
−→AB〉1/2,
=
√
−→ABTM
−→AB.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 59
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Euclidean Length with M = I
LM(−→AB) = 〈
−→AB,M
−→AB〉1/2 =
√
−→ABTM
−→AB,
=
√
(
xB − xA
yB − yA
)T (
1 00 1
) (
xB − xA
yB − yA
)
,
LE (−→AB) =
√
(xB − xA)2 + (yB − yA)2.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 60
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Metric Length with M =(
αβ
βγ
)
LM(−→AB) = 〈
−→AB,M
−→AB〉1/2 =
√
−→ABTM
−→AB,
=
√
(
xB − xA
yB − yA
)T (
α ββ γ
) (
xB − xA
yB − yA
)
,
LM(−→AB) =
(
α(xB − xA)2 + 2β(xB − xA)(yB − yA)
+γ(yB − yA)2)1/2
.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 61
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Length in a Variable Metric
In the general sense, the metric tensor M is not constant butvaries continuously for every point of space. The length of aparameterized curve γ(t) = (x(t), y(t), z(t)) , t ∈ [0, 1] isevaluated in the metric
LM(γ) =
∫ 1
0
√
(γ′(t))T M (γ(t)) γ′(t) dt,
where γ(t) is a point of the curve and γ′(t) is the tangent vectorof the curve at that point. LM(γ) is always bigger or equal to thegeodesic between the end points of the curve.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 62
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Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Area and Volume in a Metric
Area of the triangle K in a metric M:
AM(K ) =
∫
K
√
det(M) dA.
Volume of the tetrahedron K in a metric M:
VM(K ) =
∫
K
√
det(M) dV .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 63
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction 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 :
M =
(
h−21 (x , y) 0
0 h−22 (x , y)
)
, . . .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 64
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Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Example of a Metric Tensor Field
. . . where h1(x , y) is given by:
h1(x , y) =
1 − 19x/40 if x ∈ [0, 2],
20(2x−7)/3 if x ∈ ]2, 3.5],
5(7−2x)/3 if x ∈ ]3.5, 5],
15 + 4
5
(
x−52
)4if x ∈ ]5, 7], . . .
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 65
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Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Example of a Metric Tensor Field
. . . and h2(x , y) is given by:
h2(x , y) =
1 − 19y/40 if y ∈ [0, 2],
20(2y−9)/5 if y ∈ ]2, 4.5],
5(9−2y)/5 if y ∈ ]4.5, 7],
15 + 4
5
(
y−72
)4if y ∈ ]7, 9].
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 66
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Metric and Delaunay Mesh
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 67
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
What to Retain
What appears to everybody to be a skewed trianglecould be an equilateral triangle in the correspondingskewed space.
An adpated mesh is a only a regular uniform (probablyDelaunay) mesh in a skewed space.
Question 1: From where the Riemannian metric tensorcome from?
Question 2: How to build a regular uniform mesh in askewed space?
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 68
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 69
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction 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 simulatedannealing that converges iteratively towards a better mesh.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 70
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Le critere de Delaunay n’est pas un generateur de maillage
Le critere de Delaunay permet de relier des sommets pour formerune triangulation.
Le critere de Delaunay peut “assez facilement” se generaliser a unemetrique riemannienne.
Mais, le critere n’indique pas combien de sommets il faut genererni ou il faut les generer.
Associer un generateur de sommets a un algorithme de Delaunayest une approche constructive de la generation de maillage(approche gloutonne, sans retour arriere).
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 71
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Maillage unitaire
Un maillage de Delaunay dans la metrique n’est pasnecessairement de la bonne taille.
On veut plus qu’un maillage de Delaunay dans la metrique, on enveut un de la bonne taille, ie, dont les aretes ont une longueurunitaire avec la metrique riemannienne.
On ne peut pas y arriver de facon directe, mais par desmodifications successives.
Dans la boucle d’adaptation, pour que ca marche bien, le solveurdoit converger, le mailleur doit converger, et la boucle completesolveur-mailleur doit converger.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 72
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
La generation d’un maillage unitaire est un problemed’optimisation
Les degres de liberte sont le nombre et la position des sommets,ainsi que la connectivite entre eux.
Le probleme a une partie continue (la position des sommets) etune partie combinatoire (le nombre de sommets et la connectivite).On considere que c’est probablement un probleme NP-Complet.
On approche le maillage optimal avec une metaheuristique quis’apparente a du recuit-simule qui explore l’espace des maillagespossibles.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 73
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Methode des voisinages
Soit M l’ensemble des maillages conformes et simpliciaux quidiscretisent un domaine. On veut construire une suite de maillagesmi ∈ M telle que mi+1 est un maillage dans le voisinage de mi ettelle que la suite converge vers un maillage optimal.
Un maillage mi+1 est voisin du maillage mi si mi+1 peut-etreobtenu de mi a l’aide d’une transformation elementaire et locale.
Les operateurs de voisinage sont l’ajout ou la suppression d’unsommet, la reconnection entre les sommets avec le retournementd’un arete ou d’une face triangulaire, ou encore le deplacementd’un sommet.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 74
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Ajout d’un sommet
Le raffinement consiste a ajouter un sommet au milieu d’une aretetrop longue et a couper en deux les faces et les tetraedresadjacents.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 75
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Omission d’un sommet
Le maillage peut etre deraffine en enlevant les aretes trop courtes.Les elements autour de l’arete sont detruits et les deux sommets del’arete ne font plus qu’un.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 76
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Retournement de faces
Chaque face interne est entouree de deux tetraedres. Cette facepeut etre retournee en une arete entouree de trois tetraedres.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 77
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Retournement d’aretes
BA A
B
2
S3S3
S
S1S1
S2
S
5
4 S4
S5S
Une arete AB entouree de n tetraedres peut etre retournee en n− 2triangles qui donnent 2(n − 2) tetraedres avec les sommets A et B.Quand n augmente, le nombre de configurations retourneesaugmente exponentiellement.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 78
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Deplacement d’un sommet
x1
x5
x2
x3x4
x6
k1
k2
k3k4
k5
k6
x
Les sommets sont deplaces au “centre” de leurs voisins.Le “centre” doit etre evaluee avec la metrique riemannienne.C’est la seule methode disponible pour adapter des maillagesstructures.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 79
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Fonction cout
Pour piloter le processus d’optimisation, il faut definir une fonctioncout. Pour un simplexe donne, cette fonction mesure la conformiteen taille et en forme entre le simplexe et la metrique riemannienne.
P. Labbe, J. Dompierre, M.-G. Vallet, F. Guibault et J.-Y. Trepanier. “A
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.
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Georg Friedrich Bernhard RIEMANN
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 81
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Best Grid, 8 ICNGG
“Best Grid” a la session posterde la 8th International Confer-ence on Numerical Grid Gen-eration in Computational FieldSimulations, juin 2002, Hon-olulu, HawaI.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 82
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Meshing Mæstro, 11 IMR
“Meshing Mæstro” a la sessionposter de la 11th InternationalMeshing Roundtable, septem-bre 2002, Ithaca, New York.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 83
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
Adaptation de maillages anisotropes
En 3D, il reste du travail.
L’espace n’est pas pavable par des tetraedres reguliers.
L’integration a la CAO est cruciale.
L’algorithme doit etre robuste.
Le temps de calcul devient contraignant.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 84
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Control of Error in Numerical SimulationConclusions
Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
What to Retain
We want more than just a Delaunay mesh in theRiemannian metric. We want a Delaunay UNIT mesh inthe Riemannian metric.
Mesh adaptation is a optimisation problem with adiscrete part and a continuous part.
Our algorithm is a metaheuristic that convergesiteratively towards a better mesh by succesive localmodifications.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 85
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 86
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Lesson on Interpolation Error
For piecewise linear functions, the interpolation error iscontrolled by second order derivatives.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 87
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
L’erreur d’interpolation
a b
u
Soit u la solution exacte d’un probleme dans l’intervalle [a, b].
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 88
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Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Discretisation du domaine
Th
u
ba
Soit Th une triangulation du domaine.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 89
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
La solution interpolee Πhu
Πhu
Th
u
ba
Soit Πhu, la solution u interpolee sur l’ensemble des fonctions debase lineaires definies sur la triangulation Th.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 90
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
L’erreur d’interpolation ‖u − Πhu‖
Πhu
Th
u
ba
L’erreur d’interpolation ‖u − Πhu‖ est la difference entre lasolution exacte u et la solution interpolee Πhu.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 91
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
L’erreur d’interpolation ‖u − Πhu‖
Πhu
Th
u
ba
L’erreur d’interpolation ‖u − Πhu‖ pour des fonctions de baselineaires est dominee par la derivee seconde.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 92
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage optimal
u
Πhu
Thba
Pour un nombre donne de sommets, le maillage qui minimisel’erreur d’interpolation ‖u − Πhu‖ est celui qui concentre lessommets la ou la courbure est forte.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 93
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Erreur d’interpolation en 2D et 3D
En 2D, les derivees secondes de la solution u forment une matricehessienne
(
∂2u/∂x2 ∂2u/∂x∂y∂2u/∂y∂x ∂2u/∂y2
)
.
Si on rend la matrice hessienne definie positive, elle devient untenseur metrique.On definit ainsi un estimateur d’erreur anisotrope, qui ouvre la voiea l’adaptation de maillage anisotrope.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 94
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Exemple analytique
Le domaine Ω est le carre [0, 1]×[0, 1]. Le probleme est definicomme suit:
−∆u + k2u = 0 dans Ωu = g sur ∂Ω,
ou la condition de Dirichlet g est definie de telle sorte que lasolution analytique est donnee par
u = e−kx + e−ky .
Cette solution a des couches limites pour de grandes valeurs de k .
F. Guibault, P. Labbe et J. Dompierre. “Adaptivity Works! Controling the
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
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Solution analytique
u = e−kx + e−ky , k = 100.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 96
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillages adaptes
Gauche: Maillage uniforme de 268 sommets.
Centre: Maillage adapte isotrope de 268 sommets.
Droite: Maillage adapte anisotrope de 260 sommets.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 97
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Erreur d’interpolation
1e-06
1e-05
0.0001
0.001
0.01
0.1
1
0.01 0.1
Tot
al L
2 er
ror
1/sqrt(N)
L’erreur d’interpolation en norme L2 converge en O(h2).
Pour obtenir une erreur de 0.001, il faudrait
200 elements avec un maillage adapte anisotrope,2000 elements avec un maillage adapte isotrope,20000 elements avec un maillage uniforme.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 98
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
What to Retain
For piecewise linear functions, the interpolation error of afunction u is dominated by second order derivatives.
The hessian matrix is used to defined the metric tensor formesh adaptation.
Adapted anisotropic meshes minimize the interpolation errorfor a given number of nodes.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 99
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 100
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Lesson on Approximation Error
The approximation error is bounded by the interpolationerror.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 101
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
La solution approximee uh
uh
Thba
Soit uh, la solution approximee numeriquement sur latriangulation Th avec un resoluteur elements finis ou volumes finis.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 102
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
L’erreur d’approximation ‖u − uh‖
uh
Th
u
ba
L’erreur d’approximation ‖u − uh‖ est la difference entre lasolution exacte u et la solution numerique uh.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 103
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
L’erreur d’interpolation et d’approximation
uh
Th
u
ba
Πhu
Th
u
ba
Lemme de Cea:‖u − uh‖ < C‖u − Πhu‖
Si on controle l’erreur d’interpolation, on controle l’erreurd’approximation.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 104
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Exemple numerique
Navier-Stokes laminaire.
Profil NACA 0012.
Mach 2.0.
Reynolds 1000.
Angle d’attaque de 10 degres.
J. Dompierre, P. Labbe et F. Guibault. “Con-
trolling Approximation Error”. Second M. I. T.
Conference on Computational Fluid and Solid
Mechanics. Cambridge, MA, 2003.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 105
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillages et solutions adaptes
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 106
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Solution “exacte”
La solution “exacte” a 41 372 sommets et 81 899 triangles.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 107
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Convergence de ‖u − uh‖L2
0.001
0.01
0.1
1
0.01 0.1 1
log(‖
u−
uh‖
L2)
log(h)
ρ
♦♦
♦
♦♦
♦♦
♦
♦Mach
++
+
++
++
++
L’erreur d’approximation ‖u − uh‖ en norme L2 converge en O(h2).
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 108
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Ce qu’il faut retenir
The approximation error is bounded by the interpolationerror.
Adaptivity works! The approximation error is controlledby adapting a mesh according to the interpolation errorof a numerical solution.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 109
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 110
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
La simulation numerique certifiable?
On demande a trois utilisateurs de faire la simulation numeriquesuivante:
Cas test : Navier-Stokes laminaire autour d’un NACA 0012 aMach = 2 et Reynolds = 10000 avec trois maillages initiauxdifferents.
La simulation numerique est-elle certifiable, le resultat est-ilindependant de l’usager?
J. Dompierre, M.-G. Vallet, Y. Bourgault, M. Fortin et W. G. Habashi.
“Anisotropic Mesh Adaptation: Towards User-Independent, Mesh-Independent
and Solver-Independent CFD. Part III: Unstructured Meshes”. International
Journal for Numerical Methods in Fluids. vol. 39, p. 675–702, 2002.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 111
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage initial A
Maillage initial A et une solution initiale acceptable.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 112
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage initial B
Maillage initial B et une solution initiale grossiere.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 113
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage initial C
Maillage initial C et une solution initiale erronee.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 114
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
La simulation numerique n’est pas fiable
Trois usagers differents obtiennent trois solutions differentes...
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 115
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Evolution a partir du maillage A
Maillages et solutions aux etapes 0, 1, 2, 5 et 10.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 116
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Evolution a partir du maillage B
Maillages et solutions aux etapes 0, 1, 2, 5 et 10.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 117
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Evolution a partir du maillage C
Maillages et solutions aux etapes 0, 1, 2, 5 et 10.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 118
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Comparaison entre les solutions initiales et finales
Solutions initiales et finales pour les trois cas tests.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 119
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Superposition des trois solutions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 120
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
What to Retain
Three different users may obtain three different solutions tothe same problem.
Mesh adaptation makes the process automatic.
Automatic mesh adaptation leads to more certifiablenumerical simulations.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 121
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Comparaison de methodes numeriques
• Elements finis (NS2D)• Sylvain Boivin, UQAC• Forme primitive• Variables non
conservatives ρ, u, v , t• Lineaire pour ρ et t• ≃ quadratique pour u et v
• Implicite d’ordre 2 (Gear)avec GMRES non lineaire
• Code Fortran doubleprecision
• Volumes finis (NSC2KE)• Bijan Mohammadi, INRIA• Forme conservative• Variables conservatives ρ, ρu,ρv et ρE• Lineaire pour ρ, ρu, ρv et ρE• Schema de Roe, Oscher,cinematique d’ordre 1 et 2• Explicite Runge-Kutta 4• Code Fortran simpleprecision
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 122
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Cas test laminaire stationnaire
• NACA 0012.• Mach = 2.0.• Reynolds = 500.• Parois adiabatiques.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 123
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Comparaison elements finis – volumes finis
Sur le maillage initial.Elements finis a gauche, volumes finis a droite.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 124
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Superposition des solutions elements finis et volumes finis
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 125
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
What to Retain
The numerical scheme must be consistant with theequations to solve.
With mesh adaptation, the solution does not depend on thenumerical scheme.
Automatic mesh adaptation leads to more certifiablenumerical simulations.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 126
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Outline
1 OutlineGeneral Framework
2 Delaunay Mesh and its GeneralizationVoronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes
3 Control of Error in Numerical SimulationInterpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
4 Conclusions
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 127
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Applications of Spatial Discretization Control
Historically, CFD leads research on mesh adaptation becausethe computational domain may be large while thephenomenon to modelize may be very localized, stretchedand complex.
Mesh adaptation can be applied to other fields of numericalsimuations.
More generally, mesh adaptation can be used in anyapplication that has spatial data to represent.
Moreover, mesh adaptation is one of the computatiomalgeometry tools.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 128
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Couplage avec NSU2D
Projet avec la division d’Aerodynamiqueavancee de Bombardier Aeronautique.
Adaptation de maillages non structurestriangulaires pour le resoluteur NSU2Dde Dimitri Mavriplis.
Etude de differentes strategiesd’adaptation (non structure et hy-bride) pour des geometries simples oucomplexes.
O. Manole, P. Labbe, J. Dompierre et J.-Y.
Trepanier. “Anisotropic Hybrid Mesh Adapta-
tion Using a Metric Field”. 16th AIAA Com-
putational Fluid Dynamics Conference, Orlando,
FL, AIAA–2003–3822, 2003Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 129
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Ecoulement turbulent avec le modele Spalart-Allmaras
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 130
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage adapte triangulaire non structure anisotrope
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 131
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage adapte hybride
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 132
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Profil multiple d’un Boeing 737
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 133
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Profil multiple d’un Boeing 737
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 134
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Couplage avec TASCflow
Projet avec Thi Cong Vu dela division GE Energy, Hydrode GE Canada.
Adaptation de maillagesstructures multiblocs nonconformes.
Les solutions sont calculeespar TASCflow.
T. C. Vu, F. Guibault, J. Dompierre, P.
Labbe et R. Camarero. “Computation of
Fluid Flow in a Model Draft Tube Us-
ing Mesh Adaptive Techniques”. Pro-
ceedings of the 20th Hydraulic Machin-
ery and Systems. Charlotte, NC, 2000.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 135
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage initial et solution
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 136
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage adapte et solution adaptee
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 137
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Couplage avec CFX5
Projet avec Thi Cong Vu de GEEnergy, Hydro de GE Canada.
Adaptation de maillages hybrides(peau de prismes et cœur detetraedres) et de maillagestetraedriques non structures.
Les solutions sont calculees parCFX 5.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 138
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage hybride
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 139
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage adapte tetraedrique
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 140
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Couplage avec deMon
Projet avec Martin Lebœuf duDepartement de chimie del’Universite de Montreal.
Adaptation de maillages nonstructures tetraedriques pour lecalcul des equations de Kohn-Shampour une molecule de glycine.
Les solutions sont calculees pardeMon.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 141
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage cartesien et solution
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 142
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage non structure tetraedrique adapte et solution
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 143
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maillage non structure tetraedrique adapte et solution
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 144
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
STM virtuelle
Scanning Tunneling Microscope (STM) est la microscopie a effettunel.
Projet de STM virtuelle par Stephane Bedwani sous la directiond’Alain Rochefort.
Laboratoire de nanostructures, Ecole Polytechnique de Montrealhttp://nanostructures.phys.polymtl.ca
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 145
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Molecule de benzene sur du cuivre
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 146
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Fil moleculaire auto-assemble
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 147
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Geographic Information Systems (GIS)
Les informations geographiques sont definies sur une grillereguliere.
La representation des donnees geographiques sur un maillageadapte permet une acceleration du rendu dans des applicationsgraphiques.
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 148
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maui County (Lanai, Maui, Molokai), Hawaii
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 149
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maui County (Lanai, Maui, Molokai), Hawaii
Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 150
OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Maui County (Lanai, Maui, Molokai), Hawaii
1 442 401 sommets versus 77 510.
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Representation d’images
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Representation d’images
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Representation d’images
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Representation d’images
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Traitement d’images biomedicales
Reconstruction 3D et modelisation surfacique de structuresanatomiques par traitement d’images biomedicales.
Olivier Courchesne, sous la direction de Farida Cheriet.Laboratoire LIV4D (Laboratoire d’imagerie et de vision 4D),Ecole Polytechnique de Montreal.
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Transformation en maillage
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Segmentation du maillage
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Representation de surfaces
Donnees acquises par une camera 3D (www.inspeck.com)
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control
Courbure et adaptation surfacique
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Conclusion
Mesh adaptation is mainly developed for CFD applicationsbecause the computational domain may be large while thephenomenon to modelize may be very localized, stretchedand complex.
However, mostly any field of numerical simulation may takeadvantage of mesh adapdation to control the process ofnumerical simulation and to make it more certifiable.
Moreover, any scientific field with data defined on a discretemesh could take advantage of a better spatial discretization.
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OutlineDelaunay Mesh and its Generalization
Control of Error in Numerical SimulationConclusions
Conclusion
Even if this work is mainly about mesh adaptation, it is infact multidisciplinary and implies knowledge in mathematics,computer science and engineering.
In particular, this work needs knowledge in appliedmathematics, numerical methods, computational geometry,optimisation, metaheuristics, computer science andengineering.
A multidisciplinary work also implies collaboration betweenresearchers. It was done because it is useful and used bysomeone else.
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