Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

transcript

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

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

General Framework

Outline

4 Conclusions

General Framework

General Framework of Numerical Simulation

Mesh Generator

Solution

CAD System

Solver

Adaptor

CAD Model

General Framework

General Framework with Feedback

Mesh Generator

Solution

CAD System

Solver

Adaptor

CAD Model

General Framework

Mesh Adaptation Loop

Voronoı Diagrams and Delaunay MeshesVoronoı Diagrams and Delaunay Meshes in NatureGeneralization of the Notion of DistanceConstruction of Adapted Anisotropic Meshes

Outline

4 Conclusions

Lesson on Voronoi Diagram

The Voronoi diagrams are partitions of space based on thenotion of distance.

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.

The Perpendicular Bisector

d(P, S2)

d(P, S1)

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.

A Set of Vertices

Let S = Sii=1,...,N be a set of N vertices.

S11S2S10

S3S12S7

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.

C(Si )

The Voronoi Diagram

The set of Voronoi cells associated with all the vertices of the setof vertices is called the Voronoi diagram.

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.

What to Retain

The Voronoi diagrams are partitions of space into cells basedon the notion of distance.

Lesson on Delaunay Triangulation

A Delaunay triangulation of a set of vertices is atriangulation also based on the notion of distance.

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.

A Set of Vertices

S11S2S10

S3S12S7

Triangulation of a Set of Vertices

The same set of vertices can be triangulated in many differentfashions.

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.

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).

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).

Delaunay Triangulation: If all the simplices K of a triangulationT satisfy the empty sphere criterion, then the triangulation is saidto be a Delaunay triangulation.

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.

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 .

Duality Delaunay-Voronoı

The Voronoı diagram is the dual of the Delaunay triangulation andvice versa.

Delaunay triangulations have many regularity properties.

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.

Outline

4 Conclusions

Voronoı and Delaunay in Nature

Voronoı diagrams and Delaunay triangulations are not just amathematician’s whim, they represent structures that can be foundin nature.

Giraffe Hair Coat

A Turtle

A Pineapple

The Devil’s Tower

Dry Mud

Bee Cells

Dragonfly Wings

Fly Eyes

Pop Corn

Carbon Nanotubes

Soap Bubbles

A Geodesic Dome

Biosphere de Montreal

Streets of Paris

Roads in France

Outline

4 Conclusions

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

Nikolai Ivanovich Lobachevsky

Nikolai Ivanovich

LOBACHEVSKY, 1 decembre1792, Nizhny Novgorod — 24fevrier 1856, Kazan.

Janos Bolyai

Janos BOLYAI, 15 decembre 1802a Kolozsvar, Empire Austrichien(Cluj, Roumanie) — 27 janvier 1860a Marosvasarhely, Empire Austrichien(Tirgu-Mures, Roumanie).

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.

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.

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 .

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(

→√

(xB − xA)2 + (yB − yA)2

is a metric on IR2.

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.

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

xB − xA

yB − yA

(xB − xA)2 + (yB − yA)2.

Metric Tensor

A metric tensor M is a symmetric positive definite matrix

m11 m12

m12 m22

in 2D,

m11 m12 m13

m12 m22 m23

m13 m23 m33

in 3D.

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〉

= 〈−→AB ,M

−→AB〉1/2,

−→ABTM

−→AB.

Euclidean Length with M = I

LM(−→AB) = 〈

−→AB,M

−→AB〉1/2 =

−→ABTM

−→AB,

xB − xA

yB − yA

1 00 1

xB − xA

yB − yA

LE (−→AB) =

(xB − xA)2 + (yB − yA)2.

Metric Length with M =(

LM(−→AB) = 〈

−→AB,M

−→AB〉1/2 =

−→ABTM

−→AB,

xB − xA

yB − yA

α ββ γ

xB − xA

yB − yA

LM(−→AB) =

α(xB − xA)2 + 2β(xB − xA)(yB − yA)

+γ(yB − yA)2)1/2

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) =

(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.

Area and Volume in a Metric

Area of the triangle K in a metric M:

AM(K ) =

det(M) dA.

Volume of the tetrahedron K in a metric M:

VM(K ) =

det(M) dV .

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 :

h−21 (x , y) 0

0 h−22 (x , y)

, . . .

. . . 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

x−52

)4if x ∈ ]5, 7], . . .

. . . 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

y−72

)4if y ∈ ]7, 9].

Metric and Delaunay Mesh

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?

Outline

4 Conclusions

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.

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).

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.

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.

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.

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.

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.

Retournement de faces

Chaque face interne est entouree de deux tetraedres. Cette facepeut etre retournee en une arete entouree de trois tetraedres.

Retournement d’aretes

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.

Deplacement d’un sommet

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.

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.

Georg Friedrich Bernhard RIEMANN

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.

Meshing Mæstro, 11 IMR

“Meshing Mæstro” a la sessionposter de la 11th InternationalMeshing Roundtable, septem-bre 2002, Ithaca, New York.

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.

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.

Interpolation ErrorApproximation ErrorImpact of Mesh Adaptation on Numerical SimulationApplications of Spatial Discretization Control

Outline

4 Conclusions

Lesson on Interpolation Error

For piecewise linear functions, the interpolation error iscontrolled by second order derivatives.

L’erreur d’interpolation

Soit u la solution exacte d’un probleme dans l’intervalle [a, b].

Discretisation du domaine

Soit Th une triangulation du domaine.

La solution interpolee Πhu

Soit Πhu, la solution u interpolee sur l’ensemble des fonctions debase lineaires definies sur la triangulation Th.

L’erreur d’interpolation ‖u − Πhu‖

L’erreur d’interpolation ‖u − Πhu‖ est la difference entre lasolution exacte u et la solution interpolee Πhu.

L’erreur d’interpolation ‖u − Πhu‖

L’erreur d’interpolation ‖u − Πhu‖ pour des fonctions de baselineaires est dominee par la derivee seconde.

Maillage optimal

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.

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.

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.

Solution analytique

u = e−kx + e−ky , k = 100.

Maillages adaptes

Gauche: Maillage uniforme de 268 sommets.

Centre: Maillage adapte isotrope de 268 sommets.

Droite: Maillage adapte anisotrope de 260 sommets.

Erreur d’interpolation

0.0001

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 elements avec un maillage adapte anisotrope,2000 elements avec un maillage adapte isotrope,20000 elements avec un maillage uniforme.

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.

Outline

4 Conclusions

Lesson on Approximation Error

The approximation error is bounded by the interpolationerror.

La solution approximee uh

Soit uh, la solution approximee numeriquement sur latriangulation Th avec un resoluteur elements finis ou volumes finis.

L’erreur d’approximation ‖u − uh‖

L’erreur d’approximation ‖u − uh‖ est la difference entre lasolution exacte u et la solution numerique uh.

L’erreur d’interpolation et d’approximation

Lemme de Cea:‖u − uh‖ < C‖u − Πhu‖

Si on controle l’erreur d’interpolation, on controle l’erreurd’approximation.

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.

Maillages et solutions adaptes

Solution “exacte”

La solution “exacte” a 41 372 sommets et 81 899 triangles.

Convergence de ‖u − uh‖L2

0.01 0.1 1

log(‖

log(h)

♦♦

♦Mach

L’erreur d’approximation ‖u − uh‖ en norme L2 converge en O(h2).

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.

Outline

4 Conclusions

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.

Maillage initial A

Maillage initial A et une solution initiale acceptable.

Maillage initial B

Maillage initial B et une solution initiale grossiere.

Maillage initial C

Maillage initial C et une solution initiale erronee.

La simulation numerique n’est pas fiable

Trois usagers differents obtiennent trois solutions differentes...

Evolution a partir du maillage A

Maillages et solutions aux etapes 0, 1, 2, 5 et 10.

Evolution a partir du maillage B

Evolution a partir du maillage C

Comparaison entre les solutions initiales et finales

Solutions initiales et finales pour les trois cas tests.

Superposition des trois solutions

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.

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

Cas test laminaire stationnaire

• NACA 0012.• Mach = 2.0.• Reynolds = 500.• Parois adiabatiques.

Comparaison elements finis – volumes finis

Sur le maillage initial.Elements finis a gauche, volumes finis a droite.

Superposition des solutions elements finis et volumes finis

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.

Outline

4 Conclusions

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.

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

Ecoulement turbulent avec le modele Spalart-Allmaras

Maillage adapte triangulaire non structure anisotrope

Maillage adapte hybride

Profil multiple d’un Boeing 737

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.

Maillage initial et solution

Maillage adapte et solution adaptee

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.

Maillage hybride

Maillage adapte tetraedrique

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.

Maillage cartesien et solution

Maillage non structure tetraedrique adapte et solution

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

Molecule de benzene sur du cuivre

Fil moleculaire auto-assemble

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.

Maui County (Lanai, Maui, Molokai), Hawaii

1 442 401 sommets versus 77 510.

Representation d’images

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.

Transformation en maillage

Segmentation du maillage

Representation de surfaces

Donnees acquises par une camera 3D (www.inspeck.com)

Courbure et adaptation surfacique

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.

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.

Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

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