Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

Outline
Delaunay Mesh and its Generalization
Control of Error in Numerical Simulation
Conclusions

Generalization of Delaunay Meshes
for the Error Control
in Numerical Simulations

Julien Dompierre

Department of Mathematics and Computer Science
Laurentian University

Sudbury, October 2, 2009

Julien Dompierre Delaunay Mesh, Error Control and Numerical Simulation 1

Outline
General Framework
Conclusions

Outline
1 Outline
General Framework
2 Delaunay Mesh and its Generalization
Vorono¨ Diagrams and Delaunay Meshes
ı
Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Generalization of the Notion of Distance
Construction of Adapted Anisotropic Meshes
3 Control of Error in Numerical Simulation
Interpolation Error
Approximation Error
Impact of Mesh Adaptation on Numerical Simulation
Applications of Spatial Discretization Control
4 Conclusions

Outline
General Framework
Conclusions

Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions

Outline
General Framework
Conclusions

General Framework of Numerical Simulation

CAD System Mesh Generator Solver

CAD Model Mesh Solution

Adaptor


Outline
General Framework
Conclusions

General Framework with Feedback

CAD System Mesh Generator Solver

CAD Model Mesh Solution

Adaptor


Outline
General Framework
Conclusions

Mesh Adaptation Loop


Outline Vorono¨ Diagrams and Delaunay Meshes
ı
Delaunay Mesh and its Generalization Vorono¨ Diagrams and Delaunay Meshes in Nature
ı
Control of Error in Numerical Simulation Generalization of the Notion of Distance
Conclusions Construction of Adapted Anisotropic Meshes

Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions

ı
ı

Lesson on Voronoi Diagram

The Voronoi diagrams are partitions of space based on the
notion of distance.


ı
ı

Voronoi Diagram

Georgy Fedoseevich Vorono¨ April 28,
ı.
1868, Ukraine – November 20, 1908,
Warsaw. Nouvelles applications des
param`tres continus ` la th´orie des
e a e
formes quadratiques. Recherches sur les
parall´llo`des primitifs. Journal Reine
e e
Angew. Math, Vol 134, 1908.


ı
ı

The Perpendicular Bisector

Let S1 and S2 be two ver-
P tices in I 2 .
R The perpendi-
d(P, S1 )
S1 cular bisector M(S1 , S2 ) is the
d(P, S2 ) locus of points equidistant to
S1 and S2 . M(S1 , S2 ) =
2
S2 {P ∈ I | d(P, S1 ) = d(P, S2 )},
R
where d(·, ·) is the Euclidean
M distance between two points of
space.


ı
ı

A Set of Vertices

Let S = {Si }i=1,...,N be a set of N vertices.

S2 S11
S9 S10

S5 S6 S4 S8

S1
S7 S12 S3


ı
ı

The Voronoi Cell

Deﬁnition: The Voronoi cell C(Si ) associated to the vertex Si is
the locus of points of space which are closer to Si than any other
vertex:

C(Si ) = {P ∈ I 2 | d(P, Si ) ≤ d(P, Sj ), ∀j = i}.
R

C(Si )
Si


ı
ı

The Voronoi Diagram

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


ı
ı

Properties of the Voronoi Diagram

The Voronoi cells are polygons in 2D, polyhedra in 3D and
n-polytopes in nD.
The Voronoi cells are convex.
The Voronoi cells cover space without overlapping.


ı
ı

What to Retain

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


ı
ı

Lesson on Delaunay Triangulation

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


ı
ı

Delaunay Triangulation

Boris Nikolaevich Delone or Delau-
nay. 15 mars 1890, Saint Petersbourg
— 1980. Sur la sph`re vide. A la
e `
m´moire de Georges Voronoi, Bulletin of
e
the Academy of Sciences of the USSR,
Vol. 7, pp. 793–800, 1934.


ı
ı

A Set of Vertices

S2 S11
S9 S10

S5 S6 S4 S8

S1
S7 S12 S3


ı
ı

Triangulation of a Set of Vertices

The same set of vertices can be triangulated in many diﬀerent
fashions.

...


ı
ı


...

...


ı
ı


Among all these fashions, there is one (or maybe many)
triangulation of the convex hull of the set of vertices that is said to
be a Delaunay triangulation.


ı
ı

Empty Sphere Criterion of Delaunay

Empty sphere criterion: A simplex K satisﬁes the empty sphere
criterion if the open circumscribed ball of the simplex K is empty
(ie, does not contain any other vertex of the triangulation).

K

K


ı
ı

Violation of the Empty Sphere Criterion

A simplex K does not satisfy the empty sphere criterion if the
opened circumscribed ball of simplex K is not empty (ie, it
contains at least one vertex of the triangulation).

K

K


ı
ı


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


ı
ı

Delaunay Algorithm

The circumscribed S3
sphere of a simplex has
to be computed. S2
ρout
This amounts to
computing the center of C
a simplex.
The center is the point
at equal distance to all
d
the vertices of the
simplex.
S1
P


ı
ı

Delaunay Algorithm
How can we know if a
point P violates the
empty sphere criterion S3
for a simplex K ?
S2
The distance d
ρout
between the point P
and the center C has to C
be computed.
If the distance d is
greater than the radius d
ρ, the point P is not in
the circumscribed
sphere of the simplex S1
P
K.

ı
ı

Duality Delaunay-Vorono¨
ı

The Vorono¨ diagram is the dual of the Delaunay triangulation and
ı
vice versa.

Delaunay triangulations have many regularity properties.

ı
ı

What to Retain

The Voronoi diagram of a set of vertices is a partition of
space into cells based on the notion of distance.

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


ı
ı

Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions

ı
ı

Vorono¨ and Delaunay in Nature
ı

Vorono¨ diagrams and Delaunay triangulations are not just a
ı
mathematician’s whim, they represent structures that can be found
in nature.


ı
ı

Giraﬀe Hair Coat


ı
ı

A Turtle


ı
ı

A Pineapple


ı
ı

The Devil’s Tower


ı
ı

Dry Mud


ı
ı

Bee Cells


ı
ı

Dragonﬂy Wings


ı
ı

Fly Eyes


ı
ı

Pop Corn


ı
ı

Carbon Nanotubes


ı
ı

Soap Bubbles


ı
ı

A Geodesic Dome


ı
ı

Biosph`re de Montr´al
e e


ı
ı

Streets of Paris


ı
ı

Roads in France


ı
ı

Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions

ı
ı

The Key Point of this Lecture

For a given set of vertices, the Vorono¨ diagram and the
ı
Delaunay triangulation are partitions of space based on the
notion of distance.
The notion of distance can be generalized.
And so, the notions of Vorono¨ diagram and Delaunay
ı
triangulation can be generalized.

J. Dompierre, M.-G. Vallet, P. Labb´ and F. Guibault. “An Analysis of Simplex
e
Shape Measures for Anisotropic Meshes”. Computer Methods in Applied
Mechanics and Engineering. vol. 194, p. 4895–4914, 2005


ı
ı

Nikolai Ivanovich Lobachevsky

Nikolai Ivanovich
LOBACHEVSKY, 1 d´cembre e
1792, Nizhny Novgorod — 24
f´vrier 1856, Kazan.
e


ı
ı

J´nos Bolyai
a

´
Janos BOLYAI, 15 d´cembre 1802
e
` Kolozsv´r, Empire Austrichien
a a
(Cluj, Roumanie) — 27 janvier 1860
` Marosv´s´rhely, Empire Austrichien
a aa
(Tirgu-Mures, Roumanie).


ı
ı

Bernhard RIEMANN

Georg Friedrich Bernhard RIE-
MANN, 7 septembre 1826, Hanovre
¨
— 20 juillet 1866, Selasca. Uber die
Hypothesen welche der Geometrie zu
Grunde liegen. 10 juin 1854.


ı
ı

Non Euclidean Geometry

Riemann has generalized Euclidean geometry in the plane to
Riemannian geometry on a surface.

He has deﬁned the distance between two points on a surface as the
length of the shortest path between these two points (geodesic).

He has introduced the Riemannian metric that deﬁnes the
curvature of space.


ı
ı

Deﬁnition of a Metric

If S is any set, then the function

d : S×S → I
R

is called a metric on S if it satisﬁes
(i) d(A, B) ≥ 0 for all A, B in S;
(ii) d(A, B) = 0 if and only if A = B;
(iii) d(A, B) = d(B, A) for all A, B in S;
(iv) d(A, B) ≤ d(A, C ) + d(C , B) for all A, B, C in S.


ı
ı

The Euclidean Distance is a Metric

In the previous deﬁnition of a metric, let the set S be I 2 , the
R
function

d : I 2 ×I 2 → I
R R R
xA x
× B → (xB − xA )2 + (yB − yA )2
yA yB

is a metric on I 2 .
R


ı
ı

The Scalar Product is a Metric

Let a vectorial space with its scalar product ·, · . Then the norm
of the scalar product of the diﬀerence of two elements of the
vectorial space is a metric.

d(A, B) = B −A ,
1/2
= B − A, B − A ,
− − 1/2
→ →
= AB, AB ,
− T−
→ →
= AB AB.


ı
ı

The Scalar Product is a Metric

If the vectorial space is I 2 , then the norm of the scalar product of
R
− →
the vector AB is the Euclidean distance.

1/2 − T−
→ →
d(A, B) = B − A, B − A = AB AB,
T
xB − x A xB − x A
= ,
yB − y A yB − y A

= (xB − xA )2 + (yB − yA )2 .


ı
ı

Metric Tensor

A metric tensor M is a symmetric positive deﬁnite matrix

m11 m12
M= in 2D,
m12 m22
 
m11 m12 m13
M =  m12 m22 m23  in 3D.
m13 m23 m33


ı
ı

Metric Length

−→
The length LM (AB) of an edge between vertices A and B in the
metric M is given by
−→ − − 1/2
→ →
LM (AB) = AB, AB M ,
−→ −→
= AB, M AB 1/2 ,
− T
→ −→
= AB M AB.


ı
ı

Euclidean Length with M = I

−→ −→ −→ 1/2 − T
→ −→
LM (AB) = AB, M AB = AB M AB,
T
xB − x A 1 0 xB − x A
= ,
yB − y A 0 1 yB − y A
−→
LE (AB) = (xB − xA )2 + (yB − yA )2 .


ı
ı

αβ
Metric Length with M = βγ

−→ −→ −→ 1/2 − T
→ −→
LM (AB) = AB, M AB = AB M AB,
T
xB − x A α β xB − x A
= ,
yB − y A β γ yB − y A
−→
LM (AB) = α(xB − xA )2 + 2β(xB − xA )(yB − yA )
1/2
+γ(yB − yA )2 .


ı
ı

Length in a Variable Metric

In the general sense, the metric tensor M is not constant but
varies continuously for every point of space. The length of a
parameterized curve γ(t) = {(x(t), y (t), z(t)) , t ∈ [0, 1]} is
evaluated in the metric
1
LM (γ) = (γ ′ (t))T M (γ(t)) γ ′ (t) dt,
0

where γ(t) is a point of the curve and γ ′ (t) is the tangent vector
of the curve at that point. LM (γ) is always bigger or equal to the
geodesic between the end points of the curve.


ı
ı

Area and Volume in a Metric

Area of the triangle K in a metric M:

AM (K ) = det(M) dA.
K

Volume of the tetrahedron K in a metric M:

VM (K ) = det(M) dV .
K


ı
ı

Example of a Metric Tensor Field

This analytical test case is deﬁned in George and Borouchaki
(1997).

The domain is a [0, 7] × [0, 9] rectangle.

This test case has an anisotropic Riemannian metric deﬁned by :
−2
h1 (x, y ) 0
M= −2 ,...
0 h2 (x, y )


ı
ı


. . . where h1 (x, y ) is given by:

 1 − 19x/40
 if x ∈ [0, 2],

 (2x−7)/3
 20 if x ∈ ]2, 3.5],
h1 (x, y ) =
 5(7−2x)/3
 if x ∈ ]3.5, 5],

 1 4 x−5 4

5 + 5 2 if x ∈ ]5, 7], . . .


ı
ı


. . . and h2 (x, y ) is given by:

 1 − 19y /40
 if y ∈ [0, 2],

 (2y −9)/5
 20
 if y ∈ ]2, 4.5],
h2 (x, y ) = (9−2y )/5
 5

if y ∈ ]4.5, 7],

 1 4 y −7
 4
 + if y ∈ ]7, 9].
5 5 2


ı
ı

Metric and Delaunay Mesh


ı
ı

What to Retain

What appears to everybody to be a skewed triangle
could be an equilateral triangle in the corresponding
skewed space.
An adpated mesh is a only a regular uniform (probably
Delaunay) mesh in a skewed space.
Question 1: From where the Riemannian metric tensor
come from?
Question 2: How to build a regular uniform mesh in a
skewed space?


ı
ı

Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions

ı
ı

Lesson on Mesh Adaptation

Mesh adaptation is an optimisation problem.

The optimal mesh usually does not exist.

Our algorithm is a metaheuristic closed to simulated
annealing that converges iteratively towards a better mesh.


ı
ı

Le crit`re de Delaunay n’est pas un gń´rateur de maillage
e e e

Le crit`re de Delaunay permet de relier des sommets pour former
e
une triangulation.

Le crit`re de Delaunay peut “assez facilement” se gń´raliser ` une
e e e a
m´trique riemannienne.
e

Mais, le crit`re n’indique pas combien de sommets il faut gń´rer
e e e
ni o` il faut les gń´rer.
u e e

Associer un gń´rateur de sommets ` un algorithme de Delaunay
e e a
est une approche constructive de la gń´ration de maillage
e e
(approche gloutonne, sans retour arri`re).
e


ı
ı

Maillage unitaire

Un maillage de Delaunay dans la m´trique n’est pas
e
n´cessairement de la bonne taille.
e

On veut plus qu’un maillage de Delaunay dans la m´trique, on en
e
veut un de la bonne taille, ie, dont les arˆtes ont une longueur
e
unitaire avec la m´trique riemannienne.
e

On ne peut pas y arriver de fa¸on directe, mais par des
c
modiﬁcations successives.

Dans la boucle d’adaptation, pour que ca marche bien, le solveur
¸
doit converger, le mailleur doit converger, et la boucle compl`te
e
solveur-mailleur doit converger.


ı
ı

La g´n´ration d’un maillage unitaire est un probl`me
e e e
d’optimisation

Les degr´s de libert´ sont le nombre et la position des sommets,
e e
ainsi que la connectivit´ entre eux.
e

Le probl`me a une partie continue (la position des sommets) et
e
une partie combinatoire (le nombre de sommets et la connectivit´).
e
On consid`re que c’est probablement un probl`me NP-Complet.
e e

On approche le maillage optimal avec une m´taheuristique qui
e
s’apparente ` du recuit-simul´ qui explore l’espace des maillages
a e
possibles.


ı
ı

M´thode des voisinages
e

Soit M l’ensemble des maillages conformes et simpliciaux qui
discr´tisent un domaine. On veut construire une suite de maillages
e
mi ∈ M telle que mi+1 est un maillage dans le voisinage de mi et
telle que la suite converge vers un maillage optimal.

Un maillage mi+1 est voisin du maillage mi si mi+1 peut-ˆtre
e
obtenu de mi ` l’aide d’une transformation ´l´mentaire et locale.
a ee

Les op´rateurs de voisinage sont l’ajout ou la suppression d’un
e
sommet, la reconnection entre les sommets avec le retournement
d’un arˆte ou d’une face triangulaire, ou encore le d´placement
e e
d’un sommet.


ı
ı

Ajout d’un sommet

Le raﬃnement consiste ` ajouter un sommet au milieu d’une arˆte
a e
trop longue et ` couper en deux les faces et les t´tra`dres
a e e
adjacents.


ı
ı

Omission d’un sommet

Le maillage peut ˆtre d´raﬃn´ en enlevant les arˆtes trop courtes.
e e e e
Les ´l´ments autour de l’arˆte sont d´truits et les deux sommets de
ee e e
l’arˆte ne font plus qu’un.
e


ı
ı

Retournement de faces

Chaque face interne est entouré de deux t´tra`dres. Cette face
e e e
peut ˆtre retourné en une arˆte entouré de trois t´tra`dres.
e e e e e e


ı
ı

Retournement d’arˆtes
e

S4 S3 S4 S3

S5 S5
A A
B B

S2 S2

S1 S1

Une arˆte AB entouré de n t´tra`dres peut ˆtre retourné en n − 2
e e e e e e
triangles qui donnent 2(n − 2) t´tra`dres avec les sommets A et B.
e e
Quand n augmente, le nombre de configurations retournése
augmente exponentiellement.


ı
ı

D´placement d’un sommet
e

x4 x3
k3
k4
x k2
x5 k5
x2
k1
k6
x6 x1

Les sommets sont d´plac´s au “centre” de leurs voisins.
e e
Le “centre” doit ˆtre ´valu´e avec la m´trique riemannienne.
e e e e
C’est la seule m´thode disponible pour adapter des maillages
e
structur´s.
e

ı
ı

Fonction coˆt
u

Pour piloter le processus d’optimisation, il faut d´ﬁnir une fonction
e
coˆt. Pour un simplexe donn´, cette fonction mesure la conformit´
u e e
en taille et en forme entre le simplexe et la m´trique riemannienne.
e

P. Labb´, J. Dompierre, M.-G. Vallet, F. Guibault et J.-Y. Tr´panier. “A
e e
Universal Measure of the Conformity of a Mesh with Respect to an Anisotropic
Metric Field”. International Journal for Numerical Methods in Engineering.
vol 61, p. 2675–2695, 2004.

Y. Sirois, J. Dompierre, M.-G. Vallet et F. Guibault. “Measuring the conformity
of non-simplicial elements to an anisotropic metric ﬁeld”, International Journal
for Numerical Methods in Engineering. vol 64, p. 1944–1958, 2005.


ı
ı

Georg Friedrich Bernhard RIEMANN


ı
ı

Best Grid, 8 ICNGG

“Best Grid” ` la session poster
a
de la 8th International Confer-
ence on Numerical Grid Gen-
eration in Computational Field
Simulations, juin 2002, Hon-
olulu, Hawa¨I.


ı
ı

Meshing Mæstro, 11 IMR

“Meshing Mæstro” ` la session
a
poster de la 11th International
Meshing Roundtable, septem-
bre 2002, Ithaca, New York.


ı
ı

Adaptation de maillages anisotropes

En 3D, il reste du travail.
L’espace n’est pas pavable par des t´tra`dres r´guliers.
e e e
L’int´gration ` la CAO est cruciale.
e a
L’algorithme doit ˆtre robuste.
e
Le temps de calcul devient contraignant.


ı
ı

What to Retain

We want more than just a Delaunay mesh in the
Riemannian metric. We want a Delaunay UNIT mesh in
the Riemannian metric.
Mesh adaptation is a optimisation problem with a
discrete part and a continuous part.
Our algorithm is a metaheuristic that converges
iteratively towards a better mesh by succesive local
modiﬁcations.


Outline Interpolation Error
Delaunay Mesh and its Generalization Approximation Error
Control of Error in Numerical Simulation Impact of Mesh Adaptation on Numerical Simulation
Conclusions Applications of Spatial Discretization Control

Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions


Lesson on Interpolation Error

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



L’erreur d’interpolation

u

a b
Soit u la solution exacte d’un probl`me dans l’intervalle [a, b].
e



Discr´tisation du domaine
e

u

a Th b
Soit Th une triangulation du domaine.



La solution interpolé Πh u
e

u

Πh u

a Th b
Soit Πh u, la solution u interpolé sur l’ensemble des fonctions de
e
base lináires d´finies sur la triangulation Th .
e e



L’erreur d’interpolation u − Πh u

u

Πh u

a Th b
L’erreur d’interpolation u − Πh u est la diﬀ´rence entre la
e
solution exacte u et la solution interpol´e Πh u.
e



L’erreur d’interpolation u − Πh u

u

Πh u

a Th b
L’erreur d’interpolation u − Πh u pour des fonctions de base
lináires est dominé par la d´rivé seconde.
e e e e



Maillage optimal

u

Πh u

a Th b
Pour un nombre donn´ de sommets, le maillage qui minimise
e
l’erreur d’interpolation u − Πh u est celui qui concentre les
sommets l` o` la courbure est forte.
a u



Erreur d’interpolation en 2D et 3D

En 2D, les d´rivés secondes de la solution u forment une matrice
e e
hessienne
∂ 2 u/∂x 2 ∂ 2 u/∂x∂y
.
∂ 2 u/∂y ∂x ∂ 2 u/∂y 2
Si on rend la matrice hessienne d´finie positive, elle devient un
e
tenseur m´trique.
e
On d´finit ainsi un estimateur d’erreur anisotrope, qui ouvre la voie
e
` l’adaptation de maillage anisotrope.
a



Exemple analytique

Le domaine Ω est le carr´ [0, 1]×[0, 1]. Le probl`me est d´fini
e e e
comme suit:
−∆u + k 2 u = 0 dans Ω
u = g sur ∂Ω,
o` la condition de Dirichlet g est d´finie de telle sorte que la
u e
solution analytique est donné par
e

u = e −kx + e −ky .

Cette solution a des couches limites pour de grandes valeurs de k.

F. Guibault, P. Labb´ et J. Dompierre. “Adaptivity Works! Controling the
e
Interpolation Error in 3D”. Fifth World Congress on Computational Mechanics,
Vienna University of Technology, 2002.



Solution analytique

u = e −kx + e −ky , k = 100.



Maillages adapt´s
e

Gauche: Maillage uniforme de 268 sommets.
Centre: Maillage adapt´ isotrope de 268 sommets.
e
Droite: Maillage adapt´ anisotrope de 260 sommets.
e



Erreur d’interpolation
1

0.1

0.01

Total L2 error
0.001

0.0001

1e-05

1e-06
0.01 0.1
1/sqrt(N)

L’erreur d’interpolation en norme L2 converge en O(h2 ).
Pour obtenir une erreur de 0.001, il faudrait
200 ´l´ments avec un maillage adapt´ anisotrope,
ee e
2000 ´l´ments avec un maillage adapt´ isotrope,
ee e
20000 ´l´ments avec un maillage uniforme.
ee



What to Retain

For piecewise linear functions, the interpolation error of a
function u is dominated by second order derivatives.

The hessian matrix is used to deﬁned the metric tensor for
mesh adaptation.

Adapted anisotropic meshes minimize the interpolation error
for a given number of nodes.



Outline
1 Outline
General Framework
ı
ı
Interpolation Error
Approximation Error
4 Conclusions


Lesson on Approximation Error

The approximation error is bounded by the interpolation
error.


Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

Generalization of Delaunay Meshes for the Error Control in Numerical Simulations

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Generalization of Delaunay Meshes for the Error Control in Numerical Simulations