Mesh Quality

Mesh Quality
Julien Dompierre
julien@cerca.umontreal.ca

´
Centre de Recherche en Calcul Applique (CERCA)
´ ´
Ecole Polytechnique de Montreal

Mesh Quality – p. 1/331

Authors
• Research professionals
• Julien Dompierre
• Paul Labbé
• Marie-Gabrielle Vallet
• Professors
• François Guibault
• Jean-Yves Trépanier
• Ricardo Camarero


References – 1

J. D OMPIERRE , P. L ABBÉ ,
M.-G. VALLET, F. G UIBAULT
AND R. C AMARERO , Critères
de qualité pour les maillages
simpliciaux. in Maillage et
adaptation, Hermès, October
2001, Paris, pages 311–348.


References – 2

A. L IU and B. J OE, Relationship between
Tetrahedron Shape Measures, Bit, Vol. 34,
pages 268–287, (1994).


References – 3

P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F.
G UIBAULT and J.-Y. T RÉPANIER, A Universal
Measure of the Conformity of a Mesh with
Respect to an Anisotropic Metric Field,
Submitted to Int. J. for Numer. Meth. in Engng,
(2003).


References – 4

P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F.
G UIBAULT and J.-Y. T RÉPANIER, A Measure of
the Conformity of a Mesh to an Anisotropic
Metric, Tenth International Meshing Roundtable,
Newport Beach, CA, pages 319–326, (2001).


References – 5

P.-L. G EORGE AND H. B O -
ROUCHAKI , Triangulation de
Delaunay et maillage, appli-
cations aux éléments ﬁnis.
Hermès, 1997, Paris.
This book is available in En-
glish.


References – 6

P. J. F REY AND P.-L.
G EORGE, Maillages. Ap-
plications aux éléments ﬁnis.
Hermès, 1999, Paris.
This book is available in
English.


Table of Contents

1. Introduction 8. Non-Simplicial
2. Simplex Deﬁnition Elements
3. Degeneracies of 9. Shape Quality
Simplices Visualization
4. Shape Quality of 10. Shape Quality
Simplices Equivalence
5. Formulae for Sim- 11. Mesh Quality and
plices Optimization
6. Voronoi, Delaunay 12. Size Quality of
and Riemann Simplices
7. Shape Quality and 13. Universal Quality
Delaunay 14. Conclusions

Introduction and Justiﬁcations

We work on mesh generation, mesh adaptation
and mesh optimization.

How can we choose the conﬁguration that
produces the best triangles ? A triangle shape
quality measure is needed.

Face Flipping

produces the best tetrahedra ? A tetrahedron
shape quality measure is needed.


Edge Swapping
S4 S3 S4 S3

S5 S5
A A
B B

S2 S2

S1 S1

produces the best tetrahedra ? A tetrahedron
shape quality measure is needed.


Mesh Optimization

• Let O1 and O2 , two three-dimensional
unstructured tetrahedral mesh Optimizers.


Mesh Optimization

• What is the norm O of a mesh optimizer ?


Mesh Optimization

• What is the norm O of a mesh optimizer ?
• How can it be asserted that O1 > O2 ?


It’s Obvious !

• Let B be a benchmark.


It’s Obvious !

• Let M1 = O1 (B) be the optimized mesh
obtained with the mesh optimizer O1 .


It’s Obvious !



It’s Obvious !

• Common sense says : “The proof is in the
pudding”.


It’s Obvious !

• Common sense says : “The proof is in the
pudding”.
• If M1 > M2 then O1 > O2 .


Benchmarks for Mesh Optimization

J. D OMPIERRE, P. L ABBÉ, F. G UIBAULT and
R. C AMARERO.
Proposal of Benchmarks for 3D Unstructured
Tetrahedral Mesh Optimization.
7th International Meshing Roundtable, Dearborn,
MI, October 1998, pages 459–478.


The Trick...

• Because the norm O of a mesh optimizer is
unknown, the comparison of two optimizers is
replaced by the comparison of two meshes.


The Trick...

• What is the norm M of a mesh ?


The Trick...

• How can we assert that M1 > M2 ?


The Trick...

• How can we assert that M1 > M2 ?
• This is what you will know soon, or you
money back !


What to Retain

• This lecture is about the quality of the
elements of a mesh and the quality of a whole
mesh.


What to Retain

mesh.
• The concept of element quality is necessary
for the algorithms of egde and face swapping.


What to Retain

mesh.
• The concept of element quality is necessary
for the algorithms of egde and face swapping.
• The concept of mesh quality is necessary to
do research on mesh optimization.


Table of Contents

5. Formulae for Simplices 11. Mesh Quality and
6. Voronoi, Delaunay and Optimization
Riemann 12. Size Quality of
7. Shape Quality and Simplices
Delaunay 13. Universal Quality
14. Conclusions


Deﬁnition of a Simplex

Meshes in two and three dimensions are made of
polygons or polyhedra named elements.



The most simple amongst them, the simplices, are
those which have the minimal number of vertices.



The segment in one dimension.



The triangle in two dimensions.



The tetrahedron in three dimensions.



The hypertetrahedron in four dimensions.



The hypertetrahedron in four dimensions.
Quadrilaterals, pyramids, prisms, hexahedra and other
such aliens are named non-simplicial elements.


Deﬁnition of a d-Simplex in Rd

Let d + 1 points Pj = (p1j , p2j , . . . , pdj ) ∈ Rd , 1 ≤ j ≤ d + 1,
not in the same hyperplane, id est, such that the matrix of
order d + 1,
 
p11 p12 · · · p1,d+1
 p21 p22 · · · p2,d+1 
 
 . . 
A= . .
. ..
.
. . . 
. 

 pd1 pd2 · · · pd,d+1 
1 1 ··· 1

be invertible. The convex hull of the points Pj is named the
d-simplex of points Pj .


A Simplex Generates Rd

Any point X ∈ Rd , with Cartesian coordinates (xi )d , is
i=1
characterized by the d + 1 scalars λj = λj (X) deﬁned as
solution of the linear system
 d+1



 pij λj = xi for 1 ≤ i ≤ d,

j=1
d+1



 λj = 1,

j=1

whose matrix is A.


What to Retain

In two dimensions, the simplex is a triangle.


What to Retain

In three dimensions, the simplex is a tetrahedron.


What to Retain

The d + 1 vertices of a simplex in Rd give d vectors that
form a base of Rd .


What to Retain

The d + 1 vertices of a simplex in Rd give d vectors that
form a base of Rd .
The coordinates λj (X) of a point X ∈ Rd in the base
generated by the simplex are the barycentric
coordinates.


Table of Contents

14. Conclusions


Degeneracy of Simplices

A d-simplex made of d + 1 vertices Pj is degenerate if its
vertices are located in the same hyperplane, id est, if the
matrix A is not invertible.



A d-simplex is degenerate if its d + 1 vertices do not
generate the space Rd .



Such is the case if the d + 1 vertices are located in a
space of dimension lower than d.



A triangle is degenerate if its vertices are collinear or
collapsed.



collapsed.
A tetrahedron is degenerate if its vertices are coplanar,
collinear or collapsed.



collapsed.
collinear or collapsed.
Nota bene : Strictly speaking, accordingly to the
deﬁnition, a degenerate simplex is no longer a simplex.


Degeneracy Criterion

A d-simplex is degenerate if its matrix A is not
invertible. A matrix is not invertible if its determinant is
null.



null.
The size of a simplex is its area in two dimensions and
its volume in three dimensions.



null.
The size of a d-simplex K made of d + 1 vertices Pj is
given by
size(K) = det(A)/d!.



null.
given by

A triangle is degenerate if its area is null.



null.
given by

A triangle is degenerate if its area is null.
A tetrahedron is degenerate if its volume is null.


Taxonomy of Degenerate Simplices

This taxonomy is based on the different possible
degenerate states of the simplices.



There are three cases of degenerate triangles.



There are ten cases of degenerate tetrahedra.



There are ten cases of degenerate tetrahedra.
In this classiﬁcation, the four symbols
, , and stand for vertices of multiplicity
simple, double, triple and quadruple respectively.


1 – The Cap

Name h −→ 0 h=0

C
h
Cap A B A C B

Degenerate edges : None
Radius of the smallest circumcircle : ∞


2 – The Needle

Name h −→ 0 h=0

C
h
Needle A B A,C B

Degenerate edges : AC
Radius of the smallest circumcircle : hmax /2


3 – The Big Crunch

Name h −→ 0 h=0

C
h h B
Big A h A,B,C
Crunch

Degenerate edges : All
Radius of the smallest circumcircle : 0
The Big Crunch is the theory opposite of the Big Bang.


Degeneracy of Tetrahedra

There is one case of degeneracy resulting in four
collapsed vertices.
There are ﬁve cases of degeneracy resulting in four
collinear vertices.
There are four cases of degeneracy resulting in four
coplanar vertices.
D D

d
A C A a C
b
B B c


1 – The Fin

Name h −→ 0 h=0

D
h D
A C A C
Fin
B B

Degenerate faces : One cap
Radius of the smallest circumsphere : ∞


2 – The Cap

Name h −→ 0 h=0

D
Cap A h C A D C
B B
Degenerate faces : None


3 – The Sliver

Name h −→ 0 h=0

D
h C
Sliver A C A D
B B
Degenerate faces : None
Radius of the smallest circumsphere : rABC or ∞


4 – The Wedge

Name h −→ 0 h=0

D
h C, D
Wedge A C A
B B
Degenerate edges : CD
Degenerate faces : Two needles
Radius of the smallest circumsphere : rABC


5 – The Crystal

Name h −→ 0 h=0

D
A h
Crystal h C A B D C
B
Degenerate faces : Four caps


6 – The Spindle

Name h −→ 0 h=0

D
A h A B, D C
Spindle h C
B
Degenerate edges : BD
Degenerate faces : Two caps and two needles


7 – The Splitter

Name h −→ 0 h=0

D
h C
Splitter A A D B, C
h
B
Degenerate edges : BC
Degenerate faces : Two caps and two needles


8 – The Slat

Name h −→ 0 h=0

D
h C
Slat h A, D B, C
A
B
Degenerate edges : AD and BC
Degenerate faces : Four needles
Radius of the smallest circumsphere : hmax /2


9 – The Needle

Name h −→ 0 h=0

D
h
h hC A B, C, D
Needle A
B
Degenerate edges : BC, CD and DB
Degenerate faces : Three needles and one Big Crunch
Radius of the smallest circumsphere : hmax /2


10 – The Big Crunch

Name h −→ 0 h=0

D
Big A hh C
h
h A, B, C, D
Crunch h Bh

Degenerate edges : All
Degenerate faces : Four Big Crunches
Radius of the smallest circumsphere : 0


What to Retain

collapsed, hence if its area is null.


What to Retain

There are three cases of degeneracy of triangles.


What to Retain

collinear or collapsed, hence if its volume is null.


What to Retain

collinear or collapsed, hence if its volume is null.
There are ten cases of degeneracy of tetrahedra.


Table of Contents

14. Conclusions


Shape Quality of Simplices

An usual method used to quantify the quality of a mesh
is through the quality of the elements of that mesh.



A criterion usually used to quantify the quality of an
element is the shape measure.



A criterion usually used to quantify the quality of an
element is the shape measure.
This section is a guided tour of the shape measures
used for simplices.


The Regular Simplex

Deﬁnition : An element is regular if it maximizes its measure for
a given measure of its boundary.


The Regular Simplex

The equilateral triangle is regular because it maximizes
its area for a given perimeter.


The Regular Simplex

The equilateral triangle is regular because it maximizes
its area for a given perimeter.
The equilateral tetrahedron is regular because it
maximizes its volume for a given surface of its faces.


Simplicial Shape Measure

Deﬁnition A : A simplicial shape measure is a
continuous function that evaluates the shape of a simplex.
It must be invariant under translation, rotation, reﬂection
and uniform scaling of the simplex. A shape measure is
called valid if it is maximal only for the regular simplex and
if it is minimal for all degenerate simplices. Simplicial
shape measures are scaled to the interval [0, 1], and are 1
for the regular simplex and 0 for a degenerate simplex.


Remarks

The invariance under translation, rotation and
reﬂection means that the simplicial shape measures
must be independent of the coordinates system.


Remarks

The invariance under a valid uniform scaling means
that the simplicial shape measures must be
dimensionless (independent of the unit system).


Remarks

The invariance under a valid uniform scaling means
that the simplicial shape measures must be
dimensionless (independent of the unit system).
The continuity means that the simplicial shape
measures must change continuously in function of the
coordinates of the vertices of the simplex.


The Radius Ratio

The radius ratio of a simplex K is a shape measure deﬁned
as ρ = d ρK /rK , where ρK and rK are the radius of the
incircle and circumcircle of K (insphere and circumsphere
in 3D), and where d is the dimension of space.

K
ρK

rK


The Mean Ratio

Let R(r1 , r2 , r3 [, r4 ]) be an equilateral simplex having the
same [area|volume] than the simplex K(P1 , P2 , P3 [, P4 ]). Let
N be the matrix of transformation from R to K, i.e.
Pi = N ri + b, 1 ≤ i ≤ [3|4], where b is a translation vector.
s y K
K = NR + b
R
r
b
x


The Mean Ratio

Then, the mean ratio η of the simplex K is the ratio of the
geometric mean over the algebraic means of the
eigenvalues λ1 , λ2 [,λ3 ] of the matrix N T N .
 √ √
 2 λ1 λ2

2
4 3 SK
d 
 λ +λ = in 2D,
d
λi   1 2
2
1≤i<j≤3 Lij
i=1

η= =
d 
1
λi  3 √λ 1 λ 2 λ 3

 3
12 3 9VK2
d 

i=1  λ +λ +λ = L 2
in 3D.
1 2 3 1≤i<j≤4 ij


The Condition Number

F ORMAGGIA and P EROTTO (2000) use the inverse of the
condition number of the matrix.
min λiλ1
i
κ= = ,
max λi λd
i

if the eigenvalues are sorted in increasing order.


The Frobenius Norm

Freitag and Knupp (1999) use the Frobenius norm of the
matrix N = AW −1 to deﬁne a shape measure.

d d
κ= = ,
tr(N T N )tr((N T N )−1 ) d d
λi λ−1
i
i=1 i=1

where the λi are the eigenvalues of the tensor N T N .


The Minimum of Solid Angles

The simplicial shape measure θmin based on the minimum
of solid angles of the d-simplex is deﬁned by

θmin = α−1 min θi ,
1≤i≤d+1

The coefﬁcient α is the value of each solid angle of the
regular d-simplex, given by α = π/3 in two dimensions
√
and α = 6 arcsin 3/3 − π in three dimensions.


The sin of θmin

From a numerical point of view, a less expensive simplicial
shape measure is the sin of the minimum solid angle. This
avoids the computation of the arcsin(·) function in the
computation of θi in 2D and θi in 3D.

σmin = β −1 min σi ,
1≤i≤d+1

where σi = sin(θi ) in 2D and σi = sin(θi /2) in 3D. β is the
value of σi for all solid angles of the regular simplex, given
√ √
by β = sin(α) = 3/2 in 2D and β = sin(α/2) = 6/9 in 3D.


Face Angles

We can deﬁne a shape measure based on the minimum of
the twelve angles of the four faces of a tetrahedron. This
angle is π/3 for the regular tetrahedron.
But this shape measure is not valid according to
Deﬁnition A because it is insensitive to degenerate
tetrahedra that do not have degenerate faces (the sliver
and the cap).


Dihedral Angles

The dihedral angle is the angle between the intersection of
two adjacent faces to an edge with the perpendicular plane
of the edge.

Pj
ϕij

Pi
The minimum of the six dihedral angles ϕmin is used as a
shape measure.


Dihedral Angles

αϕmin = min ϕij = min (π − arccos (nij1 · nij2 )) ,
1≤i<j≤4 1≤i<j≤4

where nij1 and nij2 are the normal to the adjacent faces of
the edge Pi Pj , and where α = π − arccos(−1/3) is the
value of the six dihedral angles of the regular tetrahedron.
But this shape measure is not valid according to
Deﬁnition A. The smallest dihedral angles of the needle,
the spindle and the crystal can be as large as π/3.


The Interpolation Error Coefficient

In finite element, the interpolation error of a function over
an element is bounded by a coefficient times the
semi-norm of the function. This coefficient is the
ratio DK /̺K where DK is the diameter of the element K
and ̺K is the roundness of the element K.
 √ ρK
 2 3
 in 2 D,
hmax
γ=
 2√6 ρK in 3 D.

hmax


The Edge Ratio

Ratio of the smallest edge over the tallest.

r = hmin /hmax .

The edge ratio r is not a valid shape measure according to
Deﬁnition A because it does not vanish for some
degenerate simplices. In 2D, it can be as large as 1/2 for
√
the cap. In 3D, it can be as large as 2/2 for the sliver, 1/2
√
for the ﬁn, 3/3 for the cap and 1/3 for the crystal.


Other Shape Measure – 1

hmax /rK , the ratio of the diameter of the tetrahedron
over the circumradius, in B AKER, (1989). This is not a
valid shape measure.
hmin /rK , the ratio of the smallest edge of the
tetrahedron over the circumradius, in M ILLER et al
(1996). This is not a valid shape measure.
VK /rK 3 , the ratio of the volume of the tetrahedron over
the circumradius, in M ARCUM et W EATHERILL, (1995).



4 4 2 −3
VK i=1 Si ,
the ratio of the volume of the
tetrahedron over the area of its faces, in D E C OUGNY et
al (1990). The evaluation of this shape measure, and its
validity, are a complex problem for tetrahedra that
degenerate in four collinear vertices.
−3
VK 1≤i<j≤4 Lij , the ratio of the volume of the
tetrahedron over the average of its edges, in
DANNELONGUE and TANGUY (1991), Z AVATTIERI et al
(1996) and W EATHERILL et al (1993).



2
VK Lij − L12 L34 − L13 L24
1≤i<j≤4 −3/2

−L14 L23 + L2
ij
1≤i<j≤4
the ratio of the volume of the tetrahedron over a sum, at
the power three halfs, of many terms homogeneous to the
square of edge lenghts, in B ERZINS (1998).



−3
VK L2
1≤i<j≤4 ij , the ratio of the volume of the
tetrahedron over the quadratic average of the six edges,
in G RAICHEN et al (1991).
And so on... This list is surely not exhaustive.


There Exists an Inﬁnity of Shape
Measures

If µ and ν are two valid shape measures, if c, d ∈ R+ , then
µc ,
c(µ−1)/µ with c > 1,
αµc + (1 − α)ν d with α ∈ [0, 1],
µc ν d
are also valid simplicial shape measures.


What to Retain

The regular simplex is the equilateral one, ie, where all
its edges have the same length.


What to Retain

A shape measures evaluates the ratio to equilaterality.


What to Retain

A non valid shape measure does not vanish for all
degenerate simplices.


What to Retain

There exists an inﬁnity of valid shape measures.


What to Retain

There exists an inﬁnity of valid shape measures.
The goal of research is not to ﬁnd an other one way
better than the other ones.


Table of Contents

14. Conclusions


Formulae for the Triangle

A triangle is completely deﬁned by the knowledge of the
length of its three edges.

Quantities such that inradius, circumradius, angles, area,
etc, can be written in function of the edge lengths of the
triangle.

Let K be a non degenerate triangle of vertices P1 , P2
and P3 . The lengths of the edges Pi Pj of K are
denoted Lij = Pj − Pi , 1 ≤ i < j ≤ 3.


The Half-Perimeter

The half-perimeter pK is given by

(L12 + L13 + L23 )
pK = .
2


Heron’s Formula

The area SK of a triangle can also be written in function of
the edge lengths with Heron’s formula :
2
SK = pK (pK − L12 )(pK − L13 )(pK − L23 ).


Radius of the Incircle

The radius ρK of the incircle of the triangle K is given by

SK
ρK = .
pK


Radius of the Circumscribed Circle

The radius rK of the circumcircle of the triangle K is given
by
L12 L13 L23
rK = .
4SK


Element Diameter

The diameter of an element is the biggest Euclidean
distance between two points of an element. For a triangle,
this is also the length of the biggest edge hmax

hmax = max(L12 , L13 , L23 ),

The length of the smallest edge is denoted hmin

hmin = min(L12 , L13 , L23 ).


Solid Angle

The angle θi at vertex Pi of triangle K is the arc length
obtained by projecting the edge of the triangle opposite
to Pi on a unitary circle centerered at Pi . The angle can be
written in function of the edge lengths as
−1
θi = arcsin 2SK Lij Lik .
j,k=i
1≤j<k≤3


Formulae for the Tetrahedron

A tetrahedron is completely deﬁned by the knowledge of
the length of its six edges.

Quantities such that inradius, circumradius, angles,
volume, etc, can be written in function of the edge lengths
of the tetrahedron.


Formulae for the Tetrahedron

Let K be a non degenerate tetrahedron of vertices P1 , P2 ,
P3 and P4 . The lengths of the edges Pi Pj of K are denoted
Lij = Pj − Pi , 1 ≤ i < j ≤ 4. The area of the four faces of
the tetrahedron, △P2 P3 P4 , △P1 P3 P4 , △P1 P2 P4
and △P1 P2 P3 , are denoted by S1 , S2 , S3 and S4 . Finally, VK
is the volume of the tetrahedron K.


3D “Heron’s” Formula

Let a, b, c, e, f and g be the length of the six edges of the
tetrahedron such that the edges a, b and c are connected
to the same vertex, and such that e is the opposite edge of
a, f is opposite of b and g is the opposite of c. The volume
VK is then
2
144VK = 4a2 b2 c2
+ (b2 + c2 − e2 ) (c2 + a2 − f 2 ) (a2 + b2 − g 2 )
2 2
− a2 (b2 + c2 − e2 ) − b2 (c2 + a2 − f 2 )
2
− c2 (a2 + b2 − g 2 ) .


Radius of the Insphere

The radius ρK of the insphere of the tetrahedron K is given
by
3VK
ρK = .
S1 + S2 + S3 + S4


Radius of the Circumsphere

The radius rK of the circumsphere of the tetrahedron K is
given by

(a + b + c)(a + b − c)(a + c − b)(b + c − a)
rK = .
24VK
where a = L12 L34 , b = L13 L24 and c = L14 L23 are the
product of the length of the opposite edges of K (two
edges are opposite if they do not share a vertex.


Element Diameter

The diameter of an element is the biggest Euclidean
distance between two points of an element. For a
tetrahedron, this is also the length of the biggest edge hmax

hmax = max(L12 , L13 , L14 , L23 , L24 , L34 ),

The length of the smallest edge is denoted hmin

hmin = min(L12 , L13 , L14 , L23 , L24 , L34 ).


Solid Angle

The solid angle θi at vertex Pi of the tetrahedron K, is the
area of the spherical sector obtained by projecting the face
of the tetrahedron opposite to Pi on a unitary sphere
centerered at Pi .
P4

P1 θ1 P3

P2


Solid angle

L IU and J OE (1994) gave a formula to compute the solid
angle in function of edge lengths :
−1/2
θi = 2 arcsin 12VK (Lij + Lik )2 − L2
jk .
j,k=i
1≤j<k≤4


Table of Contents

plices Optimization

Which Is the Most Beautiful Triangle ?



A



A B


If You Chose the Triangle A...


If You Chose the Triangle A...

A
You are wrong !


If You Chose the Triangle B...


If You Chose the Triangle B...

B
You are wrong again !



A B
None of these answers !


Which Is the Most Beautiful Woman ?



A



A B


You Probably chose...


You Probably chose...

A B
Woman A.

And if One Asked these Gentlemen...


These Gentlemen Would Choose...


These Gentlemen Would Choose...

A B
Woman B.

Which Is the Most Beautiful Woman...

There is no absolute answer because the
question is incomplete.
One did not specify who was going to judge the
candidates, which was the scale of evaluation,
which were the measurements used, etc.



A B



A B
The question is incomplete : It misses a way of
measuring the quality of a triangle.


Voronoi Diagram

Georgy Fedoseevich VORO -
NOÏ . April 28, 1868, Ukraine
– November 20, 1908, War-
saw. Nouvelles applications
des paramètres continus à
la théorie des formes qua-
dratiques. Recherches sur
les parallélloèdes primitifs.
Journal Reine Angew. Math,
Vol 134, 1908.


The Perpendicular Bisector

Let S1 and S2 be two
vertices in R2 . The
perpendicular bisec-
d(P, S1 ) P tor M (S1 , S2 ) is the
S1
d(P, S2 ) locus of points equi-
distant to S1 and S2 .
S2 M (S1 , S2 ) = {P ∈
M R2 | d(P, S1 ) = d(P, S2 )},
where d(·, ·) is the Eucli-
dean distance between
two points of space.


A Cloud of Vertices

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

S2 S11
S9 S10
S5 S6 S4 S8
S1
S7 S12 S3


The Voronoi Cell

Deﬁnition : The Voronoi cell C(Si ) associated to
the vertex Si is the locus of points of space which
is closer to Si than any other vertex :
C(Si ) = {P ∈ R2 | d(P, Si ) ≤ d(P, Sj ), ∀j = i}.

C(Si )
Si


The Voronoi Diagram

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


Properties of the Voronoi Diagram

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


What to Retain

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


Delaunay Triangulation

Boris Nikolaevich D ELONE or
D ELAUNAY. 15 mars 1890,
Saint Petersbourg — 1980.
Sur la sphère vide. À la mé-
moire de Georges Voronoi,
Bulletin of the Academy of
Sciences of the USSR, Vol. 7,
pp. 793–800, 1934.


Triangulation of a cloud of Points

The same cloud of points can be triangulated in
many different fashions.

...


Triangulation of a Cloud of Points
...

...



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


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


Violation of the Empty Sphere Criterio

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



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.


Delaunay Algorithm

The circumscri-
bed sphere of a
simplex has to be S3
computed.
S2
This amounts to ρout
computing the cen- C
ter of a simplex.
The center is the
point at equal dis-
tance to all the
vertices of the sim- S1
plex.

Delaunay Algorithm

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


What to Retain

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

A Delaunay triangulation of a cloud of points
is a triangulation based on the notion of
distance.


Duality Delaunay-Voronoï

The Voronoï diagram is the dual of the Delaunay
triangulation and vice versa.


Voronoï and Delaunay in Nature

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


Voronoï and Delaunay In Nature


A Turtle


A Pineapple


The Devil’s Tower


Dry Mud


Bee Cells


Dragonﬂy Wings


Pop Corn


Fly Eyes


Carbon Nanotubes


Soap Bubbles


A Geodesic Dome


Biosphère de Montréal


Streets of Paris


Roads in France


Where Is this Guy Going ? ! !

A simplicial shape measure is an evaluation
of the ratio to equilarity.



The Voronoï diagram of a cloud of points is a
notion of distance.



notion of distance.
distance.



notion of distance.
distance.
The notion of distance can be generalized.



notion of distance.
distance.
The notion of distance can be generalized.
The notions of shape measure, of Voronoï
diagram and of Delaunay triangulation Quality –be
Mesh
can p. 129/331

Nikolai Ivanovich Lobachevsky

N IKOLAI I VANOVICH
LOBACHEVSKY, 1
décembre 1792, Nizhny
Novgorod — 24 février
1856, Kazan.


János Bolyai

J ÁNOS BOLYAI, 15 dé-
cembre 1802 à Kolozsvár,
Empire Austrichien (Cluj,
Roumanie) — 27 janvier
1860 à Marosvásárhely,
Empire Austrichien (Tirgu-
Mures, Roumanie).


Bernhard RIEMANN

G EORG F RIEDRICH B ERN -
HARD RIEMANN, 7 sep-
tembre 1826, Hanovre — 20
juillet 1866, Selasca. Über die
Hypothesen welche der Geo-
metrie zu Grunde liegen. 10
juin 1854.


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.


The Metric in the Merriam-Webster


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(x, y) ≥ 0 for all x, y in S ;
(ii) d(x, y) = 0 if and only if x = y ;
(iii) d(x, y) = d(y, x) for all x, y in S ;
(iv) d(x, y) ≤ d(x, z) + d(z, y) for all x, y, z in S.


The Euclidean Distance is a Metric

In the previous deﬁnition of a metric, let the set S
be I 2 , the function
R
d : I 2 ×I 2 → I
R R R
x1 x2
× → (x2 − x1 )2 + (y2 − y1 )2
y1 y2

is a metric on I 2 .
R


Metric Space


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.


The Scalar Product is a Metric

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

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

= (xB − xA )2 + (yB − yA )2 .


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


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 ,
−→ − 1/2
→
= AB, M AB ,
− T −
→ →
= AB M AB.


Euclidean Length with M = I

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


αβ
Metric Length with M = βγ

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


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.

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


Metric and Delaunay Mesh


Which is the Best Triangle ?

A B
The question is incomplete. The way to measure
the quality of the triangle is missing.



A B


Example of an Adapted Mesh

Adapted mesh and solution for a transonic
visquous compressible ﬂow with Mach 0.85 and
Reynolds = 5 000.

Zoom on Boundary Layer–Shock


What to Retain

Beauty, quality and shape are relative
notions.


What to Retain

notions.
We ﬁrst need to deﬁne what we want in order
to evaluate what we obtained.


What to Retain

notions.
“What we want” is written in the form of metric
tensors.


What to Retain

notions.
“What we want” is written in the form of metric
tensors.
A shape measure is a measure of the
equilarity of a simplex in this metric.


Shape Measure in a Metric

First method (constant metric)

For a simplex K, evaluate the metric tensor at
several points (Gaussian points) and ﬁnd an
averaged metric tensor.
Take this averaged metric tensor as constant
over the whole simplex and evaluate the shape
measure using this metric.



Second method (constant metric)

For a simplex K, evaluate the metric tensor at
one point (Gaussian point) and take the metric
as constant over the whole simplex. Evaluate the
shape measure using this metric.
Repeat this operation at several points and
average the shape measures.
This is what is done at INRIA.



Third methode (variable metric)

Express the shape measure as a fonction of
edge lengths only.
Evaluate the length of the edges in the metric
and compute the shape measure with these
lengths.
This is what is done in OORT.



Fourth method (variable metric)

Express the shape measure in function of the
length of the edges, the area and the volumes.
Evaluate the lengths, the area and the volume in
the metric.



Fifth method (variable metric)
Know how to evaluate quantities such as the
radius of the inscribed circle, of the
circumscribed circle, the solid angle, etc, in a
metric.
In the general sense, the triangular inequality is
not veriﬁed in a variable metric. Neither is the
sum of the angles equal to 180 degrees, etc.
The evaluation of a shape measure in a variable
metric in all its generality is an opened problem.
For the moment, it is approximated.

Table of Contents

plices Optimization

Shape Measures and Delaunay Critero
Delaunay meshes have several smoothness
properties.
The Delaunay mesh minimizes the maximum value of
all the element circumsphere radii.
When the circumsphere center of all simplices of a
mesh lie in their respective simplex, then the mesh is a
Delaunay mesh.
In a Delaunay mesh, the sum of all squared edge
lengths weighted by the volume of elements sharing that
edge is minimal.


3D-Delaunay Mesh and Degeneracy

In three dimensions, it is well known that
Delaunay meshes can include slivers which are
degenerate elements.

Why ?

How to avoid them ?


Empty Sphere Criterion of Delaunay

The empty sphere criterion of Delaunay is not a
shape measure, but it can be used like a shape
measure in an edge swapping algorithm.


Edge Swapping and θmin Shape Measu
During edge swapping, using the empty sphere
criterion (Delaunay criterion)
⇐⇒
Using the θmin shape measure (maximize the
minimum of the angles).

θ3 θ3 θ6
θ1 θ2 θ1 θ5
θ4 θ6
θ5 θ2 θ4


What to Retain

The empty sphere criterion of Delaunay is not
a shape measure but it can be used as a
shape measure.


What to Retain

shape measure.
In two dimensions, in the edge swapping
algorithm (Lawson’s method), the empty
sphere criterion of Delaunay is equivalent to
the θmin shape measure.


What to Retain

shape measure.
In two dimensions, in the edge swapping
algorithm (Lawson’s method), the empty
sphere criterion of Delaunay is equivalent to
the θmin shape measure.
There is a multitude of valid shape measures,
and thus a multitude of generalizations of the
Delaunay mesh.


Delaunay and Circumscribed Sphere

As the circumscribed sphere of a tetrahedron
gets larger, there are more chances that another
vertex of the mesh happens to be in this sphere,
and the chances that this tetrahedron and the
mesh satisfy the Delaunay criterion get smaller.
As the circumscribed sphere of a tetrahedron
gets smaller, there are fewer chances that
another vertex of the mesh happens to be in this
sphere, and the chances that this tetrahedron
and the mesh satisfy the Delaunay criterion get
bigger.

Circumscribed Sphere of Infinite Radi

The tetrahedra that degenerate into a fin, into a
cap, into a crystal, into a spindle and into a
splitter
D D D
h A h
A C A h C h C
B B B
D D
A h C
h C A h h
B B
have a circumscribed sphere of infinite radius.


Circumscribed Sphere of Bounded Ra

The tetrahedra that degenerate into a sliver, into
a wedge, into a slat, into a needle and into a
Big Crunch
D D
h h D
A C A C h C
A h
B B B
D D
h
C A hh h C
hh h
A
B h Bh
have a circumscribed sphere of bounded radius.


What to Retain

The empty sphere criterion of
Delaunay is not a valid shape
measure sensitive to all the possible
degeneracies of the tetrahedron.


Circumscribed Sphere of Bounded Ra

Amongst the degenerate tetrahedra that have a
circumscribed sphere of bounded radius, the
wedge, the slat, the needle and the Big Crunch
can be eliminitated
D
h D
A C h C
A h
B B
D D
h
C A hh h C
hh h
A
B h Bh
since they have several superimposed vertices.


The Sliver

And so, ﬁnally, we come to the sliver,
D
h C
A C A D
B B
a degenerate tetrahedron having disjoint vertices
and a bounded circumscribed sphere radius,
which makes it “Delaunay-admissible”.


Non-Convex Quadrilateral

It is forbidden to swap an edge of a non-convex
quadrilateral.
S3 S3
T1
T1 T2 S2
S2 T2
S1 S4 S1 S4
S3

T1 S2
T2
S1 S4 S1 S4

Non-Convex Quadrilateral

S3 Two adjacent triangles
forming a non-convex
quadrilateral necessa-
T1 T2 rily satisfy the empty
S2 sphere criterion of
S1 S4 Delaunay.


Loss of the Convexity Property in 3D


What to Retain

The empty sphere criterion of Delaunay is
more or less a simplicial shape measure.


What to Retain

sensitive to all the possible degeneracies of
the tetrahedron.


What to Retain

sensitive to all the possible degeneracies of
the tetrahedron.
A valid shape measure, sensitive to all the
possible degeneracies of the tetrahedron,
used in an edge swapping and face swapping
algorithm should lead to a mesh that is not a
Delaunay mesh, but that is of better quality.


Table of Contents

plices Optimization

Non-Simplicial Elements

This section proposes a method to generalize
the notions of regularity, of degeneration and of
shape measure of simplices to non simplicial
elements ; i.e., to quadrilaterals in two
dimensions, to prisms and hexahedra in three
dimensions.


Non-Simplicial Elements

On Element Shape Measures for Mesh
Optimization
PAUL L ABBÉ , J ULIEN D OMPIERRE , F RANÇOIS
G UIBAULT AND R ICARDO C AMARERO
Presented at the 2nd Symposium on Trends in
Unstructured Mesh Generation, Fifth US National
Congress on Computational Mechanics, 4–6
august 1999 University of Colorado at Boulder.


Regularity Generalization

An equilateral quadrilateral, ie that has four
edges of same length, is not necessarily a
square...



square...
Déﬁnition : An element, be it simplicial or
not, is regular if it maximizes its measure for a
given measure of its boundary.



square...
The equilateral triangle is regular because it
maximizes its area for a given perimiter.



square...
The equilateral triangle is regular because it
maximizes its area for a given perimiter.
The equilateral tetrahedron is regular
because it maximizes its volume for a given
surface of its faces.

Regular Non Simplicial Elements

The regular quadrilateral is the square.
The regular hexahedron is the cube.
The regular prism is the ... regular prism ! Its
two triangular faces are equilateral triangle
whose edges measure a. Its three quadrilateral
faces are rectangles that have a base of
√
length a and a height of length a/ 3.


Quality of Non Simplicial Elements

Proposed Extension : The shape measure of a
non simplicial element is given by the minimum
shape measure of the corner simplices
constructed from each vertex of the element and
of its neighbors.


Shape Measure of a Quadrilateral

The shape measure of a quadrilateral is the
minimum of the shape measure of its four corner
triangles formed by its four vertices.
D C D C D C D C

A B A BA B BA


Shape Measure of a Prism

The shape measure of a prism is the minimum of
the shape measure of its six corner tetrahedron
formed by its six vertices.
F
D E
F C C C
D E A B A B A B
C F F F
A B D E D E D E
C
A B


Shape Measure of an Hexahedron

The shape measure of an hexahedron is the
minimum of its eight corner tetrahedron formed
by its eight vertices.
G H
E F
H G D C D C D C
E F AHB A B HB A H
D C G G G
A B
E F E F F E
C D
A B

Shape of the Corner Simplex

The corner simplices constructed for the non
simplicial elements are not regular simplices.
For the square, the four corner triangles are
isosceles right-angled triangles.
For the cube, the eight corner tetrahedra are
isosceles right-angled tetrahedra.
For the regular prism, the six corner
tetrahedra are tetrahedron with an equilateral
triangle of side a, √ a fourth perpendicular
and
edge of length a/ 3.


Shape of the Corner Simplex

Each non simplicial shape measure has to be
normalized so as to be a shape measure equal
to unit value for regular non simplicial elements.
ρ η θmin γ
√ √
2 3 3 3
Square √
1+ 2 2 4
√
1+ 2
√ √ √
18√ 1 2 arcsin(1/ 22+12 3) 3 √6
Prism √
5(7+ 13)
√
3
2
√
6 arcsin(1/ 3)−π 7+ 13
√ √ √ √ √
2 3 2 arcsin((2− 2)/(2 3))
Cube 3−1 3 2 √
6 arcsin(1/ 3)−π
3−1


Degenerate Non Simplicial Elements

Déﬁnition :A non simplicial element is
degenerate if at least one of its corner simplices
is degenerate.
If at least one of the corner simplices is more
than degenerate, meaning that it is inverted (of
negative norm), then the non simplicial element
is concave and is also considered degenerate.


Twisted Non Simplicial Elements

In three dimensions, the deﬁnition of the shape
measure of non simplicial elements has one
ﬂaw : it is not sensitive to twisted elements.
E
D F
E C C C
F D A B A B A B
C E E E
A B F D F D F D
C
A B


Twist of Quadrilateral Faces

A critera used to measure the twist of a
quadrilateral face ABCD is to consider the
dihedral angle between the triangles ABC
and ACD on one hand, and between the
triangles ABD and BCD on the other hand.
If these dihedral angles are π, then the
quadrilateral face is a plane (not twisted). The
twist in the quadrilateral increases as the angles
differ from π.


Twist of Quadrilateral Faces

Deﬁnition :Given a valid simplicial shape
measure, the twist of a quadrilateral face is equal
to the value of the shape measure for the
tetrahedron constructed by the four vertices of
the quadrilateral face.
Thus, a plane face has no twist because the four vertices
form a degenerated tetrahedron and all valid shape
measures are null.
As a quadrilateral face is twisted, its vertices move away
from coplanarity, and the shape measure of the generated
tetrahedron gets larger.

What to Retain

The shape, the degeneration, the convexity,
the concavity and the torsion can be rewritten
as a function of simplices.
An advantage of this approach is that once that
the measurement and the shape measures for
the simplices are programmed, in Euclidean as
well as with a Riemannian metric, the extension
for non simplicial elements is direct.


Table of Contents
plices Optimization


Visualizing Shape Measures

1

0.5

QK (C)
y 2
C(x, y) x
A(0, 1/2) 1

B(0, −1/2) 1 y 0

-1 3

0 -2
0
1 x
2

Position of the three vertices A, B and C of the
triangle K used to construct the contour plots of
a shape measure.


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