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.



1 1 1

y0 y0 y0

-1 -1 -1

0 1 x 2 3 0 1 x2 3 0 1 x2 3

The edge ratio r on the left. The minimum of the
solid angles θmin in the center. The interpolation
error coefﬁcient γ on the right.



1 1

y0 y0

-1 -1

0 1 x2 3 0 1 x2 3

The radius ratio ρ on the left and the mean ratio η
on the right.


Which Shape Measure is Best

1
r is not a valid shape mea-
y0 sure.
θmin and γ are continuous
-1
but not differentiable.
0 1 x2 3 ρ and η are continuous and
differentiable.
ρ is numerically unstable.
η is the least costly.
η has circular contour
lines.


3D Rendering of Shape Measures

In 3D, 5 parameters are necessary. Two are ﬁxed
and the inﬂuence of the 3 others is visualized.


Rendering Taking a Metric Into Ac-
count
1

y0
0.2 0
M=
-1 0 1

0 1 x 2 3

Mean ratio η


Rendering Taking a Metric into Ac-
count
1

y0
20 0
M=
-1 0 1

0 1 x 2

Mean ratio η


count
1

y0
0.9 0.4
M=
-1 0.4 1

0 1 x 2 3

Mean ratio η


count
1

y0
1 0
M=
-1 0 1
0 1 x2 3

Mean ratio η


What to Retain
Mean ratio η is the privileged shape measure.


What to Retain
Circular contour lines in Euclidean space
become ellipses in the general case.


What to Retain
The shape of a triangle is a quality measure
that is relative.


What to Retain
that is relative.
A good triangle in a metric tensor is not
beautiful in a different metric tensor.


What to Retain
that is relative.
A good triangle in a metric tensor is not
beautiful in a different metric tensor.
The quality of a triangle depends on the value
of the size speciﬁcation map given in the form
of a metric tensor.


Table of Contents
plices Optimization


Equivalence of Shape Measures



1
Superposition of
contour plots of
y0
simplex shape mea-
sures ρ, η, θmin et
-1 γ.

0 1 x2 3


Deﬁnition B (from L IU and J OE, 1994) : Let µ
and ν be two different simplicial shape measures
having values in the interval [0, 1]. µ is said to be
equivalent to ν if there exists positive
constants c0 , c1 , e0 and e1 such that
c0 ν e0 ≤ µ ≤ c1 ν e1 .


Optimal Bounds
In the equivalence relation of shape measures

e0 e1
c0 ν ≤ µ ≤ c1 ν ,


Optimal Bounds

e0 e1
c0 ν ≤ µ ≤ c1 ν ,
the lower bound is said to be optimal if e0 is the
smallest possible exponent,


Optimal Bounds

e0 e1
c0 ν ≤ µ ≤ c1 ν ,
the lower bound is said to be optimal if e0 is the
smallest possible exponent,

and the upper bound is said to be optimal if e1 is
the biggest possible exponent.


Tight Bounds

c0 ν e0 ≤ µ ≤ c1 ν e1 ,


Tight Bounds

c0 ν e0 ≤ µ ≤ c1 ν e1 ,

the lower bound is said to be tight if c0 is the
biggest possible constant,


Tight Bounds

c0 ν e0 ≤ µ ≤ c1 ν e1 ,

the lower bound is said to be tight if c0 is the
biggest possible constant,
and the upper bound is said to be tight if c1 is the
smallest possible constant.


Equivalence Relation
It is indeed an equivalence relation because it is

reﬂexive,
symmetric,
transitive.


Symmetric Relation
If µ is equivalent to ν with
c0 ν e0 ≤ µ ≤ c1 ν e1 ,
then ν is equivalent to µ with
c2 µe2 ≤ ν ≤ c3 µe3 ,
−1/e1 −1/e0
where c2 = c1 , e2 = 1/e1 , c3 = c0
and e3 = 1/e0 .


Transitive Relation
If µ is equivalent to ν and if ν is equivalent to υ
with
c0 ν e0 ≤ µ ≤ c1 ν e1 and c2 υ e2 ≤ ν ≤ c3 υ e3 ,
then µ is equivalent to υ with
c4 υ e4 ≤ µ ≤ c5 υ e5
where c4 = c0 ce0 , e4 = e0 e2 , c5 = c1 ce1
2 3
and e5 = e1 e3 .


Equivalence between ρ, η and σmin
The equivalence between the tetraedron shape
measures ρ, η and σmin has been proven in L IU
and J OE, 1994, with the following conjecture on
three tight upper bounds

η 3 ≤ ρ ≤ η 3/4 , ρ4/3 ≤ η ≤ ρ1/3 ,
3/2 3/4 4/3 2/3
0.23η ≤ σmin ≤ 1.14η , 0.84σmin ≤η≤ 2.67σmin ,
2 1/2 2 1/2
0.26ρ ≤ σmin ≤ ρ , σmin ≤ρ≤ 1.94σmin .


Equivalence between η , κ, κ and γ
It can be shown that the shape measures η, κ, κ
and γ belong to the same equivalence class, at
least in two dimensions for γ.
2 2 2
γ ≤ρ≤√ γ in 2 D,
3 3
κ1/2 ≤ κ ≤ dκ1/2 in d D,
κ ≤ η ≤ dκ1/d in d D,
κ≡η in 2 D,
2/3 η 3/2 ≤ κ ≤ 3η 1/2 in 3 D.


Equivalence Classes for Shape
Measures
The equivalence relation Deﬁnition B deﬁnes
equivalence classes.


Measures
All shape measures that satisfy Deﬁnition A
that are used in practice are equivalent
according to Deﬁnition B.


Measures
Is the equivalence class of the equivalence
relation Deﬁnition B formed by all possible
simplex shape measures that satisfy
Deﬁnition A ? ? ?


Measures
Is the equivalence class of the equivalence
relation Deﬁnition B formed by all possible
simplex shape measures that satisfy
Deﬁnition A ? ? ?
No ! L IU has provided a counterexample.


Counterexample
Let µ be a shape measure that satisfies
Definition A. Then
ν = 2(µ−1)/µ
is also a shape measure. However, it cannot be
proven that µ and ν are equivalent in the sens of
Definition B since there does not exist any
constantes c0 and e0 such that c0 µeo ≤ ν when µ
tends towards zero because the exponential
asymptotic behavior of ν tends towards zero
faster than any polynomial asymptotic behavior.


What to Retain
and that are commonly used are equivalent


What to Retain
They all are sensitive to all the cases of
degeneration of the simplices.


What to Retain
They all are sensitive to all the cases of
degeneration of the simplices.
In this sense, none is better than the others.


Table of Contents

14. Conclusions


Global Quality and Optimization

The global quality of a whole mesh is evaluated via the
quality of its elements.

In practice, the comparison of two different meshes
obtained from different publications is often impossible : the
statistics presented, the shape measures and the scales
used vary from one publication to the other. Benchmarks
need to be deﬁned along with exchange standards.


Benchmark

Unit cube with a uni-
form isotropic size spe-
ciﬁcation map of 1/10.


Histogram

30
Rapport des moyennes
Rapport des rayons
25
Pourcentage des tétraèdres

20

15 Histogram of the mean
ratio η and of the radius
10 ratio ρ.
5

0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Critère de forme des tétraèdres


Histogram

30
Angle solide minimum
Angle dièdre minimum
25

20
Histogram of the mini-
15 mum of the solid angle
θmin and of the mini-
10 mum of the dihedral
angle ϕmin .
5

0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Histogram

30
Coefficient d’erreur
Rapport des arêtes
25

20

Histogram of the edge
15
ratio r and of the in-
10 terpolation error coefﬁ-
cient γ.
5

0
0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Statistics of all the Tetrahedra

min µ max σ
Radius ratio ρ 0.5151 0.9067 0.9978 0.0602
Mean ratio η 0.6559 0.9222 0.9979 0.0468
Edge ratio r 0.5696 0.7375 0.9504 0.0641
Interp. Error γ 0.4862 0.8058 0.9741 0.0709
Solid ∠ θmin 0.2962 0.7115 0.9697 0.0996
Dihedral ∠ ϕmin 0.4207 0.7657 0.9768 0.0852


Average of the Shape Measures

For a given mesh, the average depends a lot on the shape
measure used. L IU and J OE (1994) have noticed that
σmin < ρ < η.

We have noticed numerically on many meshes that

θmin < r < ϕmin < γ < ρ < η.


Average of the Shape Measures

The average, on every tetrahedra of the mesh, of a shape
measure seems to be a signiﬁcative index of the global
quality of the mesh.

Indeed, if several grids of different quality are taken and
are classiﬁed according to the average of a shape
measure, one obtains the same order, with few exceptions,
regardless of the shape measure used.


Maximum of the Shape Measures

It is not a signiﬁcative value since independently of the
shape measure and of the mesh the maximum is almost
always close to 1.

The maximum is only signiﬁcative if it is far from unit value
which is indicative of a very bad mesh.


Minimum of the Shape Measures

It is not a very significative quantity. It is significative only if
it is close to zero which is indicative of a very bad mesh.

In a series of tests, the classification of the quality of the
meshes according to the minimum of the shape measure is
chaotic. It is not advisable to characterize a whole mesh by
its worst element.


Standard Deviation of the Shape
Measure

It is a significative quantity. Small standard deviation is
indicative of good quality mesh.

In a series of tests, classification of the meshes according
to the standard deviation gives a significative classification
that is slightly chaotic.


What to Retain

Statistics on the shape of the elements of a mesh are
signiﬁcative quantities of the quality of a mesh.


What to Retain

Any valid shape measure seems to yield proper
results.


What to Retain

results.
There does not seem to be a unique quality that is
entirely indicative of the quality of a mesh.


What to Retain

results.
There does not seem to be a unique quality that is
entirely indicative of the quality of a mesh.
The average seems the most indicative quantity.


Mesh Optimization

A mesh M can be described as the set

M = m, {Xi }m , n, {Cj }n
i=1 j=1 ,

where m is the number of vertices of the mesh,
Xi = (x1i , x2i , . . . , xdi ) are the coordinates in I d of the ith
R
vertex, n is the number of simplices of the mesh, and
Cj = (c1j , c2j , . . . , cdj , cd+1,j ) is the connectivity of the jth
simplex of the mesh composed of d + 1 pointers to the
vertices of the mesh.


Optimization and Shape Measures

What is the influence of the choice of the shape measure
used in the optimization of a mesh ?

The benchmark is a triangular do-
main that is equilateral with a uni-
form and isotropic size specifica-
tion map that specifies edges of tar-
get length of 1/10 of the length of
the side of the domain. The opti-
mal mesh does exist in this special
case.


Inﬂuence of the Shape Measure

ρ η θmin

γ r

Optimization and Shape Measure

What is the influence of the choice of the shape measure
used in the optimization of a mesh ?

The benchmark is a square do-
main with a uniform and isotropic
size specification map that speci-
fies edges of 1/10 of the length of
the side of the square. The optimal
mesh does not exist in this case.


Inﬂuence of the Shape Measure

ρ η θmin

γ r Mesh Quality – p. 232/331

Influence of the Algorithm

The vertex relocation scheme is removed from the mesh
optimization process.

The benchmark is a triangular do-
main that is equilateral with a uni-
form and isotropic size specifica-
tion map that specifies edges of tar-
get length of 1/10 of the length of
the side of the domain. The optimal
mesh does exist in this case.


Inﬂuence of the Algorithm

ρ η θmin

γ r

What to Retain

If the optimal mesh exists, the mesh optimizer
converges towards the optimal mesh independently of
the shape measure used.


What to Retain

If the optimal mesh does not exist, different shape
measures will lead to different meshes. But the
difference is statistically less signiﬁcative as the
meshes become more optimized.


What to Retain

If the optimal mesh does not exist, different shape
measures will lead to different meshes. But the
difference is statistically less signiﬁcative as the
meshes become more optimized.
When the meshes are of bad quality, it is not by
changing the shape measure that they become better,
but by changing the algorithm.


Table of Contents

14. Conclusions


Size Quality of Simplices

The shape measures serve to measure the shape of
the elements of the mesh.



The shape measures are dimensionless.



The shape is one aspect of the quality of a mesh.



We seek a mesh that also respects as much as
possible the speciﬁed size of the elements.



We seek a mesh that also respects as much as
possible the speciﬁed size of the elements.
This section presents three size criteria.


Target Size of the Simplices

In C UILLIÈRE (1998), the size of the simplices is
compared to the target size.
The target size of a simplex in the reference space is
that of a unit regular simplex.
√
For a triangle, the target area is 3/4.
√
For a tetrahedron, the target volume is 2/12.

√
CK = 1 dK = √3/4 in 2D,
K 2/12 in 3D.


Size Criterion QK

The size criterion QK of the simplex K is written as :
1
QK = S det(M) dK
CK K

where S is a global scaling constant for the whole mesh.
If a simplex K is of good size according to the metric, its
size criterion QK will be of unit value.


Efﬁciency Index

Another criterion that evaluates the conformity of a mesh to
a metric is proposed by F REY and G EORGE (1999).

This criterion, contrarly to the previous one that evaluates
areas and volumes, is based on the length of the edges in
the metric.


Efficiency Index

We note Li , i = 1, · · · , na the length in the metric of the na
edges of a mesh.
The optimal length of the edges in the metric is 1.0, so that
a length of 2.0 means that the edge is two times bigger
than the specified length.
A global measure of the conformity of a mesh to the
specified size is the Efficiency Index τ
na
1
τ =1− (1 − min(Li , 1/Li ) )2 .
na i=1


Efﬁciency Index

Consider the distribution on all the edges of the mesh of
the variable τi = min(Li , 1/Li ).
Let µ = (1/na ) na τi be the average
i=1
Let σ 2 = (1/na ) na (τi − µ)2 be the standard deviation.
i=1
Then
τ = 1 − σ 2 − (µ − 1)2 .
The efﬁciency index measures both the dispersion of the
edge lengths and their proximity to the target size.


Efficiency Index

τ = 1 − σ 2 − (µ − 1)2 .

This equality shows that maximizing τ implies both
minimizing the standard deviation and bringing the average
to 1.0. The optimal value is obtained when σ = 0 and µ = 1.
This can only happen when all the edges are exactly equal
to the specified length.
The efficiency index is a good global measure of the
conformity of the length of the edges with the specified
length of the edges.


Table of Contents

14. Conclusions


A Universal Measure of Mesh Quality

Hang on to your hat...



Table of Contents
1. Introduction
2. The Metric MK of a Simplex K
3. The Speciﬁed Metric
4. The Non-Conformity EK of a Simplex K
5. The Non-Conformity ET of a Mesh T
6. Generalisation of Size Quality Measures
7. Extension to Non-Simplicial Elements
8. Final Exam
9. What to Retain


Introduction

The simplices can be of good shape without being of
good size.


Introduction

good size.
There exists quality measures for the size of the
simplices and for the mesh.


Introduction

good size.
In principle, a simplex whose edges are of unit length
in the metric is also of perfect shape in that metric.


Introduction

good size.
In practice, the meshes constructed are not exactly of
the perfect size and the simplices are composed of
edges more or less too short or too long.


Shape and Size Measures

However, the ratio of the smallest edge on largest can
√
be as large as 2/2 = 0.707 for a tetrahedron to
degenerate to a sliver.



√
be as large as 2/2 = 0.707 for a tetrahedron to
degenerate to a sliver.
This means that a simplex having edges of reasonable
size does not mean that this simplex is of reasonable
shape, since it can be degenerate.



We can do a linear combination of a shape measure
and a size measure, but this is an arbitrary choice.



We can do a linear combination of a shape measure
and a size measure, but this is an arbitrary choice.
The goal of this lecture is to introduce a universal
criterion that will measure shape and size in a single
and complete step.



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


The Metric MK of a Simplex K

How to compute the metric MK of the transformation that
transforms a simplex K into a unit equilateral element ?

Let P1 , P2 , P3 [, P4 ], the d + 1 vertices of the simplex K
in I d .
R

Let Pi Pj , 1 ≤ i < j ≤ d, the d(d + 1)/2 edges of the simplex.



In I d , d = 2 or 3, the d(d + 1)/2 components of the metric
R
are found by solving the following system of Eqs :

(Pj − Pi )T MK (Pj − Pi ) = 1 for 1 ≤ i < j ≤ d

which yields one equation per edge of the simplex.

All the edges of K measure 1 in MK .



For example in two dimensions, if the vertices of triangle K
are located at A = (xA , yA )T , B = (xB , yB )T
and C = (xC , yC )T , then this system of Eqs gives :

m11 (xB − xA )2 + 2m12 (xB − xA )(yB − yA ) + m22 (yB − yA )2 = 1,
m11 (xC − xA )2 + 2m12 (xC − xA )(yC − yA ) + m22 (yC − yA )2 = 1,
m11 (xC − xB )2 + 2m12 (xC − xB )(yC − yB ) + m22 (yC − yB )2 = 1,

which has a unique solution for all non-degenerate
triangles.



For instance, recall the triangle where vertices A and B are
located at A = (0, 1/2)T , B = (0, −1/2)T and where the
vertex C = (x, y)T free to move in the half-plane x ≥ 0.
The system of Eqs. reduces to the system
    
0 0 1 m11 1
 x2 2x(y − 2 ) (y − 1 )2   m12  =  1  ,
1
2
1 1 2
x2 2x(y + 2 ) (y + 2 ) m22 1



which yields :
 2

4y + 3 −y
 4x2 x .
MK = 
 −y


1
x
M
This metric √ K becomes identity when the vertex
C(x, y) = ( 3/2, 0)T , which corresponds to the unit
equilateral triangle.


Visualization of the Metric MK

It is usual to visualize the metric tensor as an ellipse.
Indeed, the metric tensor can be written as
MK = R−1 (θ) Λ R(θ), where the matrix Λ is the diagonal
matrix of the eigenvalues of MK , i.e., Λ = diag(λ1 , λ2 [, λ3 ]).
The eigenvalues λi are the length of the axes of the ellipse
and θ is the rotation matrix of the ellipse about the origin.



However, it is more telling to draw ellipses of size 1/ (3Λ),
this ellipse will go through the vertices of the triangle.

r=1
ℓ=1
ℓ=1
√
r = 1/ 3
ℓ=1



Ellipses of a selected
group of elements. Note in
this ﬁgure that the ellipses
pass through the vertices
of the triangle.



of the triangle.



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


The Specified Metric

A size specification map can be constructed from a
posteriori error estimators, from geometrical properties of
the domain (e.g. curvature), from user defined inputs, etc.

Isotropic size specification map (h size of the elements)
can be constructed by making the metrics diagonal
matrices whose diagonal terms are 1/h2 .


The Specified Metric MS

Whatever its origin, the size specification map contains the
information of the prescribed size and stretching of the
mesh to be built as an anisotropic metric field.

An anisotropic metric field MS is given as input.


The Average Specified Metric MS (K)

Let MS (X) be the specified Riemannian metric value at
point X. Let MS (K) be the averaged specified
Riemannian metric over a simplex K as computed by :

MS (K) = MS (X) dK dK .
K K

This integral can be approximated by a numerical
quadrature.


Visualization of MS (K)

The specified metric is de-
fined in G EORGE and B O -
ROUCHAKI (1997). It is an
analytical function that de-
fines an isotropic metric.
Note that the triangles do
not fit exactly the specified
metric.



The specified metric is de-
fined in G EORGE and B O -
ROUCHAKI (1997). It is an
analytical function that de-
fines an anisotropic metric.
metric.



Supersonic laminar vs-
cous air flow around
NACA 0012. The specified
anisotropic metric is based
on the interpolation error
(second derivatives) of the
speed field.
metric.



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


Simplex Conformity

When the metric MK of the simplex K corresponds exactly
to the averaged speciﬁed Riemannian metric MS (K) for
that simplex, the following equality holds :

MK = MS (K).

However, in practice, there is usually some discrepancy
between these two metrics and this section presents a
method to measure this discrepancy.


Simplex Conformity

This equality of metrics can be rewritten in the two
following ways :
MS −1 MK = I
and
MK −1 MS = I,
where I is the identity matrix.


Simplex Residuals

When a perfect match between what is speciﬁed and what
is realized does not happen, a residual for each of the two
previous equations yields the two following tensors :

Rs = MS −1 MK − I
and
Rb = MK −1 MS − I.
where Rs will detect the degeneration of the simplex K as
it’s volume tends to zero and Rb as it’s volume tends to
inﬁnity.


Example – Triangle ABC

Recall the triangle with two fixed vertices, one
at A = (0, 1/2)T and one at B = (0, −1/2)T , and that the
third vertex was free to move. Furthermore, if the specified
triangle is the unit equilateral triangle, then the averaged
specified Riemannian metric is equal to the identity matrix,
ie :
MS = MS −1 = I.



The residuals Rs (x, y) and Rb (x, y) can be written as
 2
  2

4y + 3 y 4y + 3 y
 4x2 −   4x2 − 1 − x 
Rs = I  x −I = ,
 y   y 
− 1 − 0
x x
 2   2 
4x 4xy 4x 4xy
 3 3   3 −1 3 
Rb =  I − I =  .
  
4xy 4y 2 4xy 4y 2
+1
3 3 3 3


Example – C(x, y) with y = 0

If the third vertex C is restricted to the axis y = 0, then all
but the ﬁrst term of these tensors vanish.
√
The two curves intersect at x = 3/2, where the residuals
become null.
20
Residual

15
Rs Rb

10

5

0

0.5 1 2 3 x


Total Residual Rt

The total residual Rt is deﬁned to be the sum of the two
residuals Rs and Rb , ie,

Rt = Rs + Rb = MS −1 MK + MK −1 MS − 2I.


The Non-Conformity EK of a
Simplex K

Definition : The non-conformity EK of a simplex K with
respect to the averaged specified Riemannian metric is
defined to be the Euclidean norm of the total residual Rt ,

EK = Rt = tr (Rt T Rt ).

The Euclidean norm of a matrix · amounts to the square
root of the sum of each term of the matrix individually
squared.



For the triangle described above with two fixed vertices
and a free vertex and for which the specified Riemannian
metric was the identity matrix, the coefficient of
non-conformity is expressed as,

2 2
4y 2 + 3 4x2 4xy y 16y 4
EK = 2
−2+ +2 − + .
4x 3 3 x 9



Logarithm base 10 of
EK when the target
metric is the identity 1
matrix. It is minimum
and equal to zero for
y0
the equilateral triangle,
and increases very ra-
pidly as the third vertex -1
moves away from the
optimal position. It is in-
0 1 x 2 3
ﬁnite for all degenerate √ T
10
triangles. MS = 01
, Xopt = 3/2, 0

Visualization of EK

The specified metric is
defined in G EORGE and
B OROUCHAKI (1997). It
is an analytical function
that defines an isotropic
metric.


Visualization of EK

The specified metric is
defined in G EORGE and
B OROUCHAKI (1997). It
is an analytical function
that define an anisotro-
pic metric.


Visualization of EK

Supersonic laminar vs-
cous air flow around
NACA 0012. The spe-
cified anisotropic metric
is based on interpola-
tion error (second deri-
vatives) of speed field.



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


The Non-Conformity ET of a Mesh T

Definition : The coefficient of non-conformity of a
mesh ET is defined as :
nK
1
ET = EKi ,
nK i=1

which is the average value of the coefficient of
non-conformity of the nK simplices of the mesh.


Properties of ET

The perfect mesh is obtained when the coefficient of
non-conformity of the mesh vanishes.
And if one simplex of the mesh degenerates, then ET
tends to infinity.
The coefficient of non-conformity of a mesh is
insensitive to compatible scaling of both the mesh and
the specified Riemannian metric.


Symmetry in Size of ET

(a) Coarse mesh (b) Perfect mesh (c) Fine mesh

If the target mesh is the middle mesh, the coefﬁcient of
non-conformity of the ﬁrst and last meshes are equivalent.


Properties of ET

It is possible to compare the quality of the mesh of two
vastly different domains, such as the mesh of a galaxy and
the mesh of a micro-circuit. In both cases, the measure
gives a comparable number that reflects the degree to
which each mesh satisfies its size specification map.
This coefficient therefore poses itself as a unique and
dimensionless measure of the non-conformity of a mesh
with respect to a size specification map given in the form of
a Riemannian metric, be it isotropic or anisotropic.



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


Generalisation of Size Quality
Measures

The non-conformity between the metric MK of a simplicial
element and the speciﬁed metric MS , ie,

MK = MS (K).

is a generalisation of the size criterion QK and the
efﬁciency index τ .


Generalisation of the Size
Criterion QK

MK (X) = MS (X),

CK = det(MK (X)) dK = det(MS (X)) dK,
K K

and then
1
QK = det(MS (X)) dK.
CK K

CK is an integral form of the conformity between the
metric MK of the simplex and the speciﬁed metric MS .


Generalisation of Efﬁciency Index τ

Let K, a simplex and AB, an edge of this simplex. Then
the pointwise conformity between the metric MK of the
simplex and the speciﬁed metric MS

MK (X) = MS (X)

can be evaluated in an integral form over the edge of the
simplex as

AB T MK (X) AB = AB T MS (X) AB
AB AB
1 = LMS (AB).


Generalisation of Efficiency Index τ

This relation
1 = LMS (AB)
can be rewritten as two residual :
R1 = 1 − LMS (AB) or R2 = 1 − 1/LMS (AB)

which is the efficiency index τ . This index is an integral
form of the conformity between the metric MK of the
simplex and the specified metric MS evaluated over the
edges of the mesh.



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


Extension to Non-Simplicial
Elements

Non-Simplicial elements are quadrilaterals in two
dimensions and prisms and hexahedra in three
dimensions.
In order to extend this measure to non-simplicial elements,
it has to be understood that the metric tensor of
non-simplicial elements is not a constant and varies for
every point of space.
In other words, the Jacobian of a simplex is constant but
the Jacobian of a non-simplicial element depends of the
point of evaluation.


Non-Simplicial Element Conformity

The conformity between the metric MK of a non-simplicial
element and the speciﬁed metric MS takes on a pointwise
nature can be rewritten as :
MK (X) = MS (X), ∀X ∈ K.


Non-Simplicial Element Conformity
Residue

The total residue Rt become a pointwise value

Rt (X) = M−1 (X)MK (X) + M−1 (X)MS (X) − 2I.
S K

Then the non-conformity EK of an element K with respect
to the speciﬁed Riemannian metric is deﬁned to be
averaged over the element K by an integration of the
Euclidean norm of the total residue Rt (X) :

K
M−1 (X)MK (X) + M−1 (X)MS (X) − 2I dK
S K
EK = .
K
dK



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


Test 1

The domain is a unit regular tri-
angle.
The size speciﬁcation map is uni-
form and isotropic.
The target edge length is 1/10.


Test 1 – Uniform Mesh

A B C


Test 1 – Uniform Mesh

A B C
ET = 0.0843 ET = 0.00 ET = 0.503


Test 2 – Isotropic Mesh

This test case is deﬁned in G EORGE and B OROUCHAKI
(1997).

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

This test case has an isotropic Riemannian metric deﬁned
by :
h−2 (x, y)
1 0
MS = −2 ,...
0 h2 (x, y)



. . . where h1 (x, y) = h2 (x, y) = h(x, y) is given by :

 1 − 19y/40 if y ∈ [0, 2],

 (2y−9)/5
 20 if y ∈ ]2, 4.5],
h(x, y) =
 5(9−2y)/5
 if y ∈ ]4.5, 7],
 1 4 y−7 4
 +
5 5 2
if y ∈ ]7, 9].



View of the size specification map as a field of tensor
metrics and view of a mesh that fits rather well these
tensor metrics.


Test 2a – Isotropic Mesh

A B C


Test 2a – Isotropic Mesh

A B C
ET = 3.18 ET = 0.104 ET = 56.2


Test 2b – Isotropic Mesh

A B C


Test 2b – Isotropic Mesh

A B C
ET = 0.104 ET = 0.929 ET = 3.18


Test 3 – Anisotropic Mesh

This test case is deﬁned in G EORGE and B OROUCHAKI
(1997).

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

This test case has an anisotropic Riemannian metric
deﬁned by :

h−2 (x, y)
1 0
MS = ,...
0 h−2 (x, y)
2



. . . 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], . . .



. . . 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) =
 5(9−2y)/5
 if y ∈ ]4.5, 7],


 1 4 y−7 4
5
+5 2 if y ∈ ]7, 9].



View of the size specification map as a field of tensor
metrics and view of a mesh that fits rather well these
tensor metrics.



A B C



A B C
ET = 0.405 ET = 2.67 ET = 0.107


Test 4 – Bernhard Riemann

The size speciﬁcation map is
deduced from an error esti-
mator based on the second
derivatives of the grey level of
the picture.



A B C



A B C
ET = 0.546 ET = 0.345 ET = 0.845


Test 5 – Flow over a Naca 0012

Supersonic laminar ﬂow at Mach 2.0, Reynolds 1000 and
an angle of attack of 10 degrees. An a posteriori error
estimator is deduced from this solution.


Test 5a – Flow over a Naca 0012

A B C


Test 5a – Flow over a Naca 0012

A B C
Speciﬁed Metric MS ET = 0.658 ET = 1160


Test 5b – Flow over a Naca 0012

A B C


Test 5b – Flow over a Naca 0012

A B C
Speciﬁed Metric MS ET = 1160 ET = 0.658


Test 5c – Flow over a Naca 0012

A B C


Test 5c – Flow over a Naca 0012

A B C
Speciﬁed Metric MS ET = 1160 ET = 0.658



Table of Contents
1. Introduction
8. Final Exam
9. What to Retain


What to Retain

This lecture presented a method to measure the
non-conformity of a simplex and of a whole mesh with
respect to a size speciﬁcation map given in the form of a
Riemannian metric.
This measure is sensitive to discrepancies in both size and
shape with respect to what is speciﬁed.
Analytical examples of the behavior were presented and
numerical examples were provided.


The Non-Conformity ET is Universal

The coefﬁcient of non-conformity of a mesh, ET , is a
universal measure in the following sense :

It is deﬁned in two and three dimensions.




It is sensitive to all simplex degeneracies.




It takes into account an Euclidean or Riemannian
metric, isotropic or anisotropic.




It is sensitive to discrepancies in shape and in size.




It is also deﬁned for non-simplicial elements.




It gives a unique number for the whole mesh.




It characterizes a whole mesh, coarse or ﬁne, in a
small or a big domain.

Mesh Optimization

This measure poses itself as a natural measure to use in
the benchmarking process. Indeed, since the measure is
able to compare two different meshes, it can help to
compare the algorithms used to produce the meshes.
This measure of the non-conformity of a mesh seems to be
an adequate cost function for mesh generation, mesh
optimization and mesh adaptation. This measure could be
used for each step such that each step minimizes the
same cost function.


Table of Contents

14. Conclusions


Conclusions

About time he ﬁnished ! ! !


Degenerate Simplices

A simplex is degenerate if its measure is null.



The degeneracy is independant of the metric.



A shape measure is valid if it is sensitive to all possible
degeneracies.



A shape measure is valid if it is sensitive to all possible
degeneracies.
A shape measure is not valid if it is not null for every
degenerate simplex.


Shape Measure

Beauty, quality and shape are relative notions.


Shape Measure

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


Shape Measure

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


Shape Measure

“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

The average of a valid shape measure on all the
simplices of the mesh seems to be a signiﬁcative index
of the global quality of the mesh.


Shape Measure

The shape measures are more or less equivalent in
assessing the quality of a mesh.


Shape Measure

The shape measures are more or less equivalent in
assessing the quality of a mesh.
The shape measures are more or less equivalent
during mesh optimization.


Size Measures

good size.


Size Measures

good size.
simplices and of the mesh.


Size Measures

good size.


Size Measures

good size.
In pratice, the meshes constructed are not exactly of
the perfect size and the simplices are composed of
edges more or less too short or too long.


Size Measures

√
be as large as 2/2 for a tetrahedron to degenerate to
a sliver.


Size Measures

√
be as large as 2/2 for a tetrahedron to degenerate to
a sliver.
This means that a simplex having edges of reasonable
size does not mean that this simplex is of reasonable
shape, since it can be degenerate.


Universal Criterion

This brings forth the problem in all its generality :
What would be a simplicial quality measure that could
measure simultaneously size and shape, that would be
sensitive to all possible degeneracies of the simplices, that
would be optimal for the régular and unitary simplex, in an
Euclidean metric or in a Riemannian metric, be it isotropic
or anisotropic, in two and in three dimensions.


Soon on your Screens !

P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F. G UIBAULT et
J.-Y. T RÉPANIER . A Measure of the Conformity of a Mesh
to an Anisotropic Metric, Tenth International Meshing
Roundtable, Newport Beach, CA, octobre 2001, pages
319–326,
has proposed such a criterion that measures the
conformity in shape and size between a mesh and the
metric that this mesh was supposed to ﬁt.


The Non-Conformity ET of a Mesh

A method to measure the non-conformity of a simplex and
of a whole mesh with respect to a size speciﬁcation map
given in the form of a Riemannian metric was given.
This measure is sensitive to discrepancies in both size and
shape with respect to what is speciﬁed.
Analytical examples of the behavior were presented and
numerical examples were provided.




It characterizes a whole mesh, coarse or ﬁne, in a
small or a big domain.

Mesh Optimization

This measure poses itself as a natural measure to use in
the benchmarking process. Indeed, since the measure is
able to compare two different meshes, it can help to
compare the algorithms used to produce the meshes.
This measure of the non-conformity of a mesh seems to be
an adequate cost function for mesh generation, mesh
optimization and mesh adaptation. This measure could be
used for each step such that each step minimizes the
same cost function.


The End


Mesh Quality

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