mesh quality

Mesh QualityJulien Dompierre

[email protected]

Centre de Recherche en Calcul Applique (CERCA)

Ecole Polytechnique de Montreal

Mesh Quality – p. 1/331

http://www.cerca.umontreal.ca/~julien

Authors

• Research professionals• Julien Dompierre• Paul Labbé• Marie-Gabrielle Vallet

• Professors• François Guibault• Jean-Yves Trépanier• Ricardo Camarero


References – 1

J. DOMPIERRE, P. LABBÉ,M.-G. VALLET, F. GUIBAULTAND R. CAMARERO, Critèresde qualité pour les maillagessimpliciaux. in Maillage etadaptation, Hermès, October2001, Paris, pages 311–348.


References – 2

A. LIU and B. JOE, Relationship betweenTetrahedron Shape Measures , Bit, Vol. 34,pages 268–287, (1994).


References – 3

P. LABBÉ, J. DOMPIERRE, M.-G. VALLET, F.GUIBAULT and J.-Y. TRÉPANIER, A UniversalMeasure of the Conformity of a Mesh withRespect to an Anisotropic Metric Field ,Submitted to Int. J. for Numer. Meth. in Engng,(2003).


References – 4

P. LABBÉ, J. DOMPIERRE, M.-G. VALLET, F.GUIBAULT and J.-Y. TRÉPANIER, A Measure ofthe Conformity of a Mesh to an AnisotropicMetric , Tenth International Meshing Roundtable,Newport Beach, CA, pages 319–326, (2001).


References – 5

P.-L. GEORGE AND H. BO-ROUCHAKI, Triangulation deDelaunay et maillage, appli-cations aux éléments finis.Hermès, 1997, Paris.This book is available in En-glish.


References – 6

P. J. FREY AND P.-L.GEORGE, Maillages. Ap-plications aux éléments finis.Hermès, 1999, Paris.This book is available inEnglish.


Table of Contents

1. Introduction2. Simplex Definition3. Degeneracies of

Simplices4. Shape Quality of

Simplices5. Formulae for Sim-

plices6. Voronoi, Delaunay

and Riemann7. Shape Quality and

Delaunay

8. Non-SimplicialElements

9. Shape QualityVisualization

10. Shape QualityEquivalence

11. Mesh Quality andOptimization

12. Size Quality ofSimplices

13. Universal Quality14. Conclusions


Introduction and Justifications

We work on mesh generation, mesh adaptationand mesh optimization.

How can we choose the configuration thatproduces the best triangles ? A triangle shapequality measure is needed.


Face Flipping

How can we choose the configuration thatproduces the best tetrahedra ? A tetrahedronshape quality measure is needed.


Edge Swapping

BA A

B

2

S3S3

S

S1S1

S2

S

5

4 S4

S5S

How can we choose the configuration thatproduces the best tetrahedra ? A tetrahedronshape quality measure is needed.


Mesh Optimization

• Let O1 and O2, two three-dimensionalunstructured 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 B be a benchmark.• Let M1 = O1(B) be the optimized mesh

obtained with the mesh optimizer O1.


It’s Obvious !


obtained with the mesh optimizer O1.• Let M2 = O2(B) be the optimized mesh

obtained with the mesh optimizer O2.


It’s Obvious !



obtained with the mesh optimizer O2.• Common sense says : “The proof is in the

pudding”.


It’s Obvious !



obtained with the mesh optimizer O2.• Common sense says : “The proof is in the

pudding”.• If ‖M1‖ > ‖M2‖ then ‖O1‖ > ‖O2‖.


Benchmarks for Mesh Optimization

J. DOMPIERRE, P. LABBÉ, F. GUIBAULT andR. CAMARERO.

Proposal of Benchmarks for 3D UnstructuredTetrahedral Mesh Optimization.

7th International Meshing Roundtable, Dearborn,MI, October 1998, pages 459–478.


The Trick...

• Because the norm ‖O‖ of a mesh optimizer isunknown, the comparison of two optimizers isreplaced 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 theelements of a mesh and the quality of a wholemesh.


What to Retain


• The concept of element quality is necessaryfor the algorithms of egde and face swapping.


What to Retain


• The concept of element quality is necessaryfor the algorithms of egde and face swapping.

• The concept of mesh quality is necessary todo research on mesh optimization.


Table of Contents



Simplices5. Formulae for Simplices6. Voronoi, Delaunay and

Riemann7. Shape Quality and

Delaunay








Definition of a Simplex

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




The most simple amongst them, the simplices, arethose 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 othersuch 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 oforder d + 1,

A =

p11 p12 · · · p1,d+1

p21 p22 · · · p2,d+1

......

. . ....

pd1 pd2 · · · pd,d+1

1 1 · · · 1

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


A Simplex Generates Rd

Any point X ∈ Rd, with Cartesian coordinates (xi)di=1, is

characterized by the d + 1 scalars λj = λj(X) defined assolution of the linear system

d+1∑

j=1

pijλj = xi for 1 ≤ i ≤ d,

d+1∑

j=1

λj = 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 thatform a base of Rd.


What to Retain



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

The coordinates λj(X) of a point X ∈ Rd in the basegenerated by the simplex are the barycentriccoordinates.


Table of Contents





Delaunay








Degeneracy of Simplices

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



A d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd.




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





A triangle is degenerate if its vertices are collinear orcollapsed.






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






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

Nota bene : Strictly speaking, accordingly to thedefinition, a degenerate simplex is no longer a simplex.


Degeneracy Criterion

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




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





The size of a d-simplex K made of d + 1 vertices Pj isgiven by

size(K) = det(A)/d!.







A triangle is degenerate if its area is null.







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 possibledegenerate 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

CaphC

A B BCA

Degenerate edges : NoneRadius of the smallest circumcircle : ∞


2 – The Needle

Name h −→ 0 h = 0

NeedlehC

A B BA,C

Degenerate edges : AC

Radius of the smallest circumcircle : hmax/2


3 – The Big Crunch

Name h −→ 0 h = 0

BigCrunch

h h

A

C

hB

A,B,C

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


Degeneracy of Tetrahedra

There is one case of degeneracy resulting in fourcollapsed vertices.There are five cases of degeneracy resulting in four

collinear vertices.There are four cases of degeneracy resulting in four

coplanar vertices.

A

B

Cc

D D

d

bA C

B

a


1 – The Fin

Name h −→ 0 h = 0

FinA

B

hD

C A

B

DC

Degenerate edges : NoneDegenerate faces : One capRadius of the smallest circumsphere : ∞


2 – The Cap

Name h −→ 0 h = 0

Cap A

Bh C

DA

B

D C

Degenerate edges : NoneDegenerate faces : NoneRadius of the smallest circumsphere : ∞


3 – The Sliver

Name h −→ 0 h = 0

Sliver A C

Dh

B

A

B

DC

Degenerate edges : NoneDegenerate faces : NoneRadius of the smallest circumsphere : rABC or ∞


4 – The Wedge

Name h −→ 0 h = 0

WedgeB

hD

A C

B

AC,D

Degenerate edges : CD

Degenerate faces : Two needlesRadius of the smallest circumsphere : rABC


5 – The Crystal

Name h −→ 0 h = 0

CrystalA

Bh

D

Ch

DA B C

Degenerate edges : NoneDegenerate faces : Four capsRadius of the smallest circumsphere : ∞


6 – The Spindle

Name h −→ 0 h = 0

Spindle AC

hh

B

DCA B,D

Degenerate edges : BD

Degenerate faces : Two caps and two needlesRadius of the smallest circumsphere : ∞


7 – The Splitter

Name h −→ 0 h = 0

Splitter AC

h

Dh

B

DA B,C

Degenerate edges : BC

Degenerate faces : Two caps and two needlesRadius of the smallest circumsphere : ∞


8 – The Slat

Name h −→ 0 h = 0

Slat hB

ChA

DA,D B,C

Degenerate edges : AD and BC

Degenerate faces : Four needlesRadius of the smallest circumsphere : hmax/2


9 – The Needle

Name h −→ 0 h = 0

NeedleB

Ah

D

Ch

h B,C,DA

Degenerate edges : BC, CD and DB

Degenerate faces : Three needles and one Big CrunchRadius of the smallest circumsphere : hmax/2


10 – The Big Crunch

Name h −→ 0 h = 0

BigCrunch

DC

B

Ah

h

h

hh

hA,B,C,D

Degenerate edges : AllDegenerate faces : Four Big CrunchesRadius of the smallest circumsphere : 0


What to Retain

A triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.


What to Retain


There are three cases of degeneracy of triangles.


What to Retain



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


What to Retain



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

There are ten cases of degeneracy of tetrahedra.


Table of Contents





Delaunay








Shape Quality of Simplices

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




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




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

This section is a guided tour of the shape measuresused for simplices.


The Regular Simplex

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


The Regular Simplex


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


The Regular Simplex


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

The equilateral tetrahedron is regular because itmaximizes its volume for a given surface of its faces.


Simplicial Shape Measure

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


Remarks

The invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent of the coordinates system.


Remarks


The invariance under a valid uniform scaling meansthat the simplicial shape measures must bedimensionless (independent of the unit system).


Remarks


The invariance under a valid uniform scaling meansthat the simplicial shape measures must bedimensionless (independent of the unit system).

The continuity means that the simplicial shapemeasures must change continuously in function of thecoordinates of the vertices of the simplex.


The Radius Ratio

The radius ratio of a simplex K is a shape measure definedas ρ = d ρK/rK , where ρK and rK are the radius of theincircle and circumcircle of K (insphere and circumspherein 3D), and where d is the dimension of space.

K

rK

ρK


The Mean Ratio

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

xb

R

ys

r

K = NR + b K


The Mean Ratio

Then, the mean ratio η of the simplex K is the ratio of thegeometric mean over the algebraic means of theeigenvalues λ1, λ2 [,λ3] of the matrix NT N .

η =

d

√d∏

i=1

λi

1d

d∑i=1

λi

=

2 2√

λ1λ2

λ1 + λ2

=4√

3 SK∑1≤i<j≤3 L2

ij

in 2D,

3 3√

λ1λ2λ3

λ1 + λ2 + λ3

=12 3

√9V 2

K∑1≤i<j≤4 L2

ij

in 3D.


The Condition Number

FORMAGGIA and PEROTTO (2000) use the inverse of thecondition number of the matrix.

κ =min

iλi

maxi

λi

=λ1

λd

,

if the eigenvalues are sorted in increasing order.


The Frobenius Norm

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

κ =d√

tr(NT N)tr((NT N)−1)=

d√(d∑

i=1

λi

) (d∑

i=1

λ−1i

) ,

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


The Minimum of Solid Angles

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

θmin = α−1 min1≤i≤d+1

θi,

The coefficient α is the value of each solid angle of theregular d-simplex, given by α = π/3 in two dimensionsand α = 6 arcsin

(√3/3

)− π in three dimensions.


The sin of θmin

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

σmin = β−1 min1≤i≤d+1

σi,

where σi = sin(θi) in 2D and σi = sin(θi/2) in 3D. β is thevalue of σi for all solid angles of the regular simplex, givenby 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 ofthe twelve angles of the four faces of a tetrahedron. Thisangle is π/3 for the regular tetrahedron.But this shape measure is not valid according toDefinition A because it is insensitive to degeneratetetrahedra that do not have degenerate faces (the sliverand the cap).


Dihedral Angles

The dihedral angle is the angle between the intersection oftwo adjacent faces to an edge with the perpendicular planeof the edge.

ϕij

Pi

Pj

The minimum of the six dihedral angles ϕmin is used as ashape measure.


Dihedral Angles

αϕmin = min1≤i<j≤4

ϕij = min1≤i<j≤4

(π − arccos (nij1 · nij2)) ,

where nij1 and nij2 are the normal to the adjacent faces ofthe edge PiPj, and where α = π − arccos(−1/3) is thevalue of the six dihedral angles of the regular tetrahedron.But this shape measure is not valid according toDefinition 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 overan element is bounded by a coefficient times thesemi-norm of the function. This coefficient is theratio DK/K where DK is the diameter of the element Kand K is the roundness of the element K.

γ =

2√

3ρK

hmax

in 2 D,

2√

6ρK

hmax

in 3 D.


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 toDefinition A because it does not vanish for somedegenerate simplices. In 2D, it can be as large as 1/2 forthe 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 tetrahedronover the circumradius, in BAKER, (1989). This is not avalid shape measure.hmin/rK , the ratio of the smallest edge of the

tetrahedron over the circumradius, in MILLER et al(1996). This is not a valid shape measure.VK/rK

3, the ratio of the volume of the tetrahedron overthe circumradius, in MARCUM et WEATHERILL, (1995).



VK4(∑4

i=1 S2i

)−3, the ratio of the volume of the

tetrahedron over the area of its faces, in DECOUGNY etal (1990). The evaluation of this shape measure, and itsvalidity, are a complex problem for tetrahedra thatdegenerate in four collinear vertices.

VK

(∑1≤i<j≤4 Lij

)−3

, the ratio of the volume of the

tetrahedron over the average of its edges, inDANNELONGUE and TANGUY (1991), ZAVATTIERI et al(1996) and WEATHERILL et al (1993).



VK

(( ∑1≤i<j≤4

Lij

)2

− L12L34 − L13L24

−L14L23 +∑

1≤i<j≤4

L2ij

)−3/2

the ratio of the volume of the tetrahedron over a sum, atthe power three halfs, of many terms homogeneous to thesquare of edge lenghts, in BERZINS (1998).



VK

(√∑1≤i<j≤4 L2

ij

)−3

, the ratio of the volume of the

tetrahedron over the quadratic average of the six edges,in GRAICHEN et al (1991).And so on... This list is surely not exhaustive.


There Exists an Infinity of ShapeMeasures

If µ and ν are two valid shape measures, if c, d ∈ R+, thenµc,c(µ−1)/µ with c > 1,αµc + (1 − α)νd with α ∈ [0, 1],µcνd

are also valid simplicial shape measures.


What to Retain

The regular simplex is the equilateral one, ie, where allits 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 alldegenerate 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 waybetter than the other ones.


Table of Contents





Delaunay








Formulae for the Triangle

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

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

Let K be a non degenerate triangle of vertices P1, P2

and P3. The lengths of the edges PiPj of K aredenoted Lij = ‖Pj − Pi‖, 1 ≤ i < j ≤ 3.


The Half-Perimeter

The half-perimeter pK is given by

pK =(L12 + L13 + L23)

2.


Heron’s Formula

The area SK of a triangle can also be written in function ofthe edge lengths with Heron’s formula :

S2K = pK(pK − L12)(pK − L13)(pK − L23).


Radius of the Incircle

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

ρK =SK

pK

.


Radius of the Circumscribed Circle

The radius rK of the circumcircle of the triangle K is givenby

rK =L12 L13 L23

4SK

.


Element Diameter

The diameter of an element is the biggest Euclideandistance 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 lengthobtained by projecting the edge of the triangle oppositeto Pi on a unitary circle centerered at Pi. The angle can bewritten in function of the edge lengths as

θi = arcsin

(2SK

( ∏

j,k 6=i

1≤j<k≤3

LijLik

)−1).


Formulae for the Tetrahedron

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

Quantities such that inradius, circumradius, angles,volume, etc, can be written in function of the edge lengthsof 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 PiPj of K are denotedLij = ‖Pj − Pi‖, 1 ≤ i < j ≤ 4. The area of the four faces ofthe tetrahedron, △P2P3P4, △P1P3P4, △P1P2P4

and △P1P2P3, 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 thetetrahedron such that the edges a, b and c are connectedto the same vertex, and such that e is the opposite edge ofa, f is opposite of b and g is the opposite of c. The volumeVK is then

144V 2K = 4a2b2c2

+ (b2 + c2 − e2) (c2 + a2 − f 2) (a2 + b2 − g2)

− a2 (b2 + c2 − e2)2 − b2 (c2 + a2 − f 2)

2

− c2 (a2 + b2 − g2)2.


Radius of the Insphere

The radius ρK of the insphere of the tetrahedron K is givenby

ρK =3VK

S1 + S2 + S3 + S4

.


Radius of the Circumsphere

The radius rK of the circumsphere of the tetrahedron K isgiven by

rK =

√(a + b + c)(a + b − c)(a + c − b)(b + c − a)

24VK

.

where a = L12L34, b = L13L24 and c = L14L23 are theproduct of the length of the opposite edges of K (twoedges are opposite if they do not share a vertex.


Element Diameter

The diameter of an element is the biggest Euclideandistance between two points of an element. For atetrahedron, 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 thearea of the spherical sector obtained by projecting the faceof the tetrahedron opposite to Pi on a unitary spherecenterered at Pi.

P4

P2

P1 P3θ1


Solid angle

LIU and JOE (1994) gave a formula to compute the solidangle in function of edge lengths :

θi = 2 arcsin

(12VK

( ∏

j,k 6=i

1≤j<k≤4

((Lij + Lik)

2 − L2jk

))−1/2).


Table of Contents






Delaunay








Which Is the Most Beautiful Triangle ?



A



A B


If You Chose the Triangle A...


If You Chose the Triangle A...

AYou are wrong !


If You Chose the Triangle B...


If You Chose the Triangle B...

BYou are wrong again !



A BNone of these answers !


Which Is the Most Beautiful Woman ?



A



A B


You Probably chose...


You Probably chose...

A BWoman A.


And if One Asked these Gentlemen...


These Gentlemen Would Choose...


These Gentlemen Would Choose...

A BWoman B.


Which Is the Most Beautiful Woman...

There is no absolute answer because thequestion is incomplete.

One did not specify who was going to judge thecandidates, which was the scale of evaluation,which were the measurements used, etc.



A B



A BThe question is incomplete : It misses a way ofmeasuring the quality of a triangle.


Voronoi Diagram

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


The Perpendicular Bisector

S1

S2

d(P, S2)

d(P, S1)

M

P

Let S1 and S2 be twovertices in R2. Theperpendicular bisec-tor M(S1, S2) is thelocus of points equi-distant to S1 and S2.M(S1, S2) = {P ∈R2 | d(P, S1) = d(P, S2)},where d(·, ·) is the Eucli-dean distance betweentwo points of space.


A Cloud of Vertices

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

S6

S11S2 S10

S4

S3S12S7

S9

S8S5

S1


The Voronoi Cell

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

C(Si) = {P ∈ R2 | d(P, Si) ≤ d(P, Sj),∀j 6= i}.

Si

C(Si)


The Voronoi Diagram

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


Properties of the Voronoi Diagram

The Voronoi cells are polygons in 2D,polyhedra in 3D and N -polytopes in ND.The Voronoi cells are convex.The Voronoi cells cover space withoutoverlapping.


What to Retain

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


Delaunay Triangulation

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


Triangulation of a cloud of Points

The same cloud of points can be triangulated inmany different fashions.

. . .


Triangulation of a Cloud of Points

. . .

. . .



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


Empty Sphere Criterion of Delaunay

Empty sphere criterion : A simplex K satisfiesthe empty sphere criterion if the opencircumscribed ball of the simplex K is empty (ie,does not contain any other vertex of thetriangulation).

K

K


Violation of the Empty Sphere Criterion

A simplex K does not satisfy the empty spherecriterion if the opened circumscribed ball ofsimplex K is not empty (ie, it contains at leastone vertex of the triangulation).

K

K



Delaunay Triangulation : If all the simplices Kof a triangulation T satisfy the empty spherecriterion, then the triangulation is said to be aDelaunay triangulation.


Delaunay Algorithm

The circumscri-bed sphere of asimplex has to becomputed.This amounts tocomputing the cen-ter of a simplex.The center is thepoint at equal dis-tance to all thevertices of the sim-plex.

S2ρout

C

S3

S1


Delaunay Algorithm

How can we know if a point P violates the emptysphere criterion for a simplex K ?

The center C and the radius ρ of thecircumscribed sphere of the simplex K has tobe computed.The distance d between the point P and thecenter C has to be computed.If the distance d is greater than the radius ρ,the point P is not in the circumscribed sphereof the simplex K.


What to Retain

The Voronoi diagram of a cloud of points is apartition of space into cells based on thenotion of distance.

A Delaunay triangulation of a cloud of pointsis a triangulation based on the notion ofdistance.


Duality Delaunay-Voronoï

The Voronoï diagram is the dual of the Delaunaytriangulation and vice versa.


Voronoï and Delaunay in Nature

Voronoï diagrams and Delaunay triangulationsare not just a mathematician’s whim, theyrepresent 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 evaluationof the ratio to equilarity.




The Voronoï diagram of a cloud of points is apartition of space into cells based on thenotion of distance.






The notion of distance can be generalized.






The notion of distance can be generalized.

The notions of shape measure, of Voronoïdiagram and of Delaunay triangulation can begeneralized.


Nikolai Ivanovich Lobachevsky

NIKOLAI IVANOVICHLOBACHEVSKY, 1décembre 1792, NizhnyNovgorod — 24 février1856, Kazan.


János Bolyai

JÁNOS BOLYAI, 15 dé-cembre 1802 à Kolozsvár,Empire Austrichien (Cluj,Roumanie) — 27 janvier1860 à Marosvásárhely,Empire Austrichien (Tirgu-Mures, Roumanie).


Bernhard RIEMANN

GEORG FRIEDRICH BERN-HARD RIEMANN, 7 sep-tembre 1826, Hanovre — 20juillet 1866, Selasca. Über dieHypothesen welche der Geo-metrie zu Grunde liegen. 10juin 1854.


Non Euclidean Geometry

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

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

He has introduced the Riemannian metric thatdefines 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 → IR

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 Sbe IR2, the function

d : IR2×IR2 → IR(x1

y1

)×

(x2

y2

)→

√(x2 − x1)2 + (y2 − y1)2

is a metric on IR2.


Metric Space


The Scalar Product is a Metric

Let a vectorial space with its scalar product 〈·, ·〉.Then the norm of the scalar product of thedifference of two elements of the vectorial spaceis a metric.

d(A, B) = ‖B − A‖,= 〈B − A, B − A〉1/2,

= 〈−→AB,−→AB〉1/2,

=

√−→AB

T −→AB.


The Scalar Product is a Metric

If the vectorial space is IR2, then the norm of thescalar product of the vector

−→AB is the Euclidean

distance.

d(A, B) = 〈B − A, B − A〉1/2 =

√−→AB

T −→AB,

=

√√√√(

xB − xA

yB − yA

)T (xB − xA

yB − yA

),

=√

(xB − xA)2 + (yB − yA)2.


Metric Tensor

A metric tensor M is a symmetric positivedefinite matrix

M =

(m11 m12

m12 m22

)in 2D,

M =

m11 m12 m13

m12 m22 m23

m13 m23 m33

in 3D.


Metric Length

The length LM(−→AB) of an edge between vertices

A and B in the metric M is given by

LM(−→AB) = 〈−→AB,

−→AB〉1/2

M ,

= 〈−→AB,M−→AB〉1/2,

=√−→

ABTM−→AB.


Euclidean Length with M = I

LM(−→AB) = 〈−→AB,M−→

AB〉1/2 =√−→

ABTM−→AB,

=

√√√√(

xB − xA

yB − yA

)T (1 0

0 1

) (xB − xA

yB − yA

)

LE(−→AB) =

√(xB − xA)2 + (yB − yA)2.


Metric Length with M =

(αβ

βγ

)

LM(−→AB) = 〈−→AB,M−→

AB〉1/2 =√−→

ABTM−→AB,

=

√√√√(

xB − xA

yB − yA

)T (α β

β γ

) (xB − xA

yB − yA

)

LE(−→AB) =

(α(xB − xA)2 + 2β(xB − xA)(yB − yA)

+γ(yB − yA)2)1/2

.


Length in a Variable Metric

In the general sense, the metric tensor M is notconstant but varies continuously for every pointof space. The length of a parameterized curveγ(t) = {(x(t), y(t), z(t)) , t ∈ [0, 1]} is evaluated inthe metric

LM(γ) =

∫ 1

0

√(γ′(t))T M (γ(t)) γ′(t) dt,

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


Area and Volume in a Metric

Area of the triangle K in a metric M :

AM(K) =

∫

K

√det(M) dA.

Volume of the tetrahedron K in a metric M :

VM(K) =

∫

K

√det(M) dV.


Metric and Delaunay Mesh


Which is the Best Triangle ?

A BThe question is incomplete. The way to measurethe quality of the triangle is missing.



A B


Example of an Adapted Mesh

Adapted mesh and solution for a transonicvisquous compressible flow with Mach 0.85 andReynolds = 5 000.


Zoom on Boundary Layer–Shock


What to Retain

Beauty, quality and shape are relativenotions.


What to Retain


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


What to Retain



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


What to Retain



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

A shape measure is a measure of theequilarity of a simplex in this metric.


Shape Measure in a Metric

First method (constant metric)

For a simplex K, evaluate the metric tensor atseveral points (Gaussian points) and find anaveraged metric tensor.

Take this averaged metric tensor as constantover the whole simplex and evaluate the shapemeasure using this metric.



Second method (constant metric)

For a simplex K, evaluate the metric tensor atone point (Gaussian point) and take the metricas constant over the whole simplex. Evaluate theshape measure using this metric.

Repeat this operation at several points andaverage the shape measures.

This is what is done at INRIA.



Third methode (variable metric)

Express the shape measure as a fonction ofedge lengths only.

Evaluate the length of the edges in the metricand compute the shape measure with theselengths.

This is what is done in OORT.



Fourth method (variable metric)

Express the shape measure in function of thelength of the edges, the area and the volumes.

Evaluate the lengths, the area and the volume inthe metric.



Fifth method (variable metric)

Know how to evaluate quantities such as theradius of the inscribed circle, of thecircumscribed circle, the solid angle, etc, in ametric.

In the general sense, the triangular inequality isnot verified in a variable metric. Neither is thesum of the angles equal to 180 degrees, etc.

The evaluation of a shape measure in a variablemetric in all its generality is an opened problem.For the moment, it is approximated.


Table of Contents






Delaunay








Shape Measures and Delaunay Criteron

Delaunay meshes have several smoothnessproperties.

The Delaunay mesh minimizes the maximum value ofall the element circumsphere radii.When the circumsphere center of all simplices of a

mesh lie in their respective simplex, then the mesh is aDelaunay mesh.In a Delaunay mesh, the sum of all squared edge

lengths weighted by the volume of elements sharing thatedge is minimal.


3D-Delaunay Mesh and Degeneracy

In three dimensions, it is well known thatDelaunay meshes can include slivers which aredegenerate elements.

Why ?

How to avoid them ?


Empty Sphere Criterion of Delaunay

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


Edge Swapping andθmin Shape Measur

During edge swapping, using the empty spherecriterion (Delaunay criterion)

⇐⇒Using the θmin shape measure (maximize theminimum of the angles).

θ5θ1

θ3

θ5

θ2θ1θ6θ4

θ4

θ3

θ2

θ6


What to Retain

The empty sphere criterion of Delaunay is nota shape measure but it can be used as ashape measure.


What to Retain


In two dimensions, in the edge swappingalgorithm (Lawson’s method), the emptysphere criterion of Delaunay is equivalent tothe θmin shape measure.


What to Retain


In two dimensions, in the edge swappingalgorithm (Lawson’s method), the emptysphere criterion of Delaunay is equivalent tothe θmin shape measure.

There is a multitude of valid shape measures,and thus a multitude of generalizations of theDelaunay mesh.


Delaunay and Circumscribed Sphere

As the circumscribed sphere of a tetrahedrongets larger , there are more chances that anothervertex of the mesh happens to be in this sphere,and the chances that this tetrahedron and themesh satisfy the Delaunay criterion get smaller.

As the circumscribed sphere of a tetrahedrongets smaller , there are fewer chances thatanother vertex of the mesh happens to be in thissphere, and the chances that this tetrahedronand the mesh satisfy the Delaunay criterion getbigger.


Circumscribed Sphere of Infinite Radius

The tetrahedra that degenerate into a fin, into acap, into a crystal, into a spindle and into asplitter

A

B

hD

C A

Bh C

DA

Bh

D

Ch

AC

hh

B

DA

Ch

Dh

Bhave a circumscribed sphere of infinite radius.


Circumscribed Sphere of Bounded Radius

The tetrahedra that degenerate into a sliver, intoa wedge, into a slat, into a needle and into aBig Crunch

A C

Dh

B B

hD

A C hB

ChA

D

BA h

DC

hh

DC

B

Ah

h

h

hhh

have a circumscribed sphere of bounded radius.


What to Retain

The empty sphere criterion ofDelaunay is not a valid shapemeasure sensitive to all the possibledegeneracies of the tetrahedron.


Circumscribed Sphere of Bounded Radius

Amongst the degenerate tetrahedra that have acircumscribed sphere of bounded radius, thewedge, the slat, the needle and the Big Crunchcan be eliminitated

B

hD

A C hB

ChA

D

BA h

DC

hh

DC

B

Ah

h

h

hhh

since they have several superimposed vertices .


The Sliver

And so, finally, we come to the sliver,

A C

Dh

B

A

B

DC

a degenerate tetrahedron having disjoint verticesand a bounded circumscribed sphere radius,which makes it “Delaunay-admissible”.


Non-Convex Quadrilateral

It is forbidden to swap an edge of a non-convexquadrilateral.

S1

S2

S1

S3

T2

T1

S4S1

S3

S4

T1T2

S2

S4

S3

T1

S4

T2

S1

S2


Non-Convex Quadrilateral

S1

S3

S2

T1

S4

T2

Two adjacent trianglesforming a non-convexquadrilateral necessa-rily satisfy the emptysphere criterion ofDelaunay.


Loss of the Convexity Property in 3D


What to Retain

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


What to Retain


The empty sphere criterion of Delaunay is notsensitive to all the possible degeneracies ofthe tetrahedron.


What to Retain


The empty sphere criterion of Delaunay is notsensitive to all the possible degeneracies ofthe tetrahedron.

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


Table of Contents






Delaunay








Non-Simplicial Elements

This section proposes a method to generalizethe notions of regularity, of degeneration and ofshape measure of simplices to non simplicialelements ; i.e., to quadrilaterals in twodimensions, to prisms and hexahedra in threedimensions.


Non-Simplicial Elements

On Element Shape Measures for MeshOptimization

PAUL LABBÉ, JULIEN DOMPIERRE, FRANÇOISGUIBAULT AND RICARDO CAMARERO

Presented at the 2nd Symposium on Trends inUnstructured Mesh Generation, Fifth US NationalCongress on Computational Mechanics, 4–6august 1999 University of Colorado at Boulder.


Regularity Generalization

An equilateral quadrilateral, ie that has fouredges of same length, is not necessarily asquare...




Définition : An element, be it simplicial ornot, is regular if it maximizes its measure for agiven measure of its boundary.





The equilateral triangle is regular because itmaximizes its area for a given perimiter.





The equilateral triangle is regular because itmaximizes its area for a given perimiter.

The equilateral tetrahedron is regularbecause it maximizes its volume for a givensurface 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 ! Itstwo triangular faces are equilateral trianglewhose edges measure a. Its three quadrilateralfaces are rectangles that have a base oflength a and a height of length a/

√3.


Quality of Non Simplicial Elements

Proposed Extension : The shape measure of anon simplicial element is given by the minimumshape measure of the corner simplicesconstructed from each vertex of the element andof its neighbors.


Shape Measure of a Quadrilateral

The shape measure of a quadrilateral is theminimum of the shape measure of its four cornertriangles formed by its four vertices.

BA AB

DC

A

D C D C D

BAB

C


Shape Measure of a Prism

The shape measure of a prism is the minimum ofthe shape measure of its six corner tetrahedronformed by its six vertices.

F

B

FD E

C

DE

E

F

ED

C

B

A

F

C

BA

FED

D

BAA

C

BC

A


Shape Measure of an Hexahedron

The shape measure of an hexahedron is theminimum of its eight corner tetrahedron formedby its eight vertices.

G

A

A B

D

G

FC C

B

D

E

H

F

AB A

A B

BH

E

D C

H

HE

D C

G

FC

G

E

D

GF

H

FE


Shape of the Corner Simplex

The corner simplices constructed for the nonsimplicial elements are not regular simplices.For the square, the four corner triangles areisosceles right-angled triangles.For the cube, the eight corner tetrahedra areisosceles right-angled tetrahedra.For the regular prism, the six cornertetrahedra are tetrahedron with an equilateraltriangle of side a, and a fourth perpendicularedge of length a/

√3.


Shape of the Corner Simplex

Each non simplicial shape measure has to benormalized so as to be a shape measure equalto unit value for regular non simplicial elements.

ρ η θmin γ

Square 21+

√2

√3

234

√3

1+√

2

Prism 18√5(7+

√13)

13√

2

2 arcsin(1/√

22+12√

3)6 arcsin(1/

√3)−π

3√

67+

√13

Cube√

3 − 1 23

3√

2 2 arcsin((2−√

2)/(2√

3))

6 arcsin(1/√

3)−π

√3 − 1


Degenerate Non Simplicial Elements

Définition :A non simplicial element isdegenerate if at least one of its corner simplicesis degenerate.

If at least one of the corner simplices is morethan degenerate, meaning that it is inverted (ofnegative norm), then the non simplicial elementis concave and is also considered degenerate.


Twisted Non Simplicial Elements

In three dimensions, the definition of the shapemeasure of non simplicial elements has oneflaw : it is not sensitive to twisted elements.

F DE

DFE

EF

D

FE

D

C

BAF A

C

BA

C

A

CD

E

BA

C

B

B


Twist of Quadrilateral Faces

A critera used to measure the twist of aquadrilateral face ABCD is to consider thedihedral angle between the triangles ABCand ACD on one hand, and between thetriangles ABD and BCD on the other hand.

If these dihedral angles are π, then thequadrilateral face is a plane (not twisted). Thetwist in the quadrilateral increases as the anglesdiffer from π.


Twist of Quadrilateral Faces

Definition :Given a valid simplicial shapemeasure, the twist of a quadrilateral face is equalto the value of the shape measure for thetetrahedron constructed by the four vertices ofthe quadrilateral face.

Thus, a plane face has no twist because the four verticesform a degenerated tetrahedron and all valid shapemeasures are null.

As a quadrilateral face is twisted, its vertices move awayfrom coplanarity, and the shape measure of the generatedtetrahedron gets larger.


What to Retain

The shape, the degeneration, the convexity,the concavity and the torsion can be rewrittenas a function of simplices.

An advantage of this approach is that once thatthe measurement and the shape measures forthe simplices are programmed, in Euclidean aswell as with a Riemannian metric, the extensionfor non simplicial elements is direct.


Table of Contents1. Introduction2. Simplex Definition3. Degeneracies of





Delaunay








Visualizing Shape Measures

yx

QK(C)C(x, y)

B(0,−1/2)

A(0, 1/2)1

0

1

0.5

y

2

1

0

-1

-2 x

32

10

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



x 3210

y

1

0

-1

x 3210

y

1

0

-1

x 3210

y

1

0

-1

The edge ratio r on the left. The minimum of thesolid angles θmin in the center. The interpolationerror coefficient γ on the right.



x 3210

y

1

0

-1

x 3210

y

1

0

-1

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


Which Shape Measure is Best

x 3210

y

1

0

-1

r is not a valid shape mea-sure.θmin and γ are continuousbut not differentiable.ρ and η are continuous anddifferentiable.ρ is numerically unstable.η is the least costly.η has circular contourlines.


3D Rendering of Shape Measures

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


Rendering Taking a Metric Into Ac-count

x 3210

y

1

0

-1

Mean ratio η

M =

(0.2 0

0 1

)


Rendering Taking a Metric into Ac-count

x 210

y

1

0

-1

Mean ratio η

M =

(20 0

0 1

)



x 3210

y

1

0

-1

Mean ratio η

M =

(0.9 0.4

0.4 1

)



x 3210

y

1

0

-1

Mean ratio η

M =

(1 0

0 1

)


What to Retain

Mean ratio η is the privileged shape measure.


What to Retain


Circular contour lines in Euclidean spacebecome ellipses in the general case.


What to Retain



The shape of a triangle is a quality measurethat is relative.


What to Retain




A good triangle in a metric tensor is notbeautiful in a different metric tensor.


What to Retain




A good triangle in a metric tensor is notbeautiful in a different metric tensor.

The quality of a triangle depends on the valueof the size specification map given in the formof a metric tensor.


Table of Contents1. Introduction2. Simplex Definition3. Degeneracies of





Delaunay








Equivalence of Shape Measures



x 3210

y

1

0

-1

Superposition ofcontour plots ofsimplex shape mea-sures ρ, η, θmin etγ.



Definition B (from LIU and JOE, 1994) : Let µand ν be two different simplicial shape measureshaving values in the interval [0, 1]. µ is said to beequivalent to ν if there exists positiveconstants c0, c1, e0 and e1 such that

c0νe0 ≤ µ ≤ c1ν

e1.


Optimal Bounds

In the equivalence relation of shape measures


e1,


Optimal Bounds



e1,

the lower bound is said to be optimal if e0 is thesmallest possible exponent,


Optimal Bounds



e1,

the lower bound is said to be optimal if e0 is thesmallest possible exponent,

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


Tight Bounds


c0 νe0 ≤ µ ≤ c1 νe1,


Tight Bounds


c0 νe0 ≤ µ ≤ c1 νe1,

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


Tight Bounds


c0 νe0 ≤ µ ≤ c1 νe1,

the lower bound is said to be tight if c0 is thebiggest possible constant,and the upper bound is said to be tight if c1 is thesmallest 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


e1,

then ν is equivalent to µ with

c2µe2 ≤ ν ≤ c3µ

e3,

where c2 = c−1/e1

1 , e2 = 1/e1, c3 = c−1/e0

0

and e3 = 1/e0.


Transitive RelationIf µ is equivalent to ν and if ν is equivalent to υwith


e1 and c2υe2 ≤ ν ≤ c3υ

e3,

then µ is equivalent to υ with

c4υe4 ≤ µ ≤ c5υ

e5

where c4 = c0ce0

2 , e4 = e0e2, c5 = c1ce1

3

and e5 = e1e3.


Equivalence between ρ, η and σmin

The equivalence between the tetraedron shapemeasures ρ, η and σmin has been proven in LIUand JOE, 1994, with the following conjecture onthree tight upper bounds

η3 ≤ ρ ≤ η3/4, ρ4/3 ≤ η ≤ ρ1/3,

0.23η3/2 ≤ σmin ≤ 1.14η3/4, 0.84σ4/3min ≤ η ≤ 2.67σ

2/3min,

0.26ρ2 ≤ σmin ≤ ρ1/2, σ2min ≤ ρ ≤ 1.94σ

1/2min.


Equivalence between η, κ, κ and γ

It can be shown that the shape measures η, κ, κand γ belong to the same equivalence class, atleast in two dimensions for γ.

2

3γ2 ≤ ρ ≤ 2√

3γ in 2 D,

κ1/2 ≤ κ ≤ dκ1/2 in d D,

κ ≤ η ≤ dκ1/d in d D,

κ ≡ η in 2 D,√2/3 η3/2 ≤ κ ≤ 3η1/2 in 3 D.


Equivalence Classes for ShapeMeasures

The equivalence relation Definition B definesequivalence classes.




All shape measures that satisfy Definition Athat are used in practice are equivalentaccording to Definition B .





Is the equivalence class of the equivalencerelation Definition B formed by all possiblesimplex shape measures that satisfyDefinition A ? ? ?





Is the equivalence class of the equivalencerelation Definition B formed by all possiblesimplex shape measures that satisfyDefinition A ? ? ?

No ! LIU has provided a counterexample.


Counterexample

Let µ be a shape measure that satisfiesDefinition A . Then

ν = 2(µ−1)/µ

is also a shape measure. However, it cannot beproven that µ and ν are equivalent in the sens ofDefinition B since there does not exist anyconstantes c0 and e0 such that c0µ

eo ≤ ν when µtends towards zero because the exponentialasymptotic behavior of ν tends towards zerofaster than any polynomial asymptotic behavior.


What to Retain

All shape measures that satisfy Definition Aand that are commonly used are equivalentaccording to Definition B .


What to Retain


They all are sensitive to all the cases ofdegeneration of the simplices.


What to Retain


They all are sensitive to all the cases ofdegeneration of the simplices.

In this sense, none is better than the others.


Table of Contents





Delaunay








Global Quality and Optimization

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

In practice, the comparison of two different meshesobtained from different publications is often impossible : thestatistics presented, the shape measures and the scalesused vary from one publication to the other. Benchmarksneed to be defined along with exchange standards.


Benchmark

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


Histogram

0

5

10

15

20

25

30

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Critère de forme des tétraèdres

Pou

rcen

tage

des

tétr

aèdr

es

Rapport des moyennesRapport des rayons

Histogram of the meanratio η and of the radiusratio ρ.


Histogram

0

5

10

15

20

25

30

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Angle solide minimumAngle dièdre minimum

Histogram of the mini-mum of the solid angleθmin and of the mini-mum of the dihedralangle ϕmin.


Histogram

0

5

10

15

20

25

30

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1


Coefficient d’erreurRapport des arêtes

Histogram of the edgeratio r and of the in-terpolation error coeffi-cient γ.


Statistics of all the Tetrahedra

min µ max σ

Radius ratio ρ 0.5151 0.9067 0.9978 0.0602Mean ratio η 0.6559 0.9222 0.9979 0.0468Edge ratio r 0.5696 0.7375 0.9504 0.0641Interp. Error γ 0.4862 0.8058 0.9741 0.0709Solid ∠ θmin 0.2962 0.7115 0.9697 0.0996Dihedral ∠ 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 shapemeasure used. LIU and JOE (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 shapemeasure seems to be a significative index of the globalquality of the mesh.

Indeed, if several grids of different quality are taken andare classified according to the average of a shapemeasure, 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 theshape measure and of the mesh the maximum is almostalways close to 1.

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


Minimum of the Shape Measures

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

In a series of tests, the classification of the quality of themeshes according to the minimum of the shape measure ischaotic. It is not advisable to characterize a whole mesh byits worst element.


Standard Deviation of the ShapeMeasure

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

In a series of tests, classification of the meshes accordingto the standard deviation gives a significative classificationthat is slightly chaotic.


What to Retain

Statistics on the shape of the elements of a mesh aresignificative quantities of the quality of a mesh.


What to Retain


Any valid shape measure seems to yield properresults.


What to Retain



There does not seem to be a unique quality that isentirely indicative of the quality of a mesh.


What to Retain



There does not seem to be a unique quality that isentirely 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}mi=1 , n, {Cj}n

j=1

},

where m is the number of vertices of the mesh,Xi = (x1i, x2i, . . . , xdi) are the coordinates in IRd of the ithvertex, n is the number of simplices of the mesh, andCj = (c1j, c2j, . . . , cdj, cd+1,j) is the connectivity of the jthsimplex of the mesh composed of d + 1 pointers to thevertices of the mesh.


Optimization and Shape Measures

What is the influence of the choice of the shape measureused 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 ofthe side of the domain. The opti-mal mesh does exist in this specialcase.


Influence of the Shape Measure

ρ η θmin

γ r


Optimization and Shape Measure

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

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


Influence of the Shape Measure

ρ η θmin

γ rMesh Quality – p. 232/331

Influence of the Algorithm

The vertex relocation scheme is removed from the meshoptimization 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 ofthe side of the domain. The optimalmesh does exist in this case.


Influence of the Algorithm

ρ η θmin

γ r


What to Retain

If the optimal mesh exists, the mesh optimizerconverges towards the optimal mesh independently ofthe shape measure used.


What to Retain


If the optimal mesh does not exist, different shapemeasures will lead to different meshes. But thedifference is statistically less significative as themeshes become more optimized.


What to Retain


If the optimal mesh does not exist, different shapemeasures will lead to different meshes. But thedifference is statistically less significative as themeshes become more optimized.

When the meshes are of bad quality, it is not bychanging the shape measure that they become better,but by changing the algorithm.


Table of Contents





Delaunay








Size Quality of Simplices

The shape measures serve to measure the shape ofthe 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 aspossible the specified size of the elements.






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

This section presents three size criteria.


Target Size of the Simplices

In CUILLIÈRE (1998), the size of the simplices iscompared 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 =

∫

K

1 dK =

{ √3/4 in 2D,√2/12 in 3D.


Size Criterion QK

The size criterion QK of the simplex K is written as :

QK = S1

CK

∫

K

√det(M) dK

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


Efficiency Index

Another criterion that evaluates the conformity of a mesh toa metric is proposed by FREY and GEORGE (1999).

This criterion, contrarly to the previous one that evaluatesareas and volumes, is based on the length of the edges inthe 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 thata length of 2.0 means that the edge is two times biggerthan the specified length.A global measure of the conformity of a mesh to thespecified size is the Efficiency Index τ

τ = 1 − 1

na

na∑

i=1

(1 − min(Li, 1/Li) )2 .


Efficiency Index

Consider the distribution on all the edges of the mesh ofthe variable τi = min(Li, 1/Li).Let µ = (1/na)

∑na

i=1 τi be the averageLet σ2 = (1/na)

∑na

i=1(τi − µ)2 be the standard deviation.Then

τ = 1 − σ2 − (µ − 1)2.

The efficiency index measures both the dispersion of theedge lengths and their proximity to the target size.


Efficiency Index

τ = 1 − σ2 − (µ − 1)2.

This equality shows that maximizing τ implies bothminimizing the standard deviation and bringing the averageto 1.0. The optimal value is obtained when σ = 0 and µ = 1.This can only happen when all the edges are exactly equalto the specified length.The efficiency index is a good global measure of theconformity of the length of the edges with the specifiedlength of the edges.


Table of Contents





Delaunay








A Universal Measure of Mesh Quality

Hang on to your hat...



Table of Contents1. Introduction2. The Metric MK of a Simplex K3. The Specified Metric4. The Non-Conformity EK of a Simplex K5. The Non-Conformity ET of a Mesh T6. Generalisation of Size Quality Measures7. Extension to Non-Simplicial Elements8. Final Exam9. What to Retain


Introduction

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


Introduction


There exists quality measures for the size of thesimplices and for the mesh.


Introduction



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


Introduction




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


Shape and Size Measures

However, the ratio of the smallest edge on largest canbe as large as

√2/2 = 0.707 for a tetrahedron to

degenerate to a sliver.




√2/2 = 0.707 for a tetrahedron to

degenerate to a sliver.

This means that a simplex having edges of reasonablesize does not mean that this simplex is of reasonableshape, since it can be degenerate.



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



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

The goal of this lecture is to introduce a universalcriterion that will measure shape and size in a singleand complete step.


The Metric MK of a Simplex K

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

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

in IRd.

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



In IRd, d = 2 or 3, the d(d + 1)/2 components of the metricare 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 Kare 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-degeneratetriangles.



For instance, recall the triangle where vertices A and B arelocated at A = (0, 1/2)T , B = (0, −1/2)T and where thevertex C = (x, y)T free to move in the half-plane x ≥ 0.The system of Eqs. reduces to the system

0 0 1

x2 2x(y − 12) (y − 1

2)2

x2 2x(y + 12) (y + 1

2)2

m11

m12

m22

=

1

1

1

,



which yields :

MK =

4y2 + 3

4x2

−y

x

−y

x1

.

This metric MK becomes identity when the vertexC(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 asMK = R−1(θ) Λ R(θ), where the matrix Λ is the diagonalmatrix of the eigenvalues of MK , i.e., Λ = diag(λ1, λ2[, λ3]).The eigenvalues λi are the length of the axes of the ellipseand 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.

ℓ = 1

ℓ = 1ℓ = 1

r = 1/√

3

r = 1



Ellipses of a selectedgroup of elements. Note inthis figure that the ellipsespass through the verticesof the triangle.


The Specified Metric

A size specification map can be constructed from aposteriori error estimators, from geometrical properties ofthe 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 diagonalmatrices whose diagonal terms are 1/h2.


The Specified Metric MS

Whatever its origin, the size specification map contains theinformation of the prescribed size and stretching of themesh 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 atpoint X. Let MS(K) be the averaged specifiedRiemannian metric over a simplex K as computed by :

MS(K) =

(∫

K

MS(X) dK

) / (∫

K

dK

).

This integral can be approximated by a numericalquadrature.


Visualization of MS(K)

The specified metric is de-fined in GEORGE and BO-ROUCHAKI (1997). It is ananalytical function that de-fines an isotropic metric.Note that the triangles donot fit exactly the specifiedmetric.



The specified metric is de-fined in GEORGE and BO-ROUCHAKI (1997). It is ananalytical function that de-fines an anisotropic metric.Note that the triangles donot fit exactly the specifiedmetric.



Supersonic laminar vs-cous air flow aroundNACA 0012. The specifiedanisotropic metric is basedon the interpolation error(second derivatives) of thespeed field.Note that the triangles donot fit exactly the specifiedmetric.


Simplex Conformity

When the metric MK of the simplex K corresponds exactlyto the averaged specified Riemannian metric MS(K) forthat simplex, the following equality holds :

MK = MS(K).

However, in practice, there is usually some discrepancybetween these two metrics and this section presents amethod to measure this discrepancy.


Simplex Conformity

This equality of metrics can be rewritten in the twofollowing ways :

MS−1MK = I

andMK

−1 MS = I,

where I is the identity matrix.


Simplex Residuals

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

Rs = MS−1MK − I

andRb = MK

−1 MS − I.

where Rs will detect the degeneration of the simplex K asit’s volume tends to zero and Rb as it’s volume tends toinfinity.


Example – Triangle ABC

Recall the triangle with two fixed vertices, oneat A = (0, 1/2)T and one at B = (0,−1/2)T , and that thethird vertex was free to move. Furthermore, if the specifiedtriangle is the unit equilateral triangle, then the averagedspecified 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

Rs = I

4y2 + 3

4x2−y

x

−y

x1

− I =

4y2 + 3

4x2− 1 −y

x

−y

x0

,

Rb =

4x2

3

4xy

3

4xy

3

4y2

3+ 1

I − I =

4x2

3− 1

4xy

3

4xy

3

4y2

3

.


Example – C(x, y) with y = 0

If the third vertex C is restricted to the axis y = 0, then allbut the first term of these tensors vanish.The two curves intersect at x =

√3/2, where the residuals

become null.

0

5

10

15

20

1

Res

idua

l

x2 30.5

Rs Rb


Total Residual Rt

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

Rt = Rs + Rb = MS−1MK + MK

−1 MS − 2I.


The Non-Conformity EK of aSimplex K

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

EK = ‖Rt‖ =√

tr (RtT Rt).

The Euclidean norm of a matrix ‖ · ‖ amounts to the squareroot of the sum of each term of the matrix individuallysquared.



For the triangle described above with two fixed verticesand a free vertex and for which the specified Riemannianmetric was the identity matrix, the coefficient ofnon-conformity is expressed as,

EK =

√(4y2 + 3

4x2− 2 +

4x2

3

)2

+ 2

(4xy

3− y

x

)2

+16y4

9.



Logarithm base 10 ofEK when the targetmetric is the identitymatrix. It is minimumand equal to zero forthe equilateral triangle,and increases very ra-pidly as the third vertexmoves away from theoptimal position. It is in-finite for all degeneratetriangles.

x 3210

y

1

0

-1

MS =(

10

01

), Xopt =

(√3/2, 0

)T


Visualization of EK

The specified metric isdefined in GEORGE andBOROUCHAKI (1997). Itis an analytical functionthat defines an isotropicmetric.


Visualization of EK

The specified metric isdefined in GEORGE andBOROUCHAKI (1997). Itis an analytical functionthat define an anisotro-pic metric.


Visualization of EK

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


The Non-Conformity ET of a Mesh T

Definition : The coefficient of non-conformity of amesh ET is defined as :

ET =1

nK

nK∑

i=1

EKi,

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


Properties of ET

The perfect mesh is obtained when the coefficient ofnon-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 andthe 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 ofnon-conformity of the first and last meshes are equivalent.


Properties of ET

It is possible to compare the quality of the mesh of twovastly different domains, such as the mesh of a galaxy andthe mesh of a micro-circuit. In both cases, the measuregives a comparable number that reflects the degree towhich each mesh satisfies its size specification map.

This coefficient therefore poses itself as a unique anddimensionless measure of the non-conformity of a meshwith respect to a size specification map given in the form ofa Riemannian metric, be it isotropic or anisotropic.


Generalisation of Size QualityMeasures

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

MK = MS(K).

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


Generalisation of the SizeCriterion QK

MK(X) = MS(X),

CK =

∫

K

√det(MK(X)) dK =

∫

K

√det(MS(X)) dK,

and then

QK =1

CK

∫

K

√det(MS(X)) dK.

CK is an integral form of the conformity between themetric 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. Thenthe pointwise conformity between the metric MK of thesimplex and the specified metric MS

MK(X) = MS(X)

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

∫

AB

√ABT MK(X) AB =

∫

AB

√ABT MS(X) AB

1 = LMS(AB).


Generalisation of Efficiency Index τ

This relation1 = 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 integralform of the conformity between the metric MK of thesimplex and the specified metric MS evaluated over theedges of the mesh.


Extension to Non-SimplicialElements

Non-Simplicial elements are quadrilaterals in twodimensions and prisms and hexahedra in threedimensions.

In order to extend this measure to non-simplicial elements,it has to be understood that the metric tensor ofnon-simplicial elements is not a constant and varies forevery point of space.

In other words, the Jacobian of a simplex is constant butthe Jacobian of a non-simplicial element depends of thepoint of evaluation.


Non-Simplicial Element Conformity

The conformity between the metric MK of a non-simplicialelement and the specified metric MS takes on a pointwisenature can be rewritten as :

MK(X) = MS(X), ∀X ∈ K.


Non-Simplicial Element ConformityResidue

The total residue Rt become a pointwise value

Rt(X) = M−1S (X)MK(X) + M−1

K (X)MS(X) − 2I.

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

EK =

∫K‖M−1

S (X)MK(X) + M−1K (X)MS(X) − 2I‖ dK∫

KdK

.


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 CET = 0.0843 ET = 0.00 ET = 0.503


Test 2 – Isotropic Mesh

This test case is defined in GEORGE and BOROUCHAKI(1997).

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

This test case has an isotropic Riemannian metric definedby :

MS =

(h−2

1 (x, y) 0

0 h−22 (x, y)

), . . .



. . . where h1(x, y) = h2(x, y) = h(x, y) is given by :

h(x, y) =

1 − 19y/40 if y ∈ [0, 2],

20(2y−9)/5 if y ∈ ]2, 4.5],

5(9−2y)/5 if y ∈ ]4.5, 7],15

+ 45

(y−72

)4if y ∈ ]7, 9].



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


Test 2a – Isotropic Mesh

A B C


Test 2a – Isotropic Mesh

A B CET = 3.18 ET = 0.104 ET = 56.2


Test 2b – Isotropic Mesh

A B C


Test 2b – Isotropic Mesh

A B CET = 0.104 ET = 0.929 ET = 3.18


Test 3 – Anisotropic Mesh

This test case is defined in GEORGE and BOROUCHAKI(1997).

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

This test case has an anisotropic Riemannian metricdefined by :

MS =

(h−2

1 (x, y) 0

0 h−22 (x, y)

), . . .



. . . where h1(x, y) is given by :

h1(x, y) =

1 − 19x/40 if x ∈ [0, 2],

20(2x−7)/3 if x ∈ ]2, 3.5],

5(7−2x)/3 if x ∈ ]3.5, 5],

15

+ 45

(x−5

2

)4if x ∈ ]5, 7], . . .



. . . and h2(x, y) is given by :

h2(x, y) =

1 − 19y/40 if y ∈ [0, 2],

20(2y−9)/5 if y ∈ ]2, 4.5],

5(9−2y)/5 if y ∈ ]4.5, 7],

15

+ 45

(y−72

)4if y ∈ ]7, 9].



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



A B C



A B CET = 0.405 ET = 2.67 ET = 0.107


Test 4 – Bernhard Riemann

The size specification map isdeduced from an error esti-mator based on the secondderivatives of the grey level ofthe picture.



A B C



A B CET = 0.546 ET = 0.345 ET = 0.845


Test 5 – Flow over a Naca 0012

Supersonic laminar flow at Mach 2.0, Reynolds 1000 andan angle of attack of 10 degrees. An a posteriori errorestimator is deduced from this solution.


Test 5a – Flow over a Naca 0012

A B C


Test 5a – Flow over a Naca 0012

A B CSpecified 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 CSpecified 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 CSpecified Metric MS ET = 1160 ET = 0.658


What to Retain

This lecture presented a method to measure thenon-conformity of a simplex and of a whole mesh withrespect to a size specification map given in the form of aRiemannian metric.

This measure is sensitive to discrepancies in both size andshape with respect to what is specified.

Analytical examples of the behavior were presented andnumerical examples were provided.


The Non-Conformity ET is Universal

The coefficient of non-conformity of a mesh, ET , is auniversal 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 Riemannianmetric, 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 asmall or a big domain.


Mesh Optimization

This measure poses itself as a natural measure to use inthe benchmarking process. Indeed, since the measure isable to compare two different meshes, it can help tocompare the algorithms used to produce the meshes.

This measure of the non-conformity of a mesh seems to bean adequate cost function for mesh generation, meshoptimization and mesh adaptation. This measure could beused for each step such that each step minimizes thesame cost function.


Table of Contents





Delaunay








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 possibledegeneracies.





A shape measure is valid if it is sensitive to all possibledegeneracies.

A shape measure is not valid if it is not null for everydegenerate simplex.


Shape Measure

Beauty, quality and shape are relative notions.


Shape Measure


We fisrt need to define what we want in order toevaluate 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 asimplex in this metric.


Shape Measure

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


Shape Measure


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


Shape Measure


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

The shape measures are more or less equivalentduring mesh optimization.


Size Measures



Size Measures


There exists quality measures for the size of thesimplices and of the mesh.


Size Measures





Size Measures




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


Size Measures


√2/2 for a tetrahedron to degenerate to

a sliver.


Size Measures


√2/2 for a tetrahedron to degenerate to

a sliver.

This means that a simplex having edges of reasonablesize does not mean that this simplex is of reasonableshape, since it can be degenerate.


Universal Criterion

This brings forth the problem in all its generality :

What would be a simplicial quality measure that couldmeasure simultaneously size and shape , that would besensitive to all possible degeneracies of the simplices, thatwould be optimal for the régular and unitary simplex, in anEuclidean metric or in a Riemannian metric, be it isotropicor anisotropic, in two and in three dimensions.


Soon on your Screens !

P. LABBÉ, J. DOMPIERRE, M.-G. VALLET, F. GUIBAULT etJ.-Y. TRÉPANIER. A Measure of the Conformity of a Meshto an Anisotropic Metric, Tenth International MeshingRoundtable, Newport Beach, CA, octobre 2001, pages319–326,

has proposed such a criterion that measures theconformity in shape and size between a mesh and themetric that this mesh was supposed to fit.


The Non-Conformity ET of a Mesh

A method to measure the non-conformity of a simplex andof a whole mesh with respect to a size specification mapgiven in the form of a Riemannian metric was given.

This measure is sensitive to discrepancies in both size andshape with respect to what is specified.

Analytical examples of the behavior were presented andnumerical examples were provided.










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


Mesh Optimization

This measure poses itself as a natural measure to use inthe benchmarking process. Indeed, since the measure isable to compare two different meshes, it can help tocompare the algorithms used to produce the meshes.

This measure of the non-conformity of a mesh seems to bean adequate cost function for mesh generation, meshoptimization and mesh adaptation. This measure could beused for each step such that each step minimizes thesame cost function.


The End


mesh quality

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