mesh quality
DESCRIPTION
Shape measures, which provide an effective quantitative mean of comparison of element shapes in a mesh, are of great relevance in many fields related to finite element analysis, but particularly in mesh adaptation. Still, while most serious works in the field of mesh adaptation directly make use of shape measures, very little work has been devoted to the actual comparison of shape measures, with the notable exceptions of Liu and Joe (1994) who have thoroughly analyzed a set of a few selected measures. While the published works present some of the standard shape measures in current use, new shape measures steadily appear in recent literature for which no analysis is available. Furthermore, no classification scheme has been proposed, and fitness of new measures is often not assessed. This lecture aims to survey a wider range of shape measures in general use, to define validity criteria for those measures and to classify then in broad categories, beginning with valid vs. invalid shape measures. The lecture also addresses issues regarding the use of shape measures in non-Euclidean spaces, such as the use of shape measures in Riemannian spaces for anisotropic mesh adaptation. The lecture summarizes important properties of simplices and introduces a classification of simplex degeneracies in two and three dimensions. I will present a wide range of shape measures, introduce shape measures validity criteria, and present a visualization scheme that helps analyze and compare shape measures to one another. Shape measures are then classified, and conclusions are drawn on the pertinence of developing new shape measures or choosing one among the currently existing ones. Mesh adaptivity is a process that generates a sequence of meshes and numerical solutions on these meshes such that the sequence converges to some goal which usually is error equirepartition whilst minimizing the computational effort by minimizing the number of vertices of the mesh. For unstructured meshes, the process of computing a mesh in the sequence can be decomposed in two steps: first, a size specification map is computed by analyzing the numerical solution; second, a mesh is computed that satisfies this size specification map. The subject of the present lecture is to offer a measure of the degree to which a mesh satisfies it\'s size specification map. More than ten years ago, Marie-Gabrielle Vallet (1990, 1991, 1992) showed that giving the size specification map using a metric tensor representation eased the generation of adapted and anisotropic meshes by combining the desired size and stretching into a single mathematical concept. Metric tensors modify the way distances are measured. The adapted and anisotropic mesh in the real Euclidean space is constructed by building a regular, isotropic and unitary mesh in the metric tensor space. The use of a metric tensor representation for the size specification map is now a widely used tool for the generation and adaptation of anisotropic meshes. It has been used in two and three dimensions, for various PDE simulations with finite element and finite volume methods, for surface discretization, graphic representation, etc. The most complete references are George and Borouchaki (1997) and Frey and George (1999) the references therein. However, the issue of metric conformity is still not clear. There is no well defined way to measure the degree to which a mesh satisfies a size specification map given in the form of a field of metric tensors. Most authors rely on two competing measures to assess the quality of their meshes with respect to a size specification map. One measure compares the simplex shape with the specified stretching. This is usually done by computing a shape criterionTRANSCRIPT
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Mesh QualityJulien Dompierre
Centre de Recherche en Calcul Applique (CERCA)
Ecole Polytechnique de Montreal
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Authors
• Research professionals• Julien Dompierre• Paul Labbé• Marie-Gabrielle Vallet
• Professors• François Guibault• Jean-Yves Trépanier• Ricardo Camarero
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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.
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References – 2
A. LIU and B. JOE, Relationship betweenTetrahedron Shape Measures , Bit, Vol. 34,pages 268–287, (1994).
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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).
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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).
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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.
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References – 6
P. J. FREY AND P.-L.GEORGE, Maillages. Ap-plications aux éléments finis.Hermès, 1999, Paris.This book is available inEnglish.
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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
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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.
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Face Flipping
How can we choose the configuration thatproduces the best tetrahedra ? A tetrahedronshape quality measure is needed.
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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.
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Mesh Optimization
• Let O1 and O2, two three-dimensionalunstructured tetrahedral mesh Optimizers.
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Mesh Optimization
• Let O1 and O2, two three-dimensionalunstructured tetrahedral mesh Optimizers.
• What is the norm ‖O‖ of a mesh optimizer ?
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Mesh Optimization
• Let O1 and O2, two three-dimensionalunstructured tetrahedral mesh Optimizers.
• What is the norm ‖O‖ of a mesh optimizer ?
• How can it be asserted that ‖O1‖ > ‖O2‖?
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It’s Obvious !
• Let B be a benchmark.
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It’s Obvious !
• Let B be a benchmark.• Let M1 = O1(B) be the optimized mesh
obtained with the mesh optimizer O1.
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It’s Obvious !
• Let B be a benchmark.• Let M1 = O1(B) be the optimized mesh
obtained with the mesh optimizer O1.• Let M2 = O2(B) be the optimized mesh
obtained with the mesh optimizer O2.
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It’s Obvious !
• Let B be a benchmark.• Let M1 = O1(B) be the optimized mesh
obtained with the mesh optimizer O1.• Let M2 = O2(B) be the optimized mesh
obtained with the mesh optimizer O2.• Common sense says : “The proof is in the
pudding”.
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It’s Obvious !
• Let B be a benchmark.• Let M1 = O1(B) be the optimized mesh
obtained with the mesh optimizer O1.• Let M2 = O2(B) be the optimized mesh
obtained with the mesh optimizer O2.• Common sense says : “The proof is in the
pudding”.• If ‖M1‖ > ‖M2‖ then ‖O1‖ > ‖O2‖.
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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.
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The Trick...
• Because the norm ‖O‖ of a mesh optimizer isunknown, the comparison of two optimizers isreplaced by the comparison of two meshes.
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The Trick...
• Because the norm ‖O‖ of a mesh optimizer isunknown, the comparison of two optimizers isreplaced by the comparison of two meshes.
• What is the norm ‖M‖ of a mesh ?
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The Trick...
• Because the norm ‖O‖ of a mesh optimizer isunknown, the comparison of two optimizers isreplaced by the comparison of two meshes.
• What is the norm ‖M‖ of a mesh ?
• How can we assert that ‖M1‖ > ‖M2‖?
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The Trick...
• Because the norm ‖O‖ of a mesh optimizer isunknown, the comparison of two optimizers isreplaced by the comparison of two meshes.
• What is the norm ‖M‖ of a mesh ?
• How can we assert that ‖M1‖ > ‖M2‖?• This is what you will know soon, or you
money back !
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What to Retain
• This lecture is about the quality of theelements of a mesh and the quality of a wholemesh.
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What to Retain
• This lecture is about the quality of theelements of a mesh and the quality of a wholemesh.
• The concept of element quality is necessaryfor the algorithms of egde and face swapping.
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What to Retain
• This lecture is about the quality of theelements of a mesh and the quality of a wholemesh.
• 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.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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Definition of a Simplex
Meshes in two and three dimensions are made ofpolygons or polyhedra named elements.
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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.
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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.
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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.
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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.
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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.
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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.
Quadrilaterals, pyramids, prisms, hexahedra and othersuch aliens are named non-simplicial elements.
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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.
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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.
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What to Retain
In two dimensions, the simplex is a triangle.
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What to Retain
In two dimensions, the simplex is a triangle.
In three dimensions, the simplex is a tetrahedron.
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What to Retain
In two dimensions, the simplex is a triangle.
In three dimensions, the simplex is a tetrahedron.
The d + 1 vertices of a simplex in Rd give d vectors thatform a base of Rd.
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What to Retain
In two dimensions, the simplex is a triangle.
In three dimensions, the simplex is a tetrahedron.
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.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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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.
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Degeneracy of Simplices
A d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd.
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Degeneracy of Simplices
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.
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Degeneracy of Simplices
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.
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Degeneracy of Simplices
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.
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Degeneracy of Simplices
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.
Nota bene : Strictly speaking, accordingly to thedefinition, a degenerate simplex is no longer a simplex.
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Degeneracy Criterion
A d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determinant isnull.
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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.
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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!.
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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.
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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 tetrahedron is degenerate if its volume is null.
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Taxonomy of Degenerate Simplices
This taxonomy is based on the different possibledegenerate states of the simplices.
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Taxonomy of Degenerate Simplices
This taxonomy is based on the different possibledegenerate states of the simplices.
There are three cases of degenerate triangles.
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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.
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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.
In this classification, the four symbols, , and stand for vertices of multiplicity
simple, double, triple and quadruple respectively.
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1 – The Cap
Name h −→ 0 h = 0
CaphC
A B BCA
Degenerate edges : NoneRadius of the smallest circumcircle : ∞
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2 – The Needle
Name h −→ 0 h = 0
NeedlehC
A B BA,C
Degenerate edges : AC
Radius of the smallest circumcircle : hmax/2
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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.
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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
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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 : ∞
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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 : ∞
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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 ∞
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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
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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 : ∞
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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 : ∞
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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 : ∞
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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
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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
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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
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What to Retain
A triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.
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What to Retain
A triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.
There are three cases of degeneracy of triangles.
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What to Retain
A triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.
There are three cases of degeneracy of triangles.
A tetrahedron is degenerate if its vertices are coplanar,collinear or collapsed, hence if its volume is null.
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What to Retain
A triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.
There are three cases of degeneracy of triangles.
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.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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Shape Quality of Simplices
An usual method used to quantify the quality of a meshis through the quality of the elements of that mesh.
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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.
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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.
This section is a guided tour of the shape measuresused for simplices.
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The Regular Simplex
Definition : An element is regular if it maximizes its measure fora given measure of its boundary.
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The Regular Simplex
Definition : An element is regular if it maximizes its measure fora given measure of its boundary.
The equilateral triangle is regular because it maximizesits area for a given perimeter.
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The Regular Simplex
Definition : An element is regular if it maximizes its measure fora given measure of its boundary.
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.
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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.
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Remarks
The invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent of the coordinates system.
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Remarks
The invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent of the coordinates system.
The invariance under a valid uniform scaling meansthat the simplicial shape measures must bedimensionless (independent of the unit system).
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Remarks
The invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent of the coordinates system.
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.
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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
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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
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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.
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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.
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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 .
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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.
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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.
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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).
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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.
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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.
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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.
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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.
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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).
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Other Shape Measure – 2
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).
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Other Shape Measure – 3
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).
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Other Shape Measure – 4
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.
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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.
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What to Retain
The regular simplex is the equilateral one, ie, where allits edges have the same length.
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What to Retain
The regular simplex is the equilateral one, ie, where allits edges have the same length.
A shape measures evaluates the ratio to equilaterality.
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What to Retain
The regular simplex is the equilateral one, ie, where allits edges have the same length.
A shape measures evaluates the ratio to equilaterality.
A non valid shape measure does not vanish for alldegenerate simplices.
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What to Retain
The regular simplex is the equilateral one, ie, where allits edges have the same length.
A shape measures evaluates the ratio to equilaterality.
A non valid shape measure does not vanish for alldegenerate simplices.
There exists an infinity of valid shape measures.
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What to Retain
The regular simplex is the equilateral one, ie, where allits edges have the same length.
A shape measures evaluates the ratio to equilaterality.
A non valid shape measure does not vanish for alldegenerate simplices.
There exists an infinity of valid shape measures.
The goal of research is not to find an other one waybetter than the other ones.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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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.
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The Half-Perimeter
The half-perimeter pK is given by
pK =(L12 + L13 + L23)
2.
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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).
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Radius of the Incircle
The radius ρK of the incircle of the triangle K is given by
ρK =SK
pK
.
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Radius of the Circumscribed Circle
The radius rK of the circumcircle of the triangle K is givenby
rK =L12 L13 L23
4SK
.
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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).
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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).
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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.
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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.
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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.
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Radius of the Insphere
The radius ρK of the insphere of the tetrahedron K is givenby
ρK =3VK
S1 + S2 + S3 + S4
.
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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.
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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).
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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
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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).
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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
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Which Is the Most Beautiful Triangle ?
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Which Is the Most Beautiful Triangle ?
A
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Which Is the Most Beautiful Triangle ?
A B
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If You Chose the Triangle A...
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If You Chose the Triangle A...
AYou are wrong !
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If You Chose the Triangle B...
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If You Chose the Triangle B...
BYou are wrong again !
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Which Is the Most Beautiful Triangle ?
A BNone of these answers !
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Which Is the Most Beautiful Woman ?
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Which Is the Most Beautiful Woman ?
A
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Which Is the Most Beautiful Woman ?
A B
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You Probably chose...
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You Probably chose...
A BWoman A.
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And if One Asked these Gentlemen...
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And if One Asked these Gentlemen...
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These Gentlemen Would Choose...
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These Gentlemen Would Choose...
A BWoman B.
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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.
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Which Is the Most Beautiful Triangle ?
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Which Is the Most Beautiful Triangle ?
A B
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Which Is the Most Beautiful Triangle ?
A BThe question is incomplete : It misses a way ofmeasuring the quality of a triangle.
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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.
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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.
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A Cloud of Vertices
Let S = {Si}i=1,...,N be a cloud of N vertices.
S6
S11S2 S10
S4
S3S12S7
S9
S8S5
S1
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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)
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The Voronoi Diagram
The set of Voronoi cells associated with all thevertices of the cloud of vertices is called theVoronoi diagram.
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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.
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What to Retain
The Voronoi diagrams are partitions of spaceinto cells based on the concept of distance.
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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.
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Triangulation of a cloud of Points
The same cloud of points can be triangulated inmany different fashions.
. . .
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Triangulation of a Cloud of Points
. . .
. . .
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Triangulation of a Cloud of Points
. . .
. . .
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Delaunay Triangulation
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.
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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
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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
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Delaunay Triangulation
Delaunay Triangulation : If all the simplices Kof a triangulation T satisfy the empty spherecriterion, then the triangulation is said to be aDelaunay triangulation.
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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
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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.
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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.
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Duality Delaunay-Voronoï
The Voronoï diagram is the dual of the Delaunaytriangulation and vice versa.
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Voronoï and Delaunay in Nature
Voronoï diagrams and Delaunay triangulationsare not just a mathematician’s whim, theyrepresent structures that can be found in nature.
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Voronoï and Delaunay In Nature
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A Turtle
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A Pineapple
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The Devil’s Tower
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Dry Mud
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Bee Cells
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Dragonfly Wings
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Pop Corn
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Fly Eyes
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Carbon Nanotubes
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Soap Bubbles
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A Geodesic Dome
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Biosphère de Montréal
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Streets of Paris
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Roads in France
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Roads in France
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Where Is this Guy Going ? ! !
A simplicial shape measure is an evaluationof the ratio to equilarity.
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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.
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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.
A Delaunay triangulation of a cloud of pointsis a triangulation based on the notion ofdistance.
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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.
A Delaunay triangulation of a cloud of pointsis a triangulation based on the notion ofdistance.
The notion of distance can be generalized.
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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.
A Delaunay triangulation of a cloud of pointsis a triangulation based on the notion ofdistance.
The notion of distance can be generalized.
The notions of shape measure, of Voronoïdiagram and of Delaunay triangulation can begeneralized.
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Nikolai Ivanovich Lobachevsky
NIKOLAI IVANOVICHLOBACHEVSKY, 1décembre 1792, NizhnyNovgorod — 24 février1856, Kazan.
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János Bolyai
JÁNOS BOLYAI, 15 dé-cembre 1802 à Kolozsvár,Empire Austrichien (Cluj,Roumanie) — 27 janvier1860 à Marosvásárhely,Empire Austrichien (Tirgu-Mures, Roumanie).
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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.
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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.
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The Metric in the Merriam-Webster
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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.
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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.
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Metric Space
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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.
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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.
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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.
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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.
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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.
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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
.
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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.
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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.
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Metric and Delaunay Mesh
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Which is the Best Triangle ?
A BThe question is incomplete. The way to measurethe quality of the triangle is missing.
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Which is the Best Triangle ?
A B
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Which is the Best Triangle ?
A B
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Example of an Adapted Mesh
Adapted mesh and solution for a transonicvisquous compressible flow with Mach 0.85 andReynolds = 5 000.
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Zoom on Boundary Layer–Shock
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What to Retain
Beauty, quality and shape are relativenotions.
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What to Retain
Beauty, quality and shape are relativenotions.
We first need to define what we want in orderto evaluate what we obtained.
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What to Retain
Beauty, quality and shape are relativenotions.
We first need to define what we want in orderto evaluate what we obtained.
“What we want” is written in the form of metrictensors.
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What to Retain
Beauty, quality and shape are relativenotions.
We first need to define what we want in orderto evaluate what we obtained.
“What we want” is written in the form of metrictensors.
A shape measure is a measure of theequilarity of a simplex in this metric.
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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.
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Shape Measure in a 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.
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Shape Measure in a Metric
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.
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Shape Measure in a Metric
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.
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Shape Measure in a 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.
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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
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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.
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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 ?
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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.
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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
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What to Retain
The empty sphere criterion of Delaunay is nota shape measure but it can be used as ashape measure.
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What to Retain
The empty sphere criterion of Delaunay is nota shape measure but it can be used as ashape measure.
In two dimensions, in the edge swappingalgorithm (Lawson’s method), the emptysphere criterion of Delaunay is equivalent tothe θmin shape measure.
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What to Retain
The empty sphere criterion of Delaunay is nota shape measure but it can be used as ashape measure.
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.
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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.
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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.
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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.
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What to Retain
The empty sphere criterion ofDelaunay is not a valid shapemeasure sensitive to all the possibledegeneracies of the tetrahedron.
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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 .
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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”.
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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
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Non-Convex Quadrilateral
S1
S3
S2
T1
S4
T2
Two adjacent trianglesforming a non-convexquadrilateral necessa-rily satisfy the emptysphere criterion ofDelaunay.
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Loss of the Convexity Property in 3D
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What to Retain
The empty sphere criterion of Delaunay ismore or less a simplicial shape measure.
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What to Retain
The empty sphere criterion of Delaunay ismore or less a simplicial shape measure.
The empty sphere criterion of Delaunay is notsensitive to all the possible degeneracies ofthe tetrahedron.
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What to Retain
The empty sphere criterion of Delaunay ismore or less a simplicial shape measure.
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.
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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
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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.
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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.
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Regularity Generalization
An equilateral quadrilateral, ie that has fouredges of same length, is not necessarily asquare...
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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.
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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.
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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 tetrahedron is regularbecause it maximizes its volume for a givensurface of its faces.
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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.
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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.
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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
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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
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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
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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.
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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
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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.
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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
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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.
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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.
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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.
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Table of Contents1. 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
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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.
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Visualizing Shape Measures
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.
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Visualizing Shape Measures
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.
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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.
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3D Rendering of Shape Measures
In 3D, 5 parameters are necessary. Two are fixedand the influence of the 3 others is visualized.
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Rendering Taking a Metric Into Ac-count
x 3210
y
1
0
-1
Mean ratio η
M =
(0.2 0
0 1
)
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Rendering Taking a Metric into Ac-count
x 210
y
1
0
-1
Mean ratio η
M =
(20 0
0 1
)
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Rendering Taking a Metric into Ac-count
x 3210
y
1
0
-1
Mean ratio η
M =
(0.9 0.4
0.4 1
)
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Rendering Taking a Metric into Ac-count
x 3210
y
1
0
-1
Mean ratio η
M =
(1 0
0 1
)
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What to Retain
Mean ratio η is the privileged shape measure.
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What to Retain
Mean ratio η is the privileged shape measure.
Circular contour lines in Euclidean spacebecome ellipses in the general case.
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What to Retain
Mean ratio η is the privileged shape measure.
Circular contour lines in Euclidean spacebecome ellipses in the general case.
The shape of a triangle is a quality measurethat is relative.
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What to Retain
Mean ratio η is the privileged shape measure.
Circular contour lines in Euclidean spacebecome ellipses in the general case.
The shape of a triangle is a quality measurethat is relative.
A good triangle in a metric tensor is notbeautiful in a different metric tensor.
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What to Retain
Mean ratio η is the privileged shape measure.
Circular contour lines in Euclidean spacebecome ellipses in the general case.
The shape of a triangle is a quality measurethat is relative.
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.
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Table of Contents1. 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
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Equivalence of Shape Measures
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Equivalence of Shape Measures
x 3210
y
1
0
-1
Superposition ofcontour plots ofsimplex shape mea-sures ρ, η, θmin etγ.
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Equivalence of Shape Measures
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.
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Optimal Bounds
In the equivalence relation of shape measures
c0νe0 ≤ µ ≤ c1ν
e1,
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Optimal Bounds
In the equivalence relation of shape measures
c0νe0 ≤ µ ≤ c1ν
e1,
the lower bound is said to be optimal if e0 is thesmallest possible exponent,
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Optimal Bounds
In the equivalence relation of shape measures
c0νe0 ≤ µ ≤ c1ν
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.
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Tight Bounds
In the equivalence relation of shape measures
c0 νe0 ≤ µ ≤ c1 νe1,
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Tight Bounds
In the equivalence relation of shape measures
c0 νe0 ≤ µ ≤ c1 νe1,
the lower bound is said to be tight if c0 is thebiggest possible constant,
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Tight Bounds
In the equivalence relation of shape measures
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.
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Equivalence Relation
It is indeed an equivalence relation because it is
reflexive,symmetric,transitive.
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Symmetric Relation
If µ is equivalent to ν with
c0νe0 ≤ µ ≤ c1ν
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.
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Transitive RelationIf µ is equivalent to ν and if ν is equivalent to υwith
c0νe0 ≤ µ ≤ c1ν
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.
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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.
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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.
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Equivalence Classes for ShapeMeasures
The equivalence relation Definition B definesequivalence classes.
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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 .
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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 ? ? ?
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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 ? ? ?
No ! LIU has provided a counterexample.
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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.
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What to Retain
All shape measures that satisfy Definition Aand that are commonly used are equivalentaccording to Definition B .
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What to Retain
All shape measures that satisfy Definition Aand that are commonly used are equivalentaccording to Definition B .
They all are sensitive to all the cases ofdegeneration of the simplices.
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What to Retain
All shape measures that satisfy Definition Aand that are commonly used are equivalentaccording to Definition B .
They all are sensitive to all the cases ofdegeneration of the simplices.
In this sense, none is better than the others.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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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.
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Benchmark
Unit cube with a uni-form isotropic size spe-cification map of 1/10.
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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.
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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
Angle solide minimumAngle dièdre minimum
Histogram of the mini-mum of the solid angleθmin and of the mini-mum of the dihedralangle ϕmin.
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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
Coefficient d’erreurRapport des arêtes
Histogram of the edgeratio r and of the in-terpolation error coeffi-cient γ.
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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
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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.
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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.
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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.
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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.
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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.
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What to Retain
Statistics on the shape of the elements of a mesh aresignificative quantities of the quality of a mesh.
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What to Retain
Statistics on the shape of the elements of a mesh aresignificative quantities of the quality of a mesh.
Any valid shape measure seems to yield properresults.
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What to Retain
Statistics on the shape of the elements of a mesh aresignificative quantities of the quality of a mesh.
Any valid shape measure seems to yield properresults.
There does not seem to be a unique quality that isentirely indicative of the quality of a mesh.
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What to Retain
Statistics on the shape of the elements of a mesh aresignificative quantities of the quality of a mesh.
Any valid shape measure seems to yield properresults.
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.
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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.
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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.
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Influence of the Shape Measure
ρ η θmin
γ r
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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.
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Influence of the Shape Measure
ρ η θmin
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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.
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Influence of the Algorithm
ρ η θmin
γ r
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What to Retain
If the optimal mesh exists, the mesh optimizerconverges towards the optimal mesh independently ofthe shape measure used.
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What to Retain
If the optimal mesh exists, the mesh optimizerconverges towards the optimal mesh independently ofthe shape measure used.
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.
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What to Retain
If the optimal mesh exists, the mesh optimizerconverges towards the optimal mesh independently ofthe shape measure used.
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.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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Size Quality of Simplices
The shape measures serve to measure the shape ofthe elements of the mesh.
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Size Quality of Simplices
The shape measures serve to measure the shape ofthe elements of the mesh.
The shape measures are dimensionless.
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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.
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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.
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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.
This section presents three size criteria.
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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.
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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.
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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.
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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 .
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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.
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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.
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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A Universal Measure of Mesh Quality
Hang on to your hat...
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A Universal Measure of Mesh Quality
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
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Introduction
The simplices can be of good shape without being ofgood size.
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Introduction
The simplices can be of good shape without being ofgood size.
There exists quality measures for the size of thesimplices and for the mesh.
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Introduction
The simplices can be of good shape without being ofgood size.
There exists quality measures for the size of thesimplices and for the mesh.
In principle, a simplex whose edges are of unit lengthin the metric is also of perfect shape in that metric.
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Introduction
The simplices can be of good shape without being ofgood size.
There exists quality measures for the size of thesimplices and for the mesh.
In principle, a simplex whose edges are of unit lengthin the metric is also of perfect shape in that metric.
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.
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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.
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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.
This means that a simplex having edges of reasonablesize does not mean that this simplex is of reasonableshape, since it can be degenerate.
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Shape and Size Measures
We can do a linear combination of a shape measureand a size measure, but this is an arbitrary choice.
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Shape and Size Measures
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.
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A Universal Measure of Mesh Quality
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
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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.
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The Metric MK of a Simplex K
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 .
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The Metric MK of a Simplex K
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.
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The Metric MK of a Simplex K
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
,
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The Metric MK of a Simplex K
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.
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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.
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Visualization of the Metric MK
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
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Visualization of the Metric MK
Ellipses of a selectedgroup of elements. Note inthis figure that the ellipsespass through the verticesof the triangle.
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Visualization of the Metric MK
Ellipses of a selectedgroup of elements. Note inthis figure that the ellipsespass through the verticesof the triangle.
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Visualization of the Metric MK
Ellipses of a selectedgroup of elements. Note inthis figure that the ellipsespass through the verticesof the triangle.
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A Universal Measure of Mesh Quality
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
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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.
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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 .
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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.
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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.
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Visualization of MS(K)
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.
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Visualization of MS(K)
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.
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A Universal Measure of Mesh Quality
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
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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.
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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.
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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.
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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.
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Example – Triangle ABC
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
.
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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
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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.
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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.
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Example – Triangle ABC
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.
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Example – Triangle ABC
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
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Visualization of EK
The specified metric isdefined in GEORGE andBOROUCHAKI (1997). Itis an analytical functionthat defines an isotropicmetric.
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Visualization of EK
The specified metric isdefined in GEORGE andBOROUCHAKI (1997). Itis an analytical functionthat define an anisotro-pic metric.
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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.
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A Universal Measure of Mesh Quality
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
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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.
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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.
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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.
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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.
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A Universal Measure of Mesh Quality
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
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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 .
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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.
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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).
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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.
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A Universal Measure of Mesh Quality
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
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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.
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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.
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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
.
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A Universal Measure of Mesh Quality
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
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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.
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Test 1 – Uniform Mesh
A B C
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Test 1 – Uniform Mesh
A B CET = 0.0843 ET = 0.00 ET = 0.503
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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)
), . . .
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Test 2 – Isotropic Mesh
. . . 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].
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Test 2 – Isotropic Mesh
View of the size specification map as a field of tensormetrics and view of a mesh that fits rather well thesetensor metrics.
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Test 2a – Isotropic Mesh
A B C
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Test 2a – Isotropic Mesh
A B CET = 3.18 ET = 0.104 ET = 56.2
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Test 2b – Isotropic Mesh
A B C
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Test 2b – Isotropic Mesh
A B CET = 0.104 ET = 0.929 ET = 3.18
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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)
), . . .
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Test 3 – Anisotropic Mesh
. . . 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], . . .
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Test 3 – Anisotropic Mesh
. . . 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].
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Test 3 – Anisotropic Mesh
View of the size specification map as a field of tensormetrics and view of a mesh that fits rather well thesetensor metrics.
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Test 3 – Anisotropic Mesh
A B C
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Test 3 – Anisotropic Mesh
A B CET = 0.405 ET = 2.67 ET = 0.107
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Test 4 – Bernhard Riemann
The size specification map isdeduced from an error esti-mator based on the secondderivatives of the grey level ofthe picture.
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Test 4 – Bernhard Riemann
A B C
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Test 4 – Bernhard Riemann
A B CET = 0.546 ET = 0.345 ET = 0.845
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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.
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Test 5a – Flow over a Naca 0012
A B C
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Test 5a – Flow over a Naca 0012
A B CSpecified Metric MS ET = 0.658 ET = 1160
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Test 5b – Flow over a Naca 0012
A B C
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Test 5b – Flow over a Naca 0012
A B CSpecified Metric MS ET = 1160 ET = 0.658
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Test 5c – Flow over a Naca 0012
A B C
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Test 5c – Flow over a Naca 0012
A B CSpecified Metric MS ET = 1160 ET = 0.658
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A Universal Measure of Mesh Quality
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
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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.
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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.
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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.
Mesh Quality – p. 317/331
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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.
Mesh Quality – p. 317/331
![Page 424: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/424.jpg)
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.
Mesh Quality – p. 317/331
![Page 425: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/425.jpg)
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.
Mesh Quality – p. 317/331
![Page 426: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/426.jpg)
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.
Mesh Quality – p. 317/331
![Page 427: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/427.jpg)
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 Quality – p. 317/331
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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.
Mesh Quality – p. 318/331
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Table of Contents
1. Introduction2. Simplex Definition3. Degeneracies of
Simplices4. Shape Quality of
Simplices5. Formulae for Simplices6. 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
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Conclusions
About time he finished ! ! !
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Degenerate Simplices
A simplex is degenerate if its measure is null.
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Degenerate Simplices
A simplex is degenerate if its measure is null.
The degeneracy is independant of the metric.
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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.
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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 not valid if it is not null for everydegenerate simplex.
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Shape Measure
Beauty, quality and shape are relative notions.
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Shape Measure
Beauty, quality and shape are relative notions.
We fisrt need to define what we want in order toevaluate what we obtained.
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Shape Measure
Beauty, quality and shape are relative notions.
We fisrt need to define what we want in order toevaluate what we obtained.
“What we want” is written in the form of metric tensors.
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Shape Measure
Beauty, quality and shape are relative notions.
We fisrt need to define what we want in order toevaluate what we obtained.
“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.
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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.
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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.
The shape measures are more or less equivalent inassessing the quality of a mesh.
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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.
The shape measures are more or less equivalent inassessing the quality of a mesh.
The shape measures are more or less equivalentduring mesh optimization.
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Size Measures
The simplices can be of good shape without being ofgood size.
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Size Measures
The simplices can be of good shape without being ofgood size.
There exists quality measures for the size of thesimplices and of the mesh.
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Size Measures
The simplices can be of good shape without being ofgood size.
There exists quality measures for the size of thesimplices and of the mesh.
In principle, a simplex whose edges are of unit lengthin the metric is also of perfect shape in that metric.
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Size Measures
The simplices can be of good shape without being ofgood size.
There exists quality measures for the size of thesimplices and of the mesh.
In principle, a simplex whose edges are of unit lengthin the metric is also of perfect shape in that metric.
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.
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Size Measures
However, the ratio of the smallest edge on largest canbe as large as
√2/2 for a tetrahedron to degenerate to
a sliver.
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Size Measures
However, the ratio of the smallest edge on largest canbe as large as
√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.
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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.
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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.
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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.
Mesh Quality – p. 328/331
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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.
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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.
Mesh Quality – p. 329/331
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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.
Mesh Quality – p. 329/331
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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.
Mesh Quality – p. 329/331
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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.
Mesh Quality – p. 329/331
![Page 456: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/456.jpg)
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.
Mesh Quality – p. 329/331
![Page 457: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/457.jpg)
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 Quality – p. 329/331
![Page 458: Mesh Quality](https://reader034.vdocuments.net/reader034/viewer/2022052307/55646033d8b42acd408b4976/html5/thumbnails/458.jpg)
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.
Mesh Quality – p. 330/331
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The End
Mesh Quality – p. 331/331
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The End
Mesh Quality – p. 331/331