Mesh Quality               Julien Dompierre          julien@cerca.umontreal.ca                                     ´Centre...
Authors•   Research professionals    • Julien Dompierre    • Paul Labbé    • Marie-Gabrielle Vallet•   Professors    • Fra...
References – 1            J. D OMPIERRE , P. L ABBÉ ,            M.-G. VALLET, F. G UIBAULT            AND R. C AMARERO , ...
References – 2A. L IU and B. J OE, Relationship betweenTetrahedron Shape Measures, Bit, Vol. 34,pages 268–287, (1994).    ...
References – 3P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F.G UIBAULT and J.-Y. T RÉPANIER, A UniversalMeasure of the Conformi...
References – 4P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F.G UIBAULT and J.-Y. T RÉPANIER, A Measure ofthe Conformity of a Me...
References – 5            P.-L. G EORGE AND H. B O -            ROUCHAKI , Triangulation de            Delaunay et maillag...
References – 6            P. J. F REY AND P.-L.            G EORGE, Maillages. Ap-            plications aux éléments finis...
Table of Contents1. Introduction        8. Non-Simplicial2. Simplex Definition     Elements3. Degeneracies of     9. Shape ...
Introduction and JustificationsWe work on mesh generation, mesh adaptationand mesh optimization.How can we choose the config...
Face FlippingHow can we choose the configuration thatproduces the best tetrahedra ? A tetrahedronshape quality measure is n...
Edge Swapping               S4   S3                 S4   S3          S5                      S5   A                       ...
Mesh Optimization •   Let O1 and O2 , two three-dimensional     unstructured tetrahedral mesh Optimizers.                 ...
Mesh Optimization •   Let O1 and O2 , two three-dimensional     unstructured tetrahedral mesh Optimizers. •   What is the ...
Mesh Optimization •   Let O1 and O2 , two three-dimensional     unstructured tetrahedral mesh Optimizers. •   What is the ...
It’s Obvious ! •   Let B be a benchmark.                             Mesh Quality – p. 14/331
It’s Obvious ! •   Let B be a benchmark. •   Let M1 = O1 (B) be the optimized mesh     obtained with the mesh optimizer O1...
It’s Obvious ! •   Let B be a benchmark. •   Let M1 = O1 (B) be the optimized mesh     obtained with the mesh optimizer O1...
It’s Obvious ! •   Let B be a benchmark. •   Let M1 = O1 (B) be the optimized mesh     obtained with the mesh optimizer O1...
It’s Obvious ! •   Let B be a benchmark. •   Let M1 = O1 (B) be the optimized mesh     obtained with the mesh optimizer O1...
Benchmarks for Mesh OptimizationJ. D OMPIERRE, P. L ABBÉ, F. G UIBAULT andR. C AMARERO.Proposal of Benchmarks for 3D Unstr...
The Trick... •   Because the norm O of a mesh optimizer is     unknown, the comparison of two optimizers is     replaced b...
The Trick... •   Because the norm O of a mesh optimizer is     unknown, the comparison of two optimizers is     replaced b...
The Trick... •   Because the norm O of a mesh optimizer is     unknown, the comparison of two optimizers is     replaced b...
The Trick... •   Because the norm O of a mesh optimizer is     unknown, the comparison of two optimizers is     replaced b...
What to Retain •   This lecture is about the quality of the     elements of a mesh and the quality of a whole     mesh.   ...
What to Retain •   This lecture is about the quality of the     elements of a mesh and the quality of a whole     mesh. • ...
What to Retain •   This lecture is about the quality of the     elements of a mesh and the quality of a whole     mesh. • ...
Table of Contents1. Introduction             8. Non-Simplicial2. Simplex Definition          Elements3. Degeneracies of    ...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.                 ...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.The most simple a...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.The most simple a...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.The most simple a...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.The most simple a...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.The most simple a...
Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements.The most simple a...
Definition of a d-Simplex in RdLet d + 1 points Pj = (p1j , p2j , . . . , pdj ) ∈ Rd , 1 ≤ j ≤ d + 1,not in the same hyperp...
A Simplex Generates RdAny point X ∈ Rd , with Cartesian coordinates (xi )d , is                                           ...
What to RetainIn two dimensions, the simplex is a triangle.                                            Mesh Quality – p. 2...
What to RetainIn two dimensions, the simplex is a triangle.In three dimensions, the simplex is a tetrahedron.             ...
What to RetainIn two dimensions, the simplex is a triangle.In three dimensions, the simplex is a tetrahedron.The d + 1 ver...
What to RetainIn two dimensions, the simplex is a triangle.In three dimensions, the simplex is a tetrahedron.The d + 1 ver...
Table of Contents1. Introduction             8. Non-Simplicial2. Simplex Definition          Elements3. Degeneracies of    ...
Degeneracy of SimplicesA d-simplex made of d + 1 vertices Pj is degenerate if itsvertices are located in the same hyperpla...
Degeneracy of SimplicesA d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd .                      ...
Degeneracy of SimplicesA d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd .Such is the case if th...
Degeneracy of SimplicesA d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd .Such is the case if th...
Degeneracy of SimplicesA d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd .Such is the case if th...
Degeneracy of SimplicesA d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd .Such is the case if th...
Degeneracy CriterionA d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determin...
Degeneracy CriterionA d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determin...
Degeneracy CriterionA d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determin...
Degeneracy CriterionA d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determin...
Degeneracy CriterionA d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determin...
Taxonomy of Degenerate SimplicesThis taxonomy is based on the different possibledegenerate states of the simplices.       ...
Taxonomy of Degenerate SimplicesThis taxonomy is based on the different possibledegenerate states of the simplices.There a...
Taxonomy of Degenerate SimplicesThis taxonomy is based on the different possibledegenerate states of the simplices.There a...
Taxonomy of Degenerate SimplicesThis taxonomy is based on the different possibledegenerate states of the simplices.There a...
1 – The Cap    Name            h −→ 0                h=0                      C                      h     Cap        A   ...
2 – The Needle   Name            h −→ 0                 h=0                  C                 h   Needle       A         ...
3 – The Big Crunch         Name         h −→ 0        h=0                       C                       h h B          Big...
Degeneracy of Tetrahedra There is one case of degeneracy resulting in fourcollapsed vertices. There are five cases of degen...
1 – The Fin   Name            h −→ 0             h=0                        D                    h                     D  ...
2 – The Cap   Name            h −→ 0             h=0                     D    Cap        A      h       C   A       D C   ...
3 – The Sliver  Name             h −→ 0               h=0                           D                                h    ...
4 – The Wedge   Name            h −→ 0              h=0                        D                            h             ...
5 – The Crystal  Name            h −→ 0                  h=0                          D              A          h  Crystal...
6 – The Spindle  Name            h −→ 0               h=0                            D              A     h               ...
7 – The Splitter Name             h −→ 0               h=0                  D                  h          C Splitter    A ...
8 – The Slat Name             h −→ 0                  h=0              D              h             C   Slat              ...
9 – The Needle  Name            h −→ 0              h=0                         D                          h              ...
10 – The Big Crunch        Name           h −→ 0         h=0                            D          Big        A hh   C    ...
What to RetainA triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.              ...
What to RetainA triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.There are thre...
What to RetainA triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.There are thre...
What to RetainA triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.There are thre...
Table of Contents1. Introduction             8. Non-Simplicial2. Simplex Definition          Elements3. Degeneracies of    ...
Shape Quality of SimplicesAn usual method used to quantify the quality of a meshis through the quality of the elements of ...
Shape Quality of SimplicesAn usual method used to quantify the quality of a meshis through the quality of the elements of ...
Shape Quality of SimplicesAn usual method used to quantify the quality of a meshis through the quality of the elements of ...
The Regular SimplexDefinition : An element is regular if it maximizes its measure for          a given measure of its bound...
The Regular SimplexDefinition : An element is regular if it maximizes its measure for          a given measure of its bound...
The Regular SimplexDefinition : An element is regular if it maximizes its measure for          a given measure of its bound...
Simplicial Shape MeasureDefinition A : A simplicial shape measure is acontinuous function that evaluates the shape of a sim...
RemarksThe invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent...
RemarksThe invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent...
RemarksThe invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent...
The Radius RatioThe radius ratio of a simplex K is a shape measure definedas ρ = d ρK /rK , where ρK and rK are the radius ...
The Mean RatioLet R(r1 , r2 , r3 [, r4 ]) be an equilateral simplex having thesame [area|volume] than the simplex K(P1 , P...
The Mean RatioThen, the mean ratio η of the simplex K is the ratio of thegeometric mean over the algebraic means of theeig...
The Condition NumberF ORMAGGIA and P EROTTO (2000) use the inverse of thecondition number of the matrix.                  ...
The Frobenius NormFreitag and Knupp (1999) use the Frobenius norm of thematrix N = AW −1 to define a shape measure.        ...
The Minimum of Solid AnglesThe simplicial shape measure θmin based on the minimumof solid angles of the d-simplex is define...
The sin of θminFrom a numerical point of view, a less expensive simplicialshape measure is the sin of the minimum solid an...
Face AnglesWe can define a shape measure based on the minimum ofthe twelve angles of the four faces of a tetrahedron. Thisa...
Dihedral AnglesThe dihedral angle is the angle between the intersection oftwo adjacent faces to an edge with the perpendic...
Dihedral Angles   αϕmin = min ϕij = min (π − arccos (nij1 · nij2 )) ,            1≤i<j≤4      1≤i<j≤4where nij1 and nij2 a...
The Interpolation Error CoefficientIn finite element, the interpolation error of a function overan element is bounded by a c...
The Edge RatioRatio of the smallest edge over the tallest.                       r = hmin /hmax .The edge ratio r is not a...
Other Shape Measure – 1 hmax /rK , the ratio of the diameter of the tetrahedronover the circumradius, in B AKER, (1989). T...
Other Shape Measure – 2     4   4      2 −3 VK      i=1   Si    ,                   the ratio of the volume of thetetrahed...
Other Shape Measure – 3                         2   VK              Lij       − L12 L34 − L13 L24         1≤i<j≤4         ...
Other Shape Measure – 4                       −3VK               L2          1≤i<j≤4 ij        , the ratio of the volume o...
There Exists an Infinity of Shape                                            MeasuresIf µ and ν are two valid shape measure...
What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length.                    ...
What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length.A shape measures eva...
What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length.A shape measures eva...
What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length.A shape measures eva...
What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length.A shape measures eva...
Table of Contents1. Introduction             8. Non-Simplicial2. Simplex Definition          Elements3. Degeneracies of    ...
Formulae for the TriangleA triangle is completely defined by the knowledge of thelength of its three edges.Quantities such ...
The Half-PerimeterThe half-perimeter pK is given by                      (L12 + L13 + L23 )                 pK =          ...
Heron’s FormulaThe area SK of a triangle can also be written in function ofthe edge lengths with Heron’s formula :        ...
Radius of the IncircleThe radius ρK of the incircle of the triangle K is given by                              SK         ...
Radius of the Circumscribed CircleThe radius rK of the circumcircle of the triangle K is givenby                          ...
Element DiameterThe diameter of an element is the biggest Euclideandistance between two points of an element. For a triang...
Solid AngleThe angle θi at vertex Pi of triangle K is the arc lengthobtained by projecting the edge of the triangle opposi...
Formulae for the TetrahedronA tetrahedron is completely defined by the knowledge ofthe length of its six edges.Quantities s...
Formulae for the TetrahedronLet K be a non degenerate tetrahedron of vertices P1 , P2 ,P3 and P4 . The lengths of the edge...
3D “Heron’s” FormulaLet a, b, c, e, f and g be the length of the six edges of thetetrahedron such that the edges a, b and ...
Radius of the InsphereThe radius ρK of the insphere of the tetrahedron K is givenby                              3VK      ...
Radius of the CircumsphereThe radius rK of the circumsphere of the tetrahedron K isgiven by             (a + b + c)(a + b ...
Element DiameterThe diameter of an element is the biggest Euclideandistance between two points of an element. For atetrahe...
Solid AngleThe solid angle θi at vertex Pi of the tetrahedron K, is thearea of the spherical sector obtained by projecting...
Solid angleL IU and J OE (1994) gave a formula to compute the solidangle in function of edge lengths :                    ...
Table of Contents1. Introduction        8. Non-Simplicial2. Simplex Definition     Elements3. Degeneracies of     9. Shape ...
Which Is the Most Beautiful Triangle ?                            Mesh Quality – p. 83/331
Which Is the Most Beautiful Triangle ?        A                            Mesh Quality – p. 83/331
Which Is the Most Beautiful Triangle ?        A                B                             Mesh Quality – p. 83/331
If You Chose the Triangle A...                             Mesh Quality – p. 84/331
If You Chose the Triangle A...         A   You are wrong !                             Mesh Quality – p. 84/331
If You Chose the Triangle B...                             Mesh Quality – p. 85/331
If You Chose the Triangle B...                            B                   You are wrong again !                       ...
Which Is the Most Beautiful Triangle ?          A               BNone of these answers !                              Mesh...
Which Is the Most Beautiful Woman ?                          Mesh Quality – p. 87/331
Which Is the Most Beautiful Woman ?       A                          Mesh Quality – p. 87/331
Which Is the Most Beautiful Woman ?       A                B                            Mesh Quality – p. 87/331
You Probably chose...                        Mesh Quality – p. 88/331
You Probably chose...           A            BWoman A.                            Mesh Quality – p. 88/331
And if One Asked these Gentlemen...                           Mesh Quality – p. 89/331
And if One Asked these Gentlemen...                           Mesh Quality – p. 89/331
These Gentlemen Would Choose...                          Mesh Quality – p. 90/331
These Gentlemen Would Choose...           A           BWoman B.                           Mesh Quality – p. 90/331
Which Is the Most Beautiful Woman...There is no absolute answer because thequestion is incomplete.One did not specify who ...
Which Is the Most Beautiful Triangle ?                            Mesh Quality – p. 92/331
Which Is the Most Beautiful Triangle ?        A                B                             Mesh Quality – p. 92/331
Which Is the Most Beautiful Triangle ?          A                          BThe question is incomplete : It misses a way o...
Voronoi Diagram           Georgy Fedoseevich VORO -           NOÏ . April 28, 1868, Ukraine           – November 20, 1908,...
The Perpendicular Bisector                         Let S1 and S2 be two                         vertices in R2 . The      ...
A Cloud of VerticesLet S = {Si }i=1,...,N be a cloud of N vertices.                      S2        S11                S9  ...
The Voronoi CellDefinition : The Voronoi cell C(Si ) associated tothe vertex Si is the locus of points of space whichis clo...
The Voronoi DiagramThe set of Voronoi cells associated with all thevertices of the cloud of vertices is called theVoronoi ...
Properties of the Voronoi Diagram The Voronoi cells are polygons in 2D, polyhedra in 3D and N -polytopes in N D. The Voron...
What to RetainThe Voronoi diagrams are partitions of spaceinto cells based on the concept of distance.                    ...
Delaunay Triangulation            Boris Nikolaevich D ELONE or            D ELAUNAY. 15 mars 1890,            Saint Peters...
Triangulation of a cloud of PointsThe same cloud of points can be triangulated inmany different fashions.                 ...
Triangulation of a Cloud of Points                 ...                 ...                            Mesh Quality – p. 10...
Triangulation of a Cloud of Points                 ...                 ...                            Mesh Quality – p. 10...
Delaunay TriangulationAmong all these fashions, there is one (or maybemany) triangulation of the convex hull of the pointc...
Empty Sphere Criterion of DelaunayEmpty sphere criterion : A simplex K satisfiesthe empty sphere criterion if the opencircu...
Violation of the Empty Sphere CriterioA simplex K does not satisfy the empty spherecriterion if the opened circumscribed b...
Delaunay TriangulationDelaunay Triangulation : If all the simplices Kof a triangulation T satisfy the empty spherecriterio...
Delaunay Algorithm  The     circumscri- bed sphere of a simplex has to be      S3 computed.                               ...
Delaunay AlgorithmHow can we know if a point P violates the emptysphere criterion for a simplex K ?   The center C and the...
What to RetainThe Voronoi diagram of a cloud of points is apartition of space into cells based on thenotion of distance.A ...
Duality Delaunay-VoronoïThe Voronoï diagram is the dual of the Delaunaytriangulation and vice versa.                      ...
Voronoï and Delaunay in NatureVoronoï diagrams and Delaunay triangulationsare not just a mathematician’s whim, theyreprese...
Voronoï and Delaunay In Nature                          Mesh Quality – p. 113/331
A Turtle           Mesh Quality – p. 114/331
A Pineapple              Mesh Quality – p. 115/331
The Devil’s Tower                    Mesh Quality – p. 116/331
Dry Mud          Mesh Quality – p. 117/331
Bee Cells            Mesh Quality – p. 118/331
Dragonfly Wings                 Mesh Quality – p. 119/331
Pop Corn           Mesh Quality – p. 120/331
Fly Eyes           Mesh Quality – p. 121/331
Carbon Nanotubes                   Mesh Quality – p. 122/331
Soap Bubbles               Mesh Quality – p. 123/331
A Geodesic Dome                  Mesh Quality – p. 124/331
Biosphère de Montréal                        Mesh Quality – p. 125/331
Streets of Paris                   Mesh Quality – p. 126/331
Roads in France                  Mesh Quality – p. 127/331
Roads in France                  Mesh Quality – p. 128/331
Where Is this Guy Going ? ! !   A simplicial shape measure is an evaluation   of the ratio to equilarity.                 ...
Where Is this Guy Going ? ! !   A simplicial shape measure is an evaluation   of the ratio to equilarity.   The Voronoï di...
Where Is this Guy Going ? ! !   A simplicial shape measure is an evaluation   of the ratio to equilarity.   The Voronoï di...
Where Is this Guy Going ? ! !   A simplicial shape measure is an evaluation   of the ratio to equilarity.   The Voronoï di...
Where Is this Guy Going ? ! !   A simplicial shape measure is an evaluation   of the ratio to equilarity.   The Voronoï di...
Nikolai Ivanovich Lobachevsky             N IKOLAI     I VANOVICH             LOBACHEVSKY,          1             décembre...
János Bolyai               J ÁNOS BOLYAI, 15 dé-               cembre 1802 à Kolozsvár,               Empire Austrichien (...
Bernhard RIEMANN         G EORG F RIEDRICH B ERN -         HARD RIEMANN, 7 sep-         tembre 1826, Hanovre — 20         ...
Non Euclidean GeometryRiemann has generalized Euclidean geometry inthe plane to Riemannian geometry on a surface.He has de...
The Metric in the Merriam-Webster                          Mesh Quality – p. 134/331
Definition of a MetricIf S is any set, then the function                    d : S×S → I                              Ris ca...
The Euclidean Distance is a MetricIn the previous definition of a metric, let the set Sbe I 2 , the function    R       d :...
Metric Space               Mesh Quality – p. 137/331
The Scalar Product is a MetricLet a vectorial space with its scalar product ·, · .Then the norm of the scalar product of t...
The Scalar Product is a MetricIf the vectorial space is I 2 , then the norm of the                          R             ...
Metric TensorA metric tensor M is a symmetric positivedefinite matrix                   m11 m12           M=               ...
Metric Length               −→The length LM (AB) of an edge between verticesA and B in the metric M is given by           ...
Euclidean Length with M = I    −→      −→    −→    1/2       − T −                                   →    →LM (AB) =   AB,...
αβMetric Length with M =                  βγ    −→      −→    −→    1/2       − T −                                   →   ...
Length in a Variable MetricIn the general sense, the metric tensor M is notconstant but varies continuously for every poin...
Area and Volume in a MetricArea of the triangle K in a metric M :           AM (K) =        det(M) dA.                    ...
Metric and Delaunay Mesh                           Mesh Quality – p. 146/331
Which is the Best Triangle ?            A                         BThe question is incomplete. The way to measurethe quali...
Which is the Best Triangle ?        A                B                               Mesh Quality – p. 148/331
Which is the Best Triangle ?        A                B                               Mesh Quality – p. 149/331
Example of an Adapted MeshAdapted mesh and solution for a transonicvisquous compressible flow with Mach 0.85 andReynolds = ...
Zoom on Boundary Layer–Shock                        Mesh Quality – p. 151/331
What to Retain  Beauty, quality and shape are relative  notions.                                   Mesh Quality – p. 152/331
What to Retain  Beauty, quality and shape are relative  notions.  We first need to define what we want in order  to evaluate...
What to Retain  Beauty, quality and shape are relative  notions.  We first need to define what we want in order  to evaluate...
What to Retain  Beauty, quality and shape are relative  notions.  We first need to define what we want in order  to evaluate...
Shape Measure in a MetricFirst method (constant metric)For a simplex K, evaluate the metric tensor atseveral points (Gauss...
Shape Measure in a MetricSecond method (constant metric)For a simplex K, evaluate the metric tensor atone point (Gaussian ...
Shape Measure in a MetricThird methode (variable metric)Express the shape measure as a fonction ofedge lengths only.Evalua...
Shape Measure in a MetricFourth method (variable metric)Express the shape measure in function of thelength of the edges, t...
Shape Measure in a MetricFifth method (variable metric)Know how to evaluate quantities such as theradius of the inscribed ...
Table of Contents1. Introduction        8. Non-Simplicial2. Simplex Definition     Elements3. Degeneracies of     9. Shape ...
Shape Measures and Delaunay CriteroDelaunay meshes have several smoothnessproperties.  The Delaunay mesh minimizes the max...
3D-Delaunay Mesh and DegeneracyIn three dimensions, it is well known thatDelaunay meshes can include slivers which aredege...
Empty Sphere Criterion of DelaunayThe empty sphere criterion of Delaunay is not ashape measure, but it can be used like a ...
Edge Swapping and θmin Shape MeasuDuring edge swapping, using the empty spherecriterion (Delaunay criterion)              ...
What to Retain  The empty sphere criterion of Delaunay is not  a shape measure but it can be used as a  shape measure.    ...
What to Retain  The empty sphere criterion of Delaunay is not  a shape measure but it can be used as a  shape measure.  In...
What to Retain  The empty sphere criterion of Delaunay is not  a shape measure but it can be used as a  shape measure.  In...
Delaunay and Circumscribed SphereAs the circumscribed sphere of a tetrahedrongets larger, there are more chances that anot...
Circumscribed Sphere of Infinite RadiThe tetrahedra that degenerate into a fin, into acap, into a crystal, into a spindle an...
Circumscribed Sphere of Bounded RaThe tetrahedra that degenerate into a sliver, intoa wedge, into a slat, into a needle an...
What to RetainThe empty sphere criterion ofDelaunay is not a valid shapemeasure sensitive to all the possibledegeneracies ...
Circumscribed Sphere of Bounded RaAmongst the degenerate tetrahedra that have acircumscribed sphere of bounded radius, the...
The SliverAnd so, finally, we come to the sliver,                  D                     h               C         A       ...
Non-Convex QuadrilateralIt is forbidden to swap an edge of a non-convexquadrilateral.               S3           S3       ...
Non-Convex Quadrilateral       S3          Two adjacent triangles                   forming a non-convex                  ...
Loss of the Convexity Property in 3D                           Mesh Quality – p. 172/331
What to Retain  The empty sphere criterion of Delaunay is  more or less a simplicial shape measure.                       ...
What to Retain  The empty sphere criterion of Delaunay is  more or less a simplicial shape measure.  The empty sphere crit...
What to Retain  The empty sphere criterion of Delaunay is  more or less a simplicial shape measure.  The empty sphere crit...
Table of Contents1. Introduction        8. Non-Simplicial2. Simplex Definition     Elements3. Degeneracies of     9. Shape ...
Non-Simplicial ElementsThis section proposes a method to generalizethe notions of regularity, of degeneration and ofshape ...
Non-Simplicial ElementsOn Element Shape Measures for MeshOptimizationPAUL L ABBÉ , J ULIEN D OMPIERRE , F RANÇOISG UIBAULT...
Regularity Generalization  An equilateral quadrilateral, ie that has four  edges of same length, is not necessarily a  squ...
Regularity Generalization  An equilateral quadrilateral, ie that has four  edges of same length, is not necessarily a  squ...
Regularity Generalization  An equilateral quadrilateral, ie that has four  edges of same length, is not necessarily a  squ...
Regularity Generalization  An equilateral quadrilateral, ie that has four  edges of same length, is not necessarily a  squ...
Regular Non Simplicial Elements  The regular quadrilateral is the square.  The regular hexahedron is the cube.  The regula...
Quality of Non Simplicial ElementsProposed Extension : The shape measure of anon simplicial element is given by the minimu...
Shape Measure of a QuadrilateralThe shape measure of a quadrilateral is theminimum of the shape measure of its four corner...
Shape Measure of a PrismThe shape measure of a prism is the minimum ofthe shape measure of its six corner tetrahedronforme...
Shape Measure of an HexahedronThe shape measure of an hexahedron is theminimum of its eight corner tetrahedron formedby it...
Shape of the Corner Simplex  The corner simplices constructed for the non simplicial elements are not regular simplices.  ...
Shape of the Corner SimplexEach non simplicial shape measure has to benormalized so as to be a shape measure equalto unit ...
Degenerate Non Simplicial ElementsDéfinition :A non simplicial element isdegenerate if at least one of its corner simplices...
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Mesh Quality

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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 criterion

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Mesh Quality

  1. 1. Mesh Quality Julien Dompierre julien@cerca.umontreal.ca ´Centre de Recherche en Calcul Applique (CERCA) ´ ´ Ecole Polytechnique de Montreal Mesh Quality – p. 1/331
  2. 2. Authors• Research professionals • Julien Dompierre • Paul Labbé • Marie-Gabrielle Vallet• Professors • François Guibault • Jean-Yves Trépanier • Ricardo Camarero Mesh Quality – p. 2/331
  3. 3. References – 1 J. D OMPIERRE , P. L ABBÉ , M.-G. VALLET, F. G UIBAULT AND R. C AMARERO , Critères de qualité pour les maillages simpliciaux. in Maillage et adaptation, Hermès, October 2001, Paris, pages 311–348. Mesh Quality – p. 3/331
  4. 4. References – 2A. L IU and B. J OE, Relationship betweenTetrahedron Shape Measures, Bit, Vol. 34,pages 268–287, (1994). Mesh Quality – p. 4/331
  5. 5. References – 3P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F.G UIBAULT and J.-Y. T RÉPANIER, A UniversalMeasure of the Conformity of a Mesh withRespect to an Anisotropic Metric Field,Submitted to Int. J. for Numer. Meth. in Engng,(2003). Mesh Quality – p. 5/331
  6. 6. References – 4P. L ABBÉ, J. D OMPIERRE, M.-G. VALLET, F.G UIBAULT and J.-Y. T RÉPANIER, A Measure ofthe Conformity of a Mesh to an AnisotropicMetric, Tenth International Meshing Roundtable,Newport Beach, CA, pages 319–326, (2001). Mesh Quality – p. 6/331
  7. 7. References – 5 P.-L. G EORGE AND H. B O - ROUCHAKI , Triangulation de Delaunay et maillage, appli- cations aux éléments finis. Hermès, 1997, Paris. This book is available in En- glish. Mesh Quality – p. 7/331
  8. 8. References – 6 P. J. F REY AND P.-L. G EORGE, Maillages. Ap- plications aux éléments finis. Hermès, 1999, Paris. This book is available in English. Mesh Quality – p. 8/331
  9. 9. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Sim- 11. Mesh Quality and plices Optimization6. Voronoi, Delaunay 12. Size Quality of and Riemann Simplices7. Shape Quality and 13. Universal Quality Delaunay 14. Conclusions Mesh Quality – p. 9/331
  10. 10. Introduction and JustificationsWe work on mesh generation, mesh adaptationand mesh optimization.How can we choose the configuration thatproduces the best triangles ? A triangle shapequality measure is needed. Mesh Quality – p. 10/331
  11. 11. Face FlippingHow can we choose the configuration thatproduces the best tetrahedra ? A tetrahedronshape quality measure is needed. Mesh Quality – p. 11/331
  12. 12. Edge Swapping S4 S3 S4 S3 S5 S5 A A B B S2 S2 S1 S1How can we choose the configuration thatproduces the best tetrahedra ? A tetrahedronshape quality measure is needed. Mesh Quality – p. 12/331
  13. 13. Mesh Optimization • Let O1 and O2 , two three-dimensional unstructured tetrahedral mesh Optimizers. Mesh Quality – p. 13/331
  14. 14. Mesh Optimization • Let O1 and O2 , two three-dimensional unstructured tetrahedral mesh Optimizers. • What is the norm O of a mesh optimizer ? Mesh Quality – p. 13/331
  15. 15. Mesh Optimization • Let O1 and O2 , two three-dimensional unstructured tetrahedral mesh Optimizers. • What is the norm O of a mesh optimizer ? • How can it be asserted that O1 > O2 ? Mesh Quality – p. 13/331
  16. 16. It’s Obvious ! • Let B be a benchmark. Mesh Quality – p. 14/331
  17. 17. It’s Obvious ! • Let B be a benchmark. • Let M1 = O1 (B) be the optimized mesh obtained with the mesh optimizer O1 . Mesh Quality – p. 14/331
  18. 18. 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 . Mesh Quality – p. 14/331
  19. 19. 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”. Mesh Quality – p. 14/331
  20. 20. 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 . Mesh Quality – p. 14/331
  21. 21. Benchmarks for Mesh OptimizationJ. D OMPIERRE, P. L ABBÉ, F. G UIBAULT andR. C AMARERO.Proposal of Benchmarks for 3D UnstructuredTetrahedral Mesh Optimization.7th International Meshing Roundtable, Dearborn,MI, October 1998, pages 459–478. Mesh Quality – p. 15/331
  22. 22. The Trick... • Because the norm O of a mesh optimizer is unknown, the comparison of two optimizers is replaced by the comparison of two meshes. Mesh Quality – p. 16/331
  23. 23. The Trick... • Because the norm O of a mesh optimizer is unknown, the comparison of two optimizers is replaced by the comparison of two meshes. • What is the norm M of a mesh ? Mesh Quality – p. 16/331
  24. 24. The Trick... • Because the norm O of a mesh optimizer is unknown, the comparison of two optimizers is replaced by the comparison of two meshes. • What is the norm M of a mesh ? • How can we assert that M1 > M2 ? Mesh Quality – p. 16/331
  25. 25. The Trick... • Because the norm O of a mesh optimizer is unknown, the comparison of two optimizers is replaced by the comparison of two meshes. • 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 ! Mesh Quality – p. 16/331
  26. 26. What to Retain • This lecture is about the quality of the elements of a mesh and the quality of a whole mesh. Mesh Quality – p. 17/331
  27. 27. What to Retain • This lecture is about the quality of the elements of a mesh and the quality of a whole mesh. • The concept of element quality is necessary for the algorithms of egde and face swapping. Mesh Quality – p. 17/331
  28. 28. What to Retain • This lecture is about the quality of the elements of a mesh and the quality of a whole mesh. • The concept of element quality is necessary for the algorithms of egde and face swapping. • The concept of mesh quality is necessary to do research on mesh optimization. Mesh Quality – p. 17/331
  29. 29. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Simplices 11. Mesh Quality and6. Voronoi, Delaunay and Optimization Riemann 12. Size Quality of7. Shape Quality and Simplices Delaunay 13. Universal Quality 14. Conclusions Mesh Quality – p. 18/331
  30. 30. Definition of a SimplexMeshes in two and three dimensions are made ofpolygons or polyhedra named elements. Mesh Quality – p. 19/331
  31. 31. Definition of a SimplexMeshes 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. Mesh Quality – p. 19/331
  32. 32. Definition of a SimplexMeshes 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. Mesh Quality – p. 19/331
  33. 33. Definition of a SimplexMeshes 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. Mesh Quality – p. 19/331
  34. 34. Definition of a SimplexMeshes 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. Mesh Quality – p. 19/331
  35. 35. Definition of a SimplexMeshes 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. Mesh Quality – p. 19/331
  36. 36. Definition of a SimplexMeshes 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. Mesh Quality – p. 19/331
  37. 37. Definition of a d-Simplex in RdLet 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,   p11 p12 · · · p1,d+1  p21 p22 · · · p2,d+1     . .  A= . . . .. . . . .  .    pd1 pd2 · · · pd,d+1  1 1 ··· 1be invertible. The convex hull of the points Pj is named thed-simplex of points Pj . Mesh Quality – p. 20/331
  38. 38. A Simplex Generates RdAny point X ∈ Rd , with Cartesian coordinates (xi )d , is i=1characterized by the d + 1 scalars λj = λj (X) defined assolution of the linear system  d+1     pij λj = xi for 1 ≤ i ≤ d,  j=1 d+1     λj = 1,  j=1whose matrix is A. Mesh Quality – p. 21/331
  39. 39. What to RetainIn two dimensions, the simplex is a triangle. Mesh Quality – p. 22/331
  40. 40. What to RetainIn two dimensions, the simplex is a triangle.In three dimensions, the simplex is a tetrahedron. Mesh Quality – p. 22/331
  41. 41. What to RetainIn 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 . Mesh Quality – p. 22/331
  42. 42. What to RetainIn 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. Mesh Quality – p. 22/331
  43. 43. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Simplices 11. Mesh Quality and6. Voronoi, Delaunay and Optimization Riemann 12. Size Quality of7. Shape Quality and Simplices Delaunay 13. Universal Quality 14. Conclusions Mesh Quality – p. 23/331
  44. 44. Degeneracy of SimplicesA 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. Mesh Quality – p. 24/331
  45. 45. Degeneracy of SimplicesA d-simplex is degenerate if its d + 1 vertices do notgenerate the space Rd . Mesh Quality – p. 25/331
  46. 46. Degeneracy of SimplicesA 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. Mesh Quality – p. 25/331
  47. 47. Degeneracy of SimplicesA 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. Mesh Quality – p. 25/331
  48. 48. Degeneracy of SimplicesA 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. Mesh Quality – p. 25/331
  49. 49. Degeneracy of SimplicesA 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. Mesh Quality – p. 25/331
  50. 50. Degeneracy CriterionA d-simplex is degenerate if its matrix A is notinvertible. A matrix is not invertible if its determinant isnull. Mesh Quality – p. 26/331
  51. 51. Degeneracy CriterionA 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. Mesh Quality – p. 26/331
  52. 52. Degeneracy CriterionA 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!. Mesh Quality – p. 26/331
  53. 53. Degeneracy CriterionA 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. Mesh Quality – p. 26/331
  54. 54. Degeneracy CriterionA 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. Mesh Quality – p. 26/331
  55. 55. Taxonomy of Degenerate SimplicesThis taxonomy is based on the different possibledegenerate states of the simplices. Mesh Quality – p. 27/331
  56. 56. Taxonomy of Degenerate SimplicesThis taxonomy is based on the different possibledegenerate states of the simplices.There are three cases of degenerate triangles. Mesh Quality – p. 27/331
  57. 57. Taxonomy of Degenerate SimplicesThis 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. Mesh Quality – p. 27/331
  58. 58. Taxonomy of Degenerate SimplicesThis 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 multiplicitysimple, double, triple and quadruple respectively. Mesh Quality – p. 27/331
  59. 59. 1 – The Cap Name h −→ 0 h=0 C h Cap A B A C BDegenerate edges : NoneRadius of the smallest circumcircle : ∞ Mesh Quality – p. 28/331
  60. 60. 2 – The Needle Name h −→ 0 h=0 C h Needle A B A,C BDegenerate edges : ACRadius of the smallest circumcircle : hmax /2 Mesh Quality – p. 29/331
  61. 61. 3 – The Big Crunch Name h −→ 0 h=0 C h h B Big A h A,B,C CrunchDegenerate edges : AllRadius of the smallest circumcircle : 0The Big Crunch is the theory opposite of the Big Bang. Mesh Quality – p. 30/331
  62. 62. Degeneracy of Tetrahedra There is one case of degeneracy resulting in fourcollapsed vertices. There are five cases of degeneracy resulting in fourcollinear vertices. There are four cases of degeneracy resulting in fourcoplanar vertices. D D d A C A a C b B B c Mesh Quality – p. 31/331
  63. 63. 1 – The Fin Name h −→ 0 h=0 D h D A C A C Fin B BDegenerate edges : NoneDegenerate faces : One capRadius of the smallest circumsphere : ∞ Mesh Quality – p. 32/331
  64. 64. 2 – The Cap Name h −→ 0 h=0 D Cap A h C A D C B BDegenerate edges : NoneDegenerate faces : NoneRadius of the smallest circumsphere : ∞ Mesh Quality – p. 33/331
  65. 65. 3 – The Sliver Name h −→ 0 h=0 D h C Sliver A C A D B BDegenerate edges : NoneDegenerate faces : NoneRadius of the smallest circumsphere : rABC or ∞ Mesh Quality – p. 34/331
  66. 66. 4 – The Wedge Name h −→ 0 h=0 D h C, D Wedge A C A B BDegenerate edges : CDDegenerate faces : Two needlesRadius of the smallest circumsphere : rABC Mesh Quality – p. 35/331
  67. 67. 5 – The Crystal Name h −→ 0 h=0 D A h Crystal h C A B D C BDegenerate edges : NoneDegenerate faces : Four capsRadius of the smallest circumsphere : ∞ Mesh Quality – p. 36/331
  68. 68. 6 – The Spindle Name h −→ 0 h=0 D A h A B, D C Spindle h C BDegenerate edges : BDDegenerate faces : Two caps and two needlesRadius of the smallest circumsphere : ∞ Mesh Quality – p. 37/331
  69. 69. 7 – The Splitter Name h −→ 0 h=0 D h C Splitter A A D B, C h BDegenerate edges : BCDegenerate faces : Two caps and two needlesRadius of the smallest circumsphere : ∞ Mesh Quality – p. 38/331
  70. 70. 8 – The Slat Name h −→ 0 h=0 D h C Slat h A, D B, C A BDegenerate edges : AD and BCDegenerate faces : Four needlesRadius of the smallest circumsphere : hmax /2 Mesh Quality – p. 39/331
  71. 71. 9 – The Needle Name h −→ 0 h=0 D h h hC A B, C, D Needle A BDegenerate edges : BC, CD and DBDegenerate faces : Three needles and one Big CrunchRadius of the smallest circumsphere : hmax /2 Mesh Quality – p. 40/331
  72. 72. 10 – The Big Crunch Name h −→ 0 h=0 D Big A hh C h h A, B, C, D Crunch h BhDegenerate edges : AllDegenerate faces : Four Big CrunchesRadius of the smallest circumsphere : 0 Mesh Quality – p. 41/331
  73. 73. What to RetainA triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null. Mesh Quality – p. 42/331
  74. 74. What to RetainA triangle is degenerate if its vertices are collinear orcollapsed, hence if its area is null.There are three cases of degeneracy of triangles. Mesh Quality – p. 42/331
  75. 75. What to RetainA 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. Mesh Quality – p. 42/331
  76. 76. What to RetainA 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. Mesh Quality – p. 42/331
  77. 77. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Simplices 11. Mesh Quality and6. Voronoi, Delaunay and Optimization Riemann 12. Size Quality of7. Shape Quality and Simplices Delaunay 13. Universal Quality 14. Conclusions Mesh Quality – p. 43/331
  78. 78. Shape Quality of SimplicesAn usual method used to quantify the quality of a meshis through the quality of the elements of that mesh. Mesh Quality – p. 44/331
  79. 79. Shape Quality of SimplicesAn 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. Mesh Quality – p. 44/331
  80. 80. Shape Quality of SimplicesAn 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. Mesh Quality – p. 44/331
  81. 81. The Regular SimplexDefinition : An element is regular if it maximizes its measure for a given measure of its boundary. Mesh Quality – p. 45/331
  82. 82. The Regular SimplexDefinition : An element is regular if it maximizes its measure for a given measure of its boundary. The equilateral triangle is regular because it maximizes its area for a given perimeter. Mesh Quality – p. 45/331
  83. 83. The Regular SimplexDefinition : An element is regular if it maximizes its measure for a given measure of its boundary. The equilateral triangle is regular because it maximizes its area for a given perimeter. The equilateral tetrahedron is regular because it maximizes its volume for a given surface of its faces. Mesh Quality – p. 45/331
  84. 84. Simplicial Shape MeasureDefinition 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. Mesh Quality – p. 46/331
  85. 85. RemarksThe invariance under translation, rotation andreflection means that the simplicial shape measuresmust be independent of the coordinates system. Mesh Quality – p. 47/331
  86. 86. RemarksThe 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). Mesh Quality – p. 47/331
  87. 87. RemarksThe 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. Mesh Quality – p. 47/331
  88. 88. The Radius RatioThe 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 ρK rK Mesh Quality – p. 48/331
  89. 89. The Mean RatioLet 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 = N ri + b, 1 ≤ i ≤ [3|4], where b is a translation vector. s y K K = NR + b R r b x Mesh Quality – p. 49/331
  90. 90. The Mean RatioThen, 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 N T N .  √ √  2 λ1 λ2  2 4 3 SK d   λ +λ = in 2D, d λi   1 2 2 1≤i<j≤3 Lij i=1  η= = d  1 λi  3 √λ 1 λ 2 λ 3   3 12 3 9VK2 d   i=1  λ +λ +λ = L 2 in 3D. 1 2 3 1≤i<j≤4 ij Mesh Quality – p. 50/331
  91. 91. The Condition NumberF ORMAGGIA and P EROTTO (2000) use the inverse of thecondition number of the matrix. min λiλ1 i κ= = , max λi λd iif the eigenvalues are sorted in increasing order. Mesh Quality – p. 51/331
  92. 92. The Frobenius NormFreitag and Knupp (1999) use the Frobenius norm of thematrix N = AW −1 to define a shape measure. d d κ= = , tr(N T N )tr((N T N )−1 ) d d λi λ−1 i i=1 i=1where the λi are the eigenvalues of the tensor N T N . Mesh Quality – p. 52/331
  93. 93. The Minimum of Solid AnglesThe simplicial shape measure θmin based on the minimumof solid angles of the d-simplex is defined by θmin = α−1 min θi , 1≤i≤d+1The coefficient α is the value of each solid angle of theregular d-simplex, given by α = π/3 in two dimensions √and α = 6 arcsin 3/3 − π in three dimensions. Mesh Quality – p. 53/331
  94. 94. The sin of θminFrom 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 min σi , 1≤i≤d+1where 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, given √ √by β = sin(α) = 3/2 in 2D and β = sin(α/2) = 6/9 in 3D. Mesh Quality – p. 54/331
  95. 95. Face AnglesWe 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). Mesh Quality – p. 55/331
  96. 96. Dihedral AnglesThe dihedral angle is the angle between the intersection oftwo adjacent faces to an edge with the perpendicular planeof the edge. Pj ϕij PiThe minimum of the six dihedral angles ϕmin is used as ashape measure. Mesh Quality – p. 56/331
  97. 97. Dihedral Angles αϕmin = min ϕij = min (π − arccos (nij1 · nij2 )) , 1≤i<j≤4 1≤i<j≤4where nij1 and nij2 are the normal to the adjacent faces ofthe edge Pi Pj , 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. Mesh Quality – p. 57/331
  98. 98. The Interpolation Error CoefficientIn 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.  √ ρK  2 3  in 2 D, hmax γ=  2√6 ρK in 3 D.  hmax Mesh Quality – p. 58/331
  99. 99. The Edge RatioRatio 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 for √the cap. In 3D, it can be as large as 2/2 for the sliver, 1/2 √for the fin, 3/3 for the cap and 1/3 for the crystal. Mesh Quality – p. 59/331
  100. 100. Other Shape Measure – 1 hmax /rK , the ratio of the diameter of the tetrahedronover the circumradius, in B AKER, (1989). This is not avalid shape measure. hmin /rK , the ratio of the smallest edge of thetetrahedron over the circumradius, in M ILLER et al(1996). This is not a valid shape measure. VK /rK 3 , the ratio of the volume of the tetrahedron overthe circumradius, in M ARCUM et W EATHERILL, (1995). Mesh Quality – p. 60/331
  101. 101. Other Shape Measure – 2 4 4 2 −3 VK i=1 Si , the ratio of the volume of thetetrahedron over the area of its faces, in D E C OUGNY etal (1990). The evaluation of this shape measure, and itsvalidity, are a complex problem for tetrahedra thatdegenerate in four collinear vertices. −3VK 1≤i<j≤4 Lij , the ratio of the volume of thetetrahedron over the average of its edges, inDANNELONGUE and TANGUY (1991), Z AVATTIERI et al(1996) and W EATHERILL et al (1993). Mesh Quality – p. 61/331
  102. 102. Other Shape Measure – 3 2 VK Lij − L12 L34 − L13 L24 1≤i<j≤4 −3/2 −L14 L23 + L2 ij 1≤i<j≤4the ratio of the volume of the tetrahedron over a sum, atthe power three halfs, of many terms homogeneous to thesquare of edge lenghts, in B ERZINS (1998). Mesh Quality – p. 62/331
  103. 103. Other Shape Measure – 4 −3VK L2 1≤i<j≤4 ij , the ratio of the volume of thetetrahedron over the quadratic average of the six edges,in G RAICHEN et al (1991). And so on... This list is surely not exhaustive. Mesh Quality – p. 63/331
  104. 104. There Exists an Infinity of Shape MeasuresIf µ and ν are two valid shape measures, if c, d ∈ R+ , then µc , c(µ−1)/µ with c > 1, αµc + (1 − α)ν d with α ∈ [0, 1], µc ν dare also valid simplicial shape measures. Mesh Quality – p. 64/331
  105. 105. What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length. Mesh Quality – p. 65/331
  106. 106. What to RetainThe regular simplex is the equilateral one, ie, where allits edges have the same length.A shape measures evaluates the ratio to equilaterality. Mesh Quality – p. 65/331
  107. 107. What to RetainThe 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. Mesh Quality – p. 65/331
  108. 108. What to RetainThe 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. Mesh Quality – p. 65/331
  109. 109. What to RetainThe 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. Mesh Quality – p. 65/331
  110. 110. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Simplices 11. Mesh Quality and6. Voronoi, Delaunay and Optimization Riemann 12. Size Quality of7. Shape Quality and Simplices Delaunay 13. Universal Quality 14. Conclusions Mesh Quality – p. 66/331
  111. 111. Formulae for the TriangleA 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 , P2and P3 . The lengths of the edges Pi Pj of K aredenoted Lij = Pj − Pi , 1 ≤ i < j ≤ 3. Mesh Quality – p. 67/331
  112. 112. The Half-PerimeterThe half-perimeter pK is given by (L12 + L13 + L23 ) pK = . 2 Mesh Quality – p. 68/331
  113. 113. Heron’s FormulaThe area SK of a triangle can also be written in function ofthe edge lengths with Heron’s formula : 2 SK = pK (pK − L12 )(pK − L13 )(pK − L23 ). Mesh Quality – p. 69/331
  114. 114. Radius of the IncircleThe radius ρK of the incircle of the triangle K is given by SK ρK = . pK Mesh Quality – p. 70/331
  115. 115. Radius of the Circumscribed CircleThe radius rK of the circumcircle of the triangle K is givenby L12 L13 L23 rK = . 4SK Mesh Quality – p. 71/331
  116. 116. Element DiameterThe 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 ). Mesh Quality – p. 72/331
  117. 117. Solid AngleThe 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 −1 θi = arcsin 2SK Lij Lik . j,k=i 1≤j<k≤3 Mesh Quality – p. 73/331
  118. 118. Formulae for the TetrahedronA 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. Mesh Quality – p. 74/331
  119. 119. Formulae for the TetrahedronLet K be a non degenerate tetrahedron of vertices P1 , P2 ,P3 and P4 . The lengths of the edges Pi Pj of K are denotedLij = Pj − Pi , 1 ≤ i < j ≤ 4. The area of the four faces ofthe tetrahedron, △P2 P3 P4 , △P1 P3 P4 , △P1 P2 P4and △P1 P2 P3 , are denoted by S1 , S2 , S3 and S4 . Finally, VKis the volume of the tetrahedron K. Mesh Quality – p. 75/331
  120. 120. 3D “Heron’s” FormulaLet 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 2 144VK = 4a2 b2 c2 + (b2 + c2 − e2 ) (c2 + a2 − f 2 ) (a2 + b2 − g 2 ) 2 2 − a2 (b2 + c2 − e2 ) − b2 (c2 + a2 − f 2 ) 2 − c2 (a2 + b2 − g 2 ) . Mesh Quality – p. 76/331
  121. 121. Radius of the InsphereThe radius ρK of the insphere of the tetrahedron K is givenby 3VK ρK = . S1 + S2 + S3 + S4 Mesh Quality – p. 77/331
  122. 122. Radius of the CircumsphereThe radius rK of the circumsphere of the tetrahedron K isgiven by (a + b + c)(a + b − c)(a + c − b)(b + c − a) rK = . 24VKwhere a = L12 L34 , b = L13 L24 and c = L14 L23 are theproduct of the length of the opposite edges of K (twoedges are opposite if they do not share a vertex. Mesh Quality – p. 78/331
  123. 123. Element DiameterThe 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 ). Mesh Quality – p. 79/331
  124. 124. Solid AngleThe 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 P1 θ1 P3 P2 Mesh Quality – p. 80/331
  125. 125. Solid angleL IU and J OE (1994) gave a formula to compute the solidangle in function of edge lengths : −1/2 θi = 2 arcsin 12VK (Lij + Lik )2 − L2 jk . j,k=i 1≤j<k≤4 Mesh Quality – p. 81/331
  126. 126. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Sim- 11. Mesh Quality and plices Optimization6. Voronoi, Delaunay 12. Size Quality of and Riemann Simplices7. Shape Quality and 13. Universal Quality Delaunay 14. Conclusions Mesh Quality – p. 82/331
  127. 127. Which Is the Most Beautiful Triangle ? Mesh Quality – p. 83/331
  128. 128. Which Is the Most Beautiful Triangle ? A Mesh Quality – p. 83/331
  129. 129. Which Is the Most Beautiful Triangle ? A B Mesh Quality – p. 83/331
  130. 130. If You Chose the Triangle A... Mesh Quality – p. 84/331
  131. 131. If You Chose the Triangle A... A You are wrong ! Mesh Quality – p. 84/331
  132. 132. If You Chose the Triangle B... Mesh Quality – p. 85/331
  133. 133. If You Chose the Triangle B... B You are wrong again ! Mesh Quality – p. 85/331
  134. 134. Which Is the Most Beautiful Triangle ? A BNone of these answers ! Mesh Quality – p. 86/331
  135. 135. Which Is the Most Beautiful Woman ? Mesh Quality – p. 87/331
  136. 136. Which Is the Most Beautiful Woman ? A Mesh Quality – p. 87/331
  137. 137. Which Is the Most Beautiful Woman ? A B Mesh Quality – p. 87/331
  138. 138. You Probably chose... Mesh Quality – p. 88/331
  139. 139. You Probably chose... A BWoman A. Mesh Quality – p. 88/331
  140. 140. And if One Asked these Gentlemen... Mesh Quality – p. 89/331
  141. 141. And if One Asked these Gentlemen... Mesh Quality – p. 89/331
  142. 142. These Gentlemen Would Choose... Mesh Quality – p. 90/331
  143. 143. These Gentlemen Would Choose... A BWoman B. Mesh Quality – p. 90/331
  144. 144. 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. Mesh Quality – p. 91/331
  145. 145. Which Is the Most Beautiful Triangle ? Mesh Quality – p. 92/331
  146. 146. Which Is the Most Beautiful Triangle ? A B Mesh Quality – p. 92/331
  147. 147. Which Is the Most Beautiful Triangle ? A BThe question is incomplete : It misses a way ofmeasuring the quality of a triangle. Mesh Quality – p. 92/331
  148. 148. Voronoi Diagram Georgy Fedoseevich VORO - NOÏ . April 28, 1868, Ukraine – November 20, 1908, War- saw. Nouvelles applications des paramètres continus à la théorie des formes qua- dratiques. Recherches sur les parallélloèdes primitifs. Journal Reine Angew. Math, Vol 134, 1908. Mesh Quality – p. 93/331
  149. 149. The Perpendicular Bisector Let S1 and S2 be two vertices in R2 . The perpendicular bisec- d(P, S1 ) P tor M (S1 , S2 ) is the S1 d(P, S2 ) locus of points equi- distant to S1 and S2 . S2 M (S1 , S2 ) = {P ∈ M R2 | d(P, S1 ) = d(P, S2 )}, where d(·, ·) is the Eucli- dean distance between two points of space. Mesh Quality – p. 94/331
  150. 150. A Cloud of VerticesLet S = {Si }i=1,...,N be a cloud of N vertices. S2 S11 S9 S10 S5 S6 S4 S8 S1 S7 S12 S3 Mesh Quality – p. 95/331
  151. 151. The Voronoi CellDefinition : 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 = i}. C(Si ) Si Mesh Quality – p. 96/331
  152. 152. The Voronoi DiagramThe set of Voronoi cells associated with all thevertices of the cloud of vertices is called theVoronoi diagram. Mesh Quality – p. 97/331
  153. 153. Properties of the Voronoi Diagram The Voronoi cells are polygons in 2D, polyhedra in 3D and N -polytopes in N D. The Voronoi cells are convex. The Voronoi cells cover space without overlapping. Mesh Quality – p. 98/331
  154. 154. What to RetainThe Voronoi diagrams are partitions of spaceinto cells based on the concept of distance. Mesh Quality – p. 99/331
  155. 155. Delaunay Triangulation Boris Nikolaevich D ELONE or D ELAUNAY. 15 mars 1890, Saint Petersbourg — 1980. Sur la sphère vide. À la mé- moire de Georges Voronoi, Bulletin of the Academy of Sciences of the USSR, Vol. 7, pp. 793–800, 1934. Mesh Quality – p. 100/331
  156. 156. Triangulation of a cloud of PointsThe same cloud of points can be triangulated inmany different fashions. ... Mesh Quality – p. 101/331
  157. 157. Triangulation of a Cloud of Points ... ... Mesh Quality – p. 102/331
  158. 158. Triangulation of a Cloud of Points ... ... Mesh Quality – p. 103/331
  159. 159. Delaunay TriangulationAmong all these fashions, there is one (or maybemany) triangulation of the convex hull of the pointcloud that is said to be a Delaunay Triangulation. Mesh Quality – p. 104/331
  160. 160. Empty Sphere Criterion of DelaunayEmpty 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 Mesh Quality – p. 105/331
  161. 161. Violation of the Empty Sphere CriterioA 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 Mesh Quality – p. 106/331
  162. 162. Delaunay TriangulationDelaunay Triangulation : If all the simplices Kof a triangulation T satisfy the empty spherecriterion, then the triangulation is said to be aDelaunay triangulation. Mesh Quality – p. 107/331
  163. 163. Delaunay Algorithm The circumscri- bed sphere of a simplex has to be S3 computed. S2 This amounts to ρout computing the cen- C ter of a simplex. The center is the point at equal dis- tance to all the vertices of the sim- S1 plex. Mesh Quality – p. 108/331
  164. 164. Delaunay AlgorithmHow can we know if a point P violates the emptysphere criterion for a simplex K ? The center C and the radius ρ of the circumscribed sphere of the simplex K has to be computed. The distance d between the point P and the center C has to be computed. If the distance d is greater than the radius ρ, the point P is not in the circumscribed sphere of the simplex K. Mesh Quality – p. 109/331
  165. 165. What to RetainThe 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. Mesh Quality – p. 110/331
  166. 166. Duality Delaunay-VoronoïThe Voronoï diagram is the dual of the Delaunaytriangulation and vice versa. Mesh Quality – p. 111/331
  167. 167. Voronoï and Delaunay in NatureVoronoï diagrams and Delaunay triangulationsare not just a mathematician’s whim, theyrepresent structures that can be found in nature. Mesh Quality – p. 112/331
  168. 168. Voronoï and Delaunay In Nature Mesh Quality – p. 113/331
  169. 169. A Turtle Mesh Quality – p. 114/331
  170. 170. A Pineapple Mesh Quality – p. 115/331
  171. 171. The Devil’s Tower Mesh Quality – p. 116/331
  172. 172. Dry Mud Mesh Quality – p. 117/331
  173. 173. Bee Cells Mesh Quality – p. 118/331
  174. 174. Dragonfly Wings Mesh Quality – p. 119/331
  175. 175. Pop Corn Mesh Quality – p. 120/331
  176. 176. Fly Eyes Mesh Quality – p. 121/331
  177. 177. Carbon Nanotubes Mesh Quality – p. 122/331
  178. 178. Soap Bubbles Mesh Quality – p. 123/331
  179. 179. A Geodesic Dome Mesh Quality – p. 124/331
  180. 180. Biosphère de Montréal Mesh Quality – p. 125/331
  181. 181. Streets of Paris Mesh Quality – p. 126/331
  182. 182. Roads in France Mesh Quality – p. 127/331
  183. 183. Roads in France Mesh Quality – p. 128/331
  184. 184. Where Is this Guy Going ? ! ! A simplicial shape measure is an evaluation of the ratio to equilarity. Mesh Quality – p. 129/331
  185. 185. Where Is this Guy Going ? ! ! A simplicial shape measure is an evaluation of the ratio to equilarity. The Voronoï diagram of a cloud of points is a partition of space into cells based on the notion of distance. Mesh Quality – p. 129/331
  186. 186. Where Is this Guy Going ? ! ! A simplicial shape measure is an evaluation of the ratio to equilarity. The Voronoï diagram of a cloud of points is a partition of space into cells based on the notion of distance. A Delaunay triangulation of a cloud of points is a triangulation based on the notion of distance. Mesh Quality – p. 129/331
  187. 187. Where Is this Guy Going ? ! ! A simplicial shape measure is an evaluation of the ratio to equilarity. The Voronoï diagram of a cloud of points is a partition of space into cells based on the notion of distance. A Delaunay triangulation of a cloud of points is a triangulation based on the notion of distance. The notion of distance can be generalized. Mesh Quality – p. 129/331
  188. 188. Where Is this Guy Going ? ! ! A simplicial shape measure is an evaluation of the ratio to equilarity. The Voronoï diagram of a cloud of points is a partition of space into cells based on the notion of distance. A Delaunay triangulation of a cloud of points is a triangulation based on the notion of distance. The notion of distance can be generalized. The notions of shape measure, of Voronoï diagram and of Delaunay triangulation Quality –be Mesh can p. 129/331
  189. 189. Nikolai Ivanovich Lobachevsky N IKOLAI I VANOVICH LOBACHEVSKY, 1 décembre 1792, Nizhny Novgorod — 24 février 1856, Kazan. Mesh Quality – p. 130/331
  190. 190. János Bolyai J ÁNOS BOLYAI, 15 dé- cembre 1802 à Kolozsvár, Empire Austrichien (Cluj, Roumanie) — 27 janvier 1860 à Marosvásárhely, Empire Austrichien (Tirgu- Mures, Roumanie). Mesh Quality – p. 131/331
  191. 191. Bernhard RIEMANN G EORG F RIEDRICH B ERN - HARD RIEMANN, 7 sep- tembre 1826, Hanovre — 20 juillet 1866, Selasca. Über die Hypothesen welche der Geo- metrie zu Grunde liegen. 10 juin 1854. Mesh Quality – p. 132/331
  192. 192. Non Euclidean GeometryRiemann 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. Mesh Quality – p. 133/331
  193. 193. The Metric in the Merriam-Webster Mesh Quality – p. 134/331
  194. 194. Definition of a MetricIf S is any set, then the function d : S×S → I Ris 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. Mesh Quality – p. 135/331
  195. 195. The Euclidean Distance is a MetricIn the previous definition of a metric, let the set Sbe I 2 , the function R d : I 2 ×I 2 → I R R R x1 x2 × → (x2 − x1 )2 + (y2 − y1 )2 y1 y2is a metric on I 2 . R Mesh Quality – p. 136/331
  196. 196. Metric Space Mesh Quality – p. 137/331
  197. 197. The Scalar Product is a MetricLet 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 , 1/2 = B − A, B − A , − − 1/2 → → = AB, AB , − T− → → = AB AB. Mesh Quality – p. 138/331
  198. 198. The Scalar Product is a MetricIf the vectorial space is I 2 , then the norm of the R − →scalar product of the vector AB is the Euclideandistance. 1/2 − T− → → d(A, B) = B − A, B − A = AB AB, T xB − xA xB − xA = , y B − yA y B − yA = (xB − xA )2 + (yB − yA )2 . Mesh Quality – p. 139/331
  199. 199. Metric TensorA metric tensor M is a symmetric positivedefinite matrix m11 m12 M= in 2D, m12 m22   m11 m12 m13   M =  m12 m22 m23  in 3D. m13 m23 m33 Mesh Quality – p. 140/331
  200. 200. Metric Length −→The length LM (AB) of an edge between verticesA and B in the metric M is given by −→ − − 1/2 → → LM (AB) = AB, AB M , −→ − 1/2 → = AB, M AB , − T − → → = AB M AB. Mesh Quality – p. 141/331
  201. 201. Euclidean Length with M = I −→ −→ −→ 1/2 − T − → →LM (AB) = AB, M AB = AB M AB, T xB − xA 1 0 xB − xA = y B − yA 0 1 y B − yA −→LE (AB) = (xB − xA )2 + (yB − yA )2 . Mesh Quality – p. 142/331
  202. 202. αβMetric Length with M = βγ −→ −→ −→ 1/2 − T − → →LM (AB) = AB, M AB = AB M AB, T xB − xA α β xB − xA = y B − yA β γ y B − yA −→LE (AB) = α(xB − xA )2 + 2β(xB − xA )(yB − yA ) 2 1/2 +γ(yB − yA ) . Mesh Quality – p. 143/331
  203. 203. Length in a Variable MetricIn 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 1 LM (γ) = (γ ′ (t))T M (γ(t)) γ ′ (t) dt, 0where 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. Mesh Quality – p. 144/331
  204. 204. Area and Volume in a MetricArea of the triangle K in a metric M : AM (K) = det(M) dA. KVolume of the tetrahedron K in a metric M : VM (K) = det(M) dV. K Mesh Quality – p. 145/331
  205. 205. Metric and Delaunay Mesh Mesh Quality – p. 146/331
  206. 206. Which is the Best Triangle ? A BThe question is incomplete. The way to measurethe quality of the triangle is missing. Mesh Quality – p. 147/331
  207. 207. Which is the Best Triangle ? A B Mesh Quality – p. 148/331
  208. 208. Which is the Best Triangle ? A B Mesh Quality – p. 149/331
  209. 209. Example of an Adapted MeshAdapted mesh and solution for a transonicvisquous compressible flow with Mach 0.85 andReynolds = 5 000. Mesh Quality – p. 150/331
  210. 210. Zoom on Boundary Layer–Shock Mesh Quality – p. 151/331
  211. 211. What to Retain Beauty, quality and shape are relative notions. Mesh Quality – p. 152/331
  212. 212. What to Retain Beauty, quality and shape are relative notions. We first need to define what we want in order to evaluate what we obtained. Mesh Quality – p. 152/331
  213. 213. What to Retain Beauty, quality and shape are relative notions. We first need to define what we want in order to evaluate what we obtained. “What we want” is written in the form of metric tensors. Mesh Quality – p. 152/331
  214. 214. What to Retain Beauty, quality and shape are relative notions. We first need to define what we want in order to evaluate what we obtained. “What we want” is written in the form of metric tensors. A shape measure is a measure of the equilarity of a simplex in this metric. Mesh Quality – p. 152/331
  215. 215. Shape Measure in a MetricFirst 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. Mesh Quality – p. 153/331
  216. 216. Shape Measure in a MetricSecond 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. Mesh Quality – p. 154/331
  217. 217. Shape Measure in a MetricThird 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. Mesh Quality – p. 155/331
  218. 218. Shape Measure in a MetricFourth 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. Mesh Quality – p. 156/331
  219. 219. Shape Measure in a MetricFifth 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. Mesh Quality – p. 157/331
  220. 220. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Sim- 11. Mesh Quality and plices Optimization6. Voronoi, Delaunay 12. Size Quality of and Riemann Simplices7. Shape Quality and 13. Universal Quality Delaunay 14. Conclusions Mesh Quality – p. 158/331
  221. 221. Shape Measures and Delaunay CriteroDelaunay meshes have several smoothnessproperties. The Delaunay mesh minimizes the maximum value of all the element circumsphere radii. When the circumsphere center of all simplices of a mesh lie in their respective simplex, then the mesh is a Delaunay mesh. In a Delaunay mesh, the sum of all squared edge lengths weighted by the volume of elements sharing that edge is minimal. Mesh Quality – p. 159/331
  222. 222. 3D-Delaunay Mesh and DegeneracyIn three dimensions, it is well known thatDelaunay meshes can include slivers which aredegenerate elements.Why ?How to avoid them ? Mesh Quality – p. 160/331
  223. 223. Empty Sphere Criterion of DelaunayThe empty sphere criterion of Delaunay is not ashape measure, but it can be used like a shapemeasure in an edge swapping algorithm. Mesh Quality – p. 161/331
  224. 224. Edge Swapping and θmin Shape MeasuDuring edge swapping, using the empty spherecriterion (Delaunay criterion) ⇐⇒Using the θmin shape measure (maximize theminimum of the angles). θ3 θ3 θ6 θ1 θ2 θ1 θ5 θ4 θ6 θ5 θ2 θ4 Mesh Quality – p. 162/331
  225. 225. What to Retain The empty sphere criterion of Delaunay is not a shape measure but it can be used as a shape measure. Mesh Quality – p. 163/331
  226. 226. What to Retain The empty sphere criterion of Delaunay is not a shape measure but it can be used as a shape measure. In two dimensions, in the edge swapping algorithm (Lawson’s method), the empty sphere criterion of Delaunay is equivalent to the θmin shape measure. Mesh Quality – p. 163/331
  227. 227. What to Retain The empty sphere criterion of Delaunay is not a shape measure but it can be used as a shape measure. In two dimensions, in the edge swapping algorithm (Lawson’s method), the empty sphere criterion of Delaunay is equivalent to the θmin shape measure. There is a multitude of valid shape measures, and thus a multitude of generalizations of the Delaunay mesh. Mesh Quality – p. 163/331
  228. 228. Delaunay and Circumscribed SphereAs 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. Mesh Quality – p. 164/331
  229. 229. Circumscribed Sphere of Infinite RadiThe tetrahedra that degenerate into a fin, into acap, into a crystal, into a spindle and into asplitter D D D h A h A C A h C h C B B B D D A h C h C A h h B Bhave a circumscribed sphere of infinite radius. Mesh Quality – p. 165/331
  230. 230. Circumscribed Sphere of Bounded RaThe tetrahedra that degenerate into a sliver, intoa wedge, into a slat, into a needle and into aBig Crunch D D h h D A C A C h C A h B B B D D h C A hh h C hh h A B h Bhhave a circumscribed sphere of bounded radius. Mesh Quality – p. 166/331
  231. 231. What to RetainThe empty sphere criterion ofDelaunay is not a valid shapemeasure sensitive to all the possibledegeneracies of the tetrahedron. Mesh Quality – p. 167/331
  232. 232. Circumscribed Sphere of Bounded RaAmongst the degenerate tetrahedra that have acircumscribed sphere of bounded radius, thewedge, the slat, the needle and the Big Crunchcan be eliminitated D h D A C h C A h B B D D h C A hh h C hh h A B h Bhsince they have several superimposed vertices. Mesh Quality – p. 168/331
  233. 233. The SliverAnd so, finally, we come to the sliver, D h C A C A D B Ba degenerate tetrahedron having disjoint verticesand a bounded circumscribed sphere radius,which makes it “Delaunay-admissible”. Mesh Quality – p. 169/331
  234. 234. Non-Convex QuadrilateralIt is forbidden to swap an edge of a non-convexquadrilateral. S3 S3 T1 T1 T2 S2 S2 T2 S1 S4 S1 S4 S3 T1 S2 T2 S1 S4 S1 S4 Mesh Quality – p. 170/331
  235. 235. Non-Convex Quadrilateral S3 Two adjacent triangles forming a non-convex quadrilateral necessa- T1 T2 rily satisfy the empty S2 sphere criterion of S1 S4 Delaunay. Mesh Quality – p. 171/331
  236. 236. Loss of the Convexity Property in 3D Mesh Quality – p. 172/331
  237. 237. What to Retain The empty sphere criterion of Delaunay is more or less a simplicial shape measure. Mesh Quality – p. 173/331
  238. 238. What to Retain The empty sphere criterion of Delaunay is more or less a simplicial shape measure. The empty sphere criterion of Delaunay is not sensitive to all the possible degeneracies of the tetrahedron. Mesh Quality – p. 173/331
  239. 239. What to Retain The empty sphere criterion of Delaunay is more or less a simplicial shape measure. The empty sphere criterion of Delaunay is not sensitive to all the possible degeneracies of the tetrahedron. A valid shape measure, sensitive to all the possible degeneracies of the tetrahedron, used in an edge swapping and face swapping algorithm should lead to a mesh that is not a Delaunay mesh, but that is of better quality. Mesh Quality – p. 173/331
  240. 240. Table of Contents1. Introduction 8. Non-Simplicial2. Simplex Definition Elements3. Degeneracies of 9. Shape Quality Simplices Visualization4. Shape Quality of 10. Shape Quality Simplices Equivalence5. Formulae for Sim- 11. Mesh Quality and plices Optimization6. Voronoi, Delaunay 12. Size Quality of and Riemann Simplices7. Shape Quality and 13. Universal Quality Delaunay 14. Conclusions Mesh Quality – p. 174/331
  241. 241. Non-Simplicial ElementsThis 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. Mesh Quality – p. 175/331
  242. 242. Non-Simplicial ElementsOn Element Shape Measures for MeshOptimizationPAUL L ABBÉ , J ULIEN D OMPIERRE , F RANÇOISG UIBAULT AND R ICARDO C AMAREROPresented at the 2nd Symposium on Trends inUnstructured Mesh Generation, Fifth US NationalCongress on Computational Mechanics, 4–6august 1999 University of Colorado at Boulder. Mesh Quality – p. 176/331
  243. 243. Regularity Generalization An equilateral quadrilateral, ie that has four edges of same length, is not necessarily a square... Mesh Quality – p. 177/331
  244. 244. Regularity Generalization An equilateral quadrilateral, ie that has four edges of same length, is not necessarily a square... Définition : An element, be it simplicial or not, is regular if it maximizes its measure for a given measure of its boundary. Mesh Quality – p. 177/331
  245. 245. Regularity Generalization An equilateral quadrilateral, ie that has four edges of same length, is not necessarily a square... Définition : An element, be it simplicial or not, is regular if it maximizes its measure for a given measure of its boundary. The equilateral triangle is regular because it maximizes its area for a given perimiter. Mesh Quality – p. 177/331
  246. 246. Regularity Generalization An equilateral quadrilateral, ie that has four edges of same length, is not necessarily a square... Définition : An element, be it simplicial or not, is regular if it maximizes its measure for a given measure of its boundary. The equilateral triangle is regular because it maximizes its area for a given perimiter. The equilateral tetrahedron is regular because it maximizes its volume for a given surface of its faces. Mesh Quality – p. 177/331
  247. 247. Regular Non Simplicial Elements The regular quadrilateral is the square. The regular hexahedron is the cube. The regular prism is the ... regular prism ! Its two triangular faces are equilateral triangle whose edges measure a. Its three quadrilateral faces are rectangles that have a base of √ length a and a height of length a/ 3. Mesh Quality – p. 178/331
  248. 248. Quality of Non Simplicial ElementsProposed 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. Mesh Quality – p. 179/331
  249. 249. Shape Measure of a QuadrilateralThe shape measure of a quadrilateral is theminimum of the shape measure of its four cornertriangles formed by its four vertices. D C D C D C D C A B A BA B BA Mesh Quality – p. 180/331
  250. 250. Shape Measure of a PrismThe shape measure of a prism is the minimum ofthe shape measure of its six corner tetrahedronformed by its six vertices. F D E F C C C D E A B A B A B C F F F A B D E D E D E C A B Mesh Quality – p. 181/331
  251. 251. Shape Measure of an HexahedronThe shape measure of an hexahedron is theminimum of its eight corner tetrahedron formedby its eight vertices. G H E F H G D C D C D CE F AHB A B HB A H D C G G GA B E F E F F E C D A B Mesh Quality – p. 182/331
  252. 252. Shape of the Corner Simplex The corner simplices constructed for the non simplicial elements are not regular simplices. For the square, the four corner triangles are isosceles right-angled triangles. For the cube, the eight corner tetrahedra are isosceles right-angled tetrahedra. For the regular prism, the six corner tetrahedra are tetrahedron with an equilateral triangle of side a, √ a fourth perpendicular and edge of length a/ 3. Mesh Quality – p. 183/331
  253. 253. Shape of the Corner SimplexEach 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 √ √ 2 3 3 3 Square √ 1+ 2 2 4 √ 1+ 2 √ √ √ 18√ 1 2 arcsin(1/ 22+12 3) 3 √6 Prism √ 5(7+ 13) √ 3 2 √ 6 arcsin(1/ 3)−π 7+ 13 √ √ √ √ √ 2 3 2 arcsin((2− 2)/(2 3)) Cube 3−1 3 2 √ 6 arcsin(1/ 3)−π 3−1 Mesh Quality – p. 184/331
  254. 254. Degenerate Non Simplicial ElementsDéfinition :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. Mesh Quality – p. 185/331
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