2.2 Adaptation de la surface
2.2.1 Anisotropie et qualit´e d’un maillage
En calcul num´erique, la notion de qualit´e d’un maillage est relative car elle d´epend de l’´equation
`
a r´esoudre (diffusion, ´ecoulements, choc etc.). Elle est souvent d´efinie par une norme sur la qualit´e individuelle des mailles.
2.2.1.1 Impact sur les solveurs num´eriques
En fait, la qualit´e du maillage influe sur la pr´ecision et la vitesse de convergence des solveurs num´eriques, outre le fait qu’il doit ˆetre une bonne approximation du domaine.
8 Jonathan Richard Shewchuk
Figure 2:A visual illustration of how large angles, but not small angles, can cause the error∇f− ∇gto explode. In each triangulation, 200 triangles are used to render a paraboloid.
35 50 65
Figure 3:As the large angle of the triangle approaches180◦, or the sliver tetrahedron becomes arbitrarily flat, the magnitude of the vertical component of∇gbecomes arbitrarily large.
sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.
Because of this sensitivity, mesh generators usually choose the shapes of elements to control�∇f−
∇g�∞, and not�f−g�∞, which can be reduced simply by using smaller elements. Section 6.1 presents quality measures that judge the shape of elements based on their fitness for interpolation.
Table 2 gives two upper bounds on�∇f− ∇g�∞over a triangle. The first upper bound is almost tight, to within a factor of two. The “weaker but simpler upper bound” of3ctrcircis not as good an indicator as the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more amenable to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any trianglet, there is a functionfsuch that�∇f− ∇g�∞=ctrcirc.
These bounds are interesting because the two-dimensional Delaunay triangulation minimizes the
max-In mechanics, is the strains.
affects discretization error in FEM.
The Importance of Approximating Gradients Accurately
f
|| f − g || 8
surface associ´ee
domaine
8 Jonathan Richard Shewchuk
Figure 2:A visual illustration of how large angles, but not small angles, can cause the error∇f − ∇g to explode. In each triangulation, 200 triangles are used to render a paraboloid.
35
Figure 3:As the large angle of the triangle approaches180◦, or the sliver tetrahedron becomes arbitrarily flat, the magnitude of the vertical component of∇gbecomes arbitrarily large.
sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.
Because of this sensitivity, mesh generators usually choose the shapes of elements to control�∇f −
∇g�∞, and not�f−g�∞, which can be reduced simply by using smaller elements. Section 6.1 presents quality measures that judge the shape of elements based on their fitness for interpolation.
Table 2 gives two upper bounds on�∇f− ∇g�∞over a triangle. The first upper bound is almost tight, to within a factor of two. The “weaker but simpler upper bound” of3ctrcirc is not as good an indicator as the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more amenable to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any trianglet, there is a functionfsuch that�∇f− ∇g�∞ =ctrcirc.
These bounds are interesting because the two-dimensional Delaunay triangulation minimizes the
max-In mechanics, is the strains.
affects discretization error in FEM.
The Importance of Approximating Gradients Accurately
f
Figure 2:A visual illustration of how large angles, but not small angles, can cause the error∇f− ∇gto explode. In each triangulation, 200 triangles are used to render a paraboloid.
35 50 65
Figure 3:As the large angle of the triangle approaches180◦, or the sliver tetrahedron becomes arbitrarily flat, the magnitude of the vertical component of∇gbecomes arbitrarily large.
sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.
Because of this sensitivity, mesh generators usually choose the shapes of elements to control�∇f−
∇g�∞, and not�f−g�∞, which can be reduced simply by using smaller elements. Section 6.1 presents quality measures that judge the shape of elements based on their fitness for interpolation.
Table 2 gives two upper bounds on�∇f− ∇g�∞over a triangle. The first upper bound is almost tight, to within a factor of two. The “weaker but simpler upper bound” of3ctrcircis not as good an indicator as the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more amenable to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any trianglet, there is a functionfsuch that�∇f− ∇g�∞=ctrcirc.
These bounds are interesting because the two-dimensional Delaunay triangulation minimizes the
max-In mechanics, is the strains.
affects discretization error in FEM.
The Importance of Approximating Gradients Accurately
f
|| f − g || 8
surface associ´ee
domaine
8 Jonathan Richard Shewchuk
Figure 2: A visual illustration of how large angles, but not small angles, can cause the error∇f − ∇g to explode. In each triangulation, 200 triangles are used to render a paraboloid.
35
Figure 3: As the large angle of the triangle approaches180◦, or the sliver tetrahedron becomes arbitrarily flat, the magnitude of the vertical component of∇gbecomes arbitrarily large.
sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.
Because of this sensitivity, mesh generators usually choose the shapes of elements to control�∇f −
∇g�∞, and not�f −g�∞, which can be reduced simply by using smaller elements. Section 6.1 presents quality measures that judge the shape of elements based on their fitness for interpolation.
Table 2 gives two upper bounds on�∇f− ∇g�∞over a triangle. The first upper bound is almost tight, to within a factor of two. The “weaker but simpler upper bound” of3ctrcircis not as good an indicator as the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more amenable to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any trianglet, there is a functionf such that�∇f− ∇g�∞=ctrcirc.
These bounds are interesting because the two-dimensional Delaunay triangulation minimizes the
max-In mechanics, is the strains.
affects discretization error in FEM.
The Importance of Approximating Gradients Accurately
f
Figure 2:A visual illustration of how large angles, but not small angles, can cause the error∇f− ∇gto explode. In each triangulation, 200 triangles are used to render a paraboloid.
35 50 65
Figure 3:As the large angle of the triangle approaches180◦, or the sliver tetrahedron becomes arbitrarily flat, the magnitude of the vertical component of∇gbecomes arbitrarily large.
sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.
Because of this sensitivity, mesh generators usually choose the shapes of elements to control�∇f−
∇g�∞, and not�f−g�∞, which can be reduced simply by using smaller elements. Section 6.1 presents quality measures that judge the shape of elements based on their fitness for interpolation.
Table 2 gives two upper bounds on�∇f− ∇g�∞over a triangle. The first upper bound is almost tight, to within a factor of two. The “weaker but simpler upper bound” of3ctrcircis not as good an indicator as the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more amenable to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any trianglet, there is a functionfsuch that�∇f− ∇g�∞=ctrcirc.
These bounds are interesting because the two-dimensional Delaunay triangulation minimizes the
max-In mechanics, is the strains.
affects discretization error in FEM.
The Importance of Approximating Gradients Accurately
f
|| f − g || 8
surface associ´ee
domaine
8 Jonathan Richard Shewchuk
Figure 2: A visual illustration of how large angles, but not small angles, can cause the error∇f − ∇g to explode. In each triangulation, 200 triangles are used to render a paraboloid.
35
Figure 3: As the large angle of the triangle approaches180◦, or the sliver tetrahedron becomes arbitrarily flat, the magnitude of the vertical component of∇gbecomes arbitrarily large.
sphere, then perturbing one of the vertices just off the equator so that the sliver has some (but not much) volume.
Because of this sensitivity, mesh generators usually choose the shapes of elements to control�∇f −
∇g�∞, and not�f −g�∞, which can be reduced simply by using smaller elements. Section 6.1 presents quality measures that judge the shape of elements based on their fitness for interpolation.
Table 2 gives two upper bounds on�∇f− ∇g�∞over a triangle. The first upper bound is almost tight, to within a factor of two. The “weaker but simpler upper bound” of3ctrcircis not as good an indicator as the stronger upper bound, but it has the advantages of being smooth almost everywhere (and therefore more amenable to numerical optimization) and faster to compute. The weaker upper bound is tight to within a factor of three: for any trianglet, there is a functionf such that�∇f− ∇g�∞=ctrcirc.
These bounds are interesting because the two-dimensional Delaunay triangulation minimizes the
max-In mechanics, is the strains.
affects discretization error in FEM.
The Importance of Approximating Gradients Accurately
f
|| f − g || 8
surface associ´ee
domaine
(3)´etir´es
Figure 2.8: Influence de l’aspect des mailles sur l’interpolation d’une solution num´erique en2D[68].
• pr´ecision: l’erreur d’interpolation locale d’une solution num´erique sur une mailleKest souvent major´ee par un facteur relatif au ratio d’aspect deK (et donc sa qualit´e) [69]. Elle d´epend de la norme utilis´ee. En norme H1 par exemple6, on a :
ku−ΠhukH1(K)≤c·q[K]· |u|H1(K), avecq[K] = h2 2r et
r: rayon du cercle inscrit deK h: le diam`etre deK
c: facteur ind´ependant deK Ainsi l’erreur deud´epend aussi de la forme de la maille, pas uniquement de la taille de ses arˆetes.
Pour se donner une id´ee intuitive, l’influence de la forme des mailles sur l’interpolation deudans
5Ainsi la surface triangul´ee n’est donc plus une vari´et´e
6En fait l’erreur d’une solution num´erique est souvent estim´ee ou major´ee par le biais d’une norme.
Ici, la norme H1 d’une fonctionf: Ω→Rest d´efinie par||f||H1(Ω)= (||f||2L2(Ω)+Pd
i=0||∂xif||2L2(Ω))1/2. Elle fait intervenir la norme L2d´efinie par : kfkL2(Ω)= (R
Ω|f(x)|2dx)1/2.
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le cas2Dest donn´ee `a la figure2.8. Elle reprise de l’exemple dans [68]. Sur cet exemple, on voit queuest mieux interpol´ee avec des mailles droites en(1), pas trop mal avec des mailles ´etir´ees en (2). Par contre, elle est pire avec des mailles plates en (3). Sinon, l’int´egration num´erique de quantit´es relatives `a un pointpd´epend de la discr´etisation du voisinage de p: sa pr´ecision d´epend du degr´e de pet des angles des mailles incidentes `a p. Par exemple, c’est le cas de la quadrature des cotangents [1] (§3.3.4) pour le calcul du laplacien sur une surface7. Ici, plus les mailles incidentes `a psont r´eguli`eres, plus pr´ecise elle sera [62].
• performance. Si la solution num´erique est calcul´ee par une m´ethode des ´el´ements finis, alors on a une ´equation AX=B`a r´esoudre, o`uA etB sont des matrices carr´ees de taillen, etnest le nombre de points du maillage. Dans ce cas, la vitesse de convergence des solveurs it´eratifs (gradient conjugu´e, Jacobi) pour la r´esolution de cette ´equation d´epend du conditionnement8 κA de la matriceA. Ainsi le temps de calcul est proportionnel `a κA. CommeA est sym´etrique, alorsκA est juste le rapport des valeurs propres maximale et minimaleλmax et λmin deA. Il se trouve queλmax (et doncκA) d´epend de la forme des mailles [68]. En effet on a :
o`u dd´esigne le degr´e maximal des points du maillage, et `i la longueur de laie arˆete deK[68].
2.2.1.2 Forme et alignement optimal
Rappelons que l’optimalit´e d’un maillageThd´epend de l’´equation `a discr´etiser. En fait, le maillage optimal celui qui suit mieux les variations de la solution num´eriqueusur le domaine ou sur la surface.
Ainsi c’est celui dont la densit´e de points et l’alignement des mailles sont adapt´ees au gradient deu.
Table 2.1: Mesures usuelles de forme d’un triangle.
nomenclature formulea valeurs id´eale scaled jacobian [71] max[`|J|
i`j] [−1,1] 2
- `minet`max: longueurs minimale et maximale des arˆetes de K.
- retR: rayons des cercles inscrits et circonscrits `a K.
- ni: vecteur normal `a l’arˆeteeide K.
- vi: vecteur reliant le barycentre de K au milieu de l’arˆeteei. - J : matrice jacobienne de K avecJ= (AB, ~~ AC)en 2D.
7En fait, il s’agit de l’op´erateur de l’op´erateur delaplace-beltramiqui g´en´eralise le laplacien au cas des vari´et´es.
Il est impliqu´e dans les noyaux de param´etrisation, de segmentation, de reconstruction ou de lissage de vari´et´es. Elle consiste en une quadrature de l’int´egrale de la divergence du gradient au voisinageSdu pointp normalis´ee par l’aire deS, et implique les angles oppos´ees `a chaque arˆete incidente `ap.
8Le conditionnement d’une matriceAest le nombreκA=kAk−1kAk, o`uk·kd´esigne la norme defrobenius.
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44 2.2. Adaptation de la surface
forme. Une synth`ese des mesures de forme d’un simplexe est donn´ee `a la table 2.1. Elles sont bas´ees sur les angles, les arˆetes, l’aire ou encore la matrice jacobienne associ´ee `a une maille. Une analyse comparative de certaines d’entre elles est donn´ee dans [70], elles sont n´eanmoins plus ou moins ´equivalentes dans le sens o`u elles visent `a discriminer les simplexes plats ou ´etir´es des simplexes quasi-´equilat´eraux. Pour une mesure de forme normalis´ee q telle que 1 correspond au triangle id´eal, l’adaptation consiste `a maximiser la normekqkp∈N. Sur cette table,
• min-angleest la mesure la plus simple et la plus intuitive. Ici, le remaillage vise `a construire une triangulation dedelaunaydu volume ou de la surface (§2.1.1). Elle convient si la solution num´erique est suffisamment r´eguli`ere (´equations elliptiques par exemple), en vertu de lacondition d’angle sur la convergence de fonctions quadratiques [72]. N´eanmoins elle ne tient pas compte de la taille des simplexes d´efinies par le diam`etre9 hdu maillageTh.
• lesedge-radii-aspect ratioquantifient l’aplatissement des mailles, tandis que leskewnesset l’ortho-gonalitymesurent leur d´eviation angulaire par rapport `a un simplexe ´equilat´eral.
• lesscaled-jacobianetfrobenius ratiosont bas´ees sur la matrice jacobienneJ deKqui encode les in-formations relatives au volume, la forme et l’orientation du simplexe. En effet une d´ecomposition en valeurs singuli`eres de J permet de retrouver les matrices associ´ees `a ces quantit´es [73] §3.
Elles mesurent la d´eviation deKpar rapport au simplexe id´eal sur ces crit`eres.
En pratique, des impl´ementations num´eriquement robustes de ces mesures existent au sein de bib-lioth`eques commeCGAL [9] ouVerdict[64].
optimalit´e. In fine, toutes ces mesures sont purement g´eom´etriques et d´ecorrel´ees de la solu-tion num´erique u. Elles d´efinissent la triangulation id´eale comme celle constitu´ee de mailles
quasi-´equilat´erales. En fait, ce n’est vrai que siuvarie de mani`ereisotrope, c’est-`a-dire uniform´ement dans toutes les directions de Ω (´equation de la chaleur par exemple). Dans le cas anisotrope, une triangu-lation id´eale est celle dont la r´epartition des points, la forme et l’´etirement des mailles sont adapt´ees
`a l’erreur de la solution num´erique, comme sur les deux exemples de la figure 2.9. En fait il s’agit de la triangulation dont la densit´e des points croit selon la courbure locale de u, et dont les mailles sont align´ees avec les directions de la hessienne10deusur chaque point11. Ainsi une bonne mesure de qualit´e doit encoder trois crit`eres :
• une taille d’arˆetes qui d´epend des valeurs propres de la hessienne deuen ses sommets.
• un alignement par rapport aux vecteurs propres de la hessienne deuen ses sommets.
• un facteur de gradation born´e12.
En fait, elle est souvent issue de l’une des mesures de la table2.1, mais n´eanmoins modifi´ee pour tenir compte des informations fournies par un estimateur d’erreur bas´e sur les hessiennes deu[74–77].
based on aL1 Lpnorm space-time error estimate done in 2011 [116]. Right, an anisotropic mesh adaptation based on theLp Lpnorm space-time error estimate presented in this article done in 2014. We clearly see the progress of the results with a lot more physics captured in the later simulations. The quality of the adapted anisotropic meshes has also been especially improved. This is due to the simultaneous advances in the error estimate, the interpolation stage, the flow solver, and the adaptive local remesher.
Figure 16: 3D blast in a city. Top, adapted surface meshes and, bottom, density iso-values on the surface. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
Figure 17: 3D blast in a city. Top, cut in the adapted volume meshes and, bottom, corresponding density iso-values. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
31
based on a L 1 L p norm space-time error estimate done in 2011 [116]. Right, an anisotropic mesh adaptation based on the L p L p norm space-time error estimate presented in this article done in 2014. We clearly see the progress of the results with a lot more physics captured in the later simulations. The quality of the adapted anisotropic meshes has also been especially improved. This is due to the simultaneous advances in the error estimate, the interpolation stage, the flow solver, and the adaptive local remesher.
Figure 16: 3D blast in a city. Top, adapted surface meshes and, bottom, density iso-values on the surface. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
Figure 17: 3D blast in a city. Top, cut in the adapted volume meshes and, bottom, corresponding density iso-values. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
31
(1)densit´e d’onde de choc (surface)
based on aL1 Lpnorm space-time error estimate done in 2011 [116]. Right, an anisotropic mesh adaptation based on theLp Lpnorm space-time error estimate presented in this article done in 2014. We clearly see the progress of the results with a lot more physics captured in the later simulations. The quality of the adapted anisotropic meshes has also been especially improved. This is due to the simultaneous advances in the error estimate, the interpolation stage, the flow solver, and the adaptive local remesher.
Figure 16: 3D blast in a city. Top, adapted surface meshes and, bottom, density iso-values on the surface. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
Figure 17: 3D blast in a city. Top, cut in the adapted volume meshes and, bottom, corresponding density iso-values. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
31
based on a L 1 L p norm space-time error estimate done in 2011 [116]. Right, an anisotropic mesh adaptation based on the L p L p norm space-time error estimate presented in this article done in 2014. We clearly see the progress of the results with a lot more physics captured in the later simulations. The quality of the adapted anisotropic meshes has also been especially improved. This is due to the simultaneous advances in the error estimate, the interpolation stage, the flow solver, and the adaptive local remesher.
Figure 16: 3D blast in a city. Top, adapted surface meshes and, bottom, density iso-values on the surface. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
Figure 17: 3D blast in a city. Top, cut in the adapted volume meshes and, bottom, corresponding density iso-values. From left to right, meshes and solutions at sub-intervals 43, 86, and 128, respectively.
31
(2)densit´e d’onde de choc (volume) Figure 2.9: Exemples de triangulations optimales en m´ecanique des fluides [78].
9Le diam`etre d’un triangle ou d’un maillage est la longueur de sa plus longue arˆete.
10La matrice hessienne Hu = ∇2u de u sur un point p est une matrice sym´etrique d´efinie positive d´ecrivant la variation du gradient deu, et donc les coefficients sont form´es par les d´eriv´ees secondes deu.
11Il a ´et´e montr´e dans [69] que l’erreur d’interpolation de u en norme H1 d´ecroit en fonction de l’angle entre l’orientation d’une mailleKet celle deu. De plus, si les mailles sont align´ees avecuet que leur ratio d’aspect est born´e par|λ1/λ2|1/2, avecλiles valeurs propres deHu, alors l’erreur deune d´epend plus de l’angle maximal deTh.
12Il s’agit d’une borne sur la variation de tailles entre arˆetes incidentes. En calcul num´erique, il est important d’avoir une variation graduelle des tailles d’arˆetes sur le domaine ou la surface, afin de permettre des interpolations plus stables.
c2018.HOBYRAKOTOARIVELO