Analyse numérique
Construction dʼun champ continu de métriques
[From discrete to continuous metric fields]
Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 639-644.

Adaptive computation using adaptive meshes is now recognized as essential for solving complex PDE problems. This computation requires, at each step, the definition of a continuous metric field to govern the generation of the adapted meshes. In practice, via an appropriate a posteriori error estimation, metrics are calculated at the vertices of the computational domain mesh. In order to obtain a continuous metric field, the discrete field is interpolated in the whole domain mesh. In this Note, a new method for interpolating discrete metric fields, based on a so-called “natural decomposition” of metrics, is introduced. The proposed method is based on known matrix decompositions and is computationally robust and efficient. Some qualitative comparisons with classical methods are made to show the relevance of this methodology.

Lʼintérêt du calcul adaptatif pour résoudre des problèmes complexes dʼEDP est pleinement reconnu de nos jours. Un tel calcul demande, à chaque étape, la définition dʼun champ continu de métriques pour gouverner la génération de maillages adaptés. En pratique, via un estimateur dʼerreur a posteriori, des métriques sont calculées aux sommets du maillage du domaine de calcul. Pour obtenir un champ continu de métriques, le champ discret est interpolé dans tout le maillage du domaine. Dans cette Note, une nouvelle méthode pour interpoler ces champs, fondée sur une décomposition dite « naturelle » des métriques, est introduite. La méthode proposée, basée sur des décompositions connues de matrices, met en jeu des algorithmes robustes et efficaces. Quelques comparaisons qualitatives avec des méthodes classiques sont réalisées afin de montrer la pertinence de cette méthodologie.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.07.009
Laug, Patrick 1; Borouchaki, Houman 2

1 INRIA Paris-Rocquencourt, GAMMA3 joint project-team, BP 105, 78153 Le Chesnay cedex, France
2 University of Technology of Troyes, GAMMA3 joint project-team, BP 2060, 10010 Troyes cedex, France
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Laug, Patrick; Borouchaki, Houman. Construction dʼun champ continu de métriques. Comptes Rendus. Mathématique, Volume 351 (2013) no. 15-16, pp. 639-644. doi : 10.1016/j.crma.2013.07.009. http://www.numdam.org/articles/10.1016/j.crma.2013.07.009/

[1] Alauzet, F. Size gradation control of anisotropic meshes, Finite Elem. Anal. Des., Volume 46 (2010) no. 1–2, pp. 181-202

[2] Arsigny, V.; Fillard, P.; Pennec, X.; Ayache, N. Log-Euclidean metrics for fast and simple calculus on diffusion tensors, Magn. Reson. Med., Volume 56 (2006), pp. 411-421

[3] dʼAzevedo, E.F.; Simpson, B. On optimal triangular meshes for minimizing the gradient error, Numer. Math., Volume 59 (1991), pp. 321-348

[4] Babuska, I.; Aziz, K. On the angle condition in the finite element method, SIAM J. Numer. Anal., Volume 73 (1976) no. 2, pp. 214-226

[5] Berzins, M. Mesh quality: a function of geometry, error estimates or both?, Eng. Comput., Volume 15 (1999), pp. 236-247

[6] Borouchaki, H.; George, P.L.; Hecht, F.; Laug, P.; Saltel, E. Delaunay mesh generation governed by metric specifications – Part I: Algorithms, Finite Elem. Anal. Des., Volume 25 (1997) no. 1–2, pp. 61-83

[7] Borouchaki, H.; George, P.L.; Mohammadi, B. Delaunay mesh generation governed by metric specifications – Part II: Application examples, Finite Elem. Anal. Des., Volume 25 (1997) no. 1–2, pp. 85-109

[8] Frey, P.J.; George, P.L. Mesh Generation, Application to Finite Elements, Wiley, 2008

[9] George, P.L.; Borouchaki, H. Delaunay Triangulation and Meshing, Application to Finite Elements, Hermes, 1998

[10] George, P.L.; Borouchaki, H.; Frey, P.J.; Laug, P.; Saltel, E. Mesh generation and mesh adaptivity, Encyclopedia of Computational Mechanics, Volume 1: Fundamentals, Wiley, 2004, pp. 497-523 (Chap. 17)

[11] Löhner, R. Adaptive remeshing for transient problems, Comput. Methods Appl. Mech. Eng., Volume 75 (1989) no. 1–3, pp. 195-214

[12] Michal, T.; Krakos, J. Anisotropic mesh adaptation through edge primitive operations, 50th AIAA Aerospace Sciences Meeting, 2012 (AIAA 2012-0159)

[13] Peraire, J.; Vahdati, M.; Morgan, K.; Zienkiewicz, O.C. Adaptive remeshing for compressible flow computations, J. Comput. Phys., Volume 72 (1987), pp. 449-466

[14] Rippa, S. Long and thin triangles can be good for linear interpolation, SIAM J. Numer. Anal., Volume 29 (1992) no. 1, pp. 257-270

[15] Thompson, J.F.; Soni, B.K.; Weatherill, N.P. Handbook of Grid Generation, CRC, 1998

[16] Topping, B.H.V.; Muylle, J.; Ivanyi, P.; Putanowicz, R.; Cheng, B. Finite Element Mesh Generation, Saxe-Coburg, 2004

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