In this note, we present a smooth extension method for the simulation of the motion of immersed rigid bodies. It is a method of the fictitious domain type, which uses Cartesian meshes and recovers the optimal order of the error by finding a smooth extension of the exact solution defined in the domain with holes. We first present the method with a Poisson problem and show next how it can be adapted to the case of immersed rigid bodies. Finally, the method is validated in both the scalar and the vector cases.
Nous présentons dans cette note une méthode de prolongement régulier pour simuler le mouvement de particules rigides immergées dans un fluide incompressible. Cʼest une méthode de type domaine fictif sur maillage cartésien permettant de retrouver lʼordre optimal de lʼerreur en espace, en trouvant un prolongement régulier de la solution exacte définie sur le domaine perforé. Nous présentons tout dʼabord la méthode sur un problème scalaire, puis nous lʼadaptons au cas des équations de Stokes incompressibles et des particules rigides. Elle est ensuite validée sur différents cas de test.
Accepted:
Published online:
Benoit Fabrèges 1; Loïc Gouarin 2; Bertrand Maury 2
@article{CRMATH_2013__351_9-10_361_0, author = {Benoit Fabr\`eges and Lo{\"\i}c Gouarin and Bertrand Maury}, title = {A smooth extension method}, journal = {Comptes Rendus. Math\'ematique}, pages = {361--366}, publisher = {Elsevier}, volume = {351}, number = {9-10}, year = {2013}, doi = {10.1016/j.crma.2013.05.011}, language = {en}, }
Benoit Fabrèges; Loïc Gouarin; Bertrand Maury. A smooth extension method. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 361-366. doi : 10.1016/j.crma.2013.05.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.05.011/
[1] Sobolev Spaces, Elsevier, 2003
[2] Control approach to fictitious domain methods. Application to fluid dynamics and electro-magnetics (D.E. Keyes; T.F. Chan; G. Meuran; J.S. Scroggs; R.G. Voigt, eds.), Fourth International Symposium on Domain Decomposition Methods for Partial Differential Equations, 1991
[3] Une analyse de la méthode des domaines fictifs pour le problème de Helmholtz extérieur, RAIRO – Modél. Math. Anal. Numér., Volume 27 (1993) no. 3, pp. 251-288
[4] A domain embedding method using the optimal distributed control and a fast algorithm, Numer. Algorithms, Volume 36 (2004) no. 2, pp. 95-112
[5] Approximate imposition of boundary conditions in immersed boundary methods, Int. J. Numer. Methods Eng., Volume 80 (2009) no. 11, pp. 1379-1405
[6] A symmetric method for weakly imposing Dirichlet boundary conditions in embedded finite element meshes, Int. J. Numer. Methods Eng., Volume 90 (2012) no. 5, pp. 636-658
[7] Analysis of the fully discrete fat boundary method, Numer. Math., Volume 118 (2011) no. 1, pp. 48-77
[8] An unfitted finite element method, based on Nitscheʼs method, for elliptic interface problems, Comput. Methods Appl. Math., Volume 191 (2002) no. 47–48, pp. 5537-5552
[9] A penalty method for the simulation of fluid–rigid body interaction (Éric Cancès; Jean-Frédéric Gerbeau, eds.), ESAIM Proceedings, vol. 14, September 2005, pp. 201-212
[10] A fat boundary method for the Poisson problem in a domain with holes, J. Sci. Comput., Volume 16 (2001) no. 3, pp. 319-339
[11] Étude de quelques méthodes de domaine fictif et applications aux ondes électromagnétiques, University Paris-6, Jussieu, 1998 (PhD thesis)
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