no. 8
Numerical Analysis
Stabilized explicit coupling for fluid–structure interaction using Nitsche's method
[Couplage explicite stabilisé en interaction fluide–structure avec la méthode de Nitsche]

Présenté par : Olivier Pironneau

Burman, Erik1 ; Fernández, Miguel A.2
1 École Polytechnique Fédérale de Lausanne, IACS, CH-1015 Lausanne, Switzerland
2 INRIA, REO team, Rocquencourt BP 105, 78153 Le Chesnay cedex, France
Comptes Rendus. Mathématique, Tome 345 (2007) no. 8, pp. 467-472.

Dans cette Note on propose un schéma de couplage explicite stabilisé basé sur la méthode de Nitsche. Le schéma est stable indépendamment du rapport de densités fluide et structure. Des expériences numériques montrent qu'on peut obtenir une précision optimale en temps après quelques itérations d'un algorithme de Résidus Corrigés.

In this Note we propose a stabilized explicit coupling scheme for fluid–structure interaction based on Nitsche's method. The scheme is stable irrespective of the fluid–solid density ratio. Numerical experiments show that optimal time accuracy can be obtained by performing a few defect-correction iterations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2007.09.010
Burman, Erik 1 ; Fernández, Miguel A. 2

1 École Polytechnique Fédérale de Lausanne, IACS, CH-1015 Lausanne, Switzerland
2 INRIA, REO team, Rocquencourt BP 105, 78153 Le Chesnay cedex, France 
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Burman, Erik; Fernández, Miguel A. Stabilized explicit coupling for fluid–structure interaction using Nitsche's method. Comptes Rendus. Mathématique, Tome 345 (2007) no. 8, pp. 467-472. doi : 10.1016/j.crma.2007.09.010. http://www.numdam.org/articles/10.1016/j.crma.2007.09.010/

[1] Becker, R.; Hansbo, P.; Stenberg, R. A finite element method for domain decomposition with non-matching grids, M2AN Math. Model. Numer. Anal., Volume 37 (2003) no. 2, pp. 209-225

[2] E. Burman, M.A. Fernández, Stabilization of explicit coupling in fluid–structure interaction with fluid incompressibility, in preparation

[3] Burman, E.; Hansbo, P. A unified stabilized method for Stokes' and Darcy's equations, J. Comput. Appl. Math., Volume 198 (2007), pp. 35-51

[4] Burman, E.; Zunino, P. A domain decomposition method based on interior penalties for advection–diffusion–reaction problems, SIAM J. Numer. Anal., Volume 44 (2006), pp. 1612-1638

[5] Causin, P.; Gerbeau, J.-F.; Nobile, F. Added-mass effect in the design of partitioned algorithms for fluid–structure problems, Comput. Methods Appl. Mech. Engrg., Volume 194 (2005) no. 42–44, pp. 4506-4527

[6] Farhat, C.; van der Zee, K.; Geuzaine, Ph. Provably second-order time-accurate loosely-coupled solution algorithms for transient nonlinear aeroelasticity, Comput. Methods Appl. Mech. Engrg., Volume 195 (2006) no. 17–18, pp. 1973-2001

[7] Fernández, M.A.; Gerbeau, J.F.; Grandmont, C. A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid, Int. J. Numer. Methods Engrg., Volume 69 (2007) no. 4, pp. 794-821

[8] Fernández, M.A.; Moubachir, M. A Newton method using exact Jacobians for solving fluid–structure coupling, Comp. & Struct., Volume 83 (2005), pp. 127-142

[9] Gerbeau, J.-F.; Vidrascu, M. A quasi-Newton algorithm based on a reduced model for fluid–structure interactions problems in blood flows, Math. Model. Numer. Anal., Volume 37 (2003) no. 4, pp. 631-648

[10] Hansbo, P.; Hermansson, J. Nitsche's method for coupling non-matching meshes in fluid–structure vibration problems, Comput. Mech., Volume 32 (2003) no. 1–2, pp. 134-139

[11] Hansbo, P.; Hermansson, J.; Svedberg, T. Nitsche's method combined with space–time finite elements for ALE fluid–structure interaction problems, Comput. Methods Appl. Mech. Engrg., Volume 193 (2004), pp. 4195-4206

[12] F. Hecht, O. Pironneau, A. Le Hyaric, K. Ohtsuka, FreeFem++ v. 2.11. User's Manual, University of Paris 6

[13] Le Tallec, P.; Mouro, J. Fluid structure interaction with large structural displacements, Comput. Methods Appl. Mech. Engrg., Volume 190 (2001), pp. 3039-3067

[14] Mok, D.P.; Wall, W.A.; Ramm, E. Accelerated iterative substructuring schemes for instationary fluid–structure interaction (Bathe, K.J., ed.), Computational Fluid and Solid Mechanics, Elsevier, 2001, pp. 1325-1328

[15] Nitsche, J. Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind, Abh. Math. Sem. Univ. Hamburg, Volume 36 (1971), pp. 9-15

[16] F. Nobile, Numerical approximation of fluid–structure interaction problems with application to haemodynamics, PhD thesis, EPFL, Switzerland, 2001

[17] Stetter, H.J. The defect correction principle and discretization methods, Numer. Math., Volume 29 (1978), pp. 425-443

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