Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T10:33:46.792Z Has data issue: false hasContentIssue false
  • English
  • Français

Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation

Published online by Cambridge University Press:  05 June 2008

Guillaume Legendre  and
Takéo Takahashi
Guillaume Legendre
Affiliation:
Centro de Modelamiento Matemático – FONDAP, UMI 2807 CNRS-Universidad de Chile, Casilla 170 – Correo 3, Santiago, Chile. glegendre@dim.uchile.cl. : CEREMADE, UMR CNRS 7534, Université de Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775 Paris Cedex 16, France.
Takéo Takahashi
Affiliation:
Institut de Mathématiques Élie Cartan de Nancy, Université de Nancy-CNRS-INRIA, BP 239, 54506 Vandœuvre-lès-Nancy Cedex, France. takeo.takahashi@iecn.u-nancy.fr
Get access

Abstract

We propose a numerical scheme to compute the motion of a two-dimensional rigid body in a viscous fluid. Our method combines the method of characteristics with a finite element approximation to solve an ALE formulation of the problem. We derive error estimates implying the convergence of the scheme.

Keywords

Fluid-structure interactionincompressible Navier-Stokes equationsarbitrary Lagrangian EulerianLagrange-Galerkin method.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 42 , Issue 4 , July 2008 , pp. 609 - 644
Copyright
© EDP Sciences, SMAI, 2008

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Achdou, Y. and Guermond, J.-L., Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 37 (2000) 799–826. CrossRef
V.I. Arnold, Ordinary Differential Equations. Springer-Verlag, Berlin, Germany (1992).
S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15. Springer-Verlag, New York, USA (1994).
P.G. Ciarlet, Mathematical Elasticity, Vol. I: Three-Dimensional Elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam, Netherlands (1988).
Ciarlet, P.G. and Raviart, P.-A., Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg. 1 (1972) 217–249. CrossRef
Donea, J., Giuliani, S. and Halleux, J.P., An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Engrg. 33 (1982) 689–723. CrossRef
Duarte, F., Gormaz, R. and Natesan, S., Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries. Comput. Methods Appl. Mech. Engrg. 193 (2004) 4819–4836. CrossRef
Farhat, C., Lesoinne, M. and Maman, N., Mixed explicit/implicit time integration of coupled aeroelastic problems: three-field formulation, geometric conservation and distributed solution. Internat. J. Numer. Methods Fluids 21 (1995) 807–835 CrossRef
Fernández, M.A., Gerbeau, J.-F. and Grandmont, C., A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid. Internat. J. Numer. Methods Engrg. 69 (2007) 794–821. CrossRef
Formaggia, L. and Nobile, F., A stability analysis for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999) 105–132.
Gastaldi, L., A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001) 123–156.
Girault, V., López, H. and Maury, B., One time-step finite element discretization of the equation of motion of two fluid flows. Numer. Methods Partial Differ. Equ. 22 (2005) 680–707. CrossRef
Glowinski, R., Pan, T.-W., Hesla, T.I., Joseph, D.D. and Periaux, J., A distributed Lagrange multiplier/fictitious domain method for the simulation of flow around moving rigid bodies: application to particulate flow. Comput. Methods Appl. Mech. Engrg. 184 (2000) 241–267. CrossRef
C. Grandmont and Y. Maday, Fluid-structure interaction: a theoretical point of view, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London (2003) 1–22.
Grandmont, C., Guimet, V. and Maday, Y., Numerical analysis of some decoupling techniques for the approximation of the unsteady fluid structure interaction. Math. Models Methods Appl. Sci. 11 (2001) 1349–1377. CrossRef
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985).
Direct, H.H. Hu simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996) 335–352.
Hughes, T.J.R., Liu, W.K. and Zimmermann, T.K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Engrg. 29 (1981) 329–349. CrossRef
Inoue, I. and Wakimoto, M., On existence of solutions of the Navier-Stokes equation in a time dependent domain. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 24 (1977) 303–319.
Janela, J., Lefebvre, A. and Maury, B., A penalty method for the simulation of fluid-rigid body interaction. ESAIM: Proc. 14 (2005) 115–123. CrossRef
Lenoir, M., Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562–580. CrossRef
Maury, B., Characteristics ALE method for the unsteady 3D Navier-Stokes equations with a free surface. Int. J. Comput. Fluid Dyn. 6 (1996) 175–188. CrossRef
Maury, B., Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys. 156 (1999) 325–351. CrossRef
Maury, B. and Glowinski, R., Fluid-particle flow: a symmetric formulation. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1079–1084. CrossRef
J. Nitsche, Finite element approximations for solving the elastic problem, in Computing methods in applied sciences and engineering (Second Internat. Sympos., Versailles, 1975), Part 1, Lecture Notes in Econom. and Math. Systems 134, Springer-Verlag, Berlin, Germany (1976) 154–167.
Pironneau, O., On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38 (1982) 309–332. CrossRef
Quaini, A. and Quarteroni, A., A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method. Math. Models Methods Appl. Sci. 17 (2007) 957–983. CrossRef
Rannacher, R., On finite element approximation of general boundary value problems in nonlinear elasticity. Calcolo 17 (1980) 175–193. CrossRef
San Martín, J., Scheid, J.-F., Takahashi, T. and Tucsnak, M., Convergence of the Lagrange-Galerkin method for the equations modelling the motion of a fluid-rigid system. SIAM J. Numer. Anal. 43 (2005) 1539–1571.
J. San Martín, L. Smaranda and T. Takahashi, Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time. Prépublication de l'Institut Élie Cartan de Nancy 17 (2006) http://hal.archives-ouvertes.fr/hal-00275223/.
Süli, E., Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53 (1988) 459–483. CrossRef
Takahashi, T., Analysis of strong solutions for the equations modelling the motion of a rigid-fluid system in a bounded domain. Adv. Differential Equations 8 (2003) 1499–1532.