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.
Mots clés : fluid-structure interaction, incompressible Navier-Stokes equations, arbitrary lagrangian eulerian, Lagrange-Galerkin method
@article{M2AN_2008__42_4_609_0, author = {Legendre, Guillaume and Takahashi, Tak\'eo}, title = {Convergence of a {Lagrange-Galerkin} method for a fluid-rigid body system in {ALE} formulation}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {609--644}, publisher = {EDP-Sciences}, volume = {42}, number = {4}, year = {2008}, doi = {10.1051/m2an:2008020}, mrnumber = {2437776}, zbl = {1142.76032}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an:2008020/} }
TY - JOUR AU - Legendre, Guillaume AU - Takahashi, Takéo TI - Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2008 SP - 609 EP - 644 VL - 42 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2008020/ DO - 10.1051/m2an:2008020 LA - en ID - M2AN_2008__42_4_609_0 ER -
%0 Journal Article %A Legendre, Guillaume %A Takahashi, Takéo %T Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation %J ESAIM: Modélisation mathématique et analyse numérique %D 2008 %P 609-644 %V 42 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2008020/ %R 10.1051/m2an:2008020 %G en %F M2AN_2008__42_4_609_0
Legendre, Guillaume; Takahashi, Takéo. Convergence of a Lagrange-Galerkin method for a fluid-rigid body system in ALE formulation. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 4, pp. 609-644. doi : 10.1051/m2an:2008020. http://www.numdam.org/articles/10.1051/m2an:2008020/
[1] Convergence analysis of a finite element projection/Lagrange-Galerkin method for the incompressible Navier-Stokes equations. SIAM J. Numer. Anal. 37 (2000) 799-826. | MR | Zbl
and ,[2] Ordinary Differential Equations. Springer-Verlag, Berlin, Germany (1992). | MR
,[3] The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15. Springer-Verlag, New York, USA (1994). | MR | Zbl
and ,[4] Vol. I: Three-Dimensional Elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam, Netherlands (1988). | MR | Zbl
, ,[5] Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg. 1 (1972) 217-249. | MR | Zbl
and ,[6] An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Engrg. 33 (1982) 689-723. | Zbl
, and ,[7] Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries. Comput. Methods Appl. Mech. Engrg. 193 (2004) 4819-4836. | MR | Zbl
, and ,[8] 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 | MR | Zbl
, and ,[9] 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. | MR
, and ,[10] A stability analysis for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999) 105-132. | MR | Zbl
and ,[11] A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001) 123-156. | MR | Zbl
,[12] One time-step finite element discretization of the equation of motion of two fluid flows. Numer. Methods Partial Differ. Equ. 22 (2005) 680-707. | MR | Zbl
, and ,[13] 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. | MR | Zbl
, , , and ,[14] Fluid-structure interaction: a theoretical point of view, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London (2003) 1-22. | MR
and ,[15] Numerical analysis of some decoupling techniques for the approximation of the unsteady fluid structure interaction. Math. Models Methods Appl. Sci. 11 (2001) 1349-1377. | MR
, and ,[16] Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985). | MR | Zbl
,[17] Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996) 335-352. | Zbl
,[18] Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Engrg. 29 (1981) 329-349. | MR | Zbl
, and ,[19] 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. | MR | Zbl
and ,[20] A penalty method for the simulation of fluid-rigid body interaction. ESAIM: Proc. 14 (2005) 115-123. | MR | Zbl
, and ,[21] Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562-580. | MR | Zbl
,[22] Characteristics ALE method for the unsteady 3D Navier-Stokes equations with a free surface. Int. J. Comput. Fluid Dyn. 6 (1996) 175-188.
,[23] Direct simulations of 2D fluid-particle flows in biperiodic domains. J. Comput. Phys. 156 (1999) 325-351. | MR | Zbl
,[24] Fluid-particle flow: a symmetric formulation. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1079-1084. | MR | Zbl
and ,[25] 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. | MR | Zbl
,[26] On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38 (1982) 309-332. | MR | Zbl
,[27] A semi-implicit approach for fluid-structure interaction based on an algebraic fractional step method. Math. Models Methods Appl. Sci. 17 (2007) 957-983. | MR
and ,[28] On finite element approximation of general boundary value problems in nonlinear elasticity. Calcolo 17 (1980) 175-193. | MR | Zbl
,[29] 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. | MR | Zbl
, , and ,[30] 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/.
, and ,[31] Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53 (1988) 459-483. | MR | Zbl
,[32] 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. | MR | Zbl
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