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.
@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: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {42},
year = {2008},
pages = {609-644},
doi = {10.1051/m2an:2008020},
mrnumber = {2437776},
zbl = {1142.76032},
language = {en},
url = {http://dml.mathdoc.fr/item/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: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 42 (2008) pp. 609-644. doi : 10.1051/m2an:2008020. http://gdmltest.u-ga.fr/item/M2AN_2008__42_4_609_0/
[1] and , 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 1740383 | Zbl 0966.76041
[2] , Ordinary Differential Equations. Springer-Verlag, Berlin, Germany (1992). | MR 1162307
[3] and , The Mathematical Theory of Finite Element Methods, Texts in Applied Mathematics 15. Springer-Verlag, New York, USA (1994). | MR 1278258 | Zbl 0804.65101
[4] , , Vol. I: Three-Dimensional Elasticity, Studies in Mathematics and its Applications 20. North-Holland Publishing Co., Amsterdam, Netherlands (1988). | MR 936420 | Zbl 0648.73014
[5] and , Interpolation theory over curved elements, with applications to finite element methods. Comput. Methods Appl. Mech. Engrg. 1 (1972) 217-249. | MR 375801 | Zbl 0261.65079
[6] , and , An arbitrary Lagrangian-Eulerian finite element method for transient dynamic fluid-structure interactions. Comput. Methods Appl. Mech. Engrg. 33 (1982) 689-723. | Zbl 0508.73063
[7] , and , Arbitrary Lagrangian-Eulerian method for Navier-Stokes equations with moving boundaries. Comput. Methods Appl. Mech. Engrg. 193 (2004) 4819-4836. | MR 2097758 | Zbl 1112.76388
[8] , and , 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 1368695 | Zbl 0865.76038
[9] , and , 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 2284413
[10] and , A stability analysis for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 7 (1999) 105-132. | MR 1699243 | Zbl 0942.65113
[11] , A priori error estimates for the Arbitrary Lagrangian Eulerian formulation with finite elements. East-West J. Numer. Math. 9 (2001) 123-156. | MR 1836870 | Zbl 0988.65082
[12] , and , 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 2212232 | Zbl 1089.76032
[13] , , , and , 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 1764191 | Zbl 0970.76057
[14] and , Fluid-structure interaction: a theoretical point of view, in Fluid-structure interaction, Innov. Tech. Ser., Kogan Page Sci., London (2003) 1-22. | MR 2076281
[15] , and , 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 1859827 | Zbl pre01882825
[16] , Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics 24. Pitman (Advanced Publishing Program), Boston, USA (1985). | MR 775683 | Zbl 0695.35060
[17] , Direct simulation of flows of solid-liquid mixtures. Int. J. Multiphase Flow 22 (1996) 335-352. | Zbl 1135.76442
[18] , and , Lagrangian-Eulerian finite element formulation for incompressible viscous flows. Comput. Methods Appl. Mech. Engrg. 29 (1981) 329-349. | MR 659925 | Zbl 0482.76039
[19] and , 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 481649 | Zbl 0381.35066
[20] , and , A penalty method for the simulation of fluid-rigid body interaction. ESAIM: Proc. 14 (2005) 115-123. | MR 2226806 | Zbl 1079.76043
[21] , Optimal isoparametric finite elements and error estimates for domains involving curved boundaries. SIAM J. Numer. Anal. 23 (1986) 562-580. | MR 842644 | Zbl 0605.65071
[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 1727335 | Zbl 0958.76045
[24] and , Fluid-particle flow: a symmetric formulation. C. R. Acad. Sci. Paris Sér. I Math. 324 (1997) 1079-1084. | MR 1451255 | Zbl 0880.76045
[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 455781 | Zbl 0346.65058
[26] , On the transport-diffusion algorithm and its applications to the Navier-Stokes equations. Numer. Math. 38 (1982) 309-332. | MR 654100 | Zbl 0505.76100
[27] and , 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 2334549 | Zbl pre05176130
[28] , On finite element approximation of general boundary value problems in nonlinear elasticity. Calcolo 17 (1980) 175-193. | MR 615816 | Zbl 0468.73091
[29] , , and , 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 2182139 | Zbl 1099.76037
[30] , and , 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/.
[31] , Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations. Numer. Math. 53 (1988) 459-483. | MR 951325 | Zbl 0637.76024
[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 2029294 | Zbl 1101.35356