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An arbitrary Lagrangian-Eulerian finite element method for solving three-dimensional free surface flows. (English) Zbl 0948.76043
Summary: This paper discusses numerical solution of unsteady three-dimensional free surface flows. The governing equilibrium equations are written in the framework of the arbitrary Lagrangian-Eulerian kinematic description. The corresponding variational formulation is established afterwards. Since the variational problems are nonlinear with respect to the moving coordinates, we derive a second-order approximate variational problem after a consistent linearization of the referential motion. Stability of the discrete formulations is ensured with the help of a new stabilization method. A robust preconditioned GMRES algorithm is then used to solve the resulting set of nonlinear equations. Finally, the computational algorithms are assessed through numerical studies of various problems: a large sloshing flow in a three-dimensional reservoir, a discharge flow from a reservoir, simulation of a liquid vortex produced inside a cylindrical container with a disk rotating at the bottom, and a three-dimensional practical hydraulic problem.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76B07 Free-surface potential flows for incompressible inviscid fluids
76B47 Vortex flows for incompressible inviscid fluids
76B10 Jets and cavities, cavitation, free-streamline theory, water-entry problems, airfoil and hydrofoil theory, sloshing
Software:
ILUT
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References:
[1] Cruyer, C.W., A bibliography of free boundary problems, ()
[2] Nichols, B.D.; Hirt, C.W., J. comput. phys., (1971)
[3] Nichols, B.D.; Hirt, C.W., ()
[4] Miyata, H., Finite difference simulation of breaking waves, J. comput. phys., 65, 179-214, (1986) · Zbl 0591.76024
[5] Harlow, F.H.; Welch, J.E., Phys. fluids, 8, 21-82, (1965)
[6] Hirt, C.W.; Nichols, B.D., Volume of fluids method for the dynamics of free boundaries, J. comput. phys., 39, 201-225, (1981) · Zbl 0462.76020
[7] Soulaimani, A., Nouveaux aspects de l’application de la méthode des éléments finis en hydrodynamique, ()
[8] Hirt, C.W.; Amsden, A.A.; Cook, J.L., An arbitrary Lagrangian-Eulerian computing method of all speeds, J. comput. phys., 14, 227-253, (1974) · Zbl 0292.76018
[9] Donéa, J., Arbitrary Lagrangian-Eulerian finite element methods, Comput. methods transient anal., 1, 473-516, (1983)
[10] Hughes, T.J.R.; Liu, W.K.; Zimmermann, T.K., Lagrangian-Eulerian finite element formulation for incompressible viscous flows, (), 329-349, Aug. 7-11, 1978 · Zbl 0482.76039
[11] Van Goethem, G., Description mixte d’Euler Lagrange et modèle d’eléments finisà domaine variable, ()
[12] Lynch, D.R.; Gray, W.G., Finite element simulation of flow in deforming regions, J. comput. phys., 36, 135-153, (1980) · Zbl 0433.76018
[13] Soulaimani, A., Contribution à la résolution numérique des problèmes hydrodynamiques à surface libre, ()
[14] Soulaimani, A.; Fortin, M.; Ouellet, Y.; Dhatt, G., Finite element simulation of two- and three-dimensional free surface flows, Comput. methods appl. mech. engrg., 3, 265-296, (1991) · Zbl 0761.76037
[15] Ogawa, S.; Ishiguro, T., A method for computing flow fields around moving bodies, J. comput. phys., 69, 49-68, (1987) · Zbl 0614.76023
[16] Brezzi, F., On the existence, uniqueness and approximations of saddle point problems arising from Lagrangian multipliers, RAIRO anal. numer., 2, 129-151, (1974) · Zbl 0338.90047
[17] Babuska, I., Error bounds for finite element method, Numer. math., 16, 322-333, (1971) · Zbl 0214.42001
[18] Hughes, T.J.R.; Franca, L.P.; Hulbert, G.M., A new finite element formulation for computational fluid dynamics: VIII. the Galerkin-least-squares method for advective-diffusive equations, Comput. methods appl. mech. engrg., 73, 173-189, (1989) · Zbl 0697.76100
[19] Arnold, D.N.; Brezzi, F.; Fortin, M., A stable finite element for the Stokes equations, Calcolo, 21, 337-344, (1984) · Zbl 0593.76039
[20] Soulaimani, A.; Fortin, M.; Ouellet, Y.; Dhatt, G.; Bertrand, F., Simple continuous pressure elements for two- and three-dimensional incompressible flows, Comput. methods appl. mech. engrg., 62, 47-69, (1987) · Zbl 0608.76024
[21] Brezzi, F.; Bristeau, M.O.; Franca, L.P.; Mallet, M.; Rogé, G., A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. method appl. mech. engrg., 96, 117-129, (1992) · Zbl 0756.76044
[22] Baiocchi, C.; Brezzi, F.; Franca, L.P., Virtual bubbles and Galerkin-least-squares type methods (ga.L.S.), Comput. methods appl. mech. engrg., 105, 125-141, (1993) · Zbl 0772.76033
[23] Saad, Y.; Schultz, M.H., GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. statist. comput., 856-869, (1986) · Zbl 0599.65018
[24] Douglas, J.; Wang, J., An absolutely stabilized finite element method for the Stokes problem, Math. comput., 52, 495-508, (1989) · Zbl 0669.76051
[25] Franca, L.P.; Frey, S., Stabilized finite element methods: II. the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 99, 209-233, (1992) · Zbl 0765.76048
[26] Pierre, R., Optimal selection of the bubble function in the stabilization of the P1-P1 element for the Stokes problem, SIAM J. numer. anal., 32, 4, (1995) · Zbl 0833.76037
[27] Habashi, W.G.; Robichaud, M.; Nguyen, V.N.; Ghali, W.S.; Fortin, M.; Liu, J.W.H., Large-scale computational fluid dynamkics by the finite element method, Int. J. numer. methods fluids, 1083-1105, (1994) · Zbl 0807.76034
[28] Dutto, L.C.; Habashi, W.G.; Robichaud, M.P.; Fortin, M., A method or finite element parallel viscous compressible flow calculations, Int. J. numer. methods fluids, 19, 275-294, (1994) · Zbl 0815.76042
[29] Saad, Y., ILUT: a dual threshold incomplete ILU factorization, Numer. linear algebra applic., 1, 387-402, (1994) · Zbl 0838.65026
[30] Vatistas, G.H., A note on liquid vortex sloshing and Kelvin’s equilibria, J. fluid. mech., 217, 241-248, (1990)
[31] Kozel, V.; Vatistas, G.H.; Wang, J., The influence of inertial waves on the structure of vortex cores, (), 92
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