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Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics. (English) Zbl 1156.76039
Summary: This work presents a multi-dimensional cell-centered unstructured finite volume scheme for the solution of multimaterial compressible fluid flows written in Lagrangian formalism. This formulation is considered in the arbitrary Lagrangian-Eulerian framework with the constraint that the mesh velocity and the fluid velocity coincide. The link between the vertex velocity and the fluid motion is obtained by a formulation of the momentum conservation in a class of multi-scale encased volumes around mesh vertices. The vertex velocity is derived with a nodal Riemann solver constructed in such a way that the mesh motion and the face fluxes are compatible. Finally, the resulting scheme conserves both momentum and total energy, and it satisfies a semi-discrete entropy inequality. The numerical results obtained for some classical 2D and 3D hydrodynamic test cases show the robustness and accuracy of the proposed algorithm.

76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
Full Text: DOI
[1] Abgrall, R.; Loubére, R.; Ovadia, J., A Lagrangian discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, Int. J. numer. meth. fluids, 44, 645-663, (2004) · Zbl 1067.76591
[2] F.L. Adessio, D.E. Carroll, K.K. Dukowicz, J.N. Johnson, B.A. Kashiwa, M.E. Maltrud, H.M. Ruppel, Caveat: a computer code for fluid dynamics problems with large distortion and internal slip, Technical Report LA-10613-MS, Los Alamos National Laboratory, 1986.
[3] T.J. Barth, Numerical methods for conservation laws on structured and unstructured meshes, Technical Report, VKI Lecture Series, 2003.
[4] T.J. Barth, D.C. Jespersen, The design and application of upwind schemes on unstructured meshes, in: AIAA Paper 89-0366, 27th Aerospace Sciences Meeting, Reno, Nevada, 1989.
[5] D.E. Burton, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Technical Report UCRL-JC-118306, Lawrence Livermore National Laboratory, 1994.
[6] Campbell, J.C.; Shashkov, M.J., A tensor artificial viscosity using a mimetic finite difference algorithm, J. comput. phys., 172, 4, 739-765, (2001) · Zbl 1002.76082
[7] Campbell, J.C.; Shashkov, M.J., A compatible Lagrangian hydrodynamics algorithm for unstructured grids, Selçuk J. appl. math., 4, 53-70, (2003) · Zbl 1150.76433
[8] Caramana, E.J.; Shashkov, M.J.; Whalen, P.P., Formulations of artificial viscosity for multidimensional shock wave computations, J. comput. phys., 144, 70-97, (1998) · Zbl 1392.76041
[9] Caramana, E.J.; Burton, D.E.; Shashkov, M.J.; Whalen, P.P., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. comput. phys., 146, 227-262, (1998) · Zbl 0931.76080
[10] Caramana, E.J.; Rousculp, C.L.; Burton, D.E., A compatible, energy and symmetry preserving Lagrangian hydrodynamics algorithm in three-dimensional Cartesian geometry, J. comput. phys., 157, 89-119, (2000) · Zbl 0961.76049
[11] Caramana, E.J.; Shashkov, M.J., Elimination of artificial grid distortion and hourglass – type motions by means of Lagrangian subzonal masses and pressures, J. comput. phys., 142, 521-561, (1998) · Zbl 0932.76068
[12] Després, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. ration. mech. anal., 178, 327-372, (2005) · Zbl 1096.76046
[13] Dukowicz, J.K., A general non-iterative Riemann solver for godunov’s method, J. comput. phys., 61, 119-137, (1984) · Zbl 0629.76074
[14] Dukowicz, J.K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. comput. phys., 99, 115-134, (1992) · Zbl 0743.76058
[15] Dukowicz, J.K.; Cline, M.C.; Addessio, F.S., A general topology method, J. comput. phys., 82, 29-63, (1989) · Zbl 0665.76032
[16] S.K. Godunov, A. Zabrodine, M. Ivanov, A. Kraiko, G. Prokopov, Résolution numérique des problémes multidimensionnels de la dynamique des gaz, Mir, 1979.
[17] Hirt, C.W.; Amsden, A.; Cook, J.L., An arbitrary-lagrangian – eulerian computing method for all flow speeds, J. comput. phys., 14, 227-253, (1974) · Zbl 0292.76018
[18] Hui, W.H.; Li, P.Y.; Li, Z.W., A unified coordinate system for solving the two-dimensional Euler equations, J. comput. phys., 153, 596-637, (1999) · Zbl 0969.76061
[19] J.R. Kamm, F.X. Timmes, On efficient generation of numerically robust Sedov solutions, Technical Report LA-UR-07-2849, Los Alamos National Laboratory, 2007.
[20] Loubére, R.; Shashkov, M.J., A subcell remapping method on staggered polygonal grids for arbitrary – lagrangian – eulerian methods, J. comput. phys., 23, 155-160, (2004)
[21] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for compressible flow problems, SIAM J. sci. comp., 29, 4, 1781-1824, (2007) · Zbl 1251.76028
[22] Nkonga, B., On the conservative and accurate CFD approximations for moving meshes and moving boundaries, Comput. methods appl. mech. eng., 190, 13-14, 1801-1825, (2000) · Zbl 1010.76063
[23] Nkonga, B.; Guillard, H., Godunov type method on non-structured meshes for three-dimensional moving boundary problems, Comput. methods appl. mech. eng., 113, 1-2, 183-204, (1994) · Zbl 0846.76060
[24] Noh, W.F., Errors for calculations of strong shocks using artificial viscosity and an artificial heat flux, J. comput. phys., 72, 78-120, (1987) · Zbl 0619.76091
[25] Richtmyer, R.D.; Morton, K.W., Difference methods for initial-value problems, (1967), John Wiley · Zbl 0155.47502
[26] Scovazzi, G., Stabilized shock hydrodynamics: II. design and physical interpretation of the SUPG operator for Lagrangian computations, Comput. methods appl. mech. eng., 196, 966-978, (2007) · Zbl 1120.76332
[27] Scovazzi, G.; Christon, M.A.; R Hughes, T.J.; Shadid, J.N., Stabilized shock hydrodynamics: I. A Lagrangian method, Comput. methods appl. mech. eng., 196, 923-966, (2007) · Zbl 1120.76334
[28] Scovazzi, G.; Love, E.; Shashkov, M.J., Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: theoretical framework and two-dimensional computations, Comput. methods appl. mech. eng., 197, 1056-1079, (2008) · Zbl 1169.76396
[29] G. Scovazzi, E.L. Love, M. Shashkov, A multi-scale Q1/P0 approach to Lagrangian shock hydrodynamics, Technical Report SAND2007-1423, Sandia National Laboratories, 2007. · Zbl 1169.76396
[30] Tadmor, E., Entropy stability theory for difference approximations of nonlinear conservation laws and related time-dependent problems, Acta numer., 12, 451-512, (2003) · Zbl 1046.65078
[31] Venkatakrishnan, V., Convergence to steady state solutions of the Euler equations on unstructured grids with limiters, J. comput. phys., 118, 120-130, (1995) · Zbl 0858.76058
[32] von Neumann, J.; Richtmyer, R.D., A method for the numerical calculations of hydrodynamical shocks, J. appl. phys., 21, 232-238, (1950) · Zbl 0037.12002
[33] Wilkins, M.L., Calculation of elastic plastic flow, Methods comput. phys., 3, 211-263, (1964)
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