A cell-centered Lagrangian hydrodynamics scheme on general unstructured meshes in arbitrary dimension. (English) Zbl 1168.76029

Summary: We describe a cell-centered Godunov scheme for Lagrangian gas dynamics on general unstructured meshes in arbitrary dimension. The construction of the scheme is based upon the definition of some geometric vectors which are defined on a moving mesh. The finite volume solver is node-based and compatible with mesh displacement. We also discuss boundary conditions. Numerical results on basic 3D tests show the efficiency of this approach. We consider a quasi-incompressible test problem for which our nodal solver gives very good results if compared with other Godunov solvers. We briefly discuss the compatibility with ALE and/or AMR techniques, and detail the coefficients of the isoparametric element in the appendix.


76M12 Finite volume methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
Full Text: DOI


[1] F.L. Addessio, J.R. Baumgardner, Dukowicz, N.L.J.K., Johnson, B.A. Kashiwa, R.M. Rauenzahn, C. Zemach. CAVEAT: A Computer Code for Fluid Dynamics Problems with Large Distortion and Internal Slip. Technical report, Los Alamos National Laboratory LA-10613, 1990.
[2] Anderson, R.W.; Elliott, N.S.; Pember, R.B., An arbitrary lagrangian – eulerian method with adaptive mesh refinement for the solution of Euler equations, J. comput. phys., 199, 2, (2005) · Zbl 1126.76348
[3] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. method appl. mech. eng., 99, 235-394, (1992) · Zbl 0763.73052
[4] D.E Burton, Multidimensional discretization of conservation laws for unstructured polyhedral grids. Technical Report, Lawrence Livermore National Laboratory, UCRL-JC-118306, 1994.
[5] J. Campbell, M.J. Shashkov, The compatible Lagrangian hydrodynamics algorithm on unstructured grids. Technical Report, Los Alamos National Labs, LA-UR-00-323, 2000. · Zbl 1150.76433
[6] 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
[7] Caramana, E.J.; Roulscup, 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
[8] Caramana, E.J.; Loubere, R., Curl-Q: a vorticity damping artificial viscosity for Lagrangian hydrodynamics calculations, J. comput. phys., 215, 2, 385-391, (2006) · Zbl 1173.76380
[9] 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
[10] Cheng, J.; Shu, C.W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. comput. phys., 227, 1567-1596, (2007) · Zbl 1126.76035
[11] R.B. Christensen, Godunov methods on a staggered mesh, an improved artificial viscosity. Tech. Rep. UCRL-JC 105269 LLNL, 1990.
[12] Després, B.; Mazeran, C., Symmetrization of Lagrangian gas dynamics and Lagrangian solvers, Comptes rendus académie des sciences (Paris), 331, 475-480, (2003) · Zbl 1293.76089
[13] Després, B.; Mazeran, C., Lagrangian gas dynamics in 2D and Lagrangian systems, Arch. rat. mech. anal., 178, 327-372, (2005) · Zbl 1096.76046
[14] Dukowicz, J.K., Efficient volume computation for three-dimensional hexahedral cells, J. comput. phys., 74, 493-496, (1988) · Zbl 0644.65019
[15] Dukowicz, J.K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. comput. phys., 99, (1992) · Zbl 0743.76058
[16] Flanaghan, D.P.; Belytshko, T., A uniform strain hexaedron and quadrilateral and orthogonal hourglass control, Int. J. numer. method eng., 17, 679-706, (1982)
[17] Godunov, S., A difference scheme for numerical computation of discontinuous solution of hydrodynamic equations, Math. sib., 47, (1959)
[18] Godunov, S., Reminiscences about difference schemes, J. comput. phys., 153, 6-25, (1999) · Zbl 0936.65106
[19] Hui, W.H., Unified coordinate system in computational fluid dynamics, Commun. comput. phys., 2, 577-610, (2006)
[20] J.R. Leveque, Numerical methods for conservation laws. Lectures in Mathematics, Birkhauser, 1991. · Zbl 0847.65053
[21] Loubere, R.; Ovadia, J.; Abgrall, R., A Lagrangian discontinuous Galerkin type method on unstructured meshes to solve hydrodynamics problems, Int. J. numer. method fluids, (2000) · Zbl 1067.76591
[22] Maire, P.H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for 2D compressible flow problems, SIAM. J. sci. comput., 29, (2007) · Zbl 1251.76028
[23] P.H. Maire, A high-order cell centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 2009, online. · Zbl 1156.76434
[24] P.H. Maire, B. Nkonga, Multi-scale Godunov type method for cell-centered discrete Lagrangian hydrodynamics, J. Comput. Phys., 2008, online (october). · Zbl 1156.76039
[25] C. Mazeran, Sur la structure mathématique et l’approximation numérique de l’hydrodynamique lagrangienne bidimensionnelle. Ph.D. Thesis, Université de Bordeaux I, November 2007.
[26] von Neumann, J.; Richtmyer, R.D., A method for the calculation of hydrodynamics shocks, J. appl. phys., 21, 232-237, (1950) · Zbl 0037.12002
[27] Pracht, W.E., Calculating three-dimensional fluid flows at all speeds with an eulerian – lagrangian computing mesh, J. comput. phys., 17, 132-159, (1975) · Zbl 0294.76016
[28] B. Scheurer, Quelques schémas numériques pour l’hydrodynamique Lagrangienne. Technical Report, Commissariat à l’Energie Atomique, CEA-R-5942, 2000.
[29] Scovazzi, G., A discourse on Galilean invariance, SUPG stabilization, and the variational multi-scale framework, Comput. method appl. mech. eng., 54, 6-8, 1108-1132, (2007) · Zbl 1120.76333
[30] Scovazzi, G., Galilean invariance and stabilized methods for compressible flows, Int. J. numer. method fluids, 54, 6-8, 757-778, (2007) · Zbl 1207.76094
[31] Scovazzi, G., Stabilized shock hydrodynamics: II. design and physical interpretation of the SUPG operator for Lagrangian computations, Comput. method appl. mech. eng., 196, 4-6, 967-978, (2007) · Zbl 1120.76332
[32] Scovazzi, G.; Christon, M.A.; Hugues, T.J.R.; Shadid, J.N., Stabilized shock hydrodynamics: I.A Lagrangian method, Comput. method appl. mech. eng., 196, 4-6, 923-966, (2007) · Zbl 1120.76334
[33] Scovazzi, G.; Love, E.; Shashkov, M.J., Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: theoretical framework and two-dimensional computations, Comput. method appl. mech. eng., 197, 1056-1079, (2008) · Zbl 1169.76396
[34] Shashkov, M., Conservative finite difference methods on general grids, (1996), CRC Press · Zbl 0844.65067
[35] Sod, G.A., A survey of finite difference methods for systems of nonlinear conservation laws, J. comput. phys., 27, 1-31, (1978) · Zbl 0387.76063
[36] L.M. Taylor, D.P. Flanaghan, PRONTO 3D, A three-dimensional transient solid dynamics program. Technical Report, Sandia National Laboratories Sand87-1912, 1989.
[37] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics, (1997), Springer · Zbl 0888.76001
[38] Whalen, P.P., Algebraic limitations on two-dimensional hydrodynamics simulations, J. comput. phys., 124, 46-54, (1996) · Zbl 0849.76079
[39] Wilkins, M., Use of artificial viscosity in multidimensional shock wave problems, J. comput. phys., 3, 36, 281-303, (1980) · Zbl 0436.76040
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.