zbMATH — the first resource for mathematics

A discontinuous Galerkin discretization for solving the two-dimensional gas dynamics equations written under total Lagrangian formulation on general unstructured grids. (English) Zbl 1349.76278
Summary: Based on the total Lagrangian kinematical description, a discontinuous Galerkin (DG) discretization of the gas dynamics equations is developed for two-dimensional fluid flows on general unstructured grids. Contrary to the updated Lagrangian formulation, which refers to the current moving configuration of the flow, the total Lagrangian formulation refers to the fixed reference configuration, which is usually the initial one. In this framework, the Lagrangian and Eulerian descriptions of the kinematical and the physical variables are related by means of the Piola transformation. Here, we describe a cell-centered high-order DG discretization of the physical conservation laws. The geometrical conservation law, which governs the time evolution of the deformation gradient, is solved by means of a finite element discretization. This approach allows to satisfy exactly the Piola compatibility condition. Regarding the DG approach, it relies on the use of a polynomial space approximation which is spanned by a Taylor basis. The main advantage in using this type of basis relies on its adaptability regardless the shape of the cell. The numerical fluxes at the cell interfaces are computed employing a node-based solver which can be viewed as an approximate Riemann solver. We present numerical results to illustrate the robustness and the accuracy up to third-order of our DG method. First, we show its ability to accurately capture geometrical features of a flow region employing curvilinear grids. Second, we demonstrate the dramatic improvement in symmetry preservation for radial flows.

76M10 Finite element methods applied to problems in fluid mechanics
76N15 Gas dynamics (general theory)
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
82C40 Kinetic theory of gases in time-dependent statistical mechanics
PDF BibTeX Cite
Full Text: DOI
[1] Abgrall, R.; Loubère, R.; Ovadia, J., A Lagrangian discontinuous Galerkin-type method on unstructured meshes to solve hydrodynamics problems, Int. J. Numer. Methods Fluids, 44, 645-663, (2004) · Zbl 1067.76591
[2] Barlow, A. J., A compatible finite element multi-material ALE hydrodynamics algorithm, Int. J. Numer. Methods Fluids, 56, 953-964, (2008) · Zbl 1169.76030
[3] Barlow, A. J., A high order cell centred dual grid Lagrangian Godunov scheme, Comput. Fluids, 83, 15-24, (2013) · Zbl 1290.76078
[4] Bazilevs, Y.; Akkerman, I.; Benson, D. J.; Scovazzi, G.; Shashkov, M. J., Isogeometric analysis of Lagrangian hydrodynamics, J. Comput. Phys., 243, 224-243, (2013) · Zbl 1349.76178
[5] Boscheri, W.; Dumbser, M., Arbitrary-Lagrangian-Eulerian one-step weno finite volume schemes on unstructured triangular meshes, Commun. Comput. Phys., 14, 1174-1206, (2013) · Zbl 1388.65075
[6] Boutin, B.; Deriaz, E.; Hoch, P.; Navaro, P., Extension of ALE methodology to unstructured conical meshes, ESAIM Proc., 7, 1-10, (2011) · Zbl 1235.76079
[7] Breil, J.; Hallo, L.; Maire, P.-H.; Olazabal-Loumé, M., Hydrodynamic instabilities in axisymmetric geometry self-similar models and numerical simulations, Laser Part. Beams, 23, 155-160, (2005)
[8] Burton, D. E.; Carney, T. C.; Morgan, N. R.; Sambasivan, S. K.; Shashkov, M. J., A cell-centered Lagrangian Godunov-like method for solid dynamics, Comput. Fluids, 83, 33-47, (2013) · Zbl 1290.76095
[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.; Shashkov, M. J., Elimination of artificial grid distorsion and hourglass-type motions by means of Lagrangian subzonal masses and pressures, J. Comput. Phys., 142, 521-561, (1998) · Zbl 0932.76068
[11] Carré, G.; Delpino, S.; Després, B.; Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension, J. Comput. Phys., 228, 5160-5183, (2009) · Zbl 1168.76029
[12] Cheng, J.; Shu, C.-W., A high order ENO conservative Lagrangian type scheme for the compressible Euler equations, J. Comput. Phys., 227, 2, 1567-1596, (2007) · Zbl 1126.76035
[13] Cheng, J.; Shu, C.-W., A third-order conservative Lagrangian type scheme on curvilinear meshes for the compressible Euler equations, Commun. Comput. Phys., 4, 1008-1024, (2008) · Zbl 1364.76111
[14] Cockburn, B.; Hou, S.; Shu, C.-W., The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws IV: the multidimensional case, Math. Comput., 54, 545-581, (1990) · Zbl 0695.65066
[15] Cockburn, B.; Lin, S.-Y.; Shu, C.-W., TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems, J. Comput. Phys., 84, 90-113, (1989) · Zbl 0677.65093
[16] Després, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Ration. Mech. Anal., 178, 327-372, (2005) · Zbl 1096.76046
[17] Dobrev, V. A.; Ellis, T. E.; Kolev, Tz. V.; Rieben, R. N., Curvilinear finite elements for Lagrangian hydrodynamics, Int. J. Numer. Methods Fluids, 65, 11-12, 1295-1310, (2011) · Zbl 1255.76075
[18] Dobrev, V. A.; Ellis, T. E.; Kolev, Tz. V.; Rieben, R. N., High order curvilinear finite elements for Lagrangian hydrodynamics, SIAM J. Sci. Comput., 34, 606-641, (2012)
[19] Dobrev, V. A.; Ellis, T. E.; Kolev, Tz. V.; Rieben, R. N., High order curvilinear finite elements for axisymmetric Lagrangian hydrodynamics, Comput. Fluids, 58-69, (2013) · Zbl 1290.76061
[20] Dukowicz, J. K.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99, 115-134, (1992) · Zbl 0743.76058
[21] Dumbser, M.; Boscheri, W., High-order unstructured Lagrangian one-step weno finite volume schemes for non-conservative hyperbolic systems: applications to compressible multi-phase flows, Comput. Fluids, 86, 405-432, (2013) · Zbl 1290.76081
[22] Germain, P., Mécanique, vol. I, (1986), Ellipses
[23] Goad, W. B., WAT: A numerical method for two-dimensional unsteady fluid flow, (1960), Los Alamos National Laboratory, Technical Report LAMS 2365
[24] Gresho, P., On the theory of semi-implicit projection methods for viscous incompressible flow and its implementation via finite-element method that also introduces a nearly consistent mass matrix. part 2: applications, Int. J. Numer. Methods Fluids, (1990) · Zbl 0712.76035
[25] Gurtin, M. E.; Fried, E.; Anand, L., The mechanics and thermodynamics of continua, (2009), Cambridge University Press
[26] Jia, Z.; Zhang, S., A new high-order discontinuous Galerkin spectral finite element for Lagrangian gas dynamics in two-dimensions, J. Comput. Phys., 230, 7, 2496-2522, (2011) · Zbl 1316.76049
[27] Kamm, J. R.; Timmes, F. X., On efficient generation of numerically robust sedov solutions, (2007), Los Alamos National Laboratory, Technical Report LA-UR-07-2849
[28] Kidder, R. E., Laser-driven compression of hollow shells: power requirements and stability limitations, Nucl. Fusion, 1, 3-14, (1976)
[29] Kluth, G.; Després, B., Discretization of hyperelasticity on unstructured mesh with a cell-centered Lagrangian scheme, J. Comput. Phys., 229, 24, 9092-9118, (2010) · Zbl 1427.74029
[30] Kolev, Tz. V.; Rieben, R. N., A tensor artificial viscosity using a finite element approach, J. Comput. Phys., 228, 8336-8366, (2010) · Zbl 1287.76166
[31] Kuzmin, D., A vertex-based hierarchical slope limiter for p-adaptative discontinuous Galerkin methods, J. Comput. Appl. Math., 233, 3077-3085, (2009) · Zbl 1252.76045
[32] Lascaux, P., Application de la méthode des éléments finis en hydrodynamique bi-dimensionnelle utilisant LES variables de Lagrange, (1972), CEA-Centre d’Etudes de Limeil, Technical Report DO 058
[33] Lascaux, P., Application of the finite element method to 2D Lagrangian hydrodynamics, (Proceedings of the International Symposium on Finite Element Methods in Flow Problems, Swansea, Wales, Jan. 7-11, 1974, (1974)), 139-152
[34] Loubère, R., Une Méthode particulaire lagrangienne de type Galerkin discontinu. application à la Mécanique des fluides et l’interaction laser/plasma, (2002), Université Bordeaux I, PhD thesis
[35] Loubère, R.; Maire, P.-H.; Vàchal, P., 3D staggered Lagrangian hydrodynamics scheme with cell-centered Riemann solver-based artificial viscosity, Int. J. Numer. Methods Fluids, 72, 22-42, (2013)
[36] Luo, H.; Baum, J. D.; Löhner, R., A DG method based on a Taylor basis for the compressible flows on arbitrary grids, J. Comput. Phys., 227, 8875-8893, (2008) · Zbl 1391.76350
[37] Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. Comput. Phys., 228, 2391-2425, (2009) · Zbl 1156.76434
[38] Maire, P.-H., A high-order one-step sub-cell force-based discretization for cell-centered Lagrangian hydrodynamics on polygonal grids, Comput. Fluids, 46, 1, 479-485, (2011)
[39] Maire, P.-H., Contribution to the numerical modeling of inertial confinement fusion, (2011), Habilitation à Diriger des Recherches, Bordeaux University, available at
[40] Maire, P.-H.; Abgrall, R.; Breil, J.; Ovadia, J., A cell-centered Lagrangian scheme for two-dimensional compressible flow problems, SIAM J. Sci. Comput., 29, 1781-1824, (2007) · Zbl 1251.76028
[41] Maire, P.-H.; Loubère, R.; Vàchal, P., Staggered Lagrangian discretization based on cell-centered Riemann solver and associated hydrodynamics scheme, Commun. Comput. Phys., 10, 940-978, (2011) · Zbl 1373.76138
[42] Mazeran, C., Sur la structure mathématique et l’approximation numérique de l’hydrodynamique lagrangienne bidimensionelle, (2007), Université Bordeaux I, PhD thesis
[43] 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
[44] Del Pino, S., A curvilinear finite-volume method to solve compressible gas dynamics in semi-Lagrangian coordinates, C. R. Math., 348, 1027-1032, (2010) · Zbl 1426.76652
[45] Plohr, B. J.; Sharp, D. H., A conservative Eulerian formulation of the equations for elastic flows, Adv. Appl. Math., 9, 481-499, (1988) · Zbl 0663.73012
[46] 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
[47] Scovazzi, G.; Christon, M. A.; Hughes, T. J.R.; Shadid, J. N., Stabilized shock hydrodynamics: I. A Lagrangian method, Comput. Methods Appl. Mech. Eng., 196, 923-966, (2007) · Zbl 1120.76334
[48] 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
[49] Shu, C.-W.; Osher, S., Efficient implementation of essentially non-oscillatory shock-capturing schemes, J. Comput. Phys., 77, 439-471, (1988) · Zbl 0653.65072
[50] Sod, G. A., A survey of several finite difference methods for systems of non-linear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31, (1978) · Zbl 0387.76063
[51] Vilar, F., Cell-centered discontinuous Galerkin discretization for two-dimensional Lagrangian hydrodynamics, Comput. Fluids, 64, 64-73, (2012) · Zbl 1365.76129
[52] Vilar, F.; Maire, P.-H.; Abgrall, R., Cell-centered discontinuous Galerkin discretizations for two-dimensional scalar conservation laws on unstructured grids and for one-dimensional Lagrangian hydrodynamics, Comput. Fluids, 46, 1, 498-604, (2010) · Zbl 1433.76093
[53] Wilkins, M. L., Calculation of elastic-plastic flows, (Methods in Computational Physics, vol. 3, (1964), Academic Press), 211-263
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.