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Stabilized shock hydrodynamics. I: A Lagrangian method. (English) Zbl 1120.76334
Summary: A new SUPG-stabilized formulation for Lagrangian hydrodynamics of materials satisfying Mie-Grüneisen equation of state is proposed. It allows the use of simplex-type (triangular/tetrahedral) meshes as well as the more commonly used brick-type (quadrilateral/hexahedral) meshes. The proposed method yields a globally conservative formulation, in which equal-order interpolation (P1 or Q1 isoparametric finite elements) is applied to velocities, displacements, and pressure. As a direct consequence, and in contrast to traditional cell-centered multidimensional hydrocode implementations, the proposed formulation allows a natural representation of the pressure gradient on element interiors. The SUPG stabilization involves additional design requirements, specific to the Lagrangian formulation. A discontinuity capturing operator in the form of a Noh-type viscosity with artificial heat flux is used to preserve stability and smoothness of the solution in shock regions. A set of challenging shock hydrodynamics benchmark tests for the Euler equations of gas dynamics in one and two space dimensions is presented. In the two-dimensional case, computations performed on quadrilateral and triangular grids are analyzed and compared. These results indicate that the new formulation is a promising technology for hydrocode applications.

MSC:
76M10 Finite element methods applied to problems in fluid mechanics
76L05 Shock waves and blast waves in fluid mechanics
Software:
HE-E1GODF
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[1] Aziz, A.K.; Monk, P., Continuous finite elements in space and time for the heat equation, Math. comput., 144, 70-97, (1998)
[2] Belytschko, T.; Liu, W.K.; Moran, B., Nonlinear finite elements for continua and structures, (2000), John Wiley & Sons New York · Zbl 0959.74001
[3] Benson, D.J., A new two-dimensional flux-limited shock viscosity for impact calculations, Comput. methods appl. mech. engrg., 93, 39-95, (1991) · Zbl 0850.73050
[4] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. methods appl. mech. engrg., 99, 235-394, (1992) · Zbl 0763.73052
[5] Bochev, P.B.; Gunzburger, M.D.; Shadid, J.N., Stability of the SUPG finite element method for transient advection – diffusion problems, Comput. methods appl. mech. engrg., 193, 2301-2323, (2004) · Zbl 1067.76563
[6] Brooks, A.N.; Hughes, T.J.R., Streamline upwind/petrov – galerkin formulations for convection dominated flows with particular emphasis on the incompressible navier – stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[7] J.C. Campbell, M.J. Shashkov, A tensor artificial viscosity using a mimetic finite difference algorithm, Technical report, LA-UR-00-2290, Los Alamos National Laboratory, 2000. · Zbl 1002.76082
[8] Campbell, J.C.; Shashkov, M.J., A tensor artificial viscosity using a mimetic finite difference algorithm, J. comput. phys., 172, 739-765, (2001) · Zbl 1002.76082
[9] Caramana, E.J.; Shashkov, M.J.; Whalen, P.P., Formulations of artificial viscosity for multi-dimensional shock wave computations, J. comput. phys., 144, 70-97, (1998) · Zbl 1392.76041
[10] M.A. Christon, Pressure gradient approximations for staggered-grid finite element hydrodynamics, Int. J. Numer. Methods Fluids, in preparation.
[11] Dettmer, W.; Perić, D., An analysis of the time integration algorithms for the finite element solutions of incompressible navier – stokes equations based on a stabilised formulation, Comput. methods appl. mech. engrg., 192, 9-10, 1177-1226, (2003) · Zbl 1091.76521
[12] Estep, D.; French, D.A., Global error control for the continuous Galerkin finite element method for ordinary differential equations, Rairo, 28, 815-852, (1994) · Zbl 0822.65054
[13] French, D.A., A space – time finite element method for the wave equation, Comput. methods appl. mech. engrg., 107, 145-157, (1993) · Zbl 0787.65069
[14] French, D.A., Continuous Galerkin finite element methods for a forward-backward heat equation, Numer. methods partial differen. equat., 15, 491-506, (1999)
[15] French, D.A.; Jensen, S., Long time behaviour of arbitrary order continuous time Galerkin schemes for some one-dimensional phase transition problems, IMA J. numer. anal., 14, 421-442, (1994) · Zbl 0806.65132
[16] French, D.A.; Peterson, T.E., A continuous space – time finite element method for the wave equation, Math. comput., 65, 491-506, (1996) · Zbl 0846.65048
[17] Hauke, G., Simple stabilizing matrices for the computation of compressible flows in primitive variables, Comput. methods appl. mech. engrg., 190, 6881-6893, (2001) · Zbl 0996.76047
[18] Hauke, G.; Hughes, T.J.R., A unified approach to compressible and incompressible flows, Comput. methods appl. mech. engrg., 113, 389-396, (1994) · Zbl 0845.76040
[19] Hauke, G.; Hughes, T.J.R., A comparative study of different sets of variables for solving compressible and incompressible flows, Comput. methods appl. mech. engrg., 153, 1-44, (1998) · Zbl 0957.76028
[20] Hughes, T.J.R., The finite element method: linear static and dynamic finite element analysis, (1987), Prentice-Hall Englewood Cliffs, NJ, (Dover reprint, 2000)
[21] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid-scale models, bubbles and the origin of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[22] Hughes, T.J.R.; Engel, G.; Mazzei, L.; Larson, M., The continuous Galerkin method is locally conservative, J. comput. phys., 163, 2, 467-488, (2000) · Zbl 0969.65104
[23] Hughes, T.J.R.; Feijóo, G.R.; Mazzei, L.; Quincy, J.-B., The variational multiscale method – a paradigm for computational mechanics, Comput. methods appl. mech. engrg., 166, 3-24, (1998) · Zbl 1017.65525
[24] 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
[25] Hughes, T.J.R.; Franca, L.P.; Mallet, M., A new finite element formulation for computational fluid dynamics: VI. convergence analysis of the generalized SUPG formulation for linear time-dependent multidimensional advective – diffusive systems, Comput. methods appl. mech. engrg., 63, 97-112, (1987) · Zbl 0635.76066
[26] Hughes, T.J.R.; Mallet, M., A new finite element formulation for computational fluid dynamics: III. the generalized streamline operator for multidimensional advective – diffusive systems, Comput. methods appl. mech. engrg., 58, 305-328, (1986) · Zbl 0622.76075
[27] Hughes, T.J.R.; Tezduyar, T.E., Finite element methods for first-order hyperbolic system with particular emphasis on the compressible Euler equations, Comput. methods appl. mech. engrg., 45, 217-284, (1984) · Zbl 0542.76093
[28] Hulme, B.L., Discrete Galerkin and related one-step methods for ordinary differential equations, Math. comput., 26-120, 881-891, (1972) · Zbl 0272.65056
[29] Jamet, P., Stability and convergence of a generalized crank – nicolson scheme on a variable mesh for the heat equation, SIAM J. numer. anal., 17, 4, 530-539, (1980) · Zbl 0454.65073
[30] Johnson, C.; Nävert, U.; Pitkäranta, J., Finite element methods for linear hyperbolic problems, Comput. methods appl. mech. engrg., 45, 285-312, (1984) · Zbl 0526.76087
[31] Johnson, C.; Szepessy, A., On the convergence of a finite element method for a nonlinear hyperbolic conservation law, Math. comput., 49, 427-444, (1987) · Zbl 0634.65075
[32] Johnson, C.; Szepessy, A.; Hasnbo, P., On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws, Math. comput., 54, 107-129, (1990) · Zbl 0685.65086
[33] Klaas, O.; Maniatty, A.; Shephard, M.S., A stabilized mixed finite element method for finite elasticity. formulation for linear displacement and pressure interpolation, Comput. methods appl. mech. engrg., 180, 65-79, (1999) · Zbl 0959.74066
[34] Kuropatenko, V.F., On difference methods for the equations of hydrodynamics, () · Zbl 0885.76055
[35] R. Loubère, Investigation of triangular meshes for compressible Lagrangian hydrodynamics, Technical report, LA-UR-05-2937, Los Alamos National Laboratory, May 2005.
[36] Maniatty, A.; Liu, Y., Stabilized finite element method for viscoplastic flow: formulation with state variable evolution, Int. J. numer. methods engrg., 56, 185-209, (2003) · Zbl 1116.74431
[37] Maniatty, A.; Liu, Y.; Klaas, O.; Shephard, M.S., Stabilized finite element method for viscoplastic flow: formulation and a simple progressive solution strategy, Comput. methods appl. mech. engrg., 190, 4609-4625, (2001) · Zbl 1059.74056
[38] Maniatty, A.; Liu, Y.; Klaas, O.; Shephard, M.S., Higher order stabilized finite element method for hyperelastic finite deformation, Comput. methods appl. mech. engrg., 191, 1491-1503, (2002) · Zbl 1098.74704
[39] L.G. Margolin, A centered artificial viscosity for cells with large aspect ratios, Technical report, UCRL-53882, Lawrence Livermore National Laboratory, 1988.
[40] Marsden, J.E.; Hughes, T.J.R., Mathematical foundations of elasticity, (1994), Dover Mineola, NY
[41] Noh, W.F., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. comput. phys., 72, 78-120, (1987) · Zbl 0619.76091
[42] G. Scovazzi, Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations, Comput. Methods Appl. Mech. Engrg., in press, doi:10.1016/j.cma.2006.08.009. · Zbl 1120.76332
[43] G. Scovazzi, Multiscale methods in science and engineering. PhD thesis, Mechanical Engineering Department, Stanford University, September 2004. Available from: <http://www.cs.sandia.gov/ gscovaz/pubs.html>.
[44] Sedov, L.I., Similarity and dimensional methods in mechanics, (1959), Academic Press New York · Zbl 0121.18504
[45] Shakib, F.; Hughes, T.J.R., A new finite element formulation for computational fluid dynamics: IX. Fourier analysis of space – time Galerkin/least-squares algorithms, Comput. methods appl. mech. engrg., 87, 35-58, (1991) · Zbl 0760.76051
[46] Shakib, F.; Hughes, T.J.R.; Johan, Z., A new finite element formulation for computational fluid dynamics: X. the compressible Euler and navier – stokes equations, Comput. methods appl. mech. engrg., 89, 141-219, (1991) · Zbl 0838.76040
[47] Sod, G.A., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. comput. phys., 27, 1-31, (1978) · Zbl 0387.76063
[48] Szepessy, A., Convergence of a shock-capturing streamline diffusion finite element method for a scalar conservation law in two space dimensions, Math. comput., 53, 527-545, (1989) · Zbl 0679.65072
[49] Tezduyar, T.E., Computation of moving boundaries and interfaces and stabilization parameters, Int. J. numer. methods fluids, 43, 555-575, (2003) · Zbl 1032.76605
[50] Tezduyar, T.E., Finite element methods for fluid dynamics with moving boundaries and interfaces, () · Zbl 0848.76036
[51] Tezduyar, T.E.; Senga, M., Stabilization and shock-capturing parameters in SUPG formulation of compressible flows, Comput. methods appl. mech. engrg., 195, 1621-1632, (2006) · Zbl 1122.76061
[52] Toro, E.F., Riemann solvers and numerical methods for fluid dynamics: A practical introduction, (1999), Springer-Verlag Berlin, Heidelberg · Zbl 0923.76004
[53] von Neumann, J.; Richtmyer, R.D., A method for the numerical computation of hydrodynamic shocks, J. appl. phys., 21, 232-237, (1950) · Zbl 0037.12002
[54] Wilkins, M.L., Use of artificial viscosity in multidimensional shock wave problems, J. comput. phys., 36, 281-303, (1979)
[55] Woodward, P.R., Trade-offs in designing explicit hydrodynamics schemes for vector computers, ()
[56] Woodward, P.R.; Colella, P., The numerical simulation of two-dimensional fluid flow with strong shocks, J. comput. phys., 54, 115-173, (1984) · Zbl 0573.76057
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