A Godunov-like point-centered essentially Lagrangian hydrodynamic approach. (English) Zbl 1351.76071
Summary: We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshes do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH and recent Lagrangian SGH. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes that remove a volume error in the PCH discretization. A 2-stage Runge-Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
References:
| [1] | Addessio, F.; Baumgardner, J.; Dukowicz, J.; Johnson, N.; Kashiwa, B.; Rauenzahn, R.; Zemach, C., CAVEAT: a computer code for fluid dynamic problems with large distortion and internal slip (1990), Los Alamos National Laboratory Report LA-10613-MS-REV.1 |
| [2] | Addessio, F.; Cline, M.; Dukowicz, J., A general topology, Godunov method, Comput. Phys. Commun., 48, 65-73 (1988) |
| [3] | Barlow, A.; Roe, P., A cell centred Lagrangian Godunov scheme for shock hydrodynamics, Comput. Fluids, 46, 133-136 (2013) · Zbl 1431.76006 |
| [4] | Barlow, A., A high order cell centred dual grid Lagrangian Godunov scheme, Comput. Fluids, 83, 15-24 (2013) · Zbl 1290.76078 |
| [5] | Barth, T., An introduction to recent developments in theory and numerics for conservation laws, (Kroner, D.; Ohlberger, M.; Rohde, C., Lecture Notes in Computational Science and Engineering (1998), Springer), 274-275 |
| [6] | Burbeau-Augoula, A., A node-centered artificial viscosity method for two-dimensional Lagrangian hydrodynamics calculations on a staggered grid, Commun. Comput. Phys., 8, 877-900 (2010) · Zbl 1364.76110 |
| [7] | Burton, D., Temporary Quadrilateral Subzoning, Lawrence Livermore National Laboratory Notes (1992) |
| [8] | Burton, D.; Carney, T.; Morgan, N.; Sambasivan, S.; Shashkov, M., A cell centered Lagrangian Godunov-like method of solid dynamics, Comput. Fluids, 83, 33-47 (2013) · Zbl 1290.76095 |
| [9] | Burton, D., Multidimensional discretization of conservation laws for unstructured polyhedral grids (1994), Lawrence Livermore National Laboratory, UCRL-JC-118306 |
| [10] | Caramana, E.; Shashkov, M., Elimination of artificial grid distortion and hourglass-type motions by means of Lagrangian subzonal masses and pressures, J. Appl. Phys., 142, 521-561 (1998) · Zbl 0932.76068 |
| [11] | Caramana, E.; Burton, D.; Shashkov, M.; Whalen, P. P., The construction of compatible hydrodynamic algorithms utilizing conservation of total energy, J. Appl. Phys., 146, 227-262 (1998) · Zbl 0931.76080 |
| [12] | Caramana, E.; Rousculp, C.; Burton, D., A compatible energy and symmetry preserving Lagrangian hydrodynamics algorithm in three dimensional cartesian geometry, J. Comput. Phys., 157, 89-119 (2000) · Zbl 0961.76049 |
| [13] | Christensen, R., Godunov methods on staggered mesh an improved artificial viscosity (1990), Lawrence Livermore Report UCRL-JC-105269 |
| [14] | Clark, R., The evolution of HOBO, Comput. Phys. Commun., 48, 61-64 (1988) |
| [15] | Crowley, W., Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics (1971), Springer-Verlag: Springer-Verlag New York/Berlin · Zbl 0215.58001 |
| [16] | Crowley, W., Free-Lagrangian methods for compressible hydrodynamics in two space dimensions, (Proceedings of the First International Conference on Free-Lagrange Methods (1985), Springer-Verlag: Springer-Verlag New York/Tokyo), 1-21 · Zbl 0581.76075 |
| [17] | Cooper, P., Explosive Engineering, 188-189 (1996), Wiley-VCH |
| [18] | Davis, S., Simplified second-order Godunov-type methods, SIAM J. Sci. Stat. Comput., 3, 445-473 (1988) · Zbl 0645.65050 |
| [19] | Despres, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. Ration. Mech. Anal., 178, 327-372 (2005) · Zbl 1096.76046 |
| [20] | Despres, B.; Lagoutière, F., Numerical resolution of a two-component compressible fluid model with interfaces, Prog. Comput. Fluid Dyn., 7, 295-310 (2007) · Zbl 1152.76443 |
| [21] | Doebling, S., Impact of numerical treatment of shocks on verification assessment of the FLAG Lagrangian hydrodynamics code using the Sedov problem (2014), Los Alamos National Laboratory Report |
| [22] | Dobrev, V.; Ellis, T.; Kolev, T.; Rieben, R., Curvillinear finite elements for Lagrangian hydrodynamics, Int. J. Numer. Methods Fluids, 65, 1295-1310 (2011) · Zbl 1255.76075 |
| [23] | Dukowicz, J., A general, non-iterative Riemann solver for Gounov’s method, J. Comput. Phys., 61, 119-137 (1985) · Zbl 0629.76074 |
| [24] | Dukowicz, J.; Meltz, B., Vorticity errors in multidimensional Lagrangian codes, J. Comput. Phys., 99, 115 (1992) · Zbl 0743.76058 |
| [25] | Esmond, M.; Thurber, A., One dimensional Lagrangian hydrocode development (2013), LA-UR-13-26506 |
| [26] | Flanagan, D.; Belytschko, T., A uniform strain hexahedron and quadrilateral with orthogonal hourglass control, Int. J. Numer. Methods Eng., 17, 679-706 (1981) · Zbl 0478.73049 |
| [27] | Fritts, M.; Boris, J., The Lagrangian solution of transient problems in hydrodynamics using a triangular mesh, J. Comput. Phys., 31, 173-215 (1979) · Zbl 0403.76033 |
| [28] | Gittings, M., TRIX: a free-Lagrangian hydroxide, (Proceedings of the Advances in the Free-Lagrange Method Conference (1990), Springer-Verlag: Springer-Verlag New York/Berlin), 28-36 |
| [29] | Godunov, S.; Zabordine, A.; Ivanov, M.; Kraiko, A.; Prokopov, G., Résolution numérique des problèmes multidimensionnels de la dynamique des gaz (1979), Mir · Zbl 0421.65056 |
| [30] | Godunov, S., Reminiscences about difference schemes, J. Comput. Phys., 153, 6-25 (1999) · Zbl 0936.65106 |
| [31] | Hirt, C.; Nichols, B., Volume-of-fluid (VOF) method for dynamics of free boundaries, J. Comput. Phys., 39, 201-225 (1981) · Zbl 0462.76020 |
| [32] | Loubère, R., Investigation of triangular meshes for compressible Lagrangian hydrodynamics (2005), LA-UR-05-2937 |
| [33] | Loubère, R.; Maire, P.; Vachal, P., A second-order compatible staggered Lagrangian hydrodynamics scheme using a cell-centered multidirectional Riemann solver, Proc. Comput. Sci., 1, 1, 1931-1939 (2010) · Zbl 1432.76206 |
| [34] | Loubère, R.; Maire, P.; Váchal, P., 3D staggered Lagrangian hydrodynamics with cell-centered Riemann solver-based artificial viscosity, Int. J. Numer. Methods Fluids, 72, 22-42 (2013) · Zbl 1455.76164 |
| [35] | Loubère, R.; Maire, P.; Váchal, P., Formulation of a staggered two-dimensional Lagrangian scheme by means of cell-centered approximately Riemann solver, (Numerical Mathematics and Advanced Applications (2009)), 617-625 · Zbl 1432.76206 |
| [36] | Loubère, R., Contribution au domaine des mèthodes numèriques Lagrangiennes et arbitrary-Lagrangian-Eulerian (2013), University of Toulouse, Habilitation à Diriger Des Recherche |
| [37] | Maire, P.; 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 |
| [38] | Maire, P., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured mesh, J. Comput. Phys., 228, 7, 6882-6915 (2009) · Zbl 1261.76021 |
| [39] | Maire, P.; Loubère, R.; Vachal, P., Staggered Lagrangian discretization based on cell-centered Riemann solver and associated hydrodynamics scheme, Commun. Comput. Phys., 10, 940-978 (2011) · Zbl 1373.76138 |
| [40] | Margolin, L.; Pyun, J., A method for treating hourglass patterns (1987), LA-UR-87-439 |
| [41] | Margolin, L., A centered artificial viscosity for cells with large aspect ratios (1988), UCRL-S3882 |
| [42] | Marsh, S., LASL Shock Hugoniot Data (1980), University of California Press |
| [43] | Morgan, N., A dissipation model for staggered grid Lagrangian hydrodynamics, Comput. Fluids, 83, 48-57 (2013) · Zbl 1290.76103 |
| [44] | Morgan, N.; Kenamond, M.; Burton, D.; Carney, T.; Ingraham, D., A contact surface algorithm for cell-centered Lagrangian hydrodynamics, J. Comput. Phys., 250, 527-554 (2013) |
| [45] | Morgan, N.; Lipnikov, K.; Burton, D.; Kenamond, M., A Lagrangian staggered grid Godunov-like approach for hydrodynamics, J. Comput. Phys., 259, 568-597 (2014) · Zbl 1349.76365 |
| [46] | Morgan, N.; Waltz, J.; Burton, D.; Canfield, T.; Risinger, L.; Wohlbier, J.; Charest, M., A Godunov-like point-centered ALE finite element hydrodynamic approach, (MultiMat 2013. MultiMat 2013, San Francisco, CA (2013)) |
| [47] | Morgan, N., A new liquid-vapor phase transition technique for the level set method (2005), Georgia Institute of Technology, Ph.D. thesis |
| [48] | Noh, W., Errors for calculations of strong shocks using an artificial viscosity and an artificial heat flux, J. Appl. Phys., 72, 78-120 (1987) · Zbl 0619.76091 |
| [49] | Noh, W.; Woodward, P., SLIC (simple line interface calculation), (Lecture Notes in Physics, vol. 59 (1976), Springer), 330-340 · Zbl 0382.76084 |
| [50] | Osher, S.; Fedkiw, R., Level set methods: an overview and some recent results, J. Comput. Phys., 169, 463-502 (2001) · Zbl 0988.65093 |
| [51] | Scovazzi, G.; Shadid, J.; Love, E.; Rider, W., A conservative nodal variational multiscale method for Lagrangian shock hydrodynamics, Comput. Methods Appl. Mech. Eng., 199, 3059-3100 (2010) · Zbl 1225.76204 |
| [52] | Scovazzi, G., Lagrangian shock hydrodynamics on tetrahedral meshes: a stable and accurate variational multi scale approach, J. Comput. Phys., 231, 8029-8069 (2013) |
| [53] | Sahota, M., An explicit-implicit solution of the hydrodynamic and radiation equations, (Proceedings of the Advances in the Free-Lagrange Method Conference (1990), Springer-Verlag: Springer-Verlag New York/Berlin), 28-36 |
| [54] | Sedov, L., Similarity and Dimensional Methods in Mechanics (1959), Academic Press · Zbl 0121.18504 |
| [55] | Sod, G., A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys., 27, 1-31 (1978) · Zbl 0387.76063 |
| [56] | Tryggvason, G.; Bunner, B.; Esmaeeli, A.; Juric, D.; Al-Rawahi, A.; Tauber, W.; Han, J.; Nas, S.; Jan, Y., A front-tracking method for the computations of multiphase flow, J. Comput. Phys., 169, 708-759 (2001) · Zbl 1047.76574 |
| [57] | 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 |
| [58] | von Neumann, J.; Richtmyer, R., A method for the calculation of hydrodynamics shocks, J. Appl. Phys., 21, 232-237 (1950) · Zbl 0037.12002 |
| [59] | Waltz, J., Microfluidics simulation using adaptive unstructured grids, Int. J. Numer. Methods Fluids, 46, 939-960 (2004) · Zbl 1060.76580 |
| [60] | Waltz, J.; Canfield, T.; Morgan, N.; Risinger, L.; Wohlbier, J., Verification of a three-dimensional unstructured finite element method using analytic and manufactured solutions, Comput. Fluids, 81, 57-67 (2013) |
| [61] | Waltz, J.; Morgan, N.; Canfield, T.; Charest, M.; Risinger, L.; Wohlbier, J., A three-dimensional finite element arbitrary Lagrangian-Eulerian method for shock hydrodynamics on instructed grids, Comput. Fluids, 92, 172-187 (2014) · Zbl 1391.76364 |
| [62] | Wilkins, M., Calculation of elastic-plastic flow, (Methods in Computational Physics, vol. 3 (1964), Academic Press: Academic Press New York), 211-263 |
| [63] | Wilkins, M., Use of artificial viscosity in multidimensional shock wave problems, J. Comput. Phys., 36, 281-303 (1980) · Zbl 0436.76040 |
| [64] | Youngs, D., Time-dependent multi-material flow with large fluid distortion, (Numerical Methods for Fluid Dynamics (1982), Academic Press), 273-285 · Zbl 0537.76071 |
| [65] | Youngs, D., An interface tracking method for a 3D Eulerian hydrodynamics code (1984), AWE technical report 44/92/25 |
| [66] | Zukas, J.; Walters, W., Explosive Effects and Applications, 105 (1998), Springer |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
