Metric-based mesh adaptation for 2D Lagrangian compressible flows. (English) Zbl 1391.76536

Summary: We present a method to compute compressible flows in 2D. It uses two steps: a Lagrangian step and a metric-based triangular mesh adaptation step. Computational mesh is locally adapted according to some metric field that depends on physical or geometrical data. This mesh adaptation step embeds a conservative remapping procedure to satisfy consistency with Euler equations. The whole method is no more Lagrangian.
After describing mesh adaptation patterns, we recall the metric formalism. Then, we detail an appropriate remapping procedure which is first-order and relies on exact intersections.
We give some hints about the parallel implementation. Finally, we present various numerical experiments which demonstrate the good properties of the algorithm.


76M25 Other numerical methods (fluid mechanics) (MSC2010)
76M12 Finite volume methods applied to problems in fluid mechanics
65M08 Finite volume methods for initial value and initial-boundary value problems involving PDEs
65M50 Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs
65Y05 Parallel numerical computation
76N15 Gas dynamics (general theory)


ReALE; Trix
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. meth. fluids, 44, 645-663, (2004) · Zbl 1067.76591
[2] F. Alauzet, P.J. Frey, Estimation d’erreur géométrique et métriques anisotropes pour l’adaptation de maillage. Partie I: aspects théoriques, Technical Report 4753, INRIA, 2003.
[3] Alauzet, F.; Frey, P.-J., Anisotropic mesh adaptation for CFD computations, Comput. meth. appl. mech. eng., 194, 48-49, 5068-5082, (2005) · Zbl 1092.76054
[4] Alauzet, F.; Li, X.; Seol, E.S.; Shephard, M.S., Parallel anisotropic 3d mesh adaptation by mesh modification, Eng. comput., 21, 3, 247-258, (2006)
[5] Anderson, R.W.; Elliott, N.S.; Pember, R.B., An arbitrary Lagrangian-Eulerian method with adaptive mesh refinement for the solution of the Euler equations, J. comput. phys., 199, 2, 598-617, (2004) · Zbl 1126.76348
[6] Bailey, D.; Berndt, M.; Kucharik, M.; Shashkov, M., Reduced-dissipation remapping of velocity in staggered arbitrary lagrangian – eulerian methods, J. comput. appl. math., 233, 12, 3148-3156, (2010), Finite Element Methods in Engineering and Science (FEMTEC 2009) · Zbl 1252.76059
[7] B.G. Baumgart, A polyhedron representation for computer vision, in: AFIPS National Computer Conference, 1975, pp. 589-596.
[8] Benson, D.J., Computational methods in Lagrangian and Eulerian hydrocodes, Comput. meth. appl. mech. eng., 99, 235-394, (1992) · Zbl 0763.73052
[9] Bourouchaki, H.; George, P.-L.; Hecht, F.; Laug, P.; Saltel, E., Delaunay mesh generation governed by metric specifications, Finite elem. anal. design, 25, 61-83, (1997) · Zbl 0897.65076
[10] D.E. Burton, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Lawrence Livermore National Laboratory preprint UCRL-JC-118306, 22 August 1994.
[11] Campbell, J.C.; Shashkov, M.J., A compatible Lagrangian hydrodynamics algorithm for unstructured grids, Selcuk J. appl. math., 4, 2, 53-70, (2003) · Zbl 1150.76433
[12] Caramana, E.J.; Burton, D.E.; Shashkov, M.J., The construction of compatible hydrodynamics algorithms utilizing conservation of total energy, J. comput. phys., 146, 1, 227-262, (1998) · Zbl 0931.76080
[13] Carré, G.; Del Pino, S.; Després, B.; Labourasse, E., A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension, J. comput. phys., 228, 14, 5160-5183, (2009) · Zbl 1168.76029
[14] Chacon-Rebollo, T.; Hou, T.-Y., A Lagrangian finite element method for the 2-D Euler equations, Commun. pure appl. math., XLIII, 735-767, (1990) · Zbl 0705.76059
[15] Cherfils, C.; Hermeline, F., Diagonal swap procedure and characterization of 2D-Delaunay triangulations, Math. model. numer. anal., 24, 5, 613-626, (1990) · Zbl 0711.65099
[16] Crowley, W.P., Free-Lagrange methods for compressible hydrodynamics in two space dimensions, (), 1-21 · Zbl 0581.76075
[17] Després, B.; Mazeran, C., Lagrangian gas dynamics in two dimensions and Lagrangian systems, Arch. rat. mech. anal., 178, 327-372, (2005) · Zbl 1096.76046
[18] C. Dobrzynski, Adaptation de Maillage anisotrope 3D et application l’aéro-thermique des bâtiments, PhDi thesis, Université Pierre et Marie Curie—Paris VI, Novembre 2005.
[19] Dobrzynski, C.; Frey, P.J.; Mohammadi, B.; Pironneau, O., Fast and accurate simulations of air-cooled structures, Comput. meth. appl. mech. eng., 195, 23-24, 3168-3180, (2006) · Zbl 1176.76062
[20] Dyadechko, V.; Shashkov, M., Reconstruction of multi-material interfaces from moment data, J. comput. phys., 227, 5361-5384, (2008) · Zbl 1220.76048
[21] Eltgroth, P.G., Asynchronous 3D free Lagrange code, (), 76-77
[22] M.W. Evans, F.H. Harlow, The Particle-In-Cell method for hydrodynamics calculations, Technical report, Los Alamos Scientific Laboratory, 1956.
[23] Frey, P.J.; George, P.-L., Mesh generation, application to finite element, (2000), Hermés Science Publications Paris, Oxford
[24] Fritts, M., Three-dimensional algorithms for grid restructuring in free-Lagrange calculations, (), 122-144
[25] ()
[26] Gittings, M.L., TRIX: a free-Lagrangian hydrocode, (), 28-36
[27] Grospellier, G.; Lelandais, B., The arcane development framework, (), 1-11
[28] Haas, J.-F.; Sturtevant, B., Interaction of weak-shock waves, J. fluid. mech., (1987)
[29] Hirt, C.W.; Amsden, A.A.; Cook, J.L., An arbitrary lagrangian – eulerian computing method for all flow speeds, J. comput. phys., 14, 41-76, (1974) · Zbl 0292.76018
[30] Hirt, C.W.; Nichols, B.D., Volume of fluid (VOF) method for the dynamics of free boundaries, J. comput. phys., 39, 1, 201-225, (1981) · Zbl 0462.76020
[31] P. Hoch, An Arbitrary Lagrangian-Eulerian strategy to solve compressible fluid flows, <http://hal.archives-ouvertes.fr/hal-00366858/PDF/ale2d.pdf>, March 2009.
[32] P. Hoch, Semi-conformal polygonal mesh adaptation seen as discontinuous grid velocity formulation for ALE simulations, MULTI-MAT’09, <http://www.eucentre.it/media/presentazioni_congresso/hoch.pdf>, September 2009.
[33] Loubére, R.; Maire, P.-H.; Shashkov, M.J.; Breil, J.; Galera, S., Reale: a reconnection-based arbitrary-lagrangian – eulerian method, J. comput. phys., 229, 4724-4761, (2010) · Zbl 1305.76067
[34] Loubére, R.; Shashkov, M.J., A subcell remapping method on staggered polygonal grids for arbitrary-lagrangian – eulerian, J. comput. phys., 209, 105-138, (2005) · Zbl 1329.76236
[35] Maire, P.-H., A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes, J. comput. phys., 228, 7, 2391-2425, (2009) · Zbl 1156.76434
[36] Maire, P.-H.; Nkonga, B., Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics, J. comput. phys., 228, 799-821, (2009) · Zbl 1156.76039
[37] Mäntylä, M., An introduction to solid modeling, (1988), Computer Science Press
[38] Morrell, J.M.; Sweby, P.K.; Barlow, A., A cell by cell anisotropic adaptive mesh ALE scheme for the numerical solution of the Euler equations, J. comput. phys., 226, 1, 1152-1180, (2005) · Zbl 1310.76096
[39] (), June
[40] van Leer, B., Towards the ultimate conservative difference scheme, V. A second-order sequel to godunov’s method, J. comput. phys., 32, 1, 101-136, (1979) · Zbl 1364.65223
[41] VanderHeyden, W.B.; Kashiwa, B.A., Compatible fluxes for Van leer advection, J. comput. phys, 146, 1, 1-28, (1998) · Zbl 0932.76058
[42] Youngs, D.L., Time-dependent multi-material flow with large fluid distortion, () · Zbl 0537.76071
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