ReALE: A reconnection-based arbitrary-Lagrangian-Eulerian method

Journal of Computational Physics

Volumn 229, Issue 12, 2010, Pages 4724-4761

  Loubère, Raphaël ^a   Maire, Pierre Henri ^b   Shashkov, Mikhail ^c   Breil, Jérôme ^b   Galera, Stéphane ^b  

^a INSTITUT DE MATHÉMATIQUES DE TOULOUSE   (France)

^b CENTRE LASERS INTENSES ET APPLICATIONS   (France)

^c LOS ALAMOS NATIONAL LABORATORY   (United States)

Author keywords

Arbitrary Lagrangian Eulerian; Cell centered scheme; Compressible flow; Lagrangian hydrodynamics; Mesh reconnection; Multi dimensional unstructured polygonal mesh; Staggered scheme; Voronoi mesh

Indexed keywords

COMPRESSIBLE FLOW; LAGRANGE MULTIPLIERS; MESH GENERATION;

ARBITRARY LAGRANGIAN EULERIAN; CELL-CENTERED SCHEME; LAGRANGIAN HYDRODYNAMICS; MESH RECONNECTION; MULTI DIMENSIONAL; MULTI-DIMENSIONAL UNSTRUCTURED POLYGONAL MESH; POLYGONAL MESHES; STAGGERED SCHEMES; VORONOI; VORONOI MESH;

NUMERICAL METHODS;

EID: 77951532030     PISSN: 00219991     EISSN: 10902716     Source Type: Journal    
DOI: 10.1016/j.jcp.2010.03.011     Document Type: Article

References (129)

1. F.L. Adessio, D.E. Carroll, K.K. Dukowicz, J.N. Johnson, B.A. Kashiwa, M.E. Maltrud, H.M. Ruppel, CAVEAT: a computer code for fluid dynamics problems with large distortion and internal slip, Technical Report LA-10613-MS, Los Alamos National Laboratory, 1986.
2. Ahn H.T., Shashkov M. Multi-material interface reconstruction on generalized polyhedral meshes. J. Comput. Phys. 2007, 226:2096-2132.
3. 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. 2004, 199:598-617.
4. Andrews M.J. Accurate computation of convective transport in transient two-phase flow. Int. J. Numer. Meth. Fluid 1995, 21(3):205-222.
5. N.V. Ardelyan, V.I. Bychkov, K.V. Kosmachevskii, M.N. Sablin, Numerical simulating 2D plasma dynamics problems on the base of implicit free-Lagrange method, in: Proceedings of WEHSFF 2007 (ECCOMAS Thematic Conference) - WEST-EAST HIGH SPEED FLOW FIELD CONFERENCE, November 19-22, 2007, Moscow, Russia, 2007. Available on web: <>. http://wehsff.imamod.ru/pages/s7.htm.
6. Ardelyan N.V., Kosmatchevskii K.V. Implicit free-Lagrange method for computing two-dimensional magnetogasdynamic flows. Comput. Math. Model. 1995, 6(4):209-224.
7. Aurenhammer F. Voronoi diagrams - a survey of fundamental geometric data structures. ACM Comput. Surveys 1991, 23(3):345-405.
8. F. Aurenhammer, R. Klein, Voronoi diagrams, Technical Report 198, Fern Universität Hagen, Department of Computer Science, Germany, 1996. Available on web: <>. http://www.pi6.fernuni-hagen.de/publ/tr198.pdf.
9. Baines M.J. Moving Finite Elements 1994, Oxford Science Publications, Clarendon Press, Oxford.
10. Ball G.J. A free-Lagrange method for unsteady compressible flow: simulation of a confined cylindrical blast wave. Shock Waves 1996, 5:311-325.
11. G.J. Ball, Sliding interfaces in a 2D free-Lagrange Godunov scheme on a Voronoi mesh, in: Workshop on Numerical Methods for Multi-material Flows, Institut Henri Poincaré, Paris, September 23-25, 2002. Available on web: <>. http://www.ann.jussieu.fr/~despres/ABSTRACT/gjball_abs.pdf.
12. Ball G.J., Howell B.P., Leighton T.G., Schofield M.J. Shock-induced collapse of a cylindrical air cavity in water: a free-Lagrange simulation. Shock Waves 2000, 10:265-276.
13. Barabanov R.A., Butnev O.I., Volkov S.G., Jogov B.M., Pronin V.A. Method Meduza-3D for computation of three-dimensional problems of gas dynamics on irregular meshes (In Russian). Quest. Atomic Sci. Tech. Ser.: Math. Mod. Phys. Process. 2005, 2:15-26.
14. A.J. Barlow, ALE and AMR mesh refinement techniques for multi-material hydrodynamics problems, ICFD Workshop on Mesh Refinement Techniques 7th December 2005, <>. http://www.icfd.rdg.ac.uk/Workshops/AMR/meshreftechf.pdf.
15. A.J. Barlow, Challenges and recent progress in developing numerical methods for multi-material ALE hydrocodes, ICFD 25 Year Anniversary Conference 15-16th September 2008, Oxford University, <>. http://www.icfd.rdg.ac.uk/ICFD25/Talks/ABarlow.pdf.
16. Barlow A.J. A compatible finite element multi-material ALE hydrodynamics algorithm. Int. J. Numer. Meth. Fluid 2008, 56:953-964.
17. T.J. Barth, Numerical methods for gasdynamic systems on unstructured meshes, in: D. Kroner, M. Ohlberger, C. Rohde (Eds.), An Introduction to Recent Developments in Theory and Numerics for Conservation Laws, Proceedings of the International School on Theory and Numerics for Conservation Laws, Lecture Notes in Computational Science and Engineering, Springer, Berlin, 1997, pp. 195-284.
18. Bauer A.L., Burton D.E., Caramana E.J., LoubFre R., Shashkov M.J., Whalen P.P. The internal consistency, stability, and accuracy of the discrete, compatible formulation of Lagrangian hydrodynamics. J. Comput. Phys. 2006, 218(2):572-593.
19. Benson D.J. An efficient, accurate, simple ALE method for nonlinear finite element programs. Comput. Meth. Appl. Mech. Eng. 1989, 72:305-350.
20. Benson D.J. Computational methods in Lagrangian and Eulerian hydrocodes. Comput. Meth. Appl. Mech. Eng. 1992, 99:235-394.
21. Benson D.J. Momentum advection on a staggered mesh. J. Comput. Phys. 1992, 100:143-162.
22. Bower A.F. Applied Mechanics of Solids 2009, CRC Press.
23. J.U. Brackbill, J.J. Monaghan (Eds.), Proceedings of the workshop on particle methods in fluid dynamics and plasma physics, Comput. Phys. Commun. 8(1), 1987.
24. Brackbill J.U., Saltzman J.S. Adaptive zoning for singular problems in two dimensions. J. Comput. Phys. 1982, 46:342-368.
25. Burton D.E. Exact conservation of energy and momentum in staggered-grid hydrodynamics with arbitrary connectivity. Advances in the Free-Lagrange Method 1990, Springer-Verlag, New York.
26. D.E. Burton, Consistent finite-volume discretization of hydrodynamics conservation laws for unstructured grids, Report UCRL-JC-118788, Lawrence Livermore National Laboratory, 1994.
27. D.E. Burton, Multidimensional discretization of conservation laws for unstructured polyhedral grids, Report UCRL-JC-118306, Lawrence Livermore National Laboratory, in: Proceedings of Second International Workshop on Analytical Methods and Process Optimization in Fluid and Gas Mechanics (SAMGOP), Arzamas-16, Russia, 1994.
28. Campbell J.C., Shashkov M.J. A tensor artificial viscosity using a mimetic finite difference algorithm. J. Comput. Phys. 2001, 172(4):739-765.
29. Campbell J.C., Shashkov M.J. A compatible Lagrangian hydrodynamics algorithm for unstructured grids. Seltuk J. Appl. Math. 2003, 4(2):53-57.
30. 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. 1998, 146(1):227-262.
31. Caramana E.J., Loubère R. " Curl-q": a vorticity damping artificial viscosity for Lagrangian hydrodynamics calculations. J. Comput. Phys. 2006, 215(2):385-391.
32. 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. 1998, 142:521-561.
33. Caramana E.J., Shashkov M.J., Whalen P.P. Formulations of artificial viscosity for multidimensional shock wave computations. J. Comput. Phys. 1998, 144:70-97.
34. Carré G., Del Pino S., DesprTs B., Labourasse E. A cell-centered Lagrangian hydrodynamics scheme in arbitrary dimension. J. Comput. Phys. 2009, 228(14):5160-5183.
35. R.A. Clark, Compressible Lagrangian hydrodynamics without Lagrangian cells, Report LAUR-85-1882, Los Alamos National Laboratory, 1985.
36. Clark R.A. The evolution of HOBO. Comput. Phys. Commun. 1988, 48:61-64.
37. M.C. Cline, J.K. Dukowicz, F.L. Addessio, CAVEAT-GT: a general topology version of the caveat code, Technical Report LA-11812-MS, Los Alamos National Laboratory, 1990.
38. Colella P. A direct Eulerian MUSCL scheme for gas dynamics. SIAM J. Stat. Comput. 1985, 6(1):104-117.
39. W.P. Crowley, FLAG: a free-Lagrange method for numerically simulating hydrodynamic flows in two dimensions, in: M. Holt (Ed.), Proceedings of the Second International Conference on Numerical Methods in Fluid Dynamics, vol. 8, September 15-19, 1970, University of California, Berkeley, Lecture Notes in Physics, Springer, Berlin, 1971, pp. 37-43.
40. DesprTs B., Mazeran C. Lagrangian gas dynamics in two dimensions and Lagrangian systems. Arch. Rational Mech. Anal. 2005, 178:327-372.
41. Du J., Fix B., Glimm J., Li X., Li Y., Wu L. A simple package for front tracking. J. Comput. Phys. 2006, 213(2):613-628.
42. Du Q., Emelianenko M., Gunzburger M. Convergence of the Lloyd algorithm for computing centroidal Voronoi tesselations. SAIM J. Numer. Anal. 2006, 44(1):102-119.
43. Du Q., Faber V., Gunzburger M. Centroidal Voronoi tesselations: applications and algorithms. SIAM Rev. 1999, 41:637-676.
44. Du Q., Gunzburger M. Grid generation and optimization based on centroidal Voronoi tesselations. Appl. Math. Comput. 2002, 133:591-607.
45. Dukowicz J.K., Cline M.C., Addessio F.S. A general topology Godunov method. J. Comput. Phys. 1989, 82:29-63.
46. Dukowicz J.K., Meltz B. Vorticity errors in multidimensional Lagrangian codes. J. Comput. Phys. 1992, 99:115-134.
47. Dukowicz J.K. A simplified adaptive mesh technique derived from the moving finite element method. J. Comput. Phys. 1984, 56:324-342.
48. Dukowicz J.K., Baumgardner J. Incremental remapping as a transport/advection algorithm. J. Comput. Phys. 2000, 160:318-335.
49. Dukowicz J.K., Kodis J.W. Accurate conservative remapping (rezoning) for arbitrary-Lagrangian-Eulerian computations. SIAM J. Stat. Comput. 1987, 8:305-321.
50. Dyachenko V.F. The free point method for problems of continuous media. USSR Comp. Math. Math. Phys. 1965, 5(4):680-688.
51. Dyachenko V.F. Some new method for numerically solving nonstationary gas dynamics problems. Comput. Meth. Appl. Mech. Eng. 1973, 2(3):265-277.
52. Dyadechko V., Shashkov M. Reconstruction of multi-material interfaces from moment data. J. Comput. Phys. 2008, 227:5361-5384.
53. Farhat C., Geuzaine P., Grandmont C. The discrete geometric conservation law and the nonlinear stability of ALE schemes for the solution of the flow problems on the moving grids. J. Comput. Phys. 2001, 174:669-694.
54. A.P. Favorskij, M. Yu. Shashkov, A.V. Solovjov, V.F. Tishkin, The Dirichlet particle method for the simulation of 2-D gas dynamic flows with strong deformations, in: A.A. Samarskii, M.P.Sapagovas (Eds.), Mathematical Modelling and Applied Mathematics, Proceedings of IMACS International Conference, Moscow/USSR June 18-23, 1990, pp. 153-162 (Elsevier Science Publishers North Holland, Amsterdam, 1992).
55. The Free-Lagrange Method: Proceedings of the First International Conference of Free-Lagrange Methods. Lecture Notes in Physics 1985, vol. 238. Springer-Verlag, New York. M.J. Fritts, W.P. Crowley, H. Trease (Eds.).
56. R.R. Giddings, Mesh movement via optimal transportation, in: Conference on Numerical Methods for Multi-material Flows Czech Technical University in Prague on September 10-14, 2007. Available on web: <>. http://www-troja.fjfi.cvut.cz/~multimat07/prezentace.html.
57. Giuliani S. An algorithm for continuous rezoning of the hydrodynamic grid in arbitrary-Lagrangian-Eulerian computer codes. Nucl. Eng. Des. 1982, 72:205-212.
58. Glimm J., Grove J., Li X.-L., Shyue nad K.-M., Zhang Q., Zeng Y. Three-dimensional front tracking. SIAM J. Sci. Comput. 1998, 19:703-727.
59. Glimm J., Grove J.W., Li X.L., Oh W., Sharp D.H. A critical analysis of Rayleigh-Taylor growth rates. J. Comput. Phys. 2001, 169:652-677.
60. Godunov S.K., Zabrodine A., Ivanov M., Kraiko A., Prokopov G. RTsolution numTrique des problFmes multidimensionnels de la dynamique des gaz 1979, Mir.
61. Grandy J. Conservative remapping and region overlays by intersecting arbitrary polyhedra. J. Comput. Phys. 1999, 148(2):133-466.
62. Haas J.-F., Sturtevant B. Interaction of weak-shock waves. J. Fluid Mech. 1987, 181:41-76.
63. Hazins V.M., Svetsov V.V. A conservative stable smoothness-enhancing free-Lagrangian method. J. Comput. Phys. 1993, 105(2):187-198.
64. Hirt C.W., Amsden A.A., Cook J.L. An arbitrary-Lagrangian-Eulerian computing method for all flow speeds. J. Comput. Phys. 1974, 14. (reprinted in J. Comput. Phys. 135 (1997) 203-216).
65. P. Hoch, Mesh quality and conservative projection in Lagrangian compressible hydrodynamic, in: Conference on Numerical Methods for Multi-material Fluid Flows, Czech Technical University in Prague on September 10-14, 2007, <>. http://www-troja.fjfi.cvut.cz/multimat07/presentations/tuesday/Rebourcet_H.
66. Hoover W.G. Smooth Particle Applied Mechanics: The State of the Art 2006, World Scientific Publishing Co. Pvt. Ltd.
67. Howell B.P., Ball G.J. Damping of mesh-induced errors in free-Lagrange simulations or Richmayer-Meshkov instability. Shock Waves 2000, 10:253-264.
68. Howell B.P., Ball G.J. A free-Lagrange augmented Godunov method for simulation of elastic-plastic solids. J. Comput. Phys. 2002, 175:128-167.
69. Huang Y., Quin H., Wang D. Centroidal Voronoi tesselation-based finite element superconvergence. Int. J. Numer. Meth. Eng. 2008, 76:1819-1839.
70. Kershaw D.S., Prasad M.K., Shaw M.J., Milovich J.L. 3D unstructured mesh ALE hydrodynamics with the upwind discontinuous finite element method. Comput. Meth. Appl. Mech. Eng. 1998, 158:81-116.
71. Kjellgren P., Hyvarien J. An arbitrary-Lagrangian-Eulerian finite element method. Comput. Mech. 1998, 21:81-90.
72. Klein R., Lingas A. A linear-time randomized algorithm for the bounded Voronoi diagram of a simple polygon. Int. J. Comput. Geomet. Appl. 1996, 6(3):263-278.
73. G. Kluth, B. Desprès, Discretization of the hyperelasticity on unstructured mesh with cell-centered scheme, Technical Report R09063, Laboratoire Jacques-Louis Lions, Universitè Pierre et Marie Curie, 2009. Available on web: <>. http://www.ann.jussieu.fr/publications/2009/R09063.html.
74. Knupp P., Margolin L.G., Shashkov M. Reference Jacobian optimization-based rezone strategies for arbitrary-Lagrangian-Eulerian methods. J. Comput. Phys. 2002, 176:93-128.
75. Kuchaři{dotless}́k M., Garimella R., Schofield S., Shashkov M. A comparative study of interface reconstruction methods for multi-material ALE simulations. J. Comput. Phys. 2010, 229(7):2432-2452.
76. Kuchaři{dotless}́k M., Shashkov M. Extension of efficient, swept-integration-based conservative remapping method for meshes with changing connectivity. Int. J. Numer. Meth. Fluid 2007, 56(8):1359-1365.
77. Kuchaři{dotless}́k M., Shashkov M., Wendroff B. An efficient linearity-and-bound-preserving remapping methods. J. Comput. Phys. 2003, 188:462-471.
78. Kull H.J. Theory of the Rayleigh-Taylor instability. Phys. Rep. 1991, 206:197-325.
79. Lipnikov K., Shashkov M. The error-minimization-based strategy for moving mesh methods. Commun. Comput. Phys. 2006, 1(1):53-81.
80. Liu G.R. Mesh Free Methods. Moving Beyond the Finite Element Method 2003, CRC Press.
81. Lohner R., Yang C. Improved ALE mesh velocities for moving bodies. Commun. Numer. Meth. Eng. 1996, 12:599-608.
82. R. Loubère, First steps into ale inc(ubator) - version 2.0.0, Technical Report, Los Alamos National Laboratory Report LAUR-04-8840, 2004.
83. Loubère R., Caramana E.J. The force/work differencing of exceptional points in the discrete, compatible formulation of Lagrangian hydrodynamics. J. Comput. Phys. 2006, 216(1):1-18.
84. Loubère R., Shashkov M. A subcell remapping method on staggered polygonal grids for arbitrary-Lagrangian-Eulerian methods. J. Comput. Phys. 2004, 204(23):155-160.
85. Loubère R., Staley M., Wendroff B. The repair paradigm: new algorithms and applications to compressible flow. J. Comput. Phys. 2005, 385-404.
86. Maire P.-H. A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry. J. Comput. Phys. 2009, 228:6882-6915.
87. Maire P.-H. A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes. J. Comput. Phys. 2009, 228:2391-2425.
88. Maire P.-H., Abgrall R., Breil J., Ovadia J. A cell-centered Lagrangian scheme for compressible flow problems. SIAM J. Sci. Comput. 2007, 29(4):1781-1824.
89. Maire P.-H., Breil J. A second-order cell-centered Lagrangian scheme for two-dimensional compressible flow problems. Int. J. Numer. Meth. Fluid 2008, 56:1417-1423.
90. Maire P.-H., Breil J., Galera S. A cell-centered arbitrary-Lagrangian-Eulerian (ALE) method. Int. J. Numer. Meth. Fluid 2008, 56:1161-1166.
91. Maire P.-H., Nkonga B. Multi-scale Godunov-type method for cell-centered discrete Lagrangian hydrodynamics. J. Comput. Phys. 2009, 228:799-821.
92. Margolin L.G., Shashkov M., Smolarkiewicz P. A discrete operator calculus for finite difference approximations. Comp. Meth. Appl. Mech. Eng. 2000, 187:365-383.
93. Margolin L.G. Introduction to " An arbitrary-Lagrangian-Eulerian computing method for all flow speeds" J. Comput. Phys. 1997, 135:198-202.
94. Margolin L.G., Shashkov M. Second-order sign-preserving conservative interpolation (remapping) on general grids. J. Comput. Phys. 2003, 184(1):266-298.
95. Mikhailova N.V., Tishkin V.F., Tyurina N.N., Favorskii A.P., Shashkovm M.Y. Numerical modelling of two-dimensional gas-dynamic flows on a variable-structure mesh. USSR Comput. Math. Math. Phys. 1988, 26(5):74-84.
96. Okabe A., Boots B., Sugihara K., Chiu S.N. Spatial Tesselations. Concepts and Applications of Voronoi Diagrams 2000, Wiley.
97. Orlova T.I., Svetsov V.V. Deep penetration of projectile material into the planet during the impact. Earth Moon Planets 1995, 71(3):255-263.
98. O'Rourke P.J., Sahota M.S. A variable explicit/implicit numerical method for calculating advection on unstructured meshes. J. Comput. Phys. 1998, 143:312-345.
99. Pasta J.R., Ulam S. Heuristic numerical work in some problems of hydrodynamics. Math. Tables Aids Comput. 1959, XIII(65):1-12.
100. Peery J.S., Carroll D.E. Multi-material ALE methods in unstructured grids. Comput. Meth. Appl. Mech. Eng. 2000, 187:591-619.
101. R. Pember, R. Anderson, Comparison of direct Eulerian Godunov and Lagrange plus remap artificial viscosity schemes for compressible flow, Technical Report 2001-2644, AIAA, 2001.
102. Pushkina I.G., Tishkin V.F. Adaptive grids from Dirichlet cells for mathematical physics problems: a methodology for grid generation, examples. Math. Model. 2000, 12(3):97-109.
103. Quirk J., Karni S. On the dynamics of a shock-bubble interaction. J. Fluid Mech. 1996, 318:129-163.
104. B. Rebourcet, Comments on the filtering of numerical instabilities in Lagrangian hydrocodes, in: Conference on Numerical Methods for Multi-material Fluid fLows, Czech Technical University in Prague on September 10-14, 2007, <>. http://www-troja.fjfi.cvut.cz/multimat07/presentations/tuesday/Rebourcet_f.
105. Rider W.J., Kothe D.B. Reconstructing volume tracking. J. Comput. Phys. 1998, 121:112-152.
106. Rudman M. Volume tracking methods for interfacial flow calculations. Int. J. Numer. Meth. Fluid 1997, 24:671-691.
107. M.S. Sahota, H.E. Trease, A three-dimensional free-Lagrange code for multimaterial flow simulations, Report LAUR-90-3383, Los Alamos National Laboratory, 1990.
108. Scovazzi G. Stabilized shock hydrodynamics: II. Design and physical interpretation of the SUPG operator for Lagrangian computations. Comput. Meth. Appl. Mech. Eng. 2007, 196:966-978.
109. Scovazzi G., Christon M.A., Hughes T.J.R., Shadid J.N. Stabilized shock hydrodynamics: I. A Lagrangian method. Comput. Meth. Appl. Mech. Eng. 2007, 196:923-966.
110. Scovazzi G., Love E., Shashkov M.J. Multi-scale Lagrangian shock hydrodynamics on Q1/P0 finite elements: theoretical framework and two-dimensional computations. Comput. Meth. Appl. Mech. Eng. 2008, 197:1056-1079.
111. Shashkov M. Closure models for multimaterial cells in arbitrary-Lagrangian-Eulerian hydrocodes. Int. J. Numer. Meth. Fluid 2007, 56:1497-1504.
112. Shashkov M., Wendroff B. The repair paradigm and application to conservation laws. J. Comput. Phys. 2004, 198(1):265-277.
113. M.Yu. Shashkov, A.V. Solovjov, A generalization of the notion of Dirichlet cell for non-convex domain, Preprint No. 32 of Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow, Russia, 1990 (in Russian).
114. M.Yu. Shashkov, A.V. Solovjov, Numerical simulation of two-dimensional flows by the free-Lagrangian flow simulations, Report TUM-M9105, Mathematisches Institut, Technische Universität, Munchen, 1991. Available at <>. http://cnls.lanl.gov/~shashkov.
115. Sofronov I.D., Rasskazova V.V., Nesterenko L.V. The use of nonregular nets for solving two-dimensional non-stationary problems in gas dynamics. Numerical Methods in Fluid Dynamics 1984, 82-121. MIR, Moscow. N.N. Yanenko, Y. Shokin (Eds.).
116. Sokolov S.S., Panov A.I., Voropinov A.A., Novikov I.G., Sobolev I.V., Yalozo A.V. Method TIM for computation of three-dimensional problems of mechanics of continuum media using unstructured polyhedral Lagrangian meshes (in Russian). Quest. Atomic Sci. Tech. Ser.: Math. Mod. Phys. Process. 2005, 3:37-52.
117. Sokolov S.S., Voropinov A.A., Novikov I.G., Panov A.I., Sobolev I.V., Pushkarev A.A. Method TIM-2D for computation of problems of mechanics of continuum media using unstructured polygonal with arbitrary connections at the nodes (in Russian). Quest. Atomic Sci. Tech. Ser.: Math. Mod. Phys. Process. 2006, 4:29-44.
118. Solovjov A., Solovjova E., Shashkov M., Tishkin V., Favorskii A. Approximation of finite difference operators on mesh of Dirichlet cells. Differ. Equat. 1986, 22:863-872.
119. V. Springel, E pur si muove: Galiliean-invariant cosmological hydrodynamical simulations on a moving mesh, Mon. Not. R. Astron. Soc. (in press), arXiv:0901.4107, <.4107>. http://arxiv.org/pdf/0901.
120. Stoker C., Gay C., Bay F., Chenot J.-L. A velocity approach for the ALE-method applied for 2D and 3D problems. Simulation of Material Processing: Theory, Methods and Applications 1998, Balkema, Rotterdam.
121. Advances in the Free-Lagrange Method: Including Contributions on Adaptive Gridding and the Smooth Particle Hydrodynamics Method: Proceedings of the Next Free-Lagrange Conference. Lecture Notes in Physics 1990, vol. 395. Springer-Verlag, New York. H. Trease, M.J. Fritts, W.P. Crowley (Eds.).
122. Trease H.E. Three-dimensional free-Lagrangian method hydrodynamics. Comput. Phys. Commun. 1988, 48:39-50.
123. van Leer B. Towards the ultimate conservative difference scheme. J. Comput. Phys. 1979, 32:101-136.
124. VanderHeyden W.B., Kashiwa B.A. Compatible fluxes from van Leer advection. J. Comput. Phys. 1998, 146:1-28.
125. von Neumann J., Richtmyer R.D. A method for the numerical calculations of hydrodynamical shocks. J. Appl. Phys. 1950, 21:232-238.
126. Whitehurst R. A free-Lagrange method for gas dynamics. Mon. Not. R. Astron. Soc. 1995, 277:655-680.
127. Wilkins M.L. Calculation of elastic plastic flow. Math. Comput. Phys. 1964, 3.
128. A.M. Winslow, Equipotential zoning of two-dimensional meshes, Technical Report UCRL-7312, Lawrence Livermore National Laboratory, 1963.
129. Winslow A.M. Numerical solution of the quasilinear Poisson equations in a nonuniform triangle mesh. J. Comput. Phys. 1966, 1:149-172.