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Variational and momentum preservation aspects of smooth particle hydrodynamic formulations. (English) Zbl 0962.76075
Summary: We present a new variational framework for various existing smooth particle hydrodynamic (SPH) techniques, and give a new corrected SPH formulation. The linear and angular momentum preserving properties of SPH formulations are also discussed. We show that, in general, in order to preserve angular momentum, the SPH equations must correctly evaluate the gradient of linear velocity field, and hence we propose a corrected algorithm that combines kernel correction with gradient correction. The theory presented is illustrated with several examples related to simple free surface flows.

MSC:
76M28 Particle methods and lattice-gas methods
76M30 Variational methods applied to problems in fluid mechanics
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[1] Lucy, L.B., A numerical approach to the testing of the fission hypothesis, Astro. J., 82, 1013, (1977)
[2] Gingold, R.A.; Monaghan, J.J., SPH: theory and application to non-spherical stars, Mon. not. R. astron. soc., 181, 375, (1977) · Zbl 0421.76032
[3] Schussler, M.; Schmitt, D., Comments on SPH, Astron. astrophys., 97, 373, (1981)
[4] Gingold, R.A.; Monaghan, J.J., Kernel estimates as a basis for general particle methods in hydrodynamics, J. comput. phys., 46, 429, (1982) · Zbl 0487.76010
[5] Belytschko, T., Feature articles: are finite elements passe, ISACM bulletin, 7, 4, (1994)
[6] Libersky, L.D.; Petschek, A.G.; Carney, T.C.; Hipp, J.R.; Allahdadi, F.A., High strain Lagrangian hydrodynamics, J. comput. phys., 109, 67, (1993) · Zbl 0791.76065
[7] Petschek, A.G.; Libersky, L.D., Cylindrical SPH, J. comput. phys., 109, 76, (1993) · Zbl 0791.76066
[8] Johnson, G.R.; Petersen, E.H.; Stryk, R.A., Incorporation of an SPH option into the epic code for a wide range of high velocity impact computations, Int. J. impact engng., 14, 385, (1993)
[9] Stellingwerf, R.F.; Wingate, C.A., Impact modelling with SPH, Int. J. impact engng., 14, 707, (1993)
[10] Benz, W.; Asphaug, E., Simulations of brittle solids using SPH, Comput. phys. comm., 87, 253, (1995) · Zbl 0918.73335
[11] Johnson, G.R.; Stryk, R.A.; Beissell, S.R., SPH for high velocity impact computations, Comput. meth. appl. mech. engng., 139, 347, (1996) · Zbl 0895.76069
[12] Johnson, G.R.; Beissel, S.R., Normalised smoothing functions for SPH impact computations, Int. J. num. meth. engng., 39, 2725, (1996) · Zbl 0880.73076
[13] Monaghan, J.J., Simulating free surface flows with SPH, J. comp. phys., 110, 399, (1994) · Zbl 0794.76073
[14] Takeda, H.; Miyama, S.M.; Sekiya, M., Numerical simulation of viscous flow by SPH, Prog. theor. phys., 92, 939, (1994)
[15] Monaghan, J.J.; Kocharyan, A., SPH simulation of multi-phase flow, Comput. phys. comm., 87, 225, (1995) · Zbl 0923.76195
[16] Morris, J.P.; Fox, P.J.; Zhu, Y., Modelling low Reynolds number incompressible flows using SPH, J. comp. phys., 136, 214, (1997) · Zbl 0889.76066
[17] Monaghan, J.J., Why particle methods work, SIAM J. sci. stat. comput., 3, 422, (1982) · Zbl 0498.76010
[18] Monaghan, J.J., Particle methods for hydrodynamics, Comput. phys. reports, 3, 71, (1985)
[19] Monaghan, J.J., An introduction to SPH, Comput. phys. comm., 48, 89, (1988) · Zbl 0673.76089
[20] Monaghan, J.J., On the problem of penetration in particle methods, J. comp. phys., 82, 1, (1989) · Zbl 0665.76124
[21] Monaghan, J.J., Sph, Annu. rev. astron. astrophys., 30, 543, (1992)
[22] Randles, P.W.; Libersky, L.D., SPH: some recent improvements and applications, Comput. methods appl. mech. engng., 139, 375, (1996) · Zbl 0896.73075
[23] B. Hassani, J. Bonet, CSPH, Department of Civil Engng., University of Wales Swansea, Internal Report CR/900/97, 1997
[24] J. Bonet, B. Hassani, T.S.L. Lok, S. Kulasageram, CSPH - A Reproducing Kernel Meshless Method for Computational Mechanics, 5th ACME-UK conference, London, 1997
[25] T.S.L. Lok, J. Bonet, CSPH Method for Viscous Free-Surface Flow, 6th ACME-UK conference, Exeter, 1998
[26] Bonet, J.; Wood, R.D., Non-linear continuum mechanics for finite element analysis, (1997), Cambridge University Press Cambridge, UK
[27] Li, S.F.; Liu, W.K., Moving least square kernel Galerkin method (II) Fourier analysis, Comput. methods appl. mech. engng., 139, 159, (1996) · Zbl 0883.65089
[28] Liu, W.K.; Li, S.F.; Belytschko, T., Moving least square kernel Galerkin method (I) methodology and convergence, Comput. methods appl. mech. engng., 143, 113, (1997) · Zbl 0883.65088
[29] Batchelor, G.K., An introduction to fluid dynamics, (1967), Cambridge University Press Cambridge, UK · Zbl 0152.44402
[30] Martin, J.C.; Moyce, W.J., Part IV. an experimental study of the collapse of liquid columns on a rigid horizontal plane, Phil. tran. R. soc. London, 244, 312, (1952)
[31] Henderson, F.M., Open channel flow, (1966), Collier-Macmillan Publishing Company London · Zbl 0193.55002
[32] Stoker, J.J., Water waves: the mathematical theory with applications, (1957), Interscience Publishers New York · Zbl 0078.40805
[33] F.H. Harlow, A.A. Amsden, Fluid dynamics: An introductory text, Los Alamos Sci. Lab., LA-4100, 1970 · Zbl 0206.55002
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