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Implicit dynamics in the material-point method. (English) Zbl 1060.74674
Summary: A time-implicit discretization is derived and validated for the material-point method (MPM). The resulting non-linear, discrete equations are solved using Newton’s method combined with either the conjugate gradient method or the generalized minimum residual method. These Newton-Krylov solvers are implemented in a matrix-free fashion for numerical efficiency. A description of the algorithms and evaluation of their performance is presented. On all test problems, if the time step is chosen appropriately, the implicit solution technique is more efficient than an explicit method without loss of desired features in the solutions. In a dramatic example, time steps 10,000 times the explicit step size are possible for the large deformation compression of a cylindrical billet at 1.2% the computational cost.

MSC:
74S30 Other numerical methods in solid mechanics (MSC2010)
Software:
KELLEY
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[1] Harlow, F.H., The particle-in-cell computing method for fluid dynamics, Methods comput. phys., 3, 319-343, (1964)
[2] Brackbill, J.U.; Ruppel, H.M., FLIP: a method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions, J. comput. phys., 65, 314-343, (1986) · Zbl 0592.76090
[3] Brackbill, J.U.; Kothe, D.B.; Ruppel, H.M., FLIP: A low-dissipation, particle-in-cell method for fluid flow, Comput. phys. commun., 48, 25-38, (1998)
[4] Sulsky, D.; Chen, Z.; Schreyer, H.L., A particle method for history-dependent materials, Comput. meths. appl. mech. engrg., 118, 179-196, (1994) · Zbl 0851.73078
[5] York, A.R.; Sulsky, D.; Schreyer, H.L., Fluid-membrane interaction based on the material-point method, Int. J. numer. methods engrg., 48, 901-924, (2000) · Zbl 0988.76073
[6] Cummins, S.J.; Brackbill, J.U., An implicit particle-in-cell method for granular materials, J. comput. phys., 180, 506-548, (2002) · Zbl 1143.74388
[7] J.E. Guilkey, J.A. Weiss, An implicit time integration strategy for use with the material point method, in: Proceedings from the First MIT Conference on Computational Fluid and Solid Mechanics, June 12-15, 2001
[8] Sulsky, D.; Zhou, S.; Schreyer, H.L., Applications of a particle-in-cell method to solid mechanics, Comput. phys. commun., 87, 236-252, (1995) · Zbl 0918.73334
[9] Lin, C.C.; Segel, L.A., Mathematics applied to deterministic problems in the natural sciences, (1988), SIAM Philadelphia · Zbl 0664.00026
[10] Sulsky, D.; Schreyer, H.L., Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems, Comput. meths. appl. mech. engrg., 139, 409-429, (1996) · Zbl 0918.73332
[11] Schreyer, H.L.; Sulsky, D.; Zhou, S., Modeling delamination with as a strong discontinuity with the material point method, Comput. methods appl. mech. engrg., 191, 2463-2481, (2002) · Zbl 1054.74070
[12] Burgess, D.; Sulsky, D.; Brackbill, J.U., Mass matrix formulation of the FLIP particle-in-cell method, J. comput. phys., 103, 1-15, (1992) · Zbl 0761.73117
[13] Kelly, C.T., Iterative methods for linear and nonlinear equations, (1995), SIAM Philadelphia
[14] Barret, R.; Berry, M.; Chan, T.F.; Demmel, J.; Donato, R.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; van der Vorst, H., Templates for the solution of linear systems: building blocks for iterative methods, (1994), SIAM Philadelphia
[15] Saad, Y.; Schultz, M.H., GMRES: A generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 856-869, (1986) · Zbl 0599.65018
[16] Knoll, D.A.; McHugh, P.R., Enhanced nonlinear iterative techniques applied to a nonequilibrium plasma flow, SIAM J. sci. comput., 19, 291-301, (1998) · Zbl 0913.76067
[17] Knoll, D.A.; Lapenta, G.; Brackbill, J.U., A multilevel iterative field solver for implicit, kinetic, plasma simulation, J. comput. phys., 149, 377-388, (1999) · Zbl 0934.76048
[18] A. Kaul, Implicit formulation of the material point method with application to metals processing, Ph.D. Dissertation, Department of Mathematics and Statistics, University of New Mexico, December, 2000
[19] Lapenta, G.; Brackbill, J., Control of the number of particles in fluid and MHD particle-in-cell methods, Comput. phys. commun., 87, 139-254, (1995) · Zbl 0923.76194
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