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Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems. (English) Zbl 0918.73332
Summary: The material point method is an evolution of particle-in-cell methods which utilize two meshes, one a material or Lagrangian mesh defined over material of the body under consideration, and the second a spatial or Eulerian mesh defined over the computational domain. Although meshes are used, they have none of the negative aspects normally associated with conventional Eulerian or Lagrangian approaches. The advantages of both the Eulerian and Lagrangian methods are achieved by using the appropriate frame for each aspect of the computation, with a mapping between the two meshes that is performed at each step in the loading process. The numerical dissipation normally displayed by an Eulerian method because of advection is avoided by using a Lagrangian step; the mesh distortion associated with the Lagrangian method is prevented by mapping to a user-controlled mesh. Furthermore, explicit material points can be tracked through the process of deformation, thereby alleviating the need to map history variables. As a consequence, problems which have caused severe numerical difficulties with conventional methods are handled fairly routinely. Examples of such problems are the upsetting of billets and the Taylor problem of cylinders impacting a rigid wall. Numerical solutions to these problems are obtained with the material point method and where possible comparisons with experimental data and existing numerical solutions are presented.

##### MSC:
 74S30 Other numerical methods in solid mechanics (MSC2010) 74M20 Impact in solid mechanics
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##### References:
 [1] Brackbill, J.U.; Ruppel, H.M., FLIP: A method for adaptively zones, particle-in-cell calculations in two dimensions, J. comput. phys., 65, 314-343, (1986) · Zbl 0592.76090 [2] Brackbill, J.U.; Kothe, D.B.; Ruppel, H.M., FLIP: A low-dissipation, particle-in-cell method for fluid flow, Comput. phys. comm., 48, 25-38, (1988) [3] Sulsky, D.; Chen, Z.; Schreyer, H.L., A particle method for history-dependent materials, Comput. methods appl. mech. engrg., 118, 179-196, (1994) · Zbl 0851.73078 [4] Sulsky, D.; Zhou, S.-J.; Schreyer, H.L., Application of a particle-in-cell method to solid mechanics, Comput. phys. comm., 87, 236-252, (1995) · Zbl 0918.73334 [5] 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 [6] Wallace, J.M.; Brackbill, J.U.; Forslund, D.W., An implicit moment electromagnetic plasma simulation in cylindrical coordinates, J. comput. phys., 63, 434-457, (1986) · Zbl 0587.76203 [7] Taylor, G.I., The use of flat-ended projectiles for determining dynamic yield stress. part I, (), 289-299 [8] Wilkins, M.L.; Guinan, M.W., Impact of cylinders on a rigid boundary, J. appl. phys., 44, 1200-1206, (1973) [9] Predebon, W.W.; Anderson, C.E.; Walker, J.D., Inclusion of evolutionary damage measures in Eulerian wavecodes, Comput. mech., 7, 221-236, (1991) · Zbl 0735.73087 [10] Johnson, G.R.; Holmquist, T.J., Evaluation of cylinder-impact test data for constitutive model constants, J. appl. phys., 64, 3901-3910, (1988) [11] Holmquist, T.J.; Johnson, G.R., Determination of constants and comparison of results for various constitutive models, (), C3-853-C3-860 [12] Kudo, H.; Matsubara, S., Joint examination project of validity of various numerical methods for the analysis of metal forming processes, () · Zbl 0408.73037 [13] Shih, A.J.M.; Yang, H.T.Y., Experimental and finite element simulation methods for rate-dependent metal forming processes, Int. J. numer. methods engrg., 31, 345-367, (1991) · Zbl 0825.73790 [14] Taylor, L.M.; Becker, E.B., Some computational aspects of large deformation, rate-dependent plasticity problems, Comput. methods appl. mech. engrg., 41, 251-277, (1983) · Zbl 0509.73046 [15] Tuǧcu, P., Thermomechanical analysis of upsetting of a cylindrical billet, Comput. struct., 58, 1-12, (1996) [16] Simo, J.C.; Armero, F.; Taylor, R.L., Improved versions of assumed enhanced strain tri-linear elements for 3D finite deformation problems, Comput. methods appl. mech. engrg., 110, 359-386, (1993) · Zbl 0846.73068
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