D. Sulsky et Hl. Schreyer, AXISYMMETRICAL FORM OF THE MATERIAL POINT METHOD WITH APPLICATIONS TOUPSETTING AND TAYLOR IMPACT PROBLEMS, Computer methods in applied mechanics and engineering, 139(1-4), 1996, pp. 409-429
The material point method is an evolution of particle-in-cell methods
which utilize two meshes, one a material or Lagrangian mesh defined ov
er material of the body under consideration, and the second a spatial
or Eulerian mesh defined over the computational domain. Although meshe
s are used, they have none of the negative aspects normally associated
with conventional Eulerian or Lagrangian approaches. The advantages o
f both the Eulerian and Lagrangian methods are achieved by using the a
ppropriate frame for each aspect of the computation, with a mapping be
tween the two meshes that is performed at each step in the loading pro
cess. The numerical dissipation normally displayed by an Eulerian meth
od because of advection is avoided by using a Lagrangian step; the mes
h distortion associated with the Lagrangian method is prevented by map
ping 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 h
ave 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 an
d existing numerical solutions are presented.