Axisymmetric form of the material point method with applications to upsetting and Taylor impact problems

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.