Local Approximation Study of Johnson's
                  Method for 2D Elasticity Problems

                  V.G. Ganzha*) and E.V. Vorozhtsov

             Institute of Theoretical and Applied Mechanics,
          Russian Academy of Sciences, Novosibirsk 630090, Russia
          -----------------------
          *) At the moment, the University of Kassel, Kassel D 34127, Germany

    The Lagrangian finite-difference method, which was developed by G.R. Johnson
in 1976 for the numerical solution of two-dimensional high velocity impact
problems within the framework of the elasticity and plasticity theories, has
gained a widespread acceptance. We investigate the local approximation of this
method with the aid of the computer algebra systems. The problem under
consideration involves tremendous intermediate expressions. We present an
algorithm of decomposition of a complex approximation problem into a finite
set of simpler subproblems whose union provides the solution of the original
approximation problem. The results obtained with the aid of this algorithm
show that the Johnson's method has no approximation in the case when the
accelerations of the nodes are computed on the basis of four triangular
elements surrounding a given node of the triangular mesh. In the case of six
elements and of the uniform triangular mesh, the approximation takes place
only if the lumped mass in the given node is equal to two thirds of the sum of
masses of the six elements having their vertices at the given node. In this
particular case, the Johnson's method has the second order of approximation
in space and the first order of approximation in time. But in the case of
irregular, distorted Lagrangian mass elements the Johnson's method has no
approximation of the original elasticity equations also in the case of the
six triangular elements surrounding the given node of the mesh. It is shown
that the first-order approximation in the case of irregular elements takes
place only in the case of small deviations of the shapes of the elements from
their regular shapes. The validity of the obtained results was checked by
using two different computer algebra systems: REDUCE and Maple.
It follows from the results obtained that the Johnson's method as applied to
high-velocity impact problems should be augmented by an algorithm for the
local mesh rezoning in the subdomains of strong distortions of the Lagrangian
mass elements to ensure the local approximation.