  We develop a computational method based on an Eulerian field called the
"reference map", which relates the current location of a material point to its
initial. The reference map can be discretized to permit finite-difference
simulation of large solid-like deformations, in any dimension, of the type that
might otherwise require the finite element method. The generality of the method
stems from its ability to easily compute kinematic quantities essential to
solid mechanics, that are elusive outside of Lagrangian frame. This
introductory work focuses on large-strain, hyperelastic materials. After a
brief review of hyperelasticity, a discretization of the method is presented
and some non-trivial elastic deformations are simulated, both static and
dynamic. The method's accuracy is directly verified against known analytical
solutions.
