  In this paper we present a conservative cell-centered Lagrangian finite
volume scheme for the solution of the hyper-elasticity equations on
unstructured multidimensional grids. The starting point of the new method is
the Eucclhyd scheme, which is here combined with the a posteriori
Multidimensional Optimal Order Detection (MOOD) limiting strategy to ensure
robustness and stability at shock waves with piece-wise linear spatial
reconstruction. The ADER (Arbitrary high order schemes using DERivatives)
approach is adopted to obtain second-order of accuracy in time as well. This
method has been tested in an hydrodynamics context and the present work aims at
extending it to the case of hyper-elasticity models. Such models are presented
in a fully Lagrangian framework and the dedicated Lagrangian numerical scheme
is derived in terms of nodal solver, GCL compliance, subcell forces and
compatible discretization. The Lagrangian numerical method is implemented in 3D
under MPI parallelization framework allowing to handle genuinely large meshes.
A relative large set of numerical test cases is presented to assess the ability
of the method to achieve effective second order of accuracy on smooth flows,
maintaining an essentially non-oscillatory behavior and general robustness
across discontinuities and ensuring at least physical admissibility of the
solution where appropriate. Pure elastic neo-Hookean and non-linear materials
are considered for our benchmark test problems in 2D and 3D. These test cases
feature material bending, impact, compression, non-linear deformation and
further bouncing/detaching motions.
