  We present a geometric variational discretization of nonlinear elasticity in
2D and 3D in the Lagrangian description. A main step in our construction is the
definition of discrete deformation gradients and discrete Cauchy-Green
deformation tensors, which allows for the development of a general discrete
geometric setting for frame indifferent isotropic hyperelastic models. The
resulting discrete framework is in perfect adequacy with the multisymplectic
discretization of fluids proposed earlier by the authors. Thanks to the unified
discrete setting, a geometric variational discretization can be developed for
the coupled dynamics of a fluid impacting and flowing on the surface of an
hyperelastic body. The variational treatment allows for a natural inclusion of
incompressibility and impenetrability constraints via appropriate penalty
terms. We test the resulting integrators in 2D and 3D with the case of a
barotropic fluid flowing on incompressible rubber-like nonlinear models.
