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We present a geometric variational discretization of nonlinear elasticity in2D and 3D in the Lagrangian description. A main step in our construction is thedefinition of discrete deformation gradients and discrete Cauchy-Greendeformation tensors, which allows for the development of a general discretegeometric setting for frame indifferent isotropic hyperelastic models. Theresulting discrete framework is in perfect adequacy with the multisymplecticdiscretization of fluids proposed earlier by the authors. Thanks to the unifieddiscrete setting, a geometric variational discretization can be developed forthe coupled dynamics of a fluid impacting and flowing on the surface of anhyperelastic body. The variational treatment allows for a natural inclusion ofincompressibility and impenetrability constraints via appropriate penaltyterms. We test the resulting integrators in 2D and 3D with the case of abarotropic fluid flowing on incompressible rubber-like nonlinear models.

Unified discrete multisymplectic Lagrangian formulation for hyperelastic solids and barotropic fluids