  This paper formulates variational integrators for finite element
discretizations of deformable bodies with heat conduction in the form of finite
speed thermal waves. The cornerstone of the construction consists in taking
advantage of the fact that the Green-Naghdi theory of type II for
thermo-elastic solids has a Hamiltonian structure. Thus, standard techniques to
construct variational integrators can be applied to finite element
discretizations of the problem. The resulting discrete-in-time trajectories are
then consistent with the laws of thermodynamics for these systems: for an
isolated system, they exactly conserve the total entropy, and nearly exactly
conserve the total energy over exponentially long periods of time. Moreover,
linear and angular momenta are also exactly conserved whenever the exact system
does. For definiteness, we construct an explicit second-order accurate
algorithm for affine tetrahedral elements in two and three-dimensions, and
demonstrate its performance with numerical examples.
