Non-linear finite element formulation applied to thermoelectric materials under hyperbolic heat conduction model

In the present work, a three-dimensional, dynamic and non-linear finite element to simulate thermoelectric behavior under a hyperbolic heat conduction model is presented. The transport equations, which couple electric and thermal energies by the Seebeck, Peltier and Thomson effects, are analytically obtained through extended non-equilibrium thermodynamics, since the local equilibrium hypothesis is not valid under the hyperbolic model. In addition, unidimensional analytical solutions are obtained to validate the finite element formulation. Numerically, isoparametric eight-node elements with two degrees of freedom (voltage and temperature) per node are used. Non-linearities due to the temperature-dependence on the transport properties and the Joule effects are addressed with the Newton-Raphson algorithm. For the dynamic problem, HHT and Newmark-β algorithms are compared to obtain accurate results, since numerical oscillations (Gibbs phenomena) are present when the initial boundary conditions are discontinuous. The last algorithm, which is regularized by relating time steps and element sizes, provides the best results. Finally, the finite element implementation is validated, comparing the analytical and the numerical solutions, and a three-dimensional example is presented.