Finite Element Analysis of Nonlinear Fully Coupled Thermoelectric Materials

Abstract

A complete nonlinear coupled finite element algorithm for thermoelectric materials is developed and implemented within a two-dimensional finite element code. Starting from a suitable formulation of the constitutive (including Seebeck, Joule, Peltier and Thompson effects) and equilibrium equations, residual vectors and consistent tangent matrices are formulated and implemented in a standard four-node isoparametric element with two degrees of freedom per node (voltage and temperature). The nonlinearities arise due to the coupling between the electric and thermal fluxes, to the dependence of the discretized material parameters with the independent variable temperature and to a convective-type term, representing the thermal energy that electrons carry. Three examples for Bi₂Te₃ thermoelements are presented. The first two compare the numerical results with simplified, one-dimensional analytical solutions. The first example is related to a linear uncoupled evaluation, the second to a nonlinear coupled Seebeck effect. Perfect agreement is obtained between the analytical (obtained by considering the material properties constant in the second case) and numerical solutions. The first two examples show that the dependency of the material coefficents is not important for one-dimensional cases. The third example, without direct analytical solution due to the complete coupling and material nonlinearity, studies the two-dimensional Peltier effect of a thermopair.