A highly efficient semi-implicit two-field modeling for thermal contact problems with non-Fourier effects

Abstract

In this paper, we revisit the thermal interface condition of thermal contact prob- lems with a nonzero thermal resistance. We point out that continuity conditions of the flux at the interfaces, which usually adopt the form of temperature gradient, leads to a paradoxical result if one of the adjacent bodies has zero temperature gradient. Specifically, that using the temperature gradient instead of the real heat flux yields a nonphysical temperature profile and fails to depict the thermal jump in cases with zero temperature gradients. We then show that the two-field (tem- perature and flux) formalism, in lieu of the classical single-field (temperature) naturally resolves this paradox. We propose an easy-to-implement semi-implicit scheme for this two-field formulation. The thermal interface problem is formu- lated in the fashion of differential algebraic equations (DAEs), in which the inter- face conditions are treated as algebraic equations constraining the associated heat process governed by differential equations. Our method can precisely fulfill the interface condition with the magnitude of machine error (10−16) and is computationally more efficient (around 8 to 10× faster) since it only needs to solve a very small Poisson-like system. End-to-end 1D and 2D simulations are presented to demonstrate the versatility and efficacy of our new method.