The method of quasiperiodic fields for thermal conduction in periodic heterogeneous media: A theoretical analysis

In this paper, a new theoretical method for upscaling the thermal conduction equation in periodic composite materials is presented. That method is named “the method of quasiperiodic fields”, and it is based on the assumption that the temperature field is quasiperiodic at the small scale. More specifically, that method allows to predict the form of the averaged governing equations, and the values of the effective transport coefficients. The new methodology involves four steps: (i) a first level averaging, (ii) the statement of a quasiperiodic problem, (iii) the development of factorized quasiperiodic problems, and (iv) the development of the closed form of the averaged thermal conduction equation. The values of the effective thermal conductivity obtained with that new method were compared with values reported in literature, and also with values obtained from direct numerical simulations.