  Diffusion behaviors of heterogeneous materials are of paramount importance in
many engineering problems. Numerical models that take into account the internal
structure of such materials are robust but computationally very expensive. This
burden can be partially decreased by using discrete models, however even then
the practical application is limited to relatively small material volumes.
  This paper formulates a homogenization scheme for discrete diffusion models.
Asymptotic expansion homogenization is applied to distinguish between (i) the
continuous macroscale description approximated by the standard finite element
method and (ii) the fully resolved discrete mesoscale description in a local
representative volume element (RVE) of material. Both transient and
steady-state variants with nonlinear constitutive relations are discussed. In
all the cases, the resulting discrete RVE problem becomes a simple linear
steady-state problem that can be easily pre-computed. The scale separation
provides a significant reduction of computational time allowing the solution of
practical problems with a negligible error introduced mainly by the finite
element discretization at the macroscale.
