Two-scale analysis of transient diffusion problems through a homogenized enriched continuum

This paper addresses the two-scale problem underlying the enriched continuum for transient diffusion problems, which was previously developed and tested at the single scale level only (Waseem et al., Comp.Mech, 65, 2020). For a linear material model exhibiting a relaxed separation of scales, a model reduction was proposed at the micro-scale that replaces the micro-scale problem with a set of uncoupled ordinary differential equations (ODEs). At the macro-scale, the balance law, the ODEs and the macroscopic constitutive equations collectively represent an enriched continuum description. Examining different discretization techniques, distinct solution methods are presented for the macro-scale enriched continuum. Proof-of-principle examples are solved for a mass diffusion system in which species diffuse slower in the inclusion than in the matrix. The results from the enriched continuum formulation are compared with the computational transient homogenization (CTH) and direct numerical simulations (DNS). Without compromising the solution accuracy, significant computational gains are obtained through the enriched continuum approach.