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.

本文讨论了瞬态扩散问题的丰富连续体下的两尺度问题，该问题先前仅在单一尺度水平上开发和测试过（Waseem et al。，Comp.Mech，65，2020）。对于表现出尺度松弛的线性材料模型，提出了在微观尺度上进行模型简化的方法，该模型简化方法是用一组未耦合的常微分方程（ODE）代替微观尺度问题。在宏观尺度上，平衡律，ODE和宏观本构方程共同代表了丰富的连续体描述。通过研究不同的离散化技术，针对宏丰富的连续体提出了不同的求解方法。解决了质量扩散系统的原理证明示例，其中物质在夹杂物中的扩散比基质中的扩散慢。将丰富的连续体公式化的结果与计算瞬态均化（CTH）和直接数值模拟（DNS）进行比较。在不影响求解精度的情况下，通过丰富的连续谱方法可以获得可观的计算收益。