Dynamic nonlinear multicontinuum homogenization of systems with intrinsically evolving microstructure

TL;DR

This work develops a dynamic multicontinuum homogenization framework for nonlinear problems with evolving microstructure, defining Continua through a fine-scale variable such as concentration. It formulates two macroscopic schemes, Galerkin and mixed multicontinuum, by constructing multicontinuum expansions and solving local cell problems to obtain effective coefficients. The approach is tested on gravity-driven fingering and high-contrast viscous/interface problems across dual- and triple-continuum media, with both constant and heterogeneous parameters, demonstrating accurate representation of coarse-scale fields. The results indicate that macroscopic models derived within this dynamic continua framework closely reproduce reference fine-scale averages, validating the method’s potential for efficient and accurate simulations of evolving multiscale systems.

Abstract

In this paper, we propose a multicontinuum homogenization approach for nonlinear problems involving dynamically evolving multiscale media. The main idea of the proposed approach is that one of the fine-scale variables defines continua. It allows us to formulate macroscopic variables and derive new macroscopic models for nonlinear problems, where coefficients can depend on fine-scale functions. As an example, we consider a fingering problem and employ the fine-scale concentration field to define continua. We consider both Galerkin and mixed multicontinuum modeling approaches. In the former, the multicontinuum theory is applied to the pressure and concentration fields; in the latter, it is also applied to the velocity field. In both approaches, we provide multicontinuum expansions, formulate cell problems, and derive the corresponding macroscopic models. We present numerical results for model problems of gravity-driven fingering, viscous fingering, and interface flattening driven by high-contrast flow. The results show that the macroscopic models, derived with the proposed approach, can provide an accurate representation of the coarse-scale solutions.

Figures (16)

• Figure 1: A snapshot of the solution
• Figure 2: Illustration of the domain $\Omega$, coarse block $\omega$, oversampled RVE $R_\omega^+$, and RVE $R_\omega$
• Figure 3: Computational grids in the cases of the two-sided extended domain and the right-extended domain (from left to right).
• Figure 4: Initial and final fine-scale concentration distributions. Gravity-driven fingering with two continua.
• Figure 5: Multicontinuum velocity and concentration plots (from left to right) at the final time. Gravity-driven fingering with two continua.