Numerical Model Reduction of Multi-Scale Electrochemical Ion Transport

Journal article, 2026

In this paper, we develop a Numerical Model Reduction (NMR) framework for multi-scale modeling of electro-chemically coupled ion transport. Upon introducing the governing equations and employing Variationally Consistent Homogenization, a two-scale model, consisting of a macro-scale and a sub-scale part, is obtained. Instead of solving for the computationally expensive FE² simulation, where the macro-scale and sub-scale problems are solved in a nested fashion, we exploit NMR by training a surrogate model that replaces the sub-scale finite element simulations. The surrogate model is trained by performing Proper Orthogonal Decomposition on snapshots of the primary fields. Each macro-scale quadrature point is no longer occupied by a Representative Volume Element simulation; instead, it is replaced by a surrogate model that consists of a system of Ordinary Differential Equations. In this way, a computationally efficient solution scheme for solving two-scale problems is obtained.