Empirical Interscale Finite Element Method (EIFEM): a data-driven multiscale framework for nonlinear heterogeneous structures

The Empirical Interscale Finite Element Method (EIFEM) is a data-driven multiscale approach for efficient simulation of nonlinear heterogeneous and architected materials. It integrates domain decomposition, reduced-order modeling, and hyperreduction to connect fine- and coarse-scale responses with high accuracy.

Each subdomain displacement is split into interface-induced and orthogonal nonlinear “bubble” components, the latter serving as internal coarse degrees of freedom. These are parametrized by linear or nonlinear expansions (e.g., POD or neural-network-based). EIFEM uses a three-field variational formulation with Localized Lagrange Multipliers, where user-defined interface modes act as coarse-scale boundary DOFs. This formulation establishes a direct mapping between coarse displacements and fine stresses, removing the need for FE²-type nested iterations and ensuring compatibility with standard FEM codes.

Computational efficiency is achieved through the Continuous Empirical Cubature Method (CECM), which builds sparse integration rules for accurate reduced integration. Applications to nonlinear beams and metamaterial lattices demonstrate over three orders-of-magnitude reduction in unknowns and integration points while preserving strain-energy errors below 1%.