The paper introduces a spectro-hierarchical algorithm for approximating solutions to elasto-dynamic problems in periodic heterogeneous materials. This homogenization technique leverages finite-dimensional perturbation theory based on Nekhoroshev’s theorem. The method uses a harmonic hierarchy to construct functions that approximate the solution with a superexponentially small error relative to the cell dimension. The fully homogenized model is connected to the “integrable case” of perturbation theory, with all constants explicitly calculated. Additionally, a threshold for the cell dimension is established to guarantee theory validity. An application example is provided for a layered material.
Fortunati et al. (2024) studied this question.