Spectro-hierarchical homogenization scheme for elasto-dynamic problems in periodic Cauchy materials

Abstract

A spectro-hierarchical algorithm is proposed to determine an approximate solution to the elasto-dynamic problem for periodic heterogeneous materials. Such a homogenization scheme is derived by employing tools from perturbation theory in finite dimension used in the context of the celebrated Theorem of Nekhoroshev. In particular, it is shown how a classical algorithm based on a suitable hierarchy of harmonics can be implemented for the problem at hand, leading to the explicit construction of functions approximating the solution of the original problem with an error that is superexponentially small in the cell dimension. According to this approach, the fully homogenized model turns out to be naturally related to the “integrable case” of perturbation theory. Furthermore, all the featured constants are estimated explicitly. More importantly, a fully detailed bound of the threshold for the cell dimension is presented to ensure the validity of the theory. An example of application of the described theory is given in the final section for the case of a layered material.