An efficient homogenization method using the trigonometric interpolation and the fast fourier transform

Abstract

This study focuses on formulation of the Augmented Lagrangian and application of the Uzawa's algorithm to solve the homogenization problem of microscopic periodic media as in composites. Unlike in the finite element model, an equally spaced grid system associated with the microstructure domain is used instead of a finite element mesh topology. Moreover, the trigonometric interpolations for the field variables at every grid point help to handle the periodic conditions. The proposed approach is a compromise between Lagrange multiplier and penalty methods, in that it enables exact representation of constraints while using penalty terms to facilitate the iteration procedure. A typical homogenization problem will be solved using this approach. The results show good consistency with those in literatures. Effects of the grid density and the penalty parameter on the convergence have also been investigated.

Keywords: Homogenization,  Augmented Lagrangian method, trigonometric interpolation.