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. | Vietnam Journal of Mechanics, VAST, Vol. 33, No. 4 (2011), pp. 215 – 223 AN EFFICIENT HOMOGENIZATION METHOD USING THE TRIGONOMETRIC INTERPOLATION AND THE FAST FOURIER TRANSFORM Ngoc-Trung Nguyen1 , Christian Licht2 and Jin-Hwe Kweon3 Kangwon National University, Chunchon, Kangwon-do, South Korea 2 LMGC, Université de Montpéllier 2, France 3 School of Mechanical and Aerospace Engineering Gyeongsang National University, South Korea 1 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. 1. INTRODUCTION For composites of complex microstructures, there are two different scales associated with microscopic and macroscopic behaviors to deal with: the slowly varying global variables and the rapidly oscillating local variables. To model a structure of such kind of material using the finite element method (FEM) one should utilize very fine mesh density so that the details at the microscale size can be captured. That leads to a very high computational cost and sometimes it is impossible to perform the analysis due to extremely high .
