An Efficient Homogenization Method using the Trigonometric Interpolation and the Fast Fourier Transform

Ngoc-Trung Nguyen (1) , Christian Licht (2) , Jin-Hwe Kweon (3)

1 Department of Mechanical & Biomedical Engineering
2 M3 - Modélisation Mathématique en Mécanique
3 School of Mechanical and Aerospace Engineering [Korea]

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 vari- ables 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.

Mots clés: homogenization; Augmented Lagrangian method; trigonometric interpolation

Domaines: Matériaux et structures en mécanique [physics.class-ph]; Matériaux et structures en mécanique [physics.class-ph]; Mécanique des solides [physics.class-ph]; Mécanique des solides [physics.class-ph]; Mécanique des structures [physics.class-ph]; Mécanique des structures [physics.class-ph]

Ngoc-Trung Nguyen, Christian Licht, Jin-Hwe Kweon. An Efficient Homogenization Method using the Trigonometric Interpolation and the Fast Fourier Transform. Vietnam Journal of Mechanics, 2011, 33 (4), pp.1-9. ⟨10.15625/0866-7136/33/4/255⟩. ⟨hal-00727187⟩