Employing the discrete fourier transform in the analysis of multiscale problems

The idea of employing the discrete Fourier transform casted as the representative cell method for the solution of multiscale problems is illustrated. Its application in combination with analytical (structural mechanics methods, Wiener-Hopf method, integral transform methods) and numerical (finite element method, higher-order theory) methods is demonstrated. Both cases of 1-D and 2-D translational symmetry are addressed. In particular, the problems for layered, cellular, and perforated materials with and without flaws (cracks) are considered. The method is shown to be a convenient and universal analysis tool. Its numerical efficiency allowed us to solve optimization problems characterized by multiple reanalysis.