Discrete Fourier transforms and their application to stress-strain problems in composite mechanics: A convergence study

We present in this paper a method for determining the convergence characteristics of the Neumann iterative solution of a discrete version of a second-type Fredholm equation. Implemented as the so-called "equivalent inclusion problem" within the context of mechanical stress/strain analysis, it allows the modeling of elastically highly heterogeneous bodies with the aid of Discrete Fourier Transforms (DFT). A method is developed with which we can quantify pre-analysis (i.e., at iteration zero) the convergence behavior of the Neumann scheme depending on the choice of an auxiliary stiffness tensor, specifically for the linear-elastic case. It is shown that a careful choice of this tensor results in both guaranteed convergence and a smaller convergence radius for the solution. Furthermore, there is some indication that as the convergence radius decreases, the scheme may converge to a solution at a faster rate translating into an increase in computational efficiency