WIAS Preprint No. 693, (2001)

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


  • Brown, Calum M.
  • Dreyer, Wolfgang
  • Müller, Wolfgang H.

2010 Mathematics Subject Classification

  • 42B10 45A05 15A09 74E30


  • Discrete Fourier transforms, Neumann iteration, fixpoint theorem, composites


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.

Appeared in

  • Proc. R. Soc. A, 458 (2002), pp. 1-21

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