Iterative Fast Fourier Transform methods are useful for calculating thefields in composite materials and their macroscopic response. By iterating backand forth until convergence, the differential constraints are satisfied inFourier space, and the constitutive law in real space. The methods correspondto series expansions of appropriate operators and to series expansions for theeffective tensor as a function of the component moduli. It is shown that thesingularity structure of this function can shed much light on the convergenceproperties of the iterative Fast Fourier Transform methods. We look at a modelexample of a square array of conducting square inclusions for which there is anexact formula for the effective conductivity (Obnosov). Theoretically some ofthe methods converge when the inclusions have zero or even negativeconductivity. However, the numerics do not always confirm this extended rangeof convergence and show that accuracy is lost after relatively few iterations.There is little point in iterating beyond this. Accuracy improves when the gridsize is reduced, showing that the discrepancy is linked to the discretization.Finally, it is shown that none of the three iterative schemes investigatedover-performs the others for all possible microstructures and all contrasts.
展开▼
