Convergence of iterative methods based on Neumann series for composite materials: theory and practice
For researchers using iterative FFT methods for composite materials, this work clarifies convergence limitations and discretization effects, but is incremental as it confirms known issues.
The paper analyzes the convergence of iterative FFT methods for computing fields in composite materials, using an exact formula for a model system. It finds that theoretical convergence ranges (including negative conductivity) are not matched numerically due to discretization errors, and no single method outperforms others across all microstructures.
Iterative Fast Fourier Transform methods are useful for calculating the fields in composite materials and their macroscopic response. By iterating back and forth until convergence, the differential constraints are satisfied in Fourier space, and the constitutive law in real space. The methods correspond to series expansions of appropriate operators and to series expansions for the effective tensor as a function of the component moduli. It is shown that the singularity structure of this function can shed much light on the convergence properties of the iterative Fast Fourier Transform methods. We look at a model example of a square array of conducting square inclusions for which there is an exact formula for the effective conductivity (Obnosov). Theoretically some of the methods converge when the inclusions have zero or even negative conductivity. However, the numerics do not always confirm this extended range of convergence and show that accuracy is lost after relatively few iterations. There is little point in iterating beyond this. Accuracy improves when the grid size is reduced, showing that the discrepancy is linked to the discretization. Finally, it is shown that none of the three iterative schemes investigated over-performs the others for all possible microstructures and all contrasts.