FFT based iterative schemes for composite conductors with uniform boundary conditions

Abstract

In the present paper, we extend the FFT method to deal with the homogenization problem of composite conductors with uniform boundary conditions. The principle of the approach consists of applying a transformation to build a periodic problem from the solution with uniform boundary conditions. It is shown that the related periodic problem must be applied to an extended domain obtained by mirror symmetry of the unit cell. The conductivity equation must then be solved on this extended domain under an applied periodic polarization field as a loading parameter. Illustrations are provided and the effective conductivity obtained with FFT is compared to finite element solutions to validate the approach. The proposed method can be applied to microstructure geometries of all kinds, including cells obtained through imaging devices.