  We present a computational framework for two-scale asymptotic homogenization
to determine the intrinsic magnetic permeability of composites. To this end,
considering linear magnetostatics, both vector and scalar potential
formulations are used. Our homogenization algorithm for solving the cell
problem is based on the displacement method presented in Lukkassen et al. 1995,
Composites Engineering, 5(5), 519-531. We propose the use of the meridional
eccentricity of the permeability tensor ellipsoid as an anisotropy index
quantifying the degree of directionality in the linear magnetic response. As
application problems, 2D regular and random microstructures with overlapping
and nonoverlapping monodisperse disks, all of which are periodic, are
considered. We show that, for the vanishing corrector function, the derived
effective magnetic permeability tensor gives the (lower) Reuss and (upper)
Voigt bounds with the vector and scalar potential formulations, respectively.
Our results with periodic boundary conditions show an excellent agreement with
analytical solutions for regular composites, whereas, for random heterogeneous
materials, their convergence with volume element size is fast. Predictions for
material systems with monodisperse overlapping disks for a given inclusion
volume fraction provide the highest magnetic permeability with the most
increased inclusion interaction. In contrast, the disk arrangements in regular
square lattices result in the lowest magnetic permeability and inadequate
inclusion interaction. Such differences are beyond the reach of the isotropic
effective medium theories, which use only the phase volume fraction and shape
as mere statistical microstructural descriptors.
