  We present a method of calculation of the effective magnetic permeability of
magnonic metamaterials containing periodically arranged magnetic inclusions of
arbitrary shapes. The spectrum of spin wave modes confined in the inclusions is
fully taken into account. Within the scope of the proposed method, we compare
two approaches. The first approach is based on a simple semi-analytical theory
that uses the numerically calculated susceptibility tensor of an isolated
inclusion as input data. Within the second approach, micromagnetic packages
with periodic boundary conditions (PBC) are used to calculate the
susceptibility of a single 2D periodic array of such inclusions, with the whole
3D metamaterial consisting of a stack of such arrays. To calculate the
susceptibility tensor of an isolated inclusion, we have implemented and
compared two different methods: (a) a micromagnetic method, in which we have
employed three different micromagnetic packages: the finite element package
NMAG and the two finite differences packages OOMMF and MicroMagus; and (b) the
modified dynamical matrix method. To illustrate the methodology, we have
calculated the effective permeability of a metamaterial consisting of a stack
of hexagonal arrays of magnetic nanodisks in a non-magnetic matrix. The range
of geometrical parameters for which such a metamaterial is characterized by the
negative permeability has been identified. The critical comparison of the
different micromagnetic packages and the dynamical matrix method (based on the
calculation of the susceptibility tensor of an isolated inclusion) has
demonstrated that their results agree to within 3 %.
