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We present a method of calculation of the effective magnetic permeability ofmagnonic metamaterials containing periodically arranged magnetic inclusions ofarbitrary shapes. The spectrum of spin wave modes confined in the inclusions isfully taken into account. Within the scope of the proposed method, we comparetwo approaches. The first approach is based on a simple semi-analytical theorythat uses the numerically calculated susceptibility tensor of an isolatedinclusion as input data. Within the second approach, micromagnetic packageswith periodic boundary conditions (PBC) are used to calculate thesusceptibility of a single 2D periodic array of such inclusions, with the whole3D metamaterial consisting of a stack of such arrays. To calculate thesusceptibility tensor of an isolated inclusion, we have implemented andcompared two different methods: (a) a micromagnetic method, in which we haveemployed three different micromagnetic packages: the finite element packageNMAG and the two finite differences packages OOMMF and MicroMagus; and (b) themodified dynamical matrix method. To illustrate the methodology, we havecalculated the effective permeability of a metamaterial consisting of a stackof hexagonal arrays of magnetic nanodisks in a non-magnetic matrix. The rangeof geometrical parameters for which such a metamaterial is characterized by thenegative permeability has been identified. The critical comparison of thedifferent micromagnetic packages and the dynamical matrix method (based on thecalculation of the susceptibility tensor of an isolated inclusion) hasdemonstrated that their results agree to within 3 %.

High frequency permeability of magnonic metamaterials with magnetic inclusions of complex shape