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Description
The study of periodic structures (phononic crystals or locally resonant metamaterials) for noise and vibration reduction has led to a plethora of effective modelling tools. For many purposes, in particular the identification of band gaps, dispersion analysis based on the model of a unit cell is a first step in the assessment of the desired functionality. If the unit cell has a complex shape, this calculation is typically done through the wave finite element model (WFEM) approach. The unit cell is meshed and represented by a mass, stiffness, and damping matrix. Periodicity is invoked by applying the Bloch theorem to the appropriate degrees of freedom. Commercial software typically allows indirect dispersion calculation where the frequency f is calculated as a function of a chosen wave vector k. The direct determination of the dispersion (calculating k for a fixed f) is straightforward but leads to expensive polynomial eigenvalue problems.The dynamic properties of periodic media can be enhanced by including materials that are classically used for noise and vibration treatment, such as acoustic absorbers and viscoelastic media. They are represented in the frequency domain by frequency-dependent homogenized properties. Viscoelastic media are described by a complex modulus, where the real part increases with frequency and the imaginary part represents the damping. Porous absorbers can be represented by two complex and frequency-dependent properties: density and wave speed. This works shows how the WFEM of 1D and 2D periodic structures can be performed in multi-material domains, where at least one material is frequency-dependent.