Parametric model order reduction by matrix interpolation for fast reanalysis of wave properties of two-dimensional periodic structures

The analysis of wave propagation and band gaps in periodic structures with 2D or 3D cells (substructures), in two dimensions, is a common research topic, focused on a broad range of industrial applications. For instance, the dispersion diagrams and Bloch mode shapes can be obtained using the standard ω(k) method for substructures with square boundaries and pseudo-periodic conditions, and for wave vector k spanning the contour of the irreducible Brillouin zone. Among the efficient strategies to compute the dispersion diagrams is via reduced substructure models. For instance, those reduced models can be created by considering a small number of low-order Craig-Bampton modes enriched with high-order static modes to describe the internal degrees of freedom, and a small number of boundary modes to describe the interface degrees of freedom. However, for parametric analyzes involving substructure modifications, those numerical strategies may turn cumbersome as they require computing the component modes (Craig-Bampton modes and boundary modes) and the reduced substructure matrices (Galerkin projections) many times. To solve this issue, an interpolation strategy can be proposed as follows. First, the reduced substructures matrices are computed for a small set of points within a parametric space. Next, an interpolation scheme between these points is proposed to quickly assess the substructure matrices for any arbitrary parameter values. Numerical examples are presented to discuss the accuracy and efficiency of the proposed interpolation strategy, where the focus is on the analysis of 2D and 3D substructures with geometric changes, and on the creation and variation of band gaps in those periodic structures.