A reduced Bloch operator finite element method for fast calculation of elastic complex band structures

This article presents an efficient reduced formulation of the Bloch Operator Finite Element method to calculate complex band structures of periodic waveguides. The use of a Bloch operator formulation allows building and solving a Bloch eigenvalue problem along a generic wave direction, thus being not limited to the unit cell Irreducible Brillouin Zone (IBZ) edges, so that band gap directionality and material absorption in elastic and damped waveguides can be fully disclosed. The proposed Reduced-Order Modeling (ROM) exploits a small set of Bloch modes, extracted at relevant frequency locations along one or more wave directions and post-processed with a Singular Value Decomposition, to reduce the dimensions of the eigenvalue problem. The performances of the proposed numerical technique are evaluated in terms of accuracy and computational saving by analyzing a linear elastic and a damped bi-periodic stubbed plate. Results demonstrate that the reduced formulation yields accurate predictions of propagative, evanescent and complex wave solutions with a reduction in computational time of more than one order of magnitude with respect to the full model calculations. Complex band structures can thus be efficiently computed over the whole IBZ