Extended bloch mode synthesis: Ultrafast method for the computation of complex band structures in phononic media

In this article an efficient numerical technique, named Extended Bloch Mode Synthesis, is proposed for the fast calculation of the elastic complex band structures in phononic media. The Bloch Mode Synthesis approach, originally developed for reducing the computational cost for the calculation of real band structures, is here extended to evaluate also evanescent/complex near field wave solutions by solving a k(ω) Bloch eigenvalue problem. The k(ω) Bloch eigenvalue problem is built by means of a Wave Finite Element (WFE) discretization of the unit cell combined with a Component Mode Synthesis approach. The Component Mode Synthesis approach is based on the Craig Bampton modal reduction of the interior unit cell degrees of freedom and provides a basis to reduce the Bloch eigenproblem dimension allowing for a fast computation of the complex band structures. The performances of the proposed scheme in terms of band structures accuracy and computational cost saving are demonstrated for a phononic stubbed plate, a case already used in literature as a benchmark for band structures calculation. It is shown that the complex band structures computational time can be reduced by two orders of magnitude with respect to the computational time needed to solve the full model with negligible errors for both real and evanescent/complex solutions.