Superelement modelling of elastic metamaterials: Complex dispersive properties of three-dimensional structured beams and plates. (13th October 2020) - Tallarico, Domenico; Hannema, Gwenael; Miniaci, Marco; Bergamini, Andrea; Zemp, Armin; Van Damme, Bart
Abstract: In this article, we offer a fast calculation method for the dispersive properties of three-dimensional micro-structured solids with one-dimensional and two-dimensional translation symmetry. We review mathematical methods to obtain the complex dispersion diagram for such structures ( i . e . the Bloch vector as a function of frequency). This can be done by solving a Bloch-Floquet eigenvalue problem which is non-linear in the Bloch vector. Such a problem inevitably calls for reduction methods as the required number of degrees of freedom of the unit cell increases. Therefore, an asymptotic-in-frequency technique is implemented in order to approximate the dynamic stiffness matrix of the unit cell. This is done by identifying and retaining the most significant nodal degrees-of-freedom, which are used to generate a unit cell "superelement". The accuracy of the Bloch vectors and corresponding eigenvectors associated with the reduced non-linear eigenvalue problem is demonstrated by direct comparison to full-size computations and shows excellent agreement combined with considerable computing time reduction and controllable limitations. Highlights: We review of dispersion calculation and reduction methods for analyzing periodic micro-structured solids. We study an algorithmic search for a ''retained set'' of dynamically significant finite-element (FE) degrees-of-freedom We define a reduced FE mass and stiffness matrices of unit cells (a "superelement") using the retained set. We compare superelement-based complex dispersion diagrams of periodic media to full FE models. We discuss the controllable accuracy and computational efficiency of the novel method.