Superelement modelling of elastic metamaterials: Complex dispersive properties of three-dimensional structured beams and plates

• Published in: Journal of sound and vibration 2020-10, Vol.484, p.115499, Article 115499
• Main Authors: Tallarico, Domenico, Hannema, Gwenael, Miniaci, Marco, Bergamini, Andrea, Zemp, Armin, Van Damme, Bart
• Format: Article
• Language: English
• Subjects: Algorithms; Asymptotic methods; Computing time; Degrees of freedom; Dispersion; Dispersive properties; Eigenvalues; Eigenvectors; Elastic metamaterials; Elastic properties; Engineering Sciences; Materials; Mathematical analysis; Matrix; Matrix methods; Mechanical properties; Metamaterials; Numerical reduction; Reduction; Stiffness matrix; Studies; Three dimensional models; Unit cell; Vectors (mathematics)
• ISSN: 0022-460X; 1095-8568
• DOI: 10.1016/j.jsv.2020.115499

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. •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.