Multi-scale modelling for two-dimensional periodic structures using a combined mode/wave based approach

• We study wave propagation characteristics in 2D periodic beam grid.
• The proposed method combines mode-based method (CMS) with wave-based method (WFEM).
• We obtain a gain of 99% in computing time by the proposed method on reduced model.
• In the reduced model, the unit cell modes under 3∗fmax3∗fmax should be included.
• The waves at bounding frequencies of stop band are proved to be stationary.

In this article, an efficient numerical approach is proposed to study the vibration of two-dimensional periodic structures. The method combines the advantages of mode-based Component Mode Synthesis (CMS) and wave-based Wave Finite Element Method (WFEM). It begins with a modal description of a mesoscopic unit cell using CMS. Subsequently, WFEM is applied to the macroscopic structure, which is considered as a waveguide. It exploits fully the periodic propriety of the structure since only one unit cell needs to be modelled. The introduction of CMS is able to reveal the influence of local dynamics of unit cell on the global behaviour of the structure, and speed up the computation of eigen-problem. The wave-mode duality is discussed which assures the combination of the two methods. The effectiveness of the proposed approach is illustrated via an example of two-dimensional beam grid. The convergence criteria of the model reduction is given and the equivalence of cell modes and stationary waves at bounding frequencies of stop bands is verified.