A unit cell based multilevel substructuring method for fast vibration response calculations of finite metamaterial structures

Locally resonant metamaterial structures have gained significant attention across multiple engineering disciplines due to their ability to exhibit vibration stop bands not found in regular materials. These structures are composed of an assembly of unit cells, which are often discretized into large finite element models due to their sub-wavelength nature and intricate design. Moreover, due to the contribution of local dynamics of resonator modes, the overall modal density of the entire structure is proportional to the number of unit cells multiplying the number of resonator modes. Therefore, high-fidelity frequency response analyses of such large-scale structures with high modal density are typically computationally expensive, making them impractical for structural design. In order to efficiently solve these models, the multilevel substructuring method is often used for a high level of dimensional reduction while balancing the errors associated with truncated component mode synthesis. However, accurate and efficient modeling of complex dynamics of metamaterial structures containing a large number unit cells still poses challenges for conventional multilevel substructuring method. Three main issues arise in this context: (i) Block Gaussian elimination becomes inefficient for large models; (ii) Ignoring mass coupling and load information during the reduction weakens accuracy, especially around the critical stop-band frequencies; (iii) Existing error estimation is not directly applicable to frequency response analyses. This work overcomes these challenges by introducing a multilevel assembly strategy, an improved interface reduction and a heuristic truncation criterion. Doing so, these advancements facilitate efficient and accurate frequency response analyses for assemblies with many unit cells, thereby enabling the practical design of locally resonant metamaterial structures in various engineering applications.