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.

中文翻译：

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一种基于晶胞的多级子结构方法，用于有限超材料结构的快速振动响应计算

局部共振超材料结构由于能够表现出常规材料中不存在的振动停止带，因此在多个工程学科中受到了极大的关注。这些结构由晶胞组件组成，由于其亚波长性质和复杂的设计，这些晶胞通常被离散为大型有限元模型。此外，由于谐振器模式的局部动力学的贡献，整个结构的整体模态密度与晶胞数量乘以谐振器模式的数量成正比。因此，对这种具有高模态密度的大规模结构进行高保真频率响应分析通常计算成本高昂，因此对于结构设计来说不切实际。为了有效地求解这些模型，多级子结构方法通常用于高水平的降维，同时平衡与截断元件模式综合相关的误差。然而，对包含大量晶胞的超材料结构的复杂动力学进行准确高效的建模仍然对传统的多级子结构方法构成挑战。在这种情况下，出现了三个主要问题：（i） 块高斯消元法对于大型模型变得效率低下;（ii） 在减少过程中忽略质量耦合和载荷信息会削弱精度，尤其是在临界阻带频率附近;（iii） 现有的误差估计并不直接适用于频率响应分析。这项工作通过引入多级汇编策略、改进的界面缩减和启发式截断标准来克服这些挑战。 这样，这些进步有助于对具有许多晶胞的组件进行高效、准确的频率响应分析，从而在各种工程应用中实现局部谐振超材料结构的实际设计。