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.

中文翻译：

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弹性超材料的超元建模：三维结构梁和板的复杂色散特性

摘要 在本文中，我们提供了一种具有一维和二维平移对称性的三维微结构固体的色散特性的快速计算方法。我们回顾了获得此类结构的复色散图的数学方法（即作为频率函数的布洛赫向量）。这可以通过解决 Bloch 向量中非线性的 Bloch-Floquet 特征值问题来完成。随着单元格所需的自由度数的增加，这种问题不可避免地需要减少方法。因此，采用频率渐近技术来逼近晶胞的动态刚度矩阵。这是通过识别和保留最重要的节点自由度来完成的，这些节点自由度用于生成单位单元“超单元”。