基于多重多级子结构声子能带与传输特性分析

基于多重多级子结构方法提出一种快速的声子晶体能带与传输特性的计算策略. 主要思想是将声子晶体划分成多层级子结构有限元模型，在能带计算中采用静凝聚和子结构周游树技术将子结构的内部刚度阵凝聚到Bloch 边界上. 由于内部刚度阵并不随着简约波矢变化，所以这种计算策略可以大大降低求解规模并提高计算效率，并不对整体有限元模型引入近似. 在传输特性计算中同样采用该策略，由于声子晶体单胞具有周期性，所以各个单胞的系数矩阵是相同的，从而减少计算量，并且可以灵活地选择是否回代求解单胞内部自由度. 数值算例以三维局域共振型声子晶体和二维Bragg 散射型声子晶体为例，计算结果验证了这种求解策略的正确性和高效性，并适用于复杂声子晶体分析. 

BAND STRUCTURE AND TRANSMISSION CHARACTERISTICS CALCULATION OF PHONONIC CRYSTALS BASED ON MULTI-LEVEL SUBSTRUCTURE TECHNIQUE

A fast scheme based on multi-level substructure technique is proposed to analyze band structure and transmission characteristics of phononic crystals. The main idea is that the finite element model of phononic crystal is divided into several substructures by a special multi-level decomposition. For the calculation of band structure, tree travelling technique and static condensation method are used to convert the internal stiffness matrix into the Bloch boundary of unit cell. Because the internal stiffness matrix does not change along with reduced wave vector, the scheme can reduce computation budget and improve efficiency greatly, while it does not introduce approximation into traditional finite element model. For the calculation of transmission characteristics, the proposed scheme can be also used to reduce computation, because unit cell of phononic crystal is periodic which has the same coefficient matrix. Moreover, the back substitution of internal displacements of unit cell can be selected flexibly. Some examples of three-dimensional locally resonant system and two-dimensional Bragg scattering phononic crystal are analyzed. Numerical results indicate that the proposed scheme is efficient and accurate, and suitable for complex phononic crystal problems.