Abstract In this paper, the Petrov-Galerkin finite element interface method is modified to the vectorial form and applied to compute the band structure of phononic crystals with complicated scatterer geometry. Value-periodic subspace and projective grid are employed in this method. Complete mathematical model together with the continuity and equilibrium conditions on the scatterer interface as well as the Bloch-periodic boundary conditions on the unit cell are presented for these systems. We then apply this method to compute the band structure of three kinds of special phononic crystals: phononic crystal with anisotropic inclusions, phononic crystal containing piezoelectric materials and phononic crystal containing nano-piezoelectric materials. Taking advantage of asymmetric basis functions and non-body-fitted meshes, our method is suitable for the calculation and analysis of phononic crystals with complicated scatterer geometries. With plenty of numerical experiments, the accuracy and convergency of the proposed method is demonstrated. The influences of the rotation angle of anisotropic inclusion, complicated scatterer shape, filling fraction of the scatterers, and the rotation angle of the scatterers to the band structure of these three kinds of phononic crystals are investigated. This will provide a new perspective for the design and manufacture of phononic crystals with specific band structures.

中文翻译：

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用于计算各向异性和压电声子晶体能带结构的 Petrov-Galerkin 方法

摘要 本文将Petrov-Galerkin有限元界面法改进为矢量形式，并应用于计算具有复杂散射体几何形状的声子晶体的能带结构。该方法采用值周期子空间和投影网格。给出了这些系统的完整数学模型以及散射体界面上的连续性和平衡条件以及晶胞上的布洛赫周期边界条件。然后我们应用该方法计算了三种特殊声子晶体的能带结构：具有各向异性夹杂物的声子晶体、含有压电材料的声子晶体和含有纳米压电材料的声子晶体。利用非对称基函数和非贴体网格，我们的方法适用于具有复杂散射体几何形状的声子晶体的计算和分析。通过大量的数值实验，证明了所提出方法的准确性和收敛性。研究了各向异性包裹体的旋转角度、复杂的散射体形状、散射体的填充率、散射体的旋转角度对这三种声子晶体能带结构的影响。这将为具有特定能带结构的声子晶体的设计和制造提供新的视角。研究了复杂的散射体形状、散射体的填充率以及散射体的旋转角度对这三种声子晶体能带结构的影响。这将为具有特定能带结构的声子晶体的设计和制造提供新的视角。研究了复杂的散射体形状、散射体的填充率以及散射体的旋转角度对这三种声子晶体能带结构的影响。这将为具有特定能带结构的声子晶体的设计和制造提供新的视角。