A wavelet-based boundary element method is employed to calculate the band structures of two-dimensional phononic crystals, which are composed of square or triangular lattices with scatterers of arbitrary cross sections. With the aid of structural periodicity, the boundary integral equations of both the scatterer and the matrix are discretized in a unit cell. To make the curve boundary compatible, the second-order scaling functions of the B-spline wavelet on the interval are used to approximate the geometric boundaries, while the boundary variables are interpolated by scaling functions of arbitrary order. For any given angular frequency, an effective technique is given to yield matrix values related to the boundary shape. Thereafter, combining the periodic boundary conditions and interface conditions, linear eigenvalue equations related to the Bloch wave vector are developed. Typical numerical examples illustrate the superior performance of the proposed method by comparing with the conventional BEM.

采用基于小波的边界元方法计算二维声子晶体的能带结构，该声子晶体由具有任意截面散射体的正方形或三角形晶格组成。借助于结构周期性，散射体和矩阵的边界积分方程都在一个晶胞中离散化。为了使曲线边界兼容，使用区间上的B样条小波的二阶缩放函数来近似几何边界，而边界变量通过任意阶的缩放函数进行插值。对于任何给定的角频率，都给出了一种有效的技术来产生与边界形状有关的矩阵值。之后，结合周期性边界条件和界面条件，建立了与Bloch波向量有关的线性特征值方程。典型的数值示例通过与常规BEM进行比较，说明了所提出方法的优越性能。