The algorithm for calculating the band structures of two-dimensional phononic crystals (PCs) using the boundary element method (BEM) has been proposed for many years. However, it has not yet been extended to three-dimensional (3D) PCs because the fundamental solutions of 3D dynamics are complex and are related to angular frequency. In this study, the BEM is applied to calculate the band structures of 3D solid-fluid coupling PCs by combining the dual reciprocity method. The use of the dual reciprocity method can help avoid nonlinear eigenvalue problems. To address the limitation of fully populated matrices in BEM, the wavelet compression method based on B-spline wavelet on the interval is adopted. Some small matrix entries, which are generated by the vanishing moment and local support characteristics, are set to zero using the provided truncation technique. This process results in the production of sparse matrices. The constructed generalized linear eigenvalue equations are modified to tackle the ill-conditioned matrix issues arising from the distinct fundamental solutions of solids and fluids. The results show that this method is superior to the finite element method in terms of calculation efficiency. Compared to conventional BEM, the current wavelet BEM can not only generate sparse matrices but also reduce the integration calculation time when handling large-scale problems.

中文翻译：

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双互易边界元法和小波压缩法计算三维固流耦合声子晶体能带结构

使用边界元法（BEM）计算二维声子晶体（PC）能带结构的算法已经提出多年。然而，它尚未扩展到三维（3D）PC，因为3D动力学的基本解决方案很复杂并且与角频率相关。本研究采用边界元法结合对偶互易法计算3D固流耦合PCs的能带结构。使用对偶互易法可以帮助避免非线性特征值问题。针对BEM中满填充矩阵的局限性，采用基于区间B样条小波的小波压缩方法。使用所提供的截断技术将由消失矩和局部支撑特性生成的一些小矩阵条目设置为零。该过程导致稀疏矩阵的产生。修改构造的广义线性特征值方程来解决由固体和流体的不同基本解引起的病态矩阵问题。结果表明，该方法在计算效率方面优于有限元法。与传统边界元法相比，当前的小波边界元法不仅可以生成稀疏矩阵，而且在处理大规模问题时可以减少积分计算时间。