声子晶体中高维高波数弹性波传播问题是困扰计算力学、物理界的一个非常有挑战性的难题。本课题以此为应用背景发展高维高波数弹性波带隙结构的径向基函数小波方法。从理论角度深入研究径向基函数小波方法在声子晶体中高维高波数波传计算方面的有效性和优越性，讨论其数值解和收敛性，并从实验角度检验相关的理论结果。基本的研究思路是①针对高维周期声子晶体，将波动方程中的位移场和弹性参数等物理量以周期径向基函数小波展开，根据变分原理，将初始波动方程转化为矩阵本征值方程，通过求解本征值得到本征频率与波矢之间的色散关系，即能带结构。②针对高维准周期声子晶体，依据高维空间的周期性引进径向基函数小波分析技术；借助 Born-von Karman周期性条件引入超胞的概念，致使周期体系的径向基函数小波方法得以实现。本课题的目标是能够稳定、高精度、无网格、高效率地计算声子晶体中各类高维高波数弹性波传播问题。

High-dimension and high-wavenumber elastic wave propagation in phononic crystals is a very challenging problem that plagues computational mechanics, the physics community. This topic as application background to the development of radial basis function wavelet method which calculates high-dimension and high-wavenumber elastic wave band gap structures. From the theoretical point of view, we deeply study the effectiveness and superiority of the radial basis function wavelet method on computing high-dimension and high-wavenumber wave propagation in phononic crystals, discuss its numerical solution and convergence, and test the theory results from the experimental point of view. The basic idea is that ① the displacement field and elastic parameters in the wave equation are expanded on periodic radial basis function wavelet for high-dimensional phononic crystals, according to the variational principle, the original wave equation is transformed into a matrix eigenvalue equation, the dispersion relation between the intrinsic frequency and wave vector is obtained by solving the intrinsic value, and that is the band structure. ② for high-dimensional quasi-periodic phononic crystals, radial basis function wavelet method can be achieved by the introduction of radial basis function wavelet analysis techniques based on the periodicity of the high-dimensional space and the concept of the supercell with the Born-von Karman periodic conditions. The objective of this project is to steadily, high-precisionly, meshless efficiently calculate all kinds of high-dimension and high-wavenumber elastic wave propagation in phononic crystals.