本项目主要探索局部径向基函数无网格配点法在复杂几何形状的周期结构材料（声子晶体）弹性波频带结构中的应用，建立其快速有效的计算方法，从理论角度深入研究该方法在复杂几何结构的周期结构材料若干波动特性计算方面的有效性和优越性。主要内容包括：首先，通过改进虚拟点方法来解决好局部径向基函数无网格配点法在复杂几何边界上求解偏导数上所带来的不稳定性，探索二维复杂几何形状的周期结构材料弹性体波频带结构计算的新方法；在此基础上，针对固-固、固-流、流-固等混合体系，考查局部径向基函数无网格配点法在计算复杂几何形状的周期结构材料的波传问题的有效性。进一步将该方法推广应用于三维周期结构材料的体波能带结构的计算中；考虑三维复杂几何结构的声子晶体的能带结构计算；结合上述理论计算分析，通过与其他数值算法的结果进行比较，检验相关的理论结果。该项目的研究有望为三维复杂几何形状的周期材料结构提供新的快速有效的计算方法。

In this project, we develop an efficient numerical tool for the band structure computation of the elastic wave in periodic materials (phononic crystals) with a scatterer of arbitrary geometry based on local radial basis function collocation method (LRBFCM). The efficiency and stability of the LRBFCM for different elastic wave problems in periodical materials with a scatterer of arbitrary geometry shapes are presented. The project are followed by the following steps: Firstly, the improved numerical techniques based on the fictitious nodes method are proposed to deal with the instability caused by the derivative calculation in the LRBFCM at the interface of arbitrary geometry shapes. By doing this, the developed LRBFCM can be applied to the band structure of 2D phononic crystals with a scatterer of arbitrary geometry shape; Secondly, the band structure of 2D phononic crystals of different systems, such as solid-solid,solid-fluid, fluid-solid systems, are fully considered to present the efficiency of the developed LRBFCM. Based on the previous work in 2D case, the LRBFCM is further applied to the band structure calculation of 3D phononic crystals with a scatterer of arbitrary geometry shapes. The numerical results of the LRBFCM are validated by comparing with those results obtained by using other numerical methods. This project aims to offer a fast numerical tool for the analysis of the 3D periodic materials with a scatterer of arbitrary geometry.