插值型维数分裂无网格法中本质边界条件施加方法研究

本文以求解三维波动方程为例，介绍了改进的插值型维数分裂无单元Galerkin方法，推导了方程的弱形式，构造了具有插值特性的逼近函数，建立了可直接施加本质边界条件的离散方程组，研究不同本质边界条件施加方法对计算结果的影响．本文列举了三种常用的处理本质边界条件的方法：直接配点法、对角元素置大数法和对角元素化一法．选取了三个数值算例，分别采用不同的本质边界条件施加方法，分析计算结果，证明了三种施加方法的有效性，讨论了每种施加方法的优缺点，并针对问题需求选出合适的施加本质边界条件的方法．与改进的无单元Galerkin方法相比，改进的插值型维数分裂无单元Galerkin方法具有更高的计算精度和更快的计算速度．

Research on the Essential Boundary Condition Treatment of the Interpolating Dimension Splitting Meshless Method

In this paper, an improved interpolating dimension splitting element-free Galerkin method is proposed for solving the 3D wave equations. The weak form of the equation is derived. The approximation function with the interpolation characteristic is constructed. The discrete system of equations whose essential boundary conditions can be imposed directly is obtained, and the influence of different essential boundary condition treatment methods on the calculation results is discussed. Three methods of treating the boundary conditions, i.e., the direct collocation method, the method of multiplying diagonal elements by large numbers and the method of changing the diagonal elements to unities were investigated. The results of three numerical examples, each with a different essential boundary condition treatment, are discussed to verify the effectiveness of three boundary condition treatment methods and ascertain the advantages and disadvantages of each method. The selection of appropriate method to impose the essential boundary conditions is discussed according to the requirements of the specific problem. Compared with the improved element-free Galerkin method, the improved interpolating dimension splitting element-free Galerkin method has higher calculation accuracy and efficiency.