中文摘要：

针对传统有限元法求解声学问题由于刚度矩阵过硬导致较大的色散误差,以及在较高波数和网格扭曲时计算精度过低甚至错误的问题,采用移动最小二乘权函数对传统有限元法的声压梯度进行加权重构,推导了梯度移动最小二乘加权（Gradient weighted by moving least-squares,GW-MLS）的二维声学计算公式。对声压梯度的加权重构使得GW-MLS模型的刚度相对于FEM模型得以软化,刚度更接近真实模型刚度。采用与有限元法相同的方式构造质量矩阵和边界积分矢量,保证质量矩阵和边界条件的正确施加和积分精度。通过二维管道声腔模型和二维车内声腔模型算例对所提出的算法进行验证,数值分析结果表明,GW-MLS有效地减少了色散误差的影响,提高了计算精度,尤其是对较高波数和网格扭曲时表现出良好的适应性。

英文摘要：

It is well known that one key issue of analyzing acoustic problems using finite element method（FEM） is ＂numerical dispersion error＂ due to the ＂overly stiff＂ nature of the FEM.This will cause the accuracy deterioration in the solution when it comes to high wave number or irregular meshes.To overcome this problem,a gradient weighted by moving least-squares（GW-MLS） is presented for analyzing acoustic problem.The gradient of acoustic pressure is reconstructed by the weight function of moving least-squares（MLS）.And this makes the GW-MLS model much softer than the ＂overly stiff＂ FEM model.In acoustic GW-MLS,the acoustic mass matrix and the vectors of boundary integrals are constructed by the standard FEM to insure the integral accuracy of the mass matrix and the boundary conditions applied on region boundary accurately.Numerical examples including a 2D tube model and car cavity model are presented.The results demonstrate that the GW-MLS reduces the numerical dispersion error effectively.And because of this,the GW-MLS achieves higher accuracy as compared with FEM when meshes are seriously distorted especially in calculating high wave number problems.