声学数值计算的有限元-最小二乘点插值法

A Finite Element-Least Square Point Interpolation Method for Acoustic Numerical Computation

针对有限元法求解Helmholtz方程时由于数值色散导致高波数计算结果不可靠的问题,提出一种混合有限元-最小二乘点插值法(FE-LSPIM)以分析二维声学问题。该方法将问题域划分为四边形单元,应用四边形单元形函数和最小二乘点插值法进行局部逼近,继承了有限元法的单元兼容性和最小二乘点插值法的二次多项式完备性,能有效减小色散效应。数值算例表明:与标准有限元法相比,特别是针对高波数问题和不规则网格模型,FE-LSPIM具有更高的计算精度和更好的收敛性。因此,FE-LSPIM能很好地应用于二维声学问题的分析计算,具有广阔的应用前景。

The finite element method （FEM） is unreliable to compute approximate solutions of Helmholtz equation for high frequencies due to the numerical dispersion.This paper introduced a hybrid FE-LSPIM to solve the 2D acoustic problem.The quadrilateral elements were used to discretize the problem domains and the shape functions of the quadrilateral element and the least square point interpolation method were used for local approximation.The present method inherited the compatibility properties of finite element method and the quadratic polynomial completeness properties of LSPIM.The FE-LSPIM greatly reduces the numerical dispersion errors and obtaines accurate results for acoustic problems.Numerical examples were studied and the results show that the FE-LSPIM achieves more accurate results and highes convergence rates as compared with the corresponding finite elements,especially for high wave number and irregularity meshes.Hence the FE-LSPIM can be well applied in solving two-dimensional acoustic problems,which has more application foreground in practice.