二维声学数值计算的光滑有限元法

Smoothed Finite Element Method for Two-dimensional Acoustic Numerical Computation

针对声学有限元分析中四节点等参单元不规则网格计算精度低甚至结果错误的问题,将光滑有限元法(Smoothed finite element method,SFEM)应用到不规则四边形网格的声学计算中,推导出光滑有限元法分析二维声学问题的原理公式。声学SFEM将声压梯度光滑处理技术结合到标准有限元法中,对声压梯度进行分区光滑处理,用光滑声压梯度来计算声学刚度矩阵,光滑域内对形函数梯度的积分转变为域边界上形函数积分,消除了坐标变换,提高了计算效率。以二维管道声腔和车内声腔模型为数值算例,研究结果表明,与标准有限元法相比,对扭曲严重的四边形网格,声学光滑有限元法具有更高的精度和准确度,特别是高频计算时。因此,光滑有限元法可以很好地应用于二维声学不规则网格的计算中,具有广阔的应用前景。

In the acoustic finite element method（FEM）,typical problems of four-node isoparametric element are low accuracy and even wrong solution for irregular meshes in numerical implementation.This paper proposes to use the smoothed finite element method（SFEM） for the acoustic analysis of irregular quadrilateral mesh,and the formulation of SFEM is presented for the two-dimensional acoustic problem.The acoustic SFEM incorporates cell-wise smoothing operations into standard finite element method and the smoothed acoustic pressure gradient is obtained by using cell-wise smoothing operation.The acoustic stiffness matrix can be calculated by using the smoothed acoustic pressure gradient matrix and the domain integrals involving shape function gradients can be recast into boundary integrals involving only shape functions.More importantly,as no coordinate transformation is involved,it is more efficient than isoparametric element.Numerical example of a two-dimensional tube and car cavity are presented to show that the acoustic SFEM achieves higher accuracy as compared with FEM when quadrilateral meshes are seriously distorted especially in the high frequency calculation.Hence the SFEM can be well applied in solving the two-dimensional acoustic problems with very irregular meshes,and has application foreground.