基于核重构的最小二乘配点法求解Helmholtz方程  

SOLVING HELMHOLTZ EQUATION BY LEAST-SQUARE COLLOCATION METHOD BASED ON REPRODUCING KERNEL PARTICLE METHOD

作　　者：史宝军[1] 袁明武[2] 舒东伟[3] 

机构地区：[1]山东建筑工程学院机电学院,济南250014 [2]北京大学力学与工程科学系,北京100871 [3]南洋理工大学机械与宇航学院,新加坡639798

出　　处：《力学学报》2006年第1期125-129,共5页Chinese Journal of Theoretical and Applied Mechanics

基　　金：山东省自然科学基金资助项目(Y2002A04).~~

摘　　要：基于核重构思想构造近似函数,将配点法和最小二乘原理相结合对微分方程进行离散,建立了Helmholtz 方程的最小二乘配点格式,并分别研究了Helmholtz方程的波传播问题和边界层问题．通过数值算例可以发现,给出的数值计算结果非常接近于精确解,计算精度明显高于SPH法的数值结果,且随着节点数目的增加, 其精确度越来越高,具有良好的收敛性．Helmholtz equation often arises while solving boundary value problems of partial differential equation by eigen function method. In physics, Helmholtz equation represents a stationary state of vibration in the fields of mechanics, acoustics and electro-magnetics. In this paper, a least-square collocation formulation for solving Helmholtz equation with Dirichlet and Neumann boundary conditions was established. The unknown interpolated functions were first constructed based on reproducing kernel particle method and Helmholtz equation was then discretized by point collocation method. The variance errors of unknown function in each discrete point are minimized by a least-square scheme to arrive at the final solution. To verify the proposed method, a wave propagation problem and a boundary layer problem of Helmholtz equation were solved. Numerical results by the present approach are compared with exact solutions and those by smooth particle hydrodynamics （SPH） method. Numerical examples show that the present method displays better accuracy and convergence than the classical SPH method for the same density of discrete points.

关 键 词：HELMHOLTZ方程 无网格法 重构核点法 最小二乘配点格式 

分 类 号：O316[理学—一般力学与力学基础]