无网格局部径向点插值法求解Helmholtz方程

摘要

A meshless local radial point interpolation method (LRPIM) for the Helmholtz equation was presented. A radial basis function (RBF) coupled with a polynomial basis function as a trail function and a quartic spline function as a test function of the weighted residual method were used in the meshless method. The discrete equation was established by the meshless local Petrov-Galerkin method. The shape function obtained in trail function had the Kronecker Delta property, and no additional treatment was required to impose essential boundary conditions. The presented method was one of the "truly meshless" methods since it did not require any background integration cells. The numerical results showed that the meshless LRPIM for the Helmholtz equation had a number of advantages, such as conciseness, quite good accuracy and easy implementation.%采用无网格局部径向点插值法(LRPIM)求解Helmholtz方程,这种无网格方法采用径向基函数耦合多项式基函数作为近似函数,并采用四次样条函数作为加权残值法中的权函数,运用局部Petrov-Galerkin方法推导出相应的离散方程,由于所构造的形函数满足Kronecker Delta性质,可以很方便地施加本质边界条件.此方法不需要积分网格,是一种真正的无网格法.数值结果表明,LRPIM法求解Helmholtz方程具有简洁、精度高和易于实现等优点.