【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．更多还原

【Abstract】 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.更多还原