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中文核心期刊

求解Helmholtz方程基于核重构思想的最小二乘配点法

Solving helmholtz equation by least-square collocation method based on reproducing kernel particle method

  • 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.

     

    Abstract: Helmholtz equation often arises while solving boundaryvalue problems of partial differential equation by eigen function method. Inphysics, Helmholtz equation represents a stationary state of vibration inthe fields of mechanics, acoustics and electro-magnetics. In this paper, aleast-square collocation formulation for solving Helmholtz equation withDirichlet and Neumann boundary conditions was established. The unknowninterpolated functions were first constructed based on reproducing kernelparticle method and Helmholtz equation was then discretized by pointcollocation method. The variance errors of unknown function in each discretepoint are minimized by a least-square scheme to arrive at the finalsolution. To verify the proposed method, a wave propagation problem and aboundary layer problem of Helmholtz equation were solved. Numerical resultsby the present approach are compared with exact solutions and those bysmooth particle hydrodynamics (SPH) method. Numerical examples show that thepresent method displays better accuracy and convergence than the classicalSPH method for the same density of discrete points.

     

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