加权最小二乘无网格法求解亥姆霍兹方程

Meshless least-squares method for solving Helmholtz equation

在移动最小二乘近似的基础上,直接使用最小二乘法建立系统的变分公式,导出了亥姆霍兹方程的加权最小二乘无网格(MWLS)法公式.MWLS法兼有伽辽金型无网格法和配点型无网格法精度高、收敛快的优点,并且克服了伽辽金法计算量大、配点法不稳定的缺陷.通过一维算例讨论了MWLS法应用于亥姆霍兹方程时各种参数的影响以及最佳参数的选择,通过二维算例证明该方法计算效率高于无单元伽辽金法(EFGM).数值结果表明MWLS法求解亥姆霍兹方程具有效率高、精度高和稳定性好的优点.对高波数波动问题给出了精确的模拟.

On the basis of moving least-squares approximation （MLSA）,the formulation of the meshless weighted least-squares （MWLS） method to solve the Helmholtz equation is proposed. The variation formulation was constructed on the least-squares discrimination. MWLS method combines the advantages of the Galerkin method and the collocation method and overcomes the disadvantage of large calculation of the Galerkin method and instability of the collocation method. The MWLS computational parameters are chosen based on a thorough numerical study of 1-dimensional problems. Several 2-dimensional examples show that the MWLS method is more efficient than the element free Galerkin method （EFGM）. These numerical results show that the MWLS method is high efficiency,high accuracy and well stability,which gives accurate simulation of high wave number of acoustical waves.