移动最小二乘法导数近似讨论

在移动最小二乘法（moving least squares method, MLS）构造无网格形函数的数值方法中，通常采用无单元伽辽金法（element-free Galerkin method, EFG）的建议，将系数向量a参与导数运算。为探讨这种导数近似算法在更一般无网格法中的适用性和合理性，针对系数向量a是否应参与运算的问题进行讨论和数值检验。结果表明：单纯从近似意义上讲，这种将系数向量代入导数运算的算法并不具有优势；从数值方法的应用意义上讲，这种导数近似算法对数值求解，特别是强式无网格法，会带来一系列潜在不稳定的问题。建议在MLS导数近似中，系数向量a不应当参与导数运算，并提出采用一种由核基函数代替普通基函数的核近似法。

Discussion on derivative approximation for moving least squares method

In the numerical method of meshless shape function constructing with moving least squares method(MLS), the element-free Galerkin method(EFG) is adopted customarily, and the coefficient vector a usually take part in the derivative operation. To investigate the applicability and rationality of this derivative approximation algorithm applying to more general meshless method, the discussion and numerical tests are carried out which focus on whether the coefficient vector a should take part in computing or not. The results show that on approximate means only, there is no advantage if the coefficient vector takes part in computing; on the application means of the numerical method, there is a series of potential instability problems using the derivative approximation algorithm on numerical calculation, especially for the strong meshless method. It is suggested that the coefficient vector a should not be included in the derivative operation. A core approximation method is proposed in which the common basis function is replaced by core primary function.