基于径向基函数的无单元法求解力学问题误差分析

径向基函数形状参数的选择在无单元法数值计算中一直是一个热门的问题,现在已总结出许多确定形状参数的经验公式. 但还没有相关研究表明这些形状参数是如何随着影响域尺寸而变化的. 本文研究了MQ(multi-quadrics) 径向基函数中形状参数对无单元法计算误差的影响. 首先,从理论上分析了形函数导数随着形状参数值的变化趋势,和以计算点为中心节点对称布置与不对称布置的形函数导数的变化规律;然后分析了影响域尺寸对误差的影响,得到了在不同影响域尺寸下,误差随形状参数值变化的规律;在此基础上,给出了影响域范围值.

THE ERROR ANALYSIS OF ELEMENT-FREE METHOD BASED ON RADIAL BASIS FUNCTION USED IN MECHANICAL PROBLEMS

Although the choice of shape parameters in the radial PIM (point interpolation method) is a hot issue in the numerical computation based on the element-free method and some empirical formulas of the shape parameters are proposed, but how the influence domain affects these shape parameters remains to be studied. In this paper, the effect of the MQ (multi-quadrics) radial basis function's shape parameters on the errors of the element-free method is studied. The variations of the shape function's derivative with the shape parameters and the symmetrical and unsymmetrical node distributions around the evaluation point are analyzed together with the effect of the size of the influence domain on the error, and based on the analysis, the relationship between the errors and the shape parameters and the range of the suitable influence domain are obtained.