On the optimal shape parameters of radial basis functions used for 2-D meshless methods
On the optimal shape parameters of radial basis functions used for 2-D meshless methods
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DOI：
10.1016/s0045-7825(01)00419-4
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发表时间：
2002-03
影响因子：
7.2
通讯作者：
Jian-Guo Wang;Guirong Liu
中科院分区：
文献类型：
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作者：
Jian-Guo Wang;Guirong Liu
为了克服单多项式基可能存在的奇异性，提出了一种径向点无网格插值方法。径向PIM采用多重曲面（MQ）或高斯函数作为基函数。这两个径向基函数都包含形状参数。虽然形状参数的选择一直是逼近理论中的热点问题，并提出了一些经验公式，但这些形状参数对径向PIM精度的影响尚未得到研究。研究了形状参数对径向PIM数值精度的影响。通过分析系统矩阵的条件数、能量误差和节点分布的不规则性，得到了合适的形状参数范围。观察到，广泛使用的MQ和互反MQ基的形状参数甚至不接近它们的最优值。本文发现最优形状参数在MQ基下为q=1.03， R=1.42，在高斯基下为c=0.003 ~ 0.03。
A radial point interpolation meshless (or radial PIM) method was proposed by authors to overcome the possible singularity associated with only polynomial basis. The radial PIM used multiquadric (MQ) or Gaussian as basis functions. These two radial basis functions all included shape parameters. Although choice of shape parameters has been a hot topic in approximation theory and some empirical formulae were proposed, how these shape parameters affect the accuracy of the radial PIM has not been studied yet. This paper studied the effect of shape parameters on the numerical accuracy of radial PIM. A range of suitable shape parameters is obtained from the analysis of the condition number of the system matrix, error of energy and irregularity of node distribution. It is observed that the widely used shape parameters for MQ and reciprocal MQ basis are not even close to their optimums. The optimal shape parameters are found in this paper to be simply q=1.03 and R=1.42 for MQ basis and c=0.003–0.03 for Gaussian basis.