On the optimal shape parameters of radial basis functions used for 2-D meshless methods.(English)Zbl 1065.74074

Summary: A radial point interpolation meshless (or radial PIM) method was proposed by the authors [Int. J. Numer. Methods Eng. 54, No. 11, 1623–1648 (2002; Zbl 1098.74741)] 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, it has not been studied yet how these shape parameters affect the accuracy of the radial PIM.
This paper studies 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 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.

MSC:

 74S30 Other numerical methods in solid mechanics (MSC2010) 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74K20 Plates

Zbl 1098.74741
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