The Optimal Shape Parameters of Radial basis Functions used for 2-D Problem by the Local Radial Point Interpolation Method

Abstract

This paper is concerned with new formulations of local radial point interpolation Meshless (LRPIM) method for the solution of two-dimensional problems in linear elasticity. This method was proposed by authors to overcome the possible singularity associated with only polynomial basis. The RPIM shape functions will be used frequently in this paper for to augment with polynomials. The radial basis function (RBF) used the multi-quadrics (MQ), the Gaussian (EXP) and the thin plate spline (TPS) as basis functions. 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 regularity of node distribution. The optimal shape parameters are found in this paper to be simply:  for EXP, ( ) for TPS, between  for all materials.(n=55) ,  for MQ,