In this paper, a new hybrid radial basis function collocation method (HRBF-CM) is proposed to help resolve two-dimensional elastostatic symmetric problems. In the new approach, the hybrid radial basis function (HRBF) combines the infinitely smooth RBF and piecewise smooth RBF, containing two parameters (the shape parameter and the weight parameter). Discretization schemes are presented in detail. We use MATLAB to implement the HRBF-CM and produce numerical results which demonstrate the potential of this method. The new method’s accuracy is higher than that of the traditional methods, especially in the case of a more significant number of nodes. We discuss the new method’s effectiveness compared to the widely used traditional RBF and also investigate the effect of parameters on the method’s performance under the new method.

Radial basis function (RBF) has been widely used in many scientific computing and engineering applications, for instance, multidimensional scattered data interpolation and solving partial differential equations. However, the accuracy and stability of the RBF methods often strongly depend on the shape parameter. A coupled RBF (CRBF) method was proposed recently and successfully applied to solve the Poisson equation and the heat transfer equation (Appl. Math. Lett., 2019, 97: 93–98). Numerical results show that the CRBF method completely overcomes the troublesome issue of the optimal shape parameter that is a formidable obstacle to global schemes. In this paper, we further extend the CRBF method to solve the elastostatic problems. Discretization schemes are present in detail. With two elastostatic numerical examples, it is found that both numerical solutions of the CRBF method and the condition numbers of the discretized matrices are almost independent of the shape parameter. In addition, even if the traditional RBF methods take the optimal shape parameter, the CRBF method achieves better accuracy.