New approximations and collocation schemes in the finite cloud method

The finite cloud method (FCM) [Int. J. Numer. Methods in Eng. 50(10) (2001) 2373] is a meshless technique combining a fixed kernel approximation of the unknown function(s) with a point collocation discretization of the governing PDEs. The meshless approximation and the collocation discretization are the two major steps in FCM. Since the quality of the numerical solution depends on the quality of the meshless approximation functions or shape functions and the discretization scheme employed, in this paper, we propose several improvements to the construction of meshless shape functions and compare several collocation schemes within the framework of the FCM. The improvements to the shape functions are combined with various collocation schemes to solve several 2-D Poisson and elastostatic examples. The convergence characteristics of the collocation schemes are studied. The accuracy of the collocation schemes with a scattered point distribution is also investigated.