Superconvergent Gradient Smoothing Meshfree Collocation Method

The high order smoothness property of meshfree shape function makes them very suitable for strong form collocation methods.Nevertheless,due to the non-polynomial nature of meshfree shape function,the computation of high order gradients is quite complicated and costly.Meanwhile,the basis degree discrepancy and low accuracy are major issues associated with the collocation methods,including the meshfree collocation method.In order to resolve these issues,a gradient smoothing methodology is proposed to construct the high order gradients of meshfree shape function.In this method,the first step involves a general gradient smoothing procedure,where the first order smoothed gradients of meshfree shape function are constructed through a meshfree interpolation of the standard derivatives of meshfree shape function.Thereafter,the second order smoothed gradients are obtained by directly performing differentiations to the first order smoothed gradients.Based on the similar smoothing-differentiating procedures,arbitrary order smoothed gradients can be obtained efficiently.It is noticed that the high order smoothed gradients evaluated at nodal locations can be conveniently recast as a set of successive first order gradient smoothing operations on the meshfree shape function,which greatly simplifies the computation of complex conventional meshfree gradient and improves the efficiency dramatically.Subsequently,introducing the smoothed gradients into the strong form leads to a superconvergent gradient smoothing meshfree collocation method.The local truncation error analysis is adopted in this study to study the error of meshfree collocation method.The error analysis shown that the accuracy of meshfree collocation method is closely related to the consistency conditions of shape function.Meanwhile,the highest order differentiation in the governing equation dominants the truncation error.Based upon the local truncation error analysis,it is systematically proved that the proposed meshfree collocation method yields superconvergent solutions.The key ingredient attributed to this superconvergence property is that the high order smoothed gradients meet the consistency conditions which go beyond the original basis degree of meshfree approximation.Another important fact is that the present formulation enables a convergent collocation scheme when the order of basis function used in meshfree approximation is lower than the highest differentiation order of the governing equation,which is non-feasible in the conventional collocation formulation.The effectiveness of the proposed methodology is validated by typical numerical examples,including the second order potential and elasticity problems,the fourth order thin beam and plate problems,and the convection-diffusion-reaction problems.Numerical results clearly demonstrate the accuracy,superconvergence and higher efficiency of the present gradient smoothing meshfree collocation method.