Superconvergent Least Squares Recursive Gradient Meshfree Collocation Methods For Structural Vibration Analysis

Different from the finite element method,meshfree methods directly employ the nodal locations to construct arbitrary order smooth shape functions,which offer an obvious advantage to compute high order gradients of shape functions required by the collocation formulation.However,the non-polynomial characteristic of meshfree shape functions,for example,the widely used moving least squares and reproducing kernel meshfree approximants,leads to a very complex and costly high order gradient computation.Alternatively,the recursive gradient formalism offers a simple and efficient way to formulate high order smoothed meshfree gradients.The recursive gradient meshfree collocation method has shown superconvergence for steady problems,however,this method yields unphysical modes and frequencies in case of structural vibration analysis.Moreover,the basis degree discrepancy issue exists as well for the standard meshfree collocation analysis of structural vibrations.In order to resolve the above issues,a least squares recursive gradient meshfree collocation formulation has been systematically developed in this thesis,with a particular emphasis on the accuracy analysis for both second and fourth order problems,especially for the frequency accuracy of free vibrations.As for the static analysis,the least squares recursive gradient meshfree collocation approach shares the same convergence rates as the recursive gradient meshfree collocation algorithm.Regarding the free vibration analysis,the least squares recursive gradient meshfree collocation formulation can effectively eliminate the aforementioned unphysical modes and frequencies,and show superior accuracy and superconvergence compared with the standard least squares meshfree collocation method.More specifically,p th and(p+1)th orders of accuracy are achieved by the proposed least squares recursive gradient meshfree collocation formulation for even and odd degrees of basis functions,respectively.On the contrary.the standard least squares meshfree collocation method is pth and(p-1)th order accurate for second order problems and(p-2)th and(p-3)th order accurate for fourth order problems.Thus the superconvergence of the proposed least squares recursive gradient meshfree collocation formulation for free vibration analysis is consistent with the previous recursive gradient meshfree collocation scheme for static analysis.Numerical examples uniformly validate the excellent accuracy and superconvergence of the proposed least squares recursive gradient meshfree collocation formulation.