A Reproducing Kernel Gradient Smoothing Framework Toward Efficient Formulation Of Arbitrary Order Galerkin Meshfree Methods

Meshfree methods characterized by nodal discretizations provide a very flexible and straightforward way to construct arbitrary order smooth and compatible approximants,which are preferred in numerical simulations of many typical problems,for instance,large deformation analysis,high order problems and moving boundary modelling,etc.Due to their sound accuracy and stability,Galerkin meshfree methods belong to one of the most popular meshfree methods with wide applications.However,the non-polynomial nature of meshfree shape functions and the general misalignment between the irregular overlapping kernel supports and background integration cells make it very difficult to accurately evaluate the meshfree Galerkin weak form,and this situation becomes more serious when high order basis functions are employed.As a result,numerical integration has been a very challenging topic and received significant attentions in the context of Galerkin meshfree methods.This dissertation aims to develop a general reproducing kernel gradient smoothing framework toward formulating efficient Galerkin meshfree methods with explicit quadrature,for both second-and fourth-order problems.Firstly,an error analysis of Galerkin meshfree methods is presented,which reveals that the convergence of Galerkin meshfree methods with Gauss integration is limited by the quadrature rules.Subsequently,a reproducing kernel gradient smoothing framework with explicit expressions is introduced to ensure the optimal convergence of Galerkin meshfree methods.It is noted the integration constraint is embedded in the construction of reproducing kernel smoothed gradient and thus the integration consistency is an inherent property of the proposed reproducing kernel gradient smoothing methodology regardless of integration schemes.Consequently,the conventional normal low order Gauss quadrature rules in finite element analysis now can be used to properly integrate the meshfree stiffness matrix simply through replacing the standard meshfree gradients with the reproducing kernel smoothed gradients at Gauss points.Furthermore,certain explicit quadrature rules are proposed to efficiently compute the reproducing kernel smoothed gradients.The total number of sample points regarding these quadrature rules is minimized from a global point of view through making as many as sample points shared by neighboring integration cells.Finally,a series of reproducing kernel gradient smoothing Galerkin meshfree methods is presented for the second order elasticity and potential problems and the fourth order thin plate and shell problems.Moreover,the reproducing kernel gradient smoothing method is implemented into the fourth order phase filed model in order to effectively model the brittle fracture problem.The proposed reproducing kernel gradient smoothing framework is featured by optimal convergence,explicit quadrature,symmetric stiffness matrix as well as high efficiency,and works well for arbitrary order meshfree basis functions.In case of linear basis function,the present reproducing kernel gradient smoothing framework reduces to the well-established stabilized conforming nodal integration for Galerkin meshfree methods.The efficiency and accuracy of the proposed methodology is thoroughly demonstrated by numerical examples.