Consistent high order meshfree Galerkin methods and applications

Meshfree methods such as the element-free Galerkin (EFG) method have been developed to be a formidable competitor and also a beneficial complement to the traditional finite element method (FEM) which dominates engineering analysis for decades. One attractive advantage of meshfree methods is that constructing high order approximation is much more convenient than that in the finite element method (FEM). However, high order meshfree methods are computationally inefficient since a large number of integration points are required. On the other hand, the stabilized conforming nodal integration method based on strain smoothing is very efficient for linear meshfree Galerkin methods, but it cannot exploit the high convergence and accuracy of meshfree methods with high order approximation. In this work, the number of quadrature points for high order meshfree methods is remarkably reduced by correcting the nodal derivatives. Such correction is rationally developed based on the Hu-Washizu three-field variational principle. The proposed method is able to exactly pass patch tests in a consistent manner and is therefore, named as consistent high order meshfree Galerkin methods. In contrast, the traditional meshfree methods cannot exactly pass patch tests. Numerical results of elastostatic problems show that the proposed technique remarkably improves the numerical performance of high order meshfree methods in terms of accuracy, convergence, efficiency and stability. Applications of the proposed methods to thin plates and shells as well as crack problems are also presented.