A Finite Element Method Using Node-Based Interpolation

TL;DR: The tests reveal that higher order basis function together with quartic spline (QS) correlation function can be effective in alleviating shear locking difficulty, and the K-FEM has a higher chance to be accepted in practice.

Abstract: During the past two decades, a large variety of mesh-free methods have been introduced as superior alternatives to the traditional FEM. However, the acceptance in professional practices seems to be slow due to their implementation complexities. Recently, the authors proposed a very convenient implementation of Element-free Galerkin Method (EFGM) using the node-based Kriging interpolation (KI). Two key properties of KI are Kronecker delta and consistency properties. Due to the former, KI passes through all the nodes thus requiring no special treatment for boundary conditions. The consequence of the latter ensures reproduction of a linear interpolation if the basis function includes the constants and linear terms. In this study, layers of finite elements around any node are adopted as its domain of influence. This method is referred to as Kriging-based FEM (K-FEM), which can be viewed as a generalized form of FEM. Precisely, if we limit the nodal domain of influence to only one finite element layer around the node, K-FEM specializes to the traditional FEM. In this study, the K-FEM was tested with 2D elastostatic, Reissner-Mindlin's plate and shell problems. The tests have been performed to investigate various important issues, including shear locking, patch test, convergence and accuracy. The tests also reveal that higher order basis function together with quartic spline (QS) correlation function can be effective in alleviating shear locking difficulty. K-FEM passes the weak patch test and therefore its convergence is guaranteed. In all cases, exceptionally accurate displacement and stress fields can be achieved in relatively coarse meshes. In addition, the same set of Kringing interpolation functions can be used to interpolate the mesh geometry. This property is particularly useful to model curved shells. The distinctive advantage of the K-FEM is its inheritance of the computational procedure of FEM. The formulation and implementation of the method are similar to those of the standard FEM. Any existing FE code can be easily extended to K-FEM; thus, the method has a higher chance to be accepted in practice.