Meshfree Analysis Using the Generalized Meshfree (GMF) Approximation

Meshfree methods are becoming widely used in many industrial fields since the finite element method (FEM) has inherent limitations, such as mesh quality and related distortion problems, to analyze sophisticated problems under large deformation. However, the meshfree methods also have their own deficiencies, mainly the high CPU cost. Recently, the generalized mesh-free (GMF) approximation is developed to improve the efficiency and accuracy in the conventional meshfree methods for solid analysis. The GMF approximation is the integrated formulation to generate existing approximations, such as moving least square (MLS), reproducing kernel (RK), and maximum entropy (ME) approximation, as well as new approximations based on the selection of a basis function. The GMF approximation has two excellent features. The one is that the GMF approximation naturally bears the weak Kronecker-delta property at boundaries, which makes the imposition of essential boundary conditions in meshfree methods easier. The other is that the GMF approximation can be extended to higher-order approximations, which can improve the accuracy of meshfree methods. In this study, some meshfree analyses are performed by LS-DYNA® to demonstrate the performance and accuracy of the GMF approximation. The results show that the convex approximation gives better performance and accuracy than the non-convex approximation in meshfree analysis.