Development of FE-meshfree hybrid methods and their application to static and free vibration problems in 2-D solid methanics

Abstract

Meshfree methods have been extensively investigated in recent years due to their flexibility in solving practical engineering problems. As for example, they do not require a mesh to discretize the problem domain, and the approximate solution is constructed entirely in terms of a set of scattered nodes. However, meshfree methods demand a high computational effort as compared to the well established finite element (FE) method. And establishing nodal connectivity in meshfree methods is relatively difficult. Furthermore, implementing the displacement boundary conditions is cumbersome in many meshfree methods due to the lack of Kronecker delta property of the meshfree shape functions. On the other hand, the finite element method has no such difficulty, and is well established and has been widely used in engineering.Nevertheless, the finite element method generally gives less accurate results compared to meshfree methods, more so under distorted meshes. The focus of this thesis is on the development of hybrid methods that aim at synergising the merits of FE and meshfree methods and its application to 2D solids.To begin with, a FE-EFG method is proposed, which is based on the local Petrov-Galerkin approach. The method uses FE shape functions as test functions and Element Free Galerkin (EFG) shape functions as trial functions. Numerical testing reveals that the method has a good tolerance to mesh distortion but poses difficulty in applying the displacement boundary conditions as experienced in EFG method. To solve this problem, a new hybrid method called FE-PIM is developed. For the FE-PIM method, the FE shape functions and the Point Interpolation Method (PIM) shape functions are