Development of FE-meshfree hybrid methods and their application to static and free vibration problems in 2-D solid methanics
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
... hods 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 ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library II used as trial and test functions, respectively. The FE-PIM method can handle the displacement boundary conditions easily. However, the choice of suitable polynomial basis for trial function is a tricky problem which is inherited from PIM. If an inappropriate polynomial basis is chosen, it leads to interpolation failure. Furthermore, the results given by the FE-PIM are not as accurate as expected. In order to remove the above drawbacks of FE-PIM, a new hybrid method called FE-LSPIM is proposed. This method employs the concepts of partition of unity finite element methods (PU-FEM), wherein the shape functions of finite element are used for PU and that of least-square point interpolation method (LSPIM) for local approximation. Thereby the proposed method inherits the completeness properties of meshfree shape functions and compatibility properties of finite element shape functions. A QUAD4 element (called FE-LSPIM QUAD4) is developed based on this approach and its performance is studied for typical benchmark problems. The results show that this element performs better than the classical isoparametric QUAD4 element and comparable to meshfree method (EFG). In particular, the proposed element has good tolerance to mesh distortions. Next, the FE-LSPIM QUAD4 is extended to free vibration problems. Numerical examples for free vibration analyses are presented and the results are compared with those obtained using other existing solution techniques. The studies indicate that the accuracy of FE-LSPIM QUAD4 is generally better than other methods. Also, the proposed element is more tolerant to mesh distortion than the classical QUAD4 element. ATTENTION: The Singapore Copyright Act applies to the use of this document. Nanyang Technological University Library III As a further extension, FE-LSPIM QUAD4 element is applied to geometrically nonlinear problems. The geometrical nonlinear formulation is derived based on the updated Lagrangian approach. The performance of the element is studied for several numerical examples. The element is seen to give better results than the QUAD4 element of ANSYS under distorted mesh. Although the FE-LSPIM satisfies the objective of developing a method that synergises the FEM and meshfree methods, it does have demerits. Singularity of interpolation matrix, shear locking tendency and computational efficiency are some of them that are critically reviewed at the end of the thesis.