| In this thesis, two meshless methods, the element free Galerkin (EFG) method and the meshless method based on the local boundary integral equation (LBIE), are described.; The first part (Chapters II and III) of the thesis is to propose efficient methods in enforcing the essential boundary conditions in the EFG method. A modified collocation method has been proposed for enforcing the discrete essential boundary conditions in the element free Galerkin (EFG) method. The present method substantially improves the computational accuracy as compared to the previous direct collocation method reported in literature. A penalty method has also been developed for imposing the essential boundary conditions in the EFG method. The penalty method is easy to implement and can reduce computational costs while retaining high accuracy. Compared with the Lagrange multiplier method which will increase the number of unknowns and result in a non-positive system stiffness matrix, the present penalty method is able to yield a banded, symmetric and positive definite system matrix.; In the second part (Chapters IV–VI), new local boundary integral equation (LBIE) formulations, for linear and nonlinear potential problems, and for problems in linear elasticity, have been proposed, and the new meshless method based on the LBIE formulations and the moving least squares approximation has been developed. This new method is a true meshless method as no boundary and domain elements are needed in the implementation. The essential boundary conditions can be easily and directly imposed even when a non-interpolative approximation scheme such as the MLS approximation is used to approximate the trial function. Numerical examples show that the present LBIE formulation possesses a tremendous potential in solving linear and non-linear problems in computational mechanics. |
