A New Boundary-type Meshless Method

The original literature and recent developments of the meshless methods are briefly reviewed in this thesis. With emphasis on the meshless approximation and integration formulation, the fundamental principles of several meshless methods, including the Element-free Galerkin method (EFG), the Meshless Local Boundary Integral Equation (MLBIE), the Meshless Local Petrov-Galerkin method (MLPG) and the Boundary Node Method (BNM), are described in detail. Based on these meshless methods, a new boundary typemeshless method------a Hybrid Boundary Node Method (HBNM) is proposedand implemented successfully for solving problems in two-dimensional potential theory, two-dimensional linear elasticity, three-dimensional potential theory and three-dimensional linear elasticity.Combining the modified variational principle functional with the Moving Least Squares (MLS) approximation, the HBNM involves three types of independent variables, i.e., for potential problems, the potentials u|~ and normal fluxes q on the boundary and potentials u inside the domain;for elasticity problems, the displacements (u|~)_i~ and tractions (t|~)_i on the boundary and displacements u_i inside the domain. The domain variables are interpolated by classical fundamental solutions and thus allowing for the transfer of the domain integration to the boundary. The boundary variables are interpolated by MLS approximation. The main idea is to retain the dimensionality advantages of the BEM, and localize the integration domain to a regular sub-domain, as in the MLBIE, such that no mesh is needed for integration.The HBNM exploits the meshless attributes of the MLS and the localization idea from the MLBIE, and hence is a truly meshless method, namely neither mesh for interpolation nor mesh for integration is required, and as input data, it needs only the data of the distributed points on the boundary. The geometrical model created by the CAD software can be directly used, and the preprocess is very simple. Thecomputation of the unknowns in the domain is easier than by BEM, the integration on the boundary once more is not required. The numerical examples show that the presented method possesses not only the high accuracy, but also the good performance of convergence. Furthermore, the method can be applied to solve the very thin structures, the thickness in micro- or nano-scale, based on 3D elasticity. The HBNM is a kind of numerical method with some excellent characteristics. Compared with the MLBIE and MLPG, the new approach has the well-known dimension-reduction advantage of the BEM, e.g. for a 3-D object, only randomly distributed nodal points are required to be constructed on the 2-D bounding surface of the body;compared with the conventional BEM, it is a meshless method, only requires a nodal data structure on the bounding surface of the domain;compared with the BNM, it is a truly meshless method, no cells are needed either for interpolation or integration purposes.