| Meshless method is a new numerical method developed in the past decades. It requires only nodes and a description of the external and internal boundary conditions, and no element connectivity is needed totally or partly. Therefore, meshless method can be widely applied to crack propagation problems, elasto-plastic analysis of materials and three-dimensional problems. The recent development of the meshless methods is briefly reviewed in the thesis. Several meshless methods, including the Element-free Galerkin Method (EFGM), the Meshless Local Boundary Integral Equation (MLBIE), Boundary Node method (BNM) and Hybrid Boundary Node method (HBNM), are described in detail. The influence on the precision of numerical results in EFGM with basis functions choosing is discussed. Based on the above, a new boundary type meshless method —Singular Hybrid Boundary Node Method (SHBNM) is proposed and implemented successfully for solving problems in two and three dimensional linear elasticity. The dissertation includes the following contents: Firstly, the properties and characteristics of the shape functions in meshless method and their influence on the numerical results are studied. The reasons of the error which appears in this method are discussed. When EFGM uses high order polynomial basis and the nodes are ill-distributed, the shape matrix A is ill-conditioned. The influence of different basic functions on the interpolation functions and the computed accuracy are analyzed and compared. Based on the above analysis, the criterion of basis function choosing has been obtained. The numerical examples have been given to show the validity of the results. Secondly, the meshless method based on orthogonal basis with high order for solving high stress gradient problems is proposed, which is composed essential boundary conditions with the penalty method. It holds nearly all qualities of EFGM and removes many drawbacks. The method has high accuracy by using high order orthogonal basis. Therefore, it is suitable for many problems in computational mechanics. A series of numerical examples and their error analysis are carried out to show the advantages of the present method. Thirdly, combining the modified variational principle function with the Moving Least Squares (MLS) approximation and exploiting the localization idea from MLBIE, a new boundary type meshless method —Singular Hybrid Boundary Node Method is proposed. It is a truly meshless method, namely neither mesh for interpolation nor mesh for integration is required. All integrals can be easily evaluated over regular shape sub-domains. As input data, it needs only the data of the distributed nodes on the boundary. Fourthly, SHBNM is implemented successfully for solving problems in two and three dimensional linear elasticity. The relative programs are compiled. The numerical examples are presented to show the efficiency and excellent characters of the present method. Fifthly, the parameters that influence the performance of the singular hybrid boundary node method are studied by the classic elastic examples and known analytical fields. The regularities of the selection of these parameters have been presented. Sixthly, the rigid body movement method is employed in the singular hybrid boundary node method to solve the hyper-singular integrations. The source points are located on the boundary exactly and no uncertain parameters included which is a particularly useful feature for the engineering practice. Finally, an adaptive integration scheme is proposed to overcome the "boundary layer effect"which is caused by locating the source points on the boundary exactly. The numerical examples show that it is an effective method to obtain high accuracy even in the vicinity of the boundary. Numerical examples have demonstrated the accuracy and convergence of SHBNM. It adapts to solve adaptive problems, crack problems and contact problems with the number of the nodes on the surface adding or reducing conveniently.
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