Researches On The Improved Meshless Local Boundary Integral Equation Methods

The meshless(or meshfree) method has been a hot direction and the development trend of numerical methods for science and engineering problems in recent years.The local boundary integral equation(LBIE) method is a meshless method, which combines moving least-squares(MLS) approximation with the local boundary integral equation. The advantages of the LBIE method are that the method is a true meshless method, as no mesh is required either for the approximation functions of the variables, or for the numerical integration. Because the MLS is used in this method, the disadvantages of the method are its less efficiency, and can form an ill-conditioned or singular equations sometimes. To these problems, combining the new interpolation function, which has higher accuracy and stablity, with LBIE method, the improved meshless local boundary integral equation methods are presented in this dissertation.A new approximation function based on the coupling of radial basis functions and polynomial basis functions is obtained, which has the property of Delta function. Combining this new approximate function with LBIE for potential problems, the LBIE method based on radial basis functions is presented for potential problem in the paper. And then combining this new approximate function with LBIE for elasticity problems, the LBIE method based on radial basis functions is proposed for elasticity problem.To increase the precision and efficiency of the LBIE for fracture problems, by introducing the analysis solution of displacements at the tip of a crack, the approximation function based on the coupling of radial basis functions and enriched polynomial basis functions is obtained, and a new enriched LBIE method based on radial basis functions is proposed in this dissertation. This method used to solve the crack problems has greater precision and computational efficiency.With the problems of the MLS, such as ill-conditioning equations, precision and efficiency, the improved MLS approximation is combined with LBIE for potential problems, then an improved LBIE method for potential problems is presented. Furthermore, the improved MLS approximation is combined with LBIE for elasticity problems, and an improved LBIE method for elasticity problems is presented too.When simulating fractures problems with the conventional LBIE method, some problems, such as the computing time, less precision and the vibration of the solution at the tip of the crack, exist. In order to reduce these shortcomings, on the basis of enriched polynomial basis functions, weighted orthogonal basis functions are used to obtain the MLS approximation by using a Schmidt orthogonalization. In the end, the improved LBIE method for elasticity fracture is proposed.In order to show the validity of the improved LBIE methods in the dissertation, corresponding MATLAB codes of these methods have been written. Some numerical examples are provided, and the validity and efficiency of these methods are demonstrated.