Researches On The Reproducing Kernel Particle Boundary Element-Free Methods

The meshless boundary integral equation method, which combines moving least-squares approximation with the boundary integral equation method, is an important branch of meshless methods. The advantages of the meshless boundary integral equation method are that no mesh is required for the construction of the shape functions, and it can be applied easily to solve problems. And in the method, nodes are required only on the boundary of the problem domain, then dimension of the problem is decreased. Because the moving least-squares approximation is used in this method, the disadvantages of the method are its less efficiency, and that it can form ill-conditioned or singular equations sometimes, and applied the boundary conditions difficultly. To these problems, combining the reproducing kernel particle method (RKPM) and boundary integral equation method, the reproducing kernel particle boundary element-free (RKP-BEF) method is presented in this dissertation. And the coupled method is also presented by combining the RKP-BEF and FEM. Furthermore, the improved RKP-BEF for fracture mechanics is proposed based on the enriched shape functions of RKPM. The RKP-BEF method presented is a meshless method which has advantages of greater precision and computational efficiency. And the RKP-BEF method can apply the boundary conditions directly.The shape function, which has the property of Delta function, in the RKPM is discussed first. When the shape function covers a number of nodes equal to the number of monomials in its basis, the RKPM shape functions, which satisfy reproducing conditions in a finite region, has an interpolate property. The improved RKPM shape functions which have the interpolate property at arbitrary nodes are obtained by coupling of a simple function and an enrichment function. The smoothness of the shape function of RKPM is no less than that of the kernel function and the values of polynomials at interpolating points can be exactly reconstructed. The difficulties for applying essential boundary conditions can be avoided by using the interpolating shape functions.On basis of the improved reproducing kernel particle method, combining the boundary integral equation method for potential problems, the reproducing kernel particle boundary element-free (RKP-BEF) method for potential problems is presented in this paper. The formulae of discretization and corresponding discrete equations are also obtained. This RKP-BEF method is a direct meshless method of the boundary integral equation and has the advantages of the higher efficiency and computational precision.On basis of the improved reproducing kernel particle method and boundary integral equation method, the reproducing kernel particle boundary element-free method for elasticity is presented. The discrete boundary integral equations of the RKP-BEF method are obtained by considering the numerical integral schemes. Then the equations with the variables at boundary nodes are obtained, and the formulae of the displacement and stress at internal points for the RKP-BEF method are given.By combining the reproducing kernel particle boundary element-free method and finite element method, a coupled RKP-BEF/FE method for elasticity or potential problems is presented. The combined equations on the unknown quantities at nodes for the RKP-BEF/FE method are obtained. As the RKP-BEF and FEM have higher precision, the method presented here is successful.When simulating fractures problems with the conventional meshless method, some problems, such as the computing time and less precision of the solution at the tip of the crack, are existed. In order to reduce these shortcomings, the improved RKP-BEF method for elasticity fracture is presented based on the enriched shape functions of reproducing kernel particle mothed.The corresponding FORTRAN codes of the above methods have been written and the numerical examples are given in the corresponding chapters. Some engineering examples are solved by the methods in this paper, and compared with the FEM simulating results. The numerical results show that the methods in this paper have higher stability and efficiency, and can be applied easily.