Meshfree Methods For Signorini Problems Using Boundary Integral Equations

This dissertation present two boundary type meshfree methods for the numerical solution of Signorini problems.This dissertation first reviews the recent developments of Signorini problems and the meshfree method in the Section one. Then, we introduce the construction process for the meshfree shape functions of boundary point interpolation method(BPIM) and boundary radial point interpolation method(BRPIM) in the Section two. Based on the large amount of research findings, Section three describes two kinds of Signorini problems. Finally, Sections four and five present respectively the BPIM and BRPIM for Signorini problems using boundary integral equations(BIEs). Some numerical examples are also given to further illustrate the efficiency and reliability of the two algorithms.This dissertation is an effective attempt in implementing BIEs-based meshfree method to Signorini problems. In this dissertation, in order to convert the nonlinear boundary inequality constraints into BIEs, we define the projection operator. Then, the unknown boundary variables are interpolated by the point interpolation scheme and the radial point interpolation scheme respectively. The two methods are boundary type meshfree methods, which possess the superiorities of both boundary element method(BEM) in dimension reduction and meshfree method without dividing mesh. Afterwards, we establish meshfree algorithms using the explicit projection iterative schemes, which are very suitable for Signorini problems. Thus, only a nodal data structure on the boundary of a domain is required, and boundary conditions can be implemented directly and easily.Finally, in this dissertation, the accuracy and efficiency of these two algorithms are demonstrated by test problems for the shallow dam problem and by means of validation results for an annular problem with analytical solutions and the electropainting problem. The numerical results of these methods are compared in detail.