Applications Of The Method Of Fundamental Solution And The Boundary Node Method To Signorini Problems

Signorini problem is a special nonlinear boundary value problem. In this problem, the boundary potential and its normal derivative alternate on the Signorini boundary in conjunction with certain nonlinear boundary inequality conditions. Finite difference method, finite element method, boundary element method and meshless method have been successfully applied to solve the problem. In recent years, some meshless boundary integral methods, such as the method of fundamental solution(MFS), the boundary point method(BNM), the boundary point interpolation method and the boundary element-free method have been developed by combining boundary integral equations and meshless shape functions.This dissertation first introduces the research background of Signorini problem the meshless methods and reviews the recent developments of some numerical methods including the meshless methods. Then, two meshless algorithms are developed for boundary-only analysis of two-dimensional and three-dimensional Signorini problem. In the first algorithm, by using a projection technique to tackle the nonlinear Signorini boundary inequality conditions, the original Signorini problem is transformed into a sequence of linear elliptic boundary value problems and then solved by the MFS. Convergence and efficiency of the present MFS is proved theoretically and verified numerically. In the second algorithm, the BNM is developed to solve Signorini problem. Convergence and efficiency of the BNM are verified numerically.In the developed algorithms, the nonlinear boundary inequality conditions are incorporated naturally into an iterative scheme by using a projection operator. Then, the original Signorini problem is solved numerically by the MFS and the BNM. Both algorithms only require nodes on the boundary of the domain, and avoid the design of quite sophisticated and time-consuming optimization algorithms.