A Meshless Method For Signorini Problems Using Indirect Boundary Integral Equation

Signorini problems arise from many kinds of physical and industrial applications. Boundary conditions in Signorini problems contain nonlinear inequality constraints. The numerical solution of this kind of problems can be obtained using the finite difference method, the finite element method and the boundary element method. However, these numerical techniques require domain and/or its boundary meshing which is usually arduous and computationally expensive. Boundary-type meshless methods reduce the computational dimensions of the original problem by one and thus simplify the efforts involved in data preparation and CPU time. Two meshless methods for Signorini problems using indirect boundary integral equation are discussed in this dissertation.In the first chapter of this dissertation, the background of Signorini problems is introduced, and then the recent developments of Signorini problems and meshless methods are reviewed. In the second chapter, the method of fundamental solution is briefed for the numerical solution of linear boundary value problems. In the third chapter, by transforming the nonlinear Signorini boundary inequality conditions into linear Robin boundary equality conditions, a meshless method is developed for boundary-only analysis of Signorini problems. In the presented numerical method, the original nonlinear inequality problem is linearized as a sequence of linear equality problems, and then an indirect boundary integral equation meshless method, the method of fundamental solution is developed to form the discrete linear system of algebraic equations. Numerical examples are finally presented for some partial differential equations with nonlinear inequality constraints. In the fourth chapter, by transforming the Signorini boundary conditions into Neumann boundary conditions, another meshless method is developed for the numerical solution of Signorini problems.The meshless methods in this paper inherently have some desirable numerical merits such as truly meshless, boundary-only and integration-free. The numerical results show that the presented meshless methods have the merits of good convergence rate, and higher computational accuracy and efficiency over the traditional mesh-based methods such as the boundary element methods.