Generalized finite difference method for solving two-dimensional non-linear obstacle problems

Abstract

In this study, the obstacle problems, also known as the non-linear free boundary problems, are analyzed by the generalized finite difference method (GFDM) and the fictitious time integration method (FTIM). The GFDM, one of the newly-developed domain-type meshless methods, is adopted in this study for spatial discretization. Using GFDM can avoid the tasks of mesh generation and numerical integration and also retain the high accuracy of numerical results. The obstacle problem is extremely difficult to be solved by any numerical scheme, since two different types of governing equations are imposed on the computational domain and the interfaces between these two regions are unknown. The obstacle problem will be mathematically formulated as the non-linear complementarity problems (NCPs) and then a system of non-linear algebraic equations (NAEs) will be formed by using the GFDM and the Fischer–Burmeister NCP-function. Then, the FTIM, a simple and powerful solver for NAEs, is used solve the system of NAEs. The FTIM is free from calculating the inverse of Jacobian matrix. Three numerical examples are provided to validate the simplicity and accuracy of the proposed meshless numerical scheme for dealing with two-dimensional obstacle problems.