Boundary element-free method (BEFM) and its application to two-dimensional elasticity problems

In this study, we first discuss the moving least-square approximation (MLS) method. In some cases, the MLS may form an ill-conditioned system of equations so that the solution cannot be correctly obtained. Hence, in this paper, we propose an improved moving least-square approximation (IMLS) method. In the IMLS method, the orthogonal function system with a weight function is used as the basis function. The IMLS has higher computational efficiency and precision than the MLS, and will not lead to an ill-conditioned system of equations. Combining the boundary integral equation (BIE) method and the IMLS approximation method, a direct meshless BIE method, the boundary element-free method (BEFM), for two-dimensional elasticity is presented. Compared to other meshless BIE methods, BEFM is a direct numerical method in which the basic unknown quantity is the real solution of the nodal variables, and the boundary conditions can be applied easily; hence, it has higher computational precision. For demonstration purpose, selected numerical examples are given.