An adaptive orthogonal improved interpolating moving least-square method and a new boundary element-free method

Abstract

The improved interpolating moving least-square (IIMLS) method has been widely used in data fitting and meshfree methods, and the obtained shape functions have the property of the delta function, compared with those obtained by the moving least-square (MLS) method. However, the moment matrix in IIMLS may be singular or ill-conditioned because of the ill quality of the point sets used. In this paper, the weighted orthogonal basis functions are applied in IIMLS to obtain a diagonal moment matrix, which can overcome the difficulty caused by directly inversing singular or ill-conditioned matrices. However, the weighted orthogonal basis functions cannot change the nature of the singular or ill-conditioned moment matrix, since the diagonal elements of the new moment matrix may be zero or close to zero. Thus, an adaptive scheme is further employed to resolve this problem by ignoring the contribution from the zero or very small diagonal elements in the diagonal moment matrix. Combined with shifted and scaled polynomial basis functions, a stabilized adaptive orthogonal IIMLS (SAO-IIMLS) approximation is obtained. Based on this approximation, a new boundary element-free method is proposed for solving elasticity problems. Numerical results for curve fitting, surface fitting and the new boundary element-free method have shown that the proposed SAO-IIMLS approximation is accurate, stable and performs well for ill quality point sets.