Regularized moving least-square method and regularized improved interpolating moving least-square method with nonsingular moment matrices

The moving least-square (MLS) method has been popular applied in surface construction and meshless methods. However, the moment matrix in MLS method may be singular for ill quality point sets and the computation of the inverse of the singular moment matrix is difficult. To overcome this problem, a regularized moving least-square method with nonsingular moment matrix is proposed. The shape functions obtained from the regularized MLS method still do not have the delta function property and may result in difficulty for imposing boundary conditions in regularized MLS based meshless method. To overcome this problem, a regularized improved interpolating moving least-square (IIMLS) method based on the IIMLS method is also proposed. Compared with the regularized MLS method, the regularized IIMLS not only has nonsingular moment matrices, but also obtains shape functions with delta function property. Shape functions of the proposed methods are compared in 1D and 2D cases, and the methods have been applied in curve fitting, surface fitting and meshless method in numerical examples.