Conservative Taylor least squares reconstruction with application to material point methods

Abstract

Within the standard material point method (MPM), the spatial errors are partially caused by the direct mapping of material-point data to the background grid. In order to reduce these errors, we introduced a novel technique that combines the least squares method with the Taylor basis functions, called the Taylor least squares (TLS), to reconstruct functions from scattered data while preserving their integrals. The TLS technique locally approximates quantities of interest such as stress and density, and when used with a suitable quadrature rule, it conserves the total mass and linear momentum after transferring the material-point information to the grid. The integration of the technique into MPM, dual domain MPM, and B-spline MPM significantly improves the results of these methods. For the considered examples, the TLS function reconstruction technique resembles the approximation properties of highly accurate spline reconstruction while preserving the physical properties of the standard algorithm.