Conservative Taylor least squares reconstruction with application to material point methods

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 Taylor Least Squares (TLS), to reconstruct functions from scattered data. The TLS technique locally approximates quantities of interest, such as stress and density, and when used with a suitable quadrature rule, conserves the total mass and linear momentum after transferring the material-point information to the grid. For one-dimensional examples, applying the TLS approximation significantly improves the results of MPM, Dual Domain Material Point Method (DDMPM), and B-spline MPM (BSMPM). Due to its outstanding conservation properties, the TLS technique outperforms the nonconservative reconstruction techniques, such as spline reconstruction. For example, in contrast to the solution generated using the global cubic-spline interpolation, the TLS solution satisfies the boundary conditions of a two-phase benchmark. Therefore, the TLS reconstruction increases the accuracy of the material point methods, while preserving the fundamental physical properties of the standard algorithm.