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 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.