Conservative Taylor least squares reconstruction with application to material point methods

Abstract

Funding informationThe Netherlands Organisation for Scientific Research (NWO); Deltares; Royal Boskalis Westminster N. V.; Van Oord Dredging and Marine Contractors; Rijkswaterstaat; Stichting FloodControl IJkdijk SummaryWithin 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.

Comparison and unification of material-point and optimal transportation meshfree methods

“…They form as a potential remedy to the grid-crossing and quadrature errors present in MPM. The concept of BSMPM was further investigated by combining B-spline basis functions with more advanced integration techniques [20,53,57] and adopting the method to unstructured grids [15].…”

“…Moreover, the number of nodes varies between 8 and 512, while the number of material points per element remains equal to 12. The analytical solution in terms of the displacement, velocity, and stress can be found in the work of Wobbes et al [57]. For the convergence analysis, the root-mean-square (RMS) error in the displacement is computed.…”

“…The choice of BSMPM is motivated by previous studies. First of all, a number of studies have indicated that BSMPM is a viable alternative not only for MPM, but also for its more advanced versions such as dual domain material-point (DDMP) [60] and convected particledomain interpolation (CPDI) [40] methods [20,45,53,57]. Secondly, Cyron et al [13] pointed out the strong similarities between B-spline and maxent basis functions.…”

A convected particle least square interpolation material point method

“…The use of monomials in the original MLS requires an additional numerical treatment when the least square moment matrix A is singular. On the other hand, Taylor least squares method is designed to enforce the mass and momentum conservation during the mapping from the material points to the nodes. This method requires the reconstruction of material point's data to integration points rather than nodes for the numerical integration on background grid.…”

“…The least square reconstruction technique has been used to improve the accuracy of the MPM beforehand. [15][16][17][18][19][20][21] However, these formulations require an inversion of the least squares moment matrix and a solution of a system of equations. This is numerically expensive, while requiring additional numerical treatments in case the moment matrix is singular.…”