Abstract
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.
Highlights
The material point method (MPM)[1,2] is a continuum-based numerical tool to simulate problems that involve large deformations
Summary 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
Summary
The material point method (MPM)[1,2] is a continuum-based numerical tool to simulate problems that involve large deformations. Modified mapping generally improves the accuracy of the solution, the standard reconstruction techniques such as spline interpolation might lead to the loss of physical properties of the material point methods. If a sufficient number of Gauss points is defined within each element, the proposed mapping technique preserves the physical qualities of the standard MPM by conserving the mass and linear momentum of the system. The conservation property of the TLS reconstruction technique is verified by computing the total mass and momentum before and after projecting the particle information to the background grid. We observe that, when material points do not cross element boundaries in the vibrating bar problem, the TLS reconstruction technique has little influence on MPM and DDMPM but significantly improves the convergence of BSMPM.
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