A gradient-smoothed material point method for reducing cell crossing noise in large deformation problems

The material point method (MPM) is an effective numerical technique for simulating extreme events with large deformations. However, the standard MPM suffers from cell crossing noise, which arises from the use of material points as integration points and piecewise linear shape functions with a discontinuous gradient at cell boundaries. In this study, we propose a novel gradient-smoothed material point method (GS-MPM) for reducing the cell crossing noise in problems involving large deformation. The proposed method attaches an unchanged smoothing domain to each material point, and a smoothed shape function gradient is obtained via a smoothing integration over this domain. This innovative treatment remedies the numerical discontinuity associated with quantities and their increments involving gradient terms as material points move across cell boundaries, effectively eliminating the cell crossing instability. The size of the smoothing domain in GS-MPM can be adjusted for specific problems, allowing overlapping or gaps, and larger domains can improve algorithm robustness in high strain-rate problems. As the proposed method merely modifies the shape function gradient from standard MPM, it achieves high computational efficiency and retains the conservation properties of mass and momentum. The proposed GS-MPM is validated through 1D quasi-static compression tests and applied to various large deformation problems, including ring collision, column collapse, and retrogressive landslide. The accuracy, robustness, and efficiency of the GS-MPM are also highlighted.