An Improved Material Point Method with Aggregated and Smoothed Bernstein Functions

Nodal shape functions and their gradients are vital in transferring physical information within the material point method (MPM). Their continuity is related to numerical stability and accuracy, and their support domain size affects computational efficiency. In this paper, a scheme of aggregated and smoothed Bernstein functions is proposed to improve the MPM. In detail, the Bernstein polynomials are smoothed with a convolution reformation to eliminate the cell crossing error, and an aggregation strategy is implemented to cut down the node amount required for field probing. Hierarchical MPM variants are obtained with choices of original Bernstein polynomials and degrees of smoothing. Numerical examples show that mass, momentum, and energy conservations are all well met, and no cell crossing noise exists. In addition, solution accuracy and numerical stability are significantly improved in large deformation problems.