An efficient staggered grid material point method

The material point method (MPM) has demonstrated itself as an effective numerical method to simulate extreme events with large deformations. However, the original MPM suffers the cell crossing noise because it takes the material points as integration points and employs the piecewise linear grid nodal shape functions whose gradient is discontinuous on the cell boundary. A number of techniques have been developed to alleviate the cell crossing noise. In this paper, a new staggered grid material point method (SGMP) is proposed to eliminate the cell crossing noise very efficiently. The volume integrals in the weak form are evaluated by cell center quadrature instead of particle quadrature as the sum of value of the integrand at each cell center of the background grid multiplied by the corresponding quadrature weight. The physical quantities and the quadrature weights at the cell centers are reconstructed efficiently based on an auxiliary grid, which is obtained by shifting the background grid half the side length of its cell in each direction Similar to the original MPM, both grids carry no permanent information and can be reset after each time step. In addition, the SGMP evaluates the constitutive equations at the particles, just like the original MPM, to readily model the history-dependent materials. To further reduce the cell crossing noise, a continuous strain rate/vorticity field is established based on the auxiliary grid, whose values are determined by the background grid velocity gradient. The strain rate/vorticity at each particle is interpolated from the auxiliary grid nodal values. Due to the overlap of the cell centers and the corresponding auxiliary grid nodes, a very efficient implementation is established in the SGMP. Numerical studies illustrate that the SGMP is capable of eliminating the cell crossing noise with little extra computational effort and the extra cost ratio reduces as the number of the grid cells or the particles increases. (C) 2019 Elsevier B.V. All rights reserved.