A frictional contact algorithm for implicit material point method

The explicit material point method (MPM) works successfully in modeling high frequency problems, but it is very computationally expensive in simulating low frequency with small time steps or quasi-static problems. Thus, several groups have developed an implicit MPM for modeling low frequency problems. Recently, a few attempts were undertaken to investigate the contact problems using the implicit MPM but the accuracy was dissatisfactory. In this paper, an augmented Lagrange formulation for the frictional inequality constraints is introduced. A discretization of the Lagrange multiplier field based on the background grid is proposed to establish the implicit MPM framework with the contact algorithm. To reduce the complexity of the solution, the Uzawa algorithm is employed to decouple the unknown variables and the Lagrange multipliers. Finally, the resulting sequent nonlinear equations are solved by the Newton method, in which the tangential matrix is assembled explicitly. By using the compressed sparse row (CSR) technique, the total storage of the matrix can be greatly reduced. Numerical studies show that the computational efficiency and accuracy of the implicit MPM with the proposed contact algorithm are much higher than the explicit MPM.