Implicit dynamics in the material-point method Original Research Article

A time-implicit discretization is derived and validated for the material-point method (MPM). The resulting non-linear, discrete equations are solved using Newtonʹs method combined with either the conjugate gradient method or the generalized minimum residual method. These Newton–Krylov solvers are implemented in a matrix-free fashion for numerical efficiency. A description of the algorithms and evaluation of their performance is presented. On all test problems, if the time step is chosen appropriately, the implicit solution technique is more efficient than an explicit method without loss of desired features in the solutions. In a dramatic example, time steps 10,000 times the explicit step size are possible for the large deformation compression of a cylindrical billet at 1.2% the computational cost.