The explicit material point method (EMPM)[[i]] works well in simulating extreme deformation problems, where the finite element method (FEM) often encounters mesh distortion. However, the implicit material point method (IMPM)[[ii]] is more suitable than EMPM in some cases, such as cutting and upsetting. Because the time step of implicit method can be much larger and numerical oscillation also can be suppressed. In many cases, the contact problem between different bodies cannot be avoided and should be considered. The contact algorithm implemented in the explicit material point method has been successfully applied to penetration, impact, fluid-structure interaction[[iii]]. However, the contact problem in the implicit material point method has not been solved as far as we know.

A new contact algorithm based on the exponential form of augmented Lagrangian method is proposed and validated for the implicit material point method. In our method, the augmented Lagrangian method in an exponential form is employed rather than in the standard form. The standard form often encounters the discontinuity difficulty, which arises in taking the second derivative of the governing equations (such as the Newton method) and holds down the convergence rate of iterative method. However, the discontinuity can be perfectly avoided in the exponential-augmented Lagrangian method.

The constraint equations derived from the Lagrangian method belong to saddle point system. The Uzawa algorithm is introduced here and works well in solving these problem, which can be divided into two parts: the inner and outer iteration. The inner iteration, aims to obtain the trial solution where the Lagrangian multipliers are fixed, are solved using Newton method with direct solver; While the outer iteration updates the multipliers when the inner iteration is converged. If the inner and outer iteration are both converged under some certain proper criteria, the final solution is obtained.

In our test, if the time step is chosen appropriately, our solution technique can be much more efficient compared to the explicit material point method. In some numerical results, the total cost of CPU time in our contact algorithm is only 1% of that in the explicit method.