A nonsmooth multiscale method for solving frictional two-body contact problems in 2d and 3d with multigrid efficiency

We present a nonsmooth multiscale method for the numerical solution of frictional contact problems in 2d and 3d. The computational effort is comparable to that of solving linear problems. Our method does not require any regularization, neither of the nonpenetration condition nor of the friction law and can be applied to contact problems with Tresca friction as well as to contact problems with Coulomb friction. For the case of Tresca friction, the global convergence of the method is shown. For the more complicated case of Coulomb friction, we develop a nonsmooth multiscale method which can be directly applied to the corresponding variational quasi-inequality. No outer iteration is required. Moreover, our multiscale approach is general in the sense that it can be used in the context of geometric as well as algebraic multigrid methods. Nonconforming domain decomposition techniques (or mortar) methods are employed in order to enforce the transmission conditions at the interface between different bodies with nonmatching meshes. Numerical examples illustrate the high robustness and efficiency of our method.