Fast and Robust Solution Methods for Dynamic Contact Problems

Abstract

For the numerical solution of dynamic contact problems, often implicit time discretization methods are used for stability reasons [2], giving rise to a nonlinear contact problem to be solved in each time step. We use a globally convergent non-smooth multigrid method which allows for solving multibody contact problems in linear elasticity as fast as linear elliptic problems. This multigrid method can be extended to the case of frictional contact problems leading to a nonlinear method for frictional contact problems with multigrid efficiency [1]. Using nonconforming domain decomposition methods, a stable discretization for the transfer of displacements and stresses at the interface between the bodies coming into contact can be developed, which is also capable of handling complicated geometries. In the framework of a Newmark-based time discretization scheme, this method is used for the construction of an efficient implicit time discretization scheme. Due to the inequality constraints at the contact interface, the time integration of dynamic contact problems often gives rise to oscillations in displacements and stresses. To remove these oscillations, we consider a stabilization based on a global L2-projection which can be interpreted as a solution dependent correction of the velocities. The stabilization leads to an additional variational inequality to be solved in each time step, which can be done efficiently using our monotone multigrid method. We discuss the properties of the resulting method and illustrate its performance for a contact problem in biomechanics.