Stress-Based Methods for Variational Inequalities in Solid Mechanics: Finite Element Discretization and Solution by Hierarchical Optimization

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The goal of this project is the extension of the methods developed during the first phase based on the dual formulation of mostly static (quasi-)variational inequalities to time-dependent ones. Again, adding the dual variable as a separate field leads to highly accurate approximations for the stresses and for the associated surface traction forces compared to standard primal discretizations. The conservation properties of these methods are particularly advantageous for time-dependent problems.Adaptive stress-based methods will be developed which employ displacement reconstructions in the a posteriori error estimation. In combination with suitably constructed multigrid solvers, this will lead to a very efficient overall treatment of such (quasi-)variational inequalities. These methods will first be studied in the context of quasi-static problems and then extended to dynamic frictional contact. For the quasi-static case, a time-stepping approach is initially used. This has the advantage that the techniques already developed during the first phase of the SPP can be used for the spatial adaptivity. This includes the a posteriori error estimators as well as the monotone multilevel method for the stress spaces.The ultimate goal of this second project phase does, however, consist of the development of adaptive space-time discretizations for time-dependent (quasi-)variational inequalities. To this end, techniques which are already well understood for standard parabolic problems will be appropriately modified for the time-dependent frictional contact problems. A first-order system least squares functional is used as a starting point for a posteriori estimation of the space-time error and associated adaptivity. For dynamical frictional contact, more challenging stability issues need to be addressed which can be done using expertise from earlier work on that topic. Finally, space-time multigrid methods will be developed for the arising space-time finite element formulations leading to a highly efficient overall solution strategy.