The goal of this project is to exploit the huge potential of finite element approaches based on the approximation of stresses as the dual variable for variational inequalities in solid mechanics. This includes models of frictional contact as well as elasto-plastic deformations and emphasizes adaptive refinement strategies based on built-in a posteriori error estimators as well as efficient iterative solution methods for the arising non-smooth problems. Approximating stresses directly by suitable finite element spaces allows for algorithmic simplifications and improved accuracy, in particular, in association with plasticity and friction where stress components play a prominent role.

The proposed work packages build upon past and current activities on first-order system least squares methods for the Signorini contact problem carried out jointly by the two participating research groups. Our approach, studied theoretically and by numerical computations in the friction-less case, shall first be extended to Coulomb friction. Moreover, the generalization to plastic material behavior and to non-local friction laws will be the subject of a substantial part of this project. Finally, it is also planned to tackle dynamic contact problems with such a stress-based approach utilizing the advantages with respect to the improved accuracy of momentum conservation and to consider the coupling with thermal effects. The availability of a built-in a posteriori error estimator to be used in an adaptive refinement strategy is one of the advantageous features concerning the discretization. On the solver side, the explicit use of stresses makes the non-smoothness of the problems more accessible and allows the development of efficient multilevel and domain decomposition methods. Stress reconstruction techniques from displacement-pressure approximations as well as localization procedures for nonconforming elements will also be investigated in the context of frictional contact of elasto-plastic deformations. The testing of the developed methods at the benchmark examples within the SPP are a further essential part of this proposed project.