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


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 spacetime 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.


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Research Area

Modeling, problem analysis, algorithm design and convergence analysis

The focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.

Realization of algorithms, adaptive discretization and model reduction

As the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.

Incorporation of parameter dependencies and robustness

In many applications the robustness of solutions with respect to a given parameter range (uncertainty set) is highly relevant. Correspondingly, in this research area of the SPP, bi- or multilevel optimization approaches will be studied in order to robustify problem solutions against uncertain parameters.


  • member's portrait

    Prof. Rolf Krause

    Università della Svizzera italiana
    Principal Investigator
  • member's portrait

    Prof. Gerhard Starke

    Universität Duisburg-Essen
    Principal Investigator
  • member's portrait

    Bernhard Kober

    Universität Duisburg-Essen
    Research Assistant

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