Simulation and Optimization of Rate-Independent Systems with Non-Convex Energies

Description

This project is concerned with the analysis, simulation and optimization of rate-independent systems with non-convex energy. Models of this type arise in a variety of applications in solid mechanics, such as for example the brittle damage of a workpiece under the influence of external loads. Rate-independent systems governed by non-convex energies provide a bunch of mathematical challenges: there exists a variety of different solution concepts and, in neither of these concepts, the solutions are in general unique. Moreover, solutions frequently provide a substantial lack of regularity, in particular jumps in time may occur. All this is caused by a complex interplay of a non-smooth dissipation and a non-convex energy.

Within this project, we intend to analyze rate-independent systems and optimal control thereof when equipped with the concept of so-called balanced viscosity solutions. This concept is based on a viscous regularization and a subsequent passage to the limit in the viscosity parameter. Besides the existence of optimal controls, we are also interested in viscous approximations of these optimal control problems, which offers a possibility to solve these optimization problems numerically.

Our vision is to provide a general, analytically sound and numerically robust framework for the simulation and optimization of rate-independent systems with non-convex energies. Moreover, our techniques and methods shall be practically verified on the basis of real-world applications such as damage evolution in brittle materials.

Publications

No publications from this project yet.

Preprints

Christian Meyer, Stephan Walther: Optimal Control of Perfect Plasticity Part I: Stress Tracking (SPP1962-129, 01/2020, [bib])

Hannes Meinlschmidt, Christian Meyer, Stephan Walther: Optimal Control of an Abstract Evolution Variational Inequality with Application to Homogenized Plasticity (SPP1962-123, 09/2019, [bib])

Dorothee Knees, Chiara Zanini: Existence of Parameterized BV-solutions for Rate-Independent Systems with Discontinuous Loads (SPP1962-122, 09/2019, [bib])

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.

Members

  • member's portrait

    Prof. Dorothee Knees

    Universität Kassel
    Principal Investigator
  • member's portrait

    Prof. Christian Meyer

    Technische Universität Dortmund
    Principal Investigator
  • member's portrait

    Stefanie Thomas

    Universität Kassel
    Research Assistant

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