Proper orthogonal decomposition-based model reduction in nonlinear solid mechanics

The present dissertation discusses projection-based model reduction techniques to reduce the computational effort of finite element simulations in nonlinear solid mechanics preserving a sufficient accuracy. The efficiency and successful application of proper orthogonal decomposition (POD)-based model reduction is shown by means of characteristic numerical examples of nonlinear solid mechanics including hyperelastic, viscoelastic as well as elastoplastic behavior.
As examples a biomechanical simulation of a part of the nasal cavity and a simulation of a robot-based incremental sheet metal forming process are investigated. Unfortunately, the POD method is not yet able to reduce the computational cost to real-time. Furthermore, a POD reduction does not yield the required accuracy for highly nonlinear elastoplastic simulations, especially the forming process.

The main focus of the thesis is to develop model reduction approaches, which overcome those problems of POD reduction. Therefore, a model reduction strategy combining POD and sub-structuring is developed for highly nonlinear elastoplastic simulations. This new approach shows an improvement of the accuracy of the reduced simulation compared to POD. The simulation results by means of the proposed approach are in very good agreement with the reference results computed by a simulation without any reduction. To increase the reduction in computational time even to real-time the POD method is extended by an additional empirical interpolation. The presented reduction method leads to a significant speed-up for simulations, where a POD reduction already achieves a sufficient level of accuracy. The approximation error reaches the same level compared to the simulation with POD.