On Rosenbrock methods for the time integration of nearly incompressible materials and their usage for nonlinear model reduction

This paper deals with the time integration and nonlinear model reduction of nearly incompressible materials that have been discretized in space by mixed finite elements. We analyze the structure of the equations of motion and show that a differential algebraic system of index 1 with a singular perturbation term needs to be solved. In the limit case, the index may jump to index 3 and thus renders the time integration into a difficult problem. For the time integration, we apply Rosenbrock methods and study their convergence behavior for a test problem, which highlights the importance of the well-known Scholz conditions for this problem class. Numerical tests demonstrate that such linear-implicit methods are an attractive alternative to established time integration methods in structural dynamics. We also combine the simulation of nonlinear materials with a model reduction step in order to prepare the ground for including such finite element structures as components in complex vehicle dynamics applications.