A time-adaptive FE2-approach within the method of vertical lines

FE2-approaches are chosen to incorporate micro-mechanical information into macroscopical computations. Although the approach is applied for several years, there are some open theoretical aspects in the algorithmic description. In this contribution the entire concept is brought into the method of vertical lines to solve initial boundary-value problems based on the homogenization concept under consideration. This requires a clear description of known and unknown quantities in the space and time discretization procedure. Dependent on the constitutive model, systems of algebraic or differential-algebraic equations (DAE) arise. Here, we concentrate on models of evolutionary type. Frequently, the resulting DAE-systems are solved using the Backward-Euler method, which are embedded in the more general class of diagonally, implicit Runge-Kutta methods. After the time discretization the resulting systems of non-linear equations have to be solved. This can be done using various schemes to obtain more efficient computations. Here, the three-level Newton algorithm, the Newton-Raphson-Schur method, numerical differentiation and a Chord-version are derived in the context of FE2 and discussed. Finally, step-size-controlled FE2-problems are solved, highlighting the importance of time-adaptive computations, particularly, in combination with the non-linear solution scheme, where the Multilevel-Newton -Chord method shows high efficiency. This contributes to an additional reduction of the overall computational times.