Rosenbrock-type methods applied to finite element computations within finite strain viscoelasticity

In this article time-adaptive high-order Rosenbrock-type methods are applied to the system of differential–algebraic equations which results from the space-discretization using finite elements based on a constitutive model of finite strain viscoelasticity. It is shown that in this smooth problem more efficient finite element computations result in comparison to classical finite element approaches since the time integration on the basis of Rosenbrock-type methods does not lead to a system of non-linear equations. In other words, all aspects of implicit finite elements as local iterations on Gauss-point level and global equilibrium iterations do not occur. The first introduction to this approach proposed by Hartmann and Wensch [22] is extended here to the case of finite strain applications, where the geometrical non-linear deformation has an essential contribution to the non-linearities. Additionally, a clear decomposition into local (element or Gauss-point) work and global computational work using the Schur-complement is introduced to exploit the classical finite element character. Moreover, the extension to the reaction force computation, which is different to the classical approach, and the influence to mixed element formulations, here, the three-field formulation for displacements, pressure and dilatation, are discussed. The performance of various Rosenbrock-type methods is investigated and shows that for low accuracy requirements as in order one methods, the proposal yields a drastic reduction of the computational time.