High-order quasi-static finite element computations in space and time with application to finite strain viscoelasticity

On the one hand, high-order quasi-static finite elements have been developed on the basis of isogeometric analysis or hierarchical shape functions based on integrated Legendre-polynomials, which display favorable behavior in comparison to linear element formulations. If we consider constitutive models of evolutionary-type such as elastoplasticity, viscoplasticity or viscoelasticity, only first order methods are employed to integrate the ordinary-differential equations at the Gauss-point level, leading to errors caused by the time-integration. On the other hand, high-order time integration methods have been devised to treat the resulting system of differential–algebraic equations (DAE-system) following the spatial discretization. This, however, is only done using spatial discretizations with linear or quadratic hh-elements based on Lagrange polynomials or related mixed element formulations. In this article both approaches are combined using a pp-version finite element approach on the basis of hierarchical shape functions and the resulting DAE-system is solved by means of high-order one-step methods. The first step in this treatise is to apply stiffly accurate, diagonally-implicit Runge–Kutta methods combined with the Multilevel-Newton algorithm. Since the computational outlay for computing the tangential stiffness matrix is very high in the case of pp-version finite elements, Rosenbrock-type methods are applied as well, which lead to a completely iteration-free technique. We then compare the two approaches. The next step is to investigate three-dimensional examples showing the properties on the basis of a constitutive model in finite strain viscoelasticity.