Computer Methods in Applied Mechanics and Engineering
On locking and robustness in the finite element method
SIAM Journal on Numerical Analysis
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems
Locking effects in the finite element approximation of plate models
Mathematics of Computation
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computer Methods for Ordinary Differential Equations and Differential-Algebraic Equations
Computing with Hp-Adaptive Finite Elements, Vol. 2: Frontiers Three Dimensional Elliptic and Maxwell Problems with Applications
A time-adaptive fluid-structure interaction method for thermal coupling
Computing and Visualization in Science
Experimental validation of high-order time integration for non-linear heat transfer problems
Computational Mechanics
Solving Differential Equations in R
Solving Differential Equations in R
Accelerated staggered coupling schemes for problems of thermoelasticity at finite strains
Computers & Mathematics with Applications
Applied Numerical Mathematics
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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 h-elements based on Lagrange polynomials or related mixed element formulations. In this article both approaches are combined using a p-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 p-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.