A comparison of time adaptive integration methods for small and large strain viscoelasticity

In quasistatic solid mechanics the initial boundary value problem has to be solved in the space and time domain. The spatial discretization is done using finite elements. For the temporal discretization three different classes of Runge-Kutta methods are compared. These methods are diagonally implicit Runge-Kutta schemes (DIRK), linear implicit Runge-Kutta methods (Rosenbrock type methods) and half-explicit Runge-Kutta schemes (HERK). It will be shown that the application of half-explicit or linear-implicit Runge-Kutta methods can enormously reduce the computational time in particular situations. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Application of Rosenbrock-type methods to a constitutive model of polymeric material

The consistent application of the time-space discretization in the case of quasi-static structural problems based on constitutive equations of evolutionary type yields after the finite element discretization to a system of differential-algebraic-equations (DAE). In order to carry out the time discretization, time-adaptive Rosenbrock-type methods are applied to the DAE-system, which offer the possibility of a completely iteration-less procedure. This presentation shows the behavior of a new global finite element approach and compares it to the classical implicit (iterative) procedure. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Comparison of diagonal-implicit, linear-implicit and half-explicit Runge–Kutta methods in non-linear finite element analyses

The method of vertical lines in the case of quasi-static solid mechanics applying constitutive models of evolutionary-type yields after the spatial discretization by means of finite elements a system of differential-algebraic equations. It is of substantial interest how fast, accurate, and stable such computations can be carried out. Moreover, the questions are how simple the implementation can be done and how susceptible a procedure is to programming errors. In this article, this is investigated for half-explicit Runge–Kutta methods that are applied to small and finite strain viscoelasticity. The advantage of the method is given by a non-iterative scheme on element level. Additionally, it turns out that for models where linear elasticity is one ingredient in the constitutive model, the method leads to only one required LU-decomposition at the beginning of the entire computation, and in each time step, only one back-substitution step has to be carried out. This outperforms current finite element computations. Order investigations of various integration schemes and the automatic step-size behavior are studied. This new proposal is compared to a classical Backward-Euler implementation, high-order stiffly accurate diagonally implicit Runge–Kutta, and recently proposed Rosenbrock-type methods. Advantages and disadvantages of the applied schemes are discussed.     

Application of Rosenbrock-type methods to a finite element formulation based on large strain viscoelasticity

The present paper is focused on the solution of differential–algebraic equation systems (DAE), which arise in large viscoelastic deformations within the quasi–static finite element context. For this purpose linearly implicit methods of Rosenbrock–type are used, which avoid completely the solution of non–linear equations. This article investigates a possible treatment of the new global approach with respect to expense and achievable accuracy. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Applications of Rosenbrock-type methods to the p-version of finite elements

In quasistatic solid mechanics the spatial as well as the temporal domain need to be discetized. For the spatial discretization usually elements with linear shape functions are used even though it has been shown that generally the p-version of the finite elemente method yields more effective discretizations, see e.g. [1], [2]. For the temporal discretization diagonal-implicit, see e.g. [4], and especially linear-implicit Runge-Kutta schemes, see e.g. [5], [6], have for smooth problems proven to be superior to the frequently applied Backward-Euler scheme (BE). Thus an approach combining the p-version of the finite element method with linear-implicit Runge-Kutta schemes, so-called Rosenbrock-type methods, is presented. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)