#### Vol. 12, No. 1, 2017

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Time-adaptive finite element simulations of dynamical problems for temperature-dependent materials

### Matthias Grafenhorst, Joachim Rang and Stefan Hartmann

Vol. 12 (2017), No. 1, 57–91
##### Abstract

Dynamical systems in finite elements yield systems of second-order differential equations. Incorporating inelastic material properties, thermomechanical coupling and particular Dirichlet boundary conditions essentially changes the underlying mathematical problem. In this respect, we investigate the behavior of a number of subproblems such as reaction force computation, high-order time-integration, time-adaptivity, etc., which yield (depending on the underlying problem) systems of differential-algebraic equations or a mixture of systems of second-order and first-order ordinary differential equations (especially if the constitutive equations are of evolutionary-type, as in the case of viscoelasticity and viscoplasticity). The main goals are to provide higher-order time integration schemes using diagonally implicit Runge–Kutta methods and the generalized-$\alpha$ method so that they may be applied to the constitutive equations, and to apply time-adaptivity via embedded schemes so that step-sizes are chosen automatically. The constitutive equations are given by a thermoviscoplasticity model of Perzyna/Chaboche-type with nonlinear kinematic hardening.

##### Keywords
dynamics, DIRK methods, finite elements, generalized-alpha method, time-adaptivity, thermoviscoplasticity