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An accurate prediction of the temperature distribution in space and time plays an important role in many industrial applications, in particular when phase transformations are involved. In this article the thermo-physical properties of steel 51CrV4 (SAE 6150) are determined and used in numerical simulations. For the simulation of the temperature field a semi-discrete approach is used, consisting of a finite element approximation in space and a high order RungeKutta integration in time. Several adaptive high-order time integration method (stiffly accurate diagonally implicit Runge-Kutta methods) are applied and their computational efficiency is investigated. The theoretical rates of convergence are achieved for all problems, including the non-linear case. Whereas the second order accurate method of Ellsiepen with time adaptive step-size control proves to be most efficient. Further, the influence of the material model on the simulation results is studied and the numerical results are verified by experiments. The best correlation of the simulation and experimental data is achieved using temperature-dependent parameters.

On the one hand, displacement controlled processes are frequently applied in Computational Mechanics using the finite‐element method, on the other hand, there are some shortcomings regarding the study and the presentation of this particular case. In this article, we focus our theoretical considerations on quasi‐static problems with constitutive equations of evolutionary type. In this case, it is currently known that the solution procedure of global and local iterations within the “Newton‐Raphson method” is related to the method of lines, where one arrives at a system of differential‐algebraic equations after the spatial discretization by means of the finite element method. This could be solved by means of the Backward‐Euler method or more appropriately using time‐adaptive, stiffly accurate, diagonally implicit Runge‐Kutta methods in combination with the Multilevel‐Newton method. In this respect the theoretical basis of currently applied implicit non‐linear finite element analyses is extended to displacement controlled processes. In this article, the calculation of the reaction forces using the method of Lagrange multipliers and the penalty method are of special interest in view of the new DAE‐approach. Accordingly, an important objective lies in a consistent notation so that the dependence of known and unknown variables as well as the transition from local to global level becomes obvious. These investigations are of utmost importance for micro‐macro homogenization techniques in FE$^2$ approaches and the development of new global time and space‐adaptive integration methods.

The thermal coupling of a fluid and a structure is of great significance for many industrial processes. As a model for cooling processes in heat treatment of steel we consider the coupling of the compressible Navier-Stokes equations along a surface with the heat equation. A partitioned approach is considered, where different codes for the sub-problems are employed. We use a finite volume method (FVM) for the fluid and a finite element method (FEM) for the heat equation.The semi-discrete coupled system is solved using stiffly stable SDIRK methods, where on each stage an FSI problem is solved. This has a black box character, since a stage calculation corresponds to a specific Backward-Euler integration step. For the resulting method it was shown by numerical experiments in [1] that second order convergence rate is obtained. This property is used for a simple time-step control, which saves considerable computational time and guarantees a specified maximum error of time integration. Here, we will consider different norms for measuring the error.