An Eulerian-Lagrangian Concept for the Numerical Simulation of Flat Steady-State Hot Rolling Processes

For the finite element analysis of stationary flat hot rolling processes, a new and efficient mathematical model was developed. The method is based on an intermediary Eulerian-Lagrangian concept, where an Eulerian coordinate is employed in the rolling direction, while Lagrangian coordinates are used in the direction of the thickness and width of the strip. This approach yields an efficient algorithm, where the time is eliminated as an independent variable in the steady-state case. Further, the vector of independent field variables consists of a velocity component in Eulerian and of displacement components in Lagrangian directions. Due to this concept, the free surface deformations can be accounted for directly and the problems encountered with pure Eulerian and Lagrangian models now appear with reduced complexity and can thus be tackled more easily. The general formalism was applied to different practical hot rolling situations, ranging from thick slabs to ultra-thin hot strips. The new model turns out to be most efficient for the calculation of the shape of free surfaces, which can be performed directly without streamline updating techniques.