High-Performance Computing Two-Scale Finite Element Simulations of a Contact Problem Using Computational Homogenization - Virtual Forming Limit Curves for Dual-Phase Steel

The appreciated macroscopic properties of dual-phase (DP) steels strongly depend on their microstructure. Therefore, accurate finite element (FE) simulations of a deformation process of such a steel require the incorporation of the microscopic heterogeneous structure. Usually, a brute force FE discretization incorporating the microstructure is not feasible since it results in exceedingly large problem sizes. Instead, the microstructure has to be incorporated by using computational homogenization. We present a numerical two-scale approach of the Nakajima test for a DP steel, which is a well known material test in the steel industry. It can be used to derive forming limit diagrams (FLDs), which allow experts to judge the maximum formability properties of a specific type of sheet metal in the considered thickness. For the simulations, we use our software package FE2TI, which is a highly scalable implementation of the well known FE2 homogenization approach. The microstructure is represented by a representative volume element (RVE) and it is discretized separately from the macroscopic problem. We discuss the incorporation of contact constraints using a penalty formulation as well as appropriate boundary conditions. In addition, we introduce a simple load step strategy and different opportunities for the choice of an initial value for a single load step by using an interpolation polynomial. Finally, we come up with computationally derived FLDs. Although we use a computational homogenization strategy, the resulting problems on both scales can be quite large. The efficient solution of such large problems requires parallel strategies. Therefore, we consider the highly scalable nonlinear domain decomposition methods FETI-DP (Finite Element Tearing and Interconnecting - Dual-Primal) and BDDC (Balancing Domain Decomposition by Constraints). For the first time, the BDDC approach is used for the parallel solution of the macroscopic problem in a simulation of the Nakajima test. We introduce a unified framework that combines all variants of nonlinear FETI-DP and nonlinear BDDC. For the first time, we introduce a nonlinear FETI-DP variant that chooses suitable elimination sets by utilizing information from the nonlinear residual. Furthermore, we show weak scaling results for different nonlinear FETI-DP variants and several model problems