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The Nakajima test is a well-known material test from the steel and metal industry to determine the forming limit of sheet metal. It is demonstrated how FE2TI, our highly parallel scalable implementation of the computational homogenization method FE   $$^2$$  2 , can be used for the simulation of the Nakajima test. In this test, a sample sheet geometry is clamped between a blank holder and a die. Then, a hemispherical punch is driven into the specimen until material failure occurs. For the simulation of the Nakajima test, our software package FE2TI has been enhanced with a frictionless contact formulation on the macroscopic level using the penalty method. The appropriate choice of suitable boundary conditions as well as the influence of symmetry assumptions regarding the symmetric test setup are discussed. In order to be able to solve larger macroscopic problems more efficiently, the balancing domain decomposition by constraints (BDDC) approach has been implemented on the macroscopic level as an alternative to a sparse direct solver. To improve the computational efficiency of FE2TI even further, additionally, an adaptive load step approach has been implemented and different extrapolation strategies are compared. Both strategies yield a significant reduction of the overall computing time. Furthermore, a strategy to dynamically increase the penalty parameter is presented which allows to resolve the contact conditions more accurately without increasing the overall computing time too much. Numerically computed forming limit diagrams based on virtual Nakajima tests are presented.

Fully-coupled micro–macro finite element simulations of the Nakajima test using parallel computational homogenization