The FE2 method is a renown computational multiscale simulation technique for solid materials with fine-scale microstructure. It allows for the accurate prediction of the mechanical behavior of structures made of heterogeneous materials with nonlinear material behavior. However, the FE2 method leads to excessive CPU time and storage requirements, even for academic two-dimensional problems. In order to allow for realistic three-dimensional two-scale simulations, a significant reduction of the CPU and memory usage is required. For this purpose, the authors have recently proposed a reduced basis homogenization scheme based on a mixed incremental variational principle. The approach exploits the potential structure of generalized standard materials. Thereby, important speed-ups and memory savings can be achieved. Using high-performance GPUs, the reduced-basis method can be further accelerated. In the present contribution, our previous works are combined and extended to form the FE2-reduced method: the FE2R. The FE2R can be used to simulate three-dimensional structural problems with consideration of the nonlinearity and microstructure of the underlying material at acceptable computational cost. Thereby, it allows for a new level of complexity in nonlinear multiscale simulations. Numerical examples illustrate the capabilities of the chosen approach. Copyright (c) 2016 John Wiley & Sons, Ltd.