In the present contribution, we first introduce a multi-scale simulations of higher-order continua of n−th grade. Therefore, we take additional terms of the Taylor approximation within the representative volume element (RVE) into account to consider the higher- order macroscale kinematic. Ensuring an energetic consistent formulation analogously to the Hill-Mandel criterion for first-gradient materials, yields the proposed formulation including a consistent linearization, as shown in detail in [1]. For the discrete setting, we apply B-splines on both scales to match the continuity requirements. The resulting staggered solution procedure consumes heavily computational resources. The deal with this, we derive a variational formulation of the multi-scale system, using a dimensional expansion. With this, we can analyze classical FE2 schemes, improve the global convergence properties by a suitable choice of the shape functions and finally, apply a null-space reduction scheme to systematically reduce the large number of unknowns. This technique reduces the complexity for the construction of the consistent linearization dramatically even for higher-order problems. Finally, we will demonstrate the properties of our latest developments using the analytical solution of a one-dimensional system with one-dimensional RVE’s. Afterwards, we show the application on two-dimensional problems of gradient elasticity and provide an outlook on multi-scale phase-field fracture. REFERENCES [1] F. Schmidt, M. Kru ̈ger, M.-A. Keip, and C. Hesch. Computational homogenization of higher-order continua. International Journal for Numerical Methods in Engineering, 123(11):2499-2529, 2022. [2] C.Hesch, F. Schmidt and S. Schuss. Variational formulation and monolithic solution of computational homogenization methods. In preparation, 2024.