  The morphology of nanostructured materials exhibiting a polydisperse porous
space, such as aerogels, is very open porous and fine grained. Therefore, a
simulation of the deformation of a large aerogel structure resolving the
nanostructure would be extremely expensive. Thus, multi-scale or homogenization
approaches have to be considered. Here, a computational scale bridging approach
based on the FE$^2$ method is suggested, where the macroscopic scale is
discretized using finite elements while the microstructure of the open-porous
material is resolved as a network of Euler-Bernoulli beams. Here, the beam
frame based RVEs (representative volume elements) have pores whose size
distribution follows the measured values for a specific material. This is a
well-known approach to model aerogel structures. For the computational
homogenization, an approach to average the first Piola-Kirchhoff stresses in a
beam frame by neglecting rotational moments is suggested. To further overcome
the computationally most expensive part in the homogenization method, that is,
solving the RVEs and averaging their stress fields, a surrogate model is
introduced based on neural networks. The networks input is the localized
deformation gradient on the macroscopic scale and its output is the averaged
stress for the specific material. It is trained on data generated by the beam
frame based approach. The effiency and robustness of both homogenization
approaches is shown numerically, the approximation properties of the surrogate
model is verified for different macroscopic problems and discretizations.
Different (Quasi-)Newton solvers are considered on the macroscopic scale and
compared with respect to their convergence properties.
