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.

翻译：形态呈现多分散孔隙空间的纳米结构材料（如气凝胶）具有高度开孔和细晶粒特征。因此，直接解析纳米结构的大型气凝胶变形模拟将极其耗时，必须采用多尺度或均匀化方法。本研究提出基于FE$^2$方法的计算尺度桥接方案：宏观尺度采用有限元离散，而开孔材料的微观结构则用欧拉-伯努利梁网络解析。其中，基于梁框架的代表性体积单元（RVE）的孔径分布遵循特定材料的实测值——这是建模气凝胶结构的经典方法。为实施计算均匀化，本文提出一种通过忽略旋转力矩来平均梁框架中第一类Piola-Kirchhoff应力的方法。为进一步克服均匀化方法中计算量最大的环节（即求解RVE并平均其应力场），引入基于神经网络的代理模型。该网络以宏观尺度的局部变形梯度为输入，输出特定材料的平均应力，并通过梁框架方法生成的数据进行训练。数值实验证明了两种均匀化方法的效率和鲁棒性，并在不同宏观问题及离散格式下验证了代理模型的近似性能。宏观尺度采用多种（拟）牛顿求解器，并对比分析了其收敛特性。