Computational homogenization for aerogel-like polydisperse open-porous materials using neural network--based surrogate models on the microscale

TL;DR

This work tackles the computational challenge of simulating deformation in aerogel-like open-porous polydisperse materials by developing a scale-bridging FE^2 framework that uses beam-frame RVEs at the microscale. It introduces neural-network surrogates trained on beam-frame data to predict the averaged macroscopic stress from the local deformation gradient, significantly accelerating the macroscopic FE problem while preserving accuracy. The study provides 2D and 3D surrogate models, demonstrates convergence of macro solvers (Newton and BFGS), and validates the approach on complex geometries (plates with holes) and torsion tests, achieving speedups up to several thousand times in MATLAB. The results enable efficient exploration and optimization of large aerogel-like structures, making multiscale simulations of polydisperse open-porous materials practically feasible.

Abstract

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.

Figures (20)

• Figure 1: An SEM image of a biopolymer aerogel. The figure is adapted from Rege et al. DLR_biopoly_aerogel2018.
• Figure 2: Circle packing (left) and Voronoi tessellation (right) for the generation of the two-dimensional RVE.
• Figure 3: Voronoi tessellation that results from copied sphere centers. Red square in the middle has periodic boundaries.
• Figure 4: Result of three-dimensional beam frame model that is compressed in $x$ direction. Image of the deformed RVE on the left and the deformation gradient with the resulting averaged first Piola-Kirchhoff stress tensor on the right. Colors of the beams represent the Von-Mises stress values in the corresponding elements. Von-Mises stress values are given in megapascal (MPa).
• Figure 5: FE$^2$ method with an RVE modeled with beams. The macroscopic deformation gradient $\overline{F}(\bar{x})$ defines the boundary conditions of the beam frame model attached to the macroscopic integration point $\bar{x}$. The stresses in the beams after solving the RVE problem is averaged and results in the macroscopic stress $\overline{P}(\overline{F}(\bar{x}))$.