FE² Computations With Deep Neural Networks: Algorithmic Structure, Data Generation, and Implementation

Abstract

Multiscale FE² computations enable the consideration of the micro-mechanical material structure in macroscopical simulations. However, these computations are very time-consuming because of numerous evaluations of a representative volume element, which represents the microstructure. In contrast, neural networks as machine learning methods are very fast to evaluate once they are trained. In this contribution, a DNN-FE² approach is presented where deep neural networks (DNNs) are applied as a surrogate model of the representative volume element. This requires a clear description of the algorithmic FE² structure and the particular integration of deep neural networks. Further, a suitable training strategy is explained, where particular knowledge of the material behavior is considered to reduce the required amount of training data. A study of how much training data is required for reliable FE² simulations is provided with special focus on the errors compared to conventional FE² simulations. It turns out that Sobolev training increases both speed-up as well as accuracy of the prediction in comparison to using two different neural networks for stress and tangent matrix prediction. To gain a significant speed-up of the FE² computations, an efficient implementation of the trained neural network in a finite element code is provided as well. The DNN-FE² simulations are even further accelerated, when drawing on state-of-the-art high-performance computing libraries and just-in-time compilation that yield a maximum speed-up of a factor of more than 5,000 compared to a reference FE² computation. Moreover, the deep neural network surrogate model is able to overcome load-step size limitations of the RVE computations in step-size controlled computations.

Keywords: multiscale finite element computations, deep neural networks, surrogate modeling, Sobolev training, representative volume elements, step-size control