FE$${}^\textrm{ANN}$$: an efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining

Abstract

Herein, we present a new data-driven multiscale framework called FE$${}^\textrm{ANN}$$ ANN which is based on two main keystones: the usage of physics-constrained artificial neural networks (ANNs) as macroscopic surrogate models and an autonomous data mining process. Our approach allows the efficient simulation of materials with complex underlying microstructures which reveal an overall anisotropic and nonlinear behavior on the macroscale. Thereby, we restrict ourselves to finite strain hyperelasticity problems for now. By using a set of problem specific invariants as the input of the ANN and the Helmholtz free energy density as the output, several physical principles, e. g., objectivity, material symmetry, compatibility with the balance of angular momentum and thermodynamic consistency are fulfilled a priori. The necessary data for the training of the ANN-based surrogate model, i. e., macroscopic deformations and corresponding stresses, are collected via computational homogenization of representative volume elements (RVEs). Thereby, the core feature of the approach is given by a completely autonomous mining of the required data set within an overall loop. In each iteration of the loop, new data are generated by gathering the macroscopic deformation states from the macroscopic finite element simulation and a subsequently sorting by using the anisotropy class of the considered material. Finally, all unknown deformations are prescribed in the RVE simulation to get the corresponding stresses and thus to extend the data set. The proposed framework consequently allows to reduce the number of time-consuming microscale simulations to a minimum. It is exemplarily applied to several descriptive examples, where a fiber reinforced composite with a highly nonlinear Ogden-type behavior of the individual components is considered. Thereby, a rather high accuracy could be proved by a validation of the approach.

A comparative study on different neural network architectures to model inelasticity

“…This clarifies that such a network may be arbitrarily nested, but for fixed w l nm is a well-defined function of the input I. In this sense, the network as a function (10) maps the input I to the output O. In order to adapt the weights w l nm such that the predictions of the NN match the expected values, a training data set  ∶= { 1 , … ,  N ds } consisting of N ds ∈ N data tuples  𝛼 is required.…”

“…Consequently, the activations must be differentiable twice to enable an optimization. 10,11,48 Therefore, the hyperbolic tangent and the softplus activation are used for the corresponding networks in the following.…”

“…To address this issue, a relatively new approach in NN-based constitutive modeling, and scientific machine learning (ML) in general, is to integrate crucial underlying physics in either a strong or weak form. These methods, known as physics-informed, 4,5 mechanics-informed, 6,7 physics-augmented, 8,9 physics-constrained, 10 or thermodynamics-based, 11 improve extrapolation capability and allow for the use of sparse training data. Thereby, various conditions such as thermodynamic consistency, material symmetry, objectivity, or also material stability can be considered.…”

“…Meanwhile, FNNs using invariants as input and the hyperelastic potential as output are a very well established approach. [8][9][10][14][15][16][17][18] Thereby, an improved training is applied that allows calibration of the network directly by tuples of stress and strain, that is, the derivative of energy with respect to the deformation is included into the loss, which is also called Sobolev training. 19,20 Alternatively, a network previously trained to predict stress coefficients can be used to construct a pseudopotential, thus ensuring thermodynamic consistency of NN-based elastic models a posteriori.…”

Learning hyperelastic anisotropy from data via a tensor basis neural network

“…This was extended by Fuhg and Bouklas [37] to anisotropic materials, strictly enforcing known symmetries up to transverse isotropy; this work showed that this simplified learning approach led to significant generalization capabilities when the physics do not radically change outside of the training region. This approach was also utilized in Kalina et al [41] integrated in a multiscale framework with automated data-mining. In Fuhg et al [32], tensor basis NNs were utilized to discover the character of the anisotropy of the material through labeled data.…”