FE${}^\textbf{ANN}$ $-$ 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 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 (FE) 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.

Neural networks meet hyperelasticity: A guide to enforcing physics

“…To remedy this weakness, a fairly new trend in NN-based constitutive modeling, and in scientific machine learning in general [41], is to include essential underlying physics in a strong form, e.g., by using adapted network architectures, or in a weak form, e.g., by modifying the loss term for the training [34]. These types of approaches, coined as physics-informed [22], mechanics-informed [4], physics-augmented [25], physics-constrained [19], or thermodynamics-based [37], enable an improvement of the extrapolation capability and the usage of sparse training data [22,27], which is particularly important when constitutive models are to be fitted to experimental data.…”

“…However, similar to the approaches [28,31,43] applied to anisotropic problems, the elastic potential is needed directly for training within [29,46]. In the meantime, NNs using invariants as inputs and the hyperelastic potential as output, thus also being a priori thermodynamically consistent, have become a fairly established approach [12,19,24,25,30,32,33,48]. Thereby, a more sophisticated training is applied, which allows the direct calibration of the network by tuples of stress and strain, i.e., the derivative of the energy with respect to the deformation is used in the loss term.…”

“…There are several approaches for building polyconvex NNs [4,7,12,24,33,47,48,50], with the most notable technique for incorporating this condition being the use of input convex neural networks (ICNNs) originally introduced by Amos et al [3]. For the fulfillment of the growth condition, a special network architecture may be applied [19], whereas using analytical growth terms is more widely spread [24,25]. Finally, while several works introduce correction terms which ensure normalization conditions, they are either not polyconvex [9,50] or restricted to the case of nearly incompressible material behavior [33].…”

“…Regarding the introduced new family of NN-based hyperelastic models, which fulfill all of the aforementioned conditions in an exact way, we advocate for naming them as physics-augmented neural networks (PANNs). The proposed framework will be very valuable in fields, where highly flexible and at the same time physically sensible constitutive models are required, such as the simulation of microstructured materials [14,19]. For this purpose, the PANN approach is build up by extending the aforementioned model [24] by polyconvex normalization terms, followed by a detailed analytical and numerical study analyzing the overall model.…”

Machine Learning Aided Multiscale Magnetostatics

“…This motivates the development of computationally feasible approaches in this realm. A recently emerging third alternative in literature to this end is employing data-driven surrogate models devising machine learning, see, e.g., [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33] and the references therein. Deploying the training costs offline materializing simulation or experimental data, these models surpass conventional rule-based approaches by drastically reducing the computational cost required during the prediction phase [34,35,36].…”