Herein, an artificial neural network (ANN)-based approach for the efficient automated modeling and simulation of isotropic hyperelastic solids is presented. Starting from a large data set comprising deformations and corresponding stresses, a simple, physically based reduction of the problem’s dimensionality is performed in a data processing step. More specifically, three deformation type invariants serve as the input instead of the deformation tensor itself. In the same way, three corresponding stress coefficients replace the stress tensor in the output layer. These initially unknown values are calculated from a linear least square optimization problem for each data tuple. Using the reduced data set, an ANN-based constitutive model is trained by using standard machine learning methods. Furthermore, in order to ensure thermodynamic consistency, the previously trained network is modified by constructing a pseudo-potential within an integration step and a subsequent derivation which leads to a further ANN-based model. In the second part of this work, the proposed method is exemplarily used for the description of a highly nonlinear Ogden type material. Thereby, the necessary data set is collected from virtual experiments of discs with holes in pure plane stress modes, where influences of different loading types and specimen geometries on the resulting data sets are investigated. Afterwards, the collected data are used for the ANN training within the reduced data space, whereby an excellent approximation quality could be achieved with only one hidden layer comprising a low number of neurons. Finally, the application of the trained constitutive ANN for the simulation of two three-dimensional samples is shown. Thereby, a rather high accuracy could be achieved, although the occurring stresses are fully three-dimensional whereas the training data are taken from pure two-dimensional plane stress states.
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.
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.
Herein, a neural network‐based constitutive model for isotropic hyperelastic solids which makes use of a physically motivated dimensionality reduction into the invariant space is presented. In order to automatically fulfill thermodynamic consistency, gradients of the network with respect to the input quantities are considered within a customized training loop. The proposed approach is exemplarily applied to the finite element simulation of two three‐dimensional samples, while only data collected from pure two‐dimensional virtual experiments are needed for the model calibration before.
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