Automated constitutive modeling of isotropic hyperelasticity based on artificial neural networks

Abstract

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.

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

“…Probably, the most common technique is the application of artificial neural networks (ANNs), which have already been proposed in the early 90s by the pioneering work of Ghabussi et al [24]. In the last decades, ANNs have been intensively used for mechanical material modeling and simulations by means of the finite element method (FEM), e. g., in [25,26,27,28,29] among others.…”

“…algorithms. By choosing problem-specific invariant sets as the input variables, material symmetries are automatically satisfied for hyperelastic models [36,29,37]. Furthermore, the thermodynamic consistency can be fulfilled by choosing the free energy as the output quantity.…”

“…The training is done by using the SLSQP optimizer (Sequential Least Squares Programming). Thereby, the ANN is trained several times, where the parameters of the best achieved training state are stored at the end [29]. An implementation of the described workflow is realized using Python, Tensorflow and SciPy.…”

“…Now, previously unknown deformations have to be detected, whereby it is meaningful to only consider states providing additional information for the material under observation. Since the intrinsic constitutive behavior of the material lives in the space of invariants I ⊂ R (𝑛+𝐴)×1 , it is useful to perform a transformation into this space at this point [29]: To avoid that this unnecessarily inflates the training process, a Note that it is useful to save the time series of a new deformation state, i. e., all states F (𝑡 0 , 𝑡 1 , . .…”

“…With this choice, a highly nonlinear behavior of the matrix phase is achieved, cf. [29]. The parameters of the fibers are chosen in such a way, that a neo-Hookean model results.…”