Learning hyperelastic anisotropy from data via a tensor basis neural network

Fuhg, Jan N.; Bouklas, Nikolaos; Jones, Reese E.

doi:10.1016/j.jmps.2022.105022

Cited by 26 publications

(15 citation statements)

References 122 publications

(119 reference statements)

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“…For instance, an extension to further material symmetry groups [10] have to be made by integrating appropriate invariant sets into the implementation. Furthermore, to make the FE ANN approach even more general, the inclusion of a preprocessing step using tensor-basis ANNs which can discover type and orientation of the underlying anisotropy would be a valuable addition [22]. In order to incorporate additional physics, the usage of ANN-based models that account for polyconvexity is possible [45,46,72].…”

Section: Discussionmentioning

confidence: 99%

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

et al. 2023

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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.

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Section: Discussionmentioning

confidence: 99%

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

et al. 2023

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“…It has been recently showcased that the incorporation of physical constraints in data-driven modeling of elastic material behavior [24,25,27,58,26] leads to more robust models compared to earlier machine learning-based approaches. Here we briefly review using the theory of representation for tensor functions towards this goal.…”

Section: Modeling Of the Elastic Responsementioning

confidence: 99%

“…Data-driven modeling of elastic material behavior was initially dominated by "black-box" models which directly map strain to stress [17,18,19,20,21,22]. Lately, these models have been extended by including physical principles and mechanistic assumptions into the modeling process by, for example, employing the representation theorem of tensor functions [23,24,25,26] or by enforcing polyconvexity of the corresponding free energy [27,26]. Furthermore, hybrid modeling frameworks have been explored where a data-driven model locally corrects the output of a traditional phenomenological approach [28,29,30].…”

Section: Introductionmentioning

confidence: 99%

Modular machine learning-based elastoplasticity: generalization in the context of limited data

Fuhg¹,

Hamel²,

Johnson³

et al. 2022

Preprint

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The development of highly accurate constitutive models for materials that undergo path-dependent processes continues to be a complex challenge in computational solid mechanics. Challenges arise both in considering the appropriate model assumptions and from the viewpoint of data availability, verification, and validation. Recently, data-driven modeling approaches have been proposed that aim to establish stress-evolution laws that avoid user-chosen functional forms by relying on machine learning representations and algorithms. However, these approaches not only require a significant amount of data but also need data that probes the full stress space with a variety of complex loading paths. Furthermore, they rarely enforce all necessary thermodynamic principles as hard constraints. Hence, they are in particular not suitable for low-data or limited-data regimes, where the first arises from the cost of obtaining the data and the latter from the experimental limitations of obtaining labeled data, which is commonly the case in engineering applications. In this work, we discuss a hybrid framework that can work on a variable amount of data by relying on the modularity of the elastoplasticity formulation where each component of the model can be chosen to be either a classical phenomenological or a data-driven model depending on the amount of available information and the complexity of the response. The method is tested on synthetic uniaxial data coming from simulations as well as cyclic experimental data for structural materials. The discovered material models are found to not only interpolate well but also allow for accurate extrapolation in a thermodynamically consistent manner far outside the domain of the training data. This ability to extrapolate from limited data was the main reason for the early and continued success of phenomenological models and the main shortcoming in machine learning-enabled constitutive modeling approaches. Training aspects and details of the implementation of these models into Finite Element simulations are discussed and analyzed.

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…However, since ellipticity is difficult to verify and ensure, the concept of polyconvexity of the strain energy potential [5,6], which implies ellipticity and is mathematically linked to the existence and stability of solutions of the elasticity problem, is preferable for the formulation of constitutive models [39]. 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].…”

Section: Introductionmentioning

confidence: 99%

Neural networks meet hyperelasticity: A guide to enforcing physics

Linden¹,

Klein²,

Kalina³

et al. 2023

Preprint

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In the present work, a hyperelastic constitutive model based on neural networks is proposed which fulfills all common constitutive conditions by construction, and in particular, is applicable to compressible material behavior. Using different sets of invariants as inputs, a hyperelastic potential is formulated as a convex neural network, thus fulfilling symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency. In addition, a physically sensible stress behavior of the model is ensured by using analytical growth terms, as well as normalization terms which ensure the undeformed state to be stress free and with zero energy. The normalization terms are formulated for both isotropic and transversely isotropic material behavior and do not violate polyconvexity. By fulfilling all of these conditions in an exact way, the proposed physics-augmented model combines a sound mechanical basis with the extraordinary flexibility that neural networks offer. Thus, it harmonizes the theory of hyperelasticity developed in the last decades with the up-to-date techniques of machine learning. Furthermore, the non-negativity of the hyperelastic potential is numerically verified by sampling the space of admissible deformations states, which, to the best of the authors' knowledge, is the only possibility for the considered nonlinear compressible models. The applicability of the model is demonstrated by calibrating it on data generated with analytical potentials, which is followed by an application of the model to finite element simulations. In addition, an adaption of the model to noisy data is shown and its extrapolation capability is compared to models with reduced physical background. Within all numerical examples, excellent and physically meaningful predictions have been achieved with the proposed physicsaugmented neural network.

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Learning hyperelastic anisotropy from data via a tensor basis neural network

Cited by 26 publications

References 122 publications

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

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

Modular machine learning-based elastoplasticity: generalization in the context of limited data

Neural networks meet hyperelasticity: A guide to enforcing physics

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