A comparative study on different neural network architectures to model inelasticity

Abstract

The mathematical formulation of constitutive models to describe the path‐dependent, that is, inelastic, behavior of materials is a challenging task and has been a focus in mechanics research for several decades. There have been increased efforts to facilitate or automate this task through data‐driven techniques, impelled in particular by the recent revival of neural networks (NNs) in computational mechanics. However, it seems questionable to simply not consider fundamental findings of constitutive modeling originating from the last decades research within NN‐based approaches. Herein, we propose a comparative study on different feedforward and recurrent neural network architectures to model 1D small strain inelasticity. Within this study, we divide the models into three basic classes: black box NNs, NNs enforcing physics in a weak form, and NNs enforcing physics in a strong form. Thereby, the first class of networks can learn constitutive relations from data while the underlying physics are completely ignored, whereas the latter two are constructed such that they can account for fundamental physics, where special attention is paid to the second law of thermodynamics in this work. Conventional linear and nonlinear viscoelastic as well as elastoplastic models are used for training data generation and, later on, as reference. After training with random walk time sequences containing information on stress, strain, and—for some models—internal variables, the NN‐based models are compared to the reference solution, whereby interpolation and extrapolation are considered. Besides the quality of the stress prediction, the related free energy and dissipation rate are analyzed to evaluate the models. Overall, the presented study enables a clear recording of the advantages and disadvantages of different NN architectures to model inelasticity and gives guidance on how to train and apply these models.

Viscoelasticty with physics-augmented neural networks: model formulation and training methods without prescribed internal variables

“…Thus, recurrent architectures are an appealing way to model inelastic behavior, since the provision of history variables or internal variables can be avoided [17]. In particular, the development of advanced RNN cells such as long short-term memory (LSTM) [18] or gated recurrent units (GRUs) [19], which provide increased memory capacity and enable an efficient training, lead to great popularity and rapid progress in this field.…”

“…Thereby, a distinction must be made between methods with weak fulfillment of the principles by an additional term in the loss function and strong fulfillment with a priori compliance with the respective principle by constraining the architecture of the network [41]. According to the comparative study presented in Rosenkranz et al [17], the second approach is more promising since it is more efficient in terms of required data, robust and can extrapolate very well due to the high degree of incorporated physics, but involves some difficulties. The challenge here is to efficiently restrict the network without loosing too much flexibility.…”

“…Thereby, thermodynamic consistency is ensured by using a convex dissipation potential but internal variables are required for training. Several approaches based on a similar modeling strategy can be found [17,39,44,57]. In contrast to [56], however, no prior knowledge of the internal variable(s) is needed for training there.…”

“…The NN-based dissipation potential depends on ( q, ε, q) [17]. Similar to the non-equilibrium part of the free energy, the dependency of ε and q is expressed in terms of p = ε−q.…”

Parametrized polyconvex hyperelasticity with physics-augmented neural networks

“…For the construction of polyconvex potentials, several approaches exist (Chen and Guilleminot, 2022;Klein et al, 2022a;Tac et al, 2022), where the most noteworthy approaches are based on input-convex neural networks (ICNNs). Proposed by Amos et al (2017), this special network architecture has not only been successfully applied in the framework of polyconvexity, but is also very attractive in, for example, other physical applications which require convexity (Huang et al, 2021;As'ad and Farhat, 2023;Rosenkranz et al, 2023) and convex optimization (Calafiore et al, 2020). Besides this particular choice of network architecture, using invariants as strain measures ensures the fulfillment of several mechanical conditions at once, for example, objectivity and material symmetry.…”