Efficient implementation of non-linear flow law using neural network into the Abaqus Explicit FEM code

Highlights

• Development of an efficient and robust numerical Artificial Neural Network based elastoplastic constitutive law for Finite Element simulation.

• Comparison of several network architectures (number of layers, activation functions) and performance analysis.

• Analysis of neural network performance for stress and derivative evaluation.

• Methodology for implementing a neural network in the Abaqus Explicit computational code.

• Validation of the proposed implementation was done through different benchmarks.

Abstract

Machine learning techniques are increasingly used to predict material behavior in scientific applications and offer a significant advantage over conventional numerical methods. In this work, an Artificial Neural Network (ANN) model is used in a finite element formulation to define the flow law of a metallic material as a function of plastic strain ${\varepsilon }^p$, plastic strain rate ${\stackrel{\mathit{.}}{\varepsilon }}^p$ and temperature $T$. First, we present the general structure of the neural network, its operation and focus on the ability of the network to deduce, without prior learning, the derivatives of the flow law with respect to the model inputs. In order to validate the robustness and accuracy of the proposed model, we compare and analyze the performance of several network architectures with respect to the analytical formulation of a Johnson–Cook behavior law for a 42CrMo4 steel. In a second part, after having selected an Artificial Neural Network architecture with 2 hidden layers, we present the implementation of this model in the Abaqus Explicit computational code in the form of a VUHARD subroutine. The predictive capability of the proposed model is then demonstrated during the numerical simulation of two test cases: the necking of a circular bar and a Taylor impact test. The results obtained show a very high capability of the ANN to replace the analytical formulation of a Johnson–Cook behavior law in a finite element code, while remaining competitive in terms of numerical simulation time compared to a classical approach.

Introduction

Numerical simulation of forming processes, machining or the behavior of structures subjected to dynamic loads and impacts requires the use of specific material behavior laws, whose parameters are identified by tests based on Taylor impacts, Hopkinson bars or Gleeble thermomechanical simulator. The behavior laws are selected according to their availability in a finite element code or the possibility, if not available, to implement them through user subroutines. In this study, our work is based on the use of the finite element code Abaqus Explicit which offers the possibility to define user behavior laws through FORTRAN subroutines VUMAT or VUHARD [1], [2] like the work proposed by Duc-Toan et al. [3] or more recently by Ming et al. [4]. The usual procedure is to select a mathematical form of the behavior law among those available in the literature (Johnson–Cook, Zerilli Armstrong, …) and then, from the results of experimental tests, to identify via a regression method, the parameters of the selected law.

In most cases, the behavior of the material at high temperatures and strain rates is highly nonlinear, and the effects of many factors on the flow stress are also nonlinear, which reduces the accuracy of the prediction by the regression methods usually used and limits the field of application. In addition, the selection, development, and numerical implementation of such constitutive equations is time-consuming. Artificial intelligence techniques allow advances concerning the laws of behavior in order to allow a better identification of these laws. Thus, Versino et al. [5] used a Machine Learning technique based on symbolic regression for the development of data-driven constitutive model. Obtaining a flow equation, and thus its analytical derivative, allows the use of iterative solvers that employ Jacobians (i.e., the Newton–Raphson scheme) that allow a higher order of convergence. This symbolic regression technique has also been used by other authors since, such as Bomarito et al. [6], Park et al. [7] using constrained symbolic regression technique, or Nassr et al. [8] using evolutionary polynomial regression.

Given this situation, it is therefore natural to look for a method to eliminate some intermediate steps between experimental tests and numerical simulation in order to simplify the computational chain. In this perspective, recent advances in deep learning constitute a way of investigation. The basic idea is to replace the analytical formulation used to calculate the flow stress $\sigma$ of the material as a function of the plastic strain ${\varepsilon }^p$, the plastic strain rate ${\stackrel{\mathit{.}}{\varepsilon }}^p$ and the temperature $T$, by an Artificial Neural Network (ANN). This neural network is trained to reproduce the behavior of the considered material only from the experimental data resulting from the tests, ignoring any assumption on the analytical form of the assumed flow law. Consequently, it is no longer necessary to postulate an analytical form of the behavior law in order to implement it in a FEM code.

Artificial neural networks and deep learning are becoming more and more important in today’s society, and their fields of application are getting wider and wider. After a boom in the early 1990s and a decline in interest towards the end of the 20th century, neural networks are experiencing a resurgence of interest and even a huge media hype under the name of deep learning. Their use in science and physics is now widespread, notably because of the current availability of efficient tools allowing to program Artificial Neural Networks thanks to widely available libraries such as Tensorflow [9] for example. The most publicized applications of deep learning are mainly related to medical diagnosis, robotics, images and language recognition, but the applications go far beyond that and all sciences can now use these techniques, including thermomechanical numerical simulation.

Artificial neural networks can solve problems that are difficult to conceptualize using traditional computational methods. Unlike a classical approach based on a regression method, an Artificial Neural Network does not need to know the mathematical form of the model it seeks to reproduce. The Artificial Neural Network learns from the training data and can reproduce the behavior of a model from the simple knowledge of a series of input and output values with no prior assumption on their nature and their interrelations. Hornik et al. [10] have rigorously established in 1989 that feed-forward Neural Network are a class of universal approximators, extending the work proposed twenty years before by Minsky and Papert [11] where they demonstrated that the simple two-layer perceptron is incapable of usefully representing or approximating functions outside a very narrow and special class. The ANN has adjustment, memorization and anticipation capabilities, and better performances than the approach based on implementing a constitutive equation. Therefore, Artificial Neural Networks can now enable novel approaches for modeling the behavior of materials and have been successfully applied in the prediction of constitutive relationships of some metals and alloys in recent years. Applicability of ANN to model path dependent plasticity has been explored and some review of the literature can be found for example in Gorji et al. [12] concerning the use of Recurrent Neural Network, in Jamli et al. [13] concerning their application in finite element analysis of metal forming processes, or in Jiao et al. [14] concerning the applicability to meta-materials and their characterization. A distinction must be made between ANN-based hardening models and ANN-based constitutive models. Both approaches have been studied by many researchers over the last thirty years. Ghaboussi et al. [15] published a pioneering paper, in which they proposed an ANN-based constitutive model for planar concrete under monotonic biaxial loading and cyclic uniaxial loading in which they successfully predicted several loading paths in the biaxial loading condition. They improved NN architecture by introducing Adaptive NN and Autoprogressive NN in [16], [17] where the network architecture evolves during the training phase to better learn complex stress–strain behavior of materials using a global load–deflection response. The approach adopted for this study is an ANN-based hardening model for which, the evaluation of the material flow stress calculated by the ANN is combined with a Radial Return type integration scheme.

Lin et al. [18] developed a neural network to predict the flow stress of 42CrMo4 steel in hot compression tests on a Gleeble thermomechanical device and showed a very good correlation between experimental and predicted results. An extension to the numerical implementation of this approach would have been desirable. Javadi et al. [19] used a neural network to capture the behavior of complex materials using a FE model incorporating a backpropagation neural network. Lu et al. [20] presented a comparative study of the modeling of an Al–Cu–Mg–Ag alloy behavior by a constitutive equation based on the Zener–Hollomon parameter and a neural network. It also shows that the model based on the ANN proposes a better prediction than the constitutive equation. Ashtiani et al. [21] compared the prediction capabilities of an ANN against a conventional approach based on several behavior laws such as Johnson–Cook, Arrhenius and Strain compensated Arrhenius and concluded better efficiency and accuracy of the neural network in predicting the hot behavior of Al–Cu–Mg–Pb alloy. Ashtiani et al. [21] have shown that a well-trained ANN can efficiently overcome the lacks of physics coming from analytical constitutive behaviors such as the Johnson–Cook, or the Arrhenius one. Ali et al. [22] presented an ANN model coupled with a rate dependent crystal plasticity finite element method to simulate the stress–strain behavior of material and its microstructure evolution in a AA6063-T6 under simple shear and tension. Stoffel et al. [23], [24] applied ANN to complicated structural deformations of shock-wave loaded plates involving both geometrical and physical non-linearities. Li et al. [25] implemented a VUMAT subroutine for Abaqus where parameters were identified through a combination of analytical formulas and a back propagation algorithm. Recently, Huang et al. [26] developed a neural network model to predict the flow stress and the microstructure evolution of Ti–6Al–4V alloy. It also showed the superiority of this approach over an Arrhenius behavior model, especially because the ANN can predict the flow stress in the whole range of deformation. Temporal Convolutional Networks have also been applied by Abueidda et al. [27] to predict the history-dependent responses of a class of cellular materials. Thermodynamics based ANN where also proposed by Masi et al. [28] to reduce physically inconsistencies in the predictions of the NN. They demonstrate the wide applicability of TANNs for modeling elasto-plastic materials, using both hyper- and hypo-plasticity models. As presented by Knight et al. [29], evolution of Neural Network Architectures is not the only way to progress, constant evolution of the hardware architecture to implement ANN will also have to be taken into account for the next future.

In Section 2, we present the main bases of the deep learning with in details the description of the structure of the neural network and the equations which govern its functioning. As we will see in Section 4 concerning the numerical implementation of the flow law, the programming of the neural network on the finite element code Abaqus Explicit requires the determination of the $3$ derivatives of the flow stress $\sigma$ with respect to the plastic strain ${\varepsilon }^p$, the plastic strain rate ${\stackrel{\mathit{.}}{\varepsilon }}^p$ and the temperature $T$. The determination of these derivatives will be the subject of the second part of Section 2. Section 3 is devoted to a detailed presentation of the ANN learning for a Johnson–Cook type behavior law for a 42CrMo4 steel. In this section, we will show the influence of the network structure on the accuracy of the flow stress and the derivatives evaluation. Section 4 is devoted to a presentation of the numerical implementation of the neural network in the Abaqus finite element code in the form of a FORTRAN VUHARD subroutine as well as numerical test cases to validate the proposed approach. A conclusion and perspectives are finally proposed at the end.