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An artificial Neural Network (NNW) is designed to serve as a surrogate model of micro-scale simulations in the context of multi-scale analyses in solid mechanics.
The design and training methodologies of the NNW are developed in order to allow accounting for history-dependent material behaviors. On the one hand, a Recurrent Neural Network (RNN) using a Gated Recurrent Unit (GRU) is constructed, which allows mimicking the internal variables required to account for history-dependent behaviors since the RNN is self-equipped with hidden variables that have the ability of tracking loading history. On the other hand, in order to achieve accuracy under multi-dimensional non-proportional loading conditions, training of the RNN is achieved using sequential data. In particular the sequential training data are collected from finite element simulations on an elasto-plastic composite RVE subjected to random loading paths. The random loading paths are generated in a way similar to a random walking in stochastic process and allow generating data for a wide range of strain-stress states and state evolution.
The accuracy and efficiency of the RNN-based surrogate model is tested on the structural analysis of an open-hole sample subjected to several loading/unloading cycles. It is shown that a similar accuracy as with a FE2 multi-scale simulation can be reached with the RNN-based surrogate model as long as the local strain state remains in the training range, while the computational time is reduced by four orders of magnitude.

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... Similarly RNNs were used to efficiently predict non-linear hysteric behavior of a structure subjected to random load history 12 . Furthermore, due to the compatibility of the NN to mimic the non-linear material behavior at the Gauss points, the NNs were used as surrogate models in multiscale scenarios to learn history dependent constitutive laws 13,14,15 . Similarly, the viscoplastic structural response was investigated through an FFNN 16,17 and a nonlinear effective electric constitutive law for graphene polymer nanocomposites was developed 18 . ...

... Now we introduce the transformation used to update the cell state, the hidden state and the output vector in the Eqs. (14), (15) and (16) below. ...

... All rights reserved. 14,15,16,17) and gray elements (see Fig. 18) elements denote that plastic yielding occurs. To demonstrate the potential of the neural network approaches, the simulation speed and the accuracy of simulated deformations are compared to each other. ...

In the present study, new methods are proposed to replace the constitutive law and the entire tangent stiffness matrix in Finite Element Analysis by artificial neural networks (ANNs). By combining the FEM with ANN, so‐called intelligent elements are developed. Firstly, as an extension to recent trends in model‐based material law replacement, we introduce an additional loss term corresponding to the material stiffness. This training procedure is referred to as Sobolev training and ensures that the ANN learns both the function approximating the stress behavior and its first derivative (material stiffness). In a following step, we introduce three methods to replace the entire local stiffness matrix of an element by approximating its generalized force‐displacement relations. These methods also adopt ANNs with Sobolev training procedure to predict the mentioned quantities. Since neural networks (NN) are universal function approximators, they are used to extract the stiffness information for elements undergoing plastic deformation. The focus of this work is to establish a neural network‐based FEM framework (independent of NN topology) to introduce an enhanced‐material law and in a consequent step also approximate stiffness information of truss, beam, and plate elements taking physical non‐linear behavior into account.

... The RNN and gated variants, e.g., the long short-term memory (LSTM) [52] cells and the gated recurrent units (GRUs) [53,54], have been applied to path-dependent materials modeling [55], including plastic composites [56], visco-elasticity [57], and homogeneous anisotropic hardening [58]. RNN-based constitutive models have also been applied to accelerate multi-scale simulations with path-dependent characteristics [59][60][61][62][63]. Recently, Bonatti and Mohr [64] proposed a self-consistent RNN for pathdependent materials such that the model predictions converge as the loading increment is decreased. ...

Characterization and modeling of path-dependent behaviors of complex materials by phenomenological models remains challenging due to difficulties in formulating mathematical expressions and internal state variables (ISVs) governing path-dependent behaviors. Data-driven machine learning models, such as deep neural networks and recurrent neural networks (RNNs), have become viable alternatives. However, pure black-box data-driven models mapping inputs to outputs without considering the underlying physics suffer from unstable and inaccurate generalization performance. This study proposes a machine-learned physics-informed data-driven constitutive modeling approach for path-dependent materials based on the measurable material states. The proposed data-driven constitutive model is designed with the consideration of universal thermodynamics principles, where the ISVs essential to the material path-dependency are inferred automatically from the hidden state of RNNs. The RNN describing the evolution of the data-driven machine-learned ISVs follows the thermodynamics second law. To enhance the robustness and accuracy of RNN models, stochasticity is introduced to model training. The effects of the number of RNN history steps, the internal state dimension, the model complexity, and the strain increment on model performances have been investigated. The effectiveness of the proposed method is evaluated by modeling soil material behaviors under cyclic shear loading using experimental stress–strain data.

... The Artificial Intelligence (AI) algorithms are now entering within material science problems [48,77,106,107]. Sheet forming can benefit from these new approaches, saving computation time at various steps of the classical 'old fashion' way of simulations. ...

This article details the ESAFORM Benchmark 2021. The deep drawing cup of a 1 mm thick, AA 6016-T4 sheet with a strong cube texture was simulated by 11 teams relying on phenomenological or crystal plasticity approaches, using commercial or self-developed Finite Element (FE) codes, with solid, continuum or classical shell elements and different contact models. The material characterization (tensile tests, biaxial tensile tests, monotonic and reverse shear tests, EBSD measurements) and the cup forming steps were performed with care (redundancy of measurements). The Benchmark organizers identified some constitutive laws but each team could perform its own identification. The methodology to reach material data is systematically described as well as the final data set. The ability of the constitutive law and of the FE model to predict Lankford and yield stress in different directions is verified. Then, the simulation results such as the earing (number and average height and amplitude), the punch force evolution and thickness in the cup wall are evaluated and analysed. The CPU time, the manpower for each step as well as the required tests versus the final prediction accuracy of more than 20 FE simulations are commented. The article aims to guide students and engineers in their choice of a constitutive law (yield locus, hardening law or plasticity approach) and data set used in the identification, without neglecting the other FE features, such as software, explicit or implicit strategy, element type and contact model.

... Surrogate models based on machine learning methods are quite popular in material analysis and design due to their powerful approximation capabilities and remarkable computational gains. Examples include the use of feedforward neural networks (FFNN) for modeling the electrical conduction in composites with graphene sheets [39,40] or the mechanical properties of reinforced composites [41], recurrent [42] and convolutional neural networks [43,44], as well as ensemble machine learning models [45]. In [46], a surrogate modeling strategy based on FFNNs was proposed by the authors, targeted at nanoreinforced polymers within the frame of multiscale analysis. ...

This paper presents a material optimization framework for identifying optimal material typologies to improve structural performance under the presence of uncertainties. Specifically, the focus in this work is on carbon nanotube (CNT)-reinforced concrete with the optimization problem consisting in finding the optimal CNT orientation in the host material so as to minimize the total deformation of structures made up from the composite. Regarding the material modeling, a two-level approach is considered to characterize the mechanical properties of the reinforced concrete. Specifically, cement mortar enhanced with carbon nanotubes is studied at a microscale level where a Drucker-Prager plasticity model is assumed to describe its inelastic behavior. Subsequently, the reinforced mortar along with the concrete’s larger aggregates is studied at a mesoscale level using continuum micromechanics. For the analysis of structural systems comprised of this composite material, an extension of the FE2 technique, termed FE3, is employed. To overcome the immense computational demands associated with FE3, efficient neural network-based surrogates are developed to approximate the nonlinear constitutive law of the composite. In this setting, the stochastic optimization problem equates to finding the optimal orientation of CNTs in the cement mortar, so as to achieve small structural deformations with low variability, under the presence of uncertainty in the loading conditions. To solve this problem, the Covariance Matrix Adaptation Evolution Strategy is chosen herein, and even though this approach requires a massive number of model runs, it is performed at a reasonable computational cost by virtue of the elaborated surrogate modeling scheme.

... An application of ANNs as surrogate models describing the anisotropic electrical response of graphene/polymer nanocomposites is shown in [57]. Furthermore, in [58], the application of recurrent neural networks (RNNs) as surrogate models within multiscale simulations of elastoplastic problems is shown. Furthermore, the RNN-based approach is compared to full FE 2 simulations. ...

Herein, we present a new data-driven multiscale framework called FEANN 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.

... Usually, neural networks are employed for this regression. In [11][12][13], the authors used recurrent neural networks, a special type of neural network, to learn a path-dependent elasto-plastic model for a composite RVE. Additionally, in Mozaffar et al. [13], microstructure descriptors were included to predict the stresses for a class of composite RVEs. ...

Understanding structure-property relations is essential to optimally design materials for specific applications. Two-scale simulations are often employed to analyze the effect of the microstructure on a component's macroscopic properties. However, they are typically computationally expensive and infeasible in multi-query contexts such as optimization and material design. To make such analyses amenable, the microscopic simulations can be replaced by surrogate models that must be able to handle a wide range of microstructural parameters. This work focuses on extending the methodology of a previous work, where an accurate surrogate model was constructed for microstructures under varying loading and material parameters using proper orthogonal decomposition and Gaussian process regression, to treat geometrical parameters. To this end, a method that transforms different geometries onto a parent domain is presented. We propose to solve an auxiliary problem based on linear elasticity to obtain the geometrical transformations. Using these transformations, combined with the nonlinear microscopic problem, we derive a fast-to-evaluate surrogate model with the following key features: (1) the predictions of the effective quantities are independent of the auxiliary problem, (2) the predicted stress fields fulfill the microscopic balance laws and are periodic, (3) the method is non-intrusive, (4) the stress field for all geometries can be recovered, and (5) the sensitivities are available and can be readily used for optimization and material design. The proposed methodology is tested on several composite microstructures, where rotations and large variations in the shape of inclusions are considered. Finally, a two-scale example is shown, where the surrogate model achieves a high accuracy and significant speed up, demonstrating its potential in two-scale shape optimization and material design problems.

... Therefore, recent work in the mechanics literature addresses the issue of learning homogenized constitutive models from computational data [28,29,6,7] or experimental data [30]. This learning problem requires determination of maps that take as inputs functions describing microstructural properties, and leads in to the topic of operator learning. ...

Fully resolving dynamics of materials with rapidly-varying features involves expensive fine-scale computations which need to be conducted on macroscopic scales. The theory of homogenization provides an approach to derive effective macroscopic equations which eliminates the small scales by exploiting scale separation. An accurate homogenized model avoids the computationally-expensive task of numerically solving the underlying balance laws at a fine scale, thereby rendering a numerical solution of the balance laws more computationally tractable. In complex settings, homogenization only defines the constitutive model implicitly, and machine learning can be used to learn the constitutive model explicitly from localized fine-scale simulations. In the case of one-dimensional viscoelasticity, the linearity of the model allows for a complete analysis. We establish that the homogenized constitutive model may be approximated by a recurrent neural network (RNN) that captures the memory. The memory is encapsulated in the evolution of an appropriate finite set of internal variables, discovered through the learning process and dependent on the history of the strain. Simulations are presented which validate the theory. Guidance for the learning of more complex models, such as arise in plasticity, by similar techniques, is given.

... The RNN and gated variants, e.g., the long short-term memory (LSTM) [50] cells and the gated recurrent units (GRUs) [51,52], have been applied to path-dependent materials modeling [53], including plastic composites [54], visco-elasticity [55], and homogeneous anisotropic hardening [56]. RNN-based constitutive models have also been applied to accelerate multi-scale simulations with path-dependent characteristics [57,58,59,60,61]. Recently, Bonatti and Mohr [62] proposed a self-consistent RNN for path-dependent materials such that the model predictions converge as the loading increment is decreased. ...

Characterization and modeling of path-dependent behaviors of complex materials by phenomenological models remains challenging due to difficulties in formulating mathematical expressions and internal state variables (ISVs) governing path-dependent behaviors. Data-driven machine learning models, such as deep neural networks and recurrent neural networks (RNNs), have become viable alternatives. However, pure black-box data-driven models mapping inputs to outputs without considering the underlying physics suffer from unstable and inaccurate generalization performance. This study proposes a machine-learned physics-informed data-driven constitutive modeling approach for path-dependent materials based on the measurable material states. The proposed data-driven constitutive model is designed with the consideration of universal thermodynamics principles, where the ISVs essential to the material path-dependency are inferred automatically from the hidden state of RNNs. The RNN describing the evolution of the data-driven machine-learned ISVs follows the thermodynamics second law. To enhance the robustness and accuracy of RNN models, stochasticity is introduced to model training. The effects of the number of RNN history steps, the internal state dimension, the model complexity, and the strain increment on model performances have been investigated. The effectiveness of the proposed method is evaluated by modeling soil material behaviors under cyclic shear loading using experimental stress-strain data.

... As an example, a computational framework to establish a datadriven constitutive model for heterogeneous path-dependent composites has been implemented to predict the stress-strain relationships via the principal values [26], in which adopted separate data-driven models were adopted for elastic and plastic parts, respectively. A recurrent neural network-accelerated multi-scale model for elastoplastic heterogeneous materials subjected to random cyclic and non-proportional loading paths was investigated by considering a single microstructure [27]. Within the small-strain regime, both linear and non-linear elastic responses of heterogeneous microstructures were captured by feeding probabilistic descriptors as an input [28]. ...

This study presents the applicability of conventional deep recurrent neural networks (RNN) to predict path-dependent plasticity associated with material heterogeneity and anisotropy. Although the architecture of RNN possesses inductive biases toward information over time, it is still challenging to learn the path-dependent material behavior as a function of the loading path considering the change from elastic to elastoplastic regimes. Our attempt is to develop a simple machine-learning-based model that can replicate elastoplastic behaviors considering material heterogeneity and anisotropy. The basic Long-Short Term Memory Unit (LSTM) is adopted for the modeling of plasticity in the two-dimensional space by enhancing the inductive bias toward the past information through manipulating input variables. Our results find that a single LSTM based model can capture the J2 plasticity responses under both monotonic and arbitrary loading paths provided the material heterogeneity. The proposed neural network architecture is then used to model elastoplastic responses of a two-dimensional transversely anisotropic material associated with computational homogenization (FE2). It is also found that a single LSTM model can be used to accurately and effectively capture the path-dependent responses of heterogeneous and anisotropic microstructures under arbitrary mechanical loading conditions.

The prediction of precipitation is of importance in the Thua Thien Hue Province, which is affected by climate change. Therefore, this paper suggests two models, namely, the Seasonal Auto-Regressive Integrated Moving Average (SARIMA) model and the Long Short-Term Memory (LSTM) model, to predict the precipitation in the province. The input data are collected for analysis at three meteorological stations for the period 1980–2018. The two models are compared in this study, and the results showed that the LSTM model was more accurate than the SARIMA model for Hue, Aluoi, and Namdong stations for forecasting precipitation. The best forecast model is for Hue station ( = 0.94, = 0.94, = 8.15), the second-best forecast model is for Aluoi station ( = 0.89, = 0.89, = 12.72), and the lowest level forecast is for Namdong station ( = 0.89, = 0.89, = 12.81). The study result may also support stakeholderswho apply these models with future data to mitigate natural disasters in Thua Thien Hue. HIGHLIGHTS
Neural network methods of SARIMA and LSTM can improve the accuracy of forecasting of monthly precipitation in the Thua Thien Hue Province.;
The local precipitation forecast system depends heavily on the neural network using meteorological data collected from Hue, Aluoi, and Namdong stations, and these are presented.;
The Min–Max normalization method for the data is applied to improve the accuracy of the precipitation forecast of the models.;
A comparison of forecasts implemented between LSTM with NSE, R2, and RMSE is made.;
The prediction of LSTM is significantly better than SARIMA for the monthly precipitation regime.;

Although being a popular approach for the modeling of laminated composites, mesoscale constitutive models often struggle to represent material response for arbitrary load cases. A better alternative in terms of accuracy is to use the FE² technique to upscale microscopic material behavior without loss of generality, but the associated computational effort can be extreme. It is therefore interesting to explore alternative surrogate modeling strategies that maintain as much of the fidelity of FE² as possible while still being computationally efficient. In this work, three surrogate modeling approaches are compared in terms of accuracy, efficiency and calibration effort: the state-of-the-art mesoscopic plasticity model by Vogler et al. (Vogler et al., 2013), regularized feed-forward neural networks and hyper-reduced-order models obtained by combining the Proper Orthogonal Decomposition (POD) and Empirical Cubature Method (ECM) techniques. Training datasets are obtained from a Representative Volume Element (RVE) model of the composite microstructure with a number of randomly-distributed linear-elastic fibers surrounded by a matrix with pressure-dependent plasticity. The approaches are evaluated with a comprehensive set of numerical tests comprising pure stress cases and three different stress combinations relevant in the design of laminated composites. The models are assessed on their ability to accurately reproduce the training cases as well as on how well they are able to predict unseen stress combinations. Gains in execution time are compared by using the trained surrogates in the FE² model of an interlaminar shear test.

The present work aims at the identification of the effective constitutive behavior of Σ5 aluminum grain boundaries (GB) for proportional loading by using machine learning (ML) techniques. The input for the ML approach is high accuracy data gathered in challenging molecular dynamics (MD) simulations at the atomic scale for varying temperatures and loading conditions. The effective traction-separation relation is recorded during the MD simulations. The raw MD data then serves for the training of an artificial neural network (ANN) as a surrogate model of the constitutive behavior at the grain boundary. Despite the extremely fluctuating nature of the MD data and its inhomogeneous distribution in the traction-separation space, the ANN surrogate trained on the raw MD data shows a very good agreement in the average behavior without any data-smoothing or pre-processing. Further, it is shown that the trained traction-separation ANN captures important physical properties and is able to predict traction values for given separations not contained in the training data. For example, MD simulations show a transition in traction-separation behaviour from pure sliding mode under shear load to combined GB sliding and decohesion with intermediate hardening regime at mixed load directions. These changes in GB behaviour are fully captured in the ANN predictions. Furthermore, by construction, the ANN surrogate is differentiable for arbitrary separation and also temperature, such that a thermo-mechanical tangent stiffness operator can always be evaluated. The trained ANN can then serve for large-scale FE simulation as an alternative to direct MD-FE coupling which is often infeasible in practical applications.

Plasticity theory aims at describing the yield loci and work hardening of a material under general deformation states. Most of its complexity arises from the nontrivial dependence of the yield loci on the complete strain history of a material and its microstructure. This motivated 3 ingenious simplifications that underpinned a century of developments in this field: 1) yield criteria describing yield loci location; 2) associative or nonassociative flow rules defining the direction of plastic flow; and 3) effective stress–strain laws consistent with the plastic work equivalence principle. However, 2 key complications arise from these simplifications. First, finding equations that describe these 3 assumptions for materials with complex microstructures is not trivial. Second, yield surface evolution needs to be traced iteratively, i.e., through a return mapping algorithm. Here, we show that these assumptions are not needed in the context of sequence learning when using recurrent neural networks, diverting the above-mentioned complications. This work offers an alternative to currently established plasticity formulations by providing the foundations for finding history- and microstructure-dependent constitutive models through deep learning.

We develop a Bayesian Inference (BI) of the parameters of a non-linear multiscale model and of its material constitutive laws using experimental composite coupon tests as observation data. In particular we consider non-aligned Short Fibers Reinforced Polymer (SFRP) as a composite material system and Mean-Field Homogenization (MFH) as a multiscale model. Although MFH is computationally efficient, when considering non-aligned inclusions, the evaluation cost of a non-linear response for a given set of model and material parameters remains too prohibitive to be coupled with the sampling process required by the BI. Therefore, a Neural-Network (NNW) is first trained using the MFH model, and is then used as a surrogate model during the BI process, making the identification process affordable.

The mathematical description of elastoplasticity is a highly complex problem due to the possible change from elastic to elasto-plastic behavior (and vice-versa) as a function of the loading path. Advanced physics-based plasticity models usually feature numerous internal variables (often of tensorial nature) along with a set of evolution equations and complementary conditions. In the present work, an attempt is made to come up with a machine-learning based model that can replicate the predictions anisotropic Yld2000-2d model with homogeneous anisotropic hardening (HAH). For this, a series of modeling problems of increasing complexity is formulated and sequentially addressed using neural network models. It is demonstrated that basic fully-connected neural network models can capture the characteristic non-linearities in the uniaxial stress-strain response such as the Bauschinger effect, permanent softening or latent hardening. A neural network with gated recurrent units (GRUs) and fully-connected layer is proposed for the modeling of plane stress plasticity for arbitrary loading paths. After training and testing the model through comparison with the Yld2000-2d/HAH model, the recurrent neural network model is also used to model the multi-axial stress-strain response of a two-dimensional foam. Here, the comparison with the results from unit cell simulations provided another validation of the proposed data-driven modeling approach.

Substructuring is a model order reduction technique that accelerates the finite element method in solid mechanics. In this improved hybrid substructuring approach, methods from computational intelligence empower a reduced-order meta element. We propose a nonlinear and inelastic intelligent meta element for history-dependent boundary value problems. Fully compatible with conventional finite elements, it can be used to assemble larger structures. Within the intelligent meta element, a new deep neural network architecture composed of convolutions and recursions, the Time-distributed Residual U-Net (TRUNet), learns to solve the history-dependent spatial regression problem. The TRUNet automatically creates and updates the internal history variables necessary for the mechanical problem. Based on a new data generation strategy, data from a wide variety of use-cases train the neural network. An interface connects the neural network and the finite element method using a new data pre- and post-processing strategy. In three numerical demonstrations of elastoplastic continua, the intelligent meta element performs well, exhibiting low errors on a separate test dataset of several thousand samples. The intelligent reduced-order models compute considerably faster and achieve excellent approximations of the displacements, stresses, and forces.

We investigate the effects of metro systems on urban growth in China. Using a difference-in-differences combined with propensity score matching framework, we find that metro systems have an economically significant positive effect on the urban GDP growth rate and that this effect is much larger for megacities with permanent populations of more than 6.15 million. Meanwhile, cross-city metro operations tend to increase economic concentrations in larger cities and expand GDP inequalities between cities connected by a cross-city metro. Our mechanism analyses further corroborate that a higher metro density has a positive effect on urban employment growth and a negative effect on the average wage of employees, but this does not improve urban total factor productivity (TFP), suggesting that a metro system is more likely to provide consumption amenities and employment access than it is to provide production amenities.

Neural networks are universal function approximators that form the backbone of most modern machine learning based models. Starting from a conventional return-mapping scheme, the algorithmic description of von Mises plasticity with isotropic hardening is mathematically reformulated such that the relationship between the strain and stress histories may be modeled through a neural network function. In essence, the neural network provides an estimate of the instantaneous elasto-plastic tangent matrix as a function of the current stress and plastic work density. For plane stress conditions, it thus describes a non-linear mapping from ℝ4 to ℝ6. The neural network function is first developed for uniaxial stress conditions including loading histories with tension-compression reversal. Special attention is paid to the identification of the network architecture and artifacts related to overfitting. Furthermore, the performance of networks featuring the same number of total parameters, but different levels of non-linearity, is compared. It is found that a fully-connected feedforward network with five hidden layers and 15 neurons per layer can describe the plane stress plasticity problem with good accuracy. The results also show that a high density of training data (of the order of ten to hundred thousand points) is needed to obtain reasonable estimates for arbitrary loading paths with strains of up to 0.2. The final neural network model is implemented into finite element software through the user material subroutine interface. A simulation of a notched tension experiment is performed to demonstrate that the neural network model yields the same (heterogeneous) mechanical fields as a conventional J2 plasticity model. The present work demonstrates that it is feasible to describe the stress-strain response of a von Mises material through a neural network model without any explicit representation of the yield function, flow rule, hardening law or evolution constraints. It is emphasized that the demonstration of feasibility is the focus of the present work, while the assessment of potential computational advantages is deferred to future research.

FE 2 multiscale simulations of history-dependent materials are accelerated by means of a recurrent neural network (RNN) surrogate for the history-dependent micro level response. We propose a simple strategy to efficiently collect stress-strain data from the micro model, and we modify the RNN model such that it resembles a nonlinear finite element analysis procedure during training. We then implement the trained RNN model in the FE 2 scheme and employ automatic differentiation to compute the consistent tangent. The exceptional performance of the proposed model is demonstrated through a number of academic examples using strain-softening Perzyna viscoplasticity as the nonlinear material model at the micro level.

This monograph provides a concise overview of the main theoretical and numerical tools to solve homogenization problems in solids with finite elements. Starting from simple cases (linear thermal case) the problems are progressively complexified to finish with nonlinear problems. The book is not an overview of current research in that field, but a course book, and summarizes established knowledge in this area such that students or researchers who would like to start working on this subject will acquire the basics without any preliminary knowledge about homogenization. More specifically, the book is written with the objective of practical implementation of the methodologies in simple programs such as Matlab. The presentation is kept at a level where no deep mathematics are required.