Micromechanics-based surrogate models for the response of composites: A critical comparison between a classical mesoscale constitutive model, hyper-reduction and neural networks

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.Applied Mechanic

On-the-fly construction of surrogate constitutive models for concurrent multiscale mechanical analysis through probabilistic machine learning

Concurrent multiscale finite element analysis (FE2) is a powerful approach for high-fidelity modeling of materials for which a suitable macroscopic constitutive model is not available. However, the extreme computational effort associated with computing a nested micromodel at every macroscopic integration point makes FE2 prohibitive for most practical applications. Constructing surrogate models able to efficiently compute the microscopic constitutive response is therefore a promising approach in enabling concurrent multiscale modeling. This work presents a reduction framework for adaptively constructing surrogate models for FE2 based on statistical learning. The nested micromodels are replaced by a machine learning surrogate model based on Gaussian Processes (GP). The need for offline data collection is bypassed by training the GP models online based on data coming from a small set of fully-solved anchor micromodels that undergo the same strain history as their associated macroscopic integration points. The Bayesian formalism inherent to GP models provides a natural tool for online uncertainty estimation through which new observations or inclusion of new anchor micromodels are triggered. The surrogate constitutive manifold is constructed with as few micromechanical evaluations as possible by enhancing the GP models with gradient information and the solution scheme is made robust through a greedy data selection approach embedded within the conventional finite element solution loop for nonlinear analysis. The sensitivity to model parameters is studied with a tapered bar example with plasticity and the framework is further demonstrated with the elastoplastic analysis of a plate with multiple cutouts and with a crack growth example for mixed-mode bending. Although not able to handle non-monotonic strain paths in its current form, the framework is found to be a promising approach in reducing the computational cost of FE2, with significant efficiency gains being obtained without resorting to offline training.Applied Mechanic

Physically recurrent neural networks for path-dependent heterogeneous materials: Embedding constitutive models in a data-driven surrogate

Driven by the need to accelerate numerical simulations, the use of machine learning techniques is rapidly growing in the field of computational solid mechanics. Their application is especially advantageous in concurrent multiscale finite element analysis (FE2) due to the exceedingly high computational costs often associated with it and the high number of similar micromechanical analyses involved. To tackle the issue, using surrogate models to approximate the microscopic behavior and accelerate the simulations is a promising and increasingly popular strategy. However, several challenges related to their data-driven nature compromise the reliability of surrogate models in material modeling. The alternative explored in this work is to reintroduce some of the physics-based knowledge of classical constitutive modeling into a neural network by employing the actual material models used in the full-order micromodel to introduce non-linearity. Thus, path-dependency arises naturally since every material model in the layer keeps track of its own internal variables. For the numerical examples, a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material is used as the microscopic model. The network is tested in a series of challenging scenarios and its performance is compared to that of a state-of-the-art Recurrent Neural Network (RNN). A remarkable outcome of the novel framework is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with popular but data-hungry models such as RNNs. Finally, the proposed network is applied to FE2 examples to assess its robustness for application in nonlinear finite element analysis.Applied Mechanic