We demonstrate a Model Order Reduction technique for a system of nonlinear equations arising from the Finite Element Method (FEM) discretization of the three-dimensional quasistatic equilibrium equation equipped with a Perzyna viscoplasticity constitutive model. The procedure employs the Proper Orthogonal Decomposition-Galerkin (POD-G) in conjunction with the Discrete Empirical Interpolation Method (DEIM). For this purpose, we collect samples from a standard full order FEM analysis in the offline phase and cluster them using a novel kk-means clustering algorithm. The POD and the DEIM algorithms are then employed to construct a corresponding reduced order model. In the online phase, a sample from the current state of the system is passed, at each time step, to a nearest neighbor classifier in which the cluster that best describes it is identified. The force vector and its derivative with respect to the displacement vector are approximated using DEIM, and the system of nonlinear equations is projected onto a lower dimensional subspace using the POD-G. The constructed reduced order model is applied to two typical solid mechanics problems showing strain-localization (a tensile bar and a wall under compression) and a three-dimensional square-footing problem.@en

FE2 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 FE2 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.@en

We study the acceleration of the finite element method (FEM) simulations using machine learning (ML) models. Specifically, we replace computationally expensive (parts of) FEM models with efficient ML surrogates. We develop three methods to speed up FEM simulations. The primary difference between these models is their degree of intrusion into the FEM source code. Here, we enumerate them from the most to the least intrusive. In the first contribution, we tackle two bottlenecks of a FEM model equipped with a viscoplastic constitutive equation namely, solving the linear system of equations and evaluating the force vector. To tackle the former, we use a proper orthogonal decomposition (POD) method. And we tackle the latter with a discrete empirical interpolation method (DEIM).We observe that DEIM does not effectively speed up such a highly nonlinear FEM model. As a remedy, we divide the time domain into subdomains using a clustering algorithm. Then we construct a set of DEIM points for each cluster. By doing so, we manage to increase the efficiency of the POD-DEIM scheme. We, however, observe that the POD-DEIM scheme is sometimes unstable. The source of this instability is, to the extent of our knowledge, an open question. In the second contribution, we consider a FE2 scheme. The micro model is a FEM model equipped with a viscoplastic constitutive equation. The evaluation of the micro model is the computational bottleneck in this framework. Therefore, we develop a recurrent neural network as a surrogate for the micro model. In this contribution, we also propose a simple but effective sampling technique to collect stress-strain data points. The RNN model is trained based on this data. We also discuss how the RNN model becomes inaccurate when extrapolating. For these scenarios, we discuss how to improve the RNN by collecting more data and retraining. In the third contribution, we develop a surrogate for the entire FEM simulation of a multi-physics problem. Specifically, we consider the FEM simulation of the electrochemical-mechanical interactions in a Li-ion battery. We propose a variant of the convolutional neural network (CNN), namely the HydraNet. The HydraNet takes the geometry of the battery and predicts all solution fields of the FEM model. Solution fields are either output of the solver or that of the post-processing. The HydraNet accepts inputs in the form of image-like fields. We discuss how to encode the geometry of the battery into a set of image-like fields. We argue that the degree of intrusion of these methods to the FEM source code is inversely related to their industrial applicability. As a result, we believe that the first method (POD-DEIM) will mostly remain an academic contribution, while the other two could have potential industrial applications. The central use case of these methods is in a multi-query application such as uncertainty quantification, design, and real-time simulations.@en