AbstractMultiscale homogenization of woven composites requires detailed micromechanical evaluations, leading to high computational costs. Data-driven surrogate models based on neural networks address this challenge but often suffer from big data requirements, limited interpretability, and poor extrapolation capabilities. This study introduces a Hierarchical Physically Recurrent Neural Network (HPRNN) employing two levels of surrogate modeling. First, Physically Recurrent Neural Networks (PRNNs) are trained to capture the nonlinear elasto-plastic behavior of warp and weft yarns using micromechanical data. In a second scale transition, a physics-encoded meso-to-macroscale model integrates these yarn surrogates with the matrix constitutive model, embedding physical properties directly into the latent space. By adopting HPRNNs, nonphysical behavior often observed in predictions from pure data-driven recurrent neural networks and transformer networks can be avoided. This results in better generalization under complex cyclic loading conditions. The framework offers a computationally efficient and explainable solution for multiscale modeling of woven composites.

Recent advancements in Markov chain Monte Carlo (MCMC) sampling and surrogate modelling have significantly enhanced the feasibility of Bayesian analysis across engineering fields. However, the selection and integration of surrogate models and cutting-edge MCMC algorithms, often depend on ad-hoc decisions. A systematic assessment of their combined influence on accuracy and efficiency is notably lacking. The present work offers a comprehensive comparative study, employing a scalable case study in computational mechanics focused on the inference of spatially varying material parameters, that sheds light on the impact of methodological choices for surrogate modelling and sampling. We show that a priori training of the surrogate model introduces large errors in the posterior estimation even in low to moderate dimensions. We introduce a simple active learning strategy based on the path of the MCMC algorithm that is superior to all a priori trained models, and determine its training data requirements. We demonstrate that the choice of the MCMC algorithm has only a small influence on the amount of training data but no significant influence on the accuracy of the resulting surrogate model. Further, we show that the accuracy of the posterior estimation largely depends on the surrogate model, but not even a tailored surrogate guarantees convergence of the MCMC. Finally, we identify the forward model as the bottleneck in the inference process, not the MCMC algorithm. While related works focus on employing advanced MCMC algorithms, we demonstrate that the training data requirements render the surrogate modelling approach infeasible before the benefits of these gradient-based MCMC algorithms on cheap models can be reaped.

We develop graph-based surrogate models to predict strain localization and collapse load in 2D elastic-perfectly plastic porous solids. Plastic deformation is represented using a Delaunay graph, where nodes correspond to void centers, edges represent potential shear bands, and edge values encode the local integral of the plastic work rate (PWR). Two edge-regression graph neural network (GNN) models are built. The first is a purely data-driven model (DDM) that maps local geometric features – edge length and orientation – to PWR. The second is a hybrid model (HM) that augments the GNN with a mechanistic prior from a limit load analysis model (LLAM) and learns a correction to its upper-bound bias. The models are trained on a dataset of 1200 representative volume elements with 20% porosity. The DDM can reliably reconstruct shear band patterns, but its accuracy deteriorates when applied to porosity values different from that seen in training. The HM, like the LLAM, does not fully capture the spatial pattern, but it removes the LLAM systematic bias and achieves uniformly low errors in total work rate for porosities ranging from 10% to 30%. Moreover, the HM requires less training data than the DDM and is more robust across random seeds. These results show that coupling a GNN with a physics-based prior yields a fast and data-efficient surrogate that preserves accuracy in macroscopic quantities while retaining meaningful spatial information, thereby offering a practical route to predict the collapse of porous solids while accounting for the exact locations of a very large number of voids.

When using the finite element method (FEM) in inverse problems, its discretization error can produce parameter estimates that are inaccurate and overconfident. The Bayesian finite element method (BFEM) provides a probabilistic model for the epistemic uncertainty due to discretization error. In this work, we apply BFEM to various inverse problems and compare its performance to the random mesh finite element method (RM-FEM) and the statistical finite element method (statFEM), which serve as a frequentist and inference-based counterpart to BFEM. We find that by propagating this uncertainty to the posterior, BFEM can produce more accurate parameter estimates and prevent overconfidence compared to FEM. Because the BFEM covariance operator is designed to leave uncertainty only in the appropriate space, orthogonal to the FEM basis, BFEM is able to outperform RM-FEM, which does not have such a structure to its covariance. Although it is also possible to use a model misspecification formulation such as statFEM to infer the discretization error downstream rather than model it at the source, the feasibility of such an approach is contingent on the availability of sufficient data. We find that the BFEM is the most robust way to consistently propagate uncertainty due to discretization error to the posterior of a Bayesian inverse problem.

In this work, we extend a recent surrogate modeling approach, the Physically Recurrent Neural Network (PRNN), to include the effect of debonding at the fiber–matrix interface of composite materials. The core idea of the PRNN is to implement the exact material models from the micromodel into one of the layers of the network to capture path-dependent behavior implicitly. For the case of debonding, additional material points with a cohesive zone model are integrated within the network, along with the bulk points associated to the fibers and/or matrix. The limitations of the existing architecture are discussed and taken into account for the development of novel architectures that better represent the stress homogenization procedure. In the proposed layout, the history variables of cohesive points act as extra latent features that help determine the local strains of bulk points. Different architectures are evaluated starting with small training datasets. To maximize the predictive accuracy and extrapolation capabilities of the network, various configurations of bulk and cohesive points are explored, along with different training dataset types and sizes.

Bayesian system identification is increasingly used in Structural Health Monitoring (SHM) to infer unobservable parameters of a structure from sensor data. The use of spatially dense measurements, such as those from distributed fibre optic sensors, can further enhance the results of Bayesian system identification due to the large volume of data. However, this combination faces two major challenges: the computational cost of inference and the correlation structure of closely spaced data points. To overcome these difficulties, we propose a methodology that combines the recently-developed Variational Bayes Monte Carlo (VBMC) method with Gaussian process modelling of model discrepancy, and extend VBMC to enable posterior predictive calculations without additional model evaluations. We demonstrate the effectiveness of the proposed methodology on a reinforced concrete slab bridge instrumented with distributed fibre optic strain sensors and analysed using a finite element model. The main outcome is that VBMC requires fewer than 200 finite element model evaluations while producing accurate estimates, whereas a conventional MCMC method requires thousands. The application of the proposed framework provides two additional novel insights: accounting for spatial correlations improves model performance and higher measurement resolution leads to more precise parameter estimates, though with limited impact on predictive accuracy. This study advances the practical implementation of Bayesian system identification in SHM by providing both the computational efficiency and statistical framework needed for modern sensing technologies.

A micromechanical model for simulating failure of unidirectional composites under cyclic loading has been developed and tested. To efficiently pass through the loading signal, a two-scale temporal framework with adaptive stepping is proposed, with a varying step size between macro time steps, and a fixed number of equally spaced micro time steps in between. With the focus on matrix dominated failure under off-axis loading, viscoplasticity and microcracking are included in the model for the polymer matrix, while carbon fibers are modeled as elastic. For a proper representation of viscous deformation in the matrix under cyclic loading, a two-scale version of the Eindhoven Glassy Polymer constitutive model is formulated, that is based on time homogenization with an effective time increment. The failure state of the representative volume element is reached by the initiation and damaging of cohesive microcracks. Cyclic and static degradation are represented by using Dávila's fatigue damage function, which is built on top of Turon's quasi-static cohesive model. The model results are compared with available experimental data on unidirectional carbon/PEEK composites tested at different stress levels, load ratios, frequencies and off-axis angles. Plasticity controlled and crack growth controlled failure mechanisms, characteristic of the long-term response of polymeric composites, are captured by the model, as well as their distinct frequency dependence. As a limit case, the model is able to reproduce the time to failure in creep loading, where the heterogeneous microstructure and viscoplastic flow of the matrix trigger the evolution of quasi-static damage. However, for the studied material system, the present model does not accurately reproduce the load ratio dependence and the off-axis angle dependence of the crack growth controlled failure mechanism.

In this work, the uncertainty associated with the finite element discretization error is modeled following the Bayesian paradigm. First, a continuous formulation is derived, where a Gaussian process prior over the solution space is updated based on observations from a finite element discretization. To avoid the computation of intractable integrals, a second, finer, discretization is introduced that is assumed sufficiently dense to represent the true solution field. A prior distribution is assumed over the fine discretization, which is then updated based on observations from the coarse discretization. This yields a posterior distribution with a mean that serves as an estimate of the solution, and a covariance that models the uncertainty associated with this estimate. Two particular choices of prior are investigated: a prior defined implicitly by assigning a white noise distribution to the right-hand side term, and a prior whose covariance function is equal to the Green’s function of the partial differential equation. The former yields a posterior distribution with a mean close to the reference solution, but a covariance that contains little information regarding the finite element discretization error. The latter, on the other hand, yields posterior distribution with a mean equal to the coarse finite element solution, and a covariance with a close connection to the discretization error. For both choices of prior a contradiction arises, since the discretization error depends on the right-hand side term, but the posterior covariance does not. We demonstrate how, by rescaling the eigenvalues of the posterior covariance, this independence can be avoided.

In this work, a hybrid physics-based data-driven surrogate model for the microscale analysis of heterogeneous material is investigated. The proposed model benefits from the physics-based knowledge contained in the constitutive models used in the full-order micromodel by embedding the material models in a neural network. Following previous developments, this paper extends the applicability of the physically recurrent neural network (PRNN) by introducing an architecture suitable for rate-dependent materials in a finite strain framework. In this model, the homogenized deformation gradient of the micromodel is encoded into a set of deformation gradients serving as input to the embedded constitutive models. These constitutive models compute stresses, which are combined in a decoder to predict the homogenized stress, such that the internal variables of the history-dependent constitutive models naturally provide physics-based memory for the network. To demonstrate the capabilities of the surrogate model, we consider a unidirectional composite micromodel with transversely isotropic elastic fibers and elasto-viscoplastic matrix material. The extrapolation properties of the surrogate model trained to replace such micromodel are tested on loading scenarios unseen during training, ranging from different strain-rates to cyclic loading and relaxation. Speed-ups of three orders of magnitude with respect to the runtime of the original micromodel are obtained.

During the last few decades, industries such as aerospace and wind energy (among others) have been remarkably influenced by the introduction of high-performance composites. One challenge, however, for modeling and designing composites is the lack of computational efficiency of accurate high-fidelity models. For design purposes, using conventional optimization approaches typically results in cumbersome procedures due to huge dimensions of the design space and high computational expense of full-field simulations. In recent years, deep learning techniques have been found to be promising methods to increase the efficiency and robustness of a variety of algorithms in multi-scale modeling and design of composites. In this perspective paper, a short overview of the recent developments in micromechanics-based machine learning for composites is given. More importantly, existing challenges for further model enhancements and future perspectives of the field development are elaborated.

Simulating the mechanical response of advanced materials can be done more accurately using concurrent multiscale models than with single-scale simulations. However, the computational costs stand in the way of the practical application of this approach. The costs originate from microscale Finite Element (FE) models that must be solved at every macroscopic integration point. A plethora of surrogate modeling strategies attempt to alleviate this cost by learning to predict macroscopic stresses from macroscopic strains, completely replacing the microscale models. In this work, we introduce an alternative surrogate modeling strategy that allows for keeping the multiscale nature of the problem, allowing it to be used interchangeably with an FE solver for any time step. Our surrogate provides all microscopic quantities, which are then homogenized to obtain macroscopic quantities of interest. We achieve this for an elasto-plastic material by predicting full-field microscopic strains using a graph neural network (GNN) while retaining the microscopic constitutive material model to obtain the stresses. This hybrid data-physics graph-based approach avoids the high dimensionality originating from predicting full-field responses while allowing non-locality to arise. In addition, this approach introduces beneficial inductive bias to the model by encoding microscopic geometrical features. By training the GNN on a variety of meshes, it learns to generalize to unseen meshes, allowing a single model to be used for a range of microstructures. The embedded microscopic constitutive model in the GNN implicitly tracks history-dependent variables and leads to improved accuracy. While the microscopic stresses are fully dependent on the microscopic strains, we found it crucial to include both microscopic strains and stresses in the loss function. We demonstrate for several challenging scenarios that the surrogate can predict complex macroscopic stress–strain paths. As the computation time of our method scales favorably with the number of elements in the microstructure compared to the FE method, our method can significantly accelerate FE² simulations.

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.

In this work we present a hybrid physics-based and data-driven learning approach to construct surrogate models for concurrent multiscale simulations of complex material behavior. We start from robust but inflexible physics-based constitutive models and increase their expressivity by allowing a subset of their material parameters to change in time according to an evolution operator learned from data. This leads to a flexible hybrid model combining a data-driven encoder and a physics-based decoder. Apart from introducing physics-motivated bias to the resulting surrogate, the internal variables of the decoder act as a memory mechanism that allows path dependency to arise naturally. We demonstrate the capabilities of the approach by combining an FNN encoder with several plasticity decoders and training the model to reproduce the macroscopic behavior of fiber-reinforced composites. The hybrid models are able to provide reasonable predictions of unloading/reloading behavior while being trained exclusively on monotonic data. Furthermore, in contrast to traditional surrogates mapping strains to stresses, the specific architecture of the hybrid model allows for lossless dimensionality reduction and straightforward enforcement of frame invariance by using strain invariants as the feature space of the encoder.