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.

Welcome to the submodule on the Matrix Method for Statics, part of Unit 2 of CIEM5000 Course base Structural Engineering at Delft University of Technology.

This book contains the material for the course.

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.

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.

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.

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.

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.

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.

In a concurrent multiscale (FE2) modeling approach the complex microstructure of composite materials is explicitly modeled on a finer scale and nested to each integration point of the macroscale. However, such generality is often associated with exceedingly high computational costs in real-scale applications. In this work, a novel Neural Network (NN) is used as the constitutive model for the microscale to tackle that issue. Unlike conventional NNs, the proposed network employs the actual material models used in the full-order micromodel as the activation function of one of the layers. The NN's capabilities are assessed (i) for a single micromodel level, where its performance is compared to that of a Recurrent Neural Network (RNN), and (ii) for an FE2 example. A highlight of the proposed network is the ability to predict unloading/reloading behavior without ever seeing it during training, a stark contrast with highly popular but data-hungry models such as RNNs.

This paper presents BIOS (acronym for Biologically Inspired Optimization System), an object-oriented framework written in C++, aimed at heuristic optimization with a focus on Surrogate-Based Optimization (SBO) and structural problems. The use of SBO to deal with structural optimization has grown considerably in recent years due to the outstanding gain in efficiency, often with little loss in accuracy. This is especially promising when adaptive sampling techniques are used. However, many issues are yet to be addressed before SBO can be employed reliably in most optimization problems. In that sense, continuous experimentation, testing and comparison are needed, which can be more easily carried out in an existing framework. The architecture is designed to implement conventional nature inspired algorithms and Sequential Approximated Optimization (SAO). The system aims to be efficient, easy to use and extensible. The efficiency and accuracy of the system are assessed on a set of benchmarks, and on the optimization of functionally graded structures. Excellent results are obtained.

Polymers and polymer composites are negatively impacted by environmental ageing, reducing their service lifetimes. The uncertainty of the material interaction with the environment compromises their superior strength and stiffness. Validation of new composite materials and structures often involves lengthy and expensive testing programs. Therefore, modelling is an affordable alternative that can partly replace extensive testing and thus reduce validation costs. Durability prediction models are often subject to conflicting requirements of versatility and minimum experimental efforts required for their validation. Based on physical observations of composite macroproperties, engineering and phenomenological models provide manageable representations of complex mechanistic models. This review offers a systematised overview of the state-of-the-art models and accelerated testing methodologies for predicting the long-term mechanical performance of polymers and polymer composites. Accelerated testing methods for predicting static, creep, and fatig ue lifetime of various polymers and polymer composites under environmental factors’ single or coupled influence are overviewed. Service lifetimes are predicted by means of degradation rate models, superposition principles, and parametrisation techniques. This review is a continuation of the authors’ work on modelling environmental ageing of polymer composites: the first part of the review covered multiscale and modular modelling methods of environmental degradation. The present work is focused on modelling engineering mechanical properties.

In a concurrent (FE2) multiscale modeling is an increasingly popular approach for modeling complex materials. As such, it is especially suited for modeling composites, as their complex microstructure can be explicitly modeled and nested to each integration point of the macroscale. However, this generality is often associated with exceedingly high computational costs for real-scale applications. One way to tackle the issue is to employ a cheaper-to-evaluate surrogate model for the microstructure based on few observations of the high-fidelity solution. On this note, Neural Networks (NN) are by far the most popular technique in building constitutive surrogates. However, conventional NNs assume a unique mapping between strains and stresses, limiting their ability to reproduce path-dependent behavior. Moreover, their data-driven nature severely limits their ability to extrapolate away from their training spaces. To circumvent these drawbacks, the alternative explored in this work is to reintroduce some of the physics-based knowledge of the problem into the NN. This is done by employing actual material models used in the full-order micromodel as the activation function of one of the layers of the network. Thus, path-dependency arises naturally since every material model in the layer has its own internal variables. To assess its capabilities, the network is employed as the surrogate model for a composite Representative Volume Element with elastic fibers and elasto-plastic matrix material. First, for a single micromodel, the performance of the network is compared to that of a state-of-the-art Recurrent Neural Network (RNN) in a number of challenging scenarios for data-driven models. Then, the proposed framework is applied to an FE2 example and the results are compared to the full-order solution in terms of accuracy and computational cost. An important outcome of the physics-infused network is the ability to naturally predict unloading/reloading behavior without ever seeing it during training, a stark contrast with highly popular but data-hungry models such as RNN.

Service lifetimes of polymers and polymer composites are impacted by environmental ageing. The validation of new composites and their environmental durability involves costly testing programs, thus calling for more affordable and safe alternatives, and modelling is seen as such an alternative. The state-of-the-art models are systematized in this work. The review offers a comprehensive overview of the modular and multiscale modelling approaches. These approaches provide means to predict the environmental ageing and degradation of polymers and polymer composites. Furthermore, the systematization of methods and models presented herein leads to a deeper and reliable understanding of the physical and chemical principles of environmental ageing. As a result, it provides better confidence in the modelling methods for predicting the environmental durability of polymeric materials and fibre-reinforced composites.

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.

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.

This work presents a reduced-order modeling framework that precludes the need for offline training and adaptively adjusts its lower-order solution space as the analysis progresses. The analysis starts with a fully-solved step and elements are clustered based on their strain response. Elements with the highest strains are solved with a local/global approach in which degrees of freedom from elements undergoing the highest amount of nonlinearity are fully-solved and the rest is approximated by a Proper Orthogonal Decomposition (POD) reduced model with full integration. Elements belonging to the remaining clusters are subjected to a hyper-reduction step using the Empirical Cubature Method (ECM). Online error estimators are used to trigger a retraining process once the reduced solution space becomes inadequate. The performance of the framework is assessed through a series of numerical examples featuring a material model with pressure-dependent plasticity.

In this paper, a number of techniques used to accelerate the solution of finite element problems involving a large number of load cycles areexplored and applied to the micromechanical analysis of fiber-reinforced composites. The microscopic domain consists of unidirectional linear-elastic fibers embedded in a viscoelastic/viscoplastic polymeric matrix. Time homogenization is applied to divide the original equilibrium problem in macro- and microchronological parts. The size of the problem is further reduced by a combination of Proper Orthogonal Decomposition (POD) and the Empirical Cubature Method (ECM), resulting in a hyper-reduced model. A novel technique for history recovery combining Gappy Data reconstruction with a k-means clustering algorithm is proposed, as well as an adaptive strategy combining time homogenization and POD without an offline training phase. The performance of each acceleration technique is assessed and the resultant speed-ups obtained by combining them are presented.

This work investigates hygrothermal aging degradation of unidirectional glass/epoxy composite specimens through a combination of experiments and numerical modeling. Aging is performed through immersion in demineralized water. Interlaminar shear testes are performed after multiple conditioning times and after single immersion/redrying cycles. Degradation of the fiber-matrix interface is estimated using single-fiber fragmentation tests and reverse modeling combining analytical and numerical models. A fractographic analysis of specimens aged at 50°C and 65°C is performed through X-ray computed tomography. The aging process is modeled using a numerical framework combining a diffusion analysis with a concurrent multiscale model with embedded hyper-reduced micromodels. At the microscale, a pressure-dependent viscoelastic/viscoplastic model with damage is used for the resin and fiber-matrix debonding is modeled with a cohesive-zone model including friction. A comparison between numerical and experimental results is performed.