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.

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.

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 inferring the discretization error via a model misspecification component is possible as well, as is done in statFEM, 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.

The phase-field hydraulic fracture model entails a non-convex energy functional. This renders a poor convergence behaviour for monolithic solution techniques, such as the Newton–Raphson method. Consequently, researchers have adopted alternative solution techniques such as the staggered solution technique and the Newton–Raphson method with convexification via extrapolation of the phase-field. Both methods are robust. However, the former is computationally expensive and in the latter, the extrapolation itself is questionable w.r.t regularity in time. In this work, a novel dissipation-based arc-length method is proposed as a robust and computationally efficient monolithic solution technique for the phase-field hydraulic fracture model. Similar to brittle fracture in force driven mechanical problems, constant flux driven hydraulic fracture processes are also unstable. Furthermore, due to the constant flux loading in hydraulic fracturing problems, scaling of the external force is not possible. Instead, the time step-size is considered as the additional unknown, augmenting the arc-length constraint equation. The robustness and computational efficiency of the proposed arc-length method is demonstrated using numerical experiments, where comparisons are made with the staggered solver as well as the quasi-Newton BFGS method.

There is a high interest in accelerating multiscale models using data-driven surrogate modeling techniques. Creating a large training dataset encompassing all relevant load scenarios is essential for a good surrogate, yet the computational cost of producing this data quickly becomes a limiting factor. Commonly, a pre-trained surrogate is used throughout the computational domain. We introduce an alternative adaptive mixture approach that uses a fast probabilistic surrogate model as a constitutive model when possible, but resorts to the true high-fidelity model when necessary. The surrogate is thus not required to be accurate for every possible load condition, enabling a significant reduction in the data collection time. We achieve this by creating phases in the computational domain corresponding to the different models. These phases evolve using a phase-field model driven by the surrogate uncertainty. When the surrogate uncertainty becomes large, the phase-field model causes a local transition from the surrogate to the high-fidelity model, maintaining a highly accurate simulation. We discuss requirements for accuracy and numerical stability and compare the phase-field model to a local approach that does not enforce spatial smoothness in phase mixing. Using a Gaussian Process surrogate for an elasto-plastic material, we demonstrate the potential of this mixture of models to accelerate multiscale simulations.

A three-dimensional mesoscopic viscoplasticity model for simulating rate-dependent plasticity and creep in unidirectional thermoplastic composites is presented. The constitutive model is a transversely isotropic extension of an isotropic finite strain viscoplasticity model for neat polymers. Rate-dependent plasticity and creep are described by a non-Newtonian flow rule where the viscosity of the material depends on an equivalent stress measure through an Eyring-type relation. In the present formulation, transverse isotropy is incorporated by defining the equivalent stress measure and flow rule as functions of transversely isotropic stress invariants. In addition, the Eyring-type viscosity function is extended with anisotropic pressure dependence. As a result of the formulation, plastic flow in fiber direction is effectively excluded and pressure dependence of the polymer matrix is accounted for. The re-orientation of the transversely isotropic plane during plastic deformations is incorporated in the constitutive equations, allowing for an accurate large deformation response. The formulation is fully implicit and a consistent linearization of the algorithmic constitutive equations is performed to derive the consistent tangent modulus. The performance of the mesoscopic constitutive model is assessed through a comparison with a micromechanical model for carbon/PEEK, with the original isotropic viscoplastic version for the polymer matrix and with hyperelastic fibers. The micromodel is first used to determine the material parameters of the mesoscale model with a few stress–strain curves. It is demonstrated that the mesoscale model gives a similar response to the micromodel under various loading conditions. Finally, the mesoscale model is validated against off-axis experiments on unidirectional thermoplastic composite plies.

A numerical framework for simulating progressive failure under high-cycle fatigue loading is validated against experiments of composite quasi-isotropic open-hole laminates. Transverse matrix cracking and delamination are modeled with a mixed-mode fatigue cohesive zone model, covering crack initiation and propagation. Furthermore, XFEM is used for simulating transverse matrix cracks and splits at arbitrary locations. An adaptive cycle jump approach is employed for efficiently simulating high-cycle fatigue while accounting for local stress ratio variations in the presence of thermal residual stresses. The cycle jump scheme is integrated in the XFEM framework, where the local stress ratio is used to determine the insertion of cracks and to propagate fatigue damage. The fatigue cohesive zone model is based on S-N curves and requires static material properties and only a few fatigue parameters, calibrated on simple fracture testing specimens. The simulations demonstrate a good correspondence with experiments in terms of fatigue life and damage evolution.

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.

Numerical methods for delamination analysis, such as the cohesive zone method, require fracture energy as an essential input. Existing formulations rely on a phenomenological relationship that links fracture energy to the mode of fracture based on linear elastic fracture mechanics (LEFM). However, doubts exist about the applicability of LEFM. It has been demonstrated that the phenomenological relationships describing fracture energy as a function of mode-ratio are not universally valid. Computational homogenization (FE²) provides an alternative where the dissipative mechanisms can be resolved on the microscale. This paper aims to assess the suitability of a proposed discontinuous FE² framework for characterizing delamination growth under mode-II conditions by comparing it to direct numerical simulations (DNS). The impact of plasticity on effective fracture energy is evaluated for two distinct mode-II test configurations. The dissipation density from the bulk integration points within the delamination propagation zone is monitored. The findings demonstrate the FE² model's capability to accurately capture plastic energy dissipation around a growing crack. Variations in plastic dissipation are observed between the mTCT and ENF test setups, leading to differences in effective mode-II fracture energy. These nuances, unaccounted for in state-of-the-art mesoscale cohesive models, highlight the FE² framework's potential for enhancing delamination modeling.

A numerical framework for simulating progressive failure under high-cycle fatigue loading is validated against experiments of composite quasi-isotropic open-hole laminates. Transverse matrix cracking and delamination are modeled with a mixed-mode fatigue cohesive zone model, covering crack initiation and propagation. Furthermore, XFEM is used for simulating transverse matrix cracks and splits at arbitrary locations. An adaptive cycle jump approach is employed for efficiently simulating high-cycle fatigue while accounting for local stress ratio variations in the presence of thermal residual stresses. The cycle jump scheme is integrated in the XFEM framework, where the local stress ratio is used to determine the insertion of cracks and to propagate fatigue damage. The fatigue cohesive zone model is based on S-N curves and requires static material properties and only a few fatigue parameters, calibrated on simple fracture testing specimens. The simulations demonstrate a good correspondence with experiments in terms of fatigue life and damage evolution.

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.

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, 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 microscale numerical framework for modeling creep rupture in unidirectional composites under off-axis loading is presented, building on recent work on imposing off-axis loading on a representative volume element. Creep deformation of the thermoplastic polymer matrix is accounted for by means of the Eindhoven Glassy Polymer material model. Creep rupture is represented with cohesive cracks, combining an energy-based initiation criterion with a time-dependent cohesive law and a global failure criterion based on the minimum in homogenized creep strain-rate. The model is compared against experiments on carbon/PEEK composite material tested at different off-axis angles, stress levels and temperatures. Creep deformation is accurately reproduced by the model, except for small off-axis angles, where the observed difference is ascribed to macroscopic variations in the experiment. Trends in rupture time are also reproduced although quantitative rupture time predictions are not for all test cases accurate.

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 this work, a recently proposed high-cycle fatigue cohesive zone model, which covers crack initiation and propagation with limited input parameters, is embedded in a robust and efficient numerical framework for simulating progressive failure in composite laminates under fatigue loading. The fatigue cohesive zone model is enhanced with an implicit time integration scheme of the fatigue damage variable which allows for larger cycle increments and more efficient analyses. The method is combined with an adaptive strategy for determining the cycle increment based on global convergence rates. Moreover, a consistent material tangent stiffness matrix has been derived by fully linearizing the underlying mixed-mode quasi-static model and the fatigue damage update. The enhanced fatigue cohesive zone model is used to describe matrix cracking and delamination in laminates. In order to allow for matrix cracks to initiate at arbitrary locations and to avoid complex and costly mesh generation, the phantom node version of the eXtended finite element method (XFEM) is employed. For the insertion of new crack segments, an XFEM fatigue crack insertion criterion is presented, which is consistent with the fatigue cohesive zone formulation. It is shown with numerical examples that the improved fatigue damage update enhances the accuracy, efficiency and robustness of the numerical simulations significantly. The numerical framework is applied to the simulation of progressive fatigue failure in an open-hole [±45]-laminate. It is demonstrated that the numerical model is capable of accurately and efficiently simulating the complete failure process from distributed damage to localized failure.

In this paper, a micromechanical framework for modeling the rate-dependent response of unidirectional composites subjected to off-axis loading is introduced. The model is intended for a thin slice representative volume element that is oriented perpendicular to the reinforcement of the composite material. The testing conditions from a uniaxial off-axis test are achieved by a dedicated strain-rate based arclength formulation. The constraint equation of the arclength model is constructed such that the deformation state of the micromodel, as imposed in its local coordinate system, corresponds to the strain-rate applied on the material in global frame of reference. The kinematic description allows for finite strains in the material, meaning that the micromodel changes orientation during the deformation process. This geometric nonlinear effect is also included in the evaluation of external loading, ensuring that the external forces are equivalent to the applied off-axis stress in global coordinate system. Several examples are considered in order to show that the model resolves rate-dependency of the material, accounts for different off-axis loading, and captures finite strains exactly. Additionally, a small strain version of the model is derived from the general nonlinear framework. Results obtained with this simplified approach are compared to results of the large deformation framework.

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.

The Thick Level Set method (TLS) is an approach for non-local damage modeling in which the damage evolution is linked to the movement of a damage front described with the level set method. More recently, a new version of the TLS, designated as the TLSV2, has been proposed as a new concept for coupling continuum damage modeling and discrete cohesive crack modeling for failure analysis in solids. The main objective of this new framework is to profit from both modeling approaches. The continuum part allows for handling crack initiation, branching and merging, whereas the cohesive part brings the capability to handle discrete cracks with large crack opening or sliding without heavily distorted elements, and with the possibility to model stiffness recovery upon contact. In this paper, a generalized framework for the TLSV2 is introduced. Two major issues with the TLSV2 method that have not been dealt with since its inception are addressed in this study, and solutions are proposed. Firstly, the method depends on the location of skeleton curve of the level set field, on which the discontinuity in the displacement field is evaluated. The problem of locating the skeleton curve can be a complicated task, mainly because topological events may emerge as the analysis progresses, such as crack branching. The skeleton curve is determined through a combination of ball-shrinking and graph-based algorithms and then mapped onto the finite element mesh. Secondly, the cohesive forces and displacement discontinuity of the TLSV2 are modeled using the phantom node method. Furthermore, a new approach to compute the averaged values of local quantities is introduced, and model calibration is discussed. The degree of stiffness recovery under compression that is still needed for the continuum part is investigated. Numerical experiments demonstrate the accuracy and ability of the proposed model to handle simulation of failure analysis presenting complex topological crack patterns.