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.

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.

A numerical framework is presented for simulating fracture and fatigue in a T-section, cut from an overmolded thermoplastic composite panel made of CF/PEEK. The framework combines a cohesive zone model for the overmolded interface with an anisotropic viscoplasticity model for the laminate and accounts for processing effects. For high-cycle fatigue analyses, a two-scale time-homogenized version of the viscoplasticity model is derived. The numerical framework is applied to the analysis of a rib pull-off test and is used to gain insights into the influence on the short- and long-term response of two typical processing effects: out-of-plane deformations of the laminate that occur during thermoforming and non-uniform healing profiles resulting from spatially varying thermal histories. Furthermore, the effects of various modeling assumptions are studied, such as modeling the local fiber orientations of each ply in the laminate with a mesoscopic ply-by-ply approach, the effect of viscoplastic deformations in the laminate, the influence of non-uniform local stress ratios, and the effect of the boundary conditions. The analyses demonstrate that the framework is capable of efficiently simulating a large number of cycles. The simulation results show that the local wrinkles in the laminate as a result of thermoforming have a significant effect on the mechanical response, especially under cyclic loading. Moreover, accounting for viscoplastic deformations appears more important when high degrees of bonding of the overmolded interface are achieved. Finally, it is shown that changes to the boundary conditions have a significant effect on the short and long-term response of the T-section, challenging the validity of the test for characterizing fracture and fatigue properties of the overmolded interface.

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.

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.

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, 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.

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.

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.

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.

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.