Due to the inherent uncertainties in manufacturing properties and intrinsic variability of materials, the assumption of homogeneous input variables is generally not justified. As a result, stochastic forward problems have emerged as a tool to incorporate these uncertainties into numerical simulations, improving model prediction capability in structural analyses. Although most of the existing methods focus on the consideration of stochastic loading, the recently developed statFEM employs a Bayesian paradigm to incorporate data and propagate uncertainties from random physical properties in finite element models. This tool, however, is not developed for cases when Gaussian assumptions are inadequate. The present work provides a copula-based approach embedded into the statFEM methodology to propagate uncertainty from arbitrarily distributed physical properties. The random variables are defined in terms of a Gaussian Copula Process, where samples are drawn from a latent variable governed by a Gaussian Process and then brought respectively to copula and marginal spaces, producing random variables with the desired distribution while retaining the usually desired smooth Gaussian dependence in the spatial domain. The quality of the approximated results is then assessed in a simplified 1D Poisson problem by comparing with Monte Carlo sampling results for different random diffusion coefficients, demonstrating that the method is capable of providing good responses for non-Gaussian physical parameters.

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.

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.

A key challenge in structural health monitoring is the inference of material properties from measurements. The challenge is particularly acute for cases that involve spatial distributions of uncertain material parameters. These spatially distributed parameters form a random field with high stochastic dimensionality. Bayesian analysis offers a robust method to infer these parameter distributions. However, its dependence on approximations such as Markov chain Monte Carlo (MCMC) sampling, which require many model evaluations, limits the applicability of Bayesian analysis, especially when the model, e.g. a finite element model, is expensive to evaluate. To address this challenge, we investigate the applicability of the Zig-Zag sampling process, a Markov process-based sampler with linear deterministic dynamics, for inferring material parameter fields from displacements measurements.

In theory, the Zig-Zag process offers excellent mixing and low autocorrelation in high-dimensional parameter spaces. However, its application has been limited to simple distributions due to the need for a global upper bound on the gradient of the posterior, a quantity typically unavailable in non-linear Bayesian inverse problems. To address this, we employ a surrogate model to approximate the posterior gradient, allowing us to globally estimate this upper bound and simulate the process efficiently. The bias introduced by the surrogate model is then alleviated with Poisson thinning of the approximate process.

This study marks the first application of a Markov process sampler to Bayesian inference in computational mechanics, yielding promising results. Our methodology demonstrates that the Zig-Zag sampler outperforms traditional MCMC methods, particularly in terms of full model evaluations needed to reach the same accuracy in the posterior moments. Nonetheless, our findings underscore the challenges introduced by the bias of the surrogate model. We present strategies to reduce the impact of correcting for this bias on the efficiency of the sampler.

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.

Laser powder bed fusion (LPBF), a prominent metal-based additive manufacturing (AM) technique, enables the production of complex, neat-net-shape components with minimal material waste and reduced lead times. However, achieving high final product quality is challenging due to numerous process variables and intricate, nonlinear interactions introduced by LPBF’s thermal-mechanical mechanisms. This study employs a copula-based analysis using multi-physics numerical simulations to probabilistically map relationships among process and part quality variables. Dependence and tail-dependence analyses are performed to provide deeper insights into variable interactions, enabling the identification of preferred operational windows with a balanced trade-off between product quality and productivity. The developed methodology advances the understanding of uncertainty propagation in LPBF, contributing toward improved process optimization, repeatability, and reliability.

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 common challenge in methods for uncertainty quantification (e.g. uncertainty propagation, inverse modeling and data assimilation) is that they typically require many model evaluations in order to propagate the uncertainty in the inputs to the quantity of interest. Especially when the model evaluation requires solving a partial differential equation numerically, computational expenses can skyrocket. To alleviate these computational costs, projection-based model order reduction techniques (proper orthogonal decomposition, balanced truncation, reduced basis methods, etc.) are often applied, which aim to minimize the model complexity. However, these reduced-order models unavoidably introduce an approximation error, presenting the modeler with a difficult trade-off between accuracy and computational cost. Often, it is unclear what is the effect of neglecting higher-order modes on the distributions that are obtained for the quantity of interest.

In this work, we present a Bayesian formulation of projection-based reduced order models. The full-order model is endowed with a carefully chosen Gaussian prior distribution, for which each basis function of the reduced-order model functions as an observation. Performing the Bayesian conditioning yields a posterior distribution whose mean recovers the classic reduced-order model solution and whose posterior covariance can be related directly to the reduced-order model error. By modeling this error probabilistically, it can be taken into account consistently by propagating it to the quantity of interest. We present a theoretical description of the method, along with an empirical study applying our Bayesian reduced-order model to a Bayesian inverse problem.

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.

MUDE stands for Modelling, Uncertainty and Data for Engineers, a required module in the MSc programs from the faculty of Civil Engineering and Geosciences at Delft University of Technology in the Netherlands.

The current version of the MUDE Textbook can be found at mude.citg.tudelft.nl/book and the most recent "complete" version is mude.citg.tudelft.nl/book/2024. Additional information about the book and its contents can be found on the Credits Page from 2024; technical information about the book and its source code can be found in the README of the GitHub repository TUDelft-MUDE/book. General information about MUDE can be found at mude.citg.tudelft.nl.

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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.

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.

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 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.

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 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.

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.

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.