Enhancing Bayesian model updating in structural health monitoring via learnable mappings

This paper presents a methodology to move toward reliable real-time structural health monitoring (SHM). The proposed procedure relies upon surrogate modeling based on a multi-fidelity (MF) deep neural network (DNN), conceived to map damage and operational parameters onto sensor recordings. Within a stochastic framework, the MF-DNN is adopted by a Markov chain Monte Carlo sampling procedure to update the probability distribution of the structural state, conditioned on noisy observations. The MF-DNN enables to locate and possibly quantify the presence of damage, and its multi-fidelity configuration effectively blends datasets featuring different fidelities without any prior assumption. The training datasets are generated with physics-based models of the monitored structure: high fidelity (HF) and low fidelity (LF) models are considered to simulate the structural response under varying operational conditions, respectively in the presence or absence of a structural damage. The MF-DNN is a composition of a fully-connected LF-DNN, which mimics sensor recordings in the undamaged condition, and of a long short-term memory HF-DNN, which is exploited to enrich the LF approximation for the considered damaged scenarios. By framing the model updating strategy as an incremental or residual modeling problem, the MF-DNN is reported to provide numerous advantages over single-fidelity based models for SHM purposes.KeywordsBayesian model updatingDeep learningMarkov chain Monte CarloStructural health monitoringMulti-fidelity methodsReduced-order modelingReal-time damage identification

Structural Health Monitoring (SHM) is rapidly developing as a multi-disciplinary technology solution for condition assessment and performance evaluation of civil infrastructures. It consists of three parts: data collection, data processing (feature extraction/selection), and decision-making (feature classification). In this research, for effectively reducing a dimension of SHM data, various methods are proposed such as advanced feature extraction, feature subset selection using optimization algorithm, and effective surrogate model based on artificial intelligence methods. These frameworks enhance the capability of the SHM process to tackle with uncertainties and big data problem. To reach such goals, a framework based on three main blocks are proposed here: feature extraction block using wavelet pocket relative energy (WPRE), feature selection block using improved version of binary harmony search algorithm and finally feature classification block using wavelet weighted least square support vector machine (WWLS-SVM). The capability of the proposed framework is compared with various well known methods for each block. Results will be presented using metrics of precision, recall, accuracy and feature-reduction. Furthermore, to show the robustness of the proposed methods, six well-known benchmark datasets of SHM domain are studied. The results validate the suitability of the proposed methods in providing data reduction and accelerating damage detection process.

Stochastic approaches to structural health monitoring (SHM) are often inevitably limited by computational constraints. For instance, for Markov chain Monte Carlo algorithms relying upon computationally expensive finite element models it is almost infeasible to sample the probability distribution of the structural state. To provide instead real-time procedures, this work proposes a non-intrusive surrogate modeling strategy, leveraging model order reduction and artificial neural networks. By relying upon a multi-fidelity (MF) framework, a composition of deep neural networks (DNNs) is devised to map damage and operational parameters onto time-dependent sensor recordings. Such an effective strategy is able to exploit datasets characterized by different fidelity levels without any prior assumption, allowing to blend a small high-fidelity (HF) dataset with a large low-fidelity (LF) dataset, ultimately alleviating the computational burden of supervised training while ensuring the accuracy of the approximated quantities of interest. The resulting surrogate model is made of an LF-DNN, which mimics sensor recordings in the undamaged condition, and of a long short-term memory HF-DNN, which adaptively refines the approximation with the effect of damage. An HF finite element model and an LF reduced order model are adopted offline to generate labeled training data of different fidelity, respectively in the presence or absence of a structural damage. Results relevant to an L-shaped cantilever beam and a portal frame railway bridge prove that the procedure efficiently provides remarkably accurate approximations, outperforming their single-fidelity counterparts. The capability of the MF-DNN to be exploited for SHM purposes is finally shown within an automated Bayesian procedure, aimed at updating the probability distribution of the structural state conditioned on sensor recordings, in the presence of operational variability and measurement noise.

The expanding structural health monitoring (SHM) systems on bridge structures have provided an abundance of multi-source data for finite element model updating (FEMU). The SHM systems on bridges usually include surveillance cameras, vibration sensors (e.g., accelerometers, strain gauges, and displacement sensors), and sometimes a weight-in-motion (WIM) system. Currently, the majority of FEMU studies focus on identified modal parameters derived from vibration data, neglecting the incorporation of video and WIM data in the updating process, which impedes a thorough quantification of uncertainty associated with the structural parameters of interest. Therefore, this paper proposes a hierarchical Bayesian FEMU framework to comprehensively integrate a variety of information sources, including videos, WIM, and vibration data. The data features comprise the static deflections of the bridge under traffic load and modal parameters identified from acceleration measurements. The measured static deflections are extracted from raw displacement data using the locally weighted regression and smoothing scatterplots method. Computer vision-based technology is employed to pinpoint the location of vehicle load on the bridge, which is then integrated into a FEM to predict vehicle-load-induced static deflection. A two-stage Markov Chain Monte Carlo sampling approach is proposed to evaluate the high-dimensional posterior distribution efficiently. The effectiveness of the proposed method is demonstrated on a laboratory three-span bridge model. The results show that the hierarchical Bayesian FEMU can provide accurate estimation and uncertainty quantification on structural stiffness and mass parameters. The updated model accurately predicts both static deflection and modal parameters, exhibiting model-predicted variability in close alignment with the identified values for observed and unobserved responses. Remarkably, this holds true even for unseen loading conditions which are not included in the updating process. These observations validate the capability of the proposed method for multi-source data fusion and uncertainty quantification of real-world bridge structures under operational conditions.

Condition assessment and prediction of existing infrastructure systems is a major objective in civil engineering. Structural health monitoring (SHM) achieves this goal by continuous data acquisition from an array of sensors deployed on the structure of interest. SHM constructs ubiquitous damage features from the acquired data, ensuring maximum sensitivity to the onset of damage and robustness to noise and variability in environmental and operational conditions. Traditionally, SHM has used a model-based approach, wherein a high-fidelity model of a structure is constructed and studied in detail to aid engineers in detecting the onset of damage based on deviations from an undamaged model of the system. However, given the complexity of structures and the inability to perfectly model all aspects of a system, a data-driven approach becomes an attractive alternative. In data-driven approaches, a surrogate model, constructed using acquired data from a system, is substituted for a real model. Although such models do not necessarily capture all the physics of a system, they are efficient for damage detection purposes. Statistical learning algorithms aid in the construction of such surrogate models, and their use has now been extensively documented in the literature. This chapter provides a brief review of applications of statistical learning algorithms, both supervised and unsupervised, in SHM for real-time condition assessment of civil infrastructure systems.

To meet the need for reliable real-time monitoring of civil structures, safety control and optimization of maintenance operations, this paper presents a computational method for the stochastic estimation of the degradation of the load bearing structural properties. Exploiting a Bayesian framework, the procedure sequentially updates the posterior probability of the damage parameters used to describe the aforementioned degradation, conditioned on noisy sensors observations, by means of Markov chain Monte Carlo (MCMC) sampling algorithms. To enable the analysis to run in real-time or quasi real-time, the numerical model of the structure is replaced with a data-driven surrogate used to evaluate the (conditional) likelihood function. The proposed surrogate model relies on a multi-fidelity (MF) deep neural network (DNN), mapping the damage and operational parameters onto sensor recordings. The MF-DNN is shown to effectively leverage information between multiple datasets, by learning the correlations across models with different fidelities without any prior assumption, ultimately alleviating the computational burden of the supervised training stage. The low fidelity (LF) responses are approximated by relying on proper orthogonal decomposition for the sake of dimensionality reduction, and a fully connected DNN. The high fidelity signals, that feed the MCMC within the outer-loop optimization, are instead generated by enriching the LF approximations through a deep long short-term memory network. Results relevant to a specific case study demonstrate the capability of the proposed procedure to estimate the distribution of damage parameters, and prove the effectiveness of the MF scheme in outperforming a single-fidelity based method.

Electrical resistivity tomography (ERT) is commonly applied for shallow subsurface imaging. Inversion techniques generate images of the subsurface resistivity structure to interpret the data, with applications including the imaging of permafrost soils. While linearized inversion is a common method, nonlinear treatment provides advantages in terms of parametrization and model selection. However, it often incurs prohibitive computational costs. Markov Chain Monte Carlo (MCMC) methods offer nonlinear uncertainty quantification for ERT, where the computational cost is dominated by the forward model evaluations. Surrogate models advance the physics forward model with a considerable speedup; therefore, they have the potential to enable MCMC applications for inverse problems that were not previously possible. We introduce a surrogate forward model for 2D ERT based on a Fourier Neural Operator (FNO). This model leverages the FNO's capability to learn and generalize mappings between infinite-dimensional function spaces, making it particularly suitable for solving PDE-driven problems like ERT. Based on the inputs of electrode geometry and subsurface resistivity distribution, FNO predicts potentials from which apparent resistivities are computed. This process reduces evaluation times of a subsurface resistivity distribution from seconds to milliseconds with prediction errors below 5%. This efficiency gain enables applying the FNO in MCMC sampling. We show several examples of MCMC sampling results with simulated data for pole-dipole arrays and realistic subsurface models. The subsurface parametrization of resistivity considers irregular grids based on Gaussian random fields or Voronoi cells. The results demonstrate that nonlinear inversion and uncertainty quantification are computationally feasible for typical field survey scales. 

Real-time Bayesian damage identification enabled by sparse PCE-Kriging meta-modelling for continuous SHM of large-scale civil engineering structures

This research work deals with aspects concerned with delamination detection in composite structures as revealed by an approach based on vibration measurements. Variations in vibration characteristics generated in composite laminates indicate the existence of delaminations because degradation due to delamination causes reduction in flexural stiffness and strength of the material and as a result vibration parameters like natural frequency responses are changed. Hence, it is possible to monitor the variation in natural frequencies to identify the presence of delamination, and assess its size and location for online structural health monitoring (SHM). The approach to this paper, therefore, typically depends on undertaking the analysis of structural models implemented by finite element analysis (FEA). The numerical solutions using FE models known as the simulator computes the natural frequencies for the delaminated and undelaminated specimens of composite laminates. However, these FE models are computationally expensive, and surrogate (approximation) models are introduced to curtail the computational expense. The simulator is employed to solve the inverse problem using algorithms based on computational intelligence concepts. An artificial neural network (ANN) model is developed to also solve the inverse problem for delamination detection directly and to provide surrogate models integrated with optimization algorithms (the gradient-based local search and non-dominated sorting genetic algorithm-II) to contain the computationally expensive simulations by FEA. This approach is termed as surrogate assisted optimization and it is seen that the engagement of surrogate models in lieu of the FE models in the optimization loop greatly enhances the accuracy of delamination detection results within an affordable computational cost and provides control over handling different variables. Meanwhile, to aid with the building of effective surrogate models using substantial number of training datasets, K-means clustering algorithm is harnessed and this effectively reduces the large training datasets usually required for ANN training. This paper demonstrated that ANN and optimization algorithms with surrogates show immense potentialities for use in delamination damage detection scenarios. Prediction errors of the algorithms were quantified and they were shown to be satisfactory when applied to previously experimental data. The algorithms in their inverse formulations are capable of predicting accurately delamination parameters. Hence, these algorithms should be employed for application in the domain of SHM where their small computational requirements could be exploited for online damage detection.

Multifidelity methods leverage low-cost surrogate models to speed up computations and make occasional recourse to expensive high-fidelity models to establish accuracy guarantees. Because surrogate and high-fidelity models are used together, poor predictions by surrogate models can be compensated with frequent recourse to high-fidelity models. Thus, there is a trade-off between investing computational resources to improve the accuracy of surrogate models versus simply making more frequent recourse to expensive high-fidelity models; however, this trade-off is ignored by traditional modeling methods that construct surrogate models that are meant to replace high-fidelity models rather than being used together with high-fidelity models. This work considers multifidelity importance sampling and theoretically and computationally trades off increasing the fidelity of surrogate models for constructing more accurate biasing densities and the numbers of samples that are required from the high-fidelity models to compensate poor biasing densities. Numerical examples demonstrate that such context-aware surrogate models for multifidelity importance sampling have lower fidelity than what typically is set as tolerance in traditional model reduction, leading to runtime speedups of up to one order of magnitude in the presented examples.

Structural Health Monitoring (SHM) is the engineering discipline of diagnosing damage and estimating safe remaining life for structures and systems. Often, SHM is accomplished by detecting changes in measured quantities from the structure of interest; if there are no competing explanations for the changes, one infers that they are the result of damage. If the structure of interest is subject to changes in its environmental or operational conditions, one must understand the effects of these changes in order that one does not falsely claim that damage has occurred when changes in measured quantities are observed. This problem – the problem of confounding influences – is particularly pressing for civil infrastructure where the given structure is usually openly exposed to the weather and may be subject to strongly varying operational conditions. One approach to understanding confounding influences is to construct a data-based response surface model that can represent measurement variations as a function of environmental and operational variables. The models can then be used to remove environmental and operational variations so that change detection algorithms signal the occurrence of damage alone. The current paper is concerned with such response surface models in the case of SHM of bridges. In particular, classes of response surface models that can switch discontinuously between regimes are discussed. Recently, it has been shown that Gaussian Process (GP) models are an effective means of developing response surface or surrogate models. However, the GP approach runs into difficulties if changes in the latent variables cause the structure of interest to abruptly switch between regimes. A good example here, which is well known in the SHM literature, is given by the Z24 Bridge in Switzerland which completely changed its dynamical behaviour when it cooled below zero degrees Celsius as the asphalt of the deck stiffened. The solution proposed here is to adopt the recently-proposed Treed Gaussian Process (TGP) model as an alternative. The approach is illustrated here on the Z24 bridge and also on data from the Tamar Bridge in the UK which shows marked switching behaviour in certain of its dynamical characteristics when its ambient wind conditions change. It is shown that treed GPs provide an effective approach to response surface modelling and that in the Tamar case, a linear model is in fact sufficient to solve the problem.

Proper sensitivity and uncertainty analysis for complex Earth and environmental systems models may become computationally prohibitive. Surrogate models can be an alternative to enable such analyses: they are cheap-to-run statistical approximations to the simulation results of the original expensive model. Several approaches to surrogate modelling exist, all with their own challenges and uncertainties. It is crucial to correctly propagate the uncertainties related to surrogate modelling to predictions, inference and derived quantities in order to draw the right conclusions from using the surrogate model.While the uncertainty in surrogate model parameters due to limited training data (expensive simulation runs) is often accounted for, what is typically ignored is the approximation error due to the surrogate’s structure (bias in reproducing the original model predictions). Reasons are that such a full uncertainty analysis is computationally costly even for surrogates (or limited to oversimplified analytic cases), and that a comprehensive framework for uncertainty propagation with surrogate models was missing.With this contribution, we propose a fully Bayesian approach to surrogate modelling, uncertainty propagation, parameter inference, and uncertainty validation. We illustrate the utility of our approach with two synthetic case studies of parameter inference and validate our inferred posterior distributions by simulation-based calibration. For Bayesian inference, the correct propagation of surrogate uncertainty is especially relevant, because failing to account for it may lead to biased and/or overconfident parameter estimates and will spoil further interpretation in the physics’ context or application of the expensive simulation model.Consistent and comprehensive uncertainty propagation in surrogate models enables more reliable approximation of expensive simulations and will therefore be useful in various fields of applications, such as surface or subsurface hydrology, fluid dynamics, or soil hydraulics.

A method is proposed to localize preferential fluid flow pathways in porous media on the basis of time‐lapse self‐potential measurements associated with salt tracer injection upstream. This method is first tested using laboratory data. A network of nonpolarizing electrodes located is connected to a highly sensitive voltmeter used to record the resulting electrical field fluctuations occurring over time at the surface of the tank. The transport of the conductive salt plume through the permeable porous materials changes the localized streaming potential coupling coefficient associated with the advective drag of the excess charge of the pore water and is also responsible for a diffusion current associated with the salinity gradient. Monitoring of the electrical potential distribution at the ground surface can be used to localize the pulse of saline water over time and to determine its velocity. This method applies in real time and can be used to track highly localized flow pathways characterized by high permeability. Our sandbox experiment demonstrates the applicability of this new method under well‐controlled conditions with a coarse‐sand channel embedded between fine‐sand banks. A finite element model allows us to reproduce the time‐lapse electrical potential distribution over the channel, but some discrepancies were observed on the banks. Finally, we performed a numerical simulation for a synthetic case study inspired by a recently published field case study. A Markov chain Monte Carlo sampler is used to determine the permeability and the porosity of the preferential fluid flow pathway of this synthetic case study.

Structural health monitoring (SHM) systems are used to analyze the health of infrastructures such as bridges, using data from various types of sensors. While SHM systems consist of various stages, feature extraction and pattern recognition steps are the most important. Consequently, signal processing techniques in the feature extraction stage and machine learning algorithms in the pattern recognition stage play an effective role in analyzing the health of bridges. In other words, there exists a plethora of signal processing techniques and machine learning algorithms, and the selection of the appropriate technique/algorithm is guided by the limitations of each technique/algorithm. The selection also depends on the requirements of SHM in terms of damage identification level and operating conditions. This has provided the motivation to conduct a Systematic literature review (SLR) of feature extraction techniques and pattern recognition algorithms for the structural health monitoring of bridges. The existing literature reviews describe the current trends in the field with different focus aspects. However, a systematic literature review that presents an in-depth comparative study of different applications of machine learning algorithms in the field of SHM of bridges does not exist. Furthermore, there is a lack of analytical studies that investigate the SHM systems in terms of several design considerations including feature extraction techniques, analytical approaches (classification/ regression), operational functionality levels (diagnosis/prognosis) and system implementation techniques (data-driven/model-based). Consequently, this paper identifies 45 recent research practices (during 2016-2023), pertaining to feature extraction techniques and pattern recognition algorithms in SHM for bridges through an SLR process. First, the identified research studies are classified into three different categories: supervised learning algorithms, neural networks and a combination of both. Subsequently, an in-depth analysis of various machine learning algorithms is performed in each category. Moreover, the analysis of selected research studies (total = 45) in terms of feature extraction techniques is made, and 25 different techniques are identified. Furthermore, this article also explores other design considerations like analytical approaches in the pattern recognition process, operational functionality and system implementation. It is expected that the outcomes of this research may facilitate the researchers and practitioners of the domain during the selection of appropriate feature extraction techniques, machine learning algorithms and other design considerations according to the SHM system requirements.

Bayesian model updating with finite element vs surrogate models: Application to a miter gate structural system