Bayesian inference provides a powerful approach to system identification and damage assessment for structures. The application of Bayesian method is motivated by the fact that inverse problems in structural engineering, including structural health monitoring, are typically ill-conditioned and ill-posed when using noisy incomplete data because of various sources of modeling uncertainties. One should not just search for a single “optimal” value for the vector of model parameters but rather attempt to describe the whole family of plausible model parameters based on measured data using a Bayesian probabilistic framework. In this article, the fundamental principles of Bayesian analysis and computation are summarized; then a review is given of recent state-of-the-art practices of Bayesian inference in system identification and damage assessment for civil infrastructure. Discussions of the benefits and deficiencies of these approaches, as well as potentially useful avenues for future studies, are also provided. Our focus is on meeting challenges that arise from system identification and damage assessment for the civil infrastructure but our presented theories also have a considerably broader applicability for inverse problems in science and technology.

We focus on a Bayesian inference framework for finite element (FE) model updating of a long-span cable-stayed bridge using long-term monitoring data collected from a wireless sensor network (WSN). A robust Bayesian inference method is proposed which marginalizes the prediction-error precisions and applies Transitional Markov Chain Monte Carlo (TMCMC) algorithm. The proposed marginalizing error precision is compared with other two treatments of prediction-error precisions, including the constant error precisions and updating error precisions through theoretical analysis and numerical investigation based on a bridge FE model. TMCMC is employed to draw samples from the posterior probability density function (PDF) of the structural model parameters and the uncertain prediction-error precision parameters if required. It is found that the proposed Bayesian inference method with prediction-error precisions marginalized as “nuisance” parameters produces an FE model with more accurate posterior uncertainty quantification and robust modal property prediction. When applying the identified modal parameters from acceleration data collected during a one-year period from the large-scale WSN on the bridge, we choose two candidate model classes using different parameter grouping based on the clustering results from a sensitivity analysis and apply Bayes’ Theorem at the model class level. By implementing the TMCMC sampler, both the posterior distributions of the structural model parameters and the plausibility of the two model classes are characterized given the real data. Computation of the posterior probabilities over the candidate model classes provides a procedure for Bayesian model class assessment, where the computation automatically implements Bayesian Ockham razor that trades off between data-fitting and model complexity, which penalizes model classes that “over-fit” the data. The results of FE model updating and assessment based on the real data using the proposed method show that the updated FE model can successfully predict modal properties of the structural system with high accuracy.

Bayesian system identification has attracted substantial interest in recent years for inferring structural models and quantifying their uncertainties based on measured dynamic response in a structure. The relative plausibility of each structural model in a specified model class is quantified by its posterior probability from Bayes’ Theorem. The relative plausibility of each model class within a set of candidate model classes for the structure can also be assessed via Bayes’ Theorem. Computation of this posterior probability over all candidate model classes automatically applies a quantitative Ockham's razor that trades off a data-fit measure with an information-theoretic measure of model complexity, which penalizes model classes that “over-fit” the data. In this article, we first present a general Bayesian system identification framework and point out that combining it with sparse Bayesian learning (SBL) is an effective strategy to implement the Bayesian Ockham razor. Then we review our recent progress in exploring SBL with the automatic relevance determination likelihood concept to detect and quantify spatially sparse substructure stiffness reductions. To characterize the full posterior uncertainty for this problem, an improved Gibbs sampling procedure for SBL is then developed. Finally, illustrative results are provided to compare the performance and validate the capability of the presented SBL algorithms for structural system identification.

In ultrasonic guided wave–based damage detection, the propagation distance recognition of wave packets is an essential step. However, it is difficult to perform direct distance extraction from guided wave signals since the multimode, mode conversion, and dispersion effects typically lead to wave packet overlapping and distortion. In addition, the identified damage location may be incorrect due to inevitable uncertainties in the procedure of propagation distance recognition and damage localization. Motivated by these difficulties, a novel two-stage approach for propagation distance recognition and damage localization is proposed based on sparse Bayesian learning framework. In the first stage, prior knowledge of a small number of wave packets contained in a signal is exploited to sparsely represent the guided wave signal and then the corresponding propagation distance and amplitude information of each wave packet can be obtained. In the next stage, only a small number of damages occurring in a structure are exploited and a vector consisting of the propagation distances extracted from the previous stage is used to match the atoms in a pre-defined over-complete distance dictionary matrix, to achieve our goal of localizing structural damage. Both procedures of the two stages are realized by the sparse Bayesian learning algorithm, which obtains the most probable value and the corresponding uncertainty. A sampling strategy is presented to transfer the uncertainty of the propagation distance recognition to the subsequent damage localization. Finally, the effectiveness of the proposed method is validated using numerical simulation and experimental investigation on aluminum plates. The proposed method is only valid for single damage localization in the present form, but it has the potential to be extended for multiple damage localization.

Guided wave (GW) testing has become one of the most important nondestructive tools for evaluation of structural in-service degradation. Although GW testing was widely used to inspect and screen many engineering structures, there are still challenges associated with its applications, which mostly originate from the dispersive and multi-mode nature of GW signals as well as noise contamination. To deal with these challenges, an effective GW signal processing technique is introduced that enables one to accurately recover multiple modes from noisy dispersive GW signals for defect localization. To characterize the dispersion phenomenon, the chirp model is introduced first, which is derived from GW dispersion characteristics. Based on this model, an over-complete dictionary is designed by considering all possible defect locations (by discretization) and the propagation paths of GW modes caused by these defects. Then the prior knowledge that structural defects or damage typically occur in localized areas and correspondingly only a small number of modes are included in the GW signals is exploited by a robust sparse Bayesian learning (SBL) framework to reduce the ill-conditioning in the inverse problem. During the implementation of the robust SBL algorithm, irrelevant basis vectors in the dictionary are pruned out and the posterior probabilities of the small number nonzero basis coefficients corresponding to those “active” GW modes are established. After sparse representation of the GW signal, the information of propagation paths and group delays of GW modes are used to identify each GW mode and then localize defects. Illustrative results from numerical simulations and experiment studies on plate structures are presented to demonstrate the capability of the proposed method.