Bayesian Finite Element Model Research Articles

Improved hierarchical Bayesian modeling framework with arbitrary polynomial chaos for probabilistic model updating

Bayesian finite element model updating techniques have found widespread application in the structural health monitoring. The conventional Bayesian modeling framework (CBMF) identifies the posterior distribution of structural parameters using data from a single experiment. However, structural parameters exhibit variability due to changes in environmental or experimental conditions, an aspect overlooked by CBMF. In recent years, the hierarchical Bayesian modeling framework (HBMF) has gradually attracted attention. This framework incorporates an additional layer of hyperparameters and utilizes multiple sets of experimental data to describe the variability in structural parameters caused by changes in experimental conditions. Unfortunately, the increase in hyperparameters and data size leads the HBMF to face the dilemma of exceedingly high computational costs. To address this issue, this study proposes an improved hierarchical Bayesian modeling framework (IHBMF). To enhance the computational efficiency of IHBMF, the importance sampling method is employed to simplify the computation of the likelihood function for hyperparameters. Subsequently, a surrogate model based on arbitrary polynomial chaos expansion is introduced to further reduce the computational cost of IHBMF. The framework's accuracy and efficiency were validated through analysis of a simply supported beam and a steel pedestrian bridge. The results demonstrate that IHBMF not only delivers a more accurate quantification of structural parameter uncertainty than CBMF but also exhibits higher computational efficiency compared to HBMF, showcasing its superior capability in structural health monitoring applications.

Time-Domain Finite Element Model Updating for Operational Monitoring and Damage Identification of Bridges

The well-known limitations of modal system identification methods have led to a broad exploration of alternative solutions for operational monitoring and damage diagnosis of structures. This study presents a time-domain Bayesian finite element model updating approach to jointly identify the vehicular loads and finite element modeling parameters of bridges using the vibration data and the location of vehicles traversing the bridge as input. A Bayesian model updating is devised and verified through a series of case studies based on numerically simulated data from a prestressed reinforced concrete box-girder bridge model. Damage states are defined for concrete degradation and delamination, steel corrosion, and loss of prestressing force. Ten different damage scenarios, encompassing the range from minor localized to major distributed damage, are examined. The responses of the damaged bridge are simulated under random traffic scenarios. The acceleration responses, along with the location of the vehicles on the bridge, are used for jointly estimating the model parameters and vehicular loads. The estimated model parameters are then used to infer the location and extent of damage within the bridge. The results show the successful performance of the proposed approach in a numerically simulated environment.

Load Identification of High-Speed Train Crossbeam Based on Bayesian Finite Element Model Updating and Load-Strain Linear Superposition Algorithm

The load identification of the crossbeams in high-speed trains is proving to be crucial, because of the complexity of its structure and environment. In this paper, a method to identify load of the crossbeam structure based on Bayesian finite element model updating and load-strain linear superposition algorithm is proposed. Firstly, the finite element model is established. Using the numerical simulation under different loads, the relative mean error of load identification is less than 0.5% and the effect of strain field reconstruction performs well. Based on the model test system, Markov Chain Monte Carlo method is used to update the parameters of the initial finite element model. Then an experimental setup is built with 64 FBG sensors installed on the crossbeam. The load identification results of Tikhonov regularization method, the load-strain linear superposition method before and after the Bayesian finite element model updating are analyzed and compared. The results show that the proposed method can significantly improve the accuracy of load identification.

A new Bayesian finite element model updating method based on information fusion of multi-source Markov chains

In this paper, we present a new Bayesian model updating method that could overcome the problem of low sampling efficiency and over-reliance on single-chain proposal distribution of traditional Markov chain Monte Carlo algorithm. The delayed rejection and adaptive strategies are introduced in sampling process to obtain a certain number of Markov chains from different proposal distributions, which can independently adjust the variances of proposal distributions and improve the acceptance rate of candidate samples. The abnormal chain detection criterion is adopted to eliminate abnormal Markov chains. Then, the initial variances of different proposal distributions are analogous to the accuracy index of multi-source sensors in the signal domain. And the multi-source sensors grouping weighted fusion algorithm is introduced to fuse the screened Markov chains to approach the posterior probability distribution with high accuracy. The implicit relationship between the parameters to be updated and the responses of the finite element model is fitted by the Kriging surrogate model to improve the computational efficiency. The results of study cases demonstrate that the proposed method has good updating efficiency, excellent updating accuracy, and a higher acceptance rate of samples, which provides a new idea for solving the stochastic model updating.

Stiffness Modification-Based Bayesian Finite Element Model Updating to Solve Coupling Effect of Structural Parameters: Formulations

The Bayesian model updating approach (BMUA) benefits from identifying the most probable values of structural parameters and providing uncertainty quantification. However, the traditional BMUA is often used to update stiffness only with the assumption of well-known mass, which allows unidentifiable cases induced by the coupling effect of mass and stiffness to be circumvented and may not be optimal for structures experiencing damages in both mass and stiffness. In this paper, the new BMUA tailored to estimating both mass and stiffness is presented by using two measurement states (original and modified systems). A new eigenequation with a stiffness-modified system is formulated to address the coupling effect of mass and stiffness. The posterior function is treated using an asymptotic approximation method, giving the new objective functions with stiffness modification. Analytical formulations of modal parameters and structural parameters are then derived by a linear optimization method. In addition, the covariance matrix of uncertain parameters is determined by the inverse of the Hessian matrix of the objective function. The performance of the proposed BMUA is evaluated through two numerical examples in this study; a probabilistic damage estimation is also implemented. The results show the proposed BMUA is superior to the traditional one in mass and stiffness updating.

A multi‐objective framework for finite element model updating using incomplete modal measurements

Finite element (FE) model updating in multi-objective framework helps for better understanding of overall performance in updating (under various variations of weightages assigned to basic components of the objective function) along with providing scope for better judgmental selection. A FE model updating in multi-objective framework is proposed with no requirement of repeated eigen-solution along with avoiding repeated possibilities of incurring mode-pairing error (by adopting an existing framework of system mode shape). Two multi-objective optimization techniques are adopted: (a) weighted sum and (b) adaptive weighted sum methods. Moreover, a possible single best solution out of the Pareto front is identified based on minimum modal distance value and compared with Gibbs sampling technique (without mode-matching). Two examples with multiple damage cases utilized in validating the proposed approach are as follows: (a) simulated example (ASCE benchmark structure) and (b) experimental example (four storied shear frame laboratory structure). It is observed that the proposed multi-objective framework has performed well in FE model updating in case of both simulated and experimental cases. Additionally, a connection (directly relating the multi-objective weights and error variances) is established between the proposed updating methodology and an existing Bayesian updating methodology to facilitate the probabilistic damage detection in Bayesian framework. Moreover, selection of an appropriate solution (out of the Pareto front) having suitable values of multi-objective weights facilitates to estimate the suitable values of error variances (based on the proposed connection), consequently enabling an efficient Bayesian FE model updating without requirement of any assumption of error variances.

Mechanics‐based model updating for identification and virtual sensing of an offshore wind turbine using sparse measurements

Offshore wind turbines are complex systems operating in harsh environment. The dynamic demands in these systems often differ from values used in design, leading to unexpected mechanical and structural failures. This signifies the importance of remote monitoring technologies for damage diagnosis and prognosis in offshore wind turbines. This study is focused on developing mechanics-based digital twins for offshore wind turbine monitoring through a model-updating process using sparse measurement data. Digital twins can be used to estimate the system unmeasured response (i.e., virtual sensing) and to predict the remaining useful fatigue life and failure point of different structural components. A time-domain sequential Bayesian finite element model updating is proposed for mechanics-based digital twinning. This approach is formulated for application to offshore wind turbine and jointly estimates the updating model parameters and the time history of unknown input forces. A classical modal-based model updating followed by modal expansion method is also implemented for comparison. In this approach, updating model parameters are estimated to minimize the discrepancies between the identified and model-predicted modal parameters of the turbine. The performance of these two approaches are studied on a 2-MW offshore wind turbine at the Blyth wind farm in the United Kingdom. Strain response time history at mudline is estimated through both approaches and compared with actual measurements for validation. It is observed that both approaches are capable of accurate response prediction while the Bayesian approach leads to slightly better results. Furthermore, the Bayesian approach allows for identification of input loads and uncertainty quantification.

Modified Hamiltonian Monte Carlo‐based Bayesian finite element model updating of steel truss bridge

The aim of this study is to develop a Hamiltonian Monte Carlo-based algorithm for finite element model updating in the Bayesian framework. The proposed algorithm uses adaptive prior-based approach, which helps to generate the intermediate pdfs. Numerical analysis is carried out with different coefficient of variations of the prior for model updating. Guidelines for their proper selection procedure are also prescribed in this work. The efficiency of the proposed method is demonstrated using synthetic experiments and actual test data for updating the finite element model of a steel truss bridge. Finally, performance of this algorithm is compared with the standard Markov chain Monte Carlo algorithm to demonstrate its advantages.

Bayesian Finite Element Model Updating of a Long-Span Suspension Bridge Utilizing Hybrid Monte Carlo Simulation and Kriging Predictor

Bayesian model updating technique has been widely investigated and utilized in the field of finite element model (FEM) updating for its advantages in system uncertainty quantification. Most existing studies focus on numerical and experimental models. More studies on large-scale civil infrastructures based on field monitoring are still required. A case study on Bayesian FEM updating of the Runyang Suspension Bridge (RSB), a long-span suspension bridge with a main span of 1,490 m, is carried out in this paper. The Bayesian updating method is utilized to update the initial FEM of RSB, aiming to make the numerical modal properties match the field monitoring results. Two stochastic sampling algorithms, i.e., the Metropolis-Hastings (MH) algorithm and the Hybrid Monte Carlo (HMC) algorithm, are respectively investigated to show their advantages and limitations in Bayesian updating. Subsequently, based on the experimentalsamples generated by the Latin hypercube sampling algorithm, a Kriging predictor is established as a surrogate model to reduce the computational burden of model updating. Results show that the HMC algorithm could guarantee much higher acceptance rate of the sampled chain than the MH algorithm especially when the updating step size is large. In addition, combined with the Kriging predictor, Bayesian model updating method could serve as an effective and efficient tool to calibrate the FEM of large-scale civil infrastructures.

A two-stage Bayesian algorithm for finite element model updating by using ambient response data from multiple measurement setups

This study presents a two-stage Bayesian finite element model updating procedure by using acceleration response measurements obtained from multiple setups. In the presented methodology, parametric uncertainties for the modal parameters are estimated by using the Bayesian Fast Fourier Transform Approach (BFFTA). Different from the previous Bayesian methods, a block diagonal covariance matrix is modeled for prior estimation of measured modal parameters. In addition, the modelling error in the eigenvalue equations is considered as soft constraints to be updated. Numerical and experimental studies are presented to validate the proposed method. The effect of soft constraints on the identification results as well as their posterior uncertainties are investigated. According to the results, it is shown that the proposed methodology can identify the most probable finite element model parameters with high level of accuracy. In addition, the posterior uncertainties obtained by the proposed procedure are significantly small when compared to the methods that consider rigid constraints for prediction and/or modelling error.

A Bayesian model updating framework for robust seismic fragility analysis of non-isolated historic masonry towers.

Seismic assessment of existing masonry structures requires a numerical model able to both reproduce their nonlinear behaviour and account for the different sources of uncertainties; the latter have to be dealt with since the unavoidable lack of knowledge on the input parameters (material properties, geometry, boundary conditions, etc.) has a relevant effect on the reliability of the seismic response provided by the numerical approaches. The steadily increasing necessity of combining different sources of information/knowledge makes the Bayesian approach an appealing technique, not yet fully investigated for historic masonry constructions. In fact, while the Bayesian paradigm is currently employed to solve inverse problems in several sectors of the structural engineering domain, only a few studies pay attention to its effectiveness for parameter identification on historic masonry structures. This study combines a Bayesian framework with probabilistic structural analyses: starting from the Bayesian finite element model updating by using experimental data it provides the definition of robust seismic fragility curves for non-isolated masonry towers. A comparison between this method and the standard deterministic approach illustrates its benefits. This article is part of the theme issue 'Environmental loading of heritage structures'.

Bayesian Finite Element Model Updating and Assessment of Cable-Stayed Bridges Using Wireless Sensor Data.

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.

A Bayesian finite element model updating with combined normal and lognormal probability distributions using modal measurements

• A Bayesian approach combining normal and lognormal distributions is formulated in details based on maximum a posteriori . • Estimation of uncertainty and probabilistic change/damage is also developed. • Compared with maximum a posteriori based Bayesian updating using normal distribution. • Computational efficiency is compared with Markov Chain Monte Carlo (MCMC) based updating techniques. • With no need of mode matching, such updating remains possible for large structures. The present work is associated with Bayesian finite element (FE) model updating using modal measurements based on maximizing the posterior probability instead of any sampling based approach. Such Bayesian updating framework usually employs normal distribution in updating of parameters, although normal distribution has usual statistical issues while using non-negative parameters. These issues are proposed to be dealt with incorporating lognormal distribution for non-negative parameters. Detailed formulations are carried out for model updating, uncertainty-estimation and probabilistic detection of changes/damages of structural parameters using combined normal-lognormal probability distribution in this Bayesian framework. Normal and lognormal distributions are considered for eigen-system equation and structural (mass and stiffness) parameters respectively, while these two distributions are jointly considered for likelihood function. Important advantages in FE model updating (e.g. utilization of incomplete measured modal data, non-requirement of mode-matching) are also retained in this combined normal-lognormal distribution based proposed FE model updating approach. For demonstrating the efficiency of this proposed approach, a two dimensional truss structure is considered with multiple damage cases. Satisfactory performances are observed in model updating and subsequent probabilistic estimations, however level of performances are found to be weakened with increasing levels in damage scenario (as usual). Moreover, performances of this proposed FE model updating approach are compared with the typical normal distribution based updating approach for those damage cases demonstrating quite similar level of performances. The proposed approach also demonstrates better computational efficiency (achieving higher accuracy in lesser computation time) in comparison with two prominent Markov Chain Monte Carlo (MCMC) techniques (viz. Metropolis-Hastings algorithm and Gibbs sampling).

Implementation of an adaptive meta-model for Bayesian finite element model updating in time domain

This work explores the feasibility of integrating an adaptive meta-model into a finite element model updating formulation using dynamic response data. A Bayesian model updating approach based on a stochastic simulation method is considered in the present formulation. Such approach is combined with a surrogate technique and an efficient model reduction technique. In particular, an adaptive surrogate model based on kriging interpolants and a model reduction technique based on substructure coupling are implemented. The integration of these techniques into the updating process reduces the computational effort to manageable levels allowing the solution of complex problems. The effectiveness of the proposed strategy is demonstrated with three finite element model updating applications.

Probabilistic identification of simulated damage on the Dowling Hall footbridge through Bayesian finite element model updating

Summary This paper presents a probabilistic damage identification study on a full-scale structure, the Dowling Hall footbridge, through a Bayesian finite element (FE) model updating. The footbridge is located at Tufts University and is equipped with a continuous monitoring system that measures its ambient acceleration response. A set of data is recorded once every hour or when triggered by large vibrations. The modal parameters of the footbridge are extracted from each set of data and are used for FE model updating. In this study, effects of physical damage are simulated by loading a small segment of the footbridge deck with concrete blocks. The footbridge deck is divided into five segments in an FE model of the test structure, and the added mass on each segment is considered as an updating parameter. Overall, 72 sets of data are collected during the loading period, and different subsets of these data are used to find the location and extent of the damage (added mass). The adaptive Metropolis–Hastings algorithm with adaption on the proposal probability density function is successfully used to generate Markov Chains for sampling the posterior probability distributions of the five updating parameters. Effects of the number of data sets used in the identification process are investigated on the posterior probability distributions of the updating parameters. The probabilistic model updating framework accurately predicts the simulated damage and the level of confidence on the obtained results. The maximum a-posteriori estimates of damage in the probabilistic approach are found to be in good agreement with their corresponding deterministic counterparts. Copyright © 2014 John Wiley & Sons, Ltd.

Model-reduction techniques for Bayesian finite element model updating using dynamic response data

This work presents a strategy for integrating a class of model reduction techniques into a finite element model updating formulation. In particular a Bayesian model updating approach based on a stochastic simulation method is considered in the present formulation. Stochastic simulation techniques require a large number of finite element model re-analyses to be performed over the space of model parameters during the updating process. Substructure coupling techniques for dynamic analysis are proposed to reduce the computational cost involved in the dynamic re-analyses. The effectiveness of the proposed strategy is demonstrated with identification and model updating applications for finite element building models using simulated seismic response data.

Uncertainty Quantification in the Assessment of Progressive Damage in a 7-Story Full-Scale Building Slice

In this paper, Bayesian linear finite-element (FE) model updating is applied for uncertainty quantification (UQ) in the vibration-based damage assessment of a 7-story RC building slice. This structure was built and tested at full scale on the University of California at San Diego-Network for Earthquake Engineering Simulation shake table: progressive damage was induced by subjecting it to a set of historical earthquake ground motion records of increasing intensity. At each damage stage, modal characteristics, such as natural frequencies and mode shapes, were identified through low-amplitude vibration testing; these data are used in the Bayesian FE model updating scheme. To analyze the results of the Bayesian scheme and gain insight into the information contained in the data, a comprehensive uncertainty and resolution analysis is proposed and applied to the 7-story building test case. The Bayesian UQ approach and subsequent resolution analysis are shown to be effective in assessing uncertainty in FE model updating. Furthermore, it is demonstrated that the Bayesian FE model updating approach provides insight into the regularization of its often ill-posed deterministic counterpart.