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

• Published in: Applied Mathematical Modelling Vol. 61; pp. 457 - 483
• Main Authors: Das, A., Debnath, N.
• Format: Journal Article
• Language: English
• Published: New York Elsevier Inc 01.09.2018; Elsevier BV
• Subjects: Bayesian analysis; Bayesian approach; Change detection; Computational mathematics; Computer simulation; Computing time; Conditional probability; Damage detection; Finite element analysis; Finite element method; Formulations; Lognormal distribution; Markov Chain Monte Carlo; Markov chains; Mathematical models; Modal data; Model updating; Normal distribution; Parameter uncertainty; Probability distribution; Sampling; Statistical analysis; Stiffness; Structural damage; Finite element method; Bayesian approach; Markov Chain Monte Carlo; Damage detection; Model updating; Lognormal distribution
• ISSN: 0307-904X; 1088-8691; 0307-904X
• DOI: 10.1016/j.apm.2018.05.004

•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).