Chapter26 Finite Element Model Updating Using the Separable Shadow Hybrid Monte Carlo Technique I. Boulkaibet, L. Mthembu, T. Marwala, M.I. Friswell, and S. Adhikari Abstract The use of Bayesian techniques in Finite Element Model (FEM) updating has recently increased. These techniques have the ability to quantify and characterize the uncertainties of dynamic structures. In order to update a FEM, the Bayesian formulation requires the evaluation of the posterior distribution function. For large systems, this functions is either difficult (or not available) to solve in an analytical way. In such cases using sampling techniques can provide good approximations of the Bayesian posterior distribution function. The Hybrid Monte Carlo (HMC) method is a powerful sampling method for solving higher-dimensional complex problems. The HMC uses the molecular dynamics (MD) as a global Monte Carlo (MC) move to reach areas of high probability. However, the acceptance rate of HMC is sensitive to the system size as well as the time step used to evaluate MD trajectory. To overcome this, we propose the use of the Separable Shadow Hybrid Monte Carlo (S2HMC) method. This method generates samples from a separable shadow Hamiltonian. The accuracy and the efficiency of this sampling method is tested on the updating of a GARTEUR SM-AG19 structure. Keywords Bayesian • Sampling • Finite element model updating • Markov Chain Monte Carlo • Hybrid Monte Carlo method • Shadow Hybrid Monte Carlo 26.1 Introduction Finite element model (FEM) is a numerical method used to model complex engineering problems [1, 2]. FEM is often used to compute displacements, stresses and strains in complex structures under a given set of loads. Due to the uncertainties (among other approximations) associated with the process of constructing a finite element model of a structure the analytical results are different from those obtained from experimental measurements [3, 4]. Thus for practical purposes the FE model needs to be updated. In recent years the use of the Bayesian framework to build model updating techniques has shown promising results in this system identification problem [4, 6–8]. This approach allows system modelling uncertainties to be expressed in terms of probability. This can be done by representing the parameters that need to be updated as random vectors with a joint probability distribution function (pdf). This distribution function is known as the posterior distribution function. For sufficiently complex problems this pdf is not available in analytical form. This is the case for the FEM updating problem where the parameter search space is non linear and of high dimension. When an analytical solution is not available sampling methods, such as the Markov Chain Monte Carlo (MCMC), offer the only practical solution to estimating the desired posterior distribution function [4, 7, 8]. One improvement on the classic MCMC is the Hybrid Monte Carlo (HMC) sampling technique. This algorithm is able to deal with an updating vector of a large size. In the HMC the derivative of the target log-density probability is used to guide the Monte Carlo trajectory and leads towards areas of high probability [5, 7, 11, 18]. An auxiliary variable, called the momentum vector is introduced and the I. Boulkaibet ( ) • L. Mthembu • T. Marwala Electrical and Electronic Engineering Department, The Centre For Intelligent System Modelling (CISM), University of Johannesburg, PO Box 524, Auckland Park 2006, South Africa e-mail: iboulkaibet@student.uj.ac.za M.I. Friswell • S. Adhikari College of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, UK R. Allemang (ed.), Topics in Modal Analysis II, Volume 8: Proceedings of the 32nd IMAC, A Conference and Exposition on Structural Dynamics, 2014, Conference Proceedings of the Society for Experimental Mechanics Series, DOI 10.1007/978-3-319-04774-4__26, © The Society for Experimental Mechanics, Inc. 2014 267
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