Bayesian evidence for finite element model updating

Linda Mthembu, PhD student, Electrical and Information Engineering, University of the Witwatersrand, Johannesburg Private Bag 3, Johannesburg, 2050, South Africa Tshilidzi Marwala, Professor of Electrical and Information Engineering, University of the Witwatersrand, Private Bag 3, Johannesburg, 2050, South Africa Michael I. Friswell, Sir George White Professor of Aerospace Engineering, Department of Aerospace Engineering, Queens Building, University of Bristol, Bristol BS8 1TR, UK Sondipon Adhikari, Chair of Aerospace Engineering, School of Engineering, Swansea University, Singleton Park, Swansea SA2 8PP, United Kingdom

Abstract

This paper considers the problem of model selection within the context of finite element model updating. Given that a number of FEM updating models, with different updating parameters, can be designed, this paper proposes using the Bayesian evidence statistic to assess the probability of each updating model. This makes it possible then to evaluate the need for alternative updating parameters in the updating of the initial FE model. The model evidences are compared using the Bayes factor, which is the ratio of evidences. The Jeffrey’s scale is used to determine the differences in the models. The Bayesian evidence is calculated by integrating the likelihood of the data given the model and its parameters over the a priori model parameter space using the new nested sampling algorithm. The nested algorithm samples this likelihood distribution by using a hard likelihood-value constraint on the sampling region while providing the posterior samples of the updating model parameters as a by-product. This method is used to calculate the evidence of a number of plausible finite element models.

Nomenclature

θ Model Parameters D Real measured system data H Finite element model (Mathematical model) Z Evidence N Number of samples Max Maximum number of iterations L Likelihood probability π Prior Probability c Damping matrix k Stiffness matrix m Mass matrix ( )t x&& Node acceleration ( )t x& Node velocity
( )t x Node displacement w Natural frequency vector φ Mode shape vector FEM Finite element model FEMUP Finite element model updating problem PDF Probability Distribution Function MCMC Markov Chain Monte Carlo

1. Introduction

System modeling forms an important stage of many engineering design problems. The results from the model either confirm or highlight limitations of the design. An analyst is usually interested in the accuracy, confidence range and more critically the correctness of the assumed mathematical model. In this paper the model domain is structural finite element models (FEMs). These models are used to approximate the dynamics of structural systems, e.g. train chassis, aircraft fuselages, bicycle frames, civil structures etc. The finite element model updating problem (FEMUP) arises when a real system’s dynamic behavior is measured (e.g. the natural frequencies at which particular system deformations occur) and the results of the mathematical model of that system do not correspond to the measured data [6, 8]. This problem is compounded by the fact that a multitude of mathematical models of the structure, with varying levels of complexity, can be developed, leading to non-unique solutions for a particular system.

Finite element models are limited by definition; they are an approximation of a real system and will thus never produce dynamic results that are equal to the measured system’s data. The challenge is then what can be done to the initial model for it to better reflect the real system’s dynamic results? This leads to the need for intelligently improved or updated models. Specifically an automatic methodology of determining salient model parameters that require updating needs to be developed. This has to be attained whilst using realistic characteristic parameters of the system in question. Two main directions of research have been established in the area of finite element model updating (FEMU); direct and indirect (iterative) methods [8].

In the direct model updating paradigm [3, 5, 8] the model modal parameters are directly equated to the measured modal data. Model updating is then characterized by the direct updating of mass or stiffness matrix elements. This effectively constrains the modal properties and frees the system matrices for updating. This approach often results in unrealistic elements in the system matrices e.g. large and physically impossible mass elements. In the indirect or iterative model updating approach the updating problem is formulated as a ‘relaxed’ optimization problem, often approached by the use of maximum likelihood

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