Finite Element Model Updating Using Bayesian Approach

Finite element (FE) models are widely used to predict the dynamic characteristics of aerospace structures. These models often give results that differ from measured results and therefore need to be updated to match measured results. Some of the updating techniques that have been proposed to date use time, modal, frequency and timefrequency domain data [1,2]. In this paper, modal domain data is used to update the FE model. A literature review on FE updating [1] reveals that the updating problem has been mainly framed in the Maximum-likelihood framework. Even though this framework has been applied successfully in industry, it has the following shortcomings: it does not offer the user confidence intervals for solutions it gives; there is no philosophical explanation of the regularization terms that are used to control the complexity of the updated model; and it cannot handle inherent ill-conditioning and non-uniqueness of FE updating problem.

The Bayesian framework is adopted to address the shortcomings explained above. Bayesian framework has been found to offer several advantages over Maximum-likelihood methods in areas closely mirroring FE updating [3][4][5]. This paper seeks to address the following issues: (1) how prior information are incorporated into FE model updating problem; and (2) applying Bayesian framework to update FE models to match experimentally measured modal properties (i.e. natural frequencies and mode shapes) to modal properties calculated from the FE model of a beam. In this Paper Markov chain Monte Carlo (MCMC) simulation [3] is used to sample the probability of the updating parameters in light of the measured modal properties. This probability is known as the posterior probability. Metropolis algorithm [6] is used as an acceptance criterion when sampling the posterior probability.

All elastic structures may be described in terms of their distributed mass, damping and stiffness matrices. If damping terms are neglected, the dynamic equation may be written in modal domain (natural frequencies and mode shapes) for the i th mode as follows [7]:

Here [M] is the mass matrix, [K] is the stiffness matrix, ω i is the i th natural frequency, {φ} i is the i th mode shape vector and {ε} i is the i th error vector. The error vector {ε} i is equal to {0} if the system matrices [M] and [K] correspond to the modal properties. If the system matrices which are usually obtained from the FE model do not match measured modal properties ω i and {φ} i then {ε} i is a non-zero vector. In maximum-likelihood method the Euclidean norm of {ε} i is minimized in order to match the system matrices to measured modal properties. Another problem that is encountered in many practical situations is that the dimension of mode shapes does not match the dimension of system matrices. This is because measured modal co-ordinates are fewer than FE modal coordinates. To ensure compatibility between system matrices and mode shape vectors, the dimension of system matrices is reduced by using a technique called Guyan reduction method [8] to match the dimension of system matrices to the dimension of measured mode shape co-ordinates.

In this paper the Bayesian method is introduced to solve the FE updating problem based on modal properties. The fundamental rule that governs the Bayesian approach is written as follows [3]:

Here {E} is a vector of updating parameters, P({E}) is the probability distribution function of updating parameters in the absence of any data, and this is known as the prior distribution, and [D] is a matrix containing natural frequencies ω i and mode shapes {φ}

There are many areas where the likelihood distribution function has been applied and these include neural networks [3]. In neural network context, the likelihood distribution function is defined as the normalized exponent of the error function. In this paper the likelihood distribution function, P([D]|{E}), is defined as the sum of square of elements of the error vector shown in equation 1, and can be written in the same way as in neural networks as follows [3]:

Here β is the coefficient of the measured modal property data contribution to the error and is set to 1 through trial and error and ε ij is the error matrix with subscript i representing the i th modal properties and j representing the i th measurement position. The superscript F is the number of measured mode shape coordinates, N is the number of measured modes and ZD is:

It should be d that in equation 3 the error ε ij is a matrix as opposed to a vector as is the case in equation 1. This is because it takes into account of all modal co-ordinates.

The prior distribution function consists of the information that is known about the problem. In FE updating it is generally accepted that FE updating is usually valid if the model is close to the true model. In this paper, it is known that not all parameters to be updated have the same level of modeling errors. This means that some parameters are to

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