Figure 7 Probability density functions for the first frequency generated by back propagating estimated parameters. The Bayesian technique of estimating parameters for a model has the advantage of producing estimates of PDFs for all of the estimated parameters as well as a PDF that can be used to assess the quality of fit of the parameters. In addition, using a parameterized prior such as a Johnson distribution (11), the variability in the PDF of the elastic parameters could also be explored by performing a series of Bayes estimations on a family of priors generated by varying the parameters of the Johnson distribution. This was not done in this work. Another advantage to the Bayesian technique is the ability to include prior information into the parameter estimation. If probability distribution functions already existed for a similar foam, those could be used as a prior instead of the uniform distribution. The disadvantages to the Bayesian technique is the large number of runs necessary to generate the estimates of the PDFs. The only practical method to implement a Bayesian method is to have a very fast running model. Also, if there is little experimental data, the estimated distributions (posterior) for the elastic parameters will look like the assumed prior distributions. For a very large number of data points, the posterior distributions will approach the likelihood function. 6. CONCLUSIONS A comparison of methods to fit the elastic parameters Young’s modulus and Poisson’s ratio from measured acoustic data is presented. Two different algorithms are used for estimating the parameters. The first is an optimization based least squares technique and the other is a Bayesian method. The parameters are fit using two different cost functions. The differences between the fit parameters are greater between the different cost functions than between the different algorithms. However, when the probability density functions are compared, the difference are most pronounce between algorithms with different trends for the Young’s modulus and the Poisson’s ratio. The analyses show that exploring different techniques when performing parameter estimations can provide a further characterization of the uncertainty. 7. ACKOWLEDGEMENTS The author would like to thank Angel Urbina for his stimulating conversations and comments throughout the preparation of this paper. Sandia is a multi-program laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract DE-AC0494AL85000. 0.9 0.95 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 x 10 4 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 x 10-3 Young's Modulus, (psi) PDF Estimate without V Estimate with V Estimate using prior Test Data 228
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