Figure 6 KDE comparisons for Young’s modulus and Poisson’s ratio using three frequencies for estimation. Once the parameters for the elastic model and the standard error are calibrated using a Bayesian approach, the resulting samples can be forward propagated through the model to calculate the frequencies. The KDE of the frequencies can then be estimated. This provides an evaluation as to how well the estimation procedure performed. The standard error represents the error in the fit that is not consistent with the elastic model assumption, ([9]). These fit variations can be due to experimental uncertainty, measurement uncertainty, data fitting uncertainty, modeling assumptions such as linear elastic, and other epistemic sources. For the problem described in this paper, the error could also be due to the density. Here, the density was assumed to be the average of all the measured densities rather than using density as a design variable. The predicted first frequency with the error can be defined using Eq. (1) as Q U Q V V 1 1 , , 1 10 g E f E (11) The standard error is multiplied by the calculated frequency because of the manner in which the likelihood is normalized in Eq. (10). To calculate the PDFs shown in Figure 7, the parameters estimations from the Bayes procedure are evaluated in Eq. (11). The standard error is included by generating a Gaussian random variable with zero mean and a standard deviation equal to 10 V. The PDF is then estimated using the KDE. In addition, frequency estimates can be generated using the prior. For this example, each of the elastic parameters are drawn from a uniform random distribution as defined by Eq. (9). Note that the estimate without the standard error does not fully cover the test data. This is because the data has some epistemic errors in them which is covered when the standard error, σ, is included. 0.8 0.9 1 1.1 1.2 1.3 1.4 x 10 5 0 1 2 3 x 10-4 Elastic Modulus PDF Elastic Modulus Optimization Bayes 0 0.1 0.2 0.3 0.4 0.5 0 20 40 60 80 100 120 Poisson's ratio PDF Poisson's ratio Optimization Bayes 227
RkJQdWJsaXNoZXIy MTMzNzEzMQ==