This work would not have been possible without the guidance and support of many people. I would like to thank the Air Force Office of Scientific Research (Project Manager: Dr. Fariba Fahroo), the VEXTEC Corporation (Investigators: Dr. Robert Tryon, Dr. Animesh Dey), and the Department of Civil and Environmental Engineering at Vanderbilt for their financial support .
Overview
The error in a computational model prediction consists of two parts: model form error (εmodel) and solution approximation error or numerical error (εnum) [1, 4]. As we will see in Chapter 4, yc is also used to quantify the model shape error.
Objective and contributions
Sensitivity analysis methods are developed to determine the contribution of each source of error to the overall uncertainty in the model prediction. Sensitivity information is helpful in identifying dominant factors contributing to model prediction uncertainty and in guiding the allocation of resources to improve the model.
Organization of the thesis
In the Merriam-Webster dictionary, the word error is defined as "the difference between an observed or calculated value and the true value" and appears to be a deterministic quantity. We can then use the sample means and variances to construct their confidence intervals.
NUMERICAL ERROR QUANTIFICATION
Introduction
To correct a measured or calculated value, the deterministic error is added directly to it, while a random sample of the stochastic error is first generated and then added to it. In general, if the error is deterministic, it will be corrected by adding it to the corresponding quantity; if the error is stochastic, it will be accounted for by including its uncertainty in the final output through sampling.
Input error
Discretization error
A good method to approximate the actual error for model V&V is the Richardson extrapolation [17, 18]. The Richardson extrapolation method uses the finite element analysis as a black box, i.e. no other modification of the finite element code is required except to refine the mesh.
Surrogate model prediction error
In the same way we can calculate the average value of the samples in the strip. Finally, if the position of the strip varies depending on the distribution of Xi (as shown in Figure 6(b)), we can calculate the variance of ( | ) .. E i with respect to Xi, which is the main effect. The main effect sensitivity index is obtained by taking the variance of the expected values.
Thus, the distribution of the discretization error is due to randomness in the input variables.
UNCERTAINTY QUANTIFICATION ERROR
Introduction
In this chapter, we discuss the error that occurs when using a limited number of samples to estimate the mean and variance or construct the empirical CDF of a stochastic quantity. We denote the uncertainty quantification error εuq as the difference between the empirical CDF value and the “true” CDF value. In addition, the spread of εuq is studied when resampling from the empirical CDF and using the samples for further analysis.
Error in Empirical CDF
Note that the variance of this error is actually a function of x, and it goes to zero at both ends of the CDF curve. The empirical CDF of the samples with 90% confidence limits is constructed using the above method. The result is compared with the true CDF curve in Figure 2, and shows that the CDF curve lies between the confidence limits.
Error Estimation for mean and variance
An example is shown here to demonstrate how to use the proposed method to estimate the true mean and variance of given samples and how to construct confidence intervals for the estimates. To compare with the classical statistical method, suppose that n=21 samples of a normal random variable X are available, but its variance is unknown. Compared to the classical method, the PDFs of σ2X agree quite well, while the PDF of μX given by the proposed method is narrower than that given by the classical method.
Compared to the results given by the classical method, the proposed method provides a narrower estimate for the mean value.
Propagation of UQ error
This method provides a generic approach for resampling sparse data and incorporating the additional uncertainty due to UQ errors in the new samples. In this study, this method is later applied to quantify the model shape error (see Section 4).
Summary
The sensitivity analysis is done in terms of the contribution to the uncertainty in the corrected model prediction yc (defined in Eq. In fact, the variance change measure compares the variance of the samples falling within the band against the global sensitivity analysis with random variables, the samples of variables are created based on their distributions.
The scatterplots of the expected values versus the corresponding errors are shown in Figure 11.
MODEL FORM ERROR ESTIMATION
Introduction
As mentioned in section 1.1, the quantification of the model shape error requires a comparison of the model prediction with validation experiment observations yobs, taking into account the output measurement error εom. This equation is evaluated by sampling each of the three terms on the right and samples of the ε model are obtained. In addition, a UQ error occurs when calculating the model shape error since a sampling-based method is used.
Output measurement error
Model form error
Summary
The two local sensitivity analysis methods introduced in Section 5.2, which are change of variance and KL divergence, are used here to determine the contribution of each error to the uncertainty of the corrected model output yc. The results are listed in Table 3. Similarly, KL distances are calculated for each of the three errors by comparing the distributions of the samples in the strips and the overall distribution of δc. To calculate the main effect sensitivity index, the location of the strip is varied and expected value of δc is calculated within each strip.
However, the structure of the crack growth problem is not like that of the previous example due to the cycle-by-cycle analysis.
SENSITIVITY ANALYSIS
Introduction
In this chapter, a sensitivity analysis is performed to estimate the contribution of each error source to the model prediction uncertainty. Previous studies in stochastic sensitivity analysis have only considered the effect of random input variables. Because the UQ error and model shape error are only calculated after the calculation of yc, this sensitivity analysis concerns only solution approximation errors and does not include UQ error or model shape error.
To perform sensitivity analysis, a product of the model that takes into account all errors is needed, that is.
Local sensitivity analysis
- Change in variance
- Kullback-Leibler divergence
The K-L divergence is non-negative and equal to zero if and only if p(x) and q(x) are exactly equal. The K-L divergence was used in a sensitivity analysis [27] to measure the contribution of a single source to the uncertainty in the model prediction. For comparison with the variance change measure, the K-L divergence is used for estimation.
This compares the difference over the entire distributions of Y and the conditional distribution (Y |Xi =xi), where Xi is fixed at a value (usually its mean).
Global sensitivity analysis
- Main effect and total effect measures
Therefore, a more comprehensive sensitivity measure is needed that includes the main effect and the interaction effect, namely the total effect index. The total effect index becomes valuable when the sum of the individual main effect indices is not close to 1, implying the existence of strong interaction effects between variables. Because the total effect index takes into account the total contribution to output due to the input Xi, the condition is 0.
S is a necessary and sufficient condition for Xi to be insignificant, i.e. that fixing Xi at a particular value has almost no effect on the model's output.
An intuitive understanding of the sensitivity measures
If the strip is thin enough, the samples falling within the strip can be considered as the samples of (Y |Xi = xi). If PDFs are constructed for the samples of both Y and (Y |Xi =xi), then the KL divergence measure can be calculated by comparing the two distributions. Change of variance measures the local sensitivity at a particular Xi, addressing only the variance but ignoring other information.
The KL divergence compares the entire distribution, making it “global” with respect to Y, but is still local with respect to Xi, since it is calculated at some value of Xi.
Deterministic and stochastic errors in sensitivity analysis
- Example: sensitivity analysis on deterministic error and stochastic error
An approximate approach is to sample εh corresponding to samples of random inputs to the FEA model; these samples of εhare are used to construct the distribution of εh. To overcome this difficulty, an approximate approach is to sample εsu corresponding to samples of the random inputs of the surrogate model; these samples of εsu are used to construct the general distribution of εsu. Similarly, by sampling the input x, each conditional yc is sampled and their PDFs are also plotted in Figure 7.
First the input x is sampled and the standard deviation of ε3 is calculated, and then 10 samples of ε3 are generated.
Summary
For each of the 9 training points, 2 more FEM runs with 8 and 16 elements respectively (the original number of elements is 4) are performed to calculate εh. In crack growth analysis, error sensitivity analysis faces major difficulties because different sample values of the same error are introduced into the analysis in each cycle. To quantify the discretization errors in the finite element analysis using Richardson extrapolation, the finite element analysis is repeated for each of the training points with finer meshes with two levels of refinement.
The first is that the errors are clearly separated and a quantification method is developed for each of the errors, including model shape errors and three typical numerical error sources.
NUMERICAL EXAMPLES
Introduction
Quantification of numerical errors, model shape errors, and UQ errors is demonstrated in this example.
Cantilever beam
- Numerical error estimation
- Sensitivity analysis of errors
- Model form error and UQ error estimation
A vertical thin strip centered at 0 on the horizontal axis is cut. from each of the scatterplots to calculate the sensitivity indices by the variance change method and the KL divergence method. In comparison, samples of model shape errors without accounting for the quantification error of uncertainty are also generated by ignoring the sampling errors in Eq. The CDFs of model shape errors with and without accounting for uncertainty quantification errors are plotted in Figure 12 (a).
By the method proposed in Section 3.4, the mean and variance of the mean of εmodel (including the effect of εuq ) are calculated to be 2.72e-3 and 3.18e-7, respectively; and the mean and variance of the variance of εmodel (including the effect of εuq ) are calculated to be 5.75e-2 and 1.48e-7, respectively. a): PDFs for model shape error (b): PDFs for sampling errors.
Crack growth in an airplane wing spar
- Structural analysis
- Crack growth analysis
- Correction of errors
- Results and sensitivity analysis
The standard deviation of εB is assumed equal to 30% of the value of B, and the standard deviation of εP is equal to 15% of the value of P. An approximate approach to overcome this obstacle is to assume that all samples of the same error have the same value throughout all cycles. This implies that in order to obtain a more accurate prediction of the final crack size, a more accurate parameter C (ie, narrower spread) is needed.
Also, some of the considered errors, such as experimental errors and solution errors, can be further broken down into different components, related to different experiments or different computational modules.
CONCLUSION
