Quantification of Input Uncertainty Propagation Through Models of Aluminum Alloy Direct Chill Casting
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NTRODUCTION
THE direct chill (DC) casting process, depicted in Figure 1, is the most common method to produce wrought aluminum ingots and simulations are often used to gain insight into the complex nature of the process.[1–4] These predictions are typically reported as single point values, ignoring the uncertainty inherent in the modeling process. Quantifying uncertainty is important in the development of models and in the proper interpretation of their output. Leaving aside the uncertainties associated with discretization and solution methods in numerical models, there are two other types which might affect the reliability of model output.[5] The first is the epistemic uncertainty, which is a lack of complete knowledge of the system, limiting a completely accurate and robust representation of input parameters and physical behavior. This effect can be addressed by improving physical models, or by adding new physics to the model, usually guided by better or more complete experimental
KYLE FEZI is with the Fort Wayne Metals Research Products Corporation, 2300 East Cardinal Drive, Columbia City, IN 46725. Contact e-mail: [email protected] MATTHEW JOHN M. KRANE is with the School of Materials Engineering, Purdue Center for Metal Casting Research, Purdue University, 701 West Stadium Avenue, West Lafayette, IN 47906. Manuscript submitted March 21, 2018.
METALLURGICAL AND MATERIALS TRANSACTIONS A
observation. However, even if the physics of the process are well represented and the numerical techniques contribute little uncertainty, a modeler must still reckon with aleatoric uncertainty, which is the natural variability in measurements and random sampling of data. In this study, the focus is on quantifying the effect of aleatoric uncertainties on model predictions. One way to address these uncertainties is to perform a sensitivity study analyzing the response of numerical predictions to variations in select input parameters over a range.[1,6–8] This method is useful to gain a basic understanding of how various input parameters affect numerical predictions and their relative importance. It does not quantify the uncertainty of the model outputs, which determine the importance of the sensitivity analysis and confidence in the predictions. The most direct approach to aleatoric uncertainty quantification is through Monte Carlo methods.[9] This process generates probability density functions (PDFs) for output quantities by evaluating the numerical model over the entire input uncertainty range. Although Monte Carlo sampling methods are effective, they are also computationally intensive, as they may require many thousands or more numerical model evaluations. Even for simple, quickly solved models, this number can be daunting; with the more sophisticated numerical models, as referenced above or in this work, the computational expense is prohibitive. One way to reduce this cost is to limit the number of numerical simulations by constructing a surrogate function to replace the more complicated model. This surrogate is more computation
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