Quantification of Epistemic Uncertainty in Grain Attachment Models for Equiaxed Solidification
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any model intended to represent some physical system, there are uncertainties in both the values of the appropriate input parameters and the assumptions used in constructing the model itself. These different sources of uncertainty may be broadly categorized into two types: epistemic and aleatoric.[1] In many cases, different models are proposed for a specific physical phenomenon, each making use of different assumptions and representing the underlying governing equations in different ways. Epistemic uncertainties, also called reducible uncertainties, may be theoretically reduced by gaining additional knowledge of a system. For example, improved measurements of a material property or experimental validation sufficiently support the use of one model type over another. The second type is aleatoric, or irreducible, uncertainty. Aleatoric uncertainties are natural, random variations in inputs. They cannot be reduced given more knowledge of the system. An example is the uncertainty in the value of a given
A. PLOTKOWSKI is with the Purdue Center for Metal Casting Research, School of Materials Engineering, Purdue University, West Lafayette, IN, 47907, and with the Mechanical, Aerospace and Biomedical Engineering Department, University of Tennessee, Knoxville, TN, 37996. Contact e-mail: [email protected] M.J.M. KRANE is with the Purdue Center for Metal Casting Research, School of Materials Engineering, Purdue University. Manuscript submitted September 17, 2016. METALLURGICAL AND MATERIALS TRANSACTIONS B
input due to the error inherent to a particular measurement technique. The practice of understanding how these different sources of uncertainty affect the output of a model is called uncertainty quantification. Due to the ever present nature of input uncertainties, UQ has become increasingly common in fields such as heat transfer,[2] fluid mechanics,[3,4] and materials[1] modeling. To this point, however, UQ has only gained limited adoption in the solidification community.[5–8] Hardin et al.[5] used UQ for design optimization in a simple casting by predicting the probability of failure for a particular riser size, and iterating to reduce cost while minimizing casting defects. Fezi and Krane[6,9] used a non-intrusive UQ method to evaluate the uncertainty in a series of solidification models of increasing complexity, beginning with a one-dimensional wire casting model, and advancing to two-dimensional models for static castings and direct chill casting. These examples primarily considered the effect of aleatoric uncertainties (although Fezi and Krane[6] considered the effect of three different permeability models for the mushy zone). The purpose of the present work is to further demonstrate how UQ may be used within the solidification field to understand the present state of solidification modeling, to improve comparisons to experimental data, and to accelerate model development and guide corresponding experimental work. In particular, cases in which multiple possible models may be used for the same physical phenomenon are of interest. To this e
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