Research Papers: Melting and Solidification

Uncertainty Quantification in Modeling Metal Alloy Solidification

[+] Author and Article Information
Kyle Fezi

Purdue Center for Metal Casting Research,
School of Materials Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: kfezi@purdue.edu

Matthew John M. Krane

Purdue Center for Metal Casting Research,
School of Materials Engineering,
Purdue University,
West Lafayette, IN 47907
e-mail: krane@purdue.edu

Contributed by the Heat Transfer Division of ASME for publication in the JOURNAL OF HEAT TRANSFER. Manuscript received April 13, 2016; final manuscript received March 13, 2017; published online April 19, 2017. Assoc. Editor: Gennady Ziskind.

J. Heat Transfer 139(8), 082301 (Apr 19, 2017) (12 pages) Paper No: HT-16-1198; doi: 10.1115/1.4036280 History: Received April 13, 2016; Revised March 13, 2017

Numerical simulations of metal alloy solidification are used to gain insight into physical phenomena that cannot be observed experimentally. These models produce results that are used to draw conclusions about a process or alloy and often compared to experimental results. However, uncertainty in model inputs cause uncertainty in model results, which have the potential to significantly affect conclusions drawn from their predictions. As a step toward understanding the effect of uncertain inputs on solidification modeling, uncertainty quantification (UQ) and sensitivity analysis are performed on a transient model of solidification of Al–4.5 wt % Cu in a rectangular cavity. The binary alloy considered has columnar solidification morphology, and this model solves equations for momentum, temperature, and species conservation. UQ and sensitivity analysis are performed for the degree of macrosegregation and solidification time. A Smolyak sparse grid algorithm is used to select input values to construct a polynomial response surface fit to model outputs. This polynomial is then used as a surrogate for the complete solidification model to determine the sensitivities and probability density functions (PDFs) of the model outputs. Uncertain model inputs of interest include the secondary dendrite arm spacing (SDAS), heat transfer coefficient, and material properties. The most influential input parameter for predicting the macrosegregation level is the dendrite arm spacing, which also strongly depends on the choice of permeability model. Additionally, the degree of uncertainty required to produce accurate predictions depends on the outputs of interest from the model.

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Fig. 1

Schematic of the numerical domain showing the dimensions and thermal boundary conditions

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Fig. 2

Mushy zone permeability as a function of secondary dendrite arm spacing and fraction solid showing (a) three different permeability functions and (b) normalized permeability as a function of solid fraction [15]

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Fig. 3

Composition distributions of a statically cast Al–4.5 wt % alloy showing (a) the distribution as volume fraction of the ingot and (b) volume distribution function with a Weibull distribution fit and the composition specifications superimposed

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Fig. 4

Contour plots of the Cu distribution, showing counterclockwise rotating liquid flow (streamlines are thin black lines) and the mushy zone (solidus and liquidus are bold lines) at (a) 100 s, (b) 500 s, (c) 1000 s, (d) 1300 s, and (e) 1430 s

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Fig. 5

Final composition fields predicted with the three permeability models

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Fig. 6

Composition distributions of three permeability models plotted with the fit of the Weibull function and compositional specifications

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Fig. 7

Secondary dendrite arm spacing measurements showing (a) uncertainty in experimental measurements (data taken from Fig. 8 in Ref. [32]) and (b) input SDAS uncertainty

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Fig. 8

Predictions of volume fraction out of the compositional specification, from three different permeability models showing (a) the resulting uncertainty in the model predictions due to dendrite arm spacing uncertainty and (b) model predictions with the resulting surrogate model overlaid. The RMSE for each surrogate fit to the model predictions is 0.68% for KI, 4.34% for KII, and 1.66% for KIII.

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Fig. 9

Normalized Weibull deviation predictions from three different permeability models showing (a) predicted uncertainty distributions and (b) WCu surrogate models. The RMSEs for surrogate model fits are 0.58% for KI, 2.57% for KII, and 3.71% for KIII.

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Fig. 10

Surrogate models for each output of interest for the three different input uncertainty levels

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Fig. 11

Model outputs of interest predictions from three different input levels of uncertainty showing probability density functions of (a) the normalized Weibull deviation, (b) the volume fraction outside the composition specification range, and (c) the total solidification time

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Fig. 12

Mean OOI sensitivities (μ*) to the uncertain inputs for case B1. The error bars are ±2σ.

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Fig. 13

Plot of the solutal and thermal buoyancy contributions for equilibrium solidification of Al–4.5 wt % Cu

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Fig. 14

Probability density functions of the model outputs comparing the four different input uncertainty levels

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Fig. 15

Uncertainty in model predictions of the OOI for cases B1–B4

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Fig. 16

Plots of the outputs of interest sensitivities to the uncertain inputs showing μ* as the height of each bar and 2σ* are the error bars

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Fig. 17

Probability density functions of the model outputs for uncertain material properties, dendrite arm spacing, and heat transfer coefficient



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