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Research Papers: Design Automation

A Constraint Satisfaction Algorithm for the Generalized Inverse Phase Stability Problem

[+] Author and Article Information
Edgar Galvan

Design Systems Laboratory,
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: e_galvan@tamu.edu

Richard J. Malak

Associate Professor
Design Systems Laboratory,
Department of Mechanical Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: rmalak@tamu.edu

Sean Gibbons

Computational Materials Science Lab,
Department of Materials Science
and Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: sean.l.gibbons@gmail.com

Raymundo Arroyave

Associate Professor
Computational Materials Science Lab,
Department of Materials Science
and Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: rarroyave@tamu.edu

1Corresponding author.

Contributed by the Design Automation Committee of ASME for publication in the JOURNAL OF MECHANICAL DESIGN. Manuscript received March 21, 2016; final manuscript received July 25, 2016; published online October 5, 2016. Assoc. Editor: Carolyn Seepersad.

J. Mech. Des 139(1), 011401 (Oct 05, 2016) (11 pages) Paper No: MD-16-1227; doi: 10.1115/1.4034581 History: Received March 21, 2016; Revised July 25, 2016

Researchers have used the (calculation of phase diagram) CALPHAD method to solve the forward phase stability problem of mapping from specific thermodynamic conditions (material composition, temperature, pressure, etc.) to the associated phase constitution. Recently, optimization has been used to solve the inverse problem: mapping specific phase constitutions to the thermodynamic conditions that give rise to them. These pointwise results, however, are of limited value since they do not provide information about the forces driving the point to equilibrium. In this paper, we investigate the problem of mapping a desirable region in the phase constitution space to corresponding regions in the space of thermodynamic conditions. We term this problem the generalized inverse phase stability problem (GIPSP) and model the problem as a continuous constraint satisfaction problem (CCSP). In this paper, we propose a new CCSP algorithm tailored for the GIPSP. We investigate the performance of the algorithm on Fe–Ti binary alloy system using ThermoCalc with the TCFE7 database against a related algorithm. The algorithm is able to generate solutions for this problem with high performance.

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Copyright © 2017 by ASME
Topics: Stability , Algorithms
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References

Figures

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Fig. 1

Illustration of a CCSP. The function f(·) represents the nonlinear CALPHAD model that maps the thermodynamic condition space (a) to the phase constitution space (b). The query C=(C1,C2,…,C4) is defined by the user in the phase constitution space, e.g., C4≡(fβ≥c), where c is a constant. Due to the nonlinear mapping between the thermodynamic conditions space and the phase constitution space, the solution in the thermodynamic conditions space is nonconvex.

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Fig. 2

Illustration of the SVDD method for two-dimensional training data. The training data depicted in (a) are implicitly mapped to (b), an N-D feature space where a hypersphere is a good representation of the population members. The hypersphere is defined by a centroid b and radius r. Equation (9) is a test to determine whether a new point z is inside the domain description. This test defines the boundary illustrated in (c).

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Fig. 3

Comparison of SVM and SVDD on a simple classification problem, where the sampling is balanced (a) and imbalanced (b). The shading indicates the true classification, and the points are the training data.

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Fig. 4

Illustrative example depicting a possible step size and resulting endpoint from a given initial point along the boundary of the SVDD M. An aim in selecting a step size is to cross the boundary of the satisfactory region.

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Fig. 5

Phase diagram for Fe–Ti binary alloy system. The shaded regions correspond to the solution sets for test cases 1 and 2.

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Fig. 6

Algorithmic performance on test case 1 as measured by (a) precision, (b) recall, and (c) misclassification. Error bars indicate the mean values and 95% confidence interval of 30 trials.

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Fig. 7

Algorithmic performance on test case 2 as measured by (a) precision, (b) recall, and (c) misclassification. Error bars indicate the mean values and 95% confidence interval of 30 trials.

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Fig. 8

Illustration of the progression of each algorithm at the function evaluations for test case 1. The top row corresponds to the progression of the GIPSP algorithm, and the bottom row corresponds to the EDSD algorithm. The satisfactory region is shaded, the samples are the points, and the classification boundaries are the outlines.

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Fig. 9

Illustration of the progression of each algorithm at the (a) 100, (b) 250, and (c) 400, function evaluations for test case 2. The top row corresponds to the progression of the GIPSP algorithm, and the bottom row corresponds to the EDSD algorithm. The satisfactory region is shaded, the samples are the points, and the classification boundaries are the outlines.

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Fig. 10

Illustration of a single trial of the EDSD algorithm at the 400 function evaluations for test case 2. The satisfactory region is shaded, the samples are the points, and the classification boundaries are the outlines. The results in this trial illustrate a possible overestimation of the EDSD algorithm in regions of the search space far from the true solution.

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