Polynomial regression and interpolation of thermodynamic data in Al–Si–Mg–Fe system
Introduction
Solidification simulations of commercial aluminum and iron alloys play an important role in manufacturing of automobile, aerospace and architecture materials due to their ability to predict macrosegregation, dendritic growth and mechanical properties [1], [2], [3], [4]. During recent years, the CALculation of PHAse Diagrams (CALPHAD) method has been proved to be a powerful tool in making accurate predictions of the phase equilibria transition in solidification processes. It is possible to calculate solidification processes in detail by coupling CALPHAD software with different solidification models [5], [6], [7], [8]. However, the large physical memory it takes when performing repetitive calculations for complex phase equilibria makes it less efficient to simulate the solidification processes on multicomponent alloy systems with macroscopic models [9].
Several techniques for addressing this issue via indirect coupling of phase equilibrium data with Thermo-Calc software have been developed. Dore et al. [10] proposed a mapping technique, in which the phase equilibrium data of Al–Si–Mg system was combined with CALPHAD calculations. Equilibrium information at each time step during the solidification simulation was obtained for the corresponding composition condition via bilinear interpolation. Du et al. [11], [12] applied this method to multicomponent Al–Cu–Mg alloys, making the content data stored in a tabulation file. While many researchers employed regression method only for the calculation of thermodynamic information of alloys with set composition [9], [13], [14], [15], [16], this technique has been adopted to calculating liquid phase fractions and solute concentrations in multicomponent systems. Notably, Ludwig et al. [17] proposed a model which contains a non-linear system of equations of interpenetration continua to simulate solidification process while coupled with a thermodynamic program. Zhao et al. [18] developed a regression model to compute solute partition coefficients and applied it to solidification simulations for the Al–Cu–Si alloys.
In this work, the authors aim to simplify the thermodynamic descriptions by using parameterized phase diagrams to enable multicomponent solidification simulations in complex macrosegregation simulations. A software package has been developed to extract phase diagram data from the thermodynamic database and store the information in a tabulation file. Both the data structure used and the method of parameterization of the thermodynamic phase diagram are analyzed. Thermodynamic calculations are performed by applying the TQ-interface from Thermo-Calc [19], [20], which is linked to the software by using a shared library. The proposed computational strategy is a combination of polynomial regression and interpolation [21], [22] due to their much less time-consuming than the Gibbs minimization computations [9], [18]. While previous works [10], [11], [12], [18] using mapping techniques and regressions have successfully modeled the solidification of ternary systems with reduced computational time, literature information on data reduction for quaternary systems is still scarce. Therefore, the A356 alloy of Al–Si–Mg–Fe quaternary system with an established thermodynamic database was employed as an example to gain a thorough understanding of the capability of the present data reduction method in modeling the solidification process in this type of systems.
Section snippets
Numerical model for describing phase diagrams
The numerical model used in the present work is based on polynomial fitting with an incorporated bilinear interpolation [21], [22]. This method is much less computationally-intensive than the minimizing of the Gibbs energy functions, which results in a significantly smaller physical memory. The regression modeling is performed by using Matlab 7.0 software [23], which can both analyze data and fit various regression functions such as polynomial functions, power functions and exponential
Thermodynamic calculations
A FORTRAN program was developed to perform the calculations in the present model. The proposed method in this work was validated by comparing the results for local thermodynamic information as well as Scheil solidification of different compositions of the A356 alloy in the Al–Si–Mg–Fe quaternary system with the calculations by Thermo-Calc software. The calculation of binary and ternary phase data that were not validated as similar computations can be found in the literature [9], [10], [11], [12]
Conclusions
An optimized method for predicting both the liquidus and the phase equilibria has been developed. It is proposed that the thermodynamic database of A356 alloys in Al–Si–Mg–Fe system can be calculated using polynomial regression and interpolation. The main inputs to the model are alloy compositions and temperature, and the present method is proven to be computationally efficient.
This technique is validated through comparison between the present results with those calculated by the Thermo-Calc
Acknowledgments
The authors would like to gratefully acknowledge the GI Institute RWTH for supporting this work and sincerely thank Dr. Björn Pustal for valuable discussions about this project, as well as his intellectual contribution and support through the course of this research. Gratefully thanks are also due to Prof. Dr. Yong Du for his constructive advices and discussions.
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