Bayesian Uncertainty Quantification and Information Fusion in CALPHAD-based Thermodynamic Modeling

Abstract

Calculation of phase diagrams is one of the fundamental tools in alloy design---more specifically under the framework of Integrated Computational Materials Engineering. Uncertainty quantification of phase diagrams is the first step required to provide confidence for decision making in property- or performance-based design. As a manner of illustration, a thorough probabilistic assessment of the CALPHAD model parameters is performed against the available data for a Hf-Si binary case study using a Markov Chain Monte Carlo sampling approach. The plausible optimum values and uncertainties of the parameters are thus obtained, which can be propagated to the resulting phase diagram. Using the parameter values obtained from deterministic optimization in a computational thermodynamic assessment tool (in this case Thermo-Calc) as the prior information for the parameter values and ranges in the sampling process is often necessary to achieve a reasonable cost for uncertainty quantification. This brings up the problem of finding an appropriate CALPHAD model with high-level of confidence which is a very hard and costly task that requires considerable expert skill. A Bayesian hypothesis testing based on Bayes' factors is proposed to fulfill the need of model selection in this case, which is applied to compare four recommended models for the Hf-Si system. However, it is demonstrated that information fusion approaches, i.e., Bayesian model averaging and an error correlation-based model fusion, can be used to combine the useful information existing in all the given models rather than just using the best selected model, which may lack some information about the system being modelled.