Sensitivity Analysis and Formulation of the Inverse Problem in the Stochastic Approach to Modelling of Phase Transformations in Steels

Abstract

The need for a reliable prediction of the distribution of microstructural parameters in metallic materials after processing was the motivation for this work. The model describing phase transformations, which considers the stochastic character of the nucleation of the new phase, was formulated. Numerical tests of the model, including sensitivity analysis, were performed and the optimal parameters such as time step, kind of the random numbers generator (RNG) and the number of the Monte Carlo points were determined. The validation of the model requires an application of proper coefficients corresponding to the considered materials. These coefficients have to be identified through the inverse analysis, which, on the other hand, uses optimization methods and requires the formulation of the appropriate objective function. Since the model involves stochastic parameters, it is a crucial task. Therefore, in the second part of the paper, a specific form of the objective function for the inverse analysis was developed. In the first approach, an objective function based on measurements of the average parameters was used and primary optimization was performed. Various optimization methods were tested. In the second approach, the hybrid objective function, which combined measured average transformation temperatures with a measure based on histograms, was used. Since, at this stage, we do not have measurements of the distribution of microstructural features, the basic histograms were generated by the model with the coefficients obtained in the first step of the optimization. The capability of finding the optimal solution for different starting points was evaluated and various approaches were compared. The elaborated original stochastic approach to modelling the phase transformations occurring during cooling after hot forming was validated on selected carbon steel.