The methods of estimating the accuracy and stability of the algorithm for determination of values of the parameters in superplasticity models

The problem of determination of the parameters values in systems of equations, describing the superplastic deformation of materials, is considered in the paper. The classifications are carried out for systems of phenomenological equations and for the sets of experimental data commonly used in superplastic applications. We discuss typical computational features of superplasticity models and the connected problems of the approximation algorithms. Some approaches are proposed to produce more universal algorithms for determination of values for the material constants. The scheme of the algorithm for the parameters approximation as applied to the superplasticity problems, based on the least-squares technique, is suggested. The algorithm proposed is general enough, extensible and applicable to different phenomenological models. One of the main results of the paper is the introduction of the metrics for the estimation of the accuracy of reproducing the experimental data with the phenomenological model under consideration (or of the divergence for the results obtained on the base of different models). In terms of the introduced metrics, the methodology is formulated for the verification of the predictive power of the phenomenological models with the material constants determined on the base of some approximation algorithm. This technique can also be applied for the estimation of the stability of the approximation algorithm to the random errors in the experimental data when some determined phenomenological model is used. On the base of the proposed metrics for the estimation of the accuracy two models, included different constitutive relations and microstructure evolution equations are compared. It is shown that the proposed algorithm is stable, and the phenomenological models with the parameter obtained by use of this algorithm can predict correctly the superplastic behaviour beyond the area of the experimental data in use. The influence of the selection of the set of experimental curves and of the digitization pattern of data on the approximation error is described. It is shown that the capacity of the phenomenological model to reproduce the specific set of experimental data also can be estimated with the help of the metrics proposed.