Застосування інтерполяції кубічними сплайнами в інтелектуальній інформаційній системі прогнозування фазової стабільності твердих розчинів

Abstract

This research work is devoted to the use of cubic spline interpolation to solve the current problem of predicting the phase stability of solid solutions. The theory of isomorphic substitutions in Urusov crystals was used to calculate the mixing energies (interaction parameters) and critical decomposition temperatures (stability temperatures) of solid solutions. The proposed intelligent information system (IIS) automates the construction of diagrams, which allows to predict the thermodynamic stability of solid solutions. IIS also provides interactive features for user convenience. The obtained results can be useful in choosing the ratio of components in "mixed" matrices, the amount of activator in luminescent, laser and other practically important materials, as well as in matrices for immobilization of toxic and radioactive waste. The results of the application of interpolation on large segments, ie with a relatively large number of nodes, are often unsatisfactory. On the one hand, at large distances between nodes the interpolation accuracy decreases, and on the other hand, with increasing number of nodes due to the influence of high-order polynomials there are oscillations of the interpolation curve, because only in this way the curve can be forced to pass through given points. This state does not correspond to the real dependence resulting from the tabular values of the nodes. Therefore, it is proposed to use spline interpolation, which has a number of important advantages. First, it is high convergence. In contrast to Lagrange interpolation polynomials, the sequence of cubic splines on a uniform grid of nodes always converges to a continuous interpolated function. Secondly, we have minimal sensitivity to the error of the original data. Small changes in the values of the function at one or more adjacent interpolation points do not significantly affect the behavior of the spline at some distance from these points. As a consequence of the above - higher interpolation accuracy. As a result of the research the calculation of unknown coefficients for the proposed splines is formalized and the advantages of practical application of the proposed interpolation method are determined. The IIS of phase stability of solid solutions was improved using interpolation by cubic splines, as a result, the accuracy of the results was increased by 4.96%.