Багатопараметрична ідентифікація теплофізичних характеристик шляхом розв’язання внутрішньої оберненої задачі теплопровiдностi

Approaches to the identification of thermophysical characteristics, using methods for solving inverse heat conduction problems and A. N. Tikhonov’s regularization method, are developed. According to the results of the experiment, temperature-dependent coefficients of heat conductivity, heat capacity, and internal heat sources are determined. In this case, the thermophysical characteristics are approximated by Schoenberg’s cubic splines, as a result of which their identification reduces to determining unknown coefficients in the approximated dependencies. Therefore, the temperature in the body will depend on these coefficients, and it can be represented using two members of the Taylor series as a linear combination of its partial derivatives with respect to the unknown coefficients, multiplied by the increments of these coefficients. Substituting this expression into the Tikhonov functional and using the minimum property of the quadratic functional, we can reduce the solution of the problem to the solution of a system of linear equations with respect to the increments of unknown coefficients. By choosing a certain regularization parameter and some functions as an initial approximation, we can implement an iterative process in which the vector of unknown coefficients for the current iteration will be equal to the sum of the vector of the coefficients obtained in the previous iteration and the coefficient increment vector as a result of solving a system of linear equations. Such an iterative process of identifying the thermophysical characteristics for each regularization parameter makes it possible to determine the mean-square discrepancy between the resulting temperature and the temperature measured as a result of the experiment. It remains to choose the regularization parameter so that this discrepancy is within the root-mean-square measurement error. Such a search, for example, is identical to algorithms for searching roots of nonlinear equations. When checking the efficiency of using the proposed method, a number of test problems were solved for bodies with known thermophysical characteristics. An analysis of the influence of random measurement errors on the error of the identifiable thermophysical characteristics of the body being studied was carried out.