К решению нестационарных нелинейных граничных обратных задач теплопроводности

Abstract

Для решения нелинейной граничной обратной задачи теплопроводности применяется метод регуляризации А. Н. Тихонова с эффективным алгоритмом поиска регуляризирующего параметра. Искомый тепловой поток на границе по временной координате аппроксимируется сплайнами Шёнберга. Применяется метод функций влияния, для чего нелинейная задача сводится к последовательности линейных обратных задач.

Для розв’язання нелінійної оберненої граничної задачі теплопровідності застосовується метод регуляризації А. М. Тихонова з ефективним алгоритмом пошуку регуляризуючого параметра. Шуканий тепловий потік на границі по часовій координаті апроксимуєmьcя сплайнами Шьонберга. Застосовується метод функцій впливу, для чого нелінійна задача зводиться до послідовності лінійних обернених задач.

To solve the nonlinear boundary inverse heat conduction problem, two approaches are used with the regularizing method of A. N. Tikhonov, for which an effective algorithm for finding the regularizing parameter has been developed. The required functions with respect to the time coordinate are approximated by Schoenberg splines and the boundary inverse problem is reduced to the determination of the approximation coefficients. In the first approach, the temperature function is replaced by two terms of the Taylor series, depending on the approximation parameters. In this case, one must calculate the partial derivatives of the temperature function with respect to all the approximation parameters. Because of the very complicated dependence of the temperature function on the approximation parameters, the partial derivatives must be calculated using the finite difference method, which ultimately leads to the need to solve for each parameter an additional direct problem at each step of the iteration process. This leads to additional computational costs. The second approach uses the influence function method for the linearized mathematical model of the thermal process. This approach allows us to significantly reduce the time of the solution of the problem, but at the first steps of the iterative process it is necessary to take into account that the temperature field is still far from the true state and the nonlinear thermophysical characteristics that depend on this state are still far from the true values. In conclusion, it should be noted that the first approach is more universal, but for a large number of parameters leads to large computational costs. For the second approach, computational costs do not increase as much as for the first approach, but it can be used only for solving boundary inverse heat conduction problems. From this it can be concluded that for the solution of multidimensional inverse problems these two approaches are desirable to be combined.