Some analytical solutions in problems of optimization of variable thermal conductivity coefficient

A. O. Vatulyan ; S. A. Nesterov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 3, pp. 33-46

Résumé

New formulations and solutions to problems of optimization of a variable thermal conductivity coefficient for an inhomogeneous pipe and a flat wall with mixed boundary conditions are presented. The quality functionals are either the average temperature or the maximum temperature, and as a limitation – either the condition of constancy of the integral thermal conductivity coefficient, or a priori information about the change in the thermal conductivity coefficient in a known range. To solve problems for a pipe, two optimization methods are used: 1) a variational approach based on the introduction of conjugate functions and the construction of an extended Lagrange functional; 2) Pontryagin’s maximum principle. To solve the optimization problem for a flat wall under the assumption of weak material inhomogeneity, the expansion method in terms of a small physical parameter is used. As the fourth problem, optimization of the variable thermal conductivity coefficient of a non-uniform flat wall with boundary conditions of the first kind is considered. The solution to a singular optimization problem is found among broken extremals. Using specific examples, a comparison was made of the values of minimized functionals for bodies with a constant thermal conductivity coefficient and an optimal variable coefficient. The gain from optimization is estimated.