519.6 Numerical modeling of the non-stationary problem of optimal placement of heat sources of minimum power in a homogeneous medium
Khayitkulov B. K. (National University of Uzbekistan)
HEAT EQUATION, NON-STATIONARY PROBLEMS, OPTIMAL PLACEMENT, HEAT SOURCES, IMPLICIT SCHEMAS, BIG M METHOD
doi: 10.18698/2309-3684-2024-1-5566
This work is devoted to the numerical solution of the non-stationary problem of optimal placement of heat sources of minimum power. The statement of the problem requires the simultaneous fulfillment of two conditions. The first condition is to ensure that the temperature is within the limits of minimum and maximum temperatures due to the optimal placement of heat sources with a minimum power in the rectangle. The second condition is that the total power of the heat sources used for heating is minimal. This problem was studied under stationary conditions in the works of other scientists. However, the problem was not considered in the non-stationary case. Since it is difficult to find a continuous solution to the boundary value problem, we are looking for a numerical solution to the problem. It is difficult to find an integral operator with a continuous kernel (Green's function). We find the numerical value of the Green's function in the form of a matrix. A new algorithm for the numerical solution of a non-stationary optimal control problem for the placement of heat sources with a minimum power in processes described by parabolic partial differential equations is proposed. A new technique for numerical solution is proposed. A mathematical and numerical model of the processes described by the heat conduction equation with constant coefficients given for the first boundary value problem is constructed. The boundary value problem is studied for the two-dimensional case. An implicit finite difference scheme was used to solve the problem numerically. According to this scheme, a system of difference equations was created. The formed system of difference equations is reduced to a linear programming problem. The problem of linear programming is solved using the M-method. For each time value, a linear programming problem is solved. A new approach to the numerical solution of problems is proposed. A general block diagram of the algorithm for solving the non-stationary problem of optimal control of the placement of heat sources with a minimum power is given. An algorithm and software for the numerical solution of the problem have been developed. A brief description of the software is given. On specific examples, it is shown that the numerical solution of the boundary value problem is within the specified limits, the sum of optimally placed heat sources with a minimum power gives a minimum to the functional. The results of the computational experiment are visualized.
Butkovskii A.G., Malyi S.A., Andreev Yu.N. Upravlenie nagrevom metalla [Metal heating control]. Moscow, Metallurgiya Publ., 1981, 272 p.
Lions J.L. Controle optimal de systemes gouvernes par des equations aux derivees partielles. Paris, Dunod, 1968, 416 p.
Egorov A.I. Optimal'noe upravlenie teplovymi i diffuzionnymi protsessami [Optimal control of thermal and diffusion processes]. Moscow, Nauka Publ., 1978, 464 p.
Fedorenko R.P. Priblizhennoe reshenie zadach optimal'nogo upravleniya [Approximate solution of optimal control problems]. Moscow, Nauka Publ., 1978, 488 p.
Osipov O.V., Brusentsev A.G. Optimal location of heat sources inside areas with complex geometric shapes. Mathematical Models and Computer Simulations, 2019, vol. 11, no. 6, pp. 905‒913.
Akhmetzyanov A.V., Kulibanov V.N. Optimal choice of coordinates for oil well drilling. Automation and Remote Control, 2002, vol. 63, no. 11, pp. 1699‒1706.
Kapjor A., Durcansky P., Vantuch M. Effect of heat source placement on natural convection from cylindrical surfaces. Energies, 2020, vol. 13, iss. 17, pp. 1‒13.
Hsu T.H., W Ang S.G. Mixed convection in a rectangular enclosure with discrete heat sources. Numerical Heat Transfer, Part A: Applications, 2010, vol. 38, iss. 6, pp. 627‒652.
Mirskay S.Yu., Sidelnikov V.I. Efficient heating of the room as the optimal control problem. Tehniko-tehnologicheskie problemy servisa [Technical and Technological Problems of Service], 2014, no. 4(30), pp. 75–78.
Sabdenov K.O., Baytasov T.M. Optimal (energy efficient) heat supply to buildings in central heating system. Izvestija Tomskogo politehnicheskogo universiteta. Inzhiniring georesursov [News of Tomsk Polytechnic University. Geo-Resource Engineering], 2015, vol. 326, no. 8, pp. 53–60.
Khaitkulov B.Kh. Homogeneous different schemes of the problem for optimum selection of the location of heat sources in a rectangular body. Solid State Technology, 2020, vol. 63, no. 17, pp. 583‒592.
Khayitkulov B.Kh. Conservative difference schemes for the optimal selection of the location of heat sources in the rod. Matematicheskoe modelirovanie i chislennye metody [Mathematical Modeling and Computational Methods], 2020, no. 3, pp. 85‒98.
Khayitkulov B.Kh. Finite-difference method for solving non-stationary problems of convection-diffusion control. Vestnik Tomskogo gosudarstvennogo universiteta. Upravlenie, vychislitel'naya tekhnika i informatika, 2021, no. 57, pp. 45‒52.
Tukhtasinov M.T., Abduolimova G.M., Khayitkulov B.Kh. Boundary control of heat propagation in a bounded body. Bjulleten' Instituta matematiki [Bulletin of the Institute of Mathematics], 2019, no. 1, pp. 1‒10.
Egorov A.I., Znamenskaya L.N. Control of a heat conduction process with a quadratic cost functional. Computational Mathematics and Mathematical Physics, 2017, vol. 57, no. 12, pp. 2005‒2016.
Dantzig G.B. Linear programming and extensions. Princeton, Princeton University Press, 2016, 656 p.
Хайиткулов Б. Х. Численное моделирование нестационарной задачи оптимального размещения источников тепла минимальной мощности в однородной среде. Математическое моделирование и численные методы, 2024, № 1, с. 55–66.
Работа выполнена при финансовой поддержке Узбекского фонда фундаментальных исследований (проект ОТ-Ф4-33).
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