MODELING IN THE PROBLEM OF OPTIMIZING THE LOCATION OF HEATING SOURCES IN A NON-STATIONARY PROBLEM OF HEAT CONDUCTION, TAKING INTO ACCOUNT CONVECTION
Keywords:
non-stationary problems; optimal placement; heat sources; convection-diffusion equations; M-method.Abstract
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 limit of minimum and maximum temperatures due to the optimal placement of heat sources with a minimum power in the parallelepiped. 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 nonstationary 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). The numerical value of the Green's function is found in the form of a matrix. A new algorithm for the numerical solution of a nostationary 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 convection-diffusion equation given for the first boundary value problem is constructed. The boundary value problem is studied for the three-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.
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