Abstract
An algorithm for the numerical solution of the optimal speed problem with phase constraint for a parabolic equation describing the heat conduction processes in inhomogeneous media is proposed. To solve the problems with the use of first-order optimization methods and finite differences on non-uniform grids, analytical formulas are obtained for the components of the gradient of the functional with respect to controllable functions. A method is proposed for selecting initial approximations for optimal controls and a step in time at each iteration, which makes it possible to accelerate the computation process. To achieve the specified accuracy, the speed problem requires 6 iterations and . Based on the analysis of the results of numerical experiments, the influence of various parameters on the iterative process is investigated and recommendations are developed on the use of the proposed algorithm. In optimal control problems, the total number of iterations in option a by the conditional gradient method is 110 and the gradient projection method is 108. In option b, the total number of iterations is CGM – 81, GPM – 64, i. e., the total number of iterations in the optimal control problem in method b the choice of the initial approximation is much less than in variant a. The optimal speed control, obtained by both methods, is close enough to test controls. Numerical experiments are also carried out in the case when the control-optimal controls have two switching points. However, the nature of the results obtained does not change. The proposed algorithm can be used to determine the optimal regime and time of thermal conductivity processes in inhomogeneous media.
Highlights
Speed problems are of great interest, when it is necessary to realize technologically desirable temperatures of a medium with the specified accuracy in the shortest possible time
Mathematical modeling of non-isothermal filtration processes of a homogeneous incompressible fluid in an inhomogeneous formation shows [3] that the determination of the temperature distribution in a reservoir in dimen sionless variables can be reduced to the solution of the boundary value problem: 1 ∂
Let’s note that many other physical processes [4] are described by the mathematical model (1)–(4), there fore, the developed numerical method can be applied to solving analogous speed problems for these processes
Summary
The object of research is an iterative numerical method for solving problems for optimal speed with phase constraint for equations of parabolic type with variable coefficients, descri bing the processes of thermal conductivity in porous media. Mathematical modeling of non-isothermal filtration processes of a homogeneous incompressible fluid in an inhomogeneous formation shows [3] that the determination of the temperature distribution in a reservoir in dimen sionless variables can be reduced to the solution of the boundary value problem:. The mathematical formulation of the problem consists in determining such controls g(t), v(t), f (x,t) and func tions u(x,t) satisfying conditions (1)–(4), with constraints: gmin ≤ g(t) ≤ gmax , vmin ≤ v(t) ≤ vmax , fmin ≤ f (t) ≤ fmax ,. Xc is performed in the minimum time T , gmin , gmax , vmin , vmax , fmin, fmax – specified numbers characterizing the limiting possibilities of thermal sources; umax – maximum permissible temperature value. Let’s note that many other physical processes [4] are described by the mathematical model (1)–(4), there fore, the developed numerical method can be applied to solving analogous speed problems for these processes
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