Geodesic
Numerical methods for solving optimal control for Stefan problems
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2016), pp. 87-100 Cet article a éte moissonné depuis la source Math-Net.Ru

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The article describes the mathematical model and a numerical method for the calculation and optimization of temperature fields with regard to phase transformations and the nonlinear material properties. It proposes a finite-difference method and a computer program that will effectively implement the computer simulation and optimization of thermal processes during melting and crystallization of the product. Direct Stefan problem was solved on the basis of one of the options through “enthalpic” method. The solution of the dual problem is found by smoothing the concentrated heat capacity and other parameters and characteristics of a feature such as a delta function. The article deals with a number of examples of optimization problems under various restrictions: minimizing energy consumption for melting the material, finding the maximum (minimum) temperature field, as well as two-sided estimate gradient of the solution at a given point in the area. In the above case, the functions of control are the source of the bulk power density, the values of which are located in a strip of arbitrary width. The results can be used in the practice of research and design in the field of metallurgy, electrical appliances, сryogenic etc. Refs 12. Figs 7. Tables 3.
Keywords: Stefan problem, optimal control, temperature field, enthalpy method, smoothing.
Mots-clés : phase transitions 
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S. A. Nekrasov; V. S. Volkov. Numerical methods for solving optimal control for Stefan problems. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2016), pp. 87-100. http://geodesic.mathdoc.fr/item/VSPUI_2016_2_a8/

[1] Vasil'ev F. P., Optimization methods, Publ. House “Faktorial Press”, M., 2002, 824 pp. (in Russian)

[2] Zubov V. I., The general method of Lagrange multipliers and optimization of processes in continuous media, Doct. dis., RAS, M., 2002, 250 pp. (in Russian)

[3] Albu A. F., Zubov V. I., “Research objectives of optimum control the process of crystallization agent at the new formulation for object complex geometric shapes”, J. of Calcul. mathematics and mathem. physics, 54:12 (2014), 1879–1893 (in Russian) | DOI | Zbl

[4] Gukasov A. K., Gukasova E. V., “The numerical solution of optimal control problem of the phase transition boundary”, Basic Research, 2014, no. 12–11, 2325–2329 (in Russian)

[5] Krektuleva R. A., Batranin A. V., “The joint solution inverse problem of heat conduction and the problem of optimal design at technology of welding with non-consumable electrode”, Bulletin of the Tomsk Polytechnic University, 320:2 (2012), 104–109 (in Russian) | MR

[6] Mel'nikova Ju. S., “Mathematical modeling of time-dependent temperature field in two-phase media”, Science and education, 2012, no. 2 (in Russian) (accessed: 12.02.2016)

[7] Buchko N. A., Enthalpy method of numerical solutions problems of heat conduction at freeze or thawing of soil, SPbGUNTiPT (in Russian) (accessed: 12.02.2016)

[8] Vasil'ev V. I., Maksimov A. M., Petrov E. E., Tsypkin G. G., “Mathematical model of the freezing-thawing of saline frozen soil”, Journal of Applied Mechanics and Technical Physics, 36:5 (1995), 689–696 | DOI | Zbl

[9] Nekrasov S. A., Interval and bilateral methods of calculation with guaranteed accuracy of electric and magnetic systems, Doct. dis., South-Russian State Politechnical University, Novocherkassk, 2002, 310 pp. (in Russian)

[10] Nekrasov S. A., “Modeling of phase transitions of the first kind by the method of integral equations in the case of a stationary moving surface source”, Engineering and Physical Journal, 66:6 (1994), 754–757 (in Russian) | MR

[11] Nekrasov S. A., “Stefan problem. Pt I”, Differential Equations, 32:8 (1996), 1114–1121 (in Russian) | MR | Zbl

[12] Nekrasov S. A., “Stefan problem. Pt II”, Differential Equations, 32:9 (1996), 1254–1258 (in Russian) | MR | Zbl