Abstract
In this paper, we present the numerical solution of the Stefan problem to calculate the temperature of the tungsten sample heated by the laser pulse. Mathematical modeling is carried out to analyze field experiments, where an instantaneous heating of the plate to 9000K is observed due to the effect of a heat flow on its surface and subsequent cooling. The problem is characterized by nonlinear coefficients and boundary conditions. An important role is played by the evaporation of the metal from the heated surface. Basing on Samarskii’s approach, we choose to implement the method of continuous counting considering the heat conductivity equation in a uniform form in the entire domain using the Dirac delta function. The numerical method has the second-order of approximation with respect to space, the interval of smoothing of the coefficients is 5K. As a result, we obtain the temperature distributions on the surface and in the cross section of the sample during cooling.
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References
D. Apushkinskaya, Free Boundary Problems. Regularity Properties Near the Fixed Boundary, Springer, Cham (2018).
A. S. Arakcheev, D. E. Apushkinskaya, I. V. Kandaurov, A. A. Kasatov, V. V. Kurkuchekov, G. G. Lazareva, A. G. Maksimova, V. A. Popov, A. V. Snytnikov, Yu. A. Trunev, A. A. Vasilyev, and L. N. Vyacheslavov, “Two-dimensional numerical simulation of tungsten melting under pulsed electron beam,” Fusion Eng. Design, 132, 13–17 (2018).
R. V. Arutyunyan, “Integral equations of the Stefan problem and their application in modeling of soil thawing,” In: Science and Education, MGTU, Moscow, No. 10, 347–419 (2015).
P. V. Breslavskiy and V. I. Mazhukin, “Algorithm for numerical solution of the hydrodynamic version of the Stefan problem using dynamically adapting grids,” Mat. Model., 3, No. 10, 104–115 (1991).zz
B. M. Budak, E. N. Solov’eva, and A. B. Uspenskiy, “Difference method with smoothing of coefficients for solving Stefan’s problems,” Zhurn. Vych. Mat. i Mat. Fiz., 5, No. 5, 828–840 (1965).
L. A. Caffarelli, “The smoothness of the free surface in a filtration problem,” Arch. Ration. Mech. Anal., 63, 77–86 (1976).
L. A. Caffarelli, “The regularity of elliptic and parabolic free boundaries,” Bull. Am. Math. Soc., 82, 616–618 (1976).
L. A. Caffarelli, “The regularity of free boundaries in higher dimensions,” Acta Math., 139, No. 3-4, 155–184 (1977).
H. Chen, C. Min, and F. Gibou, “A numerical scheme for the Stefan problem on adaptive Cartesian grids with supralinear convergence rate,” J. Comp. Phys., 228, 5803–5818 (2009).
J. W. Davis and P. D. Smith, “ITER material properties handbook,” J. Nucl. Mater., 233, 1593–1596 (1996).
G. Duvaut, “Résolution d’un problème de Stefan (fusion d’un bloc de glace à zéro degré),” C. R. Math. Acad. Sci. Paris, 276, 1461–1463 (1973).
G. Duvaut, “Two phases Stefan problem with varying specific heat coefficients,” An. Acad. Brasil. Ciênc., 47, 377–380 (1975).
A. Friedman and D. Kinderlehrer, “A one phase Stefan problem,” Indiana Univ. Math. J., 25, No. 11, 1005–1035 (1975).
R. Groot, “Second order front tracking algorithm for Stefan problem on a regular grid,” J. Comp. Phys., 372, 956–971 (2018).
C. Y. Ho, R. W. Powell, and P. E. Liley, “Thermal conductivity of elements,” J. Phys. Chem. Ref. Data, 1, 279 (1972).
J. M. Huang, M. Shelley, and D. Stein, “A stable and accurate scheme for solving the Stefan problem coupled with natural convection using the immersed boundary smooth extension method,” J. Comp. Phys., 432, 110162 (2021).
Y. Ichikawa and N. Kikuchi, “A one-phase multidimensional Stefan problem by the method of variational inequalities,” Internat. J. Numer. Methods Engrg., 14, 1197–1220 (1979).
Y. Ichikawa and N. Kikuchi, “Numerical methods for a two-phase Stefan problem by variational inequalities,” Internat. J. Numer. Methods Engrg., 14, 1221–1239 (1979).
M. Yu. Laevskiy and A. A. Kalinkin, “Two-temperature model of a hydrate-bearing rock,” Mat. Model., 22, No. 4, 23–31 (2010).
G. Lamé and B. P. Clapeyron, “Mémoire sur la solidification par refroidissement d’un globe solide,” Ann. Chem. Phys., 47, 250–256 (1831).
G.G. Lazareva, A. S.Arakcheev, I.V. Kandaurov, A. A. Kasatov, V. V. Kurkuchekov, A. G. Maksimova, V. A. Popov, A. A. Shoshin, A. V. Snytnikov, Yu. A. Trunev, A. A. Vasilyev, and L. N. Vyacheslavov, “Calculation of heat sink around cracks formed under pulsed heat load,” J. Phys. Conf. Ser., 894, 012120 (2017).
A. M. Oberman and I. Zwiers, “Adaptive finite difference methods for nonlinear elliptic and parabolic partial differential equations with free boundaries,” J. Sci. Comput., 68, 231–251 (2012).
G. Pottlacher, “Thermal conductivity of pulse-heated liquid metals at melting and in the liquid phase,” J. Noncrystal. Solids, 250, 177–181 (1999).
A. A. Samarskii and B. D. Moiseenko, “Economical pass-through numerical scheme for multidimensional Stefan problem,” Zhurn. Vych. Mat. i Mat. Fiz., 5, No. 5, 816–827 (1965).
A. A. Samarskii and P. N. Vabishchevich, Computational Heat Transfer [in Russian], URSS, Moscow (2003).
J. Stefan, “Über die Theorie der Eisbildung, insbesondere über die Eisbildung im Polarmeere,” Sitzungsber. Österreich. Akad. Wiss. Math. Naturwiss. Kl. Abt. 2, Math. Astron. Phys. Meteorol. Tech., 98, 965–983 (1889).
S. G. Taluts, “Experimental study of thermophysical properties of transition metals and ironbased alloys at high temperatures,” Doctoral Thesis, Ekaterinburg, 2001.
L. Vyacheslavov, A. Arakcheev, A. Burdakov, I. Kandaurov, A. Kasatov, V. Kurkuchekov, K. Mekler, V. Popov, A. Shoshin, D. Skovorodin, Y. Trunev, and A. Vasilyev, “Novel electron beam based test facility for observation of dynamics of tungsten erosion under intense ELM-like heat loads,” AIP Conf. Proc., 1771, 060004 (2016).
Z.-C. Wu and Q.-C. Wang, “Numerical approach to Stefan problem in a two-region and limited space,” Thermal Sci., 16, No. 5, 1325–1330 (2012).
N. N. Yanenko, Fractional Steps Method for Solving Multidimensional Problems of Mathematical Physics [in Russian], Nauka, Novosibirsk (1967).
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 67, No. 3, In honor of the 70th anniversary of Professor V. M. Filippov, 2021.
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Apushkinskaya, D.E., Lazareva, G.G. Algorithm for the Numerical Solution of the Stefan Problem and its Application to Calculations of the Temperature of Tungsten Under Impulse Action. J Math Sci 278, 225–236 (2024). https://doi.org/10.1007/s10958-024-06916-5
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DOI: https://doi.org/10.1007/s10958-024-06916-5