THERMAL SCIENCE

International Scientific Journal

AN APPLICATION OF FINITE ELEMENT METHOD FOR A MOVING BOUNDARY PROBLEM

ABSTRACT
The Stefan problems called as moving boundary problems are defined by the heat equation on the domain 0 < x < s(t). In these problems, the position of moving boundary is determined as part of the solution. As a result, they are non-linear problems and thus have limited analytical solutions. In this study, we are going to consider a Stefan problem described as solidification problem. After using variable space grid method and boundary immobilization method, collocation finite element method is applied to the model problem. The numerical solutions obtained for the position of moving boundary are compared with the exact ones and the other numerical solutions existing in the literature. The newly obtained numerical results are more accurate than the others for the time step Δt = 0.0005, it is also seen from the tables, the numerical solutions converge to exact solutions for the larger element numbers.
KEYWORDS
PAPER SUBMITTED: 2017-06-13
PAPER REVISED: 2017-11-14
PAPER ACCEPTED: 2017-11-18
PUBLISHED ONLINE: 2018-01-07
DOI REFERENCE: https://doi.org/10.2298/TSCI170613268A
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2018, VOLUME 22, ISSUE Supplement 1, PAGES [S25 - S32]
REFERENCES
  1. Kutluay, S., Bahadir, A. R., Ozdes, A., The numerical solution of one-phase classical Stefan problem, J. Comp. Appl. Math., 81 (1997), pp. 35-44
  2. Esen, A. and Kutluay, S., An isotherm migration formulation for one-phase Stefan problem with a time dependent Neumann condition, Appl. Math. and Comput., 150 (2004), pp. 59-67
  3. Esen, A. and Kutluay, S., A numerical solution of the Stefan problem with a Neumann-type boundary condition by enthalpy method, Appl. Math. and Comput., 148 (2004), pp. 321-329
  4. Mitchell, S.L., Vynnycky, M., Finite-difference methods with increased accuracy and correct initialization for one-dimensional Stefan problems, Appl. Math. and Comput., 215 (2009), pp. 1609-1621
  5. Caldwell, J., Savovic, S., Numerical solution of Stefan problem by variable space grid and boundary immobilization method, J. Math. Sci., 13 (2002), pp. 67-79
  6. Karabenli, H., Ucar, Y. and Aksan, E.N, A Collocation Finite Element Solution for Stefan Problems with Periodic Boundary Conditions, Filomat, 30 (2016), pp. 699-709
  7. Karabenli, H., Esen, A. and Aksan, E.N., Numerical Solutions for a Stefan Problem, New Trends in Mathematical Sciences, 4 (2016), pp. 175-187
  8. Asaithambi, A., Numerical Solution of Stefan Problems Using Automatic Differentiation, Appl. Math. Comput., 189 (2007), pp. 943-948
  9. Ali, A.H.A., Gardner, G.A. and Gardner, L.R.T., A collocation solution for Burgers equation using cubic B-spline finite elements, Comput. Meth. Appl. Mech. Eng., 100 (1992), pp. 325-337
  10. Murray, W. D. and Landis, F., Numerical and Machine Solutions of Transient Heat Conduction Involving Melting or Freezing, J. Heat Transfer, 81 (1959), pp. 106-112
  11. Furzeland, R.M., A comparative study of numerical methods for moving boundary problems, J. Inst. Maths. Appl., 26 (1980), pp. 411-429
  12. Landau, H.G., Heat conduction in a melting solid, Quart. J. Appl. Math., 8 (1950), pp. 81-94
  13. Finn, W. D. L. and Varoglu, E., Finite Element Solution of the Stefan Problem, in: J.R. Whiteman (Ed.), The Mathematics of Finite Elements and Applications MAFELAP Acedemic Press, New York, 1979
  14. Rubinstein, L. I., The Stefan Problem, Trans. Math. Monographs AMS Providence Rhode Island, 1971

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