Simulation of dendritic crystal growth with thermal convection (original) (raw)

Abstract

The dendritic growth of crystals under gravity influence shows a strong dependence on convection in the liquid. The situation is modelled by the Stefan problem with a Gibbs-Thomson condition coupled with the Navier-Stokes equations in the liquid phase. A finite element method for the numerical simulation of dendritic crystal growth including convection effects is presented. It consists of a parametric finite element method for the evolution of the interface, coupled with finite element solvers for the heat equation and Navier-Stokes equations in a time dependent domain. Results from numerical simulations in two space dimensions with Dirichlet and transparent boundary conditions are included.

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References (22)

  1. ANANTH, R. & GILL, W. N. Dendritic growth with convection. J. Cryst. Growth 91, (1988) 587-598.
  2. ANANTH, R. & GILL, W. N. Self-consistent theory of dendritic growth with convection. J. Cryst. Growth 108, (1991) 173-189.
  3. B ÄNSCH, E. Local mesh refinement in 2 and 3 dimensions. Impact Comput. Sci. Eng. No. 3, (1991) 181-191.
  4. B ÄNSCH, E. Adaptive finite-element techniques for Navier-Stokes equations and other transient problems. In Adaptive Finite and Boundary Elements. Computational Mechanics Publication's, Boston, co-published with Elsevier Applied Science, London (1993).
  5. B ÄNSCH, E. Simulation of instationary, incompressible flows. Acta Math. Univ. Comenianae 67, (1998) 101-114.
  6. B ÄNSCH, E. & SCHMIDT, A. A finite element method for dendritic growth. In TAYLOR, J. E. (ed.), Computational Crystal Growers Workshop, AMS Selected Lectures in Mathematics, pp. 16-20, Providence, RI. American Mathematical Society (1992).
  7. FIG. 16. Problem 4.2.1; interface and isothermal lines at t = 2.0.
  8. BRISTEAU, M. GLOWINSKI, R., & PERIAUX, J. Numerical methods for the Navier-Stokes equations. Application to the simulation of compressible and incompressible flows. Comput. Phys. Rep. 6, (1987) 73-188.
  9. CANTOR, B. & VOGEL, A. Dendritic solidification and fluid flow. J. Cryst. Growth 41, (1977) 109-123.
  10. DZIUK, G. An algorithm for evolutionary surfaces. Numer. Math. 58, (1991) 603-611.
  11. ERIKSSON, K. & JOHNSON, C. Adaptive finite element methods for parabolic problems I: A linear model problem. SIAM J. Numer. Anal. 28, (1991) 43-77.
  12. GLICKSMAN, M. E. & HUANG, S. C. Convective heat transfer during dendritic growth. In ZIEREP J. & ORTEL, (eds), Convective Transport and Instability Phenomena, pp. 557-574. Braun, Karlsruhe (1982).
  13. GLICKSMAN, M. E., KOSS, M. B., & WINSA, E. A. Dendritic growth velocities in microgravity. Phys. Rev. Lett. 73, (1994) 573-576.
  14. GLICKSMAN, M. E. & MARSH, S. P. The dendrite, in Handbook of Cryst. Growth, D. T. J. Hurle, ed., vol. 1, North-Holland, Amsterdam, 1993.
  15. GLOWINSKI, R., PAN, T.-W., KEARSLEY, A., & PERIAUX, J. Numerical simulation and optimal shape for viscous flow by a fictitious domain method. Int. J. Numer. Methods Fluids 20, (1995) 695-711.
  16. GLOWINSKI, R., PAN, T.-W., & PERIAUX, J. A fictitious domain method for external incompressible viscous flow modeled by Navier-Stokes equations. Comput. Methods Appl. Mech. Eng. 112, (1994) 133-148.
  17. GRIEBEL, M., MERZ, W., & NEUNHOEFFER, T. Mathematical modelling and numerical simulation of freezing processes of a supercooled melt under consideration of density changes. Preprint 513, SFB 256, Bonn (1997).
  18. HEYWOOD, J. G., RANNACHER, R., & TUREK, S. Artificial boundaries and flux and pressure conditions for the incompressible Navier-Stokes equations. Int. J. Numer. Methods Fluids 22, (1996) 325-352. FIG. 17. Problem 4.2.2; interfaces after 7, 17, . . . , 157 time steps. FIG. 18. Problem 4.2.2; triangulation at t = 1.57.
  19. LANGER, J. S. Instabilities and pattern formation in crystal growth. Rev. Mod. Phys. 52, (1980) 1-28.
  20. MITCHELL, W. A comparison of adaptive refinement techniques for elliptic problems. ACM Trans. Math. Softw. 15, (1989) 326-347.
  21. SCHMIDT, A. Computation of three dimensional dendrites with finite elements. J. Comput. Phys. 125, (1996) 293-312.
  22. SCHMIDT, A. Approximation of crystalline dendrite growth in two space dimensions. Acta Math. Univ. Color plate 1: Problem 4.2.1; temperature, velocity, and interface at time t = 2.0. Color plate 2: Problem 4.2.2; temperature, velocity, and interface at time t = 1.57.