Numerical Simulation of the Single Crystal Growth Process
by
Vladimir P. Ginkin
SSC IPPE, Bondarenko sq.1, Obninsk, Kaluga region, 249020, Russia.

There are many papers on the numerical simulation of the single crystal growth process (see, for example, (1) and all the references presented there). At the same time, there is no established notion to single out the most efficient approach or algorithm for solving this problem now. An approach developed at the Institute of Physics and Power Engineering in Obninsk is described in the paper. The approach was originally proposed within the framework of a 2D conductive-radiative heat transfer problem in (2) and was further developed in (3)-(5). The essential features of the process simulated are its nonstationary character and the presence of phase changes. Therefore, considering the heat transfer problem, one has to solve the Stefan problem. Different approaches for solving the problem are known presently (6(, each of them having its advantages and disadvantages. We develop the enthalpy approach under which the nonstationary heat transfer equation is formulated and solved in variables of enthalpy. This approach provides a stable and efficient numerical algorithm for solving the Stefan problem. To describe the radiation heat transfer, the method of angular coefficients is used. The main difficulties here are caused by calculation of the angular coefficient matrix for irregular forms of the radiation surfaces including heat shields situated inside of cavities. The conductive heat mass transfer is described by the Navier-Stockes equations under the Boussinesq approximation. Our approach presupposes that the Boussinesq equations are solved in the natural variables by the control volume method according to the Patankar scheme. The equations are previously transformed to exclude the convective terms and to bring them into the divergence form by a method proposed in (7). The system of equations in enthalpy for calculational meshes in each connected combination of zones with the boundary conditions prescribed is linearized by the Newton method. The system of linear equations is solved by the conjugate direction method with preconditioning by the incomplete factorization method. The algorithm described was realized in 2D (r, z)-geometry for calculating the process of crystallization of germanium conducted by the noncrucible melting method on a device "Zona-1" under the null gravity conditions. A demonstration calculation of the process was performed. Previously, a similar calculation was performed without taking into account the influence of convection (5(. The described technique for calculating heat transfer in growing crystals from the melt has the following distinctive features: heterogeneity of the domain; the presence of radiation; the presence of convective heat-mass transfer; nonlinearity of properties, i.e. dependence of the thermophysical parameters on enthalpy; nonstationarity stipulated by the time dependence of the heat generation source and the domain geometry configuration; taking into account the heat of phase change; validity of the model for 3D calculations; possibility to increase the complexity of the model (e.g. by introduction of control magnetic field and vibration impact). The calculational stability of the technique proposed is provided by the use of enthalpy variables in solving the heat transfer equation, by the use of natural variables and the Patankar scheme in calculation of velocities and pressures, by the use of the Newton iteration process to solve the nonlinear system, and by the use of balanced monotonic neutral finite-difference schemes to discretize the space variable and the implicit scheme to discretize the time variable. A high efficiency of the technique is basically provided through the use of a special organization of calculations and through performing the inner iterations by the conjugate direction method with preconditioning of initial operators by the incomplete factorization method.

References

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Date received: May 11, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caco-36.