FINITE-ELEMENT NEWTON METHOD FOR THE ANALYSIS OF CZOCHRALSKI CRYSTAL-GROWTH WITH DIFFUSE-GRAY RADIATIVE HEAT-TRANSFER

A quasi‐steady‐state, integrated system model describing high temperature heat transfer, solidification and the action of capillarity in the Czochralski crystal growth process is solved by a finite element/Newton method. The numerical analysis couples the calculation of the temperature field in all phases and the determination of the melt/crystal and melt/gas interfaces and the crystal radius free boundaries. The analysis includes conductive heat transfer in the melt, crystal, crucible, pedestal, heater and the surrounding insulation and diffuse‐grey radiation, which couples the heat transfer between surfaces, the crystal radius and the melt/gas free boundary through the view factors. Finite element approximations are used to reduce the entire problem to a coupled set of non‐linear algebraic equations. These are solved simultaneously by Newton's method with the Jacobian matrix computed by a combination of closed form expressions and finite difference approximations. Quadratic convergence of the Newton iteration is demonstrated along with a factor of four increase in computational efficiency over a successive iteration procedure that decouples the calculation of radiation from the rest of the heat transfer model. Copyright © 1990 John Wiley & Sons, Ltd