Thermal-stress reduction for a Czochralski grown single crystal

In this paper an optimal-control approach for thermal-stress reduction inside a Czochralski-grown single crystal is presented. Using the lateral heat flux as a control variable, an optimal-control formulation for minimizing thermal stress with a given crystal shape is derived. Since the thermal stress is also affected by the lateral shape of crystals during growth, the level of the stress can be reduced by growing crystals into a suitable shape. Using the lateral shape as a control variable, a similar optimal-control formulation for stress reduction is derived. In both cases, the von Mises stress is used as an objective function for the constrained optimization problem. Euler–Lagrange equations are derived using the calculus of variations and Lagrange multipliers. Various stress-reduction strategies are explored by solving the Euler–Lagrange equations numerically.