9. February 2026

BA: Optimization of cooling processes in crystal growth

Daniel-Sebastian Nastasa

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This work develops a simulation-based framework for optimizing cooling processes in crystal growth. A two-dimensional, axisymmetric FEM model (scikit-fem) for transient heat conduction with Dirichlet/Neumann boundary conditions and a coupled thermoelastic stress computation (Hooke’s law, Cauchy stress) are employed. The underlying program originates from Jesse Schardjin’s master’s thesis and was extended in this work to meet specific requirements. For optimization of the cooling process, it was augmented with material data and additional geometries. Subsequently, mathematical models were used to design boundary temperature profiles. Their evaluation is performed via the maximum von Mises stress, considering temperature-dependent stress limits. Boundary temperature profiles are parameterized as polynomials in height and controlled using linear as well as nonlinear strategies, complemented by alternative paths in the phase space spanned by bottom temperature and gradient, a temporal reparameterization to ensure practical boundary behavior (linear at the beginning and end), as well as the calculation of heat flux as a furnace-load metric. The chosen implementations (operator formulation, coupling, and numerical procedures) follow the mathematical formulation in Markus Zenk’s work: “Comparison of cooling profiles for some semiconductor crystals derived by semi-analytical and machine learning approaches.”

The results can be summarized as follows. The linear strategy serves as a robust reference path and keeps the maximum von Mises stress consistently below the stress limit. A nonlinear temporal control that exploits the temperature dependence of the maximum allowable stress shortens the process time without exceeding limits. If gradient and bottom temperature in phase space are not guided uniformly but varied separately, savings of about 2% are achieved. Higher-order boundary polynomials enable additional stress reduction. To avoid start-up spikes and to ensure the required linearity at the beginning and end, a temporal reparameterization was successfully implemented. The heat-flux calculation yields the required furnace loads, and a parameter study shows an approximately linear relationship between end time and cooling rate and a combined material- and geometry-dependent factor, which can serve as a practical planning guideline for industrial processes.