In this article, we use numerical simulation to investigate transient temperature
phenomena during sublimation growth of SiC single crytals via physical
vapor transport (also called the modified Lely method). We consider the evolution
of temperatures at the SiC source and at the SiC seed crystal, which
are highly relevant to the quality of the grown crystals, but inaccessible to
direct measurements. The simulations are based on a transient mathematical
model for the heat transport, including heat conduction, radiation, and radio
frequency (RF) induction heating. Varying the position of the induction coil
as well as the heating power, it is shown that the measurable temperature difference
between the bottom and the top of the growth apparatus can usually
not be used as a simple indicator for the respective temperature difference
between SiC source and seed. Moreover, it is shown that there can be a time
lack of 1.5 hours between the heating of the temperature measuring points
and the heating of the interior of the SiC source.
We use a numerical optimization method to determine the control parameters
frequency, power, and coil position for the radio frequency (RF) induction
heating of the growth apparatus during sublimation growth of SiC single crystals
via physical vapor transport (PVT) (also called the modified Lely method). The
control parameters are determined to minimize a functional, tuning the radial
temperature gradient on the single crystal surface as well as the vertical temperature
gradient between SiC source and seed, both being crucial for high-quality
growth. The optimization is subject to constraints with respect to a required
temperature difference between source and seed, a required temperature range at
the seed, and an upper bound for the temperature in the entire apparatus. The
numerical computations use a stationary mathematical model for the heat transport,
including heat conduction, radiation, and RF heating to solve the forward
problem, and a Nelder-Mead method for optimization. A minimal radial temperature
gradient is found to coincide with a minimal temperature at the single
crystal surface, and a maximal temperature gradient between source and seed is
found to coincide with a low coil position.
While it is well-known that the standard integral operator K of (stationary) diffuse-gray radiation, as it occurs in the radiosity equation, is compact if the domain of radiative interaction is sufficiently regular, we show noncompactness of the operator if the domain is polyhedral. We also show that a stationary operator is never compact when reinterpreted in a transient setting. Moreover, we provide new proofs, which do not use the compactness of K, for 1 being a simple eigenvalue of K for connected enclosures, and for I-(1-e)K being invertible, provided the emissivity e does not vanish identically.
We consider a control constrained optimal control problem governed by a semilinear
elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of
diffuse-gray conductive-radiative heat transfer. The problem arises from the aim to optimize the
temperature gradient within crystal growth by the physical vapor transport (PVT) method. Based
on a minimum principle for the semilinear equation as well as L1-estimates for the weak solution,
we establish the existence of an optimal solution as well as necessary optimality conditions. The
theoretical results are illustrated by results of numerical computations.
A finite volume scheme suitable for nonlinear heat transfer in materials with anisotropic thermal conductivity is formulated, focussing on the difficulties arising from the discretization of complex domains which are typical in the simulation of industrially relevant processes. The discretization is based on unstructured constrained Delaunay triangulations of the domain. For simplicity, it is assumed that the thermal conductivity tensor has vanishing off-diagonal entries and that the anisotropy is independent of the temperature. Numerical simulations verify the accuracy of the method in two test cases where a closed-form solution is available. Further results demonstrate the effectiveness of the method in computing the heat transfer in a complex growth apparatus used in crystal growth.
Using a mathematical heat transfer model including anisotropic heat conduction, radiation, and radio frequency (RF) heating, we perform numerical computations of the temperature field in an axisymmetric growth apparatus during sublimation growth of silicon carbide (SiC) bulk single crystals by physical vapor transport (PVT) (modified Lely method). Because it is not unusual for the thermal insulation of a PVT growth apparatus to possess an anisotropic thermal conductivity, we numerically study the influence that this anisotropic thermal conductivity has on the temperature field in the growth chamber. Moreover, we also study the influence of the thickness of the insulation. Our results show that, depending on the insulation's orientation, even a moderate anisotropy in the insulation can result in temperature variations of more than 100 K at the growing crystal's surface, which should be taken into account for the simulation as well as for the design of a PVT growth apparatus.
A survey of methods from numerical linear algebra for linear constant coefficient differential-algebraic equations (DAEs) and descriptor control systems is presented. We discuss numerical methods to check the solvability properties of DAEs as well as index reduction and regularization techniques. For descriptor systems we discuss controllability and observability properties and how these can be checked numerically. These methods are based on staircase forms and derivative arrays, transformed with real orthogonal transformations that are discussed in detail. Then we use the reformulated problems in several control applications for differential-algebraic equations ranging from regular and singular linear-quadratic optimal and robust control to dissipativity checking. We discuss these applications and give a systematic overview over the theory and the numerical solution methods. In particular, we show that all these applications can be treated with a common approach that is based on the computation of eigenvalues and deflating subspaces of even matrix pencils. The unified approach allows to generalize and improve several techniques that are currently in use in systems and control.