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This paper presents a spectral multidomain method for solving the Navier‐Stokes equations in the vorticity‐stream function formulation. The algorithm is based on an extensive use of the influence matrix technique and so leads to a direct method without any iterative process. Numerical results concerning the Czochralski melt configuration are reported and compared with spectral monodomain solutions to show the advantage of the domain decomposition for such a problem which solution presents a singular behaviour.

A simple approach is developed for modelling the distribution of dopant in crystals grown by the Czochralski technique. Allowance is made for the temperature dependence of the diffusivity and for the diffusion of dopant behind the moving boundary. It is found that the effect can make a significant difference to the final dopant distribution. The diffusion process is modelled using a discrete method, previously developed for situations in which the boundary is static.

Describes a new finite element scheme for the large‐scale analysis of compressible and incompressible viscous flows. The scheme is based on a combined compressible‐ incompressible Galerkin least‐squares (GLS) space‐time variational formulation. Three‐ dimensional unstructured meshes are employed, with piecewise‐constant temporal interpolation, local time‐stepping for steady flows, and linear continuous spatial interpolation in all the variables. The scheme incorporates automatic adaptive mesh refinement, with a choice of various error indicators. It is implemented on a distributed‐memory parallel computer, and includes an automatic load‐balancing procedure. Demonstrates the ability to solve both compressible and incompressible viscous flow problems using the parallel adaptive framework via numerical examples. These include Mach 3 flow over a flat plate, and a divergence‐free buoyancy‐driven flow in a cavity. The latter is a model for the steady melt flow in a Czochralski crystal growth process.

In general, the bifurcation phenomenon of the natural convection has largely been studied. But the bifurcation of natural convection under magnetic conditions has not been studied as per the authors’ knowledge. This paper aims to investigate the changes in bifurcation phenomenon by the self-induced circular magnetic field.

The authors numerically solved the natural convection in an annulus. The SIMPLE algorithm was adopted for pressure-momenturm coupling. The Boussinesq approximation was used for numerical modeling of natural convection. Finally, the Lorentz force effect by the magnetic field was considered through the source terms in the momentum conservation equation.

It was determined that the heat-transfer rate changes by 17% owing to the applied magnetic effect, and the range of the Rayleigh number for flow bifurcation is changed by the magnetic effect. Moreover, under the strong magnetic condition, the flow bifurcation continues even at very high Ra. Previously, flow bifurcation has been understood as a flow instability phenomena, and the Lorentz force was regarded as a flow-damping effect; however, in this study, it was found that the magnetic field can boost the flow instability and induce flow bifurcation even in the Rayleigh number region where the bifurcation does not appear.

This paper is dealing with the bifurcation phenomenon in MHD natural convection problems. In the past, the electromagnetic forces were regarded as always acting to damp out the existing flows; herewith, the authors first investigated that the magnetic effect can boost the bifurcation of a kind of flow instability phenomenon.

The contactless inductive flow tomography is a procedure that enables the reconstruction of the global three-dimensional flow structure of an electrically conducting fluid by measuring the flow-induced magnetic flux density outside the melt and by subsequently solving the associated linear inverse problem. The purpose of this study is to improve the accuracy of the computation of the forward problem, since the forward solution primarily determines the accuracy of the inversion.

The tomography procedure is described by a system of coupled integral equations where the integrals contain a singularity when a source point coincides with a field point. The integrals need to be evaluated to a high degree of precision to establish an accurate foundation for the inversion. The contribution of a singular point to the value of the surface and volume integrals in the system is determined by analysing the behaviour of the fields and integrals in the close proximity of the singularity.

A significant improvement of the accuracy is achieved by applying higher order elements and by attributing special attention to the singularities inherent in the integral equations.

The contribution of a singular point to the value of the surface integrals in the system is dependent upon the geometry of the boundary at the singular point. The computation of the integrals is described in detail and the improper surface and volume integrals are shown to exist. The treatment of the singularities represents a novelty in the contactless inductive flow tomography and is the focal point of this investigation.

The present paper gives an overview of the complex mathematical modelling of industrial Czochralski (CZ) and floating‐zone (FZ) processes for the growth of large silicon single crystals from melt. Extensive numerical investigations of turbulent Si‐melt flows in large diameter CZ crucibles, global thermal calculations in growth facilities and analysis of the influence of various electromagnetic fields on CZ process are presented. For FZ process, a complex system of coupled 2D and 3D mathematical models is presented to show the possibilities of modelling from the calculation of the molten zone shape till the resistivity distribution in the grown crystal. A special developed program code is presented that is used to calculate the temperature field in the crystal including radiation exchange with reflectors, stress field due to thermal expansion and shape of the dislocated zone in the case of dislocation generation. Besides the macroscopic modelling of crystal growth processes, the crystallisation model on the atomistic level in the mean field approximation is also presented.

The multi‐domain generalized differential quadrature method is applied in this paper to simulate the flows in Czochralski crystal growth. The effect of interface treatment on the numerical solution is studied through four types of interface approximations. The performance of those four interface approximations is validated by a benchmark problem suggested by Wheeler. It is demonstrated in this study that the multi‐domain GDQ approach is an efficient method which can obtain accurate numerical solutions by using very few grid points, and the overlapped interface approximation provides the most accurate numerical results.

The paper describes numerical simulation tools for electromagnetic (EM), hydrodynamic, temperature and concentration fields in industrial Czochralski (CZ) and floating zone (FZ) single silicon crystal growth facilities under the influence of several alternating current (AC) and static DC magnetic fields. Such fields are expected to provide additional means to influence the melt behaviour, especially in the industrial growth of large diameter (200–300 mm) silicon crystals. The simulation tools are based on axisymmetric 2D models for (1) AC and DC magnetic fields in the whole crystal growth facility and (2) hydrodynamics, temperature and mass transport in the melt under the influence of the EM fields. The simulation tools are verified by comparison to temperature and velocity measurements in a laboratory CZ set‐up with eutectics InGaSn model melt and to resistivity measurements in grown silicon crystals.

The quality of crystals grown from melt depends on the flow field in the melt. To simulate melt conditions, a finite element analysis is performed on flow in a heated cavity under the driving forces of natural convection, thermocapillary effects and rotation. In addition, the gravity field is modulated to simulate a microgravitational environment. The purpose for conducting this research is to determine whether the use of baffles can effectively reduce convection and suppress temperature oscillations. The results show that the baffle is able to suppress convection and reduce the amplitude of the temperature oscillations when placed perpendicular to the modulation direction. Under crystal and crucible rotation, the results with and without baffles are similar. In all cases, baffles did not induce temperature oscillations. From this study, it can be concluded that the effects of baffles on the flow behaviour depends greatly on the direction of gravity modulation and frequency.