Selecting finite element basis functions for computation of partially facetted melt/crystal interfaces appearing during the directional growth of large-scale single crystals

摘要

A finite element method capable of combining certain aspects of facetting in models of transport phenomena occurring within directional melt-growth systems was presented in Liu et al (1999 J. Ctystal Growth 205 333). However, in certain cases this method suffers from difficulties most probably associated with the nature of the quadrilateral Lagrange biquadratic elements used in the analysis. This type of discretization yields a relatively low-order approximation of the melt/crystal interface crystallographic orientation which is also discontinuous at boundaries between elements. Here, two new finite element discretization techniques aimed at improving robustness of the method are presented and tested. In one case Lagrange cubic functions are used for representation of temperature and geometry, while in the other case Lagrange and Hermite cubic functions are used for calculations of temperature and geometry respectively. Both approaches provide a higher-order approximation of interface orientation, while in the case when Hermite functions are applied, an additional benefit in the form of a continuous (across element boundaries) interface orientation is achieved. Results from a number of tests show that both higher-order approximations provide a robust and efficient method for representation of geometry. In both cases, accuracy can be further increased with the aid of local mesh refinement in the vicinity of sharp corners on the interface. Finally, certain differences exist between the performances of these two methods. However, these seem (at this stage) minor compared to the benefits achieved relative to the previous lower-order approach. [References: 27]