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Petrov‐Galerkin methods for natural convection in directional solidification of binary alloys
Author(s) -
Adornato Peter M.,
Brown Robert A.
Publication year - 1987
Publication title -
international journal for numerical methods in fluids
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.938
H-Index - 112
eISSN - 1097-0363
pISSN - 0271-2091
DOI - 10.1002/fld.1650070802
Subject(s) - petrov–galerkin method , galerkin method , finite element method , method of mean weighted residuals , natural convection , rotational symmetry , mechanics , boundary value problem , mathematics , convection , mathematical analysis , classical mechanics , physics , thermodynamics
A Petrov‐Galerkin finite element method is presented for calculation of the steady, axisymmetric thermosolutal convection and interface morphology in a model for vertical Bridgman crystal growth of nondilute binary alloys. The Petrov‐Galerkin method is based on the formulation for biquadratic elements developed by Heinrich and Zienkiewicz and is introduced into the calculation of the velocity, temperature and concentration fields. The algebraic system is solved simultaneously for the field variables and interface shape by Newton's method. The results of the Petrov‐Galerkin method are compared critically with those of Galerkin's method using the same finite element grids. Significant improvements in accuracy are found with the Petrov‐Galerkin method only when the mesh is refined and when the formulation of the residual equations is modified to account for the mixed boundary conditions that arise at the solidification interface. Calculations for alloys with stable and unstable solute gradients show the occurrence of classical flow transitions and morphological instabilities in the solidification system.