JOURNAL OF COMPUTATIONAL PHYSICS 130, 243–255 (1997) ARTICLE NO. CP965582 An Inverse Finite Element Method for Pure and Binary Solidification Problems Alexandre I. Fedoseyev 1 and J. Iwan D. Alexander Center for Microgravity and Materials Research, University of Alabama in Huntsville, Huntsville, Alabama 35899 Received September 18, 1995; revised July 9, 1996 (a) Coordinate transformation techniques. Here the un- known is mapped onto a regular geometrical region. The A 2D axisymmetric formulation for the solution of a directional solidification problem using an inverse finite-element method resulting transformed equations are then solved on this (IFEM) is presented. An algorithm developed by A. N. Alexandrou domain using N - 1 of the N boundary conditions. The (Int. J. Numer. Methods Eng. 28, 2383, 1989) has been modified Nth condition, sometimes referred to as the distinguished and extended to include more general boundary conditions. The condition [1] is used to determine the location of the free latter includes the explicit presence of an ampoule (with a complex or moving boundary in physical space in an iterative fash- shape) that contains the solid and the melt from which it is growing. Heat transfer between the ampoule and the external environment, ion. This approach has been realized, using different solu- time-dependent thermal boundary conditions, nonmonotonic tem- tion techniques, including finite element [2–5] and, re- perature distributions, and species diffusion in the melt and crystal cently, Chebyshev spectral techniques [6]. The techniques are also taken into account. Thus, our extended formulation encom- employed to handle the mapping include Landau-type passes a wider class of solidification problems than previous IFEM transformations [6, 7] and numerically generated moving methods. Numerical experiments that illustrate the suitability of the extended IFEM are presented. In particular, we present a simulation orthogonal curvilinear systems obtained by elliptic mesh of the directional solidification of zinc cadmium telluride using generation methods (see, for example, [8]). boundary conditions corresponding to an actual experiment scenario.  1997 Academic Press (b) Enthalpy methods. Enthalpy methods, first sug- gested by Rose [9], are ‘‘fixed domain’’ based approaches to phase change problems which do not require explicit 1. INTRODUCTION tracking of the phase boundary. The method involves cast- ing the energy transport equation in conservation form Phase change free boundary problems related to solidi- [10] and defining an enthalpy function for the entire two- fication or melting represent a class of problems that re- phase domain. As solidification proceeds, the location of quire simultaneous solution of the governing partial differ- the boundary between the solid and liquid phases is then ential equations and the geometry of the domain on which determined from the enthalpy distribution and the fraction they are defined. Solution techniques for such problems of solid or liquid occupying each cell. These techniques include coordinate transformation techniques, enthalpy have been applied successfully to solidification problems methods, inverse formulation methods, phase field models for pure substances and binary compounds (see, for exam- and, more recently, the inverse finite-element method ple, [10–14]). (IFEM). All these methods, have, in one form or another, been applied to the problem of the directional solidification (c) Phase field models. The phase-field approach in- of pure substances and binary compounds (see [1]). Direc- volves the assumption that a phase-field (x, t) exists which tional solidification is a commonly used technique for the specifies the phase of the system at each point x. It is then production of semiconductor crystals and some metal assumed that the total Helmholtz free energy of the system, alloys. Essentially, the solid, usually a crystal, is grown F, is a functional dependent on , temperature, composi- by translating a melt relative to an imposed temperature tion, etc. The dependence of F on  is assumed to be of gradient such that it gradually freezes in the direction anti- a ‘‘double well’’ form [15] and is usually taken in the form parallel to the translation direction (see Fig. 1). of an explicit dependence on local gradients of . The The existing solution techniques for such problems can different models that are based on the phase field ideas be subdivided into the following classes: are reviewed by Hohenberg and Halperin [16] and, for solidification in particular, have been developed by Langer [17] and Caginalp [18]. Various studies performed using 1 On leave from the Institute for Problems in Mechanics, Russian Acad- emy of Sciences, 101 Vernadsky Av, Moscow, Russia, 117526. this approach have shown that it can successfully reproduce 243 0021-9991/97 $25.00 Copyright  1997 by Academic Press All rights of reproduction in any form reserved.