Solidification Problems by the Boundary Element Method

This paper elaborates on the analysis and design of the solidification of pure metais. In the first part of this paper, a direct analysis is presented for the motion of the solid-Iiquid freezing interface and the time-dependent temperature field. An iterative implicit algorithm has been developed for this purpose using the boundary element method (BEM) with time dependent Green’s functions and convolution integrals. Emphasis is placed on two-dimensional examples. The second part of this paper provides a methodology for the solution of an inverse design Stefan problem. A method for controlling the fluxes at the freezing front and its velocity is demonstrated. The BEM in conjunction with a sequential least squares technique are used to solve this ill-posed problem that has important technological applications. The accuracy of the method is illustrated through one-dimensional numerical examples. I, INTRODUCTION Problems of solidification of pure substances share the characteristic of an isothermal moving interface (freezing front). The freezing front motion and fluxes must be calculated as part of the solution of the phase change boundary value problem. Heat conduction is assumed in both solid and liquid phases and all thermal properties are considered temperature independent. The llux discontin~ty at any point of the interface is related to its normal velocity by the equation balancing the rate of heat flow with the energy rate required to create a fresh amount of solid per unit time (Stefan condition). A solidification problem is considered direct when the temperature or the flux on the fixed boundary of a solidifying body, with given material properties, is prescribed. There is an extensive literature on the above and related “Stefan” problems. The methods used to solve these problems can be categorized (Crank, 1984) into analytical, front-tracking> front-fixing and fixed-domain methods. The existing analytical solutions are primarily for one-dimensional problems (Crank, 1984) and two-dimensional wedge-shaped spaces (Budhia and Kreith, 1973; Rathjen and Jiji, 1971). Front-tracking methods involve finite differences or finite elements on a fixed grid (Lazaridis, 1969 ; Rao and Sastri, 1984), or on a variable space grid (Murray and Landis, 19.59) or the use of adaptive meshes (Bonnerot and Janet, 1977 ; Lynch, 1982 ; Albert and O’Neill, 1986 ; Zabaras and Ruan, 1990). An alternative formulation includes front-fixing methods (Crank and Gupta, 1975) where the moving front is fixed by a suitable choice of space coordinates. In the fixed-domain methods the problem is formulated in such a way that the interface condition becomes implicit in a new form of the equations, which applies over the whole of a fixed domain (Ralph and Bathe, 1982; Hsiao, 1985; Voller and Cross, 1981; Roose and Storrer, 1984). Integral fo~ulations for one-dimensional problems have been applied extensively in the past (Chuang and Szekely, 1972 ; Banerjee and Shaw, 1982 ; Heinlein et al., 1986; O’Neill, 1983 ; Sadegh et al., 1985). O’Neill (1983) gave a general integral formulation for quasi-static phase change problems, while Zabaras and Mukherjee (1987) extended this work to transient problems. Similar work has also been reported by Sadegh et al. (1985).