Unconditionally Stable Fully Explicit Finite Difference Solution of Solidification Problems
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TRANSIENT heat-transfer problems involving melting or solidification are generally referred to as phasechange or moving boundary problems. Sometimes, they are referred to as Stefan problems, with reference to the pioneering work of Stefan around 1890. Phase-change problems have numerous applications in such areas as the metal processing, solidification of castings, environmental engineering, making of ice, the freezing of food, the cooling of large masses of igneous rock, and the thermal energy storage system in a space station. In these processes, matter is subject to a phase change. Consequently, a boundary separating two different phases develops and moves in the matter during the process. In early years, analytical methods were the only means available to render mathematically an understanding of physical processes involving the moving boundary. Although analytical methods offer an exact solution and are mathematically elegant, due to their limitations, analytical solutions are mainly for the one-dimensional (1-D) cases of an infinite or semi-infinite region with simple initial and boundary conditions and constant thermal properties.[1] Practical solidification and melting problems are rarely 1-D, initial and boundary conditions are always complex, and thermophysical properties can vary with phases, temperatures, and concentration. With the rise of high-speed digital computers, mathematical modeling and computer simulation often become the most economical and fastest approaches to provide a broad understanding of the practical processes involving the moving boundary R. TAVAKOLI and P. DAVAMI are with the Department of Material Science and Engineering, Sharif University of Technology, Tehran, Iran. Contact e-mail: [email protected]. Manuscript submitted May 29, 2006. METALLURGICAL AND MATERIALS TRANSACTIONS B
problems. Nowadays in most engineering applications, recourse for solving the phase-change problems has been made to numerical analyses that use either finite difference or finite element of boundary element methods. The success of finite element and boundary element methods lies in their ability to handle complex geometries, but they are acknowledged to be more time consuming in terms of computing and programming. Because of their simplicity in formulation and programming, finite difference techniques are still the most popular at present.[2] There are several approaches for numerical simulation of phase-change problems (for a good review, refer to References 2 through 4). Selection between various methods can be a function of problem type, required accuracy, and available computational resource. Recently, the numerical simulation is widely used as an optimization tool in the engineering applications. One of the major problems during a computer-aided design optimization is the time taken to perform the numerical simulation. Very often, this design-simulation-evaluation cycle is repeated many times before a final, satisfactory design is obtained. This iterative cycle can be extremely time consuming because the simulat
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