Numerical models for casting solidification: Part I. The coupling of the boundary element and finite difference methods
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I.
INTRODUCTION
D U R I N G the last decade, numerical simulation o f solidification has been investigated in many fields including casting, welding, ingot-making, rapid solidification, and crystal growth. Three techniques which have been utilized were analog, finite difference, and finite element methods. With the development o fthe digital computer, the latter two have come to be used widely for numerical analyses. The finite difference method ~-4 offers advantages in regard to numerical formulation, data preparations, and computing times, but it may not be appropriate for complex geometries because of its restrictions on the element shape. The finite element method,5-11 on the other hand, has been extensively applied to fluid flow, stress and strain, and solidification problems because o f its wide variety of element shapes, but it requires much effort for formulation o f the problem and data preparation, and needs long CPU times. Recently, works 12-16 concerned with a new numerical technique, boundary element method, based on the combination of integral equations and finite element methods have been reported. In the case of domain-type techniques such as finite difference and finite element methods, the domain must be divided into small elements and a series of nodes defined, which can be intemal or on the surface o fthe body. In the boundary element method, the governing equations o f the problems under consideration are reduced to a set o f integral equations on the boundary, and the resulting integral equations are discretized in the same way as in the finite element method. Therefore, nodes are defined only on the external surface, and internal unknowns are not required as the problem mathematically has been reduced to a boundary solution. This reduction is possible by applying Green's theorem and using fundamental solutions which satisfy the governing equations. Satisfaction of boundary conditions produces a system of equations which can be solved to find the unknowns on the boundary. Once all values on the
C. P. HONG is Graduate Student, Department of Metallurgy, Faculty of 'Engineering, The University of Tokyo, Tokyo 113, Japan, on leave from Kyung-Pook National University, Taegu, Korea. T. UMEDA, Associate Professor, and Y. KIMURA, Professor, are with the Department of Metallurgy, Faculty of Engineering, The University of Tokyo, Tokyo 113,Japan. Manuscript submittedMarch 2 1 , 1983. METALLURGICAL
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boundary are known, any internal variables may easily be calculated as functions o f the boundary values. The main advantages o fthe boundary element technique are the reduction in the effective dimensionality by one which leads to an appreciable reduction in the number ofunknowns governing the problem, and resultant considerable saving in data preparation and CPU times, especially in complex, threedimensional geometries, such as shaped castings. Several investigationsl4-~7 on applications o fthe boundary element method have been reported for transient heat conduction and diffusion problems.
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