TED-AJ03-349 STABILITY OF NUMERICAL SIMULATIONS OF DENDRITIC SOLIDIFICATION
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概要
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Recent novel developments in the numerical simulation of diffusion limited dendritic growth are discussed in the context of two-dimensional simulations of solidification of pure substances, and of binary alloys. The numerical model is based on the direct solution of the energy and solute conservation equations using the finite element method. The energy equation is written in terms of temperature only and is solved on a fixed mesh of bilinear isoparametric elements. The solute conservation equation is solved only in the liquid, using a mesh of quadratic triangular elements that changes at every time step. Solute diffusion in the solid is neglected, because, for the metallic alloys of interest, it is several orders of magnitude smaller than the diffusion in the liquid. The interface is tracked on the fixed mesh of bilinear elements using a set of marker points that move according to the local interface velocity; the number of markers is allowed to vary as the interface evolves so as to maintain a predetermined level of resolution at all times. The model tracks the sharp solid-liquid interface and can resolve extremely complex geometries. In this work, only two spatial dimensions are considered; however, one of the important aspects of these models is that they can be directly extended to three dimensions. The three most important numerical difficulties encountered in the simulations of dendritic growth of binary alloys are discussed : (1) The need to accurately calculate the position and velocity of the interface as part of the solution, introducing a strong non-linearity. (2) The disparity of length scales between the thermal diffusion length and the solute diffusion length, which makes it necessary to use very fine meshes over large spatial domains, greatly increasing the cost of the calculations. (3) The instability of the solid-liquid interface, particularly at high concentrations of solute. This is reflected in the need to significantly reduce the size of hte discrete time steps, further increasing the computational cost. the study of the third of these difficulties constitutes the main objective of this paper. In directional solidification of binary alloys, the presence of the solute boundary layer with a high concentration of solute at the solid-liquid interface greatly increases the instability of the interface. The effect of this accumulation of solute at the interface on the stability of the numerical calculations is studied using the continuity condition on the heat flux across the interface and the conclusions are supported with the results from examples of numerical simulations. Results from the linear stability theory of Mullins and Sekerka [23] are used to guide the numerical calculations. The simulations presented here also serve to illustrate the usefulness and the capabilities of the numerical model. The difficulties encountered in the simulations are used to assess the current modeling capabilities and the hurdles faced to produce more powerful simulators.
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著者
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Heinrich J.
Department Of Aerospace And Mechanical Engineering The University Of Arizona
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Zhao P.
Department of Aerospace and Mechanical Engineering The University of Arizona