Level Set Methods for Modeling Laser Melting of Metals

The physical model describing heat transfer and melting taking place during and after the interaction of a laser beam with a semi-infinite metal surface is based on the classical Stefan problem with appropriately chosen boundary conditions to reflect direct selective laser sintering of metals. A level set method for solving this problem is presented in this paper. From the results of these computations, we obtain time evolution of solid-liquid interface and temperature distribution. INTRODUCTION Direct selective laser sintering of metals [1] is a process in which a high-energy laser beam directly consolidates a metal powder or powder mixture to full density. Direct selective laser sintering of metals is a complex process exhibiting multiple modes of heat, mass and momentum transfer, and chemical reaction mechanisms. Among them, melting and resolidification processes in direct SLS can have significant effect on the temperature distribution, residual stress, and final microstructure quality of the parts. The inherent complexity of this process imposes serious constraints on the complexity of the models that can be constructed to enable a fundamental understanding of the important physical mechanisms in SLS. This understanding is essential to implement effective process control [2]. There are numerous previous studies for understanding this kind of phase change problem involving moving boundaries. Above all, tracking the motion of a moving front has been of great interest for many researchers. In this paper, a convenient scheme to track moving interfaces using level set theory is extended to the analysis of the Stefan problem. This level set formulation is based on front capturing. In this formulation, the boundary of solid-liquid interface is modeled as the zero set of a smooth function defined on the entire physical domain. The boundary is then updated by solving a nonlinear equation of the Hamilton-Jacobi type on the whole domain. This level set formulation of the moving interface was introduced by Osher and Sethian [3] and was capable of computing geometric properties of highly complicated boundaries without explicitly tracking the interface [4]. Equation 1 is the level set equation given by Osher and Sethian. For certain forms of the speed function F, one obtains a standard Hamilton-Jacobi equation. Equation 1 describes the time evolution of the level set function in such a way that the zero level set of this evolving function is always identified with the propagating interface shown in Fig. 1 [5]. 0 F t (1) given ) 0 , ( t x