Non-stationary heat model for electron beam melting and refining – An economic and conservative numerical method

• The introductory section is enhanced and all symbols are listed in it.
• The literature review is updated.
• The heat model is treated by an economic and conservative locally one-dimensional numerical method.
• Section 4. Thermo-physical properties is added and the actual used functions are given.
• Criteria for improvement of the quality of the obtained material are presented.

An economic and conservative numerical method is proposed for discretization and numerical simulation of non-stationary heat model concerning electron beam melting and refining (EBMR) of metals. The numerical model and optimization problems are developed to analyze and compare experiments and numerical data and to aid in understanding and optimizing EBMR. The axis-symmetric problem is decomposed into two locally one-dimensional problems. For the two problems, implicit and absolutely stable schemes are built for which the decomposition method gives rate of convergence of order one for both the space and time variables. The obtained discrete problems lead to linear systems of equation with three-diagonal matrixes which are solved via Thomas method. Proposition for the stability and realization of Thomas method is proved for one of the two one-dimensional problems. Criteria, related to the geometry of the crystallization front, for improvement of the quality of the obtained material after EBMR are discussed. Approaches for discretization of the criteria over the numerical solution of the model are proposed. Comparison between experimental and simulation results is made and the model is validated against liquid pool depth and diameter. Through applying the developed numerical scheme and criteria, optimization of the EBMR of copper ingots is achieved. Results for the best technological regime parameters according to the chosen criteria for the investigated ranges of the e-beam power and the beam radius are given. The results indicate that the model is able to quantitatively predict the liquid pool geometry and the optimization criteria, based on the profile, are able to propose optimal process parameters.