Computational modeling offers an opportunity for a better understanding and investigation of thermal transfer mechanisms. It can be used for the optimization of the electron beam melting process and for obtaining new materials with improved characteristics that have many applications in the power industry, medicine, instrument engineering, electronics, etc. A time-dependent 3D axis-symmetrical heat model for simulation of thermal transfer in metal ingots solidified in a water-cooled crucible at electron beam melting and refining (EBMR) is developed. The model predicts the change in the temperature field in the casting ingot during the interaction of the beam with the material. A modified Pismen-Rekford numerical scheme to discretize the analytical model is developed. These equation systems, describing the thermal processes and main characteristics of the developed numerical method, are presented. In order to optimize the technological regimes, different criteria for better refinement and obtaining dendrite crystal structures are proposed. Analytical problems of mathematical optimization are formulated, discretized and heuristically solved by cluster methods. Using important for the practice simulation results, suggestions can be made for EBMR technology optimization. The proposed tool is important and useful for studying, control, optimization of EBMR process parameters and improving of the quality of the newly produced materials.

We find variational symmetries, conserved quantities and identities for several equations: envelope equation, Böcher equation, the propagation of sound waves with losses, flow of a gas with losses, and the nonlinear Schrödinger equation with losses or gains, and an electro-magnetic interaction. Most of these equations do not have a variational description with the classical variational principle and we find such a description with the generalized variational principle of Herglotz.

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